D. Youm/Physics Reports 316 (1999) 1}232
BLACK HOLES AND SOLITONS IN STRING THEORY
Donam YOUM School of Natural Sciences, Institute for Advanced Study Olden Lane, Princeton, NJ 08540, USA
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
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Black holes and solitons in string theory Donam Youm School of Natural Sciences, Institute for Advanced Study, Olden Lane, Princeton, NJ 08540, USA Received October 1997; editor: A. Schwimmer Contents 1. Outline of the review 2. Soliton and BPS state 2.1. Physical parameters of solitons 2.2. Supermultiplets of extended supersymmetry 2.3. Positive energy theorem and Nester's formalism 3. Duality symmetries 3.1. Electric-magnetic duality 3.2. Target space and strong-weak coupling dualities of heterotic string on a torus 3.3. String}string duality in six dimensions 3.4. ;-duality and eleven-dimensional supergravity 3.5. S-duality of type-IIB string 3.6. ¹-duality of toroidally compacti"ed strings 3.7. M-theory 4. Black holes in heterotic string on tori 4.1. Solution generating procedure 4.2. Static, spherically symmetric solutions in four dimensions 4.3. Rotating black holes in four dimensions 4.4. General rotating "ve-dimensional solution 4.5. Rotating black holes in higher dimensions
4 5 7 11 20 24 26
37 43 47 49 50 56 62 62 65 80 83 89
5. Black holes in N"2 supergravity theories 5.1. N"2 supergravity theory 5.2. Supersymmetric attractor and black hole entropy 5.3. Explicit solutions 5.4. Principle of a minimal central charge 5.5. Double extreme black holes 5.6. Quantum aspects of N"2 black holes 6. p-branes 6.1. Single-charged p-branes 6.2. Multi-charged p-branes 6.3. Dimensional reduction and higher dimensional embeddings 7. Entropy of black holes and perturbative string states 7.1. Black holes as string states 7.2. BPS, purely electric black holes and perturbative string states 7.3. Near-extreme black holes as string states 7.4. Black holes and fundamental strings 7.5. Dyonic black holes and chiral null model 8. D-branes and entropy of black holes 8.1. Introduction to D-branes 8.2. D-brane as black holes 8.3. D-brane counting argument Acknowledgements References
E-mail address:
[email protected] (D. Youm) 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 3 7 - X
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Abstract We review various aspects of classical solutions in string theories. Emphasis is placed on their supersymmetry properties, their special roles in string dualities and microscopic interpretations. Topics include black hole solutions in string theories on tori and N"2 supergravity theories; p-branes; microscopic interpretation of black hole entropy. We also review aspects of dualities and BPS states. 1999 Elsevier Science B.V. All rights reserved. PACS: 11.27.#d
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1. Outline of the review It is a purpose of this review to discuss recent development in black hole and soliton physics in string theories. Recent rapid and exciting development in string dualities over the last couple of years changed our view on string theories. Namely, branes and other types of classical solutions that were previously regarded as irrelevant to string theories are now understood as playing important roles in non-perturbative aspects of string theories; these solutions are required to exist within string spectrum by recently conjectured string dualities. Particularly, D-branes which are identi"ed as non-perturbative string states that carry charges in R-R sector have classical p-brane solutions in string e!ective "eld theories as their long-distance limit description. p-branes, other types of classical stringy solutions and fundamental strings are interrelated via web of recently conjectured string dualities. Much of progress has been made in constructing various p-brane and other classical solutions in string theories in an attempt to understand conjectured (non-perturbative) string dualities. We review such progress in this paper. In particular, we discuss black hole solutions in string e!ective "eld theories in details. Recent years have been active period for constructing black hole solutions in string theories. Construction of black hole solutions in heterotic string on tori with the most general charge con"gurations is close to completion. (As for rotating black holes in heterotic string on ¹, one charge degree of freedom is missing for describing the most general charge con"guration.) Also, signi"cant work has been done on a special class of black holes in N"2 supergravity theories. These solutions, called double extreme black holes, are characterized by constant scalars and correspond to the minimum energy con"gurations among extreme solutions. Among other things, study of black holes and other classical solutions in string theories is of particular interest since these allow to address long-standing problems in quantum gravity such as microscopic interpretation of black hole thermodynamics within the framework of superstring theory. In this review, we concentrate on recent remarkable progress in understanding microscopic origin of black hole entropy. Such exciting developments were prompted by construction of general class of solutions in string theories and realization that non-perturbative R-R charges are carried by D-branes. Within subset of solutions with restricted range of parameters, the Bekenstein} Hawking entropy has been successfully reproduced by stringy microscopic calculations. Since the subject reviewed in this paper is broad and rapidly developing, it would be a di$cult task to survey every aspect given limited time and space. The author made an e!ort to cover as many aspects as possible, especially emphasizing aspects of supergravity solutions, but there are still many issues missing in this paper such as stringy microscopic interpretation of black hole radiation, M(atrix) theory description of black holes and the most recent developments in N"2 black holes and p-branes. The author hopes that some of missing issues will be covered by other forthcoming review paper by Maldacena [470]. The review is organized as follows. Sections 2 and 3 are introductory sections where we discuss basic facts on solitons and string dualities which are necessary for understanding the remaining sections. In these two chapters, we especially illuminate relations between BPS solutions and string dualities. In Section 4, we summarize recently constructed general class of black hole solutions in heterotic string on tori. We show explicit generating solutions in each spacetime dimensions and discuss their properties. In Section 5, we review aspects of black holes in N"2 supergravity theories. We discuss principle of a minimal central charge, double extreme solutions and quantum corrections. In Section 6, we
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summarize recent development in p-branes. Here, we show how p-branes and other related solutions "t into string spectrum and discuss their symmetry properties under string dualities. We systematically study various single-charged p-branes and multi-charged p-branes (dyonic p-branes and intersecting p-branes) in di!erent spacetime dimensions. We also discuss black holes in type-II string on tori as special cases and their embedding to p-branes in higher dimensions. In Sections 7 and 8, we summarize the recent exciting development in microscopic interpretation of black hole entropy within the framework of string theories. We discuss Sen's calculation of statistical entropy of electrically charged black holes, Tseytlin's work on statistical entropy of dyonic black holes within chiral null model and D-brane interpretation of black hole entropy.
2. Soliton and BPS state Solitons are de"ned as time-independent, non-singular, localized solutions of classical equations of motion with "nite energy (density) in a "eld theory [106,512]. Such solutions in D spacetime dimensions are alternatively called p-branes [228,234] if they are localized in D!1!p spatial coordinates and independent of the other p spatial coordinates, where p(D!1. For example, the p"0 case (0-brane) has a characteristic of point particles and is also called a black hole; p"1 case is called a string; p"2 case is a membrane. The main concern of this paper is on the p"0 case, but we discuss the extended objects (p51) in higher dimensions as embeddings of black holes and in relations to string dualities. As non-perturbative solutions of "eld theories, solitons have properties di!erent from perturbative solutions in "eld theories. First, the mass of solitons is inversely proportional to some powers of dimensionless coupling constants in "eld theories. So, in the regime where the perturbative approximations are valid (i.e. weak-coupling limit), the mass of solitons is arbitrarily large and the soliton states decouple from the low energy e!ective theories. So, their contributions to quantum e!ects are negligible. Their contribution to full dynamics becomes signi"cant in the strong coupling regime. Second, solitons are characterized by `topological chargesa, rather than by `Noether chargesa. Whereas the Noether charges are associated with the conservation laws associated with continuous symmetry of the theory, the topological conservation laws are consequence of topological properties of the space of non-singular "nite-energy solutions. The space of non-singular "nite energy solutions is divided into several disconnected parts. It takes in"nite amount of energy to make a transition from one sector to another, i.e. it is not possible to make a transition to the other sector through continuous deformation. Third, the solitons with "xed topological charges are additionally parameterized by a "nite set of numbers called `modulia. Moduli or alternatively called collective coordinates are parameters labeling di!erent degenerate solutions with the same energy. The space of solutions of "xed energy is called moduli space. The moduli of solitons are associated with symmetries of the solutions. For example, due to the translational invariance of the Yang}Mills}Higgs Lagrangian, the monopole solution sitting at the origin has the same energy as the one at an arbitrary point in R; the associated collective coordinates are the center of mass coordinates of the monopole. In addition, there are collective coordinates associated with the gauge invariance of the theory. Note, monopole carries charge of the ;(1) gauge group which is broken from the non-Abelian (S;(2)) one at in"nity (where the Higgs "eld takes its value at the gauge symmetry breaking vacuum). Thus, only relevant gauge
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transformations of non-Abelian gauge group that relate di!erent points in moduli space are those that do not approach identity at in"nity, i.e. those that reduce to non-trivial ;(1) gauge transformations at in"nity. Another important characteristic of solitons is that they are the minimum energy con"gurations for given topological charges, i.e. the energy of solitons saturates the Bogomol'nyi bound [94,510]. The lower bound is determined by the topological charges, e.g. the winding number for strings and ;(1) gauge charge carried by black holes. The original calculation [94] of the energy bound for a soliton in #at spacetime involves taking complete square of the energy density ¹ ; the minimum RR energy is saturated if the complete square terms are zero. Solitons therefore satisfy the "rst-order di!erential equations (`complete square termsa"0), the so-called Bogomol'nyi or self-dual equations. An example is the (anti) self-dual condition F "$夹F for Yang}Mills instantons IJ IJ [68]. Another example is the magnetic monopoles [507,592] in an S;(2) Yang}Mills theory, which satisfy the "rst-order di!erential equation BG"$DGU relating the magnetic "eld BG to the Higgs "eld U. Here, the Higgs "eld takes its values at the minimum of the potential <(U), where the non-Abelian gauge group S;(2) is spontaneously broken down to the Abelian ;(1) gauge group. The energy of solitons in asymptotically #at curved spacetime is given by the ADM mass [1,20,349], i.e. a PoincareH invariant conserved energy of gravitating systems. The ADM mass is de"ned in terms of a surface integral of the conserved current JI"¹IJK over a space-like J hypersurface at spatial in"nity. Here, ¹IJ is the energy-momentum tensor density and K is J a time-like Killing vector of the asymptotic spacetime. The so-called positive-energy theorem [146,295,379,387,483,531}534,631] proves that the ADM mass of gravitating systems is always positive. In such proofs, one calculates the energy associated with a small deviation around the background spacetime and "nds it always positive, implying that the background spacetime (Minkowski or anti-De Sitter space-time) is a stable vacuum con"guration. The proof of the positive energy theorem, "rst given in [631] and re"ned covariantly in [483], involves the volume and the surface integrals (related through the Stokes theorem) of Nester's 2-form, which is de"ned in terms of a spinor and its gravitational covariant derivative. Such proofs have an advantage of being easily generalized to supergravity theories. The positive energy theorem proves that the ADM mass of gravitating systems is always positive, provided the spinor satis"es the Witten's condition and the matter stress energy tensor, if any, satis"es the dominant energy condition. One way of proving positivity of the energy of solitons in curved spacetime is by embedding the solutions into (extended) supergravity theories [295,296] as solutions to equations of motion. In this case, the Nester's form is de"ned in terms of the supersymmetry parameters and their supercovariant derivatives. Then, the surface integral yields the supercharge anticommutation relations of extended supersymmetry, i.e. the 4-momentum term plus the central charge term. The 4-momentum in the surface integral is the ADM 4-momentum [1,20] of the soliton and the central charge corresponds to the topological charge carried by the soliton [201,485,489]; the soliton behaves as if a particle carrying the corresponding 4-momentum and quantum numbers. This is a reminiscent of BPS states in extended supergravities. One can think of solitons as realizations of states in supermultiplet carrying central charges of extended supersymmetry [299,399]. In fact, for
Note, the stress-energy tensor is second order in derivatives of spacetime coordinates.
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each Killing spinor, de"ned as a spinor "eld which is covariant with respect to the supercovariant derivative, one can de"ne a conserved anticommuting supercharge, whose anticommutation relation is just the surface integral of the Nester's 2-form. The integrand of the volume integral of the Nester's 2-form yields sum of terms bilinear in supersymmetry variations of the fermionic "elds in the supergravity theory. Since such terms are positive semide"nite operators, provided the (generalized) Witten's condition [631] and the dominant energy condition for the matter stress-energy tensor are satis"ed, the terms in the surface integral have to be non-negative, leading to the inequality `(ADM mass)5(the maximum eigenvalue of the topological charge term)a. Again a reminiscent of the mass bound for the states in the BPS supermultiplet. This bound is saturated i! the supersymmetry variations of fermions are all zero. The equations obtained by setting the supersymmetry variations of fermions equal to zero are called the Killing spinor equations. These are a system of "rst-order di!erential equations satis"ed by the minimum energy con"guration among solutions with the same topological charge. Such a con"guration is a bosonic con"guration which is invariant under supersymmetry transformations and therefore is called supersymmetric. The necessary and su$cient condition for the existence of supersymmetric solution is the existence of `non-zeroa superconvariantly constant spinors, i.e. Killing spinors. Note, such Killing spinors de"ne supercharges, which act on the lowest spin state to build up supermultiplets of superalgebra. Killing spinors are Goldstone modes of broken supersymmetries; for each supersymmetry preserved, the corresponding supercharge is projected onto zero norm states, and the rest of supercharges are associated with Goldstone spinor degrees of freedom originated from broken supersymmetries. The number of supercharges which are projected onto the zero norm states is determined by the number of distinct eigenvalues of the central charge matrix. In the language of solitons, such central charge matrix is determined by the charge con"gurations of solitons. Alternatively, one can determine the number of supersymmetries preserved by the solitons from the spinor constraints, which are byproducts of the Killing spinor equations along with self-dual or the "rst-order di!erential equations. The number of constraints on the Killing spinors are again determined by the charge con"guration of the solitons. These constraints determine the number of independent spinor degrees of freedom, i.e. the number of supersymmetries preserved by the soliton. Thus, the number of supersymmetries preserved by solitons is intrinsically related to the topological charge con"gurations of solitons through either the number of eigenvalues of central matrix or the number of constraints on the Killing spinors. In the following, we elaborate on ideas discussed in the above in a more precise and concrete way, by quantifying ideas and giving some examples. First, we discuss how the physical parameters (mass, angular momenta, etc.) are de"ned from solitons. Then, we discuss the BPS multiplets of extended supersymmetry theories. Finally, we discuss positive energy theorem of general relativity and extended supergravity theories. 2.1. Physical parameters of solitons We discuss how to de"ne physical parameters (e.g. the ADM mass, angular momenta, ;(1) charges) of gravitating systems. This serves to "x our conventions for de"ning parameters of solitons. The classical solutions near the space-like in"nity can be regarded as the `imprintsa of the ADM mass, angular momenta and electric/magnetic charges of the source.
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First, we discuss the parameters of spacetime metric. The physical parameters are de"ned with reference to the background (asymptotic) spacetime. We assume that the spacetime is asymptotically Minkowski at space-like in"nity, since the solitons under consideration in this review satisfy this condition. We consider the following general form of action in D spacetime dimensions:
1 R#L , (1)
16pG" , where G" is the D-dimensional Newton's constant (related to the Plank constant i as i "8pG") , " " , and L is the matter Lagrangian density. For the signature of the metric g , we take the mostly
IJ positive convention (!#2#). From (1), one obtains the following Einstein "eld equations for gravitation: (2) G "R !g R"8pG,¹ , " IJ IJ IJ IJ where the matter stress-energy tensor ¹ is de"ned as IJ 2 R((!gL )
, ¹ , (3) IJ RgIJ (!g S" (!g d"x
where ¹ are stresses, ¹ are momentum densities and ¹ is the mass-energy density GH G (i, j"1,2, D!1). In order to measure the mass, the momenta and the angular momenta of gravitating systems, one usually goes to the external spacetime far away from the source. In this region, the gravitational "eld is weak and, therefore, the Einstein's "eld equations (2) take the form linear in the deviation h of the metric g from the #at one g (g "g #h , "h ";1). This linearize "eld equations IJ IJ IJ II IJ IJ IJ have the invariance under the in"nitesimal coordinate transformations (xIPxI#mI) h P IJ h !R m !R m , which resembles the gauge transformation of ;(1) gauge "elds. (The linearized IJ J I I J Riemann tensor, Einstein tensor, etc., are examples of invariants under this transformation.) By using this gauge invariance, one can "x the gauge by imposing the `Lorentz gaugea condition R (hIJ!gIJh? )"0. This gauge condition is left invariant under the gauge transformations ? J satisfying m? @ "0. In this gauge, the Einstein's equations take the form, which resembles the @ Maxwell's equations: 1
h "!16pG" ¹ ! g ¹ ,!16pG"¹M , (4) IJ , IJ , IJ D!2 IJ
where "R R? is the #at (D!1)-dimensional space Laplacian and ¹,¹I . ? I The linearized Einstein's equations (4) have the following general solution that resembles the retarded wave solution of the Maxwell's equations:
¹M (t!"x!y", y) 1 16pG" 16pG" IJ , , h (xG)" d"\y" ¹M d"\y IJ "x!y""\ r"\ IJ (D!3)X (D!3)X "\ "\ 16pG" xI , yI¹M d"\y#2 , # IJ r"\ X "\
(5)
In the linearized "eld theory, the spacetime vector indices are raised and lowered by the Minkowski metric g . IJ
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where 2pL> X" L n#1 C 2
is the area of SL and i, k"1,2, D!1, are spatial indices. Note, the ADM D-momentum vector PI and angular momentum tensor JIJ of the gravitating system are de"ned as
PI" ¹I d"\x, JIJ" (xI¹J!xJ¹I) d"\x .
(6)
In particular, the time component of PI is the ADM mass M, i.e. M"P. With a suitable choice of coordinate basis, one can put the spatial components JGH (i, j"1,2, D!1) of JIJ in the following form expressed in terms of the angular momenta J (k"1,2, ["\]) in each rotational plane: I 0 J !J 0 [JGH]" , (7) 0 J !J 0 \
where for the even D the last row and column have zero entries. In obtaining the general leading order expression for the metric, one chooses the rest frame (PG"0) with the origin of coordinates at the center of mass of the system (xG¹ d"\x"0). In this frame, JG"0, JGH"2xG¹H d"\x and g takes the form: IJ M 1 16pG" xI 1 16pG" , , #O dt! JIG#O dt dxG ds"! 1! r"\ r"\ X r"\ r"\ (D!2)X "\ "\ M 16pG" , d #(gravitational radiation terms) dxG dxH . # 1# r"\ GH (D!2)(D!3)X "\ (8)
Note, the leading order terms of the asymptotically Minkowski metric is time independent and is determined uniquely by the ADM mass M and the intrinsic angular momenta JGH of the source. The general action (1) contains the following kinetic term for a d-form potential 1 A " A 2 B dxI2 dxIB B d! I I with "eld strength F
"dA : B> B
1 1 d"x(!g F . S " BU 16pG" 2(d#1)! B> , We omit the dilaton factor in the kinetic term for the sake of the argument.
(9)
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Note, in this kinetic term, G" is absorbed in the action in contrast to the form of the matter term in , (1). The "eld equations and Bianchi identities of A are B (10) d夹F "2i (!)B夹J , dF "0 , B B> B> " where J is the rank d source current and 夹 denotes the Hodge-dual transformation in D spacetime B dimensions, i.e. 1 (夹A )I2I"\B, eI2I"(A ) "\B>2 " with e2"\"1. I BI B d! The soliton that carries the `Noethera electric charge Q under A is an elementary extended B B object with d-dimensional worldvolume, called (d!1)-brane, and has the electric source J coming from the p-model action of the (d!1)-brane. The `topologicala magnetic charge P I of A is carried B B by a solitonic (i.e. singularity and source free) object with dI -dimensional worldvolume, called (dI !1)-brane, where dI ,D!d!2. The `Noethera electric and the `topologicala magnetic charges of A are de"ned as B 1 (!)B夹J " 夹F , Q ,(2i " M B (2i "\B\ B> B "\B 1 " 1 PI, F . (11) B (2i B> B> 1 " These charges obey the Dirac quantization condition [482,591]:
Q PI n B B" , n3Z . 4p 2
(12)
The electric and magnetic charges of A have dimensions [Q ]"¸\ "\B\ and B B [P I ]"¸ "\B\, respectively. Electric/magnetic charges are dimensionless when D"2(d#1). B Examples are point-like particles (d"1) in D"4, strings (d"2) in D"6 [311,542] and membranes (d"3) in D"8 [311,388]. From (11), one sees that the AnsaK tze for F for the soliton that carries electric or magnetic B> charge of A are respectively given by B , (13) 夹F "(2i Q e I /X I , F "(2i P I e /X B> " B B> B> " B B> B> B> where e denotes the volume form on SL, and the electric and magnetic charges of A are de"ned L B from the asymptotic behaviors: Q PI X u B , F & B> B , (14) A& B B> (2i rB> B (2i r"\B\ " " where r is the transverse distance from the (d!1)-brane, u is the volume form for the (d!1)B brane worldvolume and X is the volume form of SB> surrounding the brane. B> From the elementary (d!1)-brane, one "nds that the electric charge Q is related to the tension B ¹ of the (d!1)-brane in the following way: B Q "(2i ¹ (!)"\BB> . (15) B " B
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Here, ¹ has dimensions [¹ ]"M¸B\ in the unit c"1 and therefore is interpreted as mass per B B unit (d!1)-brane volume. In particular, for a 0-brane (d"1) the tension ¹ is the mass. The Dirac quantization condition (12), together with (15), yields the following form of the magnetic charge P I of A : B B 2pn PI" (!)"\BB>, n3Z . (16) B (2i ¹ " B We comment on the ADM mass of (d!1)-branes. Note, in deriving (8) we assumed that the metric g depends on all the spatial coordinates. So, (8) applies only to the 0-brane type soliton (or IJ black holes). The (d!1)-branes do not depend on the (d!1) longitudinal coordinates internal to the (d!1)-brane and therefore the Laplacian in (4) is replaced by the #at (D!d!1)-dimensional one. As a consequence, in particular, the (t, t)-component of the metric has the asymptotic behavior:
16pG" M 1 , B g &! 1! . (17) RR (D!2)X < r"\B\ "\B\ B\ Here, the ADM mass M of the (d!1)-brane is de"ned as M ,¹ d"\x"< ¹ d"\Bx, B B B\ where < is the volume of the (d!1)-dimensional space internal to the (d!1)-brane. So, for B\ (d!1)-branes it is the ADM mass `densitya
M o , B " ¹ d"\Bx B < B\ that has the well-de"ned meaning. As an example, we consider the elementary BPS (d!1)-brane in D dimensions. The leading order asymptotic behavior of the (t, t)-component of the metric of (d!1)-brane carrying one unit of the d-form electric charge is
D!d!2 c" B g &! 1! , dI '0 , RR D!2 r"\B\
(18)
where c",2i ¹ /dI X I is the unit (d!1)-brane electric charge and r,(x#2#x ) B " B B> "\B\ is the radial coordinate of the transverse space. For (d!1)-branes carrying m units of the basic electric charge, c" in (18) is replaced by mc". From (17) and (18), one obtains the following ADM B B mass density of the (d!1)-brane carrying one unit of electric charge: o "¹ "(1/(2i ) "Q " . " B B B
(19)
2.2. Supermultiplets of extended supersymmetry 2.2.1. Spinors in various dimensions Before we discuss the BPS states in extended supersymmetry theories, we summarize the basic properties of spinors for each spacetime dimensions D. More details can be found, for example, in
When dI "0, e.g. a string in D"4, the metric is asymptotically logarithmically divergent. In this case, the ADM mass density is determined from volume integral of the (t, t)-component of the gravitational energy-momentum pseudo-tensor [440].
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[438,575,628,629]. We assume that there is only one time-like coordinate. The types of superPoincareH algebra satis"ed by supercharges depend on D. The superalgebra is classi"ed according to the fundamental spinor representations of the homogeneous group SO(1, D!1) and the vector representation of the automorphism group that supercharges belong to. The pattern of superalgebra repeats with D mod 8. In even D, one can de"ne c-like matrix c,gcc2c"\ which anticommutes with cI and has the property c"1 (implying g"(!1)"\ ), required for constructing a projection operator. So, the 2 " complex component Dirac spinor t, which is de"ned to transform as dt"!e RIJt (RIJ,![cI, cJ]) under the in"nitesimal Lorentz transformation, in even D IJ is decomposed into 2 inequivalent Weyl spinors t "(1#c)t and t "(1!c)t with \ > 2" \ complex components each. We discuss the reality properties of spinors. One can always "nd a matrix B satisfying RIJH"BRIJB\. B de"nes the charge conjugation operation: tPtA"Ct,B\tH .
(20)
By de"nition, the charge conjugation operator C commutes with the Lorentz generators RIJ, implying that t and tA have the same Lorentz transformation properties. If C"1, or equivalently BBH"1, the Dirac spinor t can be reduced to a pair of Majorana spinors (i.e. eigenstates of C) t "(1#C)t and t "(1!C)t. This is possible in D"2, 3, 4, 8, 9 mod 8. First, in odd D, Majorana spinors are necessarily self-conjugate under C and are always real. So, the Dirac spinors in odd D are real [pseudoreal] in D"1, 3 mod 8 [D"5, 7 mod 8]. In even D, Majorana spinors can be either complex or real depending on whether t and tA have the same or opposite helicity. Namely, since Cc"(!1)"\ cC, t and tA have the same (opposite) helicity for even (odd) (D!2)/2, i.e. D"2 mod 8 [D"4, 8 mod 8]. So, in even D, the Dirac spinors are real (complex) or pseudoreal for D"2 mod 8 (D"4, 8 mod 8) or D"6 mod 8, respectively. In particular, in D"2 mod 8, both the Weyl and the Majorana conditions are satis"ed, and therefore in this case the Dirac spinor t is called Majorana}Weyl. We saw that supercharges Q (i"1,2, N), transforming as spinors under SO(1, D!1), have G di!erent chirality and reality properties depending on D. The set +Q ,, furthermore, transforms as a G vector under an automorphism group, with i acting as a vector index. The automorphism group depends on the reality properties of +Q ,. The automorphism group is SO(N), ;Sp(N) or S;(N);;(1) for G real, pseudoreal or complex case, respectively. In D"2 mod 8 and D"6 mod 8, the pair of Weyl spinors with opposite chiralities are not related via C and therefore are independent: the automorphism groups are SO(N );SO(N ) and ;Sp(N );;Sp(N ) in D"2 mod 8 and D"6 mod 8, > > > > respectively, where N #N "N. The central charge Z'( transforms as a rank 2 tensor under the > \ automorphism group with (I, J) acting as tensor indices. In D"0, 1, 7 mod 8 [D"3, 4, 5 mod 8], the central charge has the symmetry property Z'("Z(' [Z'("!Z(']. The number N of supercharges Q' in each D is restricted by the physical requirement that particle helicities should not exceed 2 when compacti"ed to D"4 [17,69,169,274,479]. This limits the maximum D with 1 time-like coordinate and consistent supersymmetric theory to be 11 with N"1 supersymmetry, i.e. 32 supercharge degrees of freedom. This corresponds to N"8 supersymmetry in D"4 when compacti"ed on ¹. In D(11, the number of spinor degrees of freedom cannot exceed that of N"1, D"11 theory. For the pseudoreal cases, i.e. D"5, 6, 7 mod 8, only even N are possible.
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2.2.2. Central charges and super-PoincareH algebra We discuss types of central charge Z'( one would expect in the super PoincareH algebra. According to a theorem by Haag et al. [330], within a unitary theory of point-like particle interactions in D"4 the central charge can be only Lorentz scalar. However, in the presence of p-branes (p51), central charges Z'(2 N transforming as Lorentz tensors can be present in the I I superalgebra without violating unitarity of interactions [201]. In fact, as will be shown, it is the Lorentz tensor type central charges in higher dimensions that are responsible for the missing central charge degrees of freedom in lower dimensions when the higher-dimensional superalgebra is compacti"ed with an assumption that no Lorentz tensor type central charges are present [603]. The Lorentz tensor type central charges appear in the supersymmetry algebra schematically in the form: (21) +Q', Q(,"d'((CcI) P # (CcI2IN) Z'(2 N , ?@ I I ? @ ?@ I N 2 where P is D-dimensional momentum, I, J"1,2, N label supersymmetries and a, b are spinor I indices in D dimensions. Here, (CcI) in (21) is replaced by (CcIP!) for positive or negative chiral ?@ ?@ Majorana spinors Q' (e.g. type-IIB theory), where P projects on the positive or negative ! ! chirality subspace, and also similarly for (CcI2IN) . Note, Z'(2 N commute with Q' and P , but ?@ I I ? I transform as second rank tensors under the Lorentz transformation, and therefore are central with respect to supertranslation algebra, only. The number of central charge degrees of freedom is determined by the number of all the possible (I, J) in (21) [41}44]. In the sum term in (21), one has to take into account the overcounting due to the Hodge-duality between p and D!p forms (Z'(2 N&Z'(2 "\N). When p"D!p, Z'(2 N are I I I I I I self-dual or anti-self dual. (For this case, the degrees of freedom are halved.) (I, J) on Z'(2 N are I I de"ned to have the same permutation symmetry as (a, b) in cI2IN so that cI2INZ'(2 N is symmetric ?@ ?@ I I under the simultaneous exchanges of indices in the pairs (I, J) and (a, b) so that they have the same symmetry property (under the exchange of the indices) as the left-hand side of (21). Namely, only terms associated with cI or cI2IN that are either symmetric or antisymmetric under the exchange ?@ ?@ of a and b can be present on the right-hand side of (21). 2.2.3. Central charges and i-symmetry The p-form central charge Z'(2 N in (21) arises from the surface term of the Wess}Zumino (WZ) I I term in the p-brane worldvolume action [201]. Before we discuss this point, we summarize how WZ term emerges in the p-brane worldvolume action [599,600]. In the Green}Schwarz (GS) formalism [86,313,315,351] of the supersymmetric p-brane worldvolume action, one achieves manifest spacetime supersymmetry by generalizing spacetime with bosonic coordinates XI (k"0, 1,2, D!1) and global Lorentz symmetry to superspace R with coordinates Z+"(XI, h?) and super-PoincareH invariance. Here, a is a Ddimensional spacetime spinor index and the spacetime spinor h? takes an additional index I (I"1,2, N) for N-extended supersymmetry theories, i.e. h'?. Fields in the GS action are
A Majorana spinor Q is de"ned as QM "Q2C, where the bar denotes the Dirac conjugate. The positive or negative chiral spinor Q is de"ned as cQ "$Q . ! ! !
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regarded as maps from the worldvolume = to R. The worldvolume = of a p-brane has coordinates mG"(q, p ,2, p ) with worldvolume vector index i"0, 1,2, p. We denote an immersion from N = to R as : =PR. The pullback H of a form in R by induces a form in =. To generalize the bosonic p-brane worldvolume Lagrangian density L "¹ [!det(R XI(m)R XJ(m)g )] to be N G H IJ invariant under the supersymmetry transformation as well as local reparameterization and global PoincareH transformations, one introduces a supertranslation invariant D-vector-valued 1-form PI,dXI!ihM cI dh. This corresponds to the spacetime component of the left-invariant 1-form P "(PI, P?"dh?) on R. The simplest and straightforward supersymmetric generalization of the bosonic worldvolume p-model action for a p-brane is [2,85}87,201,313,315,351,377] S "¹ N
dN>m(!det(PI(m)PJ(m)g ) , (22) G H IJ 5 where ¹ is the p-brane tension and PI is the mG-component of the pullback of the 1-form PI in R by N G
, i.e. ( H P )(m)"P (m) dmG with PI"R XI!ihM cIR h and P?"R h? (R ,R/RmG). (22) is manifestly G G G G G G G invariant under the global super-PoincareH and local reparameterization transformations, but is not invariant under a local fermionic symmetry, called `i-symmetrya [3,88,131,313,377], which is essential for equivalence of the GS and NSR formalisms of the worldvolume action. To make (22) invariant under the i-symmetry, one introduces an additional term S , called `Wess}Zumino 58 (WZ) actiona, into (22). To construct the Wess}Zumino (WZ) action for a p-brane, one introduces the super-PoincareH invariant closed (p#2)-form h on R. Such closed (p#2)-forms exist only N> for restricted values of D and p. The complete listing of the values of (D, p) are found in [2]. The maximum values of allowed D and p are D "11 and p "5, which can also be determined
by the worldvolume bose-fermionic degrees of freedom matching condition discussed in the next paragraph. The super-PoincareH invariant closed (p#2)-form, in general, has the form is closed, one can locally write h in terms of h "PI2PIN> dhM c 2 N> dh. Since h I I N> N> N> a (p#1)-form b (on R) as h " db . Then, a super-PoincareH invariant WZ action for N> N> N> over = [2,85}87,201,313,315,351,377]: a p-brane is obtained by integrating b N>
. S "¹ dN>m H b N> 58 N
(23)
Note, whereas S and S are individually invariant under the local reparameterization and global 58 super-PoincareH transformations, the i-symmetry is preserved only in the complete action S "S #S . The i-symmetry gauges way the half of the degrees of freedom of the spinor h, N 58 thereby only 1/2 of spacetime supersymmetry is linearly realized as worldvolume supersymmetry [378]. To summarize, the invariance under the i-symmetry necessitates the introduction of b (on R) via the WZ term; b couples to the worldvolume of the p-branes and becomes the N> N> origin of the central charge term in the supersymmetry algebra. We comment on the allowed values of p and the number N of spacetime supersymmetry for each D. This is determined [241] by matching the worldvolume bosonic degrees N and fermionic degrees N of freedom. First, we consider the case where the worldvolume theory corresponds to $ scalar supermultiplet (with components given by scalars and spinors). By choosing the static gauge (de"ned by XI(m)"(XG(m),>K(m))"(mG,>K(m)), with i"0, 1,2, p and m"p#1,2, D!1), one "nds that the number of on-shell bosonic degrees of freedom is N "D!p!1. We denote the
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number of supersymmetries and the number of real components of the minimal spinor in D-dimensional spacetime [(p#1)-dimensional worldvolume] as N and M [n and m], respectively. Then, since the i-symmetry and the on-shell condition each halves the number of fermionic degrees of freedom, the number of on-shell fermionic degrees of freedom is N "mn"MN. The allowed $ values of N and p for each D is determined by the worldvolume supersymmetry condition N "N , $ i.e. D!p!1"mn"MN. The complete listing of values of N and p are found in [234,241]. The maximum number of D in which this condition can be satis"ed is D "11 (p"2) with M"32
and N"1. So, for other cases (D(11), MN432. Similarly, the maximum value of p for which this condition can be satis"ed is p "5. The `fundamentala super p-branes [2] that satisfy
this condition are (D, p)"(11, 2), (10, 5), (6, 3), (4, 2). The 4 sets of p-branes obtained from these `fundamentala super p-branes through double-dimensional reduction are named the octonionic, quaternionic, complex and real sequences. Note, in addition to scalars and spinors, there are also higher spin "elds on the worldvolume [241]: vectors or antisymmetric tensors. First, we consider vector supermultiplets. Since a worldvolume vector has (p!1) degrees of freedom, the worldvolume supersymmetry condition N "N becomes D!2"mn"MN. This condition intro $ duces additional points in the brane-scan. Vector supermultiplets exist only for 34p49 and the bose}fermi matching condition can be satis"ed in D"4, 6, 10, only. Second, we consider tensor worldvolume supermultiplets. In p#1"6 worldvolume dimensions, there exists a chiral (n , n )"(2, 0) tensor supermultiplet (B\ , j', '( ), I, J"1,2, 4, with a self-dual 3-form "eld > \ IJ strength, corresponding to the D"11 5-brane. The decomposition of this (2, 0) supermultiplet under (1, 0) into a tensor multiplet with 1 scalar and a hypermultiplet with 4 scalars, followed by truncation to just the tensor multiplet, leads to worldvolume theory of 5-brane in D"7. 2.2.4. Central charges and topological charges We illustrate how Lorentz tensor type central charges (associated with p-branes) arise in the supersymmetry algebra [201]. Since the action S has the manifest super-PoincareH invariance, N one can construct supercharges QG from the conserved Noether currents j associated with ? ? super-PoincareH symmetry. Whereas (22) is invariant under the super-PoincareH variation, i.e. d S "0, the integrand of the WZ action (23) is only quasi-invariant. Namely, since d b "dD 1 1 N> N for some p-form D , the integrand of S transforms by total spatial derivative: N 58 ¹ d ( H b )"d( H D ),R DG dqdp2 dpN. It is D that induces `topological N 1 N> N G N N chargea which becomes central charge in the super-PoincareH algebra. Generally, when a Lagrangian density L is quasi-invariant under some transformation, i.e. d L"R DG , the associated G Noether current jG contains an `anomalousa term DG : RL !DG . jG "d Z+ R(R Z+) G Such an anomalous term modi"es the algebra of the conserved charges Q "dNp jO to include a topological (or central) terms A . For a p-brane, the WZ action (23) gives rise to the central term in the supersymmetry algebra of the form:
A "¹ (CcI2IN) dNp jO I2IN , ?@ 2 ?@ N
(24)
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where jO I2IN is the (worldvolume) time component of the topological current density 2 jG I2IN"eGH2HNR XI(m)2R NXIN(m). So, p-form central charges in supersymmetry algebra (21) has 2 H H the following general form [44] given by the surface integral of a (p#1)-form local current J'( 2 N(x) over a space-like surface embedded in D-dimensional spacetime: II I
Z'(2 N" d"\RI J'( 2 N(x) . II I I I
(25)
Here, the (p#1)-form local current J'( 2 N(x) has contributions from individual p-branes with the II I coordinates X? (q, p ,2, p ) and charges z'( (index a labeling each p-brane): I N ?
J'( 2 N(x)" dq dp 2 dp z'(d"(x!X?(q, p ,2, p )) N ? N II I ? (26) ;R X? 2R NX? N (q, p ,2, p ) . N N I
O I This (p#1)-form current is coupled to a (p#1)-form gauge potential A'( 2 N(x) of the low energy II I e!ective supergravity in the following way:
S& d"x AII2IN(x)J'( 2 N(x) '( II I
(27) " dq dp 2 dp AII2IN(X?)R X? 2R NX? N z'( , O I N '( N I ? ? where z'( are the charges of A'( 2 N(x) carried by the ath p-brane with the coordinates X? . The ? II I I "eld equation for A'( 2 N(x) is II I (28) RHR A'( 2 N (x)"J'( 2 N(x) . II I H I I I
So, one can think of Z'(2 N as being related to charges of A'( 2 N(x) with the charge source given II I I I by p-branes with their worldvolumes coupled to A'( 2 N(x). There is a one-to-one correspondence II I between A'( 2 N(x) in the e!ective supergravity theory and Z'( 2 N in the superalgebra, i.e. there II I II I are as many central extensions as form "elds in the e!ective supergravity. 2.2.5. S-theory The maximally extended superalgebra has 32 real degrees of freedom in the set +Q', of ? supercharges, i.e. N"1 supersymmetry in D"11 or N"8 supersymmetry in D"4. So, the right-hand side of (21) has at most (32;33)/2"528 degrees of freedom; the sum of D degrees of freedom of the momentum operator PI and the degrees of freedom of central charges Z'(2 N in (21) I I has to be 528. This is the main reason for the necessity of existence of p-branes in higher dimensions [603]; N"1 supersymmetry in D"11 without central charge has only 11 degrees of freedom on the right-hand side of (21).
In the supersymmetry algebra (21), the p-brane tension ¹ is set equal to 1. N So, a p-form central charge is related to boundaries of the p-brane. For example for a string ( p"1), Z & I X (0)!X (p). I I
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The extended superalgebra (21) can be derived by compactifying either a type-A superalgebra in D"2#10 or a type-B superalgebra in D"3#10 (the so-called `S-theoriesa) [41}44], with the Lorentz symmetries SO(10, 2) and SO(9, 1) SO(2, 1), respectively. The superalgebra of the typeIIA(B) theory is obtained by compactifying the type-A(B) algebra. First, the type-A algebra has the form: (29) +Q , Q ,"c++Z #c+2+Z>2 , ?@ ++ ?@ + + ? @ where M "0 , 0, 1,2, 10 is a D"12 vector index with 2 time-like indices 0 , 0. Note, in D"12 G with 2 time-like coordinates, only gamma matrices which are (anti)symmetric are c++ and c+2+, with c+2+ being self-dual. So, one has terms involving 2-form central charge Z and ++ self-dual 6-form central charge Z>2 , without momentum operator P+ term, in (29). Generally, + + in the D(12 supersymmetry algebra compacti"ed from the type-A algebra, the spinor indices a and a of 32 spinors Q? are regarded as those of SO(c#1, 1) and SO(D!1, 1) Lorentz groups, ? respectively, where c is the number of compacti"ed dimensions from the point of view of D"10 string theory. Second, the type-B algebra has the form: (30) +Q? , Q@M M ,"cI M (cq )? @M PG #cI M IIc? @M > #cI M 2I(cq )? @M XG 2 , ? @ ?@ ?@ ?@ G I III G I I where the indices are divided into the D"10 ones a, bM "1, 2,2,16 and k"0, 1,2, 9 with the Lorentz group SO(1, 9), and the D"3 ones a , bM "1, 2 and i"0 , 1 , 2 with the Lorentz group SO(1, 2). (The barred [unbarred] indices are spinor [spacetime vector] indices.) Here, cI [q ] are G gamma matrices of the SO(1, 9) [SO(1, 2)] Cli!ord algebra and c? @M "ip? @M "e? @M . We discuss the maximal extended superalgebras that follow from the type-A algebra. First, the following N"1, D"11 superalgebra is obtained from (29) by compactifying the 0 -coordinate: +Q , Q ,"(CIC) P #(CIIC) Z #(CI2IC) X 2 , ?@ I I ?@ I I ? @ ?@ I where each term on the right-hand side emerges from the terms in (29) as
(31)
Z PP Z 66"11#55 ++ I II (32) 462"462 . Z>2 PX 2 I I + + The central charges Z and Z 2 are associated with the M2 and M5 branes, respectively. The I I II maximal extended superalgebras in D(11 are obtained by compactifying the D"11 supertranslation algebra (31) on tori. The central charge degrees of freedom in lower dimensions are counted by adding the contribution from the internal momentum P (m"1,2, 11!D) and the number of K ways of wrapping M2 and M5 branes on cycles of ¹\" in obtaining various p-branes in lower dimensions. Schematically, decompositions of the terms on the right-hand side of (31) are P "P P , Z "Z ZK ZKK , II I I I K II (33) X 2 "X 2 XK2 XKK XKKK XK2K XK2K . I I III II I I I I I The N(N!1) Lorentz scalar central charges of the (maximal) N-extended D(11 superalgebras originate from the Lorentz scalar type terms (under the SO(D!1, 1) group) on the right-hand sides of (33), i.e. P , ZKK and XK2K. In this consideration, one has to take into account equivalence K under the Hodge duality in D dimensions. N(N!1)"56 Lorentz scalar central charges of N"8 superalgebra in D"4 originate from the 7 components of P , (7;6)/(2;1)"21 terms in ZKK, K
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(7;6;5;6;5)/(5;4;3;2;1)"21 terms in XK2K and 7 terms in the Hodge dual of XK2 . I I The similar argument regarding the supertranslational algebras of type-IIB and heterotic theories can be made, and details are found, for example, in [44,603]. We saw that one has to take into account the Lorentz tensor type central charges in higher dimensions to trace the higher-dimensional origin of N(N!1) 0-form central charge degrees of freedom in N-extended supersymmetry in D(11. This supports the idea that in order for the conjectured string dualities (which mix all the electric and magnetic charges associated with N(N!1) 0-form central charges in D(10 among themselves) to be valid, one has to include not only perturbative string states but also the non-perturbative branes within the full string spectrum. In lower dimensions, these central charges are carried by 0-branes (black holes). It is a purpose of Section 4 to construct the most general black hole solutions in string theories carrying all of 0-form central charges. In Section 6.3.3, we identify the (intersecting) p-branes in higher dimensions which reduce to these black holes after dimensional reduction.
2.2.6. Central charges and moduli xelds We comment on relation of central charges to ;(1) charges and moduli "elds [7,8]. Except for special cases of D"4, N"1, 2 and D"5, N"2, scalar kinetic terms in supergravity theories are described by p-model with target space manifold given by coset space G/H. Here, a non-compact continuous group G is the duality group that acts linearly on "eld strengths HK2 N> and is on-shell I I and/or o!-shell symmetry of the action. The isotropy subgroup HLG is decomposed into the automorphism group H of the superalgebra and the group H related to the matter
multiplets: H"H H . (Note, the matter multiplets do not exist for N'4 in D"4, 5 and in
maximally extended supergravities.) The properties of supergravity theories are "xed by the coset representatives ¸ of G/H. ¸ is a function of the coordinates of G/H (i.e. scalars) and is decomposed as: ¸\"(¸ K, ¸ K) , (34) ¸"(¸KR)"(¸K , ¸K), ' ' where (A, B) and I respectively correspond to 2-fold tensor representation of H and the fundamental representation of H . Here, K runs over the dimensions of a representation of G.
The (p#2)-form strength HK kinetic terms are given in terms of the following kinetic matrix determined by ¸: . (35) NKR"¸ K¸ R !¸ K¸' ' R So, the `physicala "eld strengths of (p#1)-form potentials in supergravity theories are `dresseda with scalars through the coset representative and the (p#2)-form "eld strengths appear in the supersymmetry transformation laws dressed with the scalars. The central charges of extended superalgebra, which is encoded in the supergravity transformations rules, are expressed in terms of electric QK, "\N\GK and magnetic PK, N>HK charges of (p#2)-form "eld strengths 1 1 HK"dAK (GK"RL/RHK) and the asymptotic values of scalars in the form of the coset representative, manifesting the geometric structure of moduli space. These central charges satisfy the di!erential equations that follow from the Maurer}Cartan equations satis"ed by the coset representative. One of the consequences of these di!erential equations is that the vanishing of a subset of central charges (resulting from the requirement of
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supersymmetry preserving bosonic background) forces the covariant derivative of some other central charges to vanish, i.e. `principle of minimal central chargea [258,259]. Furthermore, from de"ning relations of the kinetic matrix of (p#2)-form "eld strengths and the symmetry properties of the symplectic section under the group G, one obtains the sum rules satis"ed by the central and matter charges. For other cases, in which the scalar manifold cannot be expressed as coset space, one can apply the similar analysis as above by using techniques of special geometry [132,270,576]. For this case, the roles of the coset representative and the Maurer}Cartan equations are played respectively by the symplectic sections and the Picard}Fuchs equations [134]. 2.2.7. BPS supermultiplets We discuss massive representations of extended superalgebras with non-zero central charges [264,330,485,489,575], i.e. the BPS states. It is convenient to go to the rest frame of states de"ned by P "(M, 0,2, 0), where M is the rest mass of the state. The little group, de"ned as a set of I transformations that leave this P invariant, consists of SO(D!1), the automorphism group and I the supertranslations. Since central charges Z'( transform as a second rank tensor under the automorphism group, only the subset of automorphism group that leaves Z'( invariant should be included in the little group. Central charges inactivate some of supercharges, reducing the size of supermultiplets. In the following, we illustrate properties of the BPS states for the D"4 case with an arbitrary number N of supersymmetries. In the Majorana representation, the central charges ;'( and <'( (I, J"1,2, N) appear in the N-extended superalgebra in D"4 in the form [264]: +Q', Q(,"(cIC) P d'(#C ;'(#(c C) <'( , (36) ? @ ?@ I ?@ ?@ where a, b"1,2, 4 are indices of Majorana spinors, C is the charge conjugation matrix C"!C2, and the supercharges Q' are in the Majorana representation. In the Majorana ? representation, ;'( and <'( are hermitian operators and antisymmetric. Alternatively, one can express the superalgebra (36) in the Weyl basis. In this basis, a 4component spinor Q (a"1,2, 4) is decomposed into left- and right-handed 2-component Weyl ? spinors: "(ip Q*')? , (37) Q'"(Q' ) , (Q' )? "e? @Q Q*' * ? *? 0 @Q Q where a, b"1, 2 and a, bQ "1, 2 are Weyl spinor indices, and e?@"e?@"(ip ) "!e "!e Q is ?@ ?@ ?@ the 2-dimensional Levi}Civita symbol. Namely, the lower and upper components of a 4-component spinor Q (a"1,2, 4) are (Q ) and e? @Q Q* Q , respectively. In this 2-component Weyl basis *@ ? *? representation, anticommutations (36) become +Q', Q*( ,"(p ) Q PId'(, ? @Q I ?@ (38) +Q', Q(,"e Z'(, +Q*' , Q*( Q ,"e Q Z'( , @ ?@ ? @ ?@ ? where Z'(,!<'(#i;'(. The central charge matrix Z'( can be brought to the block diagonal form by applying the ;(N) automorphism group: Z'("diag(z eGH, z eGH,2, z eGH)"ip ZK , ,
,
(39)
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where z (m"1,2, [N/2]) are eigenvalues of Z'( and ZK ,diag(z ,2, z ). There are extra K ,
,
0 entries in the Nth row and column in Z'( for an odd N. Further rede"ning the supercharges and making use of the reality condition satis"ed by the supercharges, one can simplify supersymmetry algebra: +SK , S*L ,"d dKLd (M!(!)@z ) , ?? @@ ?@ ?@ L +SK , SL ,"+S*K , S*L ,"0 , (40) ?? @@ ?? @@ where a, b"1, 2, a, b"1, 2, and m, n"1,2, N/2. For odd N, there are extra anticommutation relations associated with the extra 0 entries in Z'(. Since the left-hand sides of (40) are positive semide"nite operators, the rest mass M of the particles in the supermultiplet is always greater than or equal to all the eigenvalue of the central charge matrix and therefore M5max+"z ", . (41) K The state that saturates the bound (41) is called the BPS state. The supermultiplet that do not saturate (41) is called the long multiplet and is the same as that of extended superalgebra without central charges. The BPS supermultiplet is called short multiplet, since there are fewer supercharges (or raising operators) available for building up supermultiplet (since the supercharges that annihilate the supersymmetric vacuum get projected onto zero norm states). The type of supermultiplet that BPS states belong to depends on the number of distinct eigenvalues of the central charge matrix Z'(. In the following, we give examples on all the possible BPS multiplets of N"4 supersymmetry algebra. N"4 superalgebra has [N/2]"2 eigenvalues z and z . There are two types of BPS supermultiplets in the N"4 superalgebra depending on whether z are the same or di!erent. When z "z , two raising operators S and S in (40) are projected onto the zero-norm ? ? states; SK (m"1, 2) annihilate the supersymmetric vacuum state and become zero. So, of ? supersymmetry is preserved and the remaining raising operators S and S act on the lowest ? ? helicity state to generate the highest spin 1 multiplet. When z Oz , say, z 'z , the raising operator S is projected onto the zero-norm states. ? Hence, of supersymmetry is preserved and the remaining raising operators S , S and S act ? ? ? on the lowest helicity state to generate the highest spin 3/2 multiplet. 2.3. Positive energy theorem and Nester's formalism We discuss the positive mass theorem [146,295,379,483,531}534,631] of general relativity. The positive mass theorem says that the total energy, i.e. the rest mass plus potential energy plus kinetic energy, of the gravitating system is always positive with a unique zero-energy con"guration with appropriate boundary conditions at in"nity, provided that the matter stress-energy tensor ¹ satis"es the dominant energy condition IJ ¹ ;I<J50 , (42) IJ for any non-space-like vectors ;I and
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asymptotically at in"nity. The reason why one cannot make gravitational energy arbitrarily negative by shrinking the size of the gravitating object (as in the case of Newtonian gravity) is that when a system collapses beyond certain size, it forms an event horizon, which hides the inside region that has singularities and negative energy: the system appears to have positive energy to an outside observer. In general relativity, there is no intrinsic de"nition of local energy density due to the equivalence principle. So, one has to de"ne energy of a gravitating system as a global quantity de"ned in background (or asymptotic) region [1,20]. In general, conserved charges of general relativity are associated with generators of symmetries of the asymptotic region. Namely, the conserved charge is de"ned as a surface integral (with the integration surface located at in"nity) of the time component of the conserved current of the symmetry in the asymptotic region. Here, the integration surface is taken to be space-like in this section and thereby gravitational energy is of the ADM type. For a set +k , of vectors that approach the Killing vectors of the asymptotic background, one I can de"ne the conserved charge K through:
1 HIJk dR " I J 4 R
dNOH CJ? k MeIe@ dR , (43) IJM @ ? H NO 1.R where H is the total stress-energy tensor [440] including the pure gravity contributions and IJ C? "C? dxI is the connection 1-form for the metric g . For a time-like Killing vector k , K is @ I@ IJ I the ADM energy relative to the zero-energy background state. For a generic asymptotically #at spacetime, the set of conserved charges K consists of the ADM 4-momentum PI and the angularmomentum tensor JIJ, which satisfy the PoincareH algebra. (The ADM mass M is the norm of the ADM 4-momentum: M"(!PIP .) For a supersymmetric bosonic background, one can de"ne I the (conserved) supercharge [206,590] QK for the conserved current JK"aKRI (R (eaKRI)"0), I I which is de"ned in terms of a Killing spinor aK and a gravitino t satisfying the Rarita}Schwinger I equation: dx eHIk " I R
K "
1 aKeIJMNc t dR " J M N I 2 R
aKeIJMNc c t dR (44) J N IM 1.R where RI"eIJMNcc t and pIJ,[cI, cJ], etc. The generators KI and QI of the asymptotic J M N K spacetime symmetries and the supersymmetry transformation satisfy the following algebras: QK"
R
dx eaKR"
[KI , KI ]"C! KI , +QI , QI ,"f +KI , (45) ! K L KL + where C! are the same structure constants that appear in the commutator of the Killing vectors k and f are the same constants in the following relation between the Killing vectors kI and the KL Killing spinors a : K a cIa "f kI . (46) K L KL At in"nity, the current J "e H kJ is conserved, i.e. RIJ "0. Here, e is the determinant of the background metric I IJ I and kJ is now Killing vectors of the background spacetime. Here, the integration measures for the surface and volume integrals are respectively de"ned as dR " e dxM dxN and dR "e dxJ dxM dxN. IJ IJMN I IJMN For #at spacetime, one can choose the basis of aK so that f "c . KL KL
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So, these conserved charges satisfy the supersymmetry algebras [JIJ, QK]" pIJKQL, +QK, QL," cKLPI . (47) L I The third equation in (47) leads to the simple proof [207] of the positivity of energy in quantum supergravity. Since the left-hand side of this equation is a positive semide"nite operator, one has [PI, QK]"0,
\1s"+QK, QLR,"s2"(cIc)KL1s"P "s250 , (48) I where "s2 is a physical state vector. For this inequality to be satis"ed, the eigenvalues P$"P" of cIcP have to be non-negative, leading to proof of the positivity of gravitational energy. Rigorous I proof of the positive energy theorem based on original Witten's proof [631] involves the following antisymmetric tensor (the Nester's 2-form [483]): EIJ"eIJMN(e cc e! e cc e) , (49) M N N M where a Dirac spinor e is assumed to approach a constant spinor e asymptotically (ePe #O(r\)) and is the gravitational covariant derivative on a spinor. The ADM 4 I momentum P is related to the surface integral of EIJ over the surface S at the space-like in"nity in I the following way:
1 (50) P e cIe " EIJ dR . IJ I 2 1 Proof of the positive energy theorem involves the surface and the volume integrals of the Nester's 2-form, which are related by the Gauss divergence theorem: Z REIJ dS "R EIJ dR . This IJ I J 1. leads to
P e cIe " G e cIe dRJ# e (cJpIM#pIMcJ) e dR , M J I R IJ R I
(51)
where G is the Einstein tensor of the metric g . From the Einstein's equations G "¹ , one sees IJ IJ IJ IJ that the "rst term on the right-hand side of (51) is positive if ¹ satis"es the dominant energy IJ condition (42). In the coordinate system in which the x-direction is normal to R, the integrand of the second term on the right-hand side of (51) simpli"es to
2 e cpGH e"2" e"!2 cG e . G G H H G So, if the spinor e satis"es the `Witten conditiona [631]:
(52)
cG e"0 , (53) G G the second term on the right-hand side of (51) is also positive. Thus, the left-hand side of (51) is always non-negative for any e , leading to proof that energy P of a gravitating system is non-negative.
This is a necessary condition [145,494,513] for a spinor e to satisfy the Witten's condition.
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The above proof of the positive energy theorem based on the Nester's formalism can be readily generalized to gravitating con"gurations in extended supergravities. In this case, the gravitational covariant derivative on spinors in the Nester's 2-form (49) is replaced by the super-covariant I derivative K on spinors. So, the Nester's 2-form generalizes to I (54) EK IJ,2( K eCIJMe!e CIJM K e)"EIJ#HIJ , M M where EIJ is the original Nester's 2-form (49) and HIJ denotes the remaining terms, which are usually expressed in terms of gauge "elds of extended supergravities. Here, the supercovariant derivative on a supersymmetry parameter e is given by the gravitino supersymmetry transformation in bosonic background, i.e. dt " K e. The lower bound for mass is given in terms of central I I charges (coming from the HIJ term in (54)) of the extended supergravity and the bound is saturated when the gravitating con"guration is a bosonic con"guration which preserves some of supersymmetry. In the following, we discuss proof of the positive mass theorem in pure N"2, D"4 supergravity as an example [296]. The N"2, D"4 supergravity is "rst obtained in [265] by coupling the (2,3/2) gauge action to the (3/2,1) matter multiplet by means of the Noether procedure. The theory uni"es electromagnetism (spin 1 "eld) with gravity (spin 2 "eld). The theory has a manifest invariance under the O(2) symmetry, which rotates 2 gravitino into each other. In the bosonic background, the supergravity transformation of the gravitino, which de"ne the supercovariant derivative K , is I 1 (G 1 ,F cJcMc e .
K e"
! (55) dt " I JM I I (2pG I 4 (2pG , , Substituting the supercovariant derivative K e from (55) into the Nester's two-form (54), one has I the following expression for HIJ:
HIJ,4e (FIJ#ic 夹FIJ)e . The integrand of the volume integral takes the form:
(56)
(57)
EK IJ"16pG ¹ I;M#16p(G e (JI#ic JI I)e#4 K eCIJM K e , J M J , M , where ;I,e cIe is a non-spacelike 4-vector, provided e is the Killing spinor, and the electric and magnetic 4-vector currents JI and JI I are de"ned through the Maxwell's equations and the Bianchi identities as FIJ"4pJI and 夹FIJ"4pJI I. J J If we assume that the Killing spinor e approaches a constant value e as rPR, then the surface and the volume integrals are 1 e [!P cI#(G (Q#ic P)]e " [¹ I;J# e (JI#ic JI I)e]dR J I I , G R , 1 #
K eCIJM K e dR , (58) M I 4pG R J , where Q and P are the electric and magnetic charges of the gauge "eld A . I
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The "rst term on the right-hand side of (58) is always non-negative, if ¹ satis"es the IJ generalized dominant energy condition: ¹ ;I<J5G\ [(J
(61)
3. Duality symmetries Past few years have been an active period for studying string dualities [227,500,543,622,635,636]. Five di!erent string theories (E ;E and SO(32) heterotic strings, type-IIA and type-IIB strings, and type-I string with SO(32) symmetry), which were previously regarded as independent perturbative theories, are now understood as being related via web of dualities. String dualities are classi"ed into T-duality, S-duality and U-duality. T-duality (or target space duality) [306] is a perturbative duality (i.e. duality that relates theories with the same string coupling) that transforms the theory with large (small) target space volume to one with small (large) target space volume [216,425,525] or connects di!erent points in (target-space) moduli space. Under T-duality, type-IIA and type-IIB strings [193,216], and E ;E and SO(32) heterotic strings [303,480,481] are interchanged. Another examples of T-duality are the O(10!D, 26!D, Z) symmetry [308] of heterotic string on ¹\" and the O(10!D, 10!D, Z) symmetry [304,305,308,570] of type-II string on ¹\". S-duality (or strong-weak coupling duality) is a non-perturbative duality that transforms string coupling to its inverse (while moduli "elds remain "xed) and interchanges perturbative string states and non-perturbative branes. Duality that relates Type-I string and SO(32) heterotic string [505,635] is an example of S-duality. Another examples are (i) the S¸(2, Z) symmetry of type-IIB string [72,376,536,538}540]; (ii) the D"6 string}string duality between the heterotic string on ¹ [on K3] and the type-II string on K3 [on a Calabi} Yau-threefold] [52,93,226,233,235,244,257,397,435,565,624,635]; (iii) the S¸(2, Z) symmetry of N"4 heterotic string in D"4 [268,339,537,544,556}560]. U-duality [39,40,45,381], which is closely related to the D"11 theory (M-theory), is regarded as a consequence of the S¸(2, Z) S-duality of type-IIB string and T-dualities of type-II strings on a torus. Thus, U-duality is a non-perturbative duality of type-II strings which necessarily interchanges NS-NS charged state and R-R charged state.
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String dualities require existence of non-perturbative states within string spectrum, as well as the well-understood perturbative states. These non-perturbative states include smooth solitons and new types of topological objects called D-branes [498]. Such non-perturbative states are extended objects, which in a low energy limit correspond to p-branes of the e!ective "eld theories. So, string theories, which are previously known as theories of (perturbative) strings, are no longer theories of strings only, but contain objects of higher/lower spatial extends. These perturbative and nonperturbative states are interrelated via string dualities. One of important discoveries of string dualities is the conjecture that there exists more fundamental theory in higher dimensions (D'10), which reduces to all of 5 perturbative string theories in di!erent limits in moduli space when the theory is compacti"ed to lower dimensions (D410). Such fundamental theories include M-theory [227,540,543,635] in D"11, F-theory in D"12 [621], and S-theories in D"12, 13 [41}44]. M-theory is de"ned as an unknown theory in D"11 (with 1 time-like coordinate) whose low energy e!ective theory is the D"11 supergravity [158] and which becomes type-IIA theory when the extra 1 spatial coordinate is compacti"ed on S of radius R. Since the radius R of S is related to the string coupling j of type-IIA theory as R"j, the strong coupling limit (j<1) [635] of type-IIA theory is M-theory, i.e. the decompacti"cation limit (RPR) of M-theory on S. Furthermore, the evidence was given in [358] for the conjecture that M-theory compacti"ed on S/Z is E ;E heterotic string. We mentioned in the previous paragraph that type-IIA and type-IIB strings, and E ;E and SO(32) heterotic strings are related via T-duality, and SO(32) heterotic string and type-I string are related via S-duality. Thus, all of the 5 di!erent perturbative string theories are obtained from M-theory by compactifying on S or S/Z , and applying dualities. F-theory is a conjectured theory in D"12 (with 2 time-like coordinates) which is proposed in an attempt to "nd geometric interpretation of the S¸(2, Z) S-duality of type-IIB theory. Namely, the complex scalar formed by the dilaton and the R-R 0-form transforms linear-fractionally under the S¸(2, Z) transformation, just like the transformation of modulus of ¹ under the T-duality of a string theory compacti"ed on ¹. F-theory is, therefore, roughly de"ned as a D"12 theory which reduces to type-IIB theory upon compacti"cation on ¹, with the modulus of ¹ given by the complex scalar. Note, since type-IIB theory on S is equivalent to M-theory on ¹, F-theory on ¹;S is the same as M-theory on ¹. The essence of string dualities is that strong coupling limit of one theory is dual to weak coupling limit of another theory with the strongly coupled string states (dual to perturbative string states) identi"ed with branes. So, branes play an important role in understanding non-perturbative aspects of and dualities in string theories. It is one of purposes of this review to summarize the recent development in solitons and black holes in string theories. The purpose of this section is to give basic facts on dualities in supersymmetric "eld theories and superstring theories that are necessary in understanding the rests of chapters of this review. Therefore, readers are referred to other literatures, e.g. [23,227,228,317,498,499,501,504,539,540,543,605,635,636], for complete understanding of the subject. In the "rst section, we discuss the symplectic transformations in extended supergravities and moduli spaces spanned by scalars in the supermultiplets. In Section 3.2, we summarize T-duality and S-duality of heterotic string on tori. In this section, we also discuss solution generating transformations that induce electric/magnetic charges of ;(1) gauge "elds of heterotic string on tori
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when applied to a charge neutral solution. These dualities are basic transformations for constructing most general black hole solutions in heterotic string on tori. In Section 3.3, we discuss string}string duality between heterotic string on ¹ and type-II string on K3, and string}string} string triality among type-IIA, type-IIB and heterotic strings in D"6. These dualities transform black hole solutions discussed in Section 4 to type-II black holes which carry R-R charges [49,56], thereby enabling interpretation of our black hole solutions in terms of D-brane picture. In the "nal section, we summarize some aspects of M-theory and ;-duality. The generating black hole solutions of heterotic string on tori are the generating solutions for type-II string on tori, as well. Namely, when the generating solutions of heterotic strings are embedded to type-II theories (note, such generating solutions carry only charges of NS-NS sector, which is common to both heterotic and type-II theories), subsets of ;-dualities of type-II theories on tori induce the remaining ;(1) charges of type-II theories on tori [171]. 3.1. Electric-magnetic duality The electric-magnetic duality in electromagnetism was conjectured by Dirac [217] based on the observation that when electric charge and current are non-zero the Maxwell's equations lack symmetry under the exchange of the electric and magnetic "elds, or in other words under the Hodge-duality transformation of the electromagnetic "eld strength. Dirac conjectured the existence of magnetic charges [218] to remedy the situation. Magnetic charges are due to the topological defect of spacetime and are given by the "rst Chern class of the ;(1) principal "ber bundle with the base manifold given by S surrounding the monopole. The requirement of the continuity of the transition function that patches the 2 covers of the northern and southern hemispheres of S or the requirement of the unobservability of Dirac string singularity restricts magnetic charge q to be quantized [546,547,630,640}643] relative to electric charge q in the
following way through the Dirac}Schwinger}Zwanzinger (DSZ) quantization rule: n qq " (n3Z) . 4p 2
(62)
Under the duality transformation, electric and magnetic charges are interchanged and correspondingly the coupling of the electromagnetic interactions is inverted due to the relation (62). So, the weak (strong) coupling limit of one theory is described by the strong (weak) coupling limit of its dual theory. The extension of the duality idea to non-Abelian gauge theories was made possible by the discovery of the 't Hooft}Polyakov monopole solution [507,592] in non-Abelian gauge theory. In the 't Hooft}Polyakov monopole con"guration, the non-Abelian gauge group is spontaneously broken down to the Abelian one at spatial in"nity by Higgs "elds that transform as the adjoint representation of the gauge group. The magnetic charge of the 't Hooft}Polyakov monopole is determined by the second homotopy group of S formed by the symmetry breaking Higgs vacuum, i.e. the winding number around S as one wraps around S surrounding the monopole. Within this context, Montonen and Olive [476] conjectured that the spontaneously broken electric nonAbelian gauge group is dual to the spontaneously broken magnetic non-Abelian gauge group. Under this duality, the gauge coupling of one theory is inverted in its dual theory, leading to the
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27
prediction that the strong coupling limit of a gauge theory is the weak coupling limit of its dual theory [549}551]. Note, the hub of the Montonen}Olive conjecture lies in the existence of symmetry breaking Higgs "elds which transform in the adjoint representation of the non-Abelian gauge group. It is a generic feature of extended supersymmetries that scalars live in the same supermultiplet as vector "elds. So, the scalars in vector supermultiplet transform as the adjoint representation under the non-Abelian gauge group. Furthermore, supersymmetric theories obey the well-known non-renormalization theorem (See for example [37,192,267,321,375,473,485,573]). The extended supersymmetry theories have another nice feature that the states preserving some of supersymmetry, i.e. BPS states, are determined entirely by their charges and moduli. These are (degenerate) ground states of the theories (parameterized by moduli). Such BPS mass formula is invariant under dualities and degeneracy of BPS states remains unchanged under dualities. For example, electrically charged BPS states at coupling g have the same mass and degeneracy as magnetically charged states BPS states at coupling 1/g. Furthermore, the supersymmetry algebra prevents the number of degeneracy of BPS states from changing as the coupling constant is varied. Thus, it is BPS states that are suitable for testing ideas on duality symmetries. In the following, we discuss generalization of electric-magnetic duality of Maxwell's equations to N-extended supersymmetry theories and study moduli spaces spanned by scalars. 3.1.1. Symplectic transformations in extended supersymmetries In supersymmetry theories, scalars ' are taken as coordinates of the target space manifold M of non-linear p-model, which we write in general in the form [270]: L "g ( )gIJ ' ( , (63) '( I J where the covariant derivative on ' is with respect to the gauge group G that the vector I "elds AK in the theory belong to: I
'"R '#gAKk'K( ) , (64) I I I of G : with Killing vector "elds kK,k'K( ) R/R ' satisfying the Lie algebra g D kD . (65) [kK, kR]"fKR Note, g is a subalgebra of the isometry algebra of M . Here, g is the metric of M . In other words, a scalar is regarded as a map from the spacetime '( manifold to M . It turns out that the types of allowed target space manifolds formed by scalars are "xed by the number m of supercharge degrees of freedom in N-extended superalgebra [527]. When m exceeds 8, the target space manifold is "xed as a symmetric space speci"ed by the number n of vector multiplets. For example, D"4, N"8 supergravity, for which m"32, has the target space manifold E /S;(8) [159,160]; the D"4, N"4 theory, for which m"16, has target space manifold SO(6,n) S;(1,1) SO(6) SO(n) ;(1) [205,544,545,555,557]. A special case is the e!ective action of the heterotic string on ¹, which is described by the N"4 supergravity coupled to the N"4 super-Yang}Mills theory with the gauge
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group ;(1) (n"22 case). Here, SO(6,12)/SO(6) SO(12) describes (classical) moduli space of Narain torus [480,481], and S;(1,1)/;(1) is parameterized by the dilaton-axion "eld. (In Section 3.2, we discuss the Sen's approach [560] of realizing such target space manifolds within the e!ective supergravity through the dimensional reduction of the heterotic string e!ective action.) For m48, the target space manifolds are less restrictive. For a D"4, N"2 theory, corresponding to m"8, the scalar manifold is factorized into a quaternionic one and a special KaK hler manifold [576,581], which are, respectively, spanned by the scalars in the hypermultiplets and the vector multiplets. For m"4 case, e.g. D"4, N"1 theory, the target space manifold is the KaK hler manifold. Within the extended supersymmetry theories described above, one can generalize [276] the electric-magnetic duality transformations, which preserve equations of motion for the ;(1) "eld strengths. For this purpose, only relevant part of scalars is from vector supermultiplets. Such generalized electric-magnetic duality transformation is realized as follows. We consider the general form of the di!eomorphism of the scalar manifold: t: MT PMT : ' C t'( ) . (66) The map that corresponds to the isometry of the scalar manifold, i.e. t*g "g , becomes the '( '( candidate for the symmetry of the theory. General form of kinetic term for vector "elds in vector supermultiplets is [270] (67) L "cKR( )FK夹FR#hKR( )FKFR , where the "eld strengths FK of gauge "elds AK are IJ I K K K K R D (68) F ,(R A !R A #gfRDA A ) , IJ I J J I I J and 夹FK ,e FKMN is the Hodge-dual of FK . Here, the n;n matrix c ( ) generalizes the IJ IJMN IJ '( coupling constant of the conventional gauge theory and the antisymmetric matrix h ( ) is the '( generalization of the h-term [630]. The transformation properties of the gauge "elds and the complex symmetric matrix NKR( ),hKR( )!icKR( ) are determined by the symplectic embedding of the isometry group G of the scalar manifold MT as follows. We consider the following homomorphism from the group Di!(M ) of di!eomorphisms t: MT PMT to the general linear group G¸(2n, R): n : Di!(MT )PG¸(2n, R) . (69) B One introduces a 2n;1 matrix <"(夹F, 夹G)2, where 夹G,!RL/RF2. Then, the map n in (69) B is de"ned by assigning, for each element m of Di!(MT ), a 2n;2n matrix A B K n (m)" K B C D K K in GL(2n, R) which transforms < as
夹F A " R 夹G C R
夹F 夹G
C
B R D R
夹F 夹G
,
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or F> A B R " R G> C D R R
F> G>
C
F>
NF>
,
(70)
where F>,F!i夹F and G>,NF> [212,213]. The transformation law (70) on < is dictated by the requirement that the Bianchi identities and the "eld equations for vector "elds remain invariant. Under the action of the di!eomorphism m on ', N( ) also transforms. If we further require that transformed "eld G to be de"ned as 夹G "!RL /RF 2, the transformation property of NKR( ) under the di!eomorphism t on ' is "xed as the fractional linear form: N( ) C N (t( ))"[C #D N( )][A #B N( )]\ , R R R R
(71)
with n (m) now restricted to belong to Sp(2n, R)LG¸(2n, R). B When B O0OC , it is a symmetry of equations of motion only. When B "0OC , the R R R R Lagrangian is invariant up to four-divergence. When B "0"C , the Lagrangian is invariant. In R R particular, the symplectic transformations (70) and (71) with B O0 are non-perturbative, since they R necessarily induce magnetic charge from purely electric con"guration and invert N, which plays the role of the gauge coupling constant. When electric/magnetic charges are quantized, Sp(2n, R) gets broken down to Sp(2n, Z) so that the charge lattice spanned by the quantized electric and magnetic charges is preserved under the transformation (70). This is the generalization of the electric-magnetic duality symmetry to the case of D"4 supersymmetry theory with n vector "elds. 3.1.2. Symplectic embedding of homogeneous spaces When the number of supercharge degrees of freedom exceeds 8, e.g. N53 in D"4, M is a homogeneous space G/H with the isometry group G. The supersymmetry restricts the dimension of M and the number n of vector multiplets to be related to each other and, therefore, M is determined uniquely by n. In the following, we discuss the symplectic embedding of the homogeneous space and show how the gauge kinetic matrix NKR is determined. We consider the following embedding of the isometry group G of G/H into Sp(2n, R): n : GPSp(2n, R): ¸( ) C n (¸( )) . B B
(72)
Applying the following isomorphism from the real symplectic group Sp(2n, R) to the complex symplectic group ;sp(n, n),Sp(2n, C)5;(n, n): k:
A B
C D
C
¹ <* < ¹*
,
(73)
where ¹,(A!iB)#(C#iD), <,(A!iB)!(C#iD) ,
(74)
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one can de"ne the complex symplectic matrix O( ) (3;sp(n, n)), for each coset representative ¸( ) of G/H, as follows: ; ( ) ;*( ) , (75) k ) n : GP;sp(n, n): ¸( ) C O( )" B ; ( ) ;*( ) where
; ( )R; ( )!; ( )R; ( )"1, ; ( )R; ( )*!; ( )R; ( )*"0 . (76) From O( ) in (75), which is de"ned for each coset representative ¸( ) of G/H, one can de"ne the following scalar matrix [276] which has all the right properties for the gauge kinetic matrix NKR"hKR!icKR: N,i[;R #;R ]\[;R !;R ] , namely, N2"N and N transforms fractional linearly under Sp(2n, R). Speci"cally, we consider the homogeneous space of the form
(77)
SO(m, n) S;(1, 1) , ST[m, n], SO(m) SO(n) ;(1) where m is the number of graviphotons and n the number of vector multiplets. Here, S;(1, 1)/;(1) is parameterized by the axion-dilaton "eld S and SO(m, n)/[SO(m) SO(n)] is parameterized by a m;n real matrix X. In the real basis, the SO(m, n) T-duality and S¸(2, R) S-duality groups of ST[m, n] are respectively embedded into the symplectic group as:
¸ O 3Sp(2m#2n, R) n : ¸3SO(m, n) C B O (¸2)\
a b a1 bg n: 3S¸(2, R) C 3Sp(2m#2n, R) , B c d cg d1
(78)
where g is an SO(m, n) invariant metric, 1 is the (m#n);(m#n) identity matrix, and a, b, c, d3R satisfy ad!bc"1. In the complex basis, the embeddings are
(¸#g¸g) (¸!g¸g) 3;sp(m#n, m#n), n : ¸3SO(m, n) C B (¸!g¸g) (¸#g¸g) t v* Re t 1#i Im t g Re v 1!i Im v g n: 3S;(1, 1) C 3;sp(m#n, m#n) . B v t* Re v 1#i Im v g Re t 1!i Im t g
(79)
The symplectic embeddings (79) make it possible to express the gauge kinetic matrix N in terms of the scalars S and X, which parameterize ST[m, n], as follows. The coset representatives of
ST[2, n] is the only special KaK hler manifold with direct product structure [266] of this form.
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S;(1, 1)/;(1) and SO(m, n)/[SO(m);SO(n)] are respectively
1 1 \1 >1 , ¸(S), n(S) >1MM 1 \1 (1#XX2) X , ¸(X), X2 (1#X2X)
(80)
where
n(S),
4 Im S . 1#"S"#2Im S
Note, M,¸(X)¸2(X) is a symmetric SO(m, n) matrix, studied by Sen [560], which will be discussed in Section 3.2. Applying transformations (79), one obtains the following symplectic embedding of the coset representations of ST[m, n]:
; (S, X) ;*(S, X) n (¸(S))n (¸(X))" 3;sp(n#m, n#m) , (81) B B ; (S, X) ;*(S, X) where the explicit expressions for n (¸(S)) and n (¸(X)) are obtained by applying the transformations B B (79). Substituting this expression into the general formula in (77), one obtains the following gauge kinetic matrix: N"i Im Sg¸(X)¸2(X)g#Re Sg"i Im SgMg#Re Sg .
(82)
Then, the Lagrangian L #L (cf. see (63) and (67)) takes the following form that corres ponds to ST[m, n]:
1 1 L"(!g R # R SRISM ! Tr(R MRIM) E 4(Im S) I I 4
1 1 ! Im SF' (g¸g) F( IJ# Re SF' g F( eIJMN . IJ '( IJ '( MN 4 8(!g
(83)
The T-duality and S-duality of the heterotic string on ¹ [465,537,544,560] are special cases of the symplectic transformations (78) with (m, n)"(6, 22). In general, under the SO(m, n) and S¸(2, R) transformations (78), the gauge "elds and the gauge kinetic matrix transform, respectively, as (cf. (70) and (71)) F> C F> "¸F>, N C N "(¸2)\N¸\, F> C F> "aF>#bgNF>, N C N "(dN#cg)(bgN#a)\ .
(84)
Note, O(m, n) [S¸(2, R)] is a perturbative (non-perturbative) symmetry, since N is not inverted (gets inverted). S¸(2, R) is the symmetry of the equations of motion only, since electric and magnetic charges get mixed, and since this corresponds to the transformations (70) and (71) with B O0OC . R R
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3.1.3. Target space manifolds of N"2 theories Contrary to D"4, N53 theories, the scalar manifolds of N"2 theories are not necessarily expressed as homogeneous symmetric coset manifolds. Scalar manifold of the D"4, N"2 theory has the generic form M "SM HM , (85) L K where SM is a special KaK hler manifold of the complex dimension n" `the number of the vector L supermultipletsa, and the manifold HM spanned by the scalars in the hypermultiplets has the K dimension 4m, where m" `the number of the hypermultipletsa. So, the metric g ( ) of M has '( the form (86) g ( )d ' d ("g *dz? dz @*#h dqS dqT . ST '( ?@ Here, g * [h ] is the special KaK hler metric on SM [the quaternionic metric on HM ]. ST L L ?@ 3.1.3.1. Special KaK hler manifolds. N"2 super-Yang-Mills theory is described by a chiral super"eld U, which is de"ned by DM ? G U"0 (like chiral super"eld in N"1 theory), with an additional constraint: (87) D? D@ Ue "e e lDM ? IDM l@Q UM e Q , ?@ G H ?@ GI H where i"1, 2 labels supercharges of N"2 theory, a, a"1, 2 are indices of chiral spinors, and DM ? is a covariant chiral superspace derivative. The component "elds of a N"2 chiral super"eld U are a scalar X , spinors jG , ;(1) gauge "eld strength F , and auxiliary scalars > satisfying a reality IJ GH constraint > "e e l>M Il (due to the constraint (87)). The action of N"2 chiral super"elds U is GH GI H determined by an arbitrary holomorphic function F(U ) of U as dxdh F(U )#c.c., and is given, in terms of the component "elds, by (88) L"g M R X RIXM #g M jM cIR j M #Im(F F\ F\ IJ)#2 , IJ I G I where the dots denote the interaction terms involving fermions, g M "R R M K is a KaK hler metric, and F ,R R F. Note, this action is a special case of the most general coupling of N"1 chiral super"elds to N"1 Abelian vector super"elds in which the KaK hler potential K and the holomorphic kinetic term function F take the following special forms [283,572]: K(X, XM )"i[FM (XM )X !F (X)XM ] (F ,R F), F "R R F . (89) The KaK hler manifold with the KaK hler potential K(X, XM ) determined by the prepotential F(X) [161,211,212] through (89) is called the special Ka( hler manifold [113,128,198,210,212,222,576]. But there is a subclass of homogeneous special manifolds, which are classi"ed in [166]. These are S;(1,1) S;(1,n ) S;(1,1) SO(2,n ) Sp(6,R) S;(3,3) T T , , , , , S;(n );;(1) ;(1) SO(2);SO(n ) S;(3);;(1) S;(3);S;(3) ;(1) T T SOH(12) E \ , and S;(6);;(1) E ;SO(2) with the corresponding symplectic groups Sp(2n #2) respectively given by Sp(4), Sp(2n #2), Sp(2n #4), Sp(14), Sp(20), T T T Sp(32), and Sp(56).
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When N"2 chiral "elds UK (K"0, 1,2, n ) are coupled to the Weyl multiplet (with compoT nents given by vierbein, 2 gravitinos and auxiliary "elds) [210,212], the invariance under the dilatation requires F(X) to be a homogeneous function of degree 2 (so that F(X) has Weyl weight 2) [210,212]. Furthermore, the requirement of canonical gravitino kinetic term imposes one constraint on the set of scalars XK as i(XM KFK!FM RXR)"1 ,
(90)
leading to gauge "xing for dilations and the special KaK hler manifold of the dimension n . Note, the T extra chiral super"eld U is introduced to "x the dilatation gauge, to break the S-supersymmetry, and to introduce the physical ;(1) gauge "eld in the N"2 supergravity multiplet (the scalar and the spinor components of the super"eld U do not become additional physical particles). The "nal form of bosonic action describing n numbers of N"2 vector multiplets coupled to N"2 T supergravity is (91) e\L"!R#g *R z?RIz @*!Im(NKR(z, z )F>KF>RIJ) , IJ ?@ I where z? (a"1,2, n ) are the coordinates of a KaK hler space spanned by the scalars T XK (K"0, 1,2, n ) which satisfy one constraint (90) (therefore, the manifold spanned by XK has T n complex dimensions). A convenient choice for z? is the inhomogeneous coordinates called the T special coordinates: z?"X?(z)/X(z), a"1,2, n . (Note, X?(z)"z? in special coordinates in which T R(X?/X)/Rz@"d? [113,128,134,576].) Here, K and NKR (cf. the scalar matrix N in (71)) are @ determined by F(X) to be of the forms [130,161,198,211,212]: e\)X X "iZM K(z )FK(Z(z))!iZR(z)FM R(ZM (z )) , Im(FK )Im(FRP)X XP , (92) Im(F P)X XP where ZK(z)"e\) XK and ZM K(z )"e\) XM K (K"0, 1,2, n ) are holomorphic sections of the projective space PC L> [128,129,198], and FKR,RKFR(X). We give some examples of the holomorphic function F(X) of N"2 theories and the corresponding special KaK hler manifold target spaces: NKR"FM KR#2i
F"iXX,
S;(1, 1) , ;(1)
F"(X)/X,
S;(1, 1) , ;(1)
F"!4(X(X),
S;(1, 1) , ;(1)
S;(1, n) F"iX gKRX , , S;(n) ;(1) K
R
dK RXKX XR F" , X
Calabi}Yau .
XQ L SO(2, 1) SO(2, n) F"!i (X)! (X?) , ; . X SO(2) SO(2);SO(n) ?
(93)
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So far, we de"ned the special Ka( hler manifold as the KaK hler manifold with the special form of the KaK hler metric given by (92), which depends on the holomorphic prepotential F. Now, we discuss the symplectic formalism of the special KaK hler manifolds of N"2 supergravity coupled to n vector supermultiplets. For the symplectic formalism [132,133,214] of the special KaK lher manifold M, one considers the tensor bundle of the type H"SV L. Here, SVPM denotes a holomorphic #at vector bundle of rank 2n #2 with structural group Sp(2n #2, R), and LPM denotes the complex T T line bundle whose "rst Chern class equals the KaK hler form of the n -dimensional Hodge}KaK hler T manifold M. A holomorphic section of the bundle H has the form [113}115,128,134,198]:
XK
X"
K, R"0, 1,2, n , T
FR
(94)
which is de"ned for each coordinate patch ; LM of the (Hodge}KaK hler) manifold M and G transforms as a vector under the symplectic transformation Sp(2n #2, R). The Hodge}KaK hler T manifold M with a bundle H described above is called special Ka( hler, if the KaK hler potential is expressed in terms of the holomorphic section X as K"!log(i1X"XM 2)"!log[i(XM KFK!FM RXR)] ,
(95)
where
1X"XM 2,!XR
0
I
!I 0
X
denotes a symplectic inner product. One further introduces the symplectic section of the bundle H according to
¸K
<"
MR
,e) X2 .
(96)
Then, by de"nition, < satis"es the constraint [128,130,161,211,212]: 1"i1<"<M 2"i(¸M KMK!M M R¸R) ,
(97)
and is covariantly holomorphic:
*<"(R *!R *K)<"0 , ? ? ? where R ,R/Rz? and R *,R/Rz ?*. ? ?
(98)
The additional two dimensions in SV come from a vector "eld in the supergravity multiplet. Alternatively, one can de"ne special Ka( hler manifold by introducing the symplectic section < (96) satisfying the constraint (97). Then, the KaK hler potential is determined in terms of the holomorphic section X as in (95) through the constraint (97) with (96) substituted.
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One further introduces the matrix of the following form:
fK ; " <"(R #R K)<, ? (a"1,2, n ) . (99) ? ? ? ? T hR ? Then, period matrix NKR (which corresponds to the gauge kinetic matrix in the N"2 theory) is de"ned via the relations M KR f R . MM K"N M KR¸M R, hK "N ? ? Therefore, the period matrix has the following explicit form:
(100)
N M KR"hK ( f \)'R , ' where the (n #1);(n #1) matrices f K and hK in the above are de"ned as T ' ' K f h K f K" ?K , hK " ? . ' ' ¸M M M K
(101)
(102)
As a consequence, under the di!eomorphism of the base manifold M, NKR transforms fractional linearly, like the gauge kinetic matrix (cf. (71)). Note, in the above symplectic formalism of the special KaK hler manifold no reference was made on the prepotential. In fact, for some cases, the existence of the prepotential is not even guaranteed [136]. We now discuss how the concept of the prepotential emerges within the framework of symplectic formalism. Under the coordinate transformations of M, X transforms as XPX "e\DMX ,
(103)
where the factor e\D corresponds to a ;(1) KaK hler transformation (i.e. K transforms as KPK#Re f (z)) and M3Sp(2n#2, R), and NKR transforms fractional linearly just like a gauge kinetic matrix (cf. (71)). From the transformation law (103) with M"I, one can infer that XK can be regarded as homogeneous coordinates of a (n #1)-dimensional projective space at least locally T [210,212,283], since XK and e\DXK are identi"ed under the KaK hler transformations. This is possible provided the Jacobian matrix R (X@/X) (a, b"1,2, n ) is invertible [132]. In this case, ? T due to the integrability condition following from (97) and (98), the lower components FR of X are expressed as R FR" RF , RX
(104)
in terms of a homogeneous function F(X)"XKFK of degree 2 in XK. Then, one can use z?,X?/X (a"1,2, n ) as the special coordinates and the holomorphic prepotential is T F(z),(X)\F(X). In terms of F and z?, the KaK hler potential K is expressed as [210] K(z, z )"!log i[2(F!F M )!(R F#R *FM )(z?!z ?*)] . ? ? For electrically neutral theories, one can always rotate to bases where a prepotential exists [155].
(105)
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3.1.3.2. Hypergeometry. N"2 hypermultiplet consists of a doublet of 0-form spinors with left and right chiralities, and 4 real scalars, which can be locally regarded as the 4 components of a quaternion. The scalars qT (v"1,2, 4n ) in n hypermultiplets form a 4n -dimensional real & & & manifold HM [30,129,198,211,277,353] with a metric ds"h (q)dqS dqT . (106) ST This manifold is endowed with 3 complex structures JV: ¹(HM)P¹(HM) (x"1, 2, 3) satisfying the quaternionic algebra JVJW"!dVW1#eVWXJX. The metric h (q) is hermitian with respect to JV: ST h(JVX, JVY)"h(X, Y), X, Y3¹ HM . (107) From JV, one can de"ne triplet of S;(2) Lie-algebra valued HyperKaK hler forms as KV"KV dqSdqT , (108) ST where KV "h (JV)U. Supersymmetry requires the existence of a principal S;(2)-bundle ST SU T SUPHM with a connection uV. The manifold HM is de"ned by requiring that KV is covariantly closed with respect to the connection uV:
KV,dKV#eVWXuWKX"0 .
(109)
There are two types of hypergeometry: rigid (local) hypergeometry corresponding to global (local) N"2 supersymmetry is called HyperKa( hler (quaternionic). The only di!erence between the two manifolds are the structure of the SU-bundle. A HyperKa( hler manifold has the #at SU-bundle, and a quaternionic manifold has the curvature of the SU-bundle proportional to the HyperKaK hler 2-form. Here, the SU-curvature is de"ned as XV,duV#eVWXuWuX . In the quaternionic case, the curvature is XV"(1/j)KV ,
(110)
(111)
where j is a real number related to the scale of the quaternionic manifold. In the limit jPR, quaternionic manifold becomes HyperKaK hler manifold [6]. The manifold HM has the following holonomy group: Hol(HM)"S;(2) H for quaternionic manifold , Hol(HM)"1 H for HyperKaK hler manifold ,
(112)
where HLSp(2n , R). We denote the #at indices that run in the fundamental representation of & S;(2) [Sp(2n , R)] as i, j"1, 2 [a, b"1,2, 2n ]. Then, the metric of the quaternionic manifold is & & expressed in terms of the vielbein 1-form UG?"UG?(q) dqS as S h "UG?UH@C e , (113) ST S T ?@ GH where C "!C [e "!e ] is the #at Sp(2n ) [Sp(2)&S;(2)] invariant metric. The vielbein ?@ @? GH HG & UG? is covariantly constant with respect to the S;(2)-connection uV and some Sp(2n , R) Lie &
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algebra valued connection *?@"*@?:
UG?,dUG?#uV(ep e\)G UH?#*?@UGAC "0 , V H @A where pV (x"1, 2, 3) are the Pauli spin matrices. Also, UG? satis"es the reality condition:
(114)
U ,(UG?)H"e C UH@ . G? GH ?@ The curvature 2-form XV forms the representation of the quaternionic algebra:
(115)
hQRXV XW "!jdVWh #jeVWXXX , SQ RU SU SU and can be written in terms of the vielbein UG? as
(116)
XV"ijC (pVe\) U?GU@H . ?@ GH
(117)
3.2. Target space and strong-weak coupling dualities of heterotic string on a torus 3.2.1. Ewective xeld theory of heterotic string The e!ective "eld theory of massless states in heterotic string is D"10 N"1 supergravity coupled to N"1 super-Maxwell theory [74,137,142]. The massless bosonic "elds of heterotic string at a generic point of Narain lattice [480,481] are metric GK , 2-form "eld BK , gauge "elds +, +, AK ' of ;(1) and dilaton "eld U, where 04M, N49 and 14I416. The "eld strengths of + AK ' and BK are de"ned as FK ' "R AK ' !R AK ' and HK "R BK !AK ' FK ' #cyc. perms., + +, +, + , , + +,. + ,. + ,. respectively. The D"10 e!ective action [74,111,137,142] of these massless bosonic modes is HK +,.!FK ' FK ' +,] , (118) L"(1/16pG )(!GK [R K #GK +,R UR U! HK % + , +,. +, where GK ,det GK , R is the Ricci scalar of GK , and G is the D"10 gravitational constant. +, %K +, We choose the mostly positive signature convention (!##2#) for the metric GK . For the +, spacetime vector index convention, the characters (A, B,2) and (M, N,2) denote #at and curved indices, respectively. The supersymmetry transformations of the fermionic "elds, i.e. gravitino t , dilatino j and + gaugini s', are CK ,.e , dt " e!HK + + +,. dj"(CK +R UK )e!HK CK +,.e , (119) + +,. ds'"FK ' CK +,e , +, where e"R e#X CK e is the gravitational covariant derivative on a spinor e. Here, + + + ): X is the spin-connection de"ned in terms of a Zehnbein EK K (de"ned as EK g EK "GK + , +, + X ,!XI #XI !XI , where XI ,EK + EK , R EK , and curved indices are obtained ! ! ! ! ! , +! by contracting with Zehnbein. CK are gamma matrices of SO(1, 9) Cli!ord algebra +CK , CK ,"2g (those with several indices are de"ned as the antisymmetrized products of gamma matrices, e.g. CK ,(CK CK !CK CK )). In this section, we "x the D"10 gravitational constant to be G "8p.
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3.2.2. Kaluza}Klein reduction and moduli space The e!ective "eld theory of massless bosonic "elds in heterotic string on a Narain torus [480,481] at a generic point of moduli space is obtained by compactifying D"10 e!ective "eld discussed in the previous section on ¹\" [465,560]. Before we discuss the compacti"cation Ansatz, we "x our notation for indices. General indices running over D"10 are denoted by upper-case letters (A, B,2; M, N,2). The lower-case Greek letters (a,2, b; k, l,2) are for D(10 spacetime coordinates and the lower-case Latin letters (a, b,2; m, n,2) are for the internal coordinates. The #at indices are denoted by the letters in the beginning of alphabets (A, B,2; a, b,2; a, b,2) and curved indices are denoted by the letters at the latter parts of alphabets (M, N,2; k, l,2; m, n,2). The compacti"cation [143,225,247,416,434,465,528,529] on ¹\" is achieved by choosing the following Abelian Kaluza}Klein (KK) Ansatz for the D"10 metric
e?Pg #G A KA L A KG IJ KL I J I KL , (120) A LG G J KL KL where A K (k"0, 1,2, D!1; m"1,2, 10!D) are KK ;(1) gauge "elds, u,UK !ln det G KL I is the D-dimensional dilaton and a,2/(D!2). Then, the a!ective action is speci"ed by the following massless bosonic "elds: the (Einstein-frame) graviton g , the dilaton u, (36!2D) ;(1) IJ gauge "elds AG ,(AK, A, A') de"ned as A,BK #BK AL#AK ' A' and A', I I IK I IK IK KL I K I I AK ' !AK ' AK, the 2-form "eld B with the "eld strength H "R B !AG ¸ FH #cyc.perms., I K I IJ IJM I JM I GH JM and the following symmetric O(10!D, 26!D) matrix of scalars (moduli) [465,560]: GK " +,
G\
!G\C
!G\a2
M" !C2G\ G#C2G\C#a2a C2G\a2#a2 , aG\C#a
!aG\
I#aG\a2
(121)
where G,[GK ], C,[AK 'AK '#BK ] and a,[AK ' ] are de"ned in terms of the internal parts of KL K KL K L D"10 "elds. M can be expressed in terms of the following O(10!D, 26!D) matrix as M"<2< [465]:
E\ !E\C !E\a2
<"
0
E
,
(122)
I where E,[e? ], C,[AK ' AK '#BK ] and a,[AK ' ]. < plays a role of a Vielbein in the K K L KL K O(10!D, 26!D) target space. Note, M parameterizes the quotient space O(10!D, 26!D)/ [O(10!D);O(26!D)] with dimensions 26!36D#D. The dimensionality precisely matches the number of scalar "elds in the matrix M: (11!D)(10!D)/2 scalars GK , (10!D)(9!D)/2 KL scalars BK , and 16(10!D) scalars AK ' . KL K The resulting theory in D(10 corresponds to 26!D vector multiplets coupled to D(10, N-extended supergravity. The supergravity multiplet consists of graviton g , 2-form potential IJ B , 10!D graviphotons A0? (a"1,2, 10!D), dilaton u, gravitinos t? (a"1,2, N) IJ I I and dilatinos j?. The "eld content in the 26!D vector multiplets is 26!D vector "elds A*' (I"1,2, 26!D), (10!D);(26!D) scalars ?' parameterizing the coset I 0
a
0
D. Youm / Physics Reports 316 (1999) 1}232
39
O(10!D, 26!D)/[O(10!D);O(26!D)] and gauginos s?'. At the string level, the 10!D graviphotons originate from the right moving sector of the heterotic string and the 26!D photons in the vector multiplets originate from the left moving sector. In terms of the "eld strengths FG of I the ;(1)\" gauge group, the graviphoton "eld strengths F0? and matter photon "eld strengths IJ F*' are expressed as IJ F0"< ¸F , F*"< ¸F , (123) IJ 0 IJ IJ * IJ where <"(< , < )2 and the O(10!D, 26!D) invariant metric ¸ is de"ned in (127). 0 * Then, the e!ective D(10 action (in the Einstein frame) takes the form [465,560]
1 1 1 (!g R ! gIJR uR u# gIJ Tr(R M¸R M¸) L" E I J I J (D!2) 8 16pG " 1 1 ! e\?PgIIYgJJYgMMYH H ! e\?PgIIYgJJYFG (¸M¸) FH , IJM IYJYMY 4 IJ GH IYJY 12
(124)
where g,det g , R is the Ricci scalar of g , and FG "R AG !R AG are the ;(1)\" gauge IJ E IJ IJ I J J I "eld strengths. Here, the D(10 gravitational constant G is de"ned in terms of the D"10 one " G as G "(2p(a )\"G , where (a is the radius of internal circles. The Einstein-frame " metric g is related to the string-frame metric g through Weyl rescaling g"e?Pg . In terms of IJ IJ IJ IJ the graviphotons A0? and photons A0' in vector multiplets, the gauge kinetic terms take the form: I I F (¸M¸)FIJ"F02F0IJ#F*2F*IJ, due to the relation M"<2<"<2< #<2< . IJ IJ IJ 0 0 * * In particular for D"4, the supersymmetry transformations (119) of fermionic "elds in the bosonic background take the following simpli"ed form in terms of D"4 "elds [236]:
1 RS 1 1 (S F0?c?@c C?# Q?@C?@ e , dt " ! ic I ! I 4 ?@ I I S 4 I 8(2 R (S !icS ) 1 1 ! cI I (S F0?cIJC? e , dj" IJ S 2(2 4(2 1 1 cI< ¸R <2 ) C! (S F*cIJ e , ds" * I 0 IJ 2(2 (2
(125)
where Q?@"(< ¸R <2)?@ is the composite SO(6) connection and S is the axion-dilaton "eld de"ned I 0 I 0 in Section 3.2.3. Here, the D"10 gamma matrices C (A"0, 1,2, 9) are decomposed into the D(10 spacetime parts cI (k"0, 1,2, D!1) and the internal space parts C? (a"1,2, 10!D). 3.2.3. Duality symmetries The D(10 e!ective action (124) is invariant under the O(10!D, 26!D) transformations (T-duality) [465,560]: MPXMX2, AG PX AH , g Pg , uPu, B PB , I GH I IJ IJ IJ IJ We choose a "1 in most of cases.
(126)
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where X3O(10!D, 26!D), i.e. with the property: X2¸X"¸,
¸" I
0 \" 0
I
0
\" 0 0
0 I
,
(127)
\"
where I denotes the n;n identity matrix. L When electric/magnetic charges are quantized according to the Dirac}Schwinger}Zwanzinger}Witten (DSZW) quantization rule [218,546,547,630,640}643], the quantized, conserved electric a and magnetic b charge vectors live on the even, self-dual, Lorentzian lattice K [410,557]. The subset of O(10!D, 26!D, R) symmetry that preserves the lattice K is O(10!D, 26!D, Z), the so-called T-duality group of heterotic string on a torus. T-duality symmetry is a perturbative symmetry, which is proven to be exact to order by order in string coupling [307]. Under the T-duality, the charge lattice vectors transform as aP¸X¸a, bP¸X¸b .
(128)
In addition, the e!ective "eld theory has an on-shell symmetry called strong-weak coupling duality (S-duality) [268,537,544,545,556}560]. The equations of motion for (124) are invariant under the SO(1, 1) [S¸(2, R)] transformation for 54D410 [D"4]. Such transformations mix electric and magnetic charges, while transforming the dilaton in a nontrivial way. When the DSZW quantization is taken into account, such duality groups break down to integer-valued subgroups Z and S¸(2, Z) for 54D410 and D"4, respectively. These are the conjectured S-dualities in heterotic string. As an example, we discuss the D"4 S¸(2, R) symmetry. D"4 case is special for the following reason. Since the 2-form "eld strengths are self-dual under the Hodge-duality, ;(1) gauge "elds obey electric-magnetic duality transformations, which leave the Maxwell's equations and Bianchi identities invariant. Also, the "eld strength H of the 2-form potential B is Hodge-dualized to IJM IJ a pseudo-scalar W (axion): eP eIJMNR W , HIJM"! N (!g forming a complex scalar S"W#ie\P with dilaton u. The (Einstein-frame) D"4 theory has the on-shell symmetry under the S¸(2, R) transformations (S-duality) [163,165,560]: aS#b SP , MPM, g Pg , IJ IJ cS#d FG P(cW#d)FG #ce\P(M¸) 夹FH , IJ IJ GH IJ where 1 eIJMNFG , 夹FGIJ" MN 2(!g and a, b, c, d3R satisfy ad!bc"1.
(129)
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41
The instanton e!ect breaks the S¸(2, R) symmetry down to S¸(2, Z) [557,569]. The electric and magnetic `lattice charge vectorsa [560] a and b that live on an even, self-dual, Lorentzian lattice K with signature (6, 22) are given in terms of the physical electric and magnetic charges Q and P (de"ned as FG +QG/r and 夹FG +PG/r) as b,¸P and a,e\(M\Q!W b [557]. Under RP RP the S-duality, a and b transform as [410,557]
a
b
a
!b
a
!c
d
b
P
,
(130)
where a, b, c, d are integers satisfying ad!bc"1. 3.2.4. Solution generating symmetries For stationary solutions, which have the Killing time coordinate, one can further perform Abelian KK compacti"cation of the time coordinate on S. The T-duality transformation of such (D!1)-dimensional action can be applied to a known D-dimensional solution to generate new types of solutions in D dimensions with di!erent spacetime structure [99,144,290,147,148,223,251,278,336,341,393,446,466,552}554,562]. The basic idea on solution generating symmetry is as follows. If the background con"guration is time independent, then under the (time-independent) general coordinate transformations, G y and RI B y transform as vectors, where ky "1,2, D!1. In addition, B y transforms as a vector under the RI RI (time-independent) gauge transformation of the 2-form "eld. So, one can add 2 new ;(1) gauge "elds associate with G y and B y to the existing D-dimensional 36!2D ;(1) gauge "elds, forming RI RI a new multiplet of vectors Ax Gx y (ix "1,2, 38!2D) [562]. In addition, since G and AG transform as I RR R scalars under the transformations mentioned above, the scalar matrix of moduli is enlarged to a (38!2D);(38!2D) matrix [562]. Under the T-duality of the (D!1)-dimensional e!ective action, the (t, t)-component of the D-dimensional metric g mixes with scalars in the moduli matrix M and the t-component of the IJ ;(1)\" gauge "elds AG , and the (t, ky )-components of g mix with the AG and the (t, ky ) I IJ I components of B . So, unlike the D-dimensional T-duality transformation, which leaves the DIJ dimensional spacetime intact, the (D!1)-dimensional T-duality transformation can be applied to a known D-dimensional solution to generate new solutions with di!erent spacetime structure. In particular, such transformations can be imposed on charge neutral solutions to generate electrically charged (under the D-dimensional ;(1)\" gauge group) solutions: 36!2D SO(1, 1) boosts in the (D!1)-dimensional T-duality group generate electric charges of ;(1)\" gauge "eld when acted on charge neutral solutions. The (D!1)-dimensional e!ective action is [495,561,562]: x ¸x R y M x ¸x ) L"(gy e\Py [R y #gy Iy Jy R y uy R y uy #gy Iy Jy Tr(R y M E I J I J ! gy Iy Iy Ygy Jy Jy Ygy My My YHx y y y Hx y y y !gy Iy Iy Ygy Jy Jy YF x Gx y y (¸x M x ¸x ) x x F x Hx ] , IJ IJM IYJYMY GH Iy YJy Y
(131)
When acted on magnetically charged solutions, such transformations induce unphysical Taub-NUT charge [562]. This is due to the singularity of Dirac monopole.
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where ;(1) gauge "elds A x Gx y (ix "1,2, 38!2D), dilaton uy , 2-form "eld Bx y y and the metric gy y y are I IJ IJ de"ned as Ax Gy ,AGy !(g )\g y AG, 14i436!2D, 14ky 4D!1 , I RR RI R I ,(g )\g y , Ax \" ,B y #AG¸ Ax Hy , Ax \" RR RI Iy RI R GH I Iy uy ,u!ln(!g ), gy y y "g y y !(g )\g y g y , RR IJ IJ RR RI RJ Bx y y ,B y y #(g )\(g y AGy !g y AGy )¸ AH#(g )\(B y g y !B y g y ) , IJ RR RI J RJ I GH R RR RI RJ RJ RI IJ and the symmetric O(11!D, 27!D) moduli matrix is given by
(132)
M#4(g )\A A2 !2(g )A 2M¸A #4(g )\A (A ¸A ) RR R R RR R R RR R R R !2(g )\A2 (g )\ !2(g )\A2¸A RR R RR RR R R Mx " . (133) 2A2¸M !2(g )\AA2¸A g #4A2¸M¸A R RR R R RR R R #4(g )\A2(A2¸A ) #4(g )\(A2¸A ) RR R R R RR R R Here,
¸ 0 0
¸x , 0
0 1
0
1 0
is an O(11!D, 27!D) invariant matrix and A ,[AG]. R R This action has invariance under the O(11!D, 27!D) T-duality [495,562]: x Hx , uy Puy , gy y y Pgy y y , Bx y y PBx y y , (134) Mx PXx M x Xx 2, A x Gx y PXx x x A I GH Iy IJ IJ IJ IJ where Xx 3O(11!D, 27!D), i.e. Xx ¸x Xx 2"¸x . D"3 case is special since a 2-form "eld strength is dual to a scalar. So, the scalar moduli space is enlarged from O(7, 23, Z)!O(7, 23)/[O(7);O(23)] to O(8, 24, Z)!O(8, 24)/[O(8);O(24)] [561]. The O(8, 24, Z) duality symmetry are generated by D"4 S¸(2, Z) S-duality and D"3 O(7, 23, Z) T-duality [561], just as U-duality in type-II string is generated by S-duality in D"10 and T-duality in D(10 [381]. The O(8, 24, Z) symmetry transformation puts the axion-dilaton "eld on the same putting as the other moduli "elds and, therefore, is non-perturbative in nature. Since we consider stationary solution, i.e. a solution with isometry in the time direction, we compactify the time coordinate as well as other internal space coordinates on ¹ to obtain D"3 e!ective action ((124) with D"3 and now k"r, h, ; m"t, 1,2, 6). Such action has an o!-shell symmetry under the O(7, 23) transformation (126). The DSZW quantization condition breaks this symmetry to integer-valued O(7, 23, Z) subset. In D"3, one can perform the following Hodge-duality transformations to trade the D"3 ;(1) "elds AG with a set of scalars t,[tG] [561]: I (135) (!he\PhI I YhJ J Y(M¸) FH "eI J M R tG , GH IYJY M where h is the D"3 space metric and i, j"1,2, 30. Here, M is a symmetric O(7, 23) matrix IJ de"ned as in (121) but now the time component is included, and ¸ is an O(7, 23) invariant metric de"ned in (127). So, the D"3 e!ective theory is described only in terms of graviton and scalars.
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43
The D"3 e!ective action has the form [561]: (136) L"(!h[R #hI J Tr(R MLR ML)] , F I J where h,det h , R is the Ricci scalar of h . M is a symmetric O(8, 24) matrix of D"3 scalars IJ F IJ de"ned as [561]
M!ePtt2
M¸t!ePt(t2¸t) ePt2 !eP ePt2¸t . M" M ¸M t t2¸M ePt2¸t !e\P#t2¸M M !eP(t2¸t) !ePt2(t2¸t) The action is manifestly invariant under the O(8, 24) transformations [561]: ePt
MPXMX2, h Ph , IJ IJ where X3O(8, 24), i.e.
(137)
(138)
¸M 0 0
XLX2"L,
L" 0 0
0 1 .
(139)
1 0
When electric and magnetic charges are quantized according to the DSZW quantization condition, the O(8, 24) is broken down to O(8, 24, Z). Since O(8, 24, Z) is generated by the conjectured S-duality in D"4 and T-duality in D"3 (which is proven to hold order by order in string coupling), the establishing O(8, 24, Z) invariance of the full string theory is equivalent to proving the S-duality in D"4 [561]. 3.3. String-string duality in six dimensions Six dimensions is special in the duality of (d!1)-branes [234]. A (d!1)-brane in D dimensions is dual to a (dI !1)-brane (dI ,D!d!2) under the Hodge-dual transformation of "eld strengths. So, in particular the heterotic string (1-brane) in D"6 is dual to another string (dI "6!2!2"2) [226]. In fact, it was found out by Du! et al. [246,598] that the type-IIA string compacti"ed on K3 surface has the same moduli space as that of the heterotic string compacti"ed on ¹, i.e. O(4, 20, Z)/O(4, 20, R)![O(4, R);O(20, R)]. Based on these observations, it is conjectured [635] that the heterotic string on ¹ is dual to the Type-IIA string on K3 surface, the so-called string-string duality in D"6 [226,229,235,340,381,635]. The e!ective action of heterotic string compacti"ed on ¹ in the string-frame is [565,635]
1 dx (!Ge\U[R GI J R UR U! GI I YGJ J YGM M YH H S" % I J IJM IYJYMY 16pG !GI I YGJ J YFG (¸M¸) FH #GI J Tr(R M¸R M¸)] , (140) IJ GH IYJY I J where k"0,2, 5, i"1,2, 24, ¸ is an O(4, 20) invariant metric and M is a symmetric O(4, 20) matrix, i.e. M2"M and M¸M2"¸. (De"nitions of D"6 "elds in terms of the D"10 "elds are
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given in Section 3.2.2.) The "eld strengths of the ;(1) gauge "elds and the 2-form potential are FG "R AG !R AG , H "(R B #2AG ¸ FH )#cyc. perms. . IJ I J J I IJM I JM I GH JM The e!ective action for type-IIA string compacti"ed on a K3 surface is [565,635]
(141)
1 dx((!G [e\UY+R #G I J R U R U S " %Y I J 16pG ! G I I YG J J YG M M YH H #G I J Tr(R M ¸R M ¸), IJM IYJYMY I J (!G IIYG JJYF G (¸M ¸) F H ]!eIJMNOCB F G ¸ F H ) , (142) IJ GH IYJY IJ MN GH OC where now the corresponding D"6 "elds in type-IIA theory are denoted with primes. Note, the "eld strength of B is de"ned without Chern}Simmons term involving ;(1) gauge "elds IJ H "R B #cyc. perms. . (143) IJM I JM The scalars and metric are de"ned similarly as those in the e!ective action (140) of heterotic string on ¹. But since the K3 surface does not have a continuous isometry, there are no KK ;(1) gauge "elds, instead there are additional ;(1) gauge "elds arising from the 1-form A and the 3-form + A in the R-R sector. +,. These two string-frame e!ective actions are described by the same "eld degrees of freedom and have the same modular space. So, they can be identi"ed as the same action, provided we perform the following conformal transformation of the metric and the Hodge-duality transformation of the 2-form "eld [565,635]: , U "!U, G "e\UG , M "M, A ?"A? IJ I I IJ (144) U (!Ge\ HIJM"eIJMNOCH . NOC Under the string}string duality, the dilaton changes its sign, indicating that the string coupling j"e\6U7 of the dual theory is inverse of the original theory. So, a perturbative string state (weak string coupling j;1) in one theory is mapped to a non-perturbative string state (strong string coupling j<1) under the string}string duality. For example, perturbative, singular, fundamental string in one theory is mapped to non-perturbative soliton string in the other theory [565]. 3.3.1. String}string}string triality Upon toroidal compacti"cation to D"4, the D"6 string}string duality (144) interchanges the D"4 S-duality and the (¹ part of) T-duality [226], while the dilaton-axion "eld and the KaK hler structure of ¹ are interchanged. So, the axion-dilaton "eld of the string}string duality transformed theory is given by the KaK hler structure of the original theory. Note, the ¹ part of the full D"4 T-duality group, i.e. the O(2, 2, Z) S¸(2, Z);S¸(2, Z) subgroup, contains not only the S¸(2, Z) factor parameterized by the KaK hler structure of ¹ but also the other S¸(2, Z) factor parameterized by the complex structure of ¹ [215,229,570]. Namely, the e!ective D"4 theory has the S¸(2, Z);S¸(2, Z);S¸(2, Z) symmetry with each S¸(2, Z) factor respectively parameterized by the dilaton-axion "eld, the KaK hler structure and the complex structure. So, on the ground of symmetry argument, one expects another string theory whose axion-dilaton "eld is given by the complex structure of the original theory [236]. In fact, mirror symmetry [24,317,319,320,372] exchanges the
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complex structure and the KaK hler structures of an internal manifold. In particular, the mirror symmetry exchanges the type-IIA and type-IIB strings, and transforms heterotic string into itself. Thus, combining the D"6 string}string duality (which interchanges the dilaton-axion "eld and the KaK hler structure) and the Mirror symmetry (which interchanges the KaK hler structure and the complex structure), we establish the `trialitya [236] among the heterotic string on K3;¹ and the type-IIA and type-IIB strings on the Calabi}Yau-threefold. For the purpose of illustrating the triality among these three theories, we consider only the ¹ part and the NS-NS sector (which is common to the three theories) described by the following D"6 e!ective action: (145) L"(1/16pG )(!Ge\U[R #GI J R UR U! GI I YGJ J YGM M YH H ] . % I J IJM IYJYMY All the three theories with such truncation have the e!ective actions of this form. We label these three D"4 theories as F , where F"H, A, B respectively denoting the heterotic theory, the 678 type-IIA theory and the type-IIB theory, and the subscripts X, >, Z respectively are the axiondilaton "eld, the KaK hler structure and the complex structure of the theory. We can take any of these three theories as the starting point, but for the purpose of de"niteness we start with the heterotic string and impose the string}string duality and the Mirror symmetry to obtain all other "ve theories. Compacti"cation on ¹ is achieved by the following KK Ansatz for the D"6 metric:
eEg #AKALG AKG IJ I J KL I KL , (146) ALG G J KL KL where k, l"0,2, 3 are the D"4 spacetime indices and m, n"1, 2 are the internal space indices. The D"6 2-form "eld is decomposed as: G " IJ
B #(AKB !B AL) B #AKB IJ I KJ IL J IL I KL . (147) B #B AL B KJ KL J KL Here, g, g , AK, B and G are respectively the D"4 dilaton (de"ned below), Einstein-frame IJ I IJ KL metric, the KK ;(1) gauge "eld, the 2-form "eld and the internal metric. To express the D"4 e!ective action in an S¸(2, Z);S¸(2, Z) T-duality invariant form, we parameterize the internal metric and the 2-form "eld as: B " IJ
G "eM\N KL
e\M#c !c !c
1
,
B "be , KL KL
(148)
and de"ne the D"4 dilaton g and axion a as (149) e\E"e\U(det G "e\U>N, eIJMNR a"(!ge\EgINgJHgMOH , KL N NHO where H is the "eld strength of B . Then, from the above real scalars we de"ne the following IJM IJ complex scalars [215] S"S #iS ,a#ie\E , ¹"¹ #i¹ ,b#ie\N , ;"; #i; ,c#ie\M ,
(150)
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which (within the framework of the heterotic string) are respectively the dilaton}axion "eld, the KaK hler structure and the complex structure. Then, the "nal form of the D"4 e!ective action is [236]:
1 (!g R !S gIIYgJJYF2 (M M )F #gIJ Tr(R M LR M L) L" E IJ 2 3 IYJY I 2 J 2 16pG 1 gIJR SR SM , (151) # gIJTr(R M LR M L)! I 3 J 3 I I 2(S ) where M , M 3S¸(2, R) are de"ned as 2 3 ¹ ; 1 1 1 1 , , M , (152) M , 3 ; ; ";" 2 ¹ ¹ "¹" and the ;(1) gauge "elds AG (i"1,2, 4) are given by A"B , A"B , A"A, A"A. I I I I I I I I I Here, L is an S¸(2, Z) invariant metric. The action is manifestly invariant under the S¸(2, R);S¸(2, R) T-duality:
M Pu2 M u , M Pu2 M u , F P(u\ u\)F , (153) 2 2 2 2 3 3 3 3 IJ 2 3 IJ where u 3S¸(2, R) and the rest of the "elds are inert. In addition, the theory has an on-shell 2 3 S-duality symmetry:
FG FG a b IJ Pu\ IJ ; u" , (154) 1 夹FG 夹FG c d IJ IJ where a, b, c, d3Z satisfy ad!bc"1, and 夹FG is the Hodge-dual (de"ned from the action (151)) of IJ the "eld strength FG . IJ We denote the theory described by (151) as H , meaning the heterotic theory with the 123 dilaton-axion "eld, the KaK hler structure and the complex structure given respectively by S, ¹, ; de"ned in (150). Under the Mirror symmetry, the KaK hler structure and the complex structure are interchanged, and therefore we obtain H theory, i.e. the heterotic string with the KaK hler 132 structure and the complex structure now respectively given by ; and ¹ de"ned in (150); the e!ective action is (151) with ¹ and ; "elds interchanged. We call H and H as the S-strings, 123 132 meaning the string theories with the dilaton}axion "eld given by S de"ned in (150). Under the D"6 string}string duality (144), the H is transformed to A . Under the Mirror 123 213 symmetry, A is transformed to B . So, the A and B are the T-strings. 213 231 213 231 We apply string}string duality to B to obtain B . Under the Mirror symmetry, B is 231 321 321 transformed to A . Therefore, we have the ;-strings given by B and A . 312 321 312 We comment on relation of the D"4 S-duality to the D"6 string}string duality. Since e!ect of the D"6 string}string duality on the D"4 theory is to interchange the complex structure and the dilaton}axion "eld, the S¸(2, Z) subset T-duality of one theory accounts for the S-duality of the string}string duality transformed theory [226,233]. Namely, the large}small radius T-duality (RPa /R) of one theory corresponds to the strong}weak coupling duality (g/2pP2p/g) of the dual theory. In terms of transformation of ;(1) gauge "elds, one can understand this as follows [236]. Under the string}string duality, electric [magnetic] charges of 2-form ;(1) gauge "elds of aS#b SP , cS#d
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one theory is transformed to magnetic [electric] charges of 2-form ;(1) gauge "elds of the dual theory, while those of KK ;(1) "elds remain inert. Since the T-duality interchanges KK ;(1) gauge "elds and 2-form ;(1) gauge "elds (associated with the same internal coordinates), under the combined action of the D"6 string}string duality and the D"4 T-duality, electric [magnetic] charges of KK ;(1) gauge "elds and magnetic [electric] charges of 2-form ;(1) gauge "elds are exchanged, which is exactly the D"4 S-duality. Thus, since the T-duality is proven to hold order by order in string coupling, proof of the conjectured D"4 S-duality amounts to proof of the D"6 string}string duality, and vice versa. Since the string}string duality interchanges the dilaton}axion "eld with the KaK hler structure, the string coupling g/2p of one theory is transformed to the worldsheet coupling a /R of the dual theory [226]. So, string quantum corrections controlled by the string coupling in one theory correspond to the stringy classical corrections (a corrections) controlled by the worldsheet coupling in the dual theory; the a [quantum] corrections in one theory can be understood in terms of the quantum [a ] corrections of the dual theory. 3.4. U-duality and eleven-dimensional supergravity Hull and Townsend [381] conjectured that the type-II superstring theories on a torus has full symmetry of low-energy e!ective "eld theory, which is larger than the direct product of the D"10 S¸(2, Z) S-duality and the O(10!d, 10!d, Z) T-duality in D"d(10 [306]. For example, the e!ective action of type-II string on ¹, which is D"4, N"8 supergravity [159,160], has an on-shell E symmetry [160], which contains S¸(2, R);O(6, 6) as the maximal subgroup. Hull and Townsend [381] conjectured that the subgroup E (Z) (broken due to the DSZW charge quantiz ations) extends to the full string theory as a new uni"ed duality group, called U-duality. U-duality uni"es the S and ¹ dualities and mixes p-model and string coupling constants. The discrete subgroup E (Z) is the intersection of the continuous E group and the discrete symplectic Sp(28, Z) group, which transforms 28 ;(1) gauge "elds of the e!ective theory linearly: E (Z)"E 5 Sp(28, Z) [381]. Under U-duality, a set of 28;2 electric and magnetic charges transform as a vector, and all the scalars in the theory mix among themselves. Unlike other types of duality, which assigns a special role to the dilaton, under U-duality the dilaton is on the same footing as moduli. Thus, unlike T-duality which is perturbative in nature, U-duality, like S-duality [560], is non-perturbative in nature, so cannot be tested within perturbative spectrum of string theories. We list the conjectured U-duality groups in various dimensions [381]. Note, the duality group in higher dimensions is a subgroup of lower dimensional duality group, since it survives compacti"cation (cf. The duality group does not act on the Einstein-frame metric). The SO(10!d, 10!d, Z) T-duality in D"d(10 and the S¸(2, Z) S-duality in D"10 are uni"ed to U-duality for d(8. The U-duality groups are, in the descending order starting from D"7: SO(5, 5, Z), S¸(5, Z), E (Z), E (Z), E (Z), E (Z), E (Z). These U-duality groups in D"d are generated by the T-duality in D"d and n numbers of U-duality groups in D"d#1 [381], where n is the possible numbers of ways in which one can compactify from D"10 to D"d#1 and then down to D"d. In comparison, the heterotic string on ¹\B with d'3 maintains the duality group in the form (T-duality group);(S-duality group), i.e. SO(10!d, 26!d, Z);Z for 105d55 or
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SO(10!d, 26!d, Z);S¸(2, Z) for d"4. For d"3 [561] and d"2 [563], the T- and S-dualities are uni"ed to U-duality given by SO(8, 24, Z) and SO(8, 24)(Z), respectively. As we will see in Section 7.1, BPS electric states are within perturbative spectrum of heterotic string [248]; all the 28 electric charges in the heterotic string on ¹ are related through the `perturbativea O(6, 22, Z) T-duality. Also, there is non-perturbative spectrum carrying the remaining 28 magnetic charges, related to perturbative spectrum via the Z subset of the `non-pertur bativea S¸(2, Z) S-duality [486,544,558]. These magnetic charges are carried by solitons. The mass of a state in heterotic string on ¹ carrying electric [magnetic] charges of the ;(1) gauge group behaves as &1 [&1/g] in the string frame, as expected for a fundamental string [a soliton]. Q For the type-II string, only 12 of the 28 electric charges couple to perturbative string states, since the remaining 16 electric charges are R-R charges, which cannot be coupled to perturbative string states. The Z subgroup of the S¸(2, Z) S-duality group relates these perturbative states to solitonic states carrying 12 magnetic charges of the same 12 ;(1) gauge "elds. The remaining 16 electric and 16 magnetic charges of the ;(1) gauge group are carried by another type of non-perturbative states, whose mass behaves as &1/g . Thus, under the (T-duality);(S-duality) subgroup, i.e. Q SO(6, 6, Z);S¸(2, Z)LE (Z), the fundamental representation 56 (representing 28 electric and 28 magnetic charges of the D"4 ;(1) gauge group) is decomposed as 56"(12, 2);(32, 1). Here, the "rst factor (12, 2) corresponds to the 12 numbers of S¸(2, Z) doublets of perturbative and solitonic states in the NS-NS sector and the second factor (32, 1) denotes the remaining non-perturbative states, which are singlets under the S¸(2, Z) group and carries 16 electric and 16 magnetic charges of 16 ;(1) gauge "elds in the R-R sector. Since the O(6, 6, Z) T-duality group of the type-II string on ¹ mixes only NS-NS charges among themselves, it is not inconsistent that string states carry only NS-NS charges. However, it is not consistent with the conjectured U-duality, since U-duality puts all the 28;2 electric and magnetic charges of D"4 ;(1) gauge group on the same putting. In addition, as we saw in the decomposition of the 56 representation, the U-duality requires existence of additional 16#16 electric and magnetic charges in the RR-sector that transform irreducibly under the O(6, 6, Z) T-duality group. Hence, one leads to the conclusion that all the RR charges, which cannot be carried by perturbative string states or solitons, should be carried by another type of nonperturbative states. The low-energy or long-distance description of these non-perturbative string states is R-R p-branes or black holes. In the original work by Hull and Townsend [381], they show that all the R-R charged black holes can arise from extreme p-branes of the D"10 e!ective supergravity via dimensional reduction. In [498], Polchinski shows that the states carrying R-R charges can be realized within string theories without introducing exotic extended objects like p-branes. Such objects are D-branes [193,445], boundaries to which the ends of open strings (with Dirichlet boundary condition) are attached. D-branes carry one unit of R-R charges. D-branes are dynamic objects and the open string states describe their #uctuations. In the strong coupling limit, D-branes become black holes. Such identi"cation made it possible to give precise statistical explanation of black hole entropy. In Section 8, we will summarize aspects of D-branes and the recent development in D-brane calculation of black hole entropy. In the following sections, we discuss the S-duality of the type-IIB string and the T-duality of type-II string on a torus. The U-duality of type-II string on a torus is generated by these two dualities. In particular, starting from generating black holes of type-II theories on a torus, one
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obtains black holes with the general charge con"guration by applying the S-duality and the T-duality transformations. For p-branes in type-II theories, one can generate p-branes of the general charge con"guration by "rst imposing SO(1, 1) boost on a charge neutral solution to induce KK electric charge and then applying the ¹- and the S-duality transformations and/or another SO(1, 1) boosts sequentially. 3.5. S-duality of type-IIB string The type-IIB string [312,536] has the S¸(2, Z) symmetry [381]. The Z LS¸(2, Z) trans formation exchanges the NS-NS 2-form potential BK (coupled to a perturbative string state) and the R-R 2-form potential BK (coupled to a non-perturbative D-brane) while changing the sign of the dilaton (or inverting the string coupling). So, the S¸(2, Z) symmetry is a strong}weak coupling duality. It is well-known that a covariant e!ective action for type-IIB string does not exist, while only the "eld equations [376,536] exist. The only problem with construction of the covariant e!ective action is the R-R 4-form potential DK (with the self-dual 5-form "eld strength FK ), whose equation of motion FK "夹FK cannot be derived from the covariant action. So, to construct the covariant e!ective action for type-IIB string, one is forced to set DK to zero. However, it is found in [72] that one can construct the type-IIB e!ective action which gives rise to the correct "eld equations and compacti"es to the correct action for the dimensionally reduced type-IIB theories without setting DK to zero. In this approach, one keeps FK di!erent from zero in the e!ective action but eliminates the self-duality constraint. After the "eld equations are obtained from this e!ective action, the self-duality constraint is imposed in order to get the correct "eld equations for the type-IIB theory. In the string-frame, such e!ective action for type-IIB string has the form [72]:
3 1 S " dx (!GK e\U !RK #4(RU)! (HK ) '' 4 2
1 3 5 1 ! (Rs)! (HK !sHK )! FK ! seGHeDK HK GHK H , 2 4 6 96(!G
(155)
The "eld strengths of the 2-form potentials BK G and the 3-form potential DK , and their gauge transformation rules are HK G"RBK G,
dBK G"RRK G ,
FK "RDK #eGHBK GBK H, dDK "RKK !eGHRRK GBK H ,
(156)
where RK G and KK are in"nitesimal gauge transformation parameters. The S¸(2, Z) symmetry of the type-IIB theory is manifest in the e!ective action in the Einsteinframe. To go to the Einstein-frame, one Weyl-rescales the metric GK "e\ UGK . +, +,
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The resulting Einstein-frame action has the form:
1 1 3 S# " dx (!GK !RK # Tr(R MK R+MK \)! HK GMK HK H '' + GH 2 4 4
5 1 ! FK ! eGHeDK HK GHK H , 6 96(!GK
(157)
where MK is a 2;2 real matrix formed by the complex scalar jK "s#ie\U:
"jK " !Re jK 1 . MK " Im jK !Re jK 1
(158)
(157) is manifestly invariant under the S¸(2, R) transformation [376,536]:
HK HK
Pu
HK HK
, MK P(u\)2MK u\,
u3S¸(2, R) .
(159)
The S¸(2, R) transformation on jK has the usual fractional-linear form jK P(ajK #b)/(cjK #d). When the DSZW type charge quantization is taken into account, the S¸(2, R) symmetry breaks down to the S¸(2, Z) subset. 3.6. ¹-duality of toroidally compactixed strings Closed strings in D dimensions in target space background with d commuting isometries have O(d, d, Z) T-duality symmetry [304,305,308,570]. The O(d, d,Z) symmetry is a perturbative symmetry proven to hold order by order in string coupling. For heterotic string, the symmetry is enlarged to O(d, d#16, Z) due to the additional rank 16 background gauge "elds. Under the Z subset that inverts the radius of S (i.e. R Pa /R ) and interchanges winding and momentum G G modes (i.e. m n ), the type-IIA and the type-IIB theories are interchanged if odd number of circles G G are acted on by the Z transformations, while heterotic string transforms to itself. The Z transformation between the type-IIA and the type-IIB strings at the e!ective "eld theory level is understood as "eld rede"nition between type-IIA and type-IIB theories, since the compacti"cation of the type-IIA and the type-IIB supergravities leads to the same supergravity theory. We consider the bosonic string worldsheet p-model with only NS-NS sector "elds (target space metric G (X), 2-form potential B (X) and dilaton U(X)), which are common to both type-II and IJ IJ heterotic strings, turned on. The action with the (curved) D-dimensional target space has form
1 dz[(G (X)#B (X))RXIRXJ! (X)R] . S" IJ IJ 2p
(160)
Let us assume that (160) is invariant under the d commuting, compact Abelian isometries [309] dXI"ekI (i"1,2, d) along the XG-direction, where [k , k ]"0. Then, the background "elds G G H become independent of XG. First, we consider the case where the background "elds have only one isometry (d"1) along, say, the direction h"X. The T-dual pair p-model actions are obtained by gauging the Abelian isometry. Following the procedures discussed in [80,82,100}102], one obtains the action dual to
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51
(160) with the dual background "elds (with primes) related to the original ones (without primes) as 1 B G G #B B ? @, G " , G " ? , G "G ! ? @ G ? G ?@ ?@ G ?@ (161) G B #B G ? @, U "U#log G , B "G G , B "B ! ? @ ? ? ?@ ?@ G where XI"(h, X?) (a"1,2, D!1). This is the curved background generalization of the RPa /R T-duality of closed strings on S. Alternatively, the dual pair p-models are obtained by the method of chiral currents [516]. One starts with a (D#d)-dimensional p-model with d Abelian (left-handed) chiral currents JG and (right-handed) anti-chiral currents JM G [309]: S "S #S #S[X] , ">B ? 1 S " dz[RhG RM hG #RhG RM hG #2R (X)RhG RM hH #C* (X)RX?RM hG #C0 (X)RhG RM X?] , 2p * * 0 0 GH 0 * ?G * G? 0
1 dz[RhG RM hG !RhG RM hG ] , S" * 0 0 * ? 2p
(162)
1 dz[C (X)RX?RM X@!U(X)R] , S[X]" ?@ 2p where i, j"1,2, d and a, b"d#1,2, D. Here, chiral and anti-chiral currents, corresponding to the ;(1)B ;;(1)B a$ne symmetries dhG "aG (z), are * 0 * 0 * 0 JM G"RM hG #R hM H #C0 RM X? . (163) JG"RhG #R hH #C* RX?, 0 GH * G? * HG 0 ?G By gauging either a vector or an axial subgroup of ;(1)B ;;(1)B , one has the following * 0 D-dimensional dual pair p-model actions:
1 1 S!" dz E! (X?)RXI RM XJ ! !(X?)R " 2p IJ ! ! 4
1 " dz E!(X?)RhG RM hH #F0!(X?)RhG RM X?#F*!(X?)RX?RM hG GH ! ! G? ! ?G ! 2p
1 #F!(X?)RX?RM X@! !(X?)R , ?@ 4
(164)
where (XI )"(hG , X?) with k, l"1,2, D, i"1,2, d, a"d#1,2, D. Here, hG ,hG $hG ! ! ! 0 * and the upper (lower) signs in $ and G correspond to the axial (vector) gauged p-model. The
More general transformations with non-zero R-R "elds are derived in [81].
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D. Youm / Physics Reports 316 (1999) 1}232
background "elds are
E! F0! G@ , !"U#log det(I$R) , E! "G! #B! " GH IJ IJ IJ F*! F! ?H ?@ 1 E!"(I$R) (IGR)\, F!"C $ C* (IGR)\C0 , GH GI IH ?@ ?@ 2 ?G GH H@
(165)
F0!"(IGR)\C0 , F*!"$C* (I$R)\ , G? GH H? ?G ?H HG with G! [B!] denoting the symmetric (the antisymmetric) part of E!. The action S! has an " isometry under the translation in the hG -direction. 8 The dual pair actions (164) are the most general p-model with d commuting compact Abelian symmetries. S> and S\ are related under the combined operations of the sign reversal of R and C*, " " and the coordinate transformation hG P!hG (i.e. hG hG ), implying equivalence of two gaugings * * > \ at least locally. To achieve global equivalence of the two gaugings, one has to impose the same periodicity conditions on both hG and hG , i.e. hG ,hG #2p. Then, one establishes the equivalence > \ ! ! of vector and axial gauged p-model actions S! (the so-called `vector-axial dualitya [426]). " One can relate S! to the bosonic string p-model action S in (160) by identifying XI"XI . Then, " ! the background "elds in S! are related to those in S in the following way: " G "(E!#E!), B "(E!!E!) , HG GH GH HG GH GH G "(F0!#F*!), B "(F0!!F*!) , ?G G? G? ?G G? G? G "(F!#F!), B "(F!!F!) . @? ?@ ?@ @? ?@ ?@
(166)
From this, transformation rule of background "elds under T-duality that relates the dual pair bosonic string p-model actions (one related to S> and the other related to S\) follows. When " " S! have isometry along only one coordinate direction (d"1), one recovers the factorized duality " transformation (161). We discuss transformations [305,309] that relate the di!erent backgrounds (of the same action) describing the equivalent conformal "eld theory. S! have the manifest invariance under the " following O(d, d, Z) transformation E!PE! "(a( E!#bK )(c( E!#dK )\
E!
(a!E! c)F0!
"
F*!(cE!#d)\ F!!F*!(cE!#d)\cF0!
det G! 1 ,
!P ! " !# log det G! Y 2
g"
a b c
d
3O(d, d, Z) ,
,
(167)
where D;D blocks a( , bK , c( and dK of the O(D, D, Z) matrix are
a( "
a
0
0 I "\B
,
bK "
b 0 0 0
, c( "
c
0
0 0
,
dK "
d
0
0 I "\B
.
(168)
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Here, I is the n;n identity matrix. The d;d matrices E! transform fractional linearly under L O(d, d, Z): E!PE! "(aE!#b)(cE!#d)\ .
(169)
The constant E! case corresponds to the transformation in the toroidal background. The O(d, d, Z) transformation is generated by the following transformations. E Integer `Ha-parameter shift of E, i.e. E PE #H (H "!H ): GH GH GH GH HG a b I H " B s.t. H"!H2 . c d 0 I B E Homogeneous transformations of E!, F*!, F0! under G¸(d, Z), F*!PA2F*!, F0!PF0!A (A3GL(2, Z)):
a b
A2
0
0
A\
"
c
d
s.t. A3GL(2, Z) .
(170) i.e. E!PA2E!A,
(171)
E Factorized dualities D , corresponding to the inversion of the radius R of the ith circle, i.e. G G R P1/R : G G G\ B\G\ CDE CDE a b I!e e G G " s.t. e "diag(0, 2, 0, 1, 0, 2, 0). (172) G c d e I!e GF F FHF F FI G G
The maximal compact subgroup of O(d, d, Z) is O(d, Z);O(d, Z) with a group element having the form:
hG PO hG , * * *
as ">B h O #O O !O h \ P1 * 0 * 0 \ , 2 O !O O #O h h > * 0 * 0 > C0PO C0, CPC , 0
a b
1 O*#O0 O*!O0 " s.t. O , O 3O(d,Z) . * 0 2 O !O O #O c d * 0 * 0 The O(d, Z);O(d, Z) transformation naturally acts on the action S hG PO hG , 0 0 0
(173)
RPO RO2, C*PC*O2, (174) 0 * * meanwhile on the actions S! as " E!PE! "[(O #O )E!#(O !O )][(O !O )E!#(O #O )]\ , * 0 * 0 * 0 * 0 F*!PF*! "2F*![(O !O )E!#(O #O )]\ , * 0 * 0 (175) F0!PF0! "[(O #O )!E! (O !O )]F0! , * 0 * 0 F!PF! "F!!F*![(O !O )E!#(O #O )]\(O !O )F0! . * 0 * 0 * 0 We now specialize to strings in #at background (i.e. toroidal compacti"cation) to understand properties of perturbative string spectrum under T-duality. The relevant part of the worldsheet
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action is the toroidal (¹B) part:
1 1 p (176) S" dp dq (gg?@G R XGR XH#e?@B R XGR XH! (g R , GH ? @ GH ? @ 2 4p where XG&XG#2pmG and i, j"1,2, d. Here, mG is a string `winding numbera along the XGdirection. G and B can be collected into the `background matrixa E"G#B. The matrix E is GH GH a special case of E! in (165) where E! do not depend on X?. The lattice KB, which de"nes ¹B"RB/(pKB), is spanned by basis vectors e satisfying B e?e?"2G . ? G H GH G The mode expansions of XG and conjugate momenta 2pP "G XQ H#B XH are G GH GH i 1 XG(p, q)"xG#mGp#qGGH(p !B mI)# [aG (E)e\ O\N#aG (E)e\ O>N] , H HI L (2 L$ n L 1 [E2 aH (E)e\ O\N#E aH (E)e\ O>N] , (177) 2pP (p, q)"p # GH L GH L G G (2 L$ where we made analytic continuation qP!iq and the momentum zero modes p take integer G values, i.e. p "n 3Z. The equal-time canonical commutation relations [XG(p, 0), P (p , 0)]" G G H idG d(p!p ) lead to commutation relations among the oscillator modes: H [xG, p ]"idG , [aG (E), aH (E)]"[aG (E), aH (E)]"mGGHd . (178) H H L K L K K>L The Hamiltonian takes the form
1 1 p dp(P#P)" Z2M(E)Z#N#NI , H"¸ #¸I " * 0 4p 2 G!BG\B BG\ m , Z" ? , M(E)" !G\B G\ n ? where P are the left- and the right-moving momenta de"ned as * 0 P "[2pP #(G!B) XHY]eGH, P "[2pP !(G#B) XHY]eGH , *? G GH ? 0? G GH ? and the number operators of the left- and the right-moving modes are
(179)
(180)
N " aG (E)G aH (E), N " aG (E)G aH (E) . (181) * \L GH L 0 \L GH L L L Here, eGH are dual basis vectors, satisfying B e?eHH"dH and B eGHeHH"(G\)GH. The left- and ? G ? G ? ? ? the right-moving momenta zero modes p "[n2#m2(B!G)]eH, p "[n2#m2(B#G)]eH , (182) 0 * transforming as a vector under O(d, d, R), form an even self-dual Lorentzian lattice CB B [480,481], i.e. p!p"2mGn 32Z. * 0 G While CB B is preserved under O(d, d, R), the Hamiltonian zero mode H "(p#p) is invariant 0 * only under its maximal compact subgroup O(d, R);O(d, R). So, the zero-mode spectrum is
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unchanged under the O(d, R);O(d, R) subgroup, only. Note, from (182) one sees that (p , p ) and * 0 hence CB B are in one-to-one correspondence with a particular background E"G#B. Thus, the moduli space is isomorphic to O(d, d, R)/[O(d, R);O(d, R)]. Under the O(d, d, Z) transformation (169) [439], M(E)PgM(E)g2 ,
(183) a (E)P(d!cE2)\a (E ), a (E)P(d#cE)\a (E ) . L L L L So, N are manifestly invariant under O(d, d, Z); the spectrum is O(d, d, Z) invariant. The O(d, d, Z) * 0 transformation is generated [304,305,308,570] by integer H-parameter shift of E, the G¸(2, Z) transformation and the factorized duality D , as discussed above. Particularly, under G¸(d, Z), G which changes the basis of KB, EPAEA2 and (m, n)P(A2m, A\n) (A3G¸(d, Z)). In addition, the spectrum is invariant under the worldsheet parity [305] pP!p, which acts on E as BP!B. The e!ect of the worldsheet parity on the spectrum is to interchange the left-handed and the right-handed modes: p p and a a . The above transformations generate the full spectrum * 0 L L preserving symmetry group G . B A particular element g of O(d, d, Z) with a"d"0 and b"c"I, i.e. EPE\ [308,570], corresponds to vector-axial duality symmetry (165). Under this transformation, n m and the Hamiltonian (179) is manifestly invariant. When B"0, the transformation becomes GPG\, i.e. the volume inversion of ¹B. A signi"cance of EPE\ is that the gauge symmetry is enhanced to the a$ne algebra S;(2)B ;S;(2)B at a single "xed point G"I and B"0. The gauge symmetry is * 0 maximally enhanced [308] at "xed points under EPE\ modulo S¸(d, Z) and H(Z) transformations, i.e. E such that E\"M2(E#H)M (M3S¸(d, Z)). Hence, an enhanced symmetry point corresponds to an orbifold singularity point [220,221] in the moduli space under some non-trivial O(d, d, Z) transformation. At the "xed point, E takes the following form in terms of the Cartan matrix C of the rank d, semi-simple, simply laced symmetry group [252]: GH E "0 (i(j) . (184) E "C (i'j), E "C , GH GH GH GG GG Non-maximally enhanced symmetry points correspond to "xed points under factorized dualities D instead of the full inversion EPE\. A simplest but non-trivial example is the d"2 case G (i.e. compacti"cation on ¹), which we discuss in Section 4.2.2. At the "xed point E"I under EPE\, the gauge symmetry is enhanced to (S;(2);S;(2)) ;(S;(2);S;(2)) . The gauge * 0 symmetry is maximally enhanced to S;(3) ;S;(3) at the point * 0 1 1 . E" 0 1
These "xed points correspond to orbifold singularities [570] in the fundamental domain of the moduli space (parameterized by two complex coordinates of the moduli space S¸(2, R)/;(1); S¸(2, R)/;(1)).
Note, O(2, 2, R) S¸(2, R);S¸(2, R). Therefore, E, which parameterizes the moduli space O(2, 2, R)/[O(2, R) ;O(2, R)], is reparameterized by the complex coordinates o and q, each parameterizing S¸(2, R)/;(1).
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3.7. M-theory In this section, we discuss some aspects of M-theory. We illustrate how di!erent superstring theories emerge from di!erent moduli space of compacti"ed M-theory and discuss the M-theory origin of string dualities. In this picture, each of "ve di!erent string theories represents a perturbative expansion about di!erent points in moduli space of the compacti"ed M-theory. Namely, 5 perturbative string theories and uncompacti"ed M-theory are located at di!erent subsets of moduli space, and it is dualities that map one subset of moduli space to another, thereby making transition between di!erent theories. In the following, we illustrate this idea by showing how di!erent theories are achieved by taking di!erent limits of parameters of moduli space and how dualities are realized as transformations of parameters of moduli space. First, we discuss the connection between the type-II theories and M-theory. Type-IIA theory is obtained from M-theory by compactifying the extra 1 spatial coordinate on S of radius R [112,232,383,601,635]. The type-IIA and the type-IIB theories are related via T-duality [193,216]. Namely, the type-IIA theory on S of radius R is perturbatively equivalent (under T-duality) to the type-IIB theory on S of radius R "1/R . So, one can think of the S-compacti"ed type-IIB theory as ¹ compacti"ed M-theory. To understand the connection between type-II theories and M-theory, one has to compactify M-theory on ¹"S;S (with the radii of each circle given by R and R from the D"11 point of view), and compactify the type-IIA and the type-IIB theories on circles of radii R and R , respectively. Here, the radius R [R ] is measured with D"10 string frame metric of the type-IIA [the type-IIB] theory. We "rst relate parameters of the type-II theories (i.e. the radii R and the string couplings g in the type-IIA/B theories) to parameters R and R of ¹ moduli space before we Q understand the various limits in the moduli space. R is related to g as R "(g ) . As for the Q Q second circle of ¹"S;S, which is also the circle upon which the type-IIA theory is compacti"ed, the radius is measured di!erently depending on the dimensionality of spacetime. Note, we denoted the radius measured in D"11 [in D"10 by the type-IIA string-frame metric] as R [R ]. Namely, since the string-frame metric g'' (k, l"0, 1,2, 9) of the type-IIA theory is IJ related to the D"11 metric G (M, N"0, 1,2, 10) as G&e\ ( g'' , where is the IJ +, IJ dilaton of the type-IIA theory, one sees that R "R /(g ) . Furthermore, one can relate the Q string coupling g of the type-IIB theory to R and R as follows. Under the T-duality between Q the type-IIA and the type-IIB theories, the string couplings are related as g "g /R . By using Q Q other relations among parameters, one "nds that g "R /R . Q
Note, due to the no-go theorem for KK compacti"cation of the D"11 supergravity [633], it might be impossible to obtain a chiral theory like type-IIB supergravity through dimensional reduction. This no-go theorem can be circumvented to obtain the (chiral) type-IIB theory by compactifying on orbifolds (rather than manifolds) [197,358,568,639]. In the case of compacti"cation of M-theory on ¹ (which is relevant to our discussion), when the size of ¹ goes to zero at the "xed shape, one obtains `chirala type-IIB theory, due to additional massive &wrapping' modes (of membrane) which become massless [22,81,540,541]. The string coupling g is de"ned as g "e6(7. Q Q
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We discuss the various limits in the M-theory moduli space of ¹ in terms of R and R [22]: M-theory and the type-IIA, B theories are located at various limiting points in the ¹-moduli space. First, M-theory is located at (R , R )"(R,R) (i.e. the decompacti"cation limit), which is also the strong coupling limit (g "(R ) PR) of the type-IIA theory [601,635]. The (uncomQ pacti"ed) type-IIA theory, de"ned as R PR and "nite string coupling g , is located at Q (R , R )"(R, xnite), i.e. M-theory on S of radius R (R. The (uncompacti"ed) type-IIB theory can be de"ned as the limit R PR, R P0 and "nite string coupling g . In this limit, Q R "R /(g ) "R /(R g ) "R (g )\ "R\ (g )\ P0. So, in terms of para Q Q Q meters of ¹, the (uncompacti"ed) type-IIB theory corresponds to the limit in which (R , R )"(0, 0) while keeping the ratio g "R /R "nite. The value of g depends on how Q Q the limit (R , R )P(0, 0) is taken and, therefore, (R , R )"(0, 0) is not really a point in the moduli space. Note, when 2 circles in ¹"S;S are exchanged (i.e. R R ), g "R /R is inverted: Q g P1/g . Such interchange of 2 circles is a subset of more general SL(2, Z) reparameterization of Q Q ¹, which acts on the complex modulus q of ¹ fractional linearly. Thus, the reparameterization symmetry of ¹, upon which M-theory is compacti"ed, manifests in the type-IIB theory as the S¸(2, Z) S-duality [381], which acts on the complex scalar o"s#ie\( (formed by 0-form s and the dilaton ) fractional linearly. Next, we discuss string theories with N"1 supersymmetry, i.e. the E ;E and SO(32) heterotic strings and type-I string. To understand the connection between M-theory and these N"1 string theories, one has to consider the moduli space of M-theory compacti"ed on S/Z ;S, i.e. a cylinder of length ¸ and radius R. We "rst comment on the relation of type-I string theory to M-theory. One can think of the type-I theory as an &orientifold' of the type-IIB theory, namely a theory of unoriented closed string (gauged under the worldsheet parity transformation X [355,356,509]) and open string with SO(32) Chan-Paton factor [314]. To see the direct relation to M-theory, it is convenient to "rst compactify one spatial coordinate, which we call >, of the type-I theory on S and then T-dualize along the S-direction, inverting the radius of S. We call such theory as the type-I theory [193,505]. Since the dual coordinate >I is pseudo-scalar under the worldsheet parity transformation (i.e. X[>I ](q, p)"!>I (q,!p)), S is mapped under this T-duality to the orbifold S/Z with "xed points at >I "0, p. So, the type-I theory is e!ectively described by the type-IIA theory on S/Z ; closed strings wrapped around S/Z look like open string stretched between two 8-plane boundaries located at "xed points of S/Z . (These (parallel) boundaries corresponds to D 8 branes.) Second, the E ;E heterotic string theory is obtained by compactifying M-theory on the orbifold S/Z [357,358]. Namely, M-theory on S/Z of length ¸ gives rise to spacetime with two D"10 faces (the so-called `end-of-the-world 9-branesa) which are separated by a distance ¸. Each of the two faces carries an E gauge "eld of the E ;E heterotic string [357,358]. In this picture, a fundamental string of the E ;E theory is interpreted as a cylindrical M 2-brane attached between the two faces. (So, the intersection of the cylindrical M 2-brane with the faces is a circle.) The string coupling is g#"¸ and, therefore, in the strong coupling limit (g#<1) the two faces Q Q move apart far away from each other, revealing the extra 11th space dimension [357,358,601,635]. When the separation is very small (¸+0), the cylindrical M 2-brane is well approximated by a closed string in D"10. (This is the weak coupling limit g#"¸ +0 of the E ;E theory.) Q
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Finally, the SO(32) heterotic theory is related to the E ;E heterotic theory via T-duality [303,480,481], and to the type-I theory via S-duality [187,380,505,635]. A corollary of these dualities is the duality between M-theory on a cylinder S/Z ;S and the SO(32) theories on S [541]. Now, we discuss the various limits in the moduli space of M-theory on S/Z ;S in terms of parameters ¸ and R [357,358]. An obvious limit in the moduli space is the small ¸ and RPR limit, which is the uncompacti"ed E ;E heterotic string. Here, ¸ is the size of S/Z upon which M-theory is compacti"ed to lead to the E ;E heterotic theory. The string coupling of the E ;E heterotic string is g#"¸ . The second obvious limit is RP0. In this limit, the M 2-brane Q wrapped around the cylinder looks like open string stretched between the interval S/Z of the length ¸, i.e. the (uncompacti"ed) type-I open string with SO(16) Chan}Paton factor attached at each end located at the 9-plane boundary. In general, a point in the moduli space with small ¸ and "nite R corresponds to the E ;E heterotic string on S of radius R. A non-vanishing &Wilson line' around S breaks E ;E down to subgroups [634], e.g. ;(1) or SO(16);SO(16) depending on the choice of the Wilson line. In particular, the E ;E heterotic string on S with gauge group SO(16);SO(16) is obtained from the SO(32) heterotic string on S with gauge group SO(16);SO(16) by inverting the radius of S. As R is decreased to a small value, one can switch to the SO(32) heterotic string on S of inverse radius 1/R by using the T-duality between the E ;E and SO(32) heterotic strings. For this case, the string coupling of the SO(32) heterotic string is g"¸/R, which is small as long as R<¸. As Q the radius R approaches smaller value so that R becomes much smaller than ¸, one can switch to the type-I theory by applying the S-duality between the SO(32) heterotic string and the type-I string. The string coupling of the type-I theory is then given by g'"1/g"R/¸. Note, under the Q Q Z transformation that exchanges two moduli R and ¸ of the cylinder, the string couplings of the type-I and the SO(32) heterotic theories are inverted, thereby manifesting as the non-perturbative type-I/heterotic duality. In the limit (R, ¸)P(0, 0) with "xed small g'"R/¸, one has the uncompacti"ed type-I theory. Q This is understood as follows. We saw that the type-I theory, which is obtained from the type-I theory on S by inverting the radius, is the RP0 limit of M-theory on the cylinder. If we further let the length ¸ of the cylinder approach zero, then in the type-I side the radius of S, upon which the type-I theory is compacti"ed, approach in"nity (i.e. the decompacti"cation limit of the type-I theory). So far, we discussed connections among M-theory and string theories with either N"1 or N"2 supersymmetry. One can further relate N"1 and N"2 theories. For this purpose, one breaks 1/2 of supersymmetry in N"2 theories by compactifying on a manifold with non-trivial holonomy. An obvious example is the D"6 string}string duality between type-IIA theory on K3 and heterotic string on ¹ [382,635]. Both of the D"6 theories have (non-chiral) N"2 supersymmetry. A corollary of this duality is that M-theory on K3 is equivalent to heterotic string on ¹ [635], since type-IIA theory is M-theory on S. The fundamental string in heterotic string on ¹ is nothing but M 5-brane wrapped around a 4-cycle of K3 surface [226,340,602]. Furthermore, K3-compacti"ed M2-brane through direct dimensional reduction is solitonic 5-brane in the heterotic theory wrapped around a 3-cycle of ¹. Thus, it leads to the conjecture that the strong coupling limit of heterotic string on ¹ is K3-compacti"ed supermembrane in D"11.
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We point out that M-theory and string theories that are connected within the moduli space (of either ¹ for the N"2 string theories or S;S/Z for the N"1 string theories) are on an equal putting if one includes `non-perturbativea branes, as well as perturbative string states, within the spectra of the 5 superstring theories. Namely, a brane that appears in one theory is necessarily related to branes of the other theories through the dimensional reduction and/or dualities [540,541]. In particular, all the branes in the 5 string theories should have interpretations in terms of M-branes through dimensional reductions and string dualities. It turns out that p-branes obtained in this way have the right property as p-branes of string theories [601] (e.g. the tension T of branes in string-frame depend on the string coupling g as &1, &1/g Q Q and &1/g for a fundamental string, Dp-branes and solitonic 5-brane, respectively) and Q the worldvolume actions (derived from those of M-theory [77,79,86,87]) have the right forms [88,241,530,604]. In the following, we discuss the M-theory origin of branes in string theories. First, we discuss branes in the type-IIA theory. Since the type-IIA theory is M-theory compacti"ed on S, all of p-branes in type-IIA theory (i.e. a fundamental string and a solitonic 5-brane in the NS-NS sector, and Dp-branes with p"0, 2, 4, 6, 8 in the R-R sector) should be obtained in this way. In D"11, there are M 2-brane [86,250] and M 5-brane [327] which are elementary and solitonic branes carrying electric and magnetic charges of the 3-form potential, respectively. Starting from M 2-brane [86,87], one obtains either fundamental string [231,232,250] in the NS-NS sector or the D 2-brane in the R-R sector, through double or direct dimensional reduction. The fundamental string and D 4-brane obtained, respectively, from the M 2- and M 5-branes via double dimensional reduction have the right dependence of the tensions on g , i.e. &1 and &1/g , Q Q respectively. Next, a D 0-brane can be thought of as the KK momentum mode of the D"11 theory on S [601]. This state with the momentum number n along the S-direction has mass (measured in the string-frame) given by M"n/g , which is the right dependence of the mass on the string coupling Q for BPS states carrying R-R charges [635]. The integer n is the electric charge of the KK ;(1) gauge "eld associated with the S-direction, indicating that such KK state is electrically charged under the 1-form potential in the R-R sector. The n"1 KK state is interpreted as a single D0-brane and the n'1 case corresponds to the (marginal) bound state of n D 0-branes. The D 6-brane is regarded as the KK monopole [601], which is magnetically charged under the 1-form potential in the R-R sector. For the D8-brane, currently there is no interpretation in terms of the D"11 theory available yet. Second, we discuss branes in type-IIB theory. In general, branes in type-IIB theory can be obtained from those in type-IIA theory by applying the T-duality between type-IIA and type-IIB theories [29,73,81,199,310]. For example, starting from D 2-brane of type-IIA theory, one obtains D 1-brane of type-IIB theory by T-dualizing along one of coordinates with the Neumann boundary condition (i.e. along a longitudinal direction of the D 2-brane). The fundamental string and the solitonic 5-brane in the NS-NS sector are obtained from the M 2- and the M 5-branes by dimensional reduction similarly as in the type-IIA case. On the other hand, one can directly relate branes in the compacti"ed type-IIB theory to those in compacti"ed M-theory by applying equivalence between type-IIB theory on S and M-theory on ¹. In this relation, one identi"es complex modulus q of ¹ with the complex scalar o of
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(uncompacti"ed) type-IIB theory, i.e. q"o [22,538]. (This is motivated by the observation that the non-perturbative S¸(2, Z) symmetry of type-IIB theory is interpreted as the ¹ moduli group of M-theory on ¹.) First, we discuss 1-branes in (uncompacti"ed) type-IIB theory. These carry (integer valued) electric charges q and q of 2-form potentials in the NS-NS and the R-R sectors [538], respectively, and are bound states of q fundamental strings and q D-strings. This bound state is absolutely stable against decay into individual strings i! the integers q and q are relatively prime [638], due to the `tension gapa and charge conservation. In the string-frame with the vacuum expectation value 1o2"i(g )\, the tension of the (q , q ) string [538] is ¹ " Q O O (q#(g )\q¹ , where ¹ is the tension of the fundamental string. This tension formula has Q the right limiting behavior: ¹ &1 for a fundamental string and ¹ &(g )\ for a D-string Q [489]. When these 1-branes are wrapped around S of radius R , one has 0-branes in D"9 with the momentum mode m and the winding mode n around S. This 0-brane of the D"9 type-IIB theory is identi"ed with the M 2-brane wrapped around ¹. Namely, the momentum mode m [winding mode n] of the type-IIB string is interpreted in the M 2-brane language as the wrapping [the KK modes] of the M 2-brane on ¹. Through these identi"cations, one has relations between the tension of the fundamental string of (uncompacti"ed) type-IIB theory and the tension ¹+ of M 2-brane: (¹ ¸)\"(1/(2p))¹+A , which is consistent with string dualities. Here, + ¸ "2pR is the circumference of S, upon which type-IIB theory is compacti"ed, and A is the + area of ¹ measured in the D"11 metric. Direct dimensional reduction of 1-branes in type-IIB theory gives rise to strings with charges (q , q ) and the tension ¹ in D"9. This type-IIB string in D"9 is identi"ed with M 2-brane O O wrapped around a (q , q ) homology cycle of ¹ with the minimal length ¸ "2pR "q !q q". O O Such string of the compacti"ed M-theory has the tension (measured by the D"11 metric) ¹ "¸ ¹+, which is consistent with relation between ¹ and ¹+ in the previous O O O ? paragraph. Second, we discuss the D"9 p-branes related to D 3-brane [238] of (uncompacti"ed) type-IIB theory. By wrapping D 3-brane around S, one obtains 2-brane with tension ¸ ¹ in D"9. Here, ¹ is the tension of D 3-brane in D"10. This 2-brane of type-IIB theory is identi"ed with 2-brane in the ¹-compacti"ed M-theory obtained by direct dimensional reduction of M 2-brane. Such identi"cation of the two 2-branes of type-IIB and M-theory leads to relation between the tensions ¹ and ¹ of fundamental string and D 3-brane: 1 ¹ " (¹ ) . 2p When D 3-brane of type-IIB theory is compacti"ed via direct dimensional reduction on S, one has 3-brane in D"9. This 3-brane is identi"ed with M 5-brane wrapped around ¹. Such an
These electric charges (q , q ) transform linearly under SL(2, Z), while the complex scalar o transforms fractional linearly, i.e. oP(ao#b)/(co#d)(ad!bc"1).
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identi"cation leads to the correct DSZ quantization relation 1 ¹+" (¹+) 2p on tensions of M 2- and M 5-branes. Third, 5-branes in type-IIB theory carry magnetic charges (p , p ) of 2-form potentials in the R-R and the NS-NS sectors, with the tension ¹ given similarly as that of 1-branes. 4-branes with N N the tension ¸ ¹ are obtained by wrapping this 5-branes around S. The corresponding N N 4-branes in the M-theory side is M 5-brane wrapped around a (p , p ) cycle of ¹. This identi"ca tion leads to the correct expression for the tension of the type-IIB 5-brane (in the string-frame) given by ¹ "((g )\/(2p))"p !p 1o2"(¹ ). In the limit where the 5-brane carries only Q N N either the R-R or the NS-NS magnetic charge, the tension behaves as ¹ &(g )\ and Q ¹ &(g )\, as expected for solitons and D-branes. Q 5-branes in D"9 have di!erent interpretations. First, a singlet 5-brane of the type-IIB theory on S is magnetically charged under the KK ;(1) gauge "eld associated with the S-direction. Second, the S¸(2, Z) family of 5-branes of the type-IIB theory on S is charged under the doublet of 2-form ;(1) gauge "elds. The corresponding singlet 5-brane in the ¹-compacti"ed M-theory is magnetically charged under the 3-form ;(1) gauge "eld. The S¸(2, Z) multiplet of 5-branes that are matched with those of the type-IIB theory on S is magnetically charged under the doublet of KK ;(1) "elds of M-theory on ¹. As for the D"9 branes associated with D 7-brane (magnetically charged under the 0-form potential), the M-theory interpretation is not well-understood yet. In [540], it is argued that p-branes with p"7, 8, 9 in M-theory that would give rise to 7-brane in D"9 do not exist, and 6-brane in D"9 cannot be obtained from D 7-brane of type-IIB theory by the periodic array along the compact direction and also is not consistent with the D"9 type-IIB theory. Finally, we comment on branes in the SO(32) theories, i.e. the type-I and the SO(32) heterotic strings. These 2 theories are related by S-duality, which inverts the string couplings (i.e. g"1/g') Q Q and exchanges the 2-form potentials of the 2 theories (the 2-form potential is in the NS-NS sector [the R-R sector] for the SO(32) heterotic theory [the type-I theory]). The electric [magnetic] charge of the 2-form potential is carried by 1-branes [5-branes]. When this 1-brane [5-brane] is compacti"ed on S, one has either 0-brane or 1-brane [4-brane or 5-brane] in D"9 depending on whether or not these branes are wrapped around S. The M-theory origin of these D"9 branes is understood from the observation that M-theory on S/Z ;S with length ¸ and radius R is related to SO(32) theory on S. Note, while M 2-brane can wrap on S/Z , M 5-brane can wrap around S, only. So, M 5-brane compacti"ed on S/Z ;S give rise to either 4-brane or 5-brane in D"9, depending on whether M 5-brane is wrapped around S or not. These branes are identi"ed with those of the SO(32) theory. Similarly, 0-brane and 1-brane of the SO(32) theories on S are identi"ed with the M 2-brane which is "rst wrapped around S/Z and then either wrapped around S or not. Note, under the exchange of parameters ¸ and R of S/Z ;S, the SO(32) heterotic theory and the type-I theory is exchanged, while the string couplings are inverted (i.e. g"1/g'"¸/R). Q Q Thus, the roles of R and ¸ are interchanged when one identi"es p-branes of the S-dual theory on S with those of the M-theory on S/Z ;S. These identi"cations yield the tensions for 1-brane and 5-brane of the SO(32) theories with consistent limiting behavior, i.e. ¹&(g)\ and Q ¹&1 for the heterotic theory, and ¹' &(g')\ and ¹' &(g')\ for the type-I theory. Q Q
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4. Black holes in heterotic string on tori 4.1. Solution generating procedure The primary goal of this section is to generate the general black hole solutions in the e!ective theories of the heterotic string on tori by applying the solution generating transformations described in Section 3.2. In principle, D(10 black hole solutions which have the most general electric/magnetic charge con"gurations and are compatible with the conjectured no-hair theorem [125,343,344,385,386] can be constructed by imposing the SO(1, 1) boosts on charge neutral solutions, i.e. Schwarzschield or Kerr solution. Here, the SO(1, 1) boosts that generate electric charges of ;\"(1) gauge group in D(10 are contained in the O(11!D, 27!D) symmetry group (134) of the (D!1)-dimensional Lagrangian. Meanwhile, magnetic charges of the ;\"(1) gauge group and the D"5 NS-NS 2-form potential are generated by the SO(1, 1) boosts in the O(8, 24) symmetry group (138) of the D"3 action. However, it is not necessary to generate all the electric/magnetic charges by applying the SO(1,1) boosts, as we explain in the following. The D-dimensional dualities, which leave the (Einsteinframe) metric intact, can be used to remove some of charge degrees of freedom (associated with the ;\"(1) gauge group in D(10) of black holes. The general black hole solution with all the redundant charge degrees of freedom removed by the D-dimensional dualities is called the `generating solutiona, since the most general solution in the class is obtained by applying the D-dimensional dualities. Thus, one only needs to generate electric [and magnetic] charges of the generating solutions by applying the SO(1, 1) boosts in O(11!D, 27!D) [O(8, 24)] duality group on the Schwarzschield or the Kerr solution [562]. The generating solution is equivalent to the solution with the most general charge con"guration due to the conjectured string dualities. This is a reminiscence of automorphism transformations of N-extended superalgebra discussed in Section 2.2, which brings the algebra in a simple form in which only [N/2] eigenvalues of central charge matrix appear in the algebra rather than whole N(N!1) central charges. The charge assignments for the generating solution for each dimensions are: E dyonic black holes in D"4 [181]: 5 charge degrees of freedom associated with gauge "elds in the ¹ part. E black holes in D"5 [182]: a magnetic charge of the NS-NS 3-form "eld strength (or an electric charge of its Hodge-dual), and 2 electric charges of KK and 2-form ;(1) gauge "elds associated with the same internal coordinate. E black holes in D56 [184]: 2 electric charges of KK and 2-form ;(1) gauge "elds associated with the same internal coordinate. For the purpose of constructing solutions, it is convenient to choose scalar asymptotic values in the `canonical formsa [562]: M "I and u "0 [and W "0 for the D"4 case]. This \" \" is not an arbitrary choice since one can bring arbitrary scalar asymptotic values to the canonical forms by applying the following O(10!D, 26!D, R) transformation: M PM K "XM X2"I , X3O(10!D, 26!D, R) , \" \"
(185)
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and the S-duality, for example for the D"4 case, given by the S¸(2, R) transformation S PSK "(aS #b)/d"i, ad"1 . (186) The D"3 O(8, 24, R) transformation that brings an asymptotic value of modulus M (137) to the form M "I is equivalent [561] to the D"4 S¸(2, R) transformation plus the D"3 O(7, 23, R) transformation. Furthermore, one brings asymptotic values of ;(1) gauge "elds AG to zero I by applying global ;(1) gauge transformations. Then, the subset of O(11!D, 27!D) [O(8, 24)] that preserves the canonical asymptotic value M "I [M "I ] is SO(26!D, 1);SO(10!D, 1) [SO(22, 2);SO(6, 2)] [562]. \" \" There are 36!2D[2;28]SO(1, 1) boosts in SO(26!D, 1);SO(10!D, 1) [SO(22, 2);SO(6, 2)]. When applied to a charge neutral solution, these boosts in SO(26!D, 1);SO(10!D, 1) [SO(22, 2) ;SO(6, 2)] induce electric charges of the ;(1)\" gauge group in D(10 [electric and magnetic charges of the ;(1) gauge group in D"4]. The starting point of constructing the generating solution is the D-dimensional Kerr solution, parameterized by the ADM mass and ["\] angular momenta. The solution in the `Boyer} Lindquista coordinate has the form [478]: (D!2N) D ds"! dt# "\ dr#(r#l cos h#K sin h) dh D (r#l)!2N G NG #(r#l cos t #K sin t ) cos h cos t 2 cos t dt G> G G> G G\ G !2(l !K )cos h sin h cos t 2 cos t cos t sin t dh dt G> G> G\ G G G !2 (l!K )cos h cos t 2 cos t H H G\ GH ;cos t sin t 2 cos t cos t sin t dt dt G G H\ H H G H 4l l kkN k 4l kN # G [(r#l)D#2lkN] d ! G G dt d # G H G H d d , G G G G G G H D D D GH
(187)
where for E Even dimensions: "\ "\ D,a (r#l)#r k(r#l)2(r#l ) G G G\ G G ;(r#l )2(r#l ), G> "\ K ,l sin t #2#l cos t 2 cos t sin t , G G> G "\ G "\ "\ N"mr ,
(188)
While the SO(1, 1) boosts in SO(22, 1);SO(6, 1)LSO(22, 2);SO(6, 2) induce electric charges of the D"4 ;(1) gauge group, the remaining SO(1, 1) boosts in SO(22, 2);SO(6, 2)!SO(26!D, 1);SO(10!D, 1) induce magnetic charges of the ;(1) gauge group.
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and k ,sin h, k ,cos h sin t ,2 , k ,cos h cos t 2 cos t sin t , "\ "\ "\ a,cos h cos t 2 cos t , "\
(189)
E Odd dimensions: "\ D,r k(r#l)2(r#l )(r#l )2(r#l ), G G\ G> "\ G K ,l sin t #2#l cos t 2cos t sin t G G> G "\ G "\ "\ #l cos t 2cos t , "\ G "\ N"mr ,
(190)
and k ,sin h, k ,cos h sin t ,2 , k ,cos h cos t 2 cos t sin t , (191) "\ "\ "\ k ,cos h cos t 2 cos t . "\ "\ Here, the repeated indices are summed over: i, j in t [ ] run from 1 to ["\] [from 1 to ["\]]. The ADM mass and the angular momenta J are G X 2 (D!2)X "\m, J " "\ml " Ml , (192) M" G 4pG G D!2 G 8pG " " where G is the D-dimensional Newton's constant and " 2p"\ X " "\ C("\) is the area of S"\. When compacti"ed to D!1 dimensions [3 dimensions (for the 4-dimensional Kerr solution)], the transformation that generates inequivalent solutions from the Kerr solution is (SO(26!D, 1);SO(10!D, 1))/(SO(26!D);SO(10!D)) [(SO(22, 2);SO(6, 22))/(SO(22);SO(6) ;SO(2))], which has (9!D)#(25!D) [2;28#1] parameters; these parameters are interpreted as (9!D)#(25!D) electric charge degrees of freedom [2;28 electric and magnetic charge degrees of freedom plus unphysical Taub-NUT charge] introduced to the Kerr solution [562]. For the D"5 case, an additional charge associated with the NS-NS 2-form "eld is generated by an SO(1, 1) boost in O(8, 24) [182]. After the generating solutions are constructed from the SO(1, 1) boosts, the remaining charge degrees of freedom are induced (without changing the Einstein-frame spacetime) from subsets of D-dimensional (continuous) duality transformations that generate new charge con"gurations from the generating ones while keeping the canonical scalar asymptotic values intact. This is
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[SO(10!D);SO(26!D)]/[SO(9!D);SO(25!D)], which introduces (9!D)#(25!D) new charge degrees of freedom, and, for the D"4 case, SO(1)LS¸(2, R), which introduces one more charge degree of freedom. Subsequently, to obtain solutions with arbitrary scalar asymptotic values, one has to undo the transformations (185) and (186). In the following sections, we discuss the generating solutions in each dimensions. Note, due to the conjectured string}string duality between heterotic string on ¹ and type-II string on K3, these also correspond to the generating solutions of general black holes in type-II strings on K3;¹L for n"6!D"0, 1, 2. For this case, some of charges of the generating solutions can be dualized to R-R charges, rendering interpretation in terms of D-branes. It turns out that by applying ;-dualities of type-II string on tori to such generating solutions, one can generate the general class of solutions of the e!ective type-II string on tori as well [171] (see Section 6.3.2 for discussions). 4.2. Static, spherically symmetric solutions in four dimensions 4.2.1. Supersymmetric solutions In this section, we derive a general BPS spherically symmetric solution with a diagonal moduli [178]. Such a solution, after subsets of O(6, 22) and S¸(2, R) transformations are applied, satis"es one ;(1) charge constraint, missing one parameter to describe the most general BPS solutions. The solution generalizes the previously known black hole solutions in heterotic string on tori as special cases, and are shown to be exact to all orders in expansions of a [173]. At particular points in moduli space, such a solution becomes massless, enhancing not only gauge symmetry but also supersymmetry [179]. 4.2.1.1. Generating solutions. A general BPS non-rotating black hole solution with a diagonal moduli matrix is obtained by solving the Killing spinor equations. With spherically symmetric AnsaK tze for "elds and a diagonal form of moduli M, the Killing spinor equations dt "0, dj"0 + and ds'"0 (cf. (119)) are satis"ed by restricted charge con"gurations (see [178] for details on allowed charge con"gurations), which we choose without loss of generality to be P, P, Q, Q. The explicit BPS non-rotating solution with such a charge con"guration has the form [178] j"r/[(r!g P)(r!g P)(r!g Q)(r!g Q)] , . . / / R"[(r!g P)(r!g P)(r!g Q)(r!g Q)] , . . / / (r!g P)(r!g P) . . , eP" (r!g Q)(r!g Q) / /
r!g P . , g " r!g P .
r!g Q / , g " r!g Q /
(193)
g "1 (mO1, 2) , KK
where j and R are components of the metric g dxI dxJ"!j dt#j\ dr# IJ R(dh#sin h d ), g "$1 and the radial coordinate is chosen so that the horizon is at r"0. . / The requirement that the ADM mass saturates the Bogomol'nyi bound restricts choice of g . / such that g sign(P#P)"!1 and g sign(Q#Q)"!1, thus yielding non-negative . /
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ADM mass of the form ""P#P"#"Q#Q" . (194) .1 Note, the relative signs for the pairs (Q, Q) and (P, P) are not restricted but in this section we consider the case where both of pairs have the same relative signs so that the solution (193) has regular horizon. The Killing spinor e of the above BPS solution satis"es the following constraints: M
E PO0 and/or PO0: CK CK ?e"ig e, . E QO0 and/or QO0: CK CK ?e"g e. / From these constraints, one sees that purely electric (or magnetic) solutions preserve 1/2, while dyonic solutions preserve 1/4 of N"4 supersymmetry. The former and the latter con"gurations fall into vector- and hyper-supermultiplets, respectively. Since the Killing spinor equations are invariant under the O(6, 22) and S¸(2, R) transformations, one can generate new BPS solutions by applying the O(6, 22) and S¸(2, R) transformations to a known BPS solution. The [SO(6)/SO(4)];[SO(22)/SO(20)] transformation with 6 ) 5!4 ) 3 22 ) 21!20 ) 19 # "50 2 2 parameters to the solution (193) leads to a general solution with zero axion and 4#50" 54"56!2 charges. 28 electric Q and 28 magnetic P charges of such a solution satisfy the two constraints P2M Q"0 (M ,(¸M¸) $¸) . (195) ! ! The subsequent SO(2)LS¸(2, R) transformation introduces one more parameter (along with a non-trivial axion), which replaces the two constraints (195) with the following one S¸(2, R) and O(6, 22) invariant constraint on charges: P2M Q [Q2M Q!P2M P]!(#!)"0 . (196) \ > > Thus, general solution in this class has 4#50#1"55"2 ) 28!1 charge degrees of freedom. By applying the O(6, 22) and S¸(2, R) transformations to (194), one obtains the following ADM mass for general solutions preserving 1/4 of supersymmetry: M "e\P+P2M P#Q2M Q#2[(P2M P)(Q2M Q)!(P2M Q)] , . (197) .1 > > > > > This agrees with the expression (213) obtained [236] by the Nester's procedure. When magnetic P and electric Q charges are parallel in the SO(6, 22) sense, i.e. P2M Q"0, (197) becomes the > ADM mass of con"gurations preserving 1/2 of N"4 supersymmetry [34,299,337, 420}422,495,560,564]: M "e\((P2M P#Q2M Q) , > > .1 whose corresponding generating solution is purely electric or magnetic subset of (193).
(198)
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4.2.1.2. General supersymmetric solution with xve charges. The BPS solution (193) has a charge con"guration satisfying one constraint (196) when further acted on by the [SO(6)/SO(4)]; [SO(22)/SO(20)] and SO(2) transformations. So, to construct the generating solution for the most general BPS solution, which conforms to the conjectured classical `no-haira theorem, one has to introduce one more charge degree of freedom into (193). Such a generating solution was constructed in [174] by using the chiral null model approach, and has the following charge con"guration: (Q, P)"(q, P ), (Q, P)"(Q , 0) , (Q, P)"(!q, P ), (Q, P)"(Q , 0) . Explicitly, the solution has the form r j" , [(r#Q )(r#Q )(r#P )(r#P )!q[r#(P #P )]] (r#P )(r#P ) , eP" [(r#Q )(r#Q )(r#P )(r#P )!q[r#(P #P )]] q(P !P ) W" , 2(r#P )(r#P ) r#P r#Q q[r#(P #P )] , . , G " G " G "!B " r#P r#Q (r#Q )(r#P ) For this solution to have a regular horizon, the charges have to satisfy the constraints P '0, P '0, Q '0, Q '0 , Q Q !q'0, (Q Q !q)P P !q(P !P )'0 . The ADM mass has the same form as that of the 4-parameter solution (193):
(199)
(200)
(201)
M
"Q #Q #P #P , (202) "+ independent of the additional parameter q. Meanwhile, the horizon area, i.e. A,4p(j\r) , is P modi"ed in the following way due to q: A"4p[(Q Q !q)P P !q(P !P )] . (203) The following ADM mass and horizon area of BPS non-rotating black hole with general charge con"guration are obtained by applying the [SO(6)/SO(4)];[SO(22)/SO(20)] and SO(2)LS¸(2, R) transformations to (202) and (203): M "a 2k a #e\Pb2k b#e\P[(b2k b)(a2k a)!(b2k a)] , "+ > > > > > A"peP[(b2¸b)(a2¸a)!(b2¸a)] ,
(204)
where the charge lattice vectors a and b live on the even self-dual Lorentzian lattice K with signature (6, 22), a ,a#W b and k ,M $¸. Here, a and b are related to the physical ;(1) !
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charges Q and P as: (2P "¸ b , (205) (2Q "ePM (a #W b ), G GH H H G GH H where we assumed that a "2, G "a eP"eP. , The ADM mass and horizon area in (204) can be put in the S¸(2, Z) S-duality, as well as the O(6,22) T-duality, invariant forms by expressing them with new charge lattice vector *2"(*2, *2),(a2, b2) and by introducing the following S¸(2, R) invariant matrices:
M"eP
W
1
W W#e\P
, L"
0
1
!1 0
.
(206)
The "nal forms are M "(8G )\(M (*?2k *@)#[2L L (*?2k *@)(*A2k *B)] ) , "+ , ?@ > ?A @B > > (207) 1 A . "p L L (*?2¸*@)(*A2¸*B) S" 2 ?A @B 4G , These are manifestly S¸(2, R) invariant, since M and * transform under SL(2, R) as [560]
MPuMu2,
*PLuL2*,
u3S¸(2, R) .
(208)
An important observation is that while for "xed values of a and b, mass changes under the variation of moduli and string coupling, entropy remains the same as one moves in the moduli and coupling space [258}260]. The fact that entropy is independent of coupling constants and moduli is consistent with the expectation that degeneracy of BPS states is a topological quantity which is independent of vacuum scalar expectation values and the fact that entropy measures the number of generate microscopic states, which should be independent of continuous parameters. 4.2.1.3. Bogomol'nyi bound. We derive the Bogomol'nyi bound on the ADM mass of asymptotically #at con"gurations within the e!ective theory of heterotic string on ¹ [236]. For this purpose, we introduce the Nester-like 2-form [483]: EK ,e cIJMd tI , (209) IJ C M where d tI is the supersymmetry transformation of physical gravitino in D"4. Given supersymC I metry transformations (119) of fermionic "elds expressed in terms of D"4 "elds [178], the Nester's 2-form reduces to the form: 1 EK IJ"e IJMd e# e\P e (< ¸(F!icF I )IJ)?C?e#2 , M 0 2(2
(210)
where < is a vielbein de"ned in (122) and ¸ is an invariant metric of O(6, 22) given in (127). Derivation of the Bogomol'nyi bound consists of evaluating the surface integral of the Nester's 2-form (210), which is related through the Stokes theorem to the volume integral of its covariant derivative. The surface integral yields
1 1 dS eEK IJ"e P "+cI# e\P +< ¸(Q!icP),?C? e , IJ I 0 4pG R 2(2G .
(211)
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where P "+ is the ADM 4-momentum [20] of the con"guration, and Q and P are physical electric I and magnetic charges of ;(1) gauge group. The integrand of the volume integral is a positive semide"nite operator, provided spinors e satisfy the (modi"ed) Witten's condition [631] n ) K e"0 (n is the 4-vector normal to a space-like hypersurface R). Thus, the bilinear form on the right-hand side of (211) is positive semide"nite, which requires that the ADM mass M has to be greater than or equal to the largest of the following 2 eigenvalues of central charge matrix: 1 e\P[Q #P $2+Q P !(Q2 P ), ] , "Z "" 0 0 0 0 0 0 (4G )
(212)
where Q ,(2(< ¸Q) and similarly for P . This yields the Bogomol'nyi bound: 0 0 0 1 M 5 e\P[Q #P #2+Q P !(Q2 P ), ] . "+ (4G ) 0 0 0 0 0 0
(213)
This bound is saturated i! supersymmetric variations of fermionic "elds are zero, i.e. BPS con"gurations. The Bogomol'nyi bound (213) can be expressed explicitly in terms of electric Q and magnetic P charges, and asymptotic values M and u of scalars, by using the identity: ¸<2 < ¸"[¸(M#¸)¸]. For example, Q "Q2¸(M #¸)¸Q. 0 0 0 4.2.2. Singular black holes and enhancement of symmetry In perturbative heterotic string theories, gauge symmetry is enhanced to non-Abelian ones through the Halpern}Frenkel}Kac\ (HFK) mechanism [127,253,273,322}324,480]. The HFK mechanism is due to extra spin one string states which are normally massive at generic points in moduli space but become massless at particular points. These points are the "xed points under discrete subgroups of T-duality of the worldsheet theory. As shown in [382], BPS states in N"4 theories become massless at particular points in the moduli space. Since BPS multiplets in N"4 theories generically contain massive spin one states, gauge symmetry is enhanced to non-Abelian ones when the BPS states become massless. Note, the BPS states carry magnetic, as well as electric, charges and, therefore, are non-perturbative in character. When BPS multiplets with highest spin 3/2 state become massless, supersymmetry as well as gauge symmetry is enhanced, a phenomenon that is never observed within perturbative string theories. In this section, we illustrate the enhancement of symmetries in the BPS states of N"4 theories by studying massless black holes in heterotic string on ¹ [119,141,179]. 4.2.2.1. Massless black holes and symmetry enhancement. First, we consider the subset of BPS states with diagonal M and purely imaginary S [179]. We rewrite the corresponding generating solution (193) with explicit dependence on scalar asymptotic values j"r/[(r!g P )(r!g P )(r!g Q )(r!g Q )] , . . / / R"[(r!g P )(r!g P )(r!g Q )(r!g Q )] , . . / /
(214)
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where the solution now depends on the following `screeneda charges: (P , P , Q , Q ),e\P (g P, g\ P, g\ Q, g Q) (215) "(e\P g b, e\P g\ b, eP g a, eP g\ a) . Here, the quantized charge lattice vectors a and b live on the even, self-dual, Lorentzian lattice K [557]. When g are chosen to satisfy g sign(P#P)"!1 and g sign(Q#Q)"!1, the . / . / ADM mass takes the following form that saturates the BPS bound: "e\P "g b#g\ b"#eP "g a#g\ a" . (216) .1 Only electric states (b"0) with a2¸a52 are matched onto perturbative string states [248]. As for dyonic states that break 1/4 of supersymmetry (i.e. those with non-parallel electric and magnetic charge vectors, i.e. Q2M PO0, and therefore cannot be related to electric solution via the S¸(2, Z) > transformations), the consistency with the S¸(2, Z) symmetry and consistent electric limit require that the electric and the magnetic lattice vectors separately satisfy the constraints a2¸a5!2 and b2¸b5!2. These subsets of BPS states (214) become massless [48,179,406] at the "xed points under T-duality RP1/R (i.e. at the ¹ self-dual point g "1 [g "1"g ]), when a"!a"$1 [a"!a"$1 and b"!b"$1] for the case b"0 [aO0Ob]. There are also additional in"nite number of S¸(2, Z) related massless BPS states. The extra massless spin 1 states associated with a"!a"$1 [a"!a"$1 and b"!b"$1] at the self-dual points of ¹ contribute to enhancement of Abelian gauge symmetry to S;(2)[S;(2);S;(2)] non-Abelian symmetry. The extra massless spin 1 states together with generic massless ;(1) gauge "elds form the adjoint representations of the enhanced non-Abelian gauge groups. The BPS multiplet which preserves 1/4 of N"4 supersymmetry contains spin 3/2 state. Thus, additional 4 massless gravitino associated with a"!a"$1 and b"!b"$1 contribute to enhancement of supersymmetry from N"4 to N"8. Note, the in"nite number of S¸(2, Z) related massless states and enhancement of supersymmetry are not realized within perturbative string theories. These are new non-perturbative phase of string theories that are required by non-perturbative string dualities. M
4.2.2.2. Maximal gauge symmetry enhancement in the moduli space of two-torus. In this subsection, we study maximal symmetry enhancement in full moduli space of ¹ parameterized by arbitrary scalar asymptotic values [119,141]. We consider the general BPS mass formula (197). The moduli space of ¹ is parameterized by the following real matrix:
M"
G\
!G\B
!B2G\ G#B2G\B
,
(217)
where G,[G ] and B,[B ] with (m, n)"1, 2 being indices associated with ¹. Thus, the KL KL e!ective theory has the O(2, 2, R) T-duality symmetry. Since O(2, 2, R) S¸(2, R);S¸(2, R) [215,570], M is reparameterized [215] by the following 2 complex scalars, which separately
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parameterize each S¸(2, R) moduli space: (G G q"q #iq , !i , o"o #io ,(G#iB , (218) G G where G,G G !G . Here, q [o] is the complex [KaK hler] structure of ¹. Each S¸(2, Z) factor [215,305,570] is generated by two transformations, which are given, for the S¸(2, Z) factor associated with q, by 1 S : qP , O q
¹ : qPq#i , O
(219)
where o remains intact, and similarly for the S¸(2, Z) factor associated with o. In addition, the p-model corresponding to ¹ is symmetric under the `mirror symmetrya (S : qo), the world sheet parity symmetry (q, o)P(q,!o) corresponding to pP!p, and the symmetry (q, o)P (!q,!o) associated with the re#ection XP!X. In accordance with the above reparameterization of moduli space, one can express the central charges (212) of the N"4 theory in the following S¸(2, R) ;S¸(2, R) ;ZO@M invariant form [119] O M (the subscript R is omitted in ADM mass and central charges) 1 "M " , "Z "" 4(S#SM )(q#q)(o#o)
(220) M ,(a( #iSbK )PK H, M ,(a( !iSM bK )PK H , H H H H where a( ,(a, a, a,!a)2, bK ,(b, b, b,!b)2, and PK ,(1,!qo, iq, io)2. Here, the charge lattice vectors a( and bK transform under S¸(2, R) ;S¸(2, R) ;ZO@M as O M a( a( a( bK bK Pu , similary for , , , S¸(2, R) : O O a( a( a( bK bK a( a( a( bK bK Pu , similary for , , , (221) S¸(2, R) : M M a( a( a( bK bK ZO@M: a( a( , bK bK . In addition, the central charges (220) are invariant under the S¸(2, R) S-duality: 1 a( a c a( P . (222) S¸(2, R) : 1 bK b d bK
Note, the ADM mass of BPS states is given by the largest of "Z " and "Z ". First, we consider the short multiplet, i.e. the BPS multiplet with 2 central charges Z equal in magnitude. It has the highest spin 1 state and preserves 1/2 of the N"4 supersymmetry. The charge lattice vectors a and b of the short multiplet live on the S-orbit satisfying pa( "sbK (s, p3Z). G G Explicitly, we write a( and bK in the S-orbit as a( "sm , bK "pm ,
a( "sn , bK "pn ,
a( "sn , bK "pn ,
a( "!sm , bK "!pm ,
(223)
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where m [n ] corresponds to momentum [winding] number of perturbative string states when G G p"0 and s [p] denotes electric [magnetic] quantum number associated with the S-modulus. Then, the central charge (220) becomes M"(s#ipS)(m !im q#in o!n qo) . (224) In the complex moduli space parameterized by (q, o), M can vanish at "xed lines .1 [121,216,384] under the Weyl re#ections w ,S , w ,S S S , w ,¹ S ¹\ and w ,
M M M M w\= w in the T-duality group. Along these lines, the Abelian gauge group ;(1) ;;(1) ; ? @ ;(1) ;;(1) ,;(1);;(1);;(1);;(1) is enhanced to the non-Abelian ;(1);S;(2) A B group. The BPS states (labeled by a) which become massless along the "xed lines L under w are as G G follows [121,122]: E L "+q"o,: a"k "$(1, 0,!1, 0) contributing to ;(1) ;;(1) ;;(1) ;S;(2) ! @ B ?>A ?\A E L "+q"o\,: a"k "$(0, 1, 0,!1) contributing to ;(1) ;;(1) ;;(1) ;S;(2) ! ? A @>B @\B E L "+q"o!i,: a"k "$(1, 1,!1, 0) contributing to ;(1) ;;(1) ;;(1) ; ! B ?>A ?\@\A S;(2) ?>@\A E L "+q"o/(io#1),: a"k "$(1, 0,!1, 1) contributing to ;(1) ;;(1) ;;(1) ! @ ?>A ?\A\B ;S;(2) . ?\A>B At points where the lines L intersect [120,122], there are additional massless states, resulting in G the maximal enhancements of gauge symmetries: E L 5L "+q"o"1,: a"k or k contributing to ;(1) ;;(1) ;S;(2) ;S;(2) ! ! ?>A @>B ?\A @\B E L 5L 5L "+q"o"e p ,: a"k or k or k contributing to ;(1) ;;(1) ; ! ! ! ?>A ?\@\A\B S;(3) . @\B ?>@\A>B Along with the above perturbative massless states, there are accompanying in"nite massless dyonic states, so-called S-orbit pa( "sbK , related via S¸(2, Z) S duality. G G 1 Second, we consider the intermediate multiplets, i.e. the BPS multiplets with "Z "O"Z ". They have the highest spin 3/2 states and preserve 1/4 of the N"4 supersymmetry. In this case, a( and bK are not proportional, i.e. a( bK !a( bK O0. The requirement that the ADM mass is zero, i.e. G H H G "Z ""0 and *Z""Z "!"Z ""0, leads to the relations [119]: a!aqo#iao!ia"0, b!bqo#ibo!ib"0 . (225) These relations are satis"ed by the following "xed points [119,141]: E q"o"i: (a, b)"(k , k ), ! ! E q"o"e p : (a, b)"(k ,k ), 24i(j44. ! H! In addition to the above massless dyonic states, there are in"nite number of S¸(2, Z) related 1 dyonic states. Since these additional massless states belong to the highest spin supermultiplet, supersymmetry as well as gauge symmetry are enhanced [179]. 4.2.2.3. Properties of massless black holes. When both of the pairs (Q, Q) and (P, P) have the same relative signs [178], the singularity of the solution (214) is always behind or located at the event horizon at r"0, corresponding to the Reissner-NordstroK m-type horizon or null singularity,
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respectively. However, the moment at least one of the pairs has the opposite relative signs [48,179,406], there is a singularity outside of the event horizon, i.e. naked singularity r '0. Explicitly, the curvature singularity is at r"r ,max+min["P ", "P "], min["Q ", "Q "],'0. These singular solutions have an unusual property of repelling massive test particles [406]. (Note, the BPS black holes need not be massless to be able to repel massive test particles [179].) There is a stable gravitational equilibrium point for a test particle at r"r where the graviA tational force is attractive for r'r and repulsive for r(r [406]. This can also be seen by A A calculating the traversal time of the geodesic motion for a test particle with energy E, mass m and zero angular momentum along the radial coordinate r, as measured by an asymptotic observer [179]:
t(r)"
P
E dr
P j(r)(E!mj(r)
.
(226)
The minimum radius that can be reached by a test particle corresponds to r 'r for which
j(r"r )"E/m, since it takes in"nite amount of time to go beyond r"r . Here, r"r is
the singularity. Massive test particles cannot reach the singularity of singular black holes in "nite time and are re#ected back. On the other hand, classical massless particles with zero angular momentum do not feel the repulsive gravitational potential due to increasing j(r), and they reach the singularity in a "nite time. Note, for regular solutions, studied in Section 4.2.1, j41, while for singular solutions studied in this section, j51 for r small enough. Thus, for regular solutions, particles are always attracted toward the singularity. When only one charge is non-zero, the regular solution has a naked singularity at r"0; t(r"r "0) is "nite. 4.2.3. Non-extreme solutions The following non-extreme generalization [180] of the BPS solution (193) is obtained by solving the Einstein and Euler}Lagrange equations: j"r(r#2b)/[(r#P )(r#P )(r#Q )(r#Q )] , R(r)"[(r#P )(r#P )(r#Q )(r#Q )] ,
eP"
(r#P )(r#P ) , (r#Q )(r#Q )
r#P , g " r#P
W"0 ,
r#Q , g " r#Q
g "B "0 (mOn), KL KL
(227)
g "1 (mO1, 2) , KK
a' "0 , K
where b'0 measures deviation from the corresponding BPS solution and P , b$((P)#b, etc. The ADM mass is M"P #P #Q #Q !4b .
(228)
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The signs $ in the expressions for P , etc. should be chosen so that MPM as bP0. To .1 have a regular horizon, one has to choose the relative signs of both pairs (Q, Q) and (P, P) to be the same [178]. In this case, the non-extreme solution is (227) with positive signs in the expressions for P , etc. and have the ADM mass M"((P)#b#((P)#b#((Q)#b#((Q)#b , which is always compatible with the Bogomol'nyi bound:
(229)
""P"#"P"#"Q"#"Q" . .1 Such solutions always have nonzero mass.
(230)
M
4.2.3.1. Space}time structure and thermal properties. We now study spacetime properties [178,180] of the regular D"4 4-charged black hole discussed in the previous subsections. There is a spacetime singularity, i.e. the Ricci scalar R blows up, at the point r"r where R"0. The event horizon, de"ned as a location where the r"constant surface is null, is at r"r & where gPP"j"0. The horizon(s) forms, provided r 'r (time-like singularity). In some cases, & singularity and the event horizon coincide: r "r . In this case, the singularity is (i) naked & (space-like singularity) when the singularity is reachable by an outside observer (at r"r 'r ) in & a "nite a$ne time q"P dr(EPP"P& dr j\(r) and (ii) also an event horizon (null singularity) ERR P P when q"R. Thermal properties of the solution (193) are speci"ed by spacetime at the event horizon. The Hawking temperature [345,346] is de"ned by the surface gravity i at the event horizon: ¹ "i/(2p)""R j(r"r )"/(4p) . & P & Entropy S is given by the Bekenstein's formula [38,64,65,343,347]:
(231)
S";(the surface area of the event horizon)"pR(r"r ) . (232) & We classify thermal and spacetime properties according to the number of non-zero charges: E All the 4 charges non-zero: There are 2 horizons at r"0,!2b and a time-like singularity is hidden behind the inner horizon, i.e. the global spacetime is that of the non-extreme Reissner} NordstroK m black hole. The Hawking temperature is ¹ "b/(p(P P Q Q ) and the & entropy is S"p(P P Q . When bP0, spacetime is that of extreme Reissner} ,MPBQRPMK m black holes. E 3 non-zero charges: A space-like singularity is located at the inner horizon (r"!2b). For example when P"0, ¹ "b /(p(2P Q Q ) and S"p(2bP Q Q . When bP0, & the singularity coincides with the horizon at r"0. E 2 non-zero charges: A space-like singularity is at r"!2b. For example when PO0OP, ¹ "1/(2p(P P ) and S"p(4bP P . As bP0, the singularity co & incides with the horizon at r"0. E 1 non-zero charge: A space-like singularity is at r"!2b. For example when PO0, ¹ "1/(2p(2bP ) and S"p(8bP . As bP0, the singularity becomes naked. &
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4.2.4. General static spherically symmetric black holes in heterotic string on a six-torus In Section 4.2.1, we saw that a general solution obtained by applying subsets of O(6,22) and S¸(2,R) transformations on the 4-charged solution has 1 charge degree of freedom missing for describing non-rotating black holes with the most general charge con"guration. It is a purpose of this section to introduce such 1 missing charge degree of freedom by applying 2 SO(1,1) boosts (in the D"3 O(8,24) duality group) along a ¹ direction (associated with 4 non-zero charges) with a zero-Taub-NUT constraint to construct the `generating solutiona for non-extreme, nonrotating black holes in heterotic string on ¹ with the most general charge con"guration of ;(1) gauge group [181]. (See [390] for an another attempt.) So, the generating solution is parameterized by the non-extremality parameter (or the ADM mass) and 6 ;(1) charges with one zero-TaubNUT constraint. 4.2.4.1. Explicit form of the generating solution. For the purpose of constructing the generating solution in a simplest possible form, it is convenient to "rst generate the 4-charged non-extreme solution (227) with the following non-zero charges by applying 4 SO(1, 1) boosts to the Schwarzschield solution: P"2m sinh d cosh d ,P , P"2m sinh d cosh d ,P , N N N N Q"2m cosh d sinh d ,Q , Q"2m cosh d sinh d ,Q . O O O O
(233)
Only non-extreme solutions compatible with the Bogomol'nyi bound and, therefore, within spectrum of states, are those with the same relative signs for both pairs (Q , Q ) and (P , P ). For this case, PK ,2m cosh d !m"$((P )#m, etc. are given with plus signs. N As the next step, one introduces one missing charge degree of freedom by applying 2 SO(1, 1)LO(8, 24) boosts with parameters d satisfying the zero Taub-NUT condition: P tanh d !Q tan d "0 .
(234)
Assuming, without loss of generality, that Q 5P , one has, from (234), d in terms of the other parameters: cosh d "Q cosh d /D,
sinh d "P sinh d /D ,
(235)
where D,sign(Q )((Q ) cosh d !(P ) sinh d .
An additional SO(1, 1) boost along a ¹ direction on the 4-charged black hole solution necessarily induces Taub-NUT term, since the metric components g get mixed with the -component of the ;(1) gauge potential, which is IJ singular [562]. One can induce any 2 of the remaining charges in the ;(1);;(1);;(1);;(1) gauge group. But we here choose to induce P and Q. For the case Q 4P , the role of d and d are interchanged.
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The "nal form of the generating solution [181] (with zero Taub-NUT charge) is = (r#m)(r!m) , R"(X>!Z) , eP" , j" X>!Z (X>!Z) 1 R W" [D(P Q #P Q )#P Q [(P )(r#QK )!(Q )(r#PK )] P D=
;
with
P P (r!QK ) sinh d #Q Q (r#PK ) cosh d sinh d cosh d , X>!Z
X > G " , G " , (r#PK )(r#QK ) (r#PK )(r#QK ) Z G "! , (r#PK )(r#QK ) [(Q )(r#PK )!(P )(r#QK )] cosh d sinh d , B "! D(r#PK )(r#QK ) G "d , B "0 (i, jO1, 2), a' "0 GH GH GH K
(236)
X"r#[(PK #QK ) cosh d #(QK !PK ) sinh d ]r#(PK QK sinh d #QK PK cosh d ) , 1 >"r# [(P )(PK !QK ) sinh d #(Q )(PK #QK ) cosh d ]r D 1 # [(P )QK PK sinh d #(Q )QK PK cosh d ] , D 1 Z" [(P P #Q Q )r#(PK Q Q #QK P P )] cosh d sinh d , D
(237)
1 ="r# [(Q )(PK #PK ) cosh d #(P )(QK !QK ) sinh d ]r D 1 # [(Q )PK PK cosh d #(P )QK QK sinh d ] . D For the sake of simpli"cation, the coordinate is chosen so that the outer horizon is at r"m. This solution has the following non-zero charges: P"P Q /D, Q"(PK !PK !QK !QK ) cosh d sinh d , P"0, Q"(Q Q cosh d #P P sinh d )/D , P"(Q P cosh d #Q P sinh d )/D, Q"0 , P"P Q (Q !Q !P !P ) sinh d cosh d /D, Q"D ,
(238)
The BPS limit (m"0 and d PR) of this solution is related to the solution (200) via subsets of SO(2);SO(2)LO(2, 2) (¹ ¹-duality) and SO(2)LS¸(2, R) transformations.
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and the ADM mass, compatible with the BPS bound [178,236], is M
1 " [(P )(PK !QK ) sinh d #(Q )(PK #QK ) cosh d ] "+ D #(PK #QK ) cosh d #(QK !PK ) sinh d .
(239)
4.2.4.2. S- and T-duality transformations. The additional 51 charge degrees of freedom needed for parameterizing the most general ;(1) charge con"guration are introduced by [O(6);O(22)]/ [O(4);O(20)] and SO(2) transformations. The resulting general solution has the charge con"guration ; (e !e ) ; (m !m ) 1 S B S B , P "( U2 , (240) Q " U2 e #e m #m S B S B (2 ; ; 0 0
CDE e2,(Q cos c#P sin c, Q cos c#P sin c, 0, 2, 0) , S
where
CDE e2,(P sin c, Q cos c, 0, 2, 0) , B
(241)
CDE m2,(P cos c, Q sin c, 0, 2, 0) , S CDE m2,(P cos c#Q sin c, P cos c#Q sin c, 0, 2, 0) , B
c is the SO(2)LS¸(2, R) rotational angle, ; 3SO(6), ; 3SO(22), 0 is a (16;1)-matrix with zero entries and U3O(6, 22, R) brings to the basis where the O(6, 22) invariant metric (127) is diagonal. And the complex scalar S and the moduli M transform to (W cos c!sin c)#ie\P cos c S " , (W sin c#cos c)#ie\P sin c
(242) ;2 0 UMU2 U, ; 0 ;2 where W, e\P and M are the axion, the dilaton and the moduli of the generating solution (236). The `Einstein-framea metric g in (236) remains unchanged, but the `string-framea metric g is IJ IJ transformed to the most general form g "g /Im (S). IJ IJ
; M "U2 0
0
4.2.4.3. Special cases of the general solution. The generating solution (236), when supplemented by appropriate subsets of S- and T-dualities, reproduces all the previously known spherically symmetric black holes in heterotic string on ¹. Here, we give some examples.
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E Non-rotating black holes in Einstein}Maxwell-dilaton system with the gauge kinetic term e\?PF FIJ [245,281,298,354]: IJ (1) P "P "Q "Q O0 case: the Reissner}NordstroK m black hole, i.e. a"0 (2) any of 3 charges non-zero and equal: the a"1/(3 case [236] (3) only 2 magnetic (or electric) charges non-zero and equal: the a"1 case [409] (4) only 1 charge non-zero: the a"(3 case, which contains in the extreme limit the followings (i) P O0 case: KK monopole [506,299,325,574], and (ii) P O0 case: H-monopole [57,286,420}422]. E P "P and Q "Q solution with subsets of S- and T-dualities applied becomes general axion}dilaton black holes found in [83,404,410,487,486]. E The solution with Q O0OQ , when supplemented by S- and T-dualities, is the general electric black hole in heterotic string [562,564]. The S-dual counterpart is the general magnetic solution [58]. E The non-BPS extreme solution (i.e. mP0, P "Q "0, "Q "!"P "P0 and d PR, while keeping me B and ("Q "!"P ")e B as "nite constants) is related by S- and T-dualities to the non-BPS extreme KK black hole studied in [301]. 4.2.4.4. Global space-time structure and thermal properties. We classify all the possible spacetime and thermal properties of non-rotating black holes in heterotic string on ¹. These properties are determined by the 6 parameters P , Q , d and m of the generating solution (236), since the D"4 T- and S-dualities, which introduce the remaining charge degrees of freedom, do not a!ect the `Einstein-framea spacetime. We separate the solutions into non-extreme (m'0) and extreme (m"0) ones. Within each class, we analyze their properties according to the range of the other 5 parameters P , Q and d . 4.2.4.4.1. Global space-time structure. There is a spacetime singularity at r"r where R(r)"0. The event horizon(s) is located at r"r where j"0, provided r 5r . ! ! (A) Non-extreme solutions (m'0). By analyzing roots of X>!Z, one sees that a singularity is always at r 4!m. Thus, global spacetime is either that of non-extreme Reissner}NordstroK m black hole when r (!m (case with 2 horizons at r"$m) or that of Schwarzschield black hole when r "!m. X>!Z has a single root at r "!m, in which case a singularity and the inner horizon coincide at r"!m, when (a) d O0 and P "0, or (b) d "0 and at least one of P and Q is zero. X>!Z has a double root at r "!m, in which case the inner horizon disappears and a singularity forms at r"!m, when (a) d O0 and only Q is non-zero, or (b) d "0 and at least 2 of P , Q are zero.
In the following analysis, it is understood that Q O0 when d O0. When Q "0, P "0 due to initial assumption "Q "5"P ". Then, d are not constrained by (234). In this case, we have a non-extreme 4-charged solution with charges P, P, Q and Q. Such a solution is related to (236) through subsets of SO(2);SO(2)LO(2, 2)LO(6, 22) and SO(2)LS¸(2, R) transformations.
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(B) Extreme solutions (mP0). When d is "nite, the ADM mass of the generating solution always saturates the Bogomol'nyi bound as mP0, i.e. becomes BPS extreme solution. When both pairs (P , P ) and (Q , Q ) have the same relative signs, the singularity is always at r 40. Global spacetime is, therefore, that of the extreme Reissner}NordstroK m black hole when r (0 (time-like singularity), or the singularity and the horizon coincide (null singularity) when r "r "0. The latter case happens when at least one out of P , Q (and Q ) is zero with ! d O0 (with d "0). The horizon at r "0 disappears (naked-singularity) when (i) only Q is non-zero with d O0, or (ii) only one out of P , Q is non-zero with d "0. When at least one of the pairs (P , P ) and (Q , Q ) has the opposite relative sign, the singularity is outside of the horizon, i.e. r '0 (singularity is naked) [48,179,400,406]. In the case of non-BPS extreme solutions [177,181], the singularity is always behind the event horizon (r (r "0), i.e. the global spacetime of the extreme Reissner}NordstroK m black hole (time-like singularity). 4.2.4.4.2. Thermal properties. Thermal properties are speci"ed by spacetime at the (outer) horizon. So, we consider only regular solutions, which include non-extreme solutions compatible with the Bogomol'nyi bound and extreme solutions with the same relative sign for both pairs (P , P ) and (Q , Q ). The entropy S is of the form p S" "[(QK #m)(PK #m) cosh d #(PK #m)(QK !m) sinh d ] "D" ;[(Q )(QK #m)(PK #m) cosh d #(P )(QK #m)(PK !m) sinh d ] ![P P (QK #m)#Q Q (PK #m)] cosh d sinh d " ,
(243)
where PK "#(P#m, etc. Entropy increases with d , approaching in"nity ["nite value] as d PR [non-BPS extreme limit is reached]. For BPS extreme solutions, entropy is (a) non-zero and "nite, approaching in"nity as d PR, when P and Q are non-zero, and (b) always zero when at least one of P , Q (and Q ) is zero with d O0 (with d "0). The Hawking temperature ¹ ""R j(r"m)"/4p is & P m S\ . ¹ " & (2
(244)
As d increases, ¹ decreases, approaching zero. In the BPS extreme limit with at least 3 of & P ,Q non-zero, ¹ is always zero. With 2 of them non-zero, ¹ is non-zero and "nite, & & approaching zero as d PR. When only one of them (only Q ) is non-zero (for the case d O0), ¹ becomes in"nite. In the non-BPS extreme limit, ¹ is zero. & & 4.2.4.5. Duality invariant entropy. We discuss the duality invariant form of entropy of nearextreme, non-rotating black hole in heterotic string on ¹ [170].
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The (t,t)-component of metric for general N"4 spherically symmetric solutions has the form g "p(r#m)(r!m)S\(r) with S(r) given by RR S(r)"p ((r#j ) . G G
(245)
Entropy S of non-extreme solutions is given by S(r) at the outer horizon, i.e. S,S(m). Generally, j are functions of 28#28 electric and magnetic charges (240) and m (through m), G and their duality invariant forms are hard to obtain. However, for the near-extreme case, in which j are expressed to leading order in m around their BPS values j, one can obtain the T- and G G S-duality invariant entropy expression, which reads p S"p (j # m (1/j j j j #O(m) . G G H I G 2 G GHI G
(246)
Here, the T- and S-duality invariants are j ,S /p"F(¸, C)F(¸,!C) , .1 G G j ,M "(F(M , C)#(F(M ,!C) , G .1 > > G 1 j j " (Q2¸Q#P2¸P)#(F(M ,C)F(M ,!C) , > > G H 2g Q GH 1 j j j " +M (Q2¸Q#P2¸P)!(Q2M Q!P2M P) .1 > > G H I 4gM Q .1 GHI ;(Q2¸Q!P2¸P)!4(Q2¸M ¸P)(Q2M P), , >
(247)
where 1 F(M ,$C)" (Q2M Q#P2M P$C(M )) > > > > 2g Q C(M )"(4(P2M Q)#(Q2M Q!P2M P) . > > > >
(248)
4.3. Rotating black holes in four dimensions We generalize the 4-charged non-extreme solution (193) to include an angular momentum [183]. (For an another attempt, see [389]. But this solution has only 3 charge degrees of freedom and is a special case of a general solution to be discussed in this section.)
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4.3.1. Explicit solution By applying the solution generating technique discussed in the beginning of this section, one obtains the following D"4, non-extreme, rotating black hole solution [183]: (r#2m sinh d )(r#2m sinh d )#l cos h N C , g " (r#2m sinh d )(r#2m sinh d )#l cos h N C 2ml cos h( sinh d cosh d sinh d cosh d ! cosh d sinh d cosh d sinh d ) N N C C N N C C , g " (r#2m sinh d )(r#2m sinh d )#l cos h N C (r#2m sinh d )(r#2m sinh d )#l cos h N C , g " (r#2m sinh d )(r#2m sinh d )#l cos h N C 2ml cos h( sinh d cosh d cosh d sinh d ! cosh d sinh d sinh d cosh d ) N N C C N N C C , B "! (r#2m sinh d )(r#2m sinhd )#l cos h N C (r#2m sinh d )(r#2m sinh d )#l cos h N N eP" , D
r!2mr#l cos h dr ds"D ! dt# #dh # D r!2mr#l #
sin h +(r#2m sinh d )(r#2m sinh d )(r#2m sinh d ) N N C D
(r#2m sinh d )#l(1# cos h)r#=#2mlr sin h,d C 4ml ! +( cosh d cosh d cosh d cosh d ! sinh d sinh d sinh d sinh d )r N N C C N N C C D
#2m sinh d sinh d sinh d sinh d , sin h dt d , N N C C
(249)
where D,(r#2m sinh d )(r#2m sinh d )(r#2m sinh d )(r#2m sinh d ) N N C C #(2lr#=) cos h , =,2ml( sinh d # sinh d # sinh d # sinh d )r N N C C #4ml(2 cosh d cosh d cosh d cosh d sinh d sinh d sinh d sinh d N N C C N N C C !2 sinh d sinh d sinh d sinh d ! sinh d sinh d sinh d N N C C N C C !sinh d sinh d sinh d ! sinh d sinh d sinh d N C C N N C !sinh d sinh d sinh d )#l cos h . (250) N N C The axion W also varies with spatial coordinates, but since its expression turns out to be cumbersome, we shall not write here explicitly.
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The ADM mass, ;(1) charges, and angular momentum are M"2m(cosh 2d # cosh 2d #cosh 2d #cosh 2d ) , C C N N Q"2m sinh 2d , Q"2m sinh 2d , C C (251) P"2m sinh 2d , P"2m sinh 2d , N N J"8lm(cosh d cosh d cosh d cosh d !sinh d sinh d sinh d sinh d ) , C C N N C C N N where G"" and the convention of [478] is followed. , When Q"Q"P"P, all the scalars are constant, and thus the solution becomes the Kerr}Newman solution. The d "d "0 case is the generating solution of a general electric N N rotating solution [562]. The case with Q"P and Q"P is constructed in [389]. The solution (249) has the inner r and the outer r horizons at \ > (252) r "m$(m!l , ! provided m5"l". In this case, the solution has the global spacetime of the Kerr}Newman black hole with the ring singularity at r"min+Q, Q, P, P, and h"p/2. The extreme solution (r "r ) is obtained by taking the limit mP"l">. In this case, the global > \ spacetime is that of the extreme Kerr}Newman solution. The BPS limit is reached by taking mP0 and d PR while keeping meBC C N N as "nite C C N N constants so that the charges remain non-zero. When J is non-zero, i.e. lO0, the singularity is naked since the condition m5"l" for existence of regular horizon (252) is not satis"ed. To have a BPS solution with regular horizon, one has to take lP0, leading to a solution with J"0. Thus, the only regular BPS solution in D"4 is the non-rotating solution, with global spacetime of the extreme Reissner}NordstroK m black hole. This is in contrast with the D"5 3-charged solution [98,182], where one can take l to zero (so that the BPS solution has regular horizon) but the angular momenta J can be non-zero. For D'5, the regular BPS limit with non-zero angular momentum is achieved without taking l to zero if only one angular momentum is non-zero [366]. G 4.3.2. Entropy of general solution The thermal entropy of the solution (249) is [183]
1 A"16p m cosh d # sinh d #m(m!l cosh d ! sinh d S" G G G G 4G , G G G G "16p m cosh d # sinh d # m cosh d ! sinh d !J , (253) G G G G G G G G where d ,d and A" dh d (g g " > is the outer-horizon area. C C N N FF (( PP Note, the thermal entropy has the form which is sum of `left-movinga and `right-movinga contributions. Each term is symmetric in d , i.e. in the 4 charges, manifesting U-duality symmetry G [381]. On the other hand, (253) is asymmetric in J: only the right-moving term has J, which reduces
See [610] for the same result from the conformal p-model perspective.
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the right-moving contribution to the entropy. This re#ects right-moving worldsheet supersymmetry of the corresponding p-model. When J"0, the entropy becomes [180]: S"32pm cosh d , G G
(254)
which again has U-duality symmetry under the exchange of 4 charges. In the regular BPS limit as well as the extreme limit, the `right-movinga term in (253) becomes zero, however entropy has di!erent form in each case. In the regular BPS limit (J"0) [178]: S"32pm cosh d "2p(PPQQ , G G
(255)
while in the extreme limit:
S"16pm cosh d # sinh d "2p(J#PPQQ . G G G G
(256)
Entropy of a black hole with general charge con"guration in the class and with arbitrary scalar asymptotic values is independent of scalar asymptotic values when expressed in terms of the charge lattice vectors a and b, and has the S- and T-duality invariant form [183]: S"2p(J#+(a2¸a)(b2¸b)!(a2¸b), .
(257)
4.4. General rotating xve-dimensional solution We construct the most general rotating black hole in heterotic string on ¹ [182]. In D"5, black holes carry only electric charges of ;(1) gauge "elds. Since the NS-NS 3-form "eld strength H is Hodge-dual to a 2-form "eld strength in D"5 in the following way: IJM eP HIJM"! eIJMHNF , HN 2!(!g
(258)
where F is the "eld strength of a new ;(1) gauge "eld A , black holes in D"5 carry an additional IJ I charge associated with the NS-NS 2-form "eld B as well as 26 electric charges of the ;(1) gauge IJ group. Thus, the most general black hole in heterotic string on ¹, compatible with the conjectured `no-hair theorema [125,343,344,385,386], is parameterized by 27 electric charges, 2 angular momenta and the non-extremality parameter. 4.4.1. Generating solution We choose to parameterize the `generating solutiona in terms of electric charges Q, Q and Q associated with H , A and A , respectively. These charges are induced through solution IJM I I generating procedure described in Section 4.1.
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The "nal form of the generating solution is [182] r#2m sinh d #l cos h#l sin h C g " , r#2m sinh d #l cos h#l sin h C (r#2m sinh d #l cos h#l sin h) C eP" , (r#2m sinh d #l cosh#l sinh) CG G m cosh d sinh d C C A" , R r#2m sinh d #l cos h#l sin h C l sinh d sinh d cosh d !l cosh d cosh d sinhd C C C C C C, A "m sin h ( r#2m sinh d #l cos h#l sin h C l cosh d sinh d sinh d !l sinh d cosh d cosh d C C C C C C, A "m cos h ( r#2m sinh d #l cos h#l sin h C m cosh d sinh d C C A" , R r#2m sinh d #l cos h#l sin h C l cosh d sinh d coshd !l sinh d cosh d sinh d C C C C C C, A "m sin h ( r#2m sinh d #l cos h#l sin h C l sinh d cosh d sinh d !l cosh d sinh d cosh d C C C C C C, A "m cos h ( r#2m sinh d #l cos h#l sin h C l sinh d sinh d cosh d !l cosh d cosh d sinh d C C C C C C, BK "!2m sin h R( r#l cos h#l sin h#2m sinh d C l sinh d sinh d cosh d !l cosh d cosh d sinh d C C C C C C, "!2m cos h BK R( r#l cos h#l sin h#2m sinh d C 2m cosh d sinh d cos h(r#l#2m cosh d ) C C C , BK "! (( r#l cos h#l sin h#2m sinh d C (r#l cos h#l sin h)(r#l cos h#l sin h!2m) ds"DM ! dt # DM
r 4m cosh sinh # dr#dh# [l l +(r#l cosh#l sinh) (r#l)(r#l)!2mr DM !2m( sinh d sinh d # sinh d sinh d #sinh d sinh d ), C C C C C C #2m+(l#l) cosh d cosh d cosh d sinh d sinh d sinh d C C C C C C !2l l sinh d sinh d sinh d ,]d d
C C C 4m sin h [(r#l cos h#l sin h)(l cosh d cosh d cosh d ! C C C DM
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!l sinh d sinh d sinh d )#2ml sinh d sinh d sinh d ]d dt C C C C C C 4m cos h ! [(r#l cos h#l sin h)(l cosh d cosh d cosh d C C C DM !l sinh d sinh d sinh d )#2ml sinh d sinh d sinh d ] d dt C C C C C C sin h # (r#2m sinh d #l) (r#2m sinh d #l cos h#l sin h) C CG DM G #2m sin h+(l cosh d !l sinh d )(r#l cos h#l sinh) C C #4ml l cosh d cosh d cosh d sinh d sinh d sinh d C C C C C C !2m sinh d sinh d (l cosh d #l sinhd ) C C C C
!2ml sinh d (sinh d #sinh d ), d C C C
cos h # (r#2m sinh d #l) (r#2m sinh d #l cos h#l sin h) C CG DM G #2m cos h+(l cosh d !l sinh d )(r#l cos h#l sin h) C C #4ml l cosh d cosh d cosh d sinh d sinh d sinh d C C C C C C !2m sinh d sinh d (l sinh d #l cosh d ) C C C C
!2ml sinh d (sinhd #sinh d ), d , C C C
(259)
where DM ,(r#2m sinh d #l cos h#l sin h)(r#2m sinh d #l cos h#l sin h) C C ;(r#2m sinh d #l cos h#l sin h) , (260) C and the subscript E in the line element denotes the Einstein-frame. The ;(1) charges, the ADM mass and the angular momenta of the generating solution (259) (with G""p/4) are , Q"m sinh 2d , Q"m sinh 2d , Q"m sinh 2d , C C C M"m(cosh 2d #cosh 2d #cosh 2d ) C C C (261) "(m#(Q)#(m#(Q)#(m#Q , J "4m(l cosh d cosh d cosh d !l sinh d sinh d sinh d ) , C C C C C C J "4m(l cosh d cosh d cosh d !l sinh d sinh d sinh d ) . C C C C C C The solution has the outer and inner horizons at: 1 1 1 r "m! l! l$ ((l!l)#4m(m!l!l) , ! 2 2 2 provided m5("l "#"l ").
(262)
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When Q"Q"Q, the generating solution becomes the D"5 Kerr}Newman solution, since g and u become constant. The generating solution with Q"Q corresponds to the case where the D"6 dilaton u "u# log det g is constant. In this case, with a subsequent rescaling of scalar asymptotic values one obtains the static solution of [368] and rotating solution of [95]. The BPS limit with J O0 and regular event horizon is de"ned as the limit in which mP0, l P0 and d PR while keeping meBC"Q, meBC"Q, meBC"Q, C C C l /m "¸ and l /m "¸ constant. In this limit, the generating solution becomes Q A" , R r#Q
J sin h J cos h A " , A " , ( r#Q ( r#Q
Q A" , R r#Q
J sin h J cos h A " , A " , ( r#Q ( r#Q
J sin h J cos h BK "! , BK " , BK "!Q cos h , R( R( (( r#Q r#Q r#Q (r#Q) , eP" g " , r#Q [(r#Q)(r#Q)]
r dr J cos h sin h ds"DM ! dt# #dh# d d
# DM r 2DM 2Jr cos h 2Jr sin h dt d # dt d
! DM DM
1 sin h (r#Q)(r#Q)(r#Q)! J sin h d # 4 DM
1 cos h (r#Q)(r#Q)(r#Q)! J cos h d , # 4 DM
(263)
where DM ,(r#Q)(r#Q)(r#Q) . The solution is speci"ed by 3 charges and only 1 angular momentum J:
(264)
J "!J ,J"(2QQQ) (¸ !¸ ) , while its ADM mass saturates the Bogomol'nyi bound:
(265)
M
"Q#Q#Q . .1
When 1 or 3 boost parameters are negative, one has the BPS limit with J "J .
(266)
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4.4.2. ¹-duality transformation The remaining 27 electric charges (needed for parameterizing the most general charge con"guration) are introduced by the [SO(5);SO(21)]/[SO(4);SO(20)] transformation on the generating solution (259). The "nal expression for electric charges is
; (e !e ) S B Q" U2 e #e S B (2 ; 0 1
,
(267)
where
CDE CDE e2,(Q, 0, 2, 0), e2,(Q, 0, 2, 0) , B S
(268)
; 3SO(5), ; 3SO(21), 0 is a (16;1)-matrix with zero entries and U3O(5, 21, R) brings to the basis where the O(5, 21) invariant metric ¸ (127) is diagonal. And the charge Q associated with B remains unchanged. The moduli M is transformed to IJ ; 0 ;2 0 M "U2 UMU2 U, (269) 0 ; 0 ;2 where M is the moduli of the generating solution (259). The subsequent O(5, 21);SO(1, 1) transformation leads to the solution with arbitrary asymptotic values M and u .
4.4.3. Entropy of general solution The thermal entropy of the generating solution (259) is [183]
1 A"4p m+2m!(l !l ), cosh d # sinh d S" G G 4G , G G #m+2m!(l #l ), cosh d ! sinh d G G G G 1 "4p 2m cosh d # sinh d ! (J !J ) G G 16 G G 1 # 2m cosh d ! sinh d ! (J #J ) , (270) G G 16 G G where d ,d , G "p/4 and the outer horizon area is de"ned as A" C C C , dh d d (g (g g !g )" >. ( ( PP FF ( ( ( ( Note, each term is symmetric under the permutation of d (i.e. 3 charges), manifesting the G conjectured U-duality symmetry [381]. Again, as in the D"4 case, the entropy (270) is cast in the form as sum of `left-movinga and `right-movinga contributions, hinting at the possibility of statistical interpretation of each term as left- and right-moving (D-brane worldvolume) contributions to microscopic degrees of freedom. Each term now carries left- or right-moving angular momentum that could be interpreted as left- or right-moving ;(1) charge [35,36] of the N"4
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superconformal "eld theory when the generating solution (259) is transformed to a solution of type-IIA string on K3;S through the conjectured string}string duality in D"6 [635]. When J "0, the entropy rearranges itself as a single term [183,362]: G (271) S"8(2pm cosh d , G G which again has manifest symmetry under permutation of charges. We discuss duality invariant forms of the entropy and the ADM mass of non-extreme, rotating black hole with general charge con"guration (267). The entropy and the ADM mass are expressed in terms of the following T-duality invariants (obtained by applying T-duality to charges of the generating solution): QPX"(Q2M Q#(Q2M Q , > \ (272) QP>"(Q2M Q!(Q2M Q , > \ while Q remains intact under T-duality. From these 3 T-duality invariant `coordinatesa X, >, Q, one de"nes the following duality invariant `non-extreme hatteda quantities XK : G XK ,(X#m, X "(X, >, Q) . (273) G G G Duality invariant forms of the entropy and the ADM mass are
S"2p
XK #m XK #( (XK !m)!(J !J ) G G G G G G
# XK #m XK !( (XK !m)!(J #J ) , G G G G G G M"XK #>K #QK . When J "0, the duality invariant expression for entropy is G S"4p((XK #m)(>K #m)(QK #m) .
(274)
(275)
4.4.3.1. BPS limit. In the regular BPS limit, the event horizon area (270) becomes [182] (276) "4p[(QQQ)(1!(¸ !¸ ))] "4p[QQQ!J] . .1 Entropy of BPS black hole with general charge con"guration (267) and with arbitrary scalar asymptotic values depends only on (quantized) charge lattice vectors a and b [182], being a statistical quantity [174,178,258}260,441,579]: A
"4p(b(a2¸a)!J . .1 Here, a and b are related to the physical charges Q and Q as S
(277)
Q"eP M a, Q"e\P b .
(278)
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In the regular BPS limit, the ADM mass (261) becomes M
"Q#Q#Q . (279) "+ A subset of the SO(5, 21) T-duality transformation on (279) leads to the following T-duality invariant expression for ADM mass of the general D"5 black hole: "eP [a2(M #¸)a] #e\P b , "+ which has dependence on scalar asymptotic values as well as charge lattice vectors. M
(280)
4.4.3.2. Near-extreme limit. In"nitesimal deviation from the BPS limit is achieved by taking the limit in which m and l are very close to zero, and d's are very large such that charges and l /m "¸ remain as "nite, non-zero constants, and then keeping only the leading order terms in m [182]. To the leading order in m, the inner and the outer horizons are located at r +m(1!(¸#¸)$([2!(¸ #¸ )][2!(¸ !¸ )]) . ! The outer horizon area to the leading order in m is [95,182] A+4p[(QQQ)(1!(¸ !¸ )#m(QQ#QQ#QQ) ;([1!(¸ #¸ )]] . J and J are no longer equal in magnitude and opposite in sign anymore: J,(J !J )"(2QQQ) (¸ !¸ )#O(m) , 1 1 1 *J,(J #J )"m(2QQQ) # # (¸ #¸ )#O(m) , Q Q Q while the ADM mass still has the form
(281)
(282)
(283)
M"(m#(Q)#(m#(Q)#(m#Q . (284) Note, when one of the charges is taken small, e.g. QP0, as in study of the microscopic entropy near the BPS limit [95,368], the ADM mass is M"M #O(m), while the area is .1 A"A #O(m ). However, when all the charges are non-zero, the deviation from the BPS limit .1 is of the forms M"M #O(m) and A"A #O(m). .1 .1 4.5. Rotating black holes in higher dimensions We discuss rotating black holes in heterotic string on ¹\" (44D49) with general ;(1)\" electric charge con"gurations [184,449]. The generating solution is parameterized by the ADM mass M (or alternatively the non-extremality parameter m), ["\] angular momenta & J (i"1,2, ["\]), and 2 electric charges of the KK and the 2-form ;(1) gauge "elds associated G with the same compacti"ed direction, which we choose without loss of generality to be Q and Q, i.e. those associated with the "rst compacti"ed direction, as well as asymptotic values of a toroidal modulus G and the dilaton u .
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The non-trivial "elds of the generating solution are [184] N sinh d cosh d N sinh d cosh d , , A" A" R R 2N sinh d #D 2N sinh d #D Nl k sinh d cosh d Nl k sinh d cosh d , A " G G , A " G G (G (G 2N sinh d #D 2N sinh d #D 2N sinh d #D D eP" , G " , 2N sinh d #D = 2Nl k sinh d sinh d [m(sinh d #sinh d )r#D] G G B G"! , R( = B G H"!4Nl l kk sinh d sinh d cosh d cosh d GH G H (( ;[N(sinh d #sinh d )#D][2N sinh d sinh d #ND(sinh d # sinh d !1)#D]/[(D!2N)=] , D!2N dr dt# ds"D"\ "\= "\ ! = "\ (r#l)!2N G G r#l cos h#K sin h # dh D
cos h cos t 2cos t G\ (r#l cos t #K sin t )dt # G> G G> G G D l!K H cos h cos t 2cos t !2 H G\ D GH ;cos t sin t 2cos t cos t sin t dt dt G G H\ H H G H l !K G> cos h sin h cos t 2cos t cos t sin t dh dt !2 G> G\ G G G D k # G [(r#l)(2N sinh d #D)(2N sinh d #D) G D= 2Nl k cosh d cosh d G G dt d
#2lN(D!2N sinh d sinh d )] d ! G G G = # GH
4Nl l kk(D!2N sinh d sinh d ) GH G H d d , G H D=
(285)
where =,(2N sinh d #D)(2N sinh d #D) and D, K , N, k , a are de"ned separately for even and odd D in (188)}(191). G G
(286)
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The ADM mass, angular momenta and electric charges of the generating solution are X m M " "\ [(D!3)(cosh 2d #cosh 2d )#2] , & 16pG " X J " "\ml cosh d cosh d , G G 4pG " (287) X Q" "\ (D!3)m sinh 2d , 16pG " X Q" "\ (D!3)m sinh 2d . 16pG " For the canonical choice of asymptotic values G "d , i.e. compacti"cation on (10!D) self-dual GH GH circles with radius R"(a , the D-dimensional gravitational constant is G "G /(2p(a )\". " Also, the KK and the 2-form "eld ;(1) charges Q and Q are quantized as p/(a and q/(a , respectively, where p, q3Z. The outer horizon area of the generating solution is [184,366] A "2mr X cosh d cosh d , (288) " > "\ where the outer horizon r is determined by > "\
"0 . (289) (r#l)!2N G PP> G The surface gravity i at the (outer) event horizon is de"ned as i"lim > j Ij, where P P I mIm ,!j and m,R/Rt#X R/R . Here, X is the angular velocity at the (outer) horizon and is I G G G de"ned by the condition that m is null on the (outer) horizon. The surface gravity and angular velocity at the outer-horizon of the generating solution are
1 1 R (P!2N) l P G i" , X" , (290) G cosh d cosh d r #l cosh d cosh d 4N PP> > G where P, "\ (r#l). G G The generating solution has a ring-like singularity at (r, h)"(0, p/2) and thus spacetime is that of the Kerr solution. The BPS limit of (285), where the ADM mass M saturates the Bogomol'nyi bound & M 5"Q#Q" , (291) & is de"ned as the limits mP0 and d PR such that Q remain as "nite constants. For D56 with only one of l non-zero, the BPS limit is also the extreme limit [366], i.e. all the horizons G collapse to r"0 as mP0. However, with more than one l non-zero, the singularity at r"0 G becomes naked, i.e. horizons disappear.
We use the convention of [478], keeping in mind that matter Lagrangian in (124) has 1/(16pG ) prefactor. "
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5. Black holes in Nⴝ2 supergravity theories 5.1. N"2 supergravity theory 5.1.1. General matter coupled N"2 supergravity We consider the general N"2 supergravity [5,6,128,198,211,271] coupled to n vector mul tiplets and n hypermultiplets. The "eld contents are as follows. The N"2 supergravity multiplet & contains the graviton, the S;(2) doublet of gravitinos tG (the S;(2) index i"1, 2 labels two I supercharges of N"2 supergravity and k"0, 1, 2, 3 is a spacetime vector index), and the graviphoton. The N"2 vector multiplets contain ;(1) gauge "elds, doublets of gauginos j? and G scalars z? (a"1,2, n ), which span the n -dimensional special KaK hler manifold. The hypermultip lets consist of hyperinos f , f? (a"1,2, 2n ) with left and right chiralities and real scalars ? & qS (u"1,2, 4n ), which span the 4n -dimensional quaternionic manifold. The general form of the & & bosonic action is [5] L,"(!g[!R#g H(z, z ) Iz? z @H#h (q) IqS qT ?@ I ST I K R K R #i(N M KRF\ F\ IJ!NKRF> F> IJ)] , (292) IJ IJ where g H"R R HK(z, z ) is the KaK hler metric, h (q) is the quaternionic metric, ?@ ? @ ST F!K,(FK $(i/2)eIJMNFK ) are the (anti-)self-dual parts of the "eld strengths II IJ MN FK "R AK!R AK#gf KR ARA of the ;(1) gauge "elds AK (K"0, 1,2, n ) in the N"2 IJ I I J I I J I supergravity and N"2 vector multiplets, and g is the gauge coupling. Here, the gauge covariant di!erentials on the scalars are de"ned as:
z?,R z?#gAKk?K(z) , I I I
z ?H,R z ?H#gAKk?H (z ) , (293) I I I K
qS,R qS#gAKkSK(q) , I I I where k?K(z) [kSK(q)] are the holomorphic [triholomorphic] Killing vectors of the KaK hler [quaternionic] manifold (cf. see (65)). We introduce a symplectic vector of the anti-self-dual "eld strengths:
Z\,
F\K G\ R
,
(294)
where G\ M KRF\R [212]. The symplectic vector Z> of self-dual "eld strengths is the complex K ,N conjugation of (294). It is convenient to rede"ne "eld strengths FK as [93,132] ¹\,1<"Z\2"(MKF\K!¸RG\ R ) , M F\K! H¸M KG\ F\?,g?@H1;M H"Z\2"g?@H( HM K) . @ @ K @
(295)
The KaK hler potential K(z, z ) and the period matrix NKR are de"ned in terms of the holomorphic prepotential F(X) and the scalar "elds XK as in (92).
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Then, ¹\ and F\? (a"1,2, n ), respectively, correspond to the "eld strengths of the gravi-photon of the supergravity multiplet and the gauge "elds of n super-Yang-Mills multiplets. The supersymmetry transformation laws for the gravitinos, the gauginos and hyperinos in the bosonic "eld background are dt "D e #[igS g #e ¹\ ]cJeH , GI IG GH IJ GH IJ (296) dj?G"icI z?eG#eGH(F\?cIJ#k?K¸M K)e , H I IJ df "iUH@ qScIe C eG#gNG e , ? S I GH ?@ ?G where e [C ] is the #at Sp(2) [Sp(2n )] invariant matrix and UH@ is the quaternionic vielbein [30]. GH ?@ & S Here, S and NG are mass-matrices given by GH ? i S " (p )Ie PVK¸K, NG "2UG kSK¸M K , (297) GH 2 V G HI ? ?S where PVK is a triplet of real 0-form prepotentials on the quaternionic manifold. 5.1.2. BPS states The BPS states of the N"2 theory have mass equal to the central charge, which is just the graviphoton charge given by 1 Z,! 2
¹\ .
(298)
1 Thus, central charge Z is characterized by the vacuum expectation of the moduli in the symplectic vector < and the symplectic charge vector given by
Q"
PK
QR
; PK,
Re F\K, QR,
1 1 in the following way [133,136,212,550,551]
Re G\ R ,
Z"1<"Q2"(¸KQK!MRPR)"e)X X (XK(z)QK!FR(z)PR) .
(299)
(300)
Note, since two vectors < and Q transform covariantly under the symplectic transformation Sp(2n #2), the central charge and, therefore, the ADM mass M""Z" have manifest symplectic covariance. 5.2. Supersymmetric attractor and black hole entropy Since entropy is a statistical quantity de"ned as the degeneracy of microscopic states, the horizon area, which de"nes the thermal entropy, should be independent of continuous quantities like scalar asymptotic values. This is an other illustration of no-hair theorem where properties of black holes are independent of scalar hairs; all the information of scalar asymptotic values get lost at the event horizon. It was discovered in [260] within N"2 theories that this is a generic property of BPS solutions in supersymmetric theory and can be derived from supersymmetry alone [8,258,259].
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To illustrate this idea, we consider general magnetic, spherically symmetric solution in N"2 theory coupled to vector super"elds [260]. Spherically symmetric AnsaK tze are [597]: qK ds"g dxI dxJ"!e3 dt#e\3 dx, FK K" e3P , P IJ r
(301)
and the scalars (moduli) zK,XK/X (K"0, 1,2, n ) depend on r, only. With these AnsaK tze, the Killing spinor equations dt "0 and dj?G"0 yield the following GI coupled "rst-order di!erential equations [260]:
4; "!
(z Nq )(zNq )(z Nz) e3 , (zNz)(z Nz )
(302) (zNz)(z Nq )(z Nz) e3 (zKq !qK ) , (zK) "! (z Nz )(zNq ) 4 where the prime denotes the di!erentiation with respect to o,1/r, and (zNq ),zKNKRqR , etc. From (302), one obtains the following second-order ordinary di!erential equations for the moduli K "elds z :
(zNq ) ((zK) ) 1 (zNz)(z Nq )(z Nz) K #q (z ) "0 . # ln (303) K K z q !q (zNz) 2 (z Nz )(zNq ) Eq. (303) can be viewed as a geodesic equation for moduli "elds zK that determines how and (zK) " for the geodesic zK evolves as o varies from 0 to R. The initial conditions zK" M M K K motion in the phase space (with coordinates z and (z ) ) are the asymptotic values (r"1/oPR) and qK through the second equation in (302)). of zK and their derivatives (determined by zK" P K K K Given initial conditions z " and (z ) " , z evolve with o, following damped geodesic motion M M in the phase space until they run into an attractive "xed point, i.e. a point where the velocities dzK/do of zK vanish. For the special example under consideration, as we see (302), the "xed point is located at
(zK) !
(304) zK "qK /q . At the "xed point, moduli depend on ;(1) charges only, loosing all their information on the initial conditions at in"nity. From this observation, one arrives at stronger version of no-hair theorem for black holes in supersymmetry theories: black holes lose all their scalar hairs near the horizon and are characterized by discrete ;(1) charges (and angular momenta), only. Nearby the horizon, the black hole approximates to the Bertotti}Robinson geometry [92,447,515] with the topology AdS ;S"\. This geometry is conformally #at and the graviphoton "eld strength is covariantly constant. Thus, in this region the ADM mass reduces to the Bertotti}Robinson mass and the supersymmetry is completely restored [138,292,300, 398,412,453]. In the asymptotic region (rPR), the spacetime is #at and, therefore, supersymmetry is unbroken. In between these regions, solutions break fraction of supersymmetries, indicating the BPS nature. Note, the supersymmetric con"guration under consideration is a bosonic con"guration, i.e. a solution to supergravity theory with all the fermionic "elds set equal to zero. However, the
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supersymmetry parameters associated with the unbroken supersymmetries, called `anti-Killing spinorsa, generate a whole supermultiplets of solutions, i.e. the superpartners to black hole solution. To generate such solutions, one start with a bosonic con"guration and applies supersymmetry transformations iteratively with the supersymmetry parameters given by the anti-Killing spinors. Such a procedure induces fermionic "elds, as well as corrections to bosonic "elds. It was shown in [407] that when this procedure is performed on double-extreme black solutions, i.e. extreme solutions with constant scalars, in the N"2 supergravity coupled to vector and hyper multiplets, there are no corrections to the "elds in the vector and hyper multiplets. This implies that although the metric, graviphoton and gravitino receive corrections, the moduli at the "xed attractor point as functions of ;(1) charges, only, remain intact under the supersymmetry transformations which generate the fermionic partners of the supersymmetric black holes. 5.3. Explicit solutions 5.3.1. General magnetically charged solutions We discuss the general spherically symmetric, magnetic (q R "0) solutions in N"2 supergravity coupled to n vector multiplets with scalar "elds varying with the radial coordinate r [260]. The AnsaK tze for the "elds are given in (301) with the scalar "elds depending on the radial coordinate r, only. The scalar "elds and the metric components satisfy the di!erential equations (302)}(303). For the purpose of solving the di!erential equations (302)}(303), we consider the simple case with q "0. In this case, the solutions are given by e3M"e)X X \) , q? z? # K oe\) for z? "z ? , 4 z?" (305) q? K oe\) for z? "!z ? , z? #i 4
ds"!e)X? X ?X?\) dt#e\)X? X ?X?>) dx . The explicit solutions for N"2 theories with speci"c prepotentials F are obtained by substituting the corresponding KaK hler potential K into the general solution (305) [260]. 5.3.2. Dyonic solutions We generalize the magnetic black holes in Section 5.3.1 to include electric charges as well [579]. Since it would be hard to solve the resulting di!erential equations with non-zero electric and magnetic charges, we take all the moduli "elds z' to be constant. In fact, as we discuss in the subsequent sections, such class of solutions corresponds to the minimum energy con"gurations among extreme solutions and, therefore, is physically interesting. Assuming that the moduli z'"X'/X are constant, from dj?G"0 one obtains the following electric and magnetic charges of dyonic solutions:
i (q' , q)"Re CX',!C F (X ) , ' 2 '
(306)
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where C is a constant and the subscript f denotes the "xed point. With a suitable choice of KaK hler gauge, which eliminates the redundant degrees of freedom in X' by half, one can solve the 2n#2 equations in (306) to "nd the expressions for z'"X'/X in terms of quantized charges q' and q. ' From dt "0 with ;(1) "eld strengths F K '"CX'e> and G>"(C/2)F e> (e> obeys GI ' ' IJ "ie> and is normalized to give 2p after being integrated over S) substituted, one obtains the *e> IJ IJ following solution for the metric: e\3"1#(CCM /4r .
(307)
This solution has the surface area given by p A" CCM . 4
(308)
5.4. Principle of a minimal central charge At the "xed attractor point in phase space, the central charge eigenvalue is extremized with respect to moduli "elds, so-called `principle of a minimal central chargea [254,258,259,414,415]. For N'2 theories, the largest eigenvalue is extremized and the smaller central charges become zero [259] at the "xed point. Since scalar asymptotic values are expressed only in terms of ;(1) charges at the "xed point, the extremized (largest) central charge depends only on ;(1) charges, thereby becoming a candidate for describing black hole entropy. It turns out that entropy of extreme black holes for each dimension has the following universal dependence on the extremal value Z of the (largest) central charge eigenvalue regardless of the number N of supersymmetries [258,259]: A S" "p"Z "? , 4
(309)
where a"2[3/2] for D"4 [D"5]. As an example, we consider the BPS dilatonic dyon [292,409] in D"4 with the mass: ""Z""(e\P"p"#eP"q") . (310) .1 The minimum of the central charge "Z" is located at g "eP""p/q", which leads to the following correct expression for the entropy which is independent of dilaton asymptotic value: M
S"A/4"p"Z ""p"pq" . (311) One can prove the principle of a minimal central charge as follows. We consider the ungauged N"2 supergravity coupled to Abelian vector multiplets and hypermultiplets, de"ned by the Lagrangian (292) and the supersymmetry transformations (296) with g"0. Since we are interested in con"gurations at the "xed point, the derivatives of scalars are zero, i.e. R z?"0 and R qS"0, at I I These are obtained by solving dj?G"0. Thus, for N54 theories, one can determine moduli "elds (at the "xed point) in terms of ;(1) charges, by minimizing the largest eigenvalue and setting the remaining eigenvalues equal to zero.
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the horizon. So, from dj?G"0, one has F\?"0. Note, the covariant derivative of the central IJ charge de"ned in (300) is 1 Z , Z"! ? ? 2
g HF>@H"(QK ¸K!PR MR)"1; "Q2 . ?@ ? ? ?
(312) 1 Since F\?"0 is equivalent to F>?"0, the central charge Z is covariantly constant at the "xed point of the moduli space:
Z " Z"1; "Q2"0 . (313) ? ? ? It can be shown [258] that the condition (313) is equivalent to the statement that the central charge takes extremum value at the "xed point: R "Z""0 . (314) ? Thus, within ungauged general Abelian N"2 supergravity we establish that central charge is minimized at the xxed point of geodesic motion of moduli evolving with o"1/r. At the "xed point in the moduli space, the central charge is expressed in terms of the symplectic ;(1) charge vector Q and the moduli as [258]: "Z""!Q2 ) M(N) ) Q , Im N#(Re N)(Im N)\(Re N) !(Re N)(Im N)\ , M(N), !(Im N)\(Re N) (Im N)\
(315) (316)
with the moduli in the matrix M(N) taking values at the "xed attractor point. The central charge minimization condition (313) "xes the asymptotic values of the moduli in terms of Q. By using the relations 1; "<2"0"1; "<M 2 satis"ed by the general symplectic section ? ? <, one can solve (313) to express Q in terms of the moduli as [258] Q"i(ZM
(317)
or in component form: PK"2 Im(ZM ¸K),
QR"2 Im(ZM MR) .
(318)
Eq. (318) can be solved to express the moduli (at the "xed point) in terms of ;(1) charges:
1 <"! 2ZM
0
I
!I 0
) M(F)#i
I
0
0
I
)Q ,
(319)
or in terms of components: !2ZM ¸K"[iP!(Im N)\(Re N)P#(Im N)\Q]K , !2ZM MR"[iQ!((Im N)#(Re N)(Im N)\(Re N))P#(Re N)(Im N)\Q]R .
(320)
Here, M(F) is de"ned as in (316) with NKR replaced by FKR,RKFR(X). Alternatively, one can rederive the relations (317) obeyed by the moduli and ;(1) charges by a variational principle [54] associated with a potential VQ(>, >M ),!i1PM "P2!1PM #P"Q2 ,
(321)
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where
P,
>K
FR(>)
;
>K,ZM XK .
(322)
Namely, at the minimum of VQ(>,>M ), relation (317) for the minimal central charge, which can be expressed in terms of P as P!PM "iQ, is satis"ed. In particular, the entropy for D"4 black holes is rewritten as S ""Z ""i1PM "P2"">" exp[!K(z, z )]" . p
(323)
Many black holes are uplifted to intersecting p-branes. In this case, energy of black holes is sum of energies of the constituent p-branes. The minimal energy of p-branes corresponds to the ADM mass of the corresponding double-extreme black holes in lower dimensions. In taking variation of moduli to "nd the minimum energy con"guration, one has to keep the gravitational constant of lower dimensions as constant. The minimum energy of p-brane is achieved when energy contributions from each constituent p-brane are equal [403]. 5.4.1. Generalization to rotating black holes Generally, rotating black holes have naked singularity in the BPS limit. D"5 rotating black holes with 3 charges has regular BPS limit (thereby the horizon area can be de"ned), if 2 angular momenta have the same absolute values [98,182]. We discuss generalization of the principle of minimal central charge to the rotating black hole case [414]. We consider the following truncated theory of 11-dimensional supergravity compacti"ed on a Calabi}Yau three-fold [103,328,329,491]: L"(!g[!R!e PF FIJ!e\ PG GIJ#(R u)] IJ IJ I 1 eIJMNHF F B . ! IJ MN H 4(2
(324)
This corresponds to the N"2 theory with F"CKR XKXRX . The supersymmetry transforma tions of the gravitino t and the gaugino s in the bosonic background are I 1 1 e\ PG )e , dt " e# (KMN!4dMCN)(eP F ! I MN (2 MN I I 12 I (325) 1 1 CIR ue# CMN(eP F #(2e\ PG )e . ds"! I MN MN 4(3 2(3
Note, lower-dimensional gravitational constant is expressed in terms of the D"10 gravitational constant and the volume of the internal space, i.e. a modulus. It is argued in [288] that singular D"4 heterotic BPS rotating black holes can be described by regular D"5 BPS rotating black holes which are compacti"ed through generalized dimensional reduction including massive Kaluza}Klein modes.
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The model corresponds to the N"2 supergravity with the graviphoton (eP F ! MN (1/(2)e\ uG ) coupled to one vector multiplet with the vector "eld component MN (eP F #(2e\ PG ). MN MN We consider the following D"5 BPS rotating black hole solution [98,182,609] (cf. (263)) to the above theory:
r 4J sinh 4J cosh dt! d # dt ds" 1! r p(r!r) p(r!r) r \ dr!r(dh#sin h d #cos h dt) . ! 1! r
(326)
For this solution, the scalar u is constant everywhere (double-extreme): eP"j. So, from ds"0, one sees that the vector "eld in the vector multiplet vanishes, i.e. B"!(j/(2)A, from which one can express u in terms of ;(1) charges as
(2 (2 8Q e\P 夹G, Q , eP 夹F . (327) eP" $ ,j; Q , & 4p $ 16p pQ 1 1 & Furthermore, the entropy is still expressed in terms of the central charge Z at the "xed point, but modi"ed by the non-zero angular momentum J: S"p(Z !J . (328) The argument can be extended to more general rotating solutions in the N"2 supergravity coupled to n vector multiplets with the gaugino supersymmetry transformations 1 3 i (329) dj "! g (u)CIR u@ e# tK CIJFK e . ? IJ I ? 4 4 2 ?@
From dj "0, one sees that at the "xed point (R u?"0) FK "0. So, the central charge is ? I IJ extremized at the "xed point: R Z"0. ? We discuss the enhancement of supersymmetry near the horizon [138,412]. Since the vector "eld in the vector multiplet is zero for (326), the solution is e!ectively described by the pure N"2 supergravity [265] with the graviphoton FI ,((3/2)jF. The supersymmetry transformation for the gravitino is 1 dt " K e" e# (CMNC #2CMdN)FI e . I I I I I MN 4(3 The integrability condition for the Killing spinor equation dt "0 is I [ K , K ] e"RK e"0 , ? @ ?@ where the super-curvature RK for solution (326) is de"ned as ?@ (r!r) X (1#C) . RK " ?@ ?@ r
(330)
(331)
(332)
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Here, the explicit forms of matrices X , which can be found in [414], are unimportant for our ?@ purpose. At the horizon (r"r ) and at in"nity (rPR), RK "0, thereby (331) does not constraint ?@ the spinor e, i.e. supersymmetry is not broken. However, for "nite values of r outside of the event horizon, for which RK O0, e is constrained by the relation: ?@ (1#C)e"0 , (333) indicating that 1/2 of supersymmetry is broken. 5.4.2. Generalization to N'2 case The principle of minimal central charge is generalized to the N"4, 8 cases by reducing N"4, 8 theories to N"2 theories, and then by applying the formalism of N"2 theories [258]. For N54 theories, there are more than 1 central charge eigenvalues Z (i"1,2, [N/2]). The ADM mass of G the BPS con"guration is given by the max+"Z ",. When the principle of minimal central charge is G applied to this eigenvalue, the smaller eigenvalues vanish and all the scalar asymptotic values are expressed in terms of ;(1) charges, only [259]. So, the extremum of the largest central charge continues to depend on integer-valued ;(1) charges, only. The entropy of extreme black holes in each D has the universal dependence on the extreme value of the largest central charge eigenvalue: S"A/4"p"Z "?, where a"2 [3/2] for D"4 [D"5], regardless of the number N of super symmetry. 5.4.2.1. Pure N"4 supergravity. Pure N"4 theory can be regarded as N"2 supergravity coupled to one N"2 vector multiplet. This can also be regarded as either S;(2);SO(4) or S;(2); S;(4) invariant truncation of N"8 theory. The former corresponds to the N"2 theory with F(X)"!iXX and the latter has no prepotential. These two theories are related by the symplectic transformation [136]: XK "X, FK "F , XK "!F , FK "X , (334) where the hat denotes the S;(4) model [165]. The SO(4) version [162,164,194] of the D"4, N"4 supergravity action without axion is
1 dx(!g[!R#2R uRIu!(e\PFIJF #ePGI IJGI )] , I" I IJ IJ 16p
(335)
where the "eld strength GI of the SO(4) theory is related to that G of the S;(4) theory as IJ IJ i 1 GI IJ" e\PeIJMHG . MH 2 (!g The dilatino supersymmetry transformation is 1 1 dK "!cIR e # pIJ(e\(F a !e(GI b )\e( . I ' (2 IJ '( IJ '( 2 '
(336)
At the "xed point (R "0), the Killing spinor equations dK "0 "x in terms of electric and I ' magnetic charges: e\( ""q"/"p". Then, writing dK "0 at the "xed point in the form (Z ) e("0, ' '(
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one learns [409] that E pq'0 case: e non-vanishing, Z "0 and M ""Z ". "+ E pq(0 case: e non-vanishing, Z "0 and M ""Z ". "+ So, smaller eigenvalues, which correspond to broken supersymmetries, vanish and entropy is given by the largest eigenvalue at the "xed point. 5.4.2.2. N"4 supergravity coupled to n vector multiplets. The target space manifold of N"4 supergravity coupled to n vector multiplets is O(6, n ) S;(1, 1) ; O(6);O(n ) ;(1) with the "rst factor parameterized by the axion-dilaton "eld S and the second factor by the coset representatives ¸ K "(¸GHK , ¸?K) (i, j"1, 2, 3, 4, K"1,2, 6#n and a"1,2, n ) [84]. The central charge is Z "e) [¸K qK!S¸ KpK] , GH GH GH where K"!ln i(S!SM ) is the KaK hler potential for S. At the "xed point, the gaugino Killing spinor equations dj?"0 require that G K K S¸?Kp !¸? qK"0 .
(337)
(338)
The dilatino Killing spinor equation dsG"0 requires the following smaller of the central charge eigenvalues to vanish: (339) "Z ""(Z ZM GH!((Z ZM GH)!"eGHIJZ Z ") , GH GH IJ GH which "xes S at the "xed point. At the "xed point, the di!erence between two eigenvalues "Z "!"Z ""((Z ZM GH)!"eGHIJZ Z " GH GH IJ becomes independent of scalars and gives rise to the horizon area [178,236] A"4p(M ) "4p"Z ""2p(qp!(q ) p) . "+
(340)
(341)
5.4.2.3. N"8 supergravity. The consistent truncation of N"8 down to N"2 is achieved by choosing HLS;(8) such that 2 residual supersymmetries are H-singlet. Such theory corresponds to N"2 supergravity couple to 15 vector multiplets (n "15) and 10 hypermultiplets (n "10). & (This is the upper limit on the number of matter multiplets that can be coupled to N"2 supergravity.) Under N"2 reduction of N"8, S;(8) group breaks down to S;(2);S;(6), leading to the decomposition of 26 central charges Z of N"8 into (1, 1)#(2, 6)#(1, 15) under S;(2);S;(6). The S;(2) invariant part (1, 1)#(1, 15) is (Z, D Z), where Z is the N"2 central G charge. So, the horizon area is again A&"Z". For example, for type-IIA theory on ¹ /Z truncated so that only 2 electric and 2 magnetic charges are non-zero, the central charge at the "xed point is product of ;(1) charges, which is black hole horizon area.
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We consider the following truncation of N"8 supergravity:
1 1 S" dx(!g R! [(Rg)#(Rp)#(Ro)] 16pG 2
!
eE [eN>M(F )#eN\M(F )#e\N\M(F )#e\N>M(F )] . 4
(342)
This is a special case of S¹; model [236] with the real parts of complex scalars zero e\E"Im S,s,
e\N"Im ¹,t, e\M"Im ;,u .
(343)
The following black hole solution to this model is reparameterization [511] of the general solution obtained in [178]: ds"!e3dt#e\3 dx , e3"t t s s , t t t s t s e\E" , e\N" , e\M" , s s s t t s F "$dt dt, FI "$ds dt , F "$dt dt, FI "$ds dt , where "q " \ "p " \ t " eE>N>M # , s " e\E\N>M # , r r
"q " \ "p " \ t " eE\N\M # , s " e\E>N\M # , r r
(344)
(345)
and s are magnetic potentials related to FI "eE!N\M夹F . The ADM mass of (344) is 1 s u t M " stu"q "# "q "# "P "# "p " . (346) "+ 4 tu st su
By minimizing (346) with respect to s, t, u, one obtains the ADM mass at the "xed point [178]: (M ) ""q p q p " , (347) "+ and "nds that the smaller central charges are zero at the "xed point. This result is proven in general setting as follows. We consider the N"8 supersymmetry transformations [160] of gravitinos W and fermions s at the "xed point: I ! dW "D e #Z cJe , ds "Z pIJe , (348) I I IJ ! IJ !
where A"1,2,8 labels supercharges of N"8 theory. We truncate the Killing spinors as e "0, e "+e , e O0, e "e "0, , ? G
(349)
This model also corresponds to ¹ part of type-IIA theory on K ;¹ or heterotic theory on ¹;¹. See Section 3.2.2 for the explicit action.
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where 6 supersymmetries are projected onto null states. Here, we splitted the index A as A"(i, a) in accordance with the breaking of S;(8) to S;(4);S;(4). By bringing Z to a block diagonal form [264] through S;(8) transformation (See Section 2.2, for details.), one "nds that the supersymmetry variations of t , s and s vanish due to (349) and the block diagonal I? ?@A ?GH choice of Z . From ds "0, one "nds that Z "0 (i.e. Z "Z "0) and from ds "0, one G?@ ?@ GHI "nds that Z "0. So, we proved within the class of con"gurations characterized by truncation (349) that the condition for unbroken supersymmetry requires the smaller central charges to vanish. And the largest central charge at the "xed point gives the ADM mass and the horizon area. 5.4.2.4. Five-dimensional theories. N"1, N"2 and N"4 theories in D"5 [11,27,103,213, 328,329,491] have 1, 2 and 3 central charges, respectively. At the "xed point, the largest central charge is minimized and the smaller central charges vanish. The horizon area is given in terms of the central charge at the "xed point by A"4pZ . The general expressions for the (largest) central charge at the "xed point for each case are as follows: E N"1 theory: Z "(d (q)\q q , where d (q)\ is the inverse of d "d t!(z) evaluated ! at the "xed point [258]. E N"2 theory: Z "(Q Q) , where Q is a charge of the 2-form potential and Q is the & $ & $ Lorentzian (5, n ) norm of other 5#n charges [580]. E N"4 theory: Z "(q XHJq XKLq XNG) , where q is 27 quantized charges transforming GH JK LN GH under E (Z) and XGH3Sp(8) is traceless. 5.5. Double extreme black holes We discuss the most general extreme spherically symmetric black holes in N"2 theories in which all the scalars are frozen to be constant all the way from the horizon (r"0) to in"nity (rPR) [415], called double-extreme black holes. For this case, the ADM mass (or the largest central charge) takes the minimum value (related to the horizon area) and, therefore, is equal to the Bertotti}Robinson mass. Whereas all the scalars are restricted to take values determined by ;(1) charges, all the ;(1) charges can take on arbitrary values. Double-extreme black holes are also of interests since they are the minimum-energy extreme con"gurations in a moduli space for given charges. The general double-extreme solution is obtained by starting from the spherically symmetric Ansatz for metric ds"e3 dt!e\3 dx, ;(r)P0,
as rPR ,
(350)
After the "rst draft of this section is "nished, more general class of N"2 supergravity black hole solutions [60,61,521,522,520], which include general rotating black holes and Eguchi}Hanson instantons, are constructed. These solutions are entirely determined in terms of the Kahler potential and the Kahler connection of the underlying special geometry, where also the holomorphic sections are expressed in terms of harmonic functions. Such general class of solutions turns out to be very important in addressing questions related to the conifold transitions in type II superstrings on Calabi-Yau spaces, when they become massless [60].
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and assuming that all the scalars are constant everywhere (R zG"0 and R qS"0) and that I I consistency condition F\G"0 for unbroken supersymmetry is satis"ed. Since all the scalars are constant, the spacetime is that of extreme Reissner}Nordstrom solution: e\3"1#M/r .
(351)
By solving the equations of motion following from (292), one obtains FK"e3
2QK 2PK dtdr! r dhr sin h d . r r
(352)
From equations of motion along with (350)}(352), one obtains the ADM mass M in terms of the electric QK and magnetic PR charges of FK: M"!2 Im NKR(QKQR#PKPR) .
(353)
The ;(1) charges (PR, QK) are related to the symplectic charges Q"(qK , q )"(FK, GR) as: R qK 2PK " . (354) 2(Re NKR)PK!2(Im NKR)QK q R
), M is expressed as In terms of (qK , q R (Im N#Re N Im N\Re N)KR (!Re N ImN\)KR 1 M"! (qK , q K ) 2 (!Im N\ Re N)KR (Im N\)KR
qR q R
""Z"#" Z" . (355) ? From the consistency condition F\?"0 for unbroken supersymmetry, one has Z"0, which is ? with equivalent to R Z"0. So, the ADM mass of double extreme black holes is M""Z" ? ?
8 scalars constrained to take values de"ned by Z"0. By solving Z"0, one obtains the ? ? following relation between (q, q ) and the holomorphic section (¸K, MR): qK 2iZM ¸K "Re , (356) 2iZM MR q R
which can be solved to express (¸, M) in terms of (q, q ). Since the ADM mass M for double-extreme solutions obeys the stabilization equations (356), the entropy is related to the ADM mass as: S"pM? ,
(357)
where a"2[3/2] for the D"4 [D"5] solutions. 5.5.1. Moduli space and critical points We have seen that the BPS condition requires scalars at the event horizon take their "xed point values expressed in terms of quantized electric/magnetic charges and, thereby, the (largest) central
The other sum rule for Z and Z is "Z"!"Z ""!Q2M(F)Q. ? ?
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charge at the event horizon is related to the black hole entropy. In this subsection, we point out that such property of extreme black holes at the "xed point can be derived from bosonic "eld equations and regularity requirement of con"gurations near the event horizon without using supersymmetry [254]. For non-extreme con"gurations, the horizon area has non-trivial dependence on (continuous) scalar asymptotic values. We consider the following general form of Bosonic Lagrangian: L"(!g[!R#G R ?R @gIJ!kKRFK FR gIHgJM!lKRFK 夹FR gIHgJM] , (358) ?@ I J IJ HM IJ HM where FK ,R AK!R AK are Abelian "eld strengths with charges (pK, qK)"((1/4p)FK, IJ I J J I (1/4p) [kKR夹FR#lKRFR]) and kKR, lKR are moduli dependent matrices. We restrict our attention to static ansatz for the metric ds"e3 dt!e\3c dxK dxL , KL where for spherically symmetric con"gurations c c dq # (dh#sin h du) , c dxK dxL" KL sinhcq sinhcq
(359)
(360)
where q runs from !R (horizon) to 0 (spatial in"nity). The function ; satis"es the boundary conditions ;Pcq as qP!R and ;(0)"1. The equations of motion for ;(q) and ?(q) can be derived from the following 1-dimensional action L
d; d ? d @ #G " #e3<( , (p, q)) , ?@ dq dq dq
(361)
describing geodesic motion in a potential <"(p q)
k#lk\l lk\ k\l
k\
p q
and with the constraint
d; d ? d @ #G !e3<( , (p, q))"c . ?@ dq dq dq
where a constant c is related to the entropy S and temperature ¹ as c"2S¹. For non-extreme con"gurations, where scalars ? have non-zero scalar R? ( ?& ? #(R?/r)), the "rst law of thermodynamics is modi"ed [297] to i dA #X dJ#tK dqK#sK dpK!G ( )R@ d ? , dM" ?@ 8p
(362)
charges
(363)
whereas the Smarr formula remains in a standard form. R? vanish i! ? take the values which extremize M, i.e. double extreme solutions (i.e. ?(q)" ? ), provided < " < is non-negative ?@ ? @ (convexity condition). Here, is the Levi}Civita covariant derivative with respect to the scalar ? manifold metric G . (This can also be directly seen from (RM/R ?) "!G ( )R@.) For this ?@ ( N O ?@ case, ? have to extremize <, i.e. (R
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"xed values in terms of conserved charges (pK, qK). The bound A44p<(p, q,
), which is derived from the requirement of "nite event horizon area A together with the constraint (362), is saturated for the (double) extreme case. Since ;PMq as qP0, one obtains the following relation from (362): M#G R?R@!<( ? )"4S¹ , (364) ?@ which states that the total self-force on black holes due to the attractive forces of gravity and scalars is not exceeded by the repulsive self-force due to vectors. The net force on black holes vanishes only in the extreme case (c"0). In the double-extreme case, the ADM mass is given by < at the "xed point, i.e. M"<(p, q, ? ) with the "xed values ? of scalars determined by (R<( , p, q)/R ?) "0, since c"0"R?. From this, one obtains the bound on the ADM mass M(S, , (p, q))5M(S, , (p, q)). Note, these results are derived only from the requirement of regularity of con"gurations near the event horizon without using supersymmetry. We specialize to the case where the target space manifold is a KaK hler manifold spanned by complex scalars zG with KaK hler potential K: RK dzGdz H . G d ?d @" ?@ RzGRz H We consider the bosonic action of N"2 supergravity coupled to vector multiplets "!R#G M R zGR z HM gIJ#Im NKRFK FR gIHgJM# Re NKRFK 夹FR gIHgJM . (365) , GH I J IJ HM IJ HM The moduli dependent matrices kKR and lKR in (358) are given by l#ik"!N. So, < in (361) has the form <(p, q, ?)""Z(z, p, q)"#"D Z(z, p, q)", where Z is the central charge and D Z is its G G KaK hler covariant derivative. By applying properties of special geometries, one obtains the following ADM mass and scalar charges L
(366) M""Z"(z , p, q), RG"GGHM DM M ZM . H By applying the general results in the previous paragraph, one can see that at the critical points of < (R <"0), where RG"0, Z is extremized: D Z"0"DM M ZM . Since the second covariant derivative G G I of "Z" at the critical point coincides with the partial (non-covariant) second derivative, one has (RM M R "Z") "G M "Z" . So, when G M is positive (negative) at the critical point, M at the critical point GH G H GH reaches its minimum (maximum). Generally, when G M changes its sign and becomes negative, some GH sort of a phase transition occurs and the e!ective Lagrangian breaks down unless new massless states appear. 5.5.2. Examples In the following, we discuss the explicit expression for M in the metric component (351) with speci"c prepotentials. 5.5.2.1. Axion dilaton black holes. The axion-dilaton black holes in the SO(4) [S;(4)] formulation of pure N"4 supergravity can be regarded as black holes in N"2 supergravity coupled to one vector multiplet with the prepotential F"!iXX (without prepotential). The holomorphic sections and ;(1) charges of S;(4) theory [165] (with hats) are related to those of SO(4) theory
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[162,164,194] (without hats) as [133,136,212]: XK "X, FK "F , XK "!F , FK "X , (367) q( "q , q( "q, q( "!q, q( "q . First, we discuss the SO(4) case. We choose the gauge X"1. Then, the prepotential F"!iXX yields the KaK hler potential
1 X e)" z, 2(z#z ) X
and (356) can be solved to "x the moduli z in terms charges: q!iq . z" q!iq So, one has central charge in terms of ;(1) charges:
qq#q q (q#iq ) , Z"¸'q!M q' " ' ' (q)#(q ) by solving (356). This leads to the ADM of the double-extreme black hole:
(368)
(369)
M""Z"""qq#q q " . (370) The corresponding expressions in the S;(4) theory are obtained by applying the transformations (367). The moduli "eld and the ADM mass are [401,410] q( #iq( , M""Z"""q( q( !q( q( " . z" q( !iq(
(371)
5.5.2.2. N"2 heterotic vacua. The e!ective "eld theory of N"2 heterotic string is described by the N"2 theory with a prepotential [5,6]
L> X XX! (XG) . F"! X G This prepotential corresponds to the manifold [209,266]
(372)
SO(2,n) S;(1,1) ; SO(2);SO(n) ;(1) with the "rst factor parameterized by the axion}dilaton "eld S"!iX/X"!iz and the second factor being the special KaK hler manifold parameterized by n complex moduli zG"XG/X (i"2,2, n#1). S belongs to a vector multiplet and the remaining vector multiplets with the scalar components zG are associated with the ;(1) gauge "elds in the left moving sector of
From this expression for K, one sees that the real part of the moduli z has to be positive, leading to the constraint Re z""qq#q q ".
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heterotic string. In particular, the n"2 case is the S¹;-model [59,119] with the complex scalars S, ¹ and ; parameterizing each S¸(2, R) factor of the moduli group. S, ¹ and ; are related to zG as z"iS, z"i¹, z"i; ,
(373)
and, therefore, the prepotential takes the form: F"!S¹;.
(374)
It is convenient to apply a singular symplectic transformation [136] (de"ned as XP!F and F P!X) on (XK, FR) to bring it to the form [135]: XK XK " . (375) FR SgRKXK
In this basis, theory has a uniform weak coupling behavior as Im SPR and the holomorphic section satis"es the constraints 1X " X2"1F " F2"X ) F"0 .
(376)
Here, 1A " B2,AKgKRBR"AKgKRBR and A ) B,AKBK with
L
0
0
gKR" 0
L
0
0
0
!I
; L,
0 1 1 0
.
(377)
By solving (356), along with (375), one "xes S in terms of ;(1) symplectic charges and obtains the ADM mass of double extreme solution [178,236]: (1q "q 21q"q2!(q ) q) q ) q #i , S" 1q "q 2 1q "q 2 M""Z""(1q "q 21q"q2!(q ) q)"(Im S)1q "q 2 .
(378)
5.5.2.3. Cubic prepotential. We consider the following general form of cubic prepotential [161]: X?X@XA F"d , a, b, c"1,2, n . ?@A X
(379)
The n "3 case is the S¹; model [59,208]. By solving (356), along with (379), one obtains the ADM mass at the "xed point in the moduli space [571]:
4 1 (D x ?)!9[q (q ) q)!2D] , (380) M" 3 ? 3q where D,d q? q@ qA , D ,d q@ qA and D ,3D !q q. Here, x ? in (380) are real ?@A ? ?@A ? ? ? solutions to the system of equations: d x ?x @"D . ?@G G
(381)
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The moduli are "xed in terms of the symplectic ;(1) charges as q? 3 x ? x ? 3 [q (q ) q)!2D]# !i M . z?" q 2 (D x A) 2 q (D x A) A A When q "0, (356) can be solved explicitly to yield the ADM mass: D (qD?#12q) , M" 3 ?
(382)
(383)
where D ,d qA , D?@,[D\] and D?,D?@q. And the moduli z? take the following "xed ?@ ?@A ?@ @ point value in terms of the symplectic charges: D? q? z?" !i DM . 6 2
(384)
When the prepotential (379) has an extra topological term [54,374,451] L c ) J ?z? , 24 ? one only needs to apply the symplectic transformation [54,136] with the matrix
1
0
= 1
3Sp(2n #2) ,
where the non-zero components of =KR are = "c ) J /24. Then, the ADM mass is given by (380) ? ? or (383) with the symplectic charges (q , q) replaced by c J q "q! ?q? , 24 R ; q K"q (385) K !=RKq c J ? q "q! q . ? ? 24
5.5.2.4. CP(n!1,1) model. The S;(1, n) CP(n!1, 1), S;(n);;(1) model has the isometry group S;(1, n). The n"1 case is the axion}dilaton black holes [83,404,409,415]. The form of prepotential depends on the way in which S;(1, n, Z) is embedded into Sp(2n#2, Z) [523]. For the Sp(2n#2, Z) embedding
X"
A
B
C D
of M";#i<3S;(1, n) given by A";, C"g<,
B"
(386)
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where g is an S;(1, n) invariant metric, the holomorphic prepotential is i F" XRgX . 2
(387)
For this case, X transforms as a vector under S;(1, n), i.e. XPMX, and the moduli space is parameterized by "( ,2, L>)2 as "1/(> and ?"z?/(> with >"1! z?z ?. In ? general, the ADM mass of the extreme solution has the form [524]: "m !n t!Q AG" A GA M " A , .1 2(1!ttM ! AGA M G) G where
(388)
m ,q#iq , n ,iq !q, Q ,iqG !q, t,X/X, AG,XG/X . A A GA G For the following "xed-point values of the moduli "elds, which satisfy (356),
(389)
tM "n /m , A M G"Q /m , A A GA A the ADM mass M takes the minimum value [524]
(390)
g 0 1 M" ("m "!"n "!"Q ")"p(q q ) A GA 0 g 2 A
q
. (391) q For other embedding X "SXS\ of S;(1, n) into Sp(2n#2) related via the symplectic transformation S3Sp(2n#2), the ADM mass is given in terms of new symplectic ;(1) charges (qYq )2"(qq )S\ by [524] g 0 qY S2 . (392) M"p(qYq )S 0 g q
5.5.2.5. General quadratic prepotential. We discuss the case where the lower-component of < is proportional to the upper component [62]:
¸'
¸'
, (393) M R ¸) ( () where R "a !ib with real matrices a and b . Note, it is su$cient to consider the case '( '( '( '( '( where R "!ib , since the most general case with a O0 is achieved by applying the symplectic '( '( '( transformation "
1
0
3Sp(2n#2) a 1 '( on the con"guration with a "0. '( The ADM mass and entropy transform under this symplectic transformation similarly as (392).
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By solving (356) with (393), one obtains the ADM mass
(RM !R)\ (R!RM )\R i M" (qq ) RM (R!RM )\ RM (RM !R)\R 2 b\ !b\a 1 " (qq ) !ab\ b#ab\a 2
q
q
q
. (394) q The case R"!ig, where g is an S;(1, n) invariant metric, is the CP(n!1, 1) model, i.e. (394) reduces to (391). The most general extreme solution to this model has the form [597] ds"!e\3 dt#e3 dx ) dx , F' "e R HI ', G "e R H , IJ IJM M (IJ IJM M ( >,ZM ¸"i(R!RM )\(RM HI !H) ,
(395)
where
e3"i(H2 HI 2)
(RM !R)\
(R!RM )\R
RM (R!RM )\ RM (RM !R)\R
q' q HI '" hI '# , H " h # ( . ( ( r r
H HI
, (396)
The asymptotically #atness condition leads to the following constraint on hI ' and h in (396): ( (RM !R)\ (R!RM )\R h "!i . (397) (h2 hI 2) RM (R!RM )\ RM (RM !R)\R hI
5.6. Quantum aspects of N"2 black holes Supersymmetric "eld theories respect remarkable perturbative non-renormalization theorems. In N"1 theories, superpotentials are not renormalized in perturbation theory [321,375]. N54 theories are "nite to all orders in perturbative quantum corrections [473,573]. So, the classical BPS solutions in N54 theories are exact to all orders in perturbative corrections. (Cf. Some classical solutions of N"4 theories are also exact solutions [47,48,173,174,369}371,518,607,608,614,615] of conformal p-model of string theory and, therefore, exact to all orders in a -corrections.) For N"2 theories, prepotentials, which determine the Lagrangians, receive perturbative quantum corrections up to one-loop level [9,209,321,548]. Hence, one has to study quantum e!ect on prepotentials for complete understanding of solutions in N"2 theories. In the following, we study the quantum aspects of black holes in the e!ective N"2 theories of compacti"ed superstring theories. It is conjectured that the E ;E heterotic string on K3;¹ and the type-II string on a Calabi}Yau manifold are a N"2 string dual pair [13,14,25,119, 257,396,397,418]. Since the dilaton}axion "eld S belongs to a vector multiplet (hyper multiplet) of
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the heterotic (type-II) theory, moduli space of hyper multiplet (prepotential for a vector multiplet) is exact at the tree level, due to the absence of neutral perturbative couplings between vector multiplets and hypermultiplets [13,118,167,168,418,435]. Thus, applying the duality between heterotic and type-II strings, one can compute the exact prepotential for vector multiplets (hyper multiplet superpotential) of the heterotic (type-II) theory. The prepotentials of the N"2 e!ective "eld theories of these string theories contain the cubic terms: F(X)"d X?X@XA/X , (398) ?@A plus correction terms that include part of quantum corrections, instanton and topological terms which cannot be included in (398). For the type-IIA string on a Calabi}Yau three-fold, real coe$cients d are the topological intersection numbers d , J J J , where J 3H (>, Z) ?@A ?@A ? @ A ? are the KaK hler cone generators. For the heterotic string on K3;¹, d describe the classical parts ?@A and the non-exponential parts of perturbative corrections to the prepotential. The KaK hler potential associated with (398) is K(z, z )"!log(!id (z!z )?(z!z )@(z!z )A) . (399) ?@A General double-extreme black holes and a special class of extreme black holes with non-constant scalars of the N"2 theory with (398) are discussed in [50]. In the following, we study the e!ect of the quantum correction terms of the prepotential on the classical solutions [50,51,54,123,514]. 5.6.1. E ;E heterotic string on K3;¹ At generic points in moduli space, the E ;E heterotic string on K3;¹ is characterized by 65 gauge-neural hypermultiplets (20 from the K3 surface and 45 from the gauge bundle) and 19 vectors (18 from vector multiplets and 1 from the gravity multiplet). So, the moduli z? (a"1,2, n ) in the vector multiplets consist of the axion}dilaton S, the ¹ moduli ¹ and ;, and Wilson lines
"i
(400)
We denote the moduli other than S as ¹K (m"2,2, n ). The number of Wilson lines with one of ;(1) factors coming from the gravity multiplet) depends on the choice of S;(2) bundles with instanton numbers (d , d )"(12!n, 12#n) [4,116,397]. For example, the S¹; model (i.e. the complete Higgsing of the D"6 gauge group E ;E ) is possible for n"0, 1, 2. Since prepotentials of N"2 theories are not renormalized beyond one-loop perturbative levels, the prepotentials are written in the form [9,10,136,209]: F"F#F#F,. ,
(401)
The scalar components of the remaining hypermultiplets in 56 of E are not spectrum-preserving moduli, since their non-zero vacuum expectation values induce mass for some of the non-Abelian "elds, resulting in change in the spectrum of light particles in the e!ective theories.
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where F is the tree-level prepotential and F (F,.) is the one-loop (non-perturbative) correction. The classical moduli space of the E ;E heterotic string on K3;¹ is [132,135,136,255,263,266,272] SO(2, n !1) S;(1, 1) SO(2);SO(n !1) ;(1) with S residing in the separate moduli space S;(1, 1) . ;(1) The tree-level prepotential is
X LT F"! XX! (XG) "!S ¹;! (
(402)
K"!log (S#SM )!log (¹#¹M )(;#;M )! (
(404) F"!S ¹;! (
(405)
I J @
where c"c (0)f(3)/8p and e[x]"exp 2pix. Here, c (4kl!b) are the expansion coe$cients of L L particular Jacobi modular form [117] and p (¹, ;, <) is the one-loop cubic polynomial, which L depends on the particular instanton embedding n, given by [70,117,451] p (¹, ;, <)"!;!(#n)<#(1#n);<#n¹< . L
(406)
5.6.1.1. Perturbative corrections. In Section 5.5.2, we obtained the general expression for entropy (or the ADM mass of the double extreme black hole) in the tree level e!ective theory of the
This is possible for the instanton embedding with n"0, 1, 2.
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heterotic string on K3;¹. (See (378) for the tree level expression.) The entropy depends on the symplectic charge vector Q"(q q) through the full triality invariant form D"1q "q 21q"q2!(q ) q) [123] and is invariant under T-duality since the dilaton Im S and the combination 1q "q 2 remain intact under T-duality [123,136,209]. Once perturbative quantum corrections are considered, T-duality transformations get modi"ed due to the one-loop corrections to the prepotential:
F"F#F"!S ¹;! (
(409)
K where F K ,RF/RX . Note, F keeps the classical value. Whereas XK K transforms exactly the same way as in classical theory, the T-duality transformation rules of FK K get modi"ed at one-loop level due to the modi"cation of prepotential [9,121,209,338]:
XK KP;K KRXK R, FK KP[(;K 2)\]RKFK R#[(;K 2)\C]KRXK R ,
(410)
where ;3SO(2, 2#n, Z) and the symmetric integral matrix C encodes the quantum corrections. From the relation X"!FK "!iSXK , one sees that S is no longer invariant under the perturbative T-duality transformation (410), but transforms as [209] i[(;K 2)\]R (H #CRDXK D) RK SPS# . ;K KXK
(411)
Note, 1q "q 2 in the classical expression (378) is still invariant under the perturbative T-duality, but the dilaton Im S transforms under the perturbative T-duality (411). Since superstring theories are exact under T-duality order by order in perturbative corrections, one expects entropy to be invariant under T-duality. One way of making entropy to be manifestly invariant under T-duality is to introduce the invariant dilaton}axion "eld S [209,338] which do not transform under T-duality. This motivates the following conjectured expression for entropy at one-loop level [123]: S "p"Z ""p(Im S )1p "p 2 .
(412)
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The perturbative dilaton Im S is understood as follows. The perturbative prepotential (407) leads to the following perturbative KaK hler potential [209]: K"!log[(S#SM )#< (¹K,¹M K)]!log[(¹K#¹M K)g (¹L#¹M L)] , (413) %1 KL where 2(h#hM )!(¹K#¹M K)(R Kh#R M KhM ) 2 2 (414) < (¹K, ¹M K)" %1 (¹K#¹M K)g (¹L#¹M L) KL is the Green}Schwarz term [124,203,450] describing the mixing of the dilaton with the other moduli ¹K. From this expression for K, one infers that the true string perturbative coupling constant g is of the modi"ed form: 1 4p " (S#SM #< (¹K, ¹M K)) . (415) %1 2 g One can prove this conjectured form (412) of the perturbative entropy by solving (356) with (407) substituted. In the following, as examples of quantum corrections to N"2 black holes, we "nd explicit expressions of entropy for the axion-free (Re z?"0) solution with special forms of the perturbative prepotential. We assume that 2>!ip">#>M ,jO0. Note, the symplectic magnetic charge in the perturbative basis de"ned by (408) is expressed in terms of the charges pH and qR in the original basis as q( "(p, q , p,2). By solving the stability equation P!PM "iQ with the following perturbative prepotential: d >?>@>A #ic(>), >K,ZM XK , F(>)" ?@A >
(416)
one obtains the following expression for the entropy (cf. See (323)):
S (p) "!2(q !2cj) j# . p j
(417)
Here, j in (417) is obtained by solving the following equation derived from (356): d p?p@pA 3p q " ?@A #2cj, q "! d p@pA . ? j j ?@A
(418)
For the case cpO0, one can express j in terms of charges as 3pq #p?q ?, j" 6cp
(419)
from which one sees that charges satisfy the following constraint when cp"0: 3pq #p?q "0 . ? For the case c"0 and q O0, j is d p?p@pA q , j"$ ?@A
(420)
(421)
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with the sign $ determined by the condition q j(0 that the entropy should be positive. As a special case, when c"p"0 the entropy is S "2(q d p?p@pA , ?@A p
(422)
with the solution having only n #1 independent charges, i.e. q "0. In particular, with the cubic ? prepotential (>) >>> #a , F(>)"!b > > entropy (422) becomes S "2(!q (bppp!a(p)) . p
(423)
The explicit solution with non-constant scalar "elds is obtained by just substituting the symplectic charges by the associated harmonic functions in the stabilization equations. For the case where the prepotential is the general cubic prepotential, i.e. (416) with c"0, the solution has the form [50]: ds"!e\3 dt#e3 dx ) dx, e3"(H d H?H@HA , ?@A F? "e R H?, F "R (H )\, z?"iH H?e\3 , KL KLN N K K where the harmonic functions in the solution are
(424)
p? q H?"(2 h?# , H "(2 h # . r r
(425)
Here, the constants h's are constrained to satisfy the asymptotically #at spacetime condition: 4h C h?h@hA"1 , ?@A and, therefore, the ADM mass of the solution is
(426)
q 1 M" # p?h C h@hA . (427) 4h 2 ?@A When h's take the values at the horizon expressed in terms of charges as h "q /c and h?"p?/c, the ADM mass (427) reduces to the entropy S"pM" in (422).
As a special case, we consider the following prepotential corresponding to the S¹; model with the cubic quantum correction: (X) XXX #a . F" X X
(428)
For this case, the metric component has the form
q e3"4 h # r
p h! r
p h! r
p p h# #a h# . r r
(429)
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Note, the quantum correction term acts as a regulator, smoothing out singularity, for example those of massless black holes [48,49,179,318,406,408,488,578], of classical solutions. 5.6.2. Type-II string on Calabi}Yau manifolds Type-IIA string on a Calabi}Yau three-fold [115,373] gives rise to N"2 theory with h #1 vector "elds and h #1 hypermultiplets (h and h being the Hodge numbers of the three-fold), with the additional hyper multiplet and vector "eld coming from those associated with the dilaton and the gravity multiplet, respectively. The moduli in the vector multiplets consist of the KaK hlerclass moduli t? (a"1,2, n "h ), where h "dim(H (M, Z)). The general form of type-IIA prepotential is [115,373]
sf(3) 1 1 # nP ¸i e i d t? , (430) F''"! C t?t@tA! ?@A ? 2(2p) (2p) 2 B 2 BF 6 B BF ? where nP 2 F are the rational instanton numbers of genus 0 and s is the Euler number. Here, the B B prepotential is de"ned inside of the KaK hler cone p(K)"+ t?J " t?'0,, where J are the (1, 1)? ? ? forms of the Calabi}Yau 3-fold M. In the large KaK hler class moduli limit (t?PR), only the classical part in the prepotential, which is related to the intersection numbers, survives. To the general form (430) of the prepotential, one can add an extra topological term which is determined by the second Chern class c of the Calabi}Yau 3-fold: F c )J (431) F'' " ?t?; c ) J , c J . ? ? 24 + ? The e!ect of adding such term to the prepotential is the symplectic transformation corresponding to constant shift of the h-angle [630] (cf. the paragraph below (384)). In particular, the type-IIA model dual to the Heterotic string on K3;¹ with n "4 and n"2 (an S¹;< model) [117] corresponds to the compacti"cation on the Calabi}Yau three-fold P (20) [397] with h "4 and Euler number s"!372 [71]. The transformations be tween the moduli in the pair of these type-IIA and heterotic theories are
t";!2<, t"S!¹, t"¹!;, t"< ,
(432)
and some of the instanton numbers [71,117] and the Euler number of the threefold are given in terms of the quantities of the heterotic string by: nP "!2c (4kl!b), s"2c (0) . J>I I J>I>@ The cubic intersection-number part of the prepotential is
(433)
!F'' "t((t)#tt#4tt#2tt#3(t))#(t)#8(t)t#t(t)#2(t)t #8ttt#2(t)t#12t(t)#6t(t)#6(t) . (434) In the limit t"<"0 (i.e. the S¹; model of the heterotic string with n"2), the model reduces to the type-IIA string compacti"ed on P (24) with h "3 and s"!480. The linear topological term, for this case, takes the form: 23 11 c )J t " t#t#2t"S#¹# ; . 24 6 6
(435)
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5.6.2.1. Entropy formula. For black holes in type-IIA string on a Calabi}Yau 3-fold, entropy depends not only on ;(1) charges, but also on the topological quantities of the 3-fold. In the following, we consider the double-extreme black holes in the type-IIA superstring on a Calabi}Yau 3-fold with the following special form of prepotential: C >?>@>A c ) J # ?>>? , F''(>)"! ?@A 24 6>
(436)
where >?,ZM X?. Note, the prepotential is determined by the classical intersection numbers C "!6d and the expansion coe$cients c ) J "24= of the 2nd Chern class c of the ?@A ?@A ? ? 3-fold. Note, the above form of prepotential can be obtained by imposing the symplectic transformation of the following form on the prepotential without 2nd Chern class terms:
p K q R
"
0
pK
= 1
qR
1
.
(437)
The general expression for the entropy with p"q "0 and = "0 is obtained in (422). By ? ? imposing the symplectic transformation (437), one obtains the following entropy for the type-IIA theory with the prepotential (436) and the charge con"guration p"q "0: ? S "2((q != p ?)d p @p Ap B . ? @AB p
(438)
5.6.3. Higher-dimensional embedding The above D"4 black holes in string theories with the cubic prepotential arise from the compacti"cation of the following intersecting M-brane solution
1 1 du dv#H du# C H?H@HA dx#H?u , ds " ? (C H?H@HA) 6 ?@A ?@A
(439)
due to the duality [635] among the heterotic string on K3;¹, the type-II string on a Calabi}Yau 3-fold (C>) and M-theory on C>;S [103,261,262]. This solution corresponds to 3 M 5-branes (with the corresponding harmonic functions H?, a"1, 2, 3) intersecting over a 3-brane (with the spatial coordinates x), and the momentum (parameterized by the harmonic function H ) #owing along the common string. Here, the intersection of 4-cycles that each M 5-brane wraps around is determined by the parameters C and each pair of such 4-cycles intersect over the 2-dimensional ?@A line element u . ? Compactifying the internal coordinates and the common string direction, one obtains the following D"4 solution with spacetime of the extreme Reissner}Nordstrom black hole:
1 1 dt# ! HC H?H@HA dx . ds"! ?@A 6 (!HC H?H@HA ?@A
(440)
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6. p-branes The purpose of this section is to review the recent development in p-branes and other higherdimensional con"gurations in string theories. (See also [284] for another review on this subject.) Study of p-branes plays an important part in understanding non-perturbative aspects of string theories in string dualities. The recently conjectured string dualities require the existence of p-branes within string spectrum along with well-understood perturbative string states. Furthermore, microscopic interpretation of entropy, absorption and radiation rates of black holes within string theories involves embedding of black holes in higher dimensions as intersecting p-brane. This section is organized as follows. In Section 6.1, we summarize properties of single-charged p-branes. In Section 6.2, we systematically study multi-charged p-branes, which include dyonic p-branes and intersecting p-branes. In the "nal section, we review the lower-dimensional p-branes and their classi"cation. We also discuss various p-brane embeddings of black holes. 6.1. Single-charged p-branes In this section, we discuss p-branes which carry one type of charge. Such single-charged p-branes are basic constituents from which `bound statea multi-charged p-branes, such as dyonic p-branes and intersecting p-branes, are constructed. p-branes in D dimensions are de"ned as p-dimensional objects which are localized in D!1!p spatial coordinates and independent of the other p spatial coordinates, thereby having p translational spacelike isometries. Note, the allowed values of (D, p) for which supersymmetric p-branes exist are limited and can be determined by the bose-fermi matching condition [2]. (The details are discussed in Section 2.2.3.) The e!ective action for a single-charged p-brane has the form:
1 1 1 d"x (!g R! (R )! e\?N(F , (441) I (p)" N> " 2 2(p#2)! 2i " where i is the D-dimensional gravitational constant, is the D-dimensional dilaton and " F ,dA is the "eld strength of (p#1)-form potential A . Here, the parameter a(p) given N> N> N> below is determined by the requirement that the e!ective action (441) and the p-model action (443) scale in the same way [242] under the rescaling of "elds 2(p#1)(p #1) 2(p#1)(p #1) "4! , a(p)"4! D!2 p#p #2
(442)
where p ,D!p!4 corresponds to spatial dimensions of the dual brane. 6.1.1. Elementary p-branes Electric charge of (p#1)-form potential A in I (p) is carried by the `elementarya p-brane. N> " The `elementarya p-brane has a d-function singularity at the core, requiring existence of singular electric charge source for its support so that equations of motion are satis"ed everywhere. Namely, the electric charge carried by the `elementarya p-brane is a Noether charge with the Noether
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current associated with the p-brane worldvolume p-model action:
1 (p!1) (!c S "¹ dN>m ! (!ccGHR X+R X,g e?N( N># G H +, N N 2 2
1 ! eG2GN>R X+2R N>X+N>A 2 N> , G G + + (p#1)!
(443)
where X+(mG) (M"0, 1,2, D!1; i"0, 1,2, p) is the spacetime trajectory of p-branes, ¹ is N the elementary p-brane tension, c (m) (g (X)) is the worldvolume (spacetime) metric and GH +, in (443) is related to the canonical metric A "A 2 N> dX2dXN>. The metric g +, N> + + g in the Einstein-frame e!ective action (441) through the Weyl-rescaling g "e?N( N>g . +, +, +, The `elementarya p-brane is a solution to the equations of motion of the combined action I (p)#S . In particular, "eld equation and Bianchi identity of A are " N N> d夹(e\?N(F )"2i (!1)N> 夹J , dF "0 , (444) N> " N> N> where the electric charge source current J "J+2+N>dX 2dX N> is + + N> d"(x!X) J+2+N>(x)"¹ dN>m eG2GN>R X+2R N>X+N> . (445) N G G (!g
Here, 夹 denotes the Hodge-dual operator in D dimensions, i.e. 1 (夹<)+2+"\B, e+2+"< "\B>2 " + + d! with the alternating symbol e+2+" de"ned as e2"\"1. The Noether electric charge is
1 (夹J) " e\?N( 夹F , (446) "\N\ (2i N > N> +"\N\ 1 " where SN> surrounds the elementary p-brane. In solving the Euler}Lagrange equations of the combined action I (p)#S to obtain the " N elementary p-brane solution, one assumes the P ;SO(D!p!1) symmetry for the con"guraN> tion. Here, P is the (p#1)-dimensional Poincare group of the p-brane worldvolume and N> SO(D!p!1) is the orthogonal group of the transverse space. Accordingly, the spacetime coordinates are splitted into x+"(xI, yK), where k"0,2, p and m"p#1,2, D!1. Due to the P invariance, "elds are independent of xI, and SO(D!p!1) invariance further requires that N> this dependence is only through y"(d yKyL. In solving the equations, it is convenient to make KL the following static gauge choice for the spacetime bosonic coordinates X+ of the p-brane: Q" N
XI"mI, >K"constant ,
(447)
where k"0, 1,2, p (m"p#1,2, D!1) corresponds to directions internal (transverse) to the p-brane. The general Ansatz for metric with P ;SO(D!p!1) symmetry is N> ds"e Wg dxI dxJ#e Wd dyK dyL . (448) IJ KL
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By solving the Euler}Lagrange equations with these AnsaK tze, one obtains the following solution for the elementary p-brane: ds"f (y)\N > N>N >g dxI dxJ#f (y)N> N>N >d dyK dyL , IJ KL e("e(f (y)\?N , A
I2IN>
e?N ( "! e 2 N> f (y)\ , I I Ng
(449)
where Ng is the determinant of the metric g and f (y) is given by IJ
2e?N (i ¹ 1 " N 1# , p '!1 , ( pJ #1)X yNJ > N > f (y)" e?N (i ¹ " N ln y, p "!1 . 1! n
(450)
By solving the Killing spinor equations with the P ;SO(D!p!1) symmetric "eld AnsaK tze, N> one sees that the extreme `elementarya p-brane preserves 1/2 of supersymmetry with the Killing spinors satisfying the constraint: (1!CM )e"0 ,
(451)
where 1 CM , eGG2GNR X+R X+2R NX+NC 2 N , G G G ++ + (p#1)!(!c
(452)
which has properties CM "1 and Tr CM "0 that make (1$CM ) a projection operator. This orig inates from the fermionic i-symmetry of the super-p-brane action. The extreme `elementarya p-brane (449) saturates the following Bogomol'nyi bound for the mass per unit area M " d"\N\y h : N 1 i M 5 "Q "e?N( , " N (2 N
(453)
where h is the total energy-momentum pseudo-tensor of the gravity-matter system. +, 6.1.2. Solitonic p -branes The magnetic charge of A is carried by a solitonic p -brane, which is topological in nature and N> free of spacetime singularity. Since magnetic charges can be supported without source at the core, solitonic p -branes are solutions to the Euler}Lagrange equations of the e!ective action I (p) alone. " is de"ned as The `topologicala magnetic charge P of A N> N
1 P" F , N (2i N> N> 1 "
(454)
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where SN> surrounds the solitonic p -brane. The magnetic charge P is quantized relative to the N electric charge Q via a Dirac quantization condition: N n Q P N N" , n3Z . (455) 2 4n Solitonic p -brane solution has the form: ds"g(y)\N> N>N >g dxI dxJ#g(y)N > N>N >d dyK dyL , IJ KL (456) e("e( g(y)?N , F "(2i P e /X , N> " N N> N> where e is the volume form on SN> and g(y) is given by N> (2e\N>( ?NN>N >i P 1 " N . (457) g(y)"1# yN> (p#1)X N> The `solitonica p -brane (456) preserves 1/2 of supersymmetry and saturates the following Bogomol'nyi bound for the ADM mass per unit p -volume: 1 M 5 "P "e\?N( . N (2 N
(458)
Note, the sign di!erence in dependence of the mass densities (453) and (458) on the dilaton asymptotic value . So, in the limit of large , the mass density M (M ) is large (small), and vice N N versa. 6.1.3. Dual theory We consider the theory whose actions II (p ) and SI are given by (441) and(443) with p replaced by " N p "D!p!4. So, in this new theory, p -branes carry electric charge QI and p-branes carry N magnetic charge PI of (p #1)-form potential AI . N N> Since a p-brane from one theory and a p -brane from the other theory are both `elementarya (or `solitonica), it is natural to assume that these branes are dual pair describing the same physics. One assumes that the graviton and the dilaton in the pair actions are the same, but the "eld strengths are related by FI "e\?N( 夹F . (Thereby, the role of "eld equations and F and FI N> N> N> N> Bianchi identities are interchanged.) Then, it follows that since Q "PI and P "QI the Dirac N N N N quantization conditions for electric/magnetic charge pairs (Q , P ) and (QI , PI ) lead to the following N N N N quantization condition for the tensions of the dual pair `elementarya p-brane and p -brane: (459) i ¹ ¹ ""n"p . " N N By performing the Weyl-rescaling of metrics to the string-frame, one sees that the p-brane (p -brane) loop counting parameter g (g ) is N N g "e"\?N( N>, g "e"\?N ( N > , (460) N N with a(p )"!a(p). It follows that the brane loop counting parameters of the dual pair are related by (461) (g )N>"1/(g )N > . N N
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Thus, the strongly (weakly) coupled p-branes are the weakly (strongly) coupled p -branes. In particular, a string (p"1) in D"6 are dual to another string (p "6!1!4"1), thereby strongly (weakly) coupled string theory being equivalent to weakly (strongly) coupled dual string theory. Other interesting examples are membrane/"vebrane dual pair in D"11, string/"vebrane dual pair in D"10 [219,239,240], self-dual 3-branes in D"10 and self-dual 0-branes in D"4. Note, in D"11 supergravity strong and weak coupling limits do not have meaning due to absence of the dilaton. 6.1.4. Blackbranes We discuss non-extreme generalization of BPS `solitonica p -brane in Section 6.1.2. Such solution is obtained [464] by solving the Euler}Lagrange equations following from I (p) with " R;SO(p#3);E(p ) symmetric "eld AnsaK tze. The non-extreme p -brane solution is ds"!D D\N > N>N > dt#D\D?N N>\ dr > \ > \ #rD?N N> dX #DN> N>N > dxG dx , \ N> \ G e\("D?N, F "(p#1)(r r )N> e , (462) \ N> >\ N> where D ,1!(r /r)N> and i"1,2, p . The magnetic charge P and the ADM mass per unit ! ! N p -volume M of (462) are N X X (463) P " N> (p#1)(r r )N> , M " N>[(p#2)rN>!rN>] . >\ N > \ N (2i 2i " " The solution (462) has the event horizon at r"r and the inner horizon at r"r , and therefore is > \ alternatively called `blacka p -brane. The requirement of the regular event horizon, i.e. r 5r , > \ leads to the Bogomol'nyi bound (2M 5"P "e\?N( . N N In the limit r "0, and A become trivial, i.e. P "0, and spacetime reduces to the product \ N> N of (D!p )-dimensional Schwarzschield spacetime and #at RN . In the extreme limit (r "r ), the > \ symmetry is enhanced to that of the BPS p -brane, i.e. P ;SO(p#3), since g "g G G. Such RR VV N> extreme solution is related to the BPS solution (456) through the change of variable yN>"rN>!rN>. In the extreme limit, the event-horizon and the singularity completely disap\ pear, i.e. becomes soliton with geodesically complete spacetime in the region r'r "r . > \ 6.2. Multi-charged p-branes In this section, we discuss p-branes carrying more than one types of charges. These p-branes are `bound statesa of single-charged p-branes. Multi-charged p-branes are classi"ed into two categories, namely `marginala and `non-marginala con"gurations. The `marginala (BPS) bound states have zero binding energy and, therefore, the mass density M is sum of the charge densities Q of the G constituent p-branes, i.e. M" Q . Such bound states with n constituent p-branes preserve at least G G ()L of supersymmetry. The marginal bound states include intersecting and overlapping p-branes. The `non-marginala (BPS) bound states have non-zero binding energy and the mass density of the
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form M" Q. The quantized charges Q of `non-marginala bound states take relatively prime G G G integer values. In general, non-marginal p-brane bound states are obtained from single-charged p-branes or marginal p-brane bound states by applying the S¸(2, Z) electric/magnetic duality transformations and, therefore, preserve the same amount of supersymmetries as the initial p-brane con"gurations (before the S¸(2, Z) transformations). In particular, intersecting p-branes are further categorized into orthogonally intersecting p-branes and p-branes intersecting at angles. 6.2.1. Dyonic p-branes In D"2p#4, i.e. dimensions for which p"p , p-branes can carry both electric and magnetic charges of A . Examples are dyonic black holes (p"0) in D"4; dyonic strings (p"1) in D"6; N> dyonic membranes (p"2) in D"8. Such dyonic p-branes can be constructed by applying the D"2(p#2) S¸(2, Z) electric/magnetic duality transformations on single-charged p-branes. #+ These dyonic p-branes have P ;SO(D!p!1) symmetry and are characterized by one harN> monic function, just like single-charged p-branes, since the S¸(2, Z) transformations leave the Einstein-frame metric intact. In particular, in D"2 mod 4, the (p#2)-form "eld strengths satisfy a real self-duality condition F "!夹F and, thereby, electric and magnetic charges are N> N> identi"ed, i.e. Q "!P . N N The S¸(2, Z) electric/magnetic duality transformations of 2k-form "eld strength F in D"4k #+ I can generally be understood as the ¹ moduli transformations of D"(4k#2) theory compacti"ed on ¹ [311]. We consider the following D"(4k#2) action
I " dI>x (!gR#a [dCH#H 夹H] , I>
(464)
where C is a (4k#2)-form potential with the "eld strength H"dC. We compactify the action (464) on ¹ with the following AnsaK tze for the "elds 1 ("q" dy#2Re q dx dy#dx) , ds(M )"ds(M )# I> I Im q C"B dy#A dx,
H"G dy#F dx ,
(465)
where q is the moduli parameter of ¹, (x, y) are coordinates of ¹ (i.e. x&x#1 and y&y#1), and A, B [F, G] are (2k!1)-forms [2k-forms] in D"4k. By applying the self-duality condition [625] of H, one "nds that 2k-form G is an auxiliary "eld, which can be eliminated by its "eld equation as G"Im q 夹F!Re q F, and one "nds that F" dA. The "nal expression for the D"4k action is
1 dq dq #a FG , I " dIx (!g R! I 2 (Im q)
(466)
where a real constant a is, in the convention of [388], given by a"2[(2k)!]\. The complex scalar q, which is expressed in terms of real scalars o and p as q"2o#ie\N, parameterizes the target space manifold M"S¸(2, Z)!S¸(2, R)/;(1), which is the fundamental domain of S¸(2, Z) in the upper half q complex plane. Here, ;(1) is a subgroup of S¸(2, R) which preserves the vacuum
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expectation value 1q2. The "eld equations are invariant under the following S¸(2, R) electric/ #+ magnetic duality transformation of F:
a b aq#b ; A" 3S¸(2, R) . (F, G)P(F, G)A\, qP cq#d c d
(467)
Note, this S¸(2, Z) transformation is not S-duality of string theories, since p is not the D"4k #+ dilaton. (In (464), the dilaton is set to zero.) The following electric Q and magnetic P charge densities form an S¸(2, R) doublet: #+ 1 1 G, P" F. (468) Q" X X I I This S¸(2, R) transformation on single-charged (2k!2)-branes yields (2k!2)-branes which #+ carry both electric and magnetic charges of A . Such dyonic p-branes preserve 1/2 of superI\ symmetry. Charges (Q, P) and (Q , P ) of two dyonic (2k!2)-branes satisfy the generalized Nepomechie-Teitelboim quantization condition [482,591]:
QP !Q P3Z .
(469)
When such dyonic (2k!2)-branes are uplifted to D"(4k#2) through (465), the solutions become self-dual (2k!1)-branes [230,238]. The electric and magnetic charges of the dyonic (2k!2)-branes are interpreted as winding numbers of the self-dual (2k!1)-branes around the x and y directions of ¹. The dyonic (2k!2)-branes (k"1, 2) uplifted to D"11 describe p-brane which interpolates between the M 2-brane and the M 5-brane, i.e. a membrane within a 5-brane (2"2 , 5 ). + + In the following, we speci"cally discuss the k"2 case [311,388]. The associated action ((466) with k"2) is type-IIB e!ective action (155) consistently truncated and compacti"ed to D"8. The N"2, D"8 supergravity has an S¸(3, R);S¸(2, R) on-shell symmetry, whose S¸(3, Z);S¸(2, Z) subset is the conjectured U-duality symmetry of D"8 type-II string. The R-R 4-form "eld strength and its dual "eld strength transform as (1, 2) under S¸(3, R);S¸(2, R). This U-duality group contains as a subset the SO(2, 2, Z),[S¸(2, Z);S¸(2, Z)]/Z T-duality group. The S¸(2, Z) factor (in S¸(3, Z);S¸(2, Z)) is the electric/magnetic duality (467), which is a subset of `perturbativea T-duality group. All the `non-perturbativea transformations are contained in the S¸(3, Z) factor. The following dyonic membrane solution with 1q2"i is obtained by applying the ;(1)L S¸(2, R) transformation with a parameter m to purely magnetic membrane: #+ ds "H\ ds(M)#H ds(E) , F " cos m(夹 dH)# sin m dH\e(M) , sin(2m)(1!H)#2iH , (470) q" 2(sinm#H cosm) where ds(M) [ds(E)] is the metric of D"3 Minkowski space M (the "ve-dimensional Euclidean space E), e(M) is the volume form of M, 夹 is the Hodge-dual operator in E and H is a harmonic function given by (457) with p"2. Here, ;(1) is the subgroup of S¸(2, R) that #+ preserves 1q2"i. In the quantum theory, the ;(1) group breaks down to Z due to Dirac
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quantization condition, resulting in either electric or magnetic solutions when applied to purely magnetic solution. (But the solution in (470) with an arbitrary m satis"es the Euler}Lagrange equations following from (466) and, therefore, can be taken as an initial solution to which the `integer-valueda duality transformations are applied.) To generate dyonic solutions with an arbitrary 1q2 and are relevant to the quantum theory, one has to apply the full S¸(2, R) transformation to (470). The pair of electric and magnetic charges of #+ such dyonic solutions take co-prime integer values. The ADM mass density of this extreme dyonic membrane saturates the Bogomol'nyi bound: M5[e6N7(Q#21o2P)#e\6N7P] ,
(471)
and therefore 1/2 of supersymmetry is preserved. When uplifted to D"10, this dyonic membrane becomes the following self-dual 3-brane of type-IIB theory [238]: ds "H\ [ds(M)#dv]#H [ds(E)#du] , (472) F "( 夹 dH) du# dH\e(M) dv , where the coordinates (u, v) are related to the coordinates (x, y) in (465) through (y, x)"(v, u)A\ (A3S¸(2, Z)). The electric and magnetic charges of the above D"8 dyonic membrane are respectively interpreted as the winding numbers of this D"10 self-dual 3-brane around x and y directions of ¹. The D"8 dyonic membrane (470) uplifted to D"11 is a special case of orthogonally intersecting M-brane interpreted as a membrane within a 5-brane (2"2 , 5 ): + + ds "H (sinm#H cosm) [H\ ds(M)#(sinm#H cosm)\ ds(E)#ds(E)] , 1 1 3 sin 2m F" cos m 夹 dH# sin m dH\e(M)# dHe(E) . (473) 2 2 2[sin m#H cos m] This M-brane bound state interpolates between the M 5-brane and the M 2-brane as m is varied from 0 to p/2. As long as magnetic charge is non-zero (mO0), (473) is non-singular, thereby singularity of the D"8 dyonic membrane (470) is resolved by its interpretation in D"11. By compactifying an extra spatial isometry direction of (473) on S, one obtains 3 di!erent types of dyonic p-branes in type-IIA theory: (i) a membrane within a 4-brane (2"2, 4), (ii) a membrane within a 5-brane (2"2, 5) and (iii) a string within a 4-brane (1"1, 4). One can construct dyonic p-branes in DO2(p#2) by compactifying purely electric or purely magnetic p-branes down to D"2(p#2) (or D"2(p #2)), applying electric/magnetic duality transformations of p-branes (or p -branes) in D"2(p#2) (or D"2(p #2)), and then uplifting the
The above procedure can be applied to intersecting p-branes to generate p-brane bound states which interpolate between di!erent intersecting p-branes, e.g. the interpolation between the intersecting two (p#2)-branes and the intersecting two p-branes; the interpolation between the intersecting p-brane and (p#2)-brane and the intersecting (p#2)-brane and p-brane [151].
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dyonic solution to the original dimensions. In particular, the Z subset of the D"2(p#2) (or D"2(p #2)) electric/magnetic duality transformations relates electric p-brane and magnetic p-brane. Thus, the elementary p-brane in DO2(p#2) is interpreted as the electrically charged partner of magnetic p-brane, establishing electric/magnetic duality between electric `elementarya p-brane and magnetic `solitonica p-brane in DO2(p#2). To enlarge this Z electric/magnetic duality symmetry in D"2(p#2) (or D"2(p #2)) to the S¸(2, Z) symmetry so that one can generate dyonic p-branes from single-charged p-branes, one has to turn on a pseudo-scalar "eld. For example, the dyonic 5-brane in D"10 type-IIB theory is constructed in [72] by applying the S¸(2, Z) ;S¸(2, Z) transformation of the truncated type-IIB theory in D"6. (This '' #+ SO(2, 2),S¸(2, Z) ;S¸(2, Z) group is a subgroup of SO(5, 5) U-duality group of type-II string '' #+ on ¹.) There, it is found out that non-zero R-R "elds (which are related to the pseudo-scalar "eld) are needed for the solution to have both electric and magnetic charges. The type-IIB S¸(2, Z) S-duality transformation leads to dyonic p-brane whose electric and '' magnetic charges coming from di!erent sectors (NS-NS/R-R) of string theory. General dyonic solutions where form "elds carry both electric and magnetic charges are generated by additionally applying the S¸(2, Z) electric/magnetic transformation in D"6 [72]. In particular, the elec#+ tric/magnetic duality transformation that relates the solitonic 5-brane and elementary 5-brane is the product of (Z ) and (Z ) transformations. Such dyonic 5-brane solutions preserve 1/2 of '' #+ supersymmetry. We comment on generalization of dyonic p-branes discussed in this section. In [230], general dyonic p-branes within consistently truncated heterotic string on ¹, where truncated moduli "elds are parameterized by 3 complex modulus parameters ¹G (i"1, 2, 3) of 3 ¹ in ¹, is constructed. Thereby, the O(6, 22, Z) T-duality symmetry of heterotic string on ¹ is broken down to S¸(2, Z). The general class of multi-charged p-brane solution is then characterized by harmonic functions each associated with ¹G and the dilaton}axion scalar S. Such solutions break more than 1/2 of supersymmetry. With trivial S, ()L of supersymmetry is preserved for n non-trivial ¹G. With non-trivial S, additional 1/2 of supersymmetries is broken unless all of ¹G are non-trivial. With all ¹G non-trivial, () or none of supersymmetry is preserved, depending on the chirality choices. In particular, a special case of general class of solution with S and only one of ¹G non-trivial corresponds to generalization of D"6 self-dual dyonic string, when such solution is uplifted to D"6. This dyonic solution is parameterized by 2 harmonic functions, which are respectively associated with electric and magnetic charges of 2-form potential. In the self-dual limit, i.e. when the electric and magnetic charges are equal, the solution becomes the D"6 self-dual dyonic string. Within the context of non self-dual theory, the solution preserve only 1/4 of supersymmetry, whereas as a solution of self-dual theory it preserve 1/2 of supersymmetry. 6.2.2. Intersecting p-branes 6.2.2.1. Constituent p-branes. Before we discuss intersecting p-branes, we summarize various single-charged p-branes in D"10, 11, which are constituents of intersecting and overlapping p-branes. These p-branes are special cases of `elementarya p-branes and `solitonica p -branes discussed in Section 6.1. They are characterized by a harmonic function H(y) in the transverse space (with coordinates y ,2,y ) and break 1/2 of supersymmetry. D"11 supergravity has 3-form N> "\ potential. So, the basic p-branes (called M p-branes) are an electric `elementarya membrane and
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a magnetic `solitonica "vebrane: ds"HN> [H\(!dt#dx#2#dx)#(dy #2#dy )] , N N N N> c F ?"! R ?H\, a"3,2,10 (for p"2) , RV V W 2 W N c F ?2 ?" e 2 R ?H , a "6,2, 10 (for p"5) , W W G 2 ? ? W N
(474)
where c"1(!1) for (anti-) branes and harmonic function cQ N N H "1# N " y!y "\N is for M p-brane located at the +0, 1,2, p, hyperplane at yG"yG . The Killing spinor e of these M p-branes satis"es the following constraint: (475) C 2 e"ce , N where C 2 ,C C 2C is the product of #at spacetime gamma matrices associated with the N N worldvolume directions. In D"10, there are 3 types of p-branes, depending on types of charges that p-branes carry. The electric charge of NS-NS 2-form potential is carried by NS-NS strings (or fundamental strings): ds"H\(!dt#dx)#dy#2#dy, e("H\ . (476) The magnetic charge of NS-NS 2-form potential is carried by NS-NS 5-branes (or solitons): ds"!dt#dx#2#dx#H(dy#2#dy), e("H . The charges of R-R (p#1)-form potentials are carried by R-R p-branes:
(477)
ds"H\ (!dt#dx#2#dx)#H (dy #2#dy) , N N> (478) e("H\N\ , F 2 N G"R GH\ , W RV V W where p"0, 2, 4, 6 (p"1, 3, 5, 7) for R-R p-branes in type-IIA (type-IIB) theory. These R-R p-branes of the e!ective "eld theory are long distance limit of D p-branes in type-II superstring theories [193]: the transverse (longitudinal) directions of R-R p-branes correspond to coordinates with Dirichlet (Neumann) boundary conditions. The Killing spinors of D"10 p-branes satisfy one constraint. The left-moving and the rightmoving Majorana}Weyl spinors e and e have the same (opposite) chirality for the type-IIB * 0 (type-IIA) theory, i.e. C e "e [C e "e and C e "!e ]. So, spinor constraints are * 0 * 0 * * 0 0 di!erent for type-IIA/B theories: E E E E
NS-NS strings: e "C e , e "!C e * * 0 0 IIA NS-NS "vebranes: e "C 2 e , e "C 2 e 0 * * 0 IIB NS-NS "vebranes: e "C 2 e , e "!C 2 e 0 * * 0 R-R p-branes: e "C 2 e . * N 0
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Whereas in type-II superstring theories there are D p-branes with p"!1, 0,2, 9, R-R p-branes in massless e!ective "eld theories cover only range p46. So, there is no place in the massless type-II supergravities for R-R 7- and 8-branes. Although the R-R 7-brane in type-IIB theory can be related to 8-form dual of pseudo-scalar of type-IIB theory [293], it cannot be T-dualized to R-R p-branes in type-IIA theory since it is speci"c to the uncompactixed type IIB theory, only. In [76], it is proposed that D p-branes of type-II superstring theories with p'6 can be realized as R-R p-branes of massive type-II supergravity theories. In the following, we discuss R-R 7- and 8-branes in some detail, since their properties and T-duality transformation rules are di!erent from other R-R p-branes. The R-R 8-brane is coupled to 9-form potential. The introduction of 10-form "eld strength into the theory does not increase the bosonic degrees of freedom (therefore, is not ruled out by supersymmetry consideration), but leads to non-zero cosmological constant. In fact, the massive type-IIA supergravity constructed in [204] contains such cosmological constant term. It is argued [498] that the existence of the massive type-IIA supergravity with the cosmological constant is related to the existence of the 9-form potential of type-IIA theory. In [76], new massive type-IIA supergravity is formulated by introducing 9-form potential A whose 10-form "eld strength F "10dA is interpreted as the cosmological constant once Hodge-dualized. (Note, the cos mological constant m in the massive type-IIA supergravity is independent of the dilaton in the string-frame, which is typical for terms in R-R sector.) In this new formulation, the cosmological constant m is promoted to a "eld M(x) by introducing A as a Lagrange multiplier for the constraint dM"0 that M(x)"m is a constant: the "eld equation for M(x) simply determines the new "eld strength F , while the "eld equation for A implies that M(x)"m. It is conjectured in [154] that the massive type-IIA supergravity is related to hypothetical H-theory [44] in D"13 through the Scherk}Schwarz type dimensional reduction (see below for the detailed discussion), rather than to M-theory. The conjectured type-IIB Sl(2, Z) duality requires the pseudo-scalar s to be periodically identi"ed (s&s#1), which together with T-duality between massive type-IIA and type-IIB supergravities implies that the cosmological constant m is quantized in unit of the radius of type-IIB compacti"cation circle, i.e. m"n/R (n3Z). The massive type-IIA supergravity admits the following 8-brane (or domain wall) as a natural ground state solution: ds"H\ g dxI dxJ#H dy, e\("H , (479) IJ and the Killing spinor e satis"es one constraint CM e"$e. The form of harmonic function H(y), W which is linear in y, depends on M(x). When M(x)"m everywhere, H"m"y!y ", with a kink singularity at y"y . When M(x) is locally constant, the corresponding solution is interpreted as a domain wall separating regions with di!erent values of M (or cosmological constant). An example is H"!Q y#b [H"Q y#b] for y(0 [y'0], where 8-brane charges Q are \ > ! de"ned as the values of M as yP$Rand b is related to string coupling constant e( at the 8-brane core. This 8-brane solution is asymptotically left-#at (right-#at) when Q "0 (Q "0). The multi \ > 8-brane generalization can be accomplished by allowing kink singularities of H at ordered points y"y (y (2(y . L R-R 9-brane of type-IIB theory is interpreted as D"10 spacetime.
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The R-R 8-brane of massive type-IIA supergravity can be interpreted as the KK 6-brane of D"11 supergravity [76]. Namely, after the R-R 8-brane is compacti"ed to 6-brane in D"8, the 6-brane can be lifted as the R-R 6-brane of massless type-IIA supergravity, which is interpreted as the KK monopole of M-theory on S [601]. Under the T-duality, the R-R 8-brane is expected to transform to R-R 7-brane or 9-brane in type-IIB theory. First, T-duality transformation of massless type-II supergravity along a transverse direction of the R-R 8-brane leads to the product of S and D"9 Minkowski spacetime, which is 9-brane. (Note, the direct dimensional reduction requires H to be constant.) Second, T-duality transformation leading to type-IIB 7-brane is much involved and we discuss in detail in the following. T-duality transformation involving massive type-IIA supergravity requires construction of D"9 massive type-IIB supergravity. D"9 massive type-IIB supergravity is obtained from massless type-IIB supergravity through the Scherk}Schwarz type dimensional reduction procedure [528], i.e. "elds are allowed to depend on internal coordinates. This is motivated by the observation that the `StuK ckelberga type symmetry, which "xes the m-dependence of "eld strengths in type-IIA supergravity, is realized within type-IIB supergravity as a general coordinate transformation in the internal direction, which requires some of R-R "elds to depend on the internal coordinates. Namely, an axionic "eld s(x, z) (i.e. R-R 0-form "eld) is allowed to have an additional linear dependence on the internal coordinate z, i.e. s(x, z)Pmz#s(x), where x is the lowerdimensional coordinates. Since s appears always through ds in the action, the compacti"ed action has no dependence on the internal coordinate z. The result is the massive supergravity with cosmological term. The massive type-IIA supergravity compacti"ed on S through the standard KK procedure is related via T-duality to the massless type-IIB supergravity compacti"ed on S through the Scherk}Schwarz procedure. This massive T-duality transformation generalizes those of massless type-II supergravity in [81]. The explicit expression for m-dependent correction to the T-duality transformation can be found in [76]. Under the massive T-duality, the type-IIA 8-brane transforms to the type-IIB 7-brane, which is the "eld theory realization of D 7-brane of type-IIB superstring theory. By applying the T-duality of massless type-II supergravities to this 7-brane, one obtains a 6-brane of type-IIA theory, whereas transformation to 8-brane of type-IIA theory requires the application of massive T-duality. This generalized compacti"cation Ansatz for s is a special case of the generalized compacti"cation on a d-dimensional manifold M where an (n!1)-form potential A (n4d) for which nth B L\ cohomology class HL(M , R) of M is non-trivial is allowed to have an additional linear dependence B B on the (n!1)-form u (z), i.e. A (x, z)"mu (z)#standard terms [154,443]. (All the other L\ L\ L\ "elds are reduced by the standard KK procedure.) Here, du represents non-trivial nth cohomolL\ ogy of M . (In the case of compacti"cation of type-IIB theory on S, the cohomology dz is the B volume form on S.) Since A appears in the action always through dA , the lower-dimenL\ L\ sional action depends on A only through its zero mode harmonics on M , only. The constant L\ B m manifests in lower dimensions as a cosmological constant and, thereby, the compacti"ed D-dimensional action admits domain wall, i.e. (D!2)-brane, solutions. The general pattern for mass generation is as follows. First, the KK vector potentials always become massive. Second, a "eld that appears in a bilinear term in the Chern}Simons modi"cation of a higher rank "eld strength acquires mass if it is multiplied by A (with general dimensional reduction Ansatz). L\
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Third, when axionic "eld AGHI associated with D"11 3-form potential A is used for the +,. Scherk}Schwarz reduction, the lower-dimensional theory contains a topological mass term. In these mechanisms, the "elds associated with the StuK ckelburg symmetry (under which the eaten "elds undergo pure non-derivative shift symmetries) get absorbed by other "elds to become mass terms for the potentials that absorb them. The consistency of the theory requires that the "elds that are eaten should not appear in the Lagrangian. Note, whereas the original Scherk}Schwarz mechanism [528] is designed to give a mass to the gravitino, thus breaking supersymmetry, and do not generate scalar potentials, in our case a cosmological constant is generated and the full supersymmetry is preserved. In addition to single-charged p-branes, there are other supersymmetric con"gurations which are basic building blocks of p-brane bound states. These are the gravitational plane fronted wave (denoted 0 ), called `pp-wavea, and the KK monopole (denoted 0 ). Their existence within p-brane U K bound states are required by duality symmetries. First, the KK monopole in D"11 is introduced [601] in an attempt to give D"11 interpretation of type-IIA D 6-brane [367]. The KK monopole is the magnetic dual of the KK modes of D"11 theory on S, which is identi"ed as R-R 0-brane (electrically charged under the KK ;(1) gauge "eld) [635]. The D"11 KK monopole, which preserves 1/2 of supersymmetry, has the form [601]: ds "!dt#dy ) dy# depends on D and the numbers of
But the Scherk}Schwarz reduced theories have smaller symmetry than those reduced via the standard KK procedure [154].
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isometry directions of the con"guration. More general solution is obtained by replacing ="Q/"x"L by =(u, x). In particular, the choice ="f (u)xG corresponds to wave with the pro"le f propagating G G along y; an asymptotic observer, however, observes pp-wave with ="Q/"x"L and Q&1[du f (u)]2. When the Lorentz boost is imposed on a black p-brane and extreme limit is G taken, one has a bound state of a p-brane and pp-wave. 6.2.2.2. Orthogonally intersecting p-branes. From single-charged p-branes, one can construct `marginala bound states of p-branes by applying general intersection rules. The `marginala bound states are called orthogonally `overlappinga (`intersectinga) p-branes if the constituent p-branes are separated (located at the same point) in a direction transverse to all of the p-branes. We denote the con"guration where a (p#r)-brane intersects with a (p#s)-brane over a p-brane as (p"p#r, p#s) [493]. We add subscripts (NS, R, M) in this notation to specify types of charges that the constituent p-branes carry, e.g. 2 , 1 and 3 respectively denote M 2-brane, NS-NS string (or + ,1 0 fundamental string) and R-R 3-brane. The intersection rules are "rst studied in [492] in an attempt to interpret GuK ven solutions [327], and general harmonic superposition rules (which prescribe how products of powers of the harmonic functions of intersecting p-branes occur) of intersecting p-branes are formulated in [611]. (Such harmonic superposition rules of intersecting M-branes are already manifest in general overlapping M-branes constructed in [287].) Intersection rules can be independently derived from the `no forcea condition on a p-brane probe moving in another p-brane background [613]. Alternatively, one can derive general intersection rules for `marginala bound state p-branes in diverse dimensions from equations of motion, only [18]. Namely, from the equations of motion following from general D-dimensional action with one dilaton and arbitrary numbers of n -form "eld strengths F (A"1,2, N) with kinetic terms L 1 e? (F , L 2n ! one obtains the following intersection rule prescribing (p "p , p ): (p #1)(p #1) 1 ! e a e a , (482) p #1" 2 D!2 where e "#(!) when the brane A is electric (magnetic). It is interesting that the relation (482), derived from the equations of motion of the e!ective action alone, predicts D-branes (0"1 , p ) and ,1 0 D-branes for higher branes. In the following, we discuss intersection rules for a pair of branes. Intersecting con"gurations with more than two constituents are constructed by applying the intersection rules to all the possible pairs of branes. There are 3 types of intersecting p-branes: self-intersections, branes ending on branes and branes within branes. First, p-branes of the same type intersect only over (p!2)-brane (so-called (p!2) self-intersection rule), denoted as (p!2"p, p). This can be understood [492] from the fact that p-brane worldvolume theory contains a scalar (interpreted as a Goldstone mode of spontaneously broken translational invariance by the p-brane), which is Hodge-dualized to a worldvolume (p!1)-form potential that the (p!2)-brane couples to. The second type, denoted as (p!1"p, q) with q'p, is interpreted as a q-brane ending on a p-brane with (p!1)-branes being the ends of
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q-branes on the p-branes. This type of intersecting p-branes can be constructed by applying S- and T-duality transformations on M 2-brane ending on M 5-brane (1"2 , 5 ) or fundamental string + + ending on R-R p-brane (0"1 , p ). The third type, denoted as (p"p, q) with p(q, is interpreted as ,1 0 p-branes inside of the worldvolume of q-brane [224]. This type of intersecting p-branes preserves fraction of supersymmetry when the projection operators P and P (de"ning spinor constraints N O associated with the constituent D p-branes) either commutes or anticommutes [493]. If P and N P commute, then (p"p, q) preserves 1/4 of supersymmetry. This happens i! p"q mode 4. When O P and P anticommute, one has con"gurations that preserve 1/2 of supersymmetry. An example is N O 0-brane within membrane (0"0 , 2 ), which can be obtained from (2"2 , 5 ) (473) through dimen0 0 + + sional reduction on S and T-duality transformations. The spacetime coordinates of intersecting branes are divided into 3 parts: (i) the overall worldvolume coordinates mI (k"0, 1,2, d!1), which are common to all the constituent branes; (ii) the relative transverse coordinates x? (a"1,2, n), which are transverse to part of the constituent branes; (iii) the overall transverse coordinates yG (i"1,2, l), which are transverse to all of the constituent branes. Since a transverse (longitudinal) coordinate of R-R p-branes corresponds to a coordinate with Dirichlet (Neumann) boundary condition, these 3 types of coordinates respectively correspond to coordinates of open strings of the NN-, ND- (or DN-) and DD-types. Intersecting p-branes are divided into 3 types, according to the dependence of harmonic functions on these coordinates. The "rst type, for which the general intersection rules are formulated in [492,611], has all the harmonic functions depending on the overall transverse coordinates, only. For the second type, one harmonic function depends on the overall transverse coordinates and the other on the relative transverse coordinates. The third type has both harmonic functions depending on the relative transverse coordinates. For the "rst (third) type, constituent p-branes are, therefore, localized in the overall (relative) transverse directions but delocalized in the relative (overall) transverse directions. We will be mainly concerned with intersection rules for the "rst type and later we comment on the rest of types. One can construct supersymmetric (BPS) intersecting p-branes when spinor constraints associated with constituent p-branes (given in the previous paragraphs) are compatible with one another with non-zero Killing spinors. When none of spinor constraint is expressed as a combination of other spinor constraints, intersecting N number of p-branes preserve (), of supersymmetry. (Note, from now on N stands for the number of constituent p-branes, not the number of extended supersymmetries.) One can introduce an additional p-brane without breaking any more supersymmetry, if some combination of spinor constraints of existing constituent p-branes gives rise to spinor constraint of the added p-brane. First, we discuss intersecting M p-branes. Intersecting rules, studies in [492,611], are: E Two M 2-branes (M 5-branes) intersect over a 0-brane (a 3-brane), i.e. (0"2 , 2 )((3"5 , 5 )). + + + + E M 2-brane and M 5-brane intersect over a string, i.e. (1"2 , 5 ), interpreted as M 2-brane ending + + on M 5-brane over a string. The worldvolume theory is described by D"6 (0, 2) supermultiplet with bosonic "elds given by 5 scalars and a 2-form that has self-dual "eld strength, and M 2-brane charge is carried by a self-dual string inside the worldvolume theory. E One can add momentum along an isometry direction by applying an SO(1, 1) boost transformation.
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All the possible intersecting M-branes are determined by these intersection rules. One can intersect up to 8 M p-branes by applying intersection rules to each pair of constituent M p-branes: the complete classi"cation up to T-duality transformations is given in [75]. Intersecting M-brane solutions can be constructed from the harmonic superposition rules. These are "rst formulated in [611] for BPS M-branes and are generalized to the non-extreme case in [175,243] and to the rotating case in [185]. First, the harmonic superposition rules for the BPS case are as follows: E The overall conformal factor of the metric is the product of the appropriate powers of the harmonic functions associated with constituent M p-branes: (483) ds" HNG G> (y)[2] , N G where p "2, 5, and H G(y) are harmonic functions in overall transverse space (with dimension l), G N namely of (i) the form cG N for l'2 , H G(y)"1# N "y!y "l\ (ii) logarithmic form for l"2, and (iii) linear form for l"1. Here, y"(y#2#yl is the radial coordinate of the overall transverse space. So, the spacetime is asymptotically #at when l'2. E The metric is diagonal (unless there is a momentum along an isometry direction) and each component inside of [2] in (483) is the product of the inverse of harmonic functions associated with the constituent p -branes whose worldvolume coordinates include the corresponding G coordinate. E When momentum is added to an isometry direction, say m-direction, the above rules are modi"ed by the harmonic function K(y) associated with the momentum as follows: !dt#dmP!K\(y)dt#K(y)dmY , dmY ,dm#[K\(y)!1] dt .
(484)
E When l'3, one can add a KK monopole in a 3-dimensional subspace of the overall transverse space. All the harmonic functions depend only on these 3 transverse coordinates and the metric is modi"ed in the overall transverse components as follows dyGdyGPdy#2#dyl #H\(dyl $a cos h d )#H(dr#r dX) , \ \ ))
(485)
where the harmonic function H"1#a /r is associated with the KK monopole charge )) P "$a X . The 2-form "eld strength associated with the KK monopole has the form )) )) F "P e . )) E Non-zero components of 4-form "eld strength F are given by (474) for each constituent M p-branes. Now, we discuss generalization of harmonic superposition rules to the non-extreme, rotating case. The solutions get modi"ed by the harmonic functions g (y)"1#(l/y) associated with G G
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angular momentum parameters l and the following combinations of g : G G
\ a# l\ kg\ for even l,
G G G Gl, f\ ,G l l g . G for odd l l\ kg\ G G G G l
(486)
Here, a and k are de"ned in (189) and (191). The harmonic superposition rules are modi"ed in the G following way: E Harmonic functions are modi"ed as 2m sinh d 2m sinh d cosh d PH"1#fl , H"1# y l\ y l\
(487)
where the boost parameter d is associated with electric/magnetic charge or momentum 2m sinh d cosh d. E The metric components get further modi"ed as (488) dtPf dt, d dyG dyHPf \dy#y dXY l , \ GH l where f,1!fl(2m/y l\) and f ,G\ l !fl(2m/y \) are non-extremality functions. Here, dXY l denotes the angular parts of the metric. In the limit l "0, dXY l becomes #at metric \ G \ dXl of Sl\. Generally, dXY l is given by the angular metric components of rotating black \ \ hole in (compacti"ed) D"l#1 and the general rules for constructing these components are unknown yet. (The explicit forms of dXY l up to 3 M p-brane intersections with momentum \ along an isometry direction are given in [185].) E The harmonic superposition rule (484) with a momentum along an isometry direction m is modi"ed as !dt#dmPK\f dt#KdmY , dmY ,dm#[K \!1]dt ,
(489)
where d is a boost parameter associated with momentum 2m sinh d cosh d and K \,1!fl(2m sinh d/rl\)K\. E The non-zero components of the 4-form "eld strength are the same as the BPS case, when l "0. G For l O0, there are additional non-zero components associated with the induced electric/ G magnetic "elds due to rotations. The general construction rules are unknown yet; the explicit expressions up to 3 intersections with momentum along an isometry direction are given in [185]. Next, we discuss intersecting p-branes in D"10. The intersection rules are as follows: E Two fundamental strings can only be parallely oriented, i.e. 1 ""1 . ,1 ,1 E Two solitonic 5-branes orthogonally intersect over 3-spaces, i.e. (3"5 , 5 ). ,1 ,1 E A fundamental string and a solitonic 5-brane can only be parallely oriented, i.e. 1 ""5 or ,1 ,1 (1"1 , 5 ). ,1 ,1 E An R-R p-brane and an R-R q-brane intersect over n-spaces (n"p , q ) such that 00 00 p#q!2n"4.
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E A fundamental string orthogonally intersects an R-R p-brane over a point, i.e. (0"1 , p ). ,1 0 E A solitonic 5-brane intersects an R-R p-brane over n-spaces (n"5 , p ) such that p!n"1. ,1 0 These intersecting p-branes preserve 1/4 of supersymmetry except the multi-centered con"guration 1 ""1 . These intersection rules are derived in [613] by applying `no forcea condition and in [18] ,1 ,1 from the equations of motion. Alternatively, one can derive these rules from the intersection rules of M-branes by applying KK procedure and duality transformations, which we discuss in the following in details. Intersecting p-branes in D"10 can be obtained from intersecting M-branes through compacti"cation on S and duality transformations. We have the following p-branes in type-IIA theory from the compacti"cations R and R of M-branes along a longitudinal and a transverse directions, i.e. , the double and direct dimensional reductions, respectively: 0, 2 , 0 1 , 2 P 2 P ,1 + 0 +
0 4 , 5 P 0, 5 , 5 P + 0 + ,1
(490)
which can be understood from the relations of type-IIA form "elds to the D"11 3-form "eld under the standard KK procedure. Dimensional reduction of 0 and 0 in D"11 yields the following U K type-IIA con"gurations: 0, 0 , 0 P 0 6 , 0 0 , 0 P 0 P 0 U U K 0 U
0, 0 , 0 P K K
(491)
where R (R ) on 0 denotes the reduction in the direction associated with Taub-NUT term (the , K other directions) of the metric. Duality transformations further relate di!erent types of p-branes. First, we consider T-duality between type-IIA and type-IIB theories on S. T-duality (161) on type-IIA/B R-R p-branes along a tangent (transverse) direction leads to type-IIB/A R-R (p!1)branes ((p#1)-branes). Note, T-duality on a longitudinal direction introduces an additional overall transverse coordinate that harmonic functions have to depend on, which is not always guaranteed. So, the T-duality on a longitudinal direction is called `dangerousa, whereas the T-duality on a transverse direction is `safea since the resulting con"gurations are guaranteed to be solutions to the equations of motion. T-duality transformation rules of NS-NS p-branes can be inferred from T-duality transformation of "elds [81] (see also (161)) as follows. Among other things, T-duality interchanges o!-diagonal metric components g and the same components B of the NS-NS 2-form potential, where a is the I? I? T-duality transformation direction. So, when p-brane has non-trivial (k"t, a)-component of the metric or NS-NS 2-form potential, T-duality transformation is reminiscent of interchange of momentum mode (electric charge of KK ;(1) "eld g ) and winding mode (electric charge of I? NS-NS 2-form ;(1) "eld B ) under T-duality. So, pp-wave (whose linear momentum is identi"ed I? with KK electric charge) and NS-NS string (carrying electric charge of the NS-NS 2-form "eld, identi"ed as string winding mode) are interchanged when T-duality is performed along the longitudinal direction of string or the direction of pp-wave propagation. T-duality along the other directions yields the same type of solutions. Second, the (k" , a)-component of metric (NS-NS G 2-form potential), where are angular coordinates associated with rotational symmetry, corresG ponds to the Taub-NUT term (is associated with magnetic charge of a solitonic 5-brane 5 ). So, ,1 magnetic monopole (or Taub-NUT solution) and NS-NS 5-brane are interchanged when T-duality
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transformation is applied along the direction transverse to NS-NS 5-brane or along the coordinate associated with Taub-NUT term. T-duality along the other directions yields the same type of solutions. Second, the type-IIB S¸(2, Z) S-duality transformation can be used to relate NS-NS charged solutions and R-R charged solutions which are coupled to 2-form potentials. Under the Z subset transformation, NS-NS string and NS-NS 5-brane transform to R-R 1-brane and R-R 5-brane, respectively. Full S¸(2, Z) transformations on NS-NS string (NS-NS 5-brane) yields `non-marginala BPS bound states of p NS-NS strings and q R-R strings (p NS-NS 5-brane and q R-R 5-brane) with the pair of integers (p, q) relatively prime. The duality transformation rules are, therefore, summarized as follows: 2, (p#1) , 1 P 2 0 , 1 P 2, 1 , 2 (p!1) , p P p P 0 0 0 ,1 U ,1 ,1 0 2 1 , 0 P 2, 0 , 5 P 2 5 , 0 P U ,1 U U ,1 ,1
2, 0 , 5 P ,1 K
(492)
2 5 , 0 P 2, 0 , 1 1 1 , 5 1 5 . 0 P 0 ,1 K ,1 K K ,1 0 We now discuss various intersecting p-branes in D"10. First, we consider intersecting R-R p-branes in type-II theories. D-brane con"gurations are supersymmetric if the number l of coordinates of DN or ND type is the multiple of 4 [501]. (See Section 8.3.2 for details on this point.) At the level of low-energy intersecting R-R p-branes of the e!ective "eld theories, this means that solutions are supersymmetric when the number of relative transverse coordinates is the multiple of 4, i.e. n"4, 8. This can also be derived from the condition that the Killing spinor constraints e "C 2 e of the constituent R-R p-branes * N 0 are compatible with one another. When both of the harmonic functions depend only on the relative transverse coordinates, BPS con"gurations are possible for n"8 case only, and otherwise only n"4 con"gurations are BPS. Since our main concern is the intersecting R-R p-branes with all the harmonic functions depending only on the overall transverse coordinates, we concentrate on the n"4 case. Since T-duality preserves the total number n of the relative transverse coordinates, one can obtain all the intersecting 2 R-R p-branes with n"4 by applying `safea T-duality transformations to intersecting R-R 0-brane and R-R 4-brane, i.e. (0"0 , 4 ). One can further add R-R p-branes in such a way that n"4 for each pair of constituent 0 0 R-R p-branes. It is shown [75] that one can intersect up to 8 R-R p-branes which satisfy the n"4 rule for each pair: the complete classi"cation up to T-dualities is given in [75]. The explicit solutions for BPS intersecting R-R p-branes can be constructed by the following harmonic superposition rules: E The metric is diagonal, with each component having the multiplicative contribution of H\ (H ) from each constituent R-R p-brane whose worldvolume (transverse) coordinates N N include the associated coordinate. E Dilaton is given by the product of harmonic functions associated with the constituent R-R p-branes: e\(" HNII\ . I N E Non-zero components of (p#2)-form "eld strengths are given by (478) for each constituent R-R p-branes.
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As an example, solution for intersecting R-R (p#r)-brane and R-R (p#s)-brane over a p-brane, i.e. (p"p#r, p#s), is of the form:
H Q (dx?) ds "(H H )\ g dmI dmJ# N>P N>P N>Q IJ H N>Q ? H Q>P # N>Q (dx?)#(H H ) d dyG dyH , N>P N>Q GH H N>P ?Q> e\("HN>P\ HN>Q\ , N>P N>Q F 2 Q G"R GH\ , F Q>2 Q>P G"R GH\ . (493) V W RV V W W N>Q RV W N>P We discuss intersecting p-branes which contain NS-NS p-branes. First, (0"1 , p ) with 04p48, ,1 0 is nothing but open strings that end on D-brane. This type of con"gurations can be obtained by "rst compactifying (0"2 , 2 ) on S along a longitudinal direction of one of M 2-brane (resulting in + + (0"1 , 2 )), and then by sequentially applying T-duality transformations along the directions ,1 0 transverse to the NS-NS string. Second, (p!1"5 , p ) with 14p46, are interpreted as D p-brane ,1 0 ending on NS-NS 5-brane. Namely, NS-NS 5-branes act as a D-brane for D-branes. This interpretation is consistent with the observation [46] that M 5-branes are boundaries of M 2-branes. This type of intersecting branes can be constructed by "rst compactifying (1"2 , 5 ) on S along an + + overall transverse direction (resulting in (1"5 , 2 )), and then by sequentially applying T-duality ,1 0 transformations along the longitudinal directions of the NS-NS 5-brane. Third, intersecting NS-NS p-branes can be obtained in the following ways: (i) compacti"cation of (3"5 , 5 ) along an + + overall transverse direction leads to (2"5 , 5 ), (ii) compacti"cation of (1"2 , 5 ) along a relative ,1 ,1 + + transverse direction which is longitudinal to M 2-brane leads to (1"2 , 5 ), (iii) the type-IIB ,1 ,1 S-duality on (!1"1 , 1 ) yields (!1"1 , 1 ). 0 0 ,1 ,1 We comment on the case some or all of harmonic functions depend on the relative transverse coordinates [75]. These types of intersecting p-branes can be constructed by applying the general harmonic superposition rules, taking into account of dependence of harmonic functions on the relative transverse coordinates. In particular, the metric components associated with the relative coordinates (that harmonic functions depend on) have to be the same so that the equations of motion are satis"ed. First, the second type of intersecting p-branes, i.e. one harmonic function depends on the relative transverse coordinates, can be constructed from the "rst type of intersecting p-branes, i.e. all the harmonic functions depend on the overall transverse coordinates, by letting one of harmonic functions depend on the relative transverse coordinates. Thus, the classi"cation of the second type is the same as that of the "rst type. The third type of p-branes, i.e. all the harmonic functions depend on the relative transverse coordinates, have 8 relative transverse coordinates (n"8) for a pair of p-branes. It is impossible to have more than two p-branes with each pair having n"8. In D"11, the only con"guration of the third type is (1"5 , 5 ) [287]. This M-brane + + preserves () of supersymmetry, since the Killing spinor satis"es two constraints of the form (475), each corresponding to a constituent M 5-brane. By compactifying an overall transverse direction of
M 2-brane with its longitudinal coordinates given by the overall longitudinal and overall transverse coordinates of (1"5 , 5 ) can be further added without breaking any more supersymmetry. The added M 2-brane intersects the M 5+ + branes over strings and is interpreted as an M 2-brane stretched between two M 5-branes.
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(1"5 , 5 ) on S, one obtains (1"5 , 5 ), which was "rst constructed in [423]. Further application + + ,1 ,1 of the type-IIB SL(2, Z) transformation leads to (1"5 , 5 ). 0 0 The series of application of T-duality transformations, then, yield a set of overlapping 2 R-R p-branes with n"8. (Complete list can be found in [287].) These overlapping p-branes correspond, at string theory level, to D-brane bound states with 8 ND or DN directions, and therefore should be supersymmetric. When 2 R-R p-branes intersect in a point, one can add a fundamental string without breaking any more supersymmetry. This type of con"gurations is interpreted as a fundamental string stretching between two D-branes. For the case where 2 R-R p-branes intersect in a string, one can add pp-wave along the string intersection without breaking anymore supersymmetry. Another third type of intersecting p-branes in D"10 can be constructed by compactifying (1"5 , 5 ) along a relative transverse direction, resulting in (1"4 , 5 ), followed by series of + + 0 ,1 T-duality transformations along the longitudinal directions of the NS-NS 5-brane, resulting in (p!3"p , 5 ) with 34p48. One can further add R-R (p!2)-branes to these con"gurations; 0 ,1 these con"gurations are interpreted as a D (p!2)-brane stretching between D p-brane and NS-NS 5-brane. 6.2.2.3. Other variations of intersecting p-branes. So far, we discussed intersecting p-branes with p!p "0 mod 4. Existence of such classical intersecting p-branes that preserve fraction of supersymmetry is expected from the perturbative D-brane argument [501,504]. One can construct such solutions by applying the harmonic superposition rules discussed in the above. In this subsection, we discuss another type of p-brane bound states which do not follow the intersection rules discussed in the previous subsection. Such p-brane bound states contain pair of constituent p-branes with p!p "2 and still preserve fraction of supersymmetry. These p-brane con"gurations can be generated by applying dimensional reduction or T-duality along a direction at angle with a transverse and a longitudinal directions of the constituent p-branes [96,519,152]. (Hereafter, we call them as `tilteda reduction and `tilteda T-duality.) These p-brane con"gurations can also be constructed by applying `ordinarya dimensional reduction and sequence of `ordinarya duality transformations on (2"2 , 5 ) (473), which preserves 1/2 of supersymmetry. + + In fact, it con#icts with di!eomorphic invariance of the underlying theory that one has to choose speci"c directions (which are either transverse or longitudinal to the constituent p-branes) for dimensional reduction or T-duality transformations [152]. So, the existence of such new p-brane con"gurations is required by (M-theory/IIA string and IIA/IIB string) duality symmetries and di!eomorphism invariance of the underlying theories. We now discuss the basic rules of `tilteda T-duality and `tilteda dimensional reduction on constituent p-branes. Before one applies `tilteda T-duality and dimensional reduction to p-branes, One can further add a fundamental string along the string intersection without breaking any more supersymmetry. T-duality along the fundamental string direction leads to type-IIB (1"5 , 5 ) with pp-wave propagating along the string ,1 ,1 intersection. Note, the former con"guration preserves only 1/8 of supersymmetry, rather than 1/4, if regarded as a solution of type-IIB theory. It is argued [501,504] that D-brane bound state with p!p "6 is not supersymmetric and is unstable due to repulsive force. Although a solution that may be interpreted as (0"0 , 6 ) is constructed in [152], its interpretation is 00 00 ambiguous due to abnormal singularity structure of harmonic functions, and it cannot be derived from (2"2 , 5 ) though + + a chain of T-duality transformations.
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one rotates a pair of a longitudinal x and a transverse y coordinates of a (constituent) p-brane by an angle a (a di!eomorphism that mixes x and y):
x
"
y
cos a !sin a
x
sin a
y
cos a
.
(494)
(Note, x and y have to be di!eomorphic directions so that one can compactify these directions on S.) Then, one compacti"es or applies T-duality along the x -direction. These procedures preserve supersymmetry. When a"0 [a"p/2], such compacti"cation or T-duality transformation is the compacti"cation or T-duality transformation along a longitudinal direction x (a transverse direction y). Thus, as the angle a is varied from 0 to p/2, the resulting bound state interpolates between the corresponding two limiting con"gurations. First, we discuss the `tilteda reduction R of solutions in D"11. The `tilteda reduction on ? M-branes leads to type-IIA p-brane bound states, interpreted as `brane within branea: 0? (4"4 , 5 ) , 0? (1"1 , 2 ) , 5 P (495) 2 P ,1 0 + 0 ,1 + where the subscript A means type-IIA con"guration. From 0 and 0 in D"11, one obtains the K U following type-IIA bound states: 0? (0 "6 ) , 0 P 0? (0 "0 ) , 0 P (496) K U 0 U K 0 where (0 "0 ) is interpreted as a `boosteda D 0-brane. U 0 Second, we discuss `tilteda T-duality transformations ¹ . The o!-diagonal metric component ? g induced by the coordinate rotation (494) is transformed to the same component B of 2-form VYWY VYWY potential under T-duality. So, the T-transformed solution has diagonal metric and non-zero F "B #2pa F , where F is the worldvolume gauge "eld strength [445]. For R-R p-brane IJ IJ IJ IJ bound states, the corresponding perturbative D-brane now, therefore, satis"es the modi"ed boundary condition R XI!iFIR XJ"0. The induced #ux F is related to a as F "!tan a. L J J VYWY VYWY The ADM mass of the transformed con"guration is of the form M&Q#Q, a characteristic of non-threshold bound states. First, the `tilteda T-duality on type-IIA/B p-branes leads to the following type-IIB/A bound states: 2? (0 "5 ), (p#1) P 2? (p"p , (p#2) ) , 2? (0 "1 ), 5 P (497) 1 P U ,1 ,1 K ,1 0 0 0 ,1 where (0 "1 ) is simply a boosted fundamental string. The existence of the D-brane bound states U ,1 (p"p , (p#2) ) preserving 1/2 of supersymmetry is also expected from the perturbative D-brane 0 0 considerations. One can apply `tilteda T-duality transformation more than once to obtain new p-brane bound state con"gurations. For example, by applying `tilteda T-duality transformations to D 2-brane in two di!erent directions, one obtains D (4, 2, 2, 0)-brane bound state [96]. Second, `tilteda T-duality on 0 and 0 in type-IIA/B theory yields the following type-IIB/A bound states: K U 2? (0 "1 ) . 2? (0 "5 ), 0 P (498) 0 P K ,1 U U ,1 K Next, we discuss p-brane bound states obtained by "rst imposing a Lorentz boost along a transverse direction and then applying T-duality transformation or reduction along the direction of the boost: T-duality along a boost and reduction along a boost, respectively denoted as ¹ and R . T T The Lorentz boost yields non-threshold bound state of a p-brane and pp-wave. This bound state
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interpolates between extreme p-brane and plane wave, as the boost angle a (see below for its de"nition) varies from zero (no boost) to p/2 (in"nite boost). The ADM energy E of such non-threshold bound state of extreme p-brane (with mass M) and pp-wave (with momentum p) has the form E"M#p, a reminiscent of relativistic kinematic relation of a particle of the rest mass M with the linear momentum p, rendering the interpretation of such bound state as `boosteda p-brane. The Lorentz boost with velocity v/c"sin a along a transverse direction y has the form [519]: tPt "(cos a)\(t#y sin a), yPy "(cos a)\(y#t sin a) .
(499)
In the above, the angle a is related to the boost parameter b as cosh b"1/cos a. After the Lorentz boost (499), one compacti"es or applies T-duality transformation along y . First, we discuss the reduction R along a boost. Since the momentum of pp-wave manifests as T the KK electric charge after reduction along the direction of momentum #ow, the resulting bound state always involves R-R 0-brane. The following type-IIA bound states are obtained from the compacti"cation with a boost of con"gurations in D"11: 0T (0"0 , 5 ) , 0T (0"0 , 2 ) , 5 P 2 P 0 0 + 0 ,1 + 0T (0 "0 ) , 0 P 0T (0"0 , 6 ) . 0 P U 0 U K 0 0
(500)
Second, we discuss the T-duality ¹ along a boost. Under the T-duality, the linear momentum of T a pp-wave is transformed to the electric charge of the NS-NS 2-form potential [359]. So, the T-dualized con"gurations always involve a fundamental string with non-zero winding mode. We have the following type-IIB/A bound states from type-IIA/B con"gurations: 2T 1 , 1 P ,1 ,1
2T (0 "1 ), p P 2T (1"1 , (p#1) ) , 5 P ,1 K ,1 0 ,1 0
2T (0 "1 ), 0 P 2T (0 "1 , 5 ) . 0 P U U ,1 K K $ ,1
(501)
The di!eomorphic invariance and duality symmetries require new type of bound states in M-theory that preserve 1/2 of supersymmetry. These new M-theory con"gurations can be constructed by uplifting new type-IIA con"guration discussed in this subsection. For example, by uplifting (0 "5 ) or (4"4 , 6 ) , one obtains the M 5-brane and the KK monopole bound state in K ,1 0 0 D"11. Another example is the M 2-brane and the KK monopole bound state uplifted from (1"1 , 6 ) or (0 "1 ) . ,1 0 K ,1 This boost angle a can be identi"ed with the angle a of coordinate rotation in (494). Namely, the non-threshold typeIIA (q , q ) string bound state obtained from D"11 pp-wave through tilted dimensional reduction at an angle a, followed by T-duality, can also be obtained by reduction along a boost of M 2-brane with the same boost angle a, followed by T-duality transformation. One can straightforwardly apply this procedure to intersecting M-branes, followed by sequence of T-duality transformations, to construct p-brane bound states that interpolate between those that preserve 1/4 of supersymmetry and those that preserve 1/2 of supersymmetry as a is varied from 0 to p/2 [151].
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Non-threshold type-IIB (q , q ) string, obtained from R-R or NS-NS string through S¸(2, Z) duality, can be related to D"11 pp-wave compacti"ed at angle [519]. This is understood as follows. When the compacti"cation torus ¹ (parameterized by isometric coordinates (x, y)) is rectangular, the angle a of coordinate rotation de"nes the direction of (q , q ) cycle in ¹, around which the D"11 pp-wave is wrapped, as cos a"q /(q#q. (So, choice of di!erent angle a corresponds to di!erent choice of a cycle in ¹.) Starting from pp-wave propagating along x, one performs coordinate rotation (494) in the plane (x, y), where y is an isometric transverse direction of the pp-wave, and then compacti"es along y -direction, which is the direction of (q , q )-cycle of ¹ with coordinates (x, y). The resulting type-IIA con"guration is a non-threshold bound state of pp-wave and D 0-brane. Subsequent T-duality transformation along y leads to type-IIB (q , q ) string solution. This is related to the fact that the type-IIB S¸(2, Z) symmetry is the modular symmetry of the D"11 supergravity on ¹ [81,538,539] (see Section 3.7 for detailed discussion). Had we started from the bound state of M 2-brane and pp-wave along a longitudinal direction of the M 2-brane, we would end up with boosted type-IIB (q , q ) string that preserves 1/4 of supersymmetry. (The M 2-brane charge is, therefore, interpreted as a momentum of the type-IIB (q , q ) string.) On the other hand, the non-threshold type-IIB (q , q ) 5-brane can be related to M 5-brane. Namely, one "rst compacti"es M 5-brane at an angle to obtain (4"4 , 5 ) and then applies 0 ,1 T-duality transformation along the relative transverse direction to obtain type-IIB (q , q ) 5-brane bound state. Following the similar procedures, one can obtain the non-threshold type-IIB (q , q ) string from M 2-brane. 6.2.2.4. Branes intersecting at angles. In [89], it is shown that one can construct BPS D-brane bound states where the constituent D-branes intersect at angles other than the right angle. We "rst summarize formalism of [89]. Then, we discuss the corresponding classical solutions [33,55,97,151,285] in the e!ective "eld theory. In the presence of a D p-brane, two spinors e and e (corresponding to N"2 spacetime supersymmetry) of type-II string theory satisfy the constraint: N e "C e " : eIC e , (502) N G I G where e is an orthonormal frame spanning D p-brane worldvolume. For N numbers of constituent G D-branes, it is convenient to de"ne the following raising and lowering operators from gamma matrices: !iC ), aI"(C #iC ), k"1,2, N , aR"(C I I\ I I I\ which satisfy the anticommutation relations:
(503)
+aH, aR,"dH , +aH, aI,"0"+aR, aR, . (504) I I H I The lowering operators aI de"ne the `vacuuma "02, satisfying aI"02"0. Under an S;(N) rotation zGPRG zH of the complex coordinates zI,x?I#ix@I spanned by D p-branes, the raising and the H lowering operators transform as aIPRIaH, aRPRR HaR . H I I H
(505)
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One can construct intersecting D p-branes at angles in the following way. One starts from two D p-branes oriented, say, along the directions Re zG and applies S;(p) rotation zGPRG zH H to one of D p-branes. For this type of intersecting D p-branes at angles, the spinor e satis"es the constraint: N N (aR#aI)e" (RR HaR#RIaH)e . I I H H I I
(506)
So, the resulting con"guration has unbroken supersymmetries "02 and N aR"02. These two I I spinors have the same (opposite) chirality for p even (odd). One can further compactify this intersecting D p-branes at angles on a torus and then apply T-duality transformations to obtain other types of intersecting D p-branes at angles. Alternatively, one can start from (q"(p#q) , (p#q) ) and rotate one D (p#q)-brane relative to the other by applying the SO(2p) 0 0 transformation. The resulting con"guration is supersymmetric when the S;(p)LSO(2p) transformation is applied. When these intersecting D-branes are further compacti"ed on tori, the consistency of toroidal compacti"cation imposes the quantization condition for the intersecting angles h's in relation to the moduli of tori [32,89]. When intersecting D-branes at angles are compacti"ed on a manifold M, the unbroken supersymmetry should commute with the `generalizeda holonomy group (de"ned by a modi"ed connection } with torsion } due to non-zero antisymmetric tensor backgrounds) of M. Here, the spinor constraint g" eIC g de"nes an action of discrete generalized holonomy. Generally, G G I starting from an intersecting D p-brane at angles with FK "F#B"0, where F [B] is the worldvolume 2-form "eld strength [the NS-NS 2-form], one obtains a con"guration with FK O0 when T-duality is applied. For intersecting D 2-branes at angles, the necessary and su$cient condition for preserving supersymmetry is that FK is anti-self-dual [89]. Classical solution realization of intersecting D-branes at angles is "rst constructed in [97]. Starting from n parallel D 2-branes (with each constituent D 2-brane located at x"x , a"1,2, n, ? and its charge related to l '0) lying in the (y, y) plane, one rotates each constituent D 2-brane by ? an S;(2) angle a in the (y, y) and (y, y) planes, i.e. (z, z)P(e ??z, e\ ??z) where z"y#iy ? and z"y#iy. The solution in the string frame has the form: ds"(1#X
L 1 !dt# (dyH)# X +[(R )dyG]#[(R )dyH], ? ?G ?H 1#X H ?
# (dxG) , G
dt L A" X (R )dyG(R )dyH ? ?G ?H 1#X ? L ! X X sin(a !a )(dydy!dydy) , ? @ ? @ ?@ e( "(1#X;
L L X, X # X X sin(a !a ) , ? ? @ ? @ ? ?@
(507)
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where R? (a"1,2, n) are (block-diagonal) SO(4) matrices that correspond to the above mentioned rotation of constituent D 2-branes and harmonic functions
l 1 ? X (x)" ? 3 "x!x " ? are associated with constituent D 2-branes located at x"x . This con"guration preserves 1/4 of ? supersymmetry and interpolates between previously known con"gurations: (i) a "0, p/2 case is ? orthogonally oriented D 2-branes, (ii) a "a , ∀a, case is parallel n D 2-branes oriented in di!erent ? direction through S;(2) rotations, etc. The ADM mass density of (507) is the sum of those of constituent D 2-branes, i.e. A L m" l , ? 2i ? and is independent of the S;(2) rotation angles a . The physical charge density is also simply the ? sum of those of constituent D 2-branes, although the charge densities in di!erent planes (yG, yH) of the intersecting D 2-branes depend on a . ? T-duality on (507) yields other types of D-brane bound states. The T-duality along transverse directions leads to angled D p-branes with p'2. The T-duality along worldvolume directions leads to more exotic bound states of D-branes. Namely, since the constituent D 2-branes intersect with one another at angles, the worldvolume direction that one chooses for T-duality transformation is necessarily at angle with some of constituent D 2-branes. Consequently, the resulting con"guration is exotic bound state of D p-brane (pO2) and bound states of the type studied in [96] (e.g. bound state of D (p#1)-brane and D (p!1)-brane, and D (4, 2, 2, 0)-brane bound state) obtained by applying the `tilteda T-duality transformation(s) on a D p-brane. The above intersecting D-branes at angles and related con"gurations are alternatively derived by (i) applying the `tilteda boost transformation on the orthogonally intersecting two D-branes, followed by the sequence of ¹-S-¹ transformations of type-II strings [55], or (ii) applying `reduction along a boosta followed by T-duality transformations [151]. For the former case [151], the resulting con"gurations are mixed bound states of R-R branes that necessarily involves fundamental string. It is essential that one has to turn on both D-brane charges of original orthogonally intersecting D-branes and apply the S-duality between two T-duality transformations to have con"gurations where the D-branes intersect at angles. Intersecting p-branes at angles in more general setting, starting from M-branes with the #at Euclidean transverse space EL replaced by the toric hyper-KaK hler manifold M , are studied in L [285]. We "rst discuss the general formalism and then specialize to the case of intersecting p-branes in D"10, 11. 4n-dimensional toric hyper-KaK hler manifold M with a tri-holomorphic ¹L isometry has the L following general form of metric: ds "; dxG ) dxH#;GH(du #A )(du #A ) (i, j"1,2, n) , &) GH G G H H
(508)
The manifold M is tri-holomorphic i! the triplet KaK hler 2-forms X"(du #A ) dxG!; dxG;dxH are indepenL G G GH dent of u , i.e. L GX"0. G . .P
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where u (which are periodically identi"ed } to remove a coordinate singularity } u &u #2p, G G G thereby parameterizing ¹L) correspond to the ;(1) isometry directions of M and L xG"+xG " r"1, 2, 3, parameterize n copies of Euclidean spaces E. The n"1 case is Taub-NUT P space. The hyper-KaK hler condition relates n 1-forms A "dxH ) x with "eld strengths F "dA to G GH G G ; through ePQJRJ ; "FPQ "RPuQ !RQ uP . (So, a toric hyper-KaK hler metric is speci"ed by GH H IG HIG H IG I HG ; alone.) This implies that ; are harmonic functions on M , i.e. ;GHR ) R ;"0. Generally, GH GH L G H a positive-de"nite symmetric n;n matrix ;(xG) is linear combination ,+N, (509) ; ";# ; [+p,, a (+p,)] GH K GH GH + , N K of the following harmonic functions speci"ed by a set of n real numbers +p "i"1,2, n,, called G a `p-vectora, and an arbitrary 3-vector a: pp G H ; [+p,, a]" . (510) GH 2" p xI!a" I I The vacuum hyper-KaK hler manifold EL;¹L with moduli space Sl(n, Z)!Gl(n, R)/SO(n) has the metric (508) with ; "; (so, A "0). Regular non-vacuum hyper-KaK hler manifold is representGH GH G ed by harmonic functions ; [+p,, a] associated with a 3(n!1)-plane in EL de"ned by 3-vector GH equations L p xI"a. The hyper-KaK hler metric (508) is non-singular, provided +p, are coprime I I integers. The Sl(n, Z) transformation ;PS2;S (S3Sl(n, Z)) on the hyper-KaK hler metric (508) leads to another hyper-KaK hler metric with the Sl(n, Z) transformed p-vector S+p,. The angle h between two 3(n!1)-planes de"ned by two p-vectors +p, and +p , is given by cos h"p ) p /(pp . Here, the inner product is de"ned as p ) q"(;)GHp q , which is invariant under Sl(n, Z). The solution G H (508) is, therefore, speci"ed by angles and distances between mutually intersecting 3(n!1)-planes associated with harmonic functions ; [+p,, a (+p,)]. GH K The special case where *;,;!; is diagonal (i.e. the p-vectors have the form (0,2, 1,2, 0) and *; "d (1/2"xG"): there are only n intersecting 3(n!1)-planes) describes n GH GH fundamental BPS monopoles in maximally broken rank (n#1) gauge theories found in [444], thereby called LWY metric. When additionally ; is diagonal (so that ; is diagonal), n 3(n!1)planes intersect orthogonally (cos h"0) and M "M ;2;M . In this case, one can L always choose u such that F "dA and ; are related as F "夹d; , where 夹 is the Hodge-dual GH G G GG G GG on E. The hyper-KaK hler manifold M preserves fraction of supersymmetry. It admits (n#1) L covariantly constant SO(4n) spinors (in the decomposition of D-dimensional Lorentz spinor representation under the subgroup Sl(n, R);SO(4n)) if the holonomy of M is strictly Sp(n), which L corresponds to the case where 3(n!1)-planes intersect non-orthogonally, i.e. ; is non-diagonal. These covariantly constant SO(4n) spinors arise as singlets in the decomposition of the spinor representation of SO(4n) into representations of holonomy group of M , i.e. Sp(n) for this case. L The only toric hyper-KaK hler manifolds whose holonomy is a proper subgroup of Sp(n) are those corresponding to the `orthogonallya intersecting or `parallela 3(n!1)-planes. For this case, M "M ;2M (i.e. product of n Taub-NUT space) with Sp(1)L holonomy and diagonal L ; (thereby, 3(n!1)-planes intersecting orthogonally), and more supersymmetry is preserved GH since the non-singlet spinor representations of Sp(n) are further decomposed under the proper
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subgroup Sp(1)L. A trivial case corresponds to the case ; ";, i.e. the vacuum hyper-KaK hler GH GH manifold: since the holonomy group is trivial, all the supersymmetries are preserved. The starting point of general class of intersecting p-branes is the following D"11 solution, which is the product of the D"3 Minkowski space and M : ds "ds(E )#; dxG ) dxH#;GH(du #A )(du #A ) , (511) GH G G H H where i"1, 2. For a general solution with non-diagonal ; , thereby with the Sp(2) holonomy for GH M , (511) admits (n#1)"3 covariantly constant spinors. Namely, 32-component real spinor in D"11 is decomposed under Sl(2, R);SO(8) as 32P(2, 8 ) (2, 8 ). Two SO(8) spinor representaQ A tions 8 and 8 are further, respectively, decomposed under Sp(2)LSO(8) as 8 P5 1 1 1 and Q A Q 8 P4 4. So, 3/16 of supersymmetry is preserved. When ; is diagonal (so, M "M ;M and A GH 3-planes orthogonally intersect), the holonomy group is Sp(1);Sp(1). Under the Sp(1);Sp(1) subgroup, non-singlet Sp(2) spinor representations 5 and 4 are, respectively, decomposed as 5P(2, 2) (1, 1) and 4P(2, 1) (1, 2). So, 8/32"1/4 of supersymmetry is preserved. One can generalize the solution (511) to include M-branes without breaking any more supersymmetry, resulting in `generalized M-branesa, where the transverse Euclidean space is replaced by M . The harmonic functions (associated with M-branes) are independent of the ;(1) isometry L coordinates u , thereby M p-branes are delocalized in the u -directions. G G First, one can naturally include an M 2-brane to the solution (511), since the transverse space of M 2-brane has dimensions 8: ds "H\ ds(E )#H [; dxG ) dxH#;GH(du #A )(du #A )] , GH G G H H F"$u(E )dH\ ,
(512)
where u(E ) is the volume form on E , the signs $ are those of M 2-brane charge and H"H(xG) is a harmonic function (associated with M 2-brane) on M , i.e. ;GHR ) R H"0. The G H SO(1, 10) Killing spinor of this solution is decomposed into the SO(8) spinors of de"nite chiralities 8 and 8 , which are related to the signs $. So, depending on the sign of M 2-brane charge, either A Q all supersymmetries are broken or 3/16 of supersymmetry is preserved. Second, one can add M 5-branes to the solution (511) if M "M ;M , i.e. ;"diag(; (x), ; (x)). For this purpose, it is convenient to introduce 2 1-form potentials AI G (i"1, 2) with "eld strengths FI which can be related to the harmonic functions H (x) and H (x) G (associated with 2 M 5-branes) as dH "夹FI . (This is analogous to the relations d; "夹F G G G G satis"ed by the diagonal components of ; and the "eld strengths F "dA of the solution (511) G G when both *; and ; are diagonal.) Here, 夹 is the Hodge-dual on E. The explicit solution has the form: ds "(H H ) [(H H )\ ds(E )#H\[; dx ) dx#;\(du #A )] # H\[; dx ) dx#;\(du #A )]#dz] , F"[FI (du #A )#FI (du #A )]dz . The subscripts s and c denote two possible SO(8) chiralities.
(513)
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Generally with non-constant ; and H , (513) preserves 1/8 of supersymmetry, provided the proper G G relative sign of M 5-brane charges is chosen. In the following, we discuss the intersecting (overlapping) p-brane interpretation of solutions obtained via dimensional reduction and duality transformations of the D"11 solutions (511)}(513). Due to the triholomorphicity of the Killing vector "elds R/Ru , the Killing spinors G survive in these procedures. As for the p-branes associated with the harmonic functions ; , there is GH a one-to-one correspondence between 3(n!1)-planes and p-branes, and the intersection angle of p-branes is given by the angle between the corresponding p-vectors, which de"ne 3(n!1)-planes. First, we discuss intersecting (overlapping) p-branes related to (511) and (512). Since (511) is a special case of (512) with H"1, we consider p-branes related to (512), and then comment on the H"1 case. First, one compacti"es one of the ;(1) isometry directions of M , say u without loss of generality, on S, resulting in a type-IIA solution, and then applies the T-duality transformation along the other ;(1) isometry direction, i.e. the u -direction, to obtain the following type-IIB solution: ds"(det ;) H [H\(det ;)\ds(E )#(det ;)\; dxG ) dxH#H\ dz] , # GH
(514)
(det ; ; , i D"u(E )dH\ , B "A dz, q"! #i I G G ; ; where is the dilaton, q,l#ie\( (l"R-R 0-form "eld), B (i"1, 2) are 2-form potentials in G the NS-NS and R-R sectors, D is the 4-form potential, and z,u . This solution is interpreted as D 3-brane (with harmonic function H) stretching between 5-branes along the z-direction. The 5-branes in this type-IIB con"guration are speci"ed by a set of intersecting 3-planes L p xI"a I I in E. From the expression for q in (514), one sees that the Sl(2, R) transformation ;P(S\)2;S\ (S3Sl(2, R)) in M is realized in this type-IIB con"guration as the type-IIB Sl(2, R) symmetry aq#b qP cq#d of equations of motion, where
S"
a b c
d
.
The condition that Sl(2, R) is broken down to Sl(2, Z) so that M with the coprime integers +p , p , remains non-singular after the transformation is translated into the type-IIB language that the Sl(2, R) symmetry of the equations of motion is broken down to the Sl(2, Z) S-duality symmetry of type-IIB string theory. In the following we discuss particular cases of (514). We "rst consider the solution (514) with ;"diag(H (x), H (x)) and H"1. In this case, M "M ;M with holonomy Sp(1);Sp(1), thereby preserving 1/4 of supersymmetry. The corresponding solution is `orthogonallya intersecting (2"5 , 5 ): ,1 0 ds"(H H ) [(H H )\ds(E )#H\dx ) dx#H\dx ) dx#dz] , (515) # where harmonic functions H "1#(2"xG")\ (i"1, 2) are respectively associated with NS-NS G 5-brane and R-R 5-brane [285].
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A particular case of (514) with ;"1"H and a single 3-plane in E (de"ned by +p , p ,) is the bound state of NS-NS 5-brane and R-R 5-brane with charge vector (p , p ). So, the restriction that M is non-singular, i.e. +p , p , are coprime integer, manifests in the type-IIB theory that the corresponding p-brane con"guration is a non-marginal bound state of NS-NS 5-brane and R-R 5-brane. There is a correlation between the D"11 Sl(2, Z) transformation, which rotates a 3-plane in E, and the type-IIB Sl(2, Z) transformation, which rotates the charge vectors of the 2-form "eld doublet B . G The general type-IIB solution (514) with non-diagonal ; is interpreted as an arbitrary number of 5-branes intersecting or overlapping at angles. p-vectors and ; specify orientations and charges of 5-branes, and a determines distance of 5-branes from the origin. Since the corresponding M has the Sp(2) holonomy, 3/16 of supersymmetry is preserved. Next, we discuss the intersecting p-branes in type-IIA string and M-theory related to (512). The intersecting p-branes in type-IIA theory are constructed in the following way. First, one T-dualizes the type-IIB solution (514) along a direction in E to obtain the following type-IIA `generalized fundamental stringa solution, which can also be obtained from (511) by compactifying on a spatial direction in E : ds"H\ds(E )#; dxG ) dxH#;GH(du #A )(du #A ) , GH G G H H (516) B"u(E )H\, "! ln H . Subsequent T-dualities along u and u lead to the type-IIA solution: ds"H\ds(E )#; (dxG ) dxH#duGduH) , GH (517) B"A duG#u(E )H\, " ln det ;! ln H , G where B is the NS-NS 2-form potential and is the dilaton. This solution is interpreted as an arbitrary number of NS-NS 5-branes intersecting on a fundamental string (with a harmonic function H), generalizing the solutions in [423]. The case with diagonal ; represents orthogonally intersecting NS-NS 5-branes. More general case with non-diagonal ; represents intersecting NS-NS 5-branes at angles and preserves 3/16 of supersymmetry. The following solution in D"11 is obtained by uplifting the type-IIA solution (517): ds "H (det ;) [H\(det ;)\ds(E ) # (det ;)\; (dxG ) dxH#duG duH)#H\dy] , (518) GH F"[F duG#u(E )dH\]dy . G When ; is of LWY type, this solution represents parallel M 2-branes which intersect intersecting 2 M 5-branes (orthogonally when ; is diagonal as well) over a string. For more general form of ;, the solution represents the M 2-branes intersecting arbitrary numbers of M 5-branes at angles and preserves 3/16 of supersymmetry. T-duality on the type-IIA solution (517) along the fundamental string direction leads to intersecting type-IIB NS-NS 5-branes with a pp-wave along the common intersection direction. Further application of Z LS¸(2, Z) S-duality transformation leads to the following solution involving R-R 5-branes, which preserves 3/16 of supersymmetry when 5-branes intersect at
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angles: ds"(det ;) [dt dp#Hdp#; (dxG ) dxH#duG duH)] , # GH (519) B "A duG, q"i(det ; , G where B is the R-R 2-form potential. The H"1 case is classical solution realization of intersecting D-branes at angles of [89]. In [89], the condition for unbroken supersymmetry is given by the holonomy condition arising in the KK compacti"cations. This corresponds to the holonomy condition on the hyper-KaK hler manifold of [285]. Namely, intersecting R-R p-branes preserve fraction of supersymmetry if orientations of the constituent R-R p-branes are related by rotations in the Sp(2) subgroup of SO(8). This is seen by considering spinor constraints of intersecting two D 5-branes, where one D 5-brane is oriented in the (12345) 5-plane and the other D 5-brane rotated into the (16289) 5-plane by an angle h. For this con"guration, type-IIB chiral spinors e (A"1, 2) satisfy the constraints C e"e and R\(h)C R(h)e"e, where R(h)" exp+!h(C #C #C #C ), is the SO(1, 9) spinor representation of the above mentioned SO(8) rotation. (The rotational matrix R(h) is associated with an element of ;(2, H) Sp(2) that commutes with quaternionic conjugation, which is the rotation mentioned above.) It can be shown [285] that (i) for h"0, p, 1/2 of supersymmetry is preserved since the second spinor constraint is trivially satis"ed, (ii) for h"$p/2, 1/4 is preserved, and (iii) for all other values of h, 3/16 is preserved. Finally, we discuss intersecting (overlapping) p-branes related to (513). The compacti"cation on one of the ;(1) isometry directions followed by the T-duality along the other ;(1) isometry direction yields the following type-IIB solution: ds"(H H ; ; ) [(; ; H H )\ds(E )#(; H )\dx ) dx # #(; H )\dx ) dx#(H H )\dz#(; ; )\dy] , (520) H ; . B"A dz#AI dy, B "A dz#AI dy, q"i H ; This solution represents 2 NS-NS 5-branes in the planes (1, 2, 3, 4, 5) and (1, 6, 7, 8, 9) and 2 R-R 5-branes in the planes (1, 5, 6, 7, 8) and (1, 2, 3, 4, 9) intersecting orthogonally. Since the spinor constraint associated with one of the constituent p-branes is expressed as a combination of the rest three independent spinor constraints, the solution preserves () of supersymmetry. Related intersecting p-brane is constructed by applying T-duality, oxidation and dimensional reduction. One can further include additional p-branes without breaking any more supersymmetry, provided the spinor constraints of the added p-branes can be expressed as a combination of spinor constraints of the existing p-branes.
6.3. Dimensional reduction and higher dimensional embeddings The lower-dimensional (D(10) p-branes can be obtained from those in D"10, 11 through dimensional reduction. Reversely, most of lower-dimensional p-branes are related to D"10, 11 p-branes via dimensional reduction and dualities. In particular, many black holes in D(10 originate from D"10,11 p-brane bound states, which makes it possible to "nd microscopic origin
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of black hole entropy. It is purpose of this section to discuss various p-branes in D(10. We also discuss various p-brane embeddings of black holes. There are two ways of compactifying p-branes to lower dimensions. First, one can compactify along a longitudinal direction. It is called the `double dimensional reductiona (since both worldvolume and spacetime dimensions are reduced, bringing a p-brane in D dimensions to (p!1)brane in D!1 dimensions diagonally in the D versus p brane-scan) or `wrappinga of branes (around cycles of compacti"cation manifold). Since target space "elds are independent of longitudinal coordinates, one only needs to require periodicity of "elds in the compacti"cation directions. Second, one can compactify a transverse direction of a p-brane. It is called the `direct dimensional reductiona (since this takes us vertically on the bran-scan, taking a p-brane in D dimensions to a p-brane in D!1 dimensions) or `constructing periodic arraysa of p-branes (along the compacti"ed direction). Since "elds depend on transverse coordinates, direct dimensional reduction is more involved [286,422,461]. For this purpose, one takes periodic array of parallel p-branes (with the period of the size of compact manifold) along the transverse direction. Then, one takes average over the transverse coordinate, integrating over continuum of charges distributed over the transverse direction. The resulting con"guration is independent of the transverse coordinate, making it possible to apply standard Kaluza}Klein dimensional reduction. In the double [direct] dimensional reduction, the values of p [p] and D are preserved; in the direct dimensional reduction, the asymptotic behavior of the "elds (which goes as &1/"y"N ) changes. Conventionally, the direct dimensional reduction uses the zero-force property of BPS p-branes, which allows the construction of multicentered p-branes. Note, however that it is also possible to apply the vertical dimensional reduction even for non-BPS extreme p-branes [461] and non-extreme p-branes [463,442], contrary to the conventional lore. Namely, since the equations of motion (of a non-extreme, axially symmetric black (D!4)-brane in D dimensions, for the non-extreme case) can be reduced to Laplace equations in the transverse space with suitable choice of "eld AnsaK tze, one can still construct multi-center p-branes for non-BPS and non-extreme cases as well. For the non-extreme case, when an inxnite number of non-extreme p-branes are periodically arrayed along a line, the net force on each p-brane is zero and the conical singularities along the axis of periodic array act like `strutsa that hold the constituents in place. Furthermore, since the direction of periodic array is compacti"ed on S with each p-brane precisely separated by the circumference of S, the instability problem of such a con"guration can be overcome. One can also lift p-branes as another p-branes in higher-dimensions, so-called &dimensional oxidation'. First, the oxidation of a p-brane in D dimensions to a p-brane in D#1 dimensions (i.e. the reverse of the direct dimensional reduction) is never possible in the standard KK dimensional reduction, since the oxidized p-brane in D#1 dimensions has to depend on the extra transverse
Such staking-up procedure breaks down for (D!3)-branes in D dimensions, due to conical asymptotic spacetime [461]. This is also related to the fact that (D!2)-branes reside in massive supergravity, rather than ordinary massless type-II theory. Or one can promote such isometry directions as the spatial worldvolume of a p-brane, leading to an intersecting p-brane solution [287,424,461,611].
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coordinate introduced by oxidation. (This is analogous to the `dangerousa T-duality transformation.) Second, the oxidation of a p-brane in D dimensions to a (p#1)-brane in D#1 dimensions (i.e. the reverse of the double dimensional reduction) is classi"ed into two groups. A p-brane in D dimensions is called `rustya if it can be oxidized to a (p#1)-brane in D#1 dimensions. Otherwise, it is called `stainlessa. Thus, p-branes in bran scan are KK descendants of stainless p-branes in some higher dimensions. A p-brane in D dimensions is `stainlessa when (i) there is not the antisymmetric tensor in D#1 dimensions that the corresponding (p#1)-brane couples to, or (ii) the exponential prefactors a and a for (p#2)-form "eld strength kinetic terms in D#1 and "> " D dimensions do not satisfy the relation 2(p #1) . a "a ! "> " (D!2)(D!1) (This relation is satis"ed by the expression for a in (531), provided D remains unchanged in the dimensional reduction procedure.) In this section, we focus on p-branes in D(10 with only "eld strengths of the same rank turned on, comprehensively studied in [243,455}463]. A special case is black holes, which are 0-brane bound states. We also discuss their supersymmetry properties and interpretations as bound states of higher-dimensional p-branes. 6.3.1. General solutions We concentrate on p-branes in D"11 supergravity on tori. Bosonic Lagrangian of D"11 supergravity is (521) L "(!GK [R K ! FK ]#FK FK AK , % where FK "dAK is the "eld strength of the 3-form potential AK . So, such p-branes have interpreta tion in terms of M-theory or type-II string theory con"gurations. Although one can directly reduce the D"11 action down to D(11 by compactifying on ¹\" in one step, it turns out to be more convenient to reduce the action (521) one dimension at a time iteratively until one reaches D dimensions. Namely, one compacti"es 11!D times on S, making use of the following KK Ansatz: ds "e?Pds #e\"\?P(dz#A ) , "> " (522) A (x, z)"A (x)#A (x)dz , L L L\ where u is a dilatonic scalar, A "A dxI is a KK 1-form "eld, A is an n-form "eld arising from I L AK and a,1/(2(D!1)(D!2). The explicit form of resulting D(11 action can be found elsewhere [456]. The advantage of such compacti"cation procedure is that spin-0 "elds are
Such a p-brane in D dimensions should rather be viewed as a p-brane in D#1 dimensions whose charge is uniformly distributed along the extra coordinate. This is interpreted as the limit where one of charges of intersecting two p-branes in D#1 dimensions is zero [461]. Contrary to the conventional lore that all the p-branes in D(10 are obtained from those in D"10, 11 through dimensional reductions, there are stainless p-branes in D(10 which cannot be viewed as dimensional reductions of p-branes in D"10, 11. So, the conventional brane scan is modi"ed [459].
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manifestly divided into two classes in the Lagrangian. Namely, only dilatonic scalars
"( ,2,
) appear in the exponential prefactors of the n-form "eld kinetic terms. The \" dilatonic scalars originate from the diagonal components of the internal metric and are true scalars. The couplings of to n-form potentials A? are characterized by the `dilaton vectorsa a in the L ? following way: 1 (523) e\L "! ea? (F? ) . L> LU 2n! ? The complete expressions for `dilaton vectorsa, which are expressed as linear combinations of basic constant vectors, are found in [456]. On the other hand, the remaining spin-0 "elds coming from the o!-diagonal components of GK and the internal components of AK are axionic, being +, associated with constant shift symmetries, and should rather be called 0-form potentials, which couple to solitonic (D!3)-branes. (Note, there are no elementary p-branes for 1-form "eld strengths.) Up to present time, study of p-branes within the above described theory has been mostly concentrated on the case where only n-form potentials of the same rank are turned on, with the restrictions that terms related to the last term in (521) (denoted as L from now on) and the $$ `Chern}Simonsa terms in (n#1)-form "eld strengths are zero. These restrictions place constraints on possible charge con"gurations for p-branes. These constraints become non-trivial when a pbrane involves both undualized and dualized "eld strengths, i.e. when the p-brane has both electric and magnetic charges coming from di!erent "eld strengths. The former [later] type of constraint is satis"ed as long as the dualized and undualized "eld strengths have [do not have] common internal indices i, j, k. 6.3.1.1. Supersymmetry properties. The supersymmetry preserved by p-branes is determined from the Bogomol'nyi matrix M, which is de"ned by the commutator of supercharges Q " Re C !t dR per unit p-volume: ! C .
N dR &eR Me , (524) .R where N "e C !d t is the Nester's form de"ned from the supersymmetry transformation rule C ! of D"11 gravitino t : (525) N "e C !D e #e C!!e F # e C !2!e F 2 . !! ! ! ! The "rst term in N gives rise to the ADM mass density and the last two terms respectively contribute to the electric and magnetic charge density terms in M. The Bogomol'nyi matrix for the 11-dimensional supergravity on (S)\" is in the form of the ADM mass density m term plus the electric and magnetic Page charge density [490] (de"ned respectively as (1/4u ) "\L夹F and L "\L 1 (1/4u ) LF ) terms. L 1 L [Q , Q ]" C C
In particular, to set the 0-forms AGHI to zero consistently with their equations of motion, the bilinear products of "eld strengths that occur multiplied by AGHI in L should vanish. $$
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Since the Bogomol'nyi matrix is obtained from the Hermitian supercharges, its eigenvalues are non-negative. The matrix M has zero eigenvalues for each component of unbroken supersymmetry associated with the Killing spinor e satisfying d t "0. Since D"11 spinor has 32 components, C the fraction of preserved supersymmetry is k/32, where k is the number of 0 eigenvalues of M (i.e. the nullity of the matrix M) or equivalently the number of linearly independent Killing spinors. The amount of preserved supersymmetry is determined as follows. First, one calculates the ADM mass density m from the p-brane solutions. Then, one plugs m, together with the Page electric and magnetic charge densities of the p-branes, into the Bogomol'nyi matrix. The multiplicity k of 0 eigenvalues of the resulting matrix M determines the fraction of supersymmetry preserved by the corresponding p-branes. 6.3.1.2. Multi-scalar p-branes. General p-branes with more than one non-zero p-brane charges are called `multi-scalar p-branesa, since such p-branes have more than one non-trivial dilatonic scalars. The Lagrangian density has the following truncated form:
1 1 ea? (F? ) , (526) L"(!g R! (R )! N> 2 2(p#2)! ? where "eld strengths F? "dA? are de"ned without `Chern}Simonsa modi"cations. In this N> N> action, the rank p#2 of "eld strengths is assumed to not exceed D/2, namely those with p#2'D/2 are Hodge-dualized. This is justi"ed by the fact that the dual of "eld strength of an elementary (solitonic) p-brane is identical to the "eld strength of solitonic (elementary) (D!p!2)-brane, with the corresponding dilaton vector di!ering only by sign. We consider extreme p-branes with N non-zero (p#2)-form "eld strengths, each of which is either elementary or solitonic, but not both. For the simplicity of calculations, the SO(1, p!2);SO(D!p#1) symmetric metric Ansatz (448) is assumed to satisfy (p#1)A# (p #1)B"0 [457], so that the "eld equations are linear. The p-brane solutions are then determined completely by the dot products M ,a ) a of the dilaton vectors a associated with non?@ ? @ ? zero (p#1)-form "eld strengths F? (a"1,2, N). In solving the equations, it is assumed that N> M is invertible, which requires the number N of non-trivial F? to be not greater than the num?@ N> ber of the components in , i.e. N411!D. For such p-branes, only N components u ,a ) of ? ?
are non-trivial. If one further takes the Ansatz !eu #2(p#1)AJ (M\) u , then M @ ?@ @ ?@ ? takes the form: 2(p#1)(p #1) M "4d ! . ?@ ?@ D!2
(527)
The conditions on "elds that linearize the "eld equations and lead to M of the form (527) are also ?@ dictated by supersymmetry transformation rules for the BPS con"gurations. Thus, a necessary condition for `multi-scalar p-branesa to be BPS is for the dilaton vectors a associated with ? participating "eld strengths to satisfy (527). The following extreme multi-scalar p-brane solution is
When M is singular, analysis depends on the number of rescaling parameters [456]. The only new solution is the ?@ case a "0, which yields solutions with a"0 and (F?)"F/N, ∀a. ? ?
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obtained by taking further simplifying Ansatz discussed in [457]: , , e CP?\N> "H , ds" H\N > "\ dxI dxJg # HN> "\ dyK dyK , (528) ? ? IJ ? ? ? where harmonic functions 1 j H "1# ? (y,(yKyK) ? p #1 yN> are associated with p-branes carrying the Page charges P "j /4, and the "eld strengths are given, ? ? respectively, for the electric and magnetic cases by F? "dH\dN>x, F? "夹(dH\dN>x) . (529) N> ? N> ? The elementary and solitonic p-branes are related by P! . The ADM mass density is the sum of the mass densities of the constituent p-branes, i.e. m" , P . Multi-center generalization is ? ? achieved by replacing harmonic functions by [424]. j ? G H "1# ? "y!y "N > ? G G 6.3.1.3. Single-scalar p-branes. We discuss the case where the bosonic Lagrangian for 11-dimensional supergravity on (S)\" is consistently truncated to the following form with one dilatonic scalar and one (p#2)-form "eld strength [456,459]:
1 1 e?((F ) , L"(!g R! (R )! N> 2 (p#2)!
(530)
where we parameterize the exponential prefactor a in the form: 2(p#1)(p #1) a"D! . D!2
(531)
This expression for a is motivated from (422), now with an arbitrary parameter D replacing 4. By consistently truncating (526), one has (530) with (F ), (F? ) and a, given by (for the case ? N> N> M is invertible) ?@ \ a" (M\) , "a (M\) a ) . (532) ?@ ? ?@ ? @ ? @ By taking AnsaK tze which reduce equations of motion for (530) to the "rst order, one obtains [459] the `single-scalar p-branea solution with the Page charge density P"j/4:
e("H? CD,
ds"H\N > D"\ dxI dxJ g #HN> D"\ dyK dyK , IJ
where 1 (Dj H"1! . 2(p#1) rN>
(533)
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The mass density is m"j/2(D. Although inequivalent charge con"gurations give rise to the same D, i.e. the same solution, supersymmetry properties depend on charge con"gurations. Note, although one can obtain p-brane solutions (533) for any values of p and a, hence for any values of D, only speci"c values of p and a can occur in supergravity theories. The value of D is preserved in the compacti"cation process, provided no "elds are truncated. For p-branes with 1 constituent, D is always 4, as can be seen from the form of a in (442), determined by the requirement of scaling symmetry, and always 1/2 of supersymmetry is preserved. The value D"4 can also be understood from the facts that D"4 in D"11, since there is no dilaton in D"11, and the value of D is preserved in dimensional reduction not involving "eld truncations. When the "eld strength is a linear combination of more than one original "eld strengths, D(4. With all the Page charges P "j /4 of `multi-scalar p-branea (528) equal, one has `single-scalar ? ? p-branea with the Page charge P"j/4 (j"(Nj ). By substituting M in (527) into (532), one has ? ?@ D"4/N. So, `single-scalar p-branesa with D"4/N (N52) are bound states of N single-charged p-branes (with D"4) with zero binding energy, and preserve the same fraction of supersymmetry as their multi-scalar generalizations. Only `single-scalar p-branesa with D"4/N (N5Z>) and `multi-scalar p-branesa can be supersymmetric. (Non-supersymmetric p-branes in this class is related to supersymmetric ones by reversing the signs of certain charges.) And only single-scalar p-branes with D"4/N (N52) have multi-scalar generalizations. 6.3.1.4. Dyonic p-branes. In D"2(p#2), p-branes can carry both electric and magnetic charges of (p#2)-form "eld strengths. There are two types of dyonic p-branes [456]: (i) the "rst type has electric and magnetic charges coming from di!erent "eld strengths, (ii) the second type has dyonic "eld strengths. As in the multi-scalar p-brane case, the requirements that L "0 and the $$ Chern}Simons terms are zero place constraints on possible dyonic solutions in D"2(p#2)"4, 6, 8. Such restrictions rule out dyonic p-branes of the "rst type in D"6, 8. For the second type, dyonic p-brane in D"8 is special since it has non-zero 0-form potential A [388], thereby requiring non-zero source term FK FK e+,./0123, and can be obtained +,./ 0123 from purely electric/magnetic membrane by duality rotation, unlike dyonic D"6 string and D"4 0-brane of the second type. Dyonic p-branes of the second type include self-dual 3-branes in D"10 [238,367], self-dual string [242] and dyonic string [230] in D"6, and dyonic black hole in D"4 [456]. There are two possible dyonic p-branes (associate with (530)) of the second type with the Page charge densities j /4 [456]: (1) a"p#1 case (i.e. *"2p#2) with the solution G 1 1 j j , e ?(\N> "1# , e\ ?(\N> "1# a(2 rN> a(2 rN>
(534)
For p-branes with DO4, the scaling symmetry of combined worldvolume and e!ective supergravity actions is broken. The solutions for dyonic p-branes of the "rst type have the form (528) with Lagrangian (526) containing both Hodgedualized and undualized "eld strengths.
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and (2) a"0 case (i.e. D"p#1) with the solution
1 j#j 1
"0, e\N> "1# . 2 p#1 rN>
(535)
The ADM mass density for (534) is m"(1/2(D)(j #j ), whereas for (535) m"(1/2(D)(j#j. The solution (535) is invariant under electric/magnetic duality and, there fore, is equivalent to the purely elementary (j "0) or solitonic (j "0) case. For (534) with j "j , the "eld strength is self-dual and "0, thereby (534) and (535) are equivalent, but for (535) j and j are independent. When j "!j , (534) corresponds to anti-self-dual massless string with enhanced supersymmetry. Note, the solutions (534) and (535) are not restricted to those obtained from the D"11 supergravity on tori. For D"8, 6 and 4, which are relevant for the D"11 supergravity on tori, D's for (534) and (535) are respectively +6, 3,, +4, 2, and +2, 1,. So, (534) and (534) with p"2 (i.e. D"8) are excluded. 6.3.1.5. Black p-branes. We discuss non-extreme p-branes. Non-extreme p-branes are additionally parameterized by the non-extremality parameter k'0. There are two ways of constructing non-extreme p-branes. The "rst method involves a universal prescription for `blackeninga extreme p-branes, which deforms extreme solutions with [243]: k (r,"y") eD"1! rN> dtPeD dt, drPe\D dr ,
(536)
while modifying harmonic functions associated with p-branes as H"1#k sin h2d / ? rN >P1#k sinh d /rN >. The resulting non-extreme p-branes, called `type-2 non-extreme p? branesa, have an event horizon at r"r "k N >, which covers the singularity at the core r"0. > The ADM mass density has the generic form m& ((Q )#k, which is always larger than the ? ? extreme counterpart, and all the supersymmetry is broken since the Bogomol'nyi bound is not saturated. For type-2 non-extreme p-branes (with p51), the PoincareH invariance is broken down to R;EN because of the extra factor eD in the (t, t)-component of the metric. For 0-branes, the metric remains isotropic but the quantity (p#1)A#(p #1)B no longer vanishes. In the second method, the metric Ansatz (448) remains intact but instead general solution to the "eld equations is obtained [455,462] without simplifying AnsaK tze, e.g. (p#1)A#(p #1)B"0, that linearize "eld equations. (In solving the "eld equations without simplifying AnsaK tze, one encounters an additional integration constant interpreted as non-extremality parameter.) So, the resulting non-extreme p-branes, called `type-1 non-extreme p-branesa, preserve the full PoincareH invariance (in the worldvolume) of extreme p-branes. So, type-1 non-extreme p'0 solutions do not overlap with the type-2 non-extreme counterparts. But type-1 non-extreme 0-branes contain type-2 non-extreme 0-branes as a subset. The equations of motion for single-scalar p-branes and dyonic p-branes of the second type are, respectively, casted into the forms of the Liouville equation and the Toda-like equations for two
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variables, which are subject to the "rst integral constraint. The equations of motion for dyonic p-branes are solvable when a"p#1 (i.e. D"2(p#1)) or a"3(p#1) (i.e. D"4(p#1)). When a"p#1, the equations of motion are reduced to two independent Liouville equations. Since D44 in supergravity theories, only dyonic strings in D"6 and dyonic black holes in D"4 are relevant, with only dyonic strings having BPS limit. When a"3(p#1), the equations of motion are reduced to S;(3) Toda equations. Only dyonic black holes are possible in supergravity theory for this case. In the extreme limit, such black holes preserve supersymmetry when either electric or magnetic charge is zero. For multi-scalar p-branes with N "eld strengths, the equations of motion are Toda-like in general, but when the extreme limit is BPS (i.e. dilaton vectors satisfy (527)) the equations of motion become N independent Liouville equations. The requirements that non-extreme p-branes are asymptotically Minkowskian and dilatons are "nite at the event horizon (thereby the event horizon is regular) place restrictions on parameters of the solutions. 6.3.1.6. Massless p-branes. For multi-scalar p-branes and a dyonic p-brane (535) of the second type, the ADM mass density has the form m& j . So, they can be massless when some of the ? ? Page charges are negative. In this case, there are additional 0 eigenvalues of the Bogomol'nyi matrix, enhancing supersymmetry. Generally massless p-branes are ruled out if one requires the Bogomol'nyi matrix to have only non-negative eigenvalues, since the Bogomol'nyi matrix is obtained from the commutator of the Hermitian supercharges. Since some of the Page charges are negative, the massless p-branes have naked singularity. On the other hand, if one allows negative eigenvalues, one can have p-branes preserving more than 1/2 of supersymmetry and some of non-BPS multi-scalar p-branes can become supersymmetric due to the appearance of 0 eigenvalues with suitable sign choice of Page charges (but their single-scalar counterparts are non-BPS, since Page charges have to be equal in the single-scalar limit) [457]. 6.3.2. Classixcation of solutions In this subsection, we classify p-branes discussed in the previous subsection according to their supersymmetry properties. Since single-scalar p-branes and their multi-scalar generalizations preserve the same amount of supersymmetry (except for the special case of massless p-branes), the classi"cation of multi-scalar p-branes is along the same line as that of single-scalar p-branes. Single-scalar p-branes are supersymmetric only when D"4/N (N3Z>) and the dilaton vectors a (associated with the participating "eld strengths) of their multi-scalar p-brane counterparts ? satisfy relations (527). Spin-0 "elds, i.e. dilatonic scalars and 0-form "elds, form target space manifold of p-model. The target space manifold has a coset structure G/H, where GKE (R) (n"11!D) is the (realL>L valued) ;-duality group and H is a linearly realized maximal subgroup of G. Under the G-transformations, the equations of motion are invariant. When the DSZ quantization is taken
In the exceptional case of 4-scalar solution with 2-form "eld strengths in D"4, it is possible to have massless p-branes where the Bogomol'nyi matrices have no negative eigenvalues [457]. For D46, this is the case only when all the "eld strengths are Hodge-dualized to those with rank 4D/2 [458].
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into account, G and H break down to integer-valued subgroups. The subgroup G(Z) is the conjectured ;-duality group of type-II string on tori. The asymptotic values of spin-0 "elds, called `modulia, de"ne the `scalar vacuuma. The asymptotic values of dilatonic scalars and 0-form "elds are, respectively, interpreted as the `coupling constantsa and `h-anglesa of the theory. One can parameterize spin-0 "elds by a G-valued matrix <(x) which transforms under rigid G-transformation by right multiplication and under local H-transformation by left transformation: <(x)Ph(x)<(x)K\, h(x)3H and K3G. It is convenient to de"ne a new scalar matrix M,<2<, which is inert under H but transforms under G as MPKMK2. So, the ;-duality group G generally changes the `vacuuma of the theory. By applying a G-transformation, one can bring the asymptotic value of M to the canonical form M "1. The subgroup H leaves M "1 intact (i.e. H is the ;-duality little group of the scalar vacuum), thereby acting as solution classifying isotropy group (of the vacuum) that organizes the distinct solutions of the theory into families of the same energy. The integer-valued subgroup G(Z)5H is identi"ed with the Weyl group of G that transforms the set of dilaton vectors a associated with "eld strengths of the same rank as weight vectors of the irreducible representa? tions of G(Z). The ; Weyl group in D49 contains a subgroup S consisting of the permutations of the \" internal coordinates (i j), corresponding to the permutations of "eld strengths, and (for D48) the additional discrete symmetries that interchange "eld strengths and the Hodge dualized "eld strengths (namely, the interchange of "eld strength equations of motion and Bianchi identity). At the same time, the associated dilaton vectors transform in such a way that theory is invariant, respectively, by permutation or change of signs, forming an irreducible multiplet under the Weyl group. In particular, M are invariant under the ; Weyl transformations and therefore D is also ?@ preserved. Furthermore, since the Bogomol'nyi matrix M is invariant under the ; Weyl group, p-branes with the same eigenvalues of M and, therefore, the same supersymmetry property are related by the ; Weyl group. Starting with a p-brane with a set of a , one generates a ; Weyl group ? multiplet of p-branes with the same M (or same D) and the same eigenvalues of M. In the case of ?@ multi-scalar or dyonic p-branes, where the N Page charges are independent parameters, the size of the ; Weyl multiplet is larger than that of single-scalar p-brane counterparts, since the participating "eld strengths are now distinguishable. We classify p-branes according to the rank of "eld strengths that p-branes couple to [456,457]. Particularly, BPS p-branes are possible with N"1 4-form/3-form "eld strength, N44 2-form "eld strengths and N47 1-form "eld strengths. BPS p-branes with N participating "eld strengths appear in lower dimensions once they occur in some higher dimensions; the p-branes in those maximal dimensions are `stainless super p-branesa. Generally, BPS p-branes with D"4, 2, respectively preserve , , of supersymmetry, and 0-branes with D", , , which occur only in D"4, all preserve . As for the super p-branes with 4 "eld strengths, there are two inequivalent solutions: (i) those that preserve of supersymmetry (denoted D"1 ) and are coupled to 2-form In the following, we also call the real-valued group G as the ;-duality group, but the distinction between G(R) and G(Z) will be clear from the context. This permutation is a discrete subset of G(R) which acts on a "eld strength multiplet linearly.
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and 1-form "eld strengths (ii) those that preserve (denoted D"1) and are coupled to 1-form "eld strengths, only. Lastly, we show that H-transformations on black holes discussed in Section 4 generate the most general black holes in D"11 supergravity on tori [171]. 6.3.2.1. 4-form xeld strength. There is only one 4-form "eld strength in each dimension, but within the supergravity models under consideration in this section, the 4-form "eld strength exists only in D58, since those in D(8 are Hodge-dualized to lower ranks. So, no multi-scalar generalization is possible. There is a unique single-scalar p-brane, which is either elementary membrane or solitonic (D!6)-brane. In D"8, one can construct dyonic membrane of the second type (534), but it is ruled out by the constraint L "0. $$ 6.3.2.2. 3-form xeld strengths. There are 11!D 3-forms in D56, except in D"7 where there is an extra 3-form coming from the Hodge-dualization of the 4-form. The associated dilaton vectors satisfy M "2d !2(D!6)/(D!2), which are not of the form (527), and, therefore, the multi?@ ?@ scalar generalization is not possible. In fact, this expression for M yields D"2#2/N in the limit ?@ F"F/N, ∀a: supersymmetry is completely broken unless N"1 (i.e. D"4), in which case 1/2 of ? supersymmetry is preserved. In D"6, one can construct dyonic strings. Due to the constraint L "0, only dyonic strings of the second type, which are (534) and (535) with D"4 and 2, $$ respectively, are possible. 6.3.2.3. 2-Form xeld strengths. Two-form "eld strengths couple to elementary 0-branes and solitonic (D!4)-branes. Analysis of 2-form "eld strengths and 1-form "eld strengths is complicated due to their proliferation in lower dimensions. We therefore discuss only supersymmetric cases; complete classi"cation of p-branes including non-supersymmetric ones can be found in [456,457]. The dilaton vectors a associated with N participating 2-form "eld strengths satisfy (527) only for ? N44. The dimensions D in which these BPS p-branes with N"1, 2, 3, 4 2-form "eld strengths appear are, respectively, D410, 9, 5, 4. For N"1, 2, 3, the p-branes preserve 2\, of supersymmetry, and for the N"4 case, the solutions preserve 1/8. Whereas p-branes with N43 can be either purely electric/magnetic or dyonic (of the "rst type), p-branes with N"4 are intrinsically dyonic (of the "rst type). In D"4, there are 4 inequivalent BPS black holes with D"4/N (N"1, 2, 3, 4), corresponding to dilaton couplings a"(3, 1, 1/(3, 0, respectively. These black holes are interpreted as bound states of N D"5 KK black holes with a"(3 [249,511]. In D"4, dyonic 0-branes of both "rst and second types satisfy the constraint L "0. $$ Discussion on the "rst type is along the same line as the multi-scalar 0-branes. As for the second type, we have solutions (534) and (535) with a"1/(3 (i.e. N"2) and a"0 (i.e. N"4), respectively. First, a"0 case is intrinsically dyonic of the "rst type even when j "0 or j "0. Although the explicit forms of solutions are insensitive to signs of Page charges, their supersymmetry properties depend on their relative signs. Second, the supersymmetry property of a"1/(3 case is insensitive to the signs of Page charges. Supersymmetry is preserved when (i) j "0 or j "0, Note, only for these values of a, the 0-branes have a regular event horizon.
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corresponding to purely solitonic or elementary solution preserving 1/4 of supersymmetry, or (ii) j "!j , corresponding to a massless black hole preserving 1/2. 6.3.2.4. 1-form xeld strengths. 1-form "eld strengths couple to solitonic (D!3)-branes, only. The dilaton vectors satisfy (527) for N47 participating "eld strengths. N"1, 2, 3 cases occur, respectively, in D49, 8, 6, whereas N"5, 6, 7 cases occur in D"4, only. As for the N"4 case, there are 2 inequivalent BPS solutions: (i) those occurring in D46, denoted N"4 or D"1 and (ii) those occurring only in D"4, denoted N"4 or D"1. For generic values of Page charges, p-branes preserve 2\, [ ] of supersymmetry for N"1, 2, 3, 4 [N"5, 6, 7]. The N"4 case preserves . In the case D"1 , , , , p-branes are BPS or non-BPS, depending on the signs of the Page charges. However, for D"4, 2, , 1, their supersymmetry properties are independent of the Page charge signs. 6.3.2.5. Black holes in 44D49. The 0-branes in D"11 supergravity on tori with the most general charge con"gurations can be obtained by applying subsets of ;-duality transformations on the generating solutions. As in the case of black holes in heterotic string theory on tori, the set of transformations that generate the general black holes with the canonical asymptotic value of scalar matrix M "1 from the generating solutions is of the form H/H , where H is the largest subgroup of H that leaves the generating solutions intact. The H/H transformation introduces dim(H)!dim(H ) parameters, which together with the parameters of the generating solutions form the complete parameters of the most general solution. The number of ;(1) charges of the generating solutions are 5, 3, 2 for D"4, 5,56, respectively. The charge con"gurations for these generating solutions are the same as the heterotic case in Section 4, with all the charges coming from the NS-NS sector. To generate solutions with an arbitrary asymptotic value of the scalar matrix M, one additionally imposes a general (real-valued) ;-duality transformation. The `dresseda 0-brane charge ZM "< Z can be rearranged in an N;N anti-symmetric complex matrix Z (A, B"1,2, N), where N is the number of the maximal supersymmetry " in D dimensions. ZM and Z are invariant under the global G-transformation but transform " under the local H-transformation. Z appears in the supersymmetry algebra in the form " [Q , Q ]"C Z . In general, Z is splitted into blocks of ,;, submatrices. Two diagonal ? @ ?@ " " blocks Z correspond to NS-NS charges and two o!-diagonal blocks represent R-R charges. 0 * By applying the H-transformation Z PZ "hZ h2 (h3H), one can bring the matrix Z into " " " " the skew-diagonal form with complex skew eigenvalues j (i"1,2, N/2). These eigenvalues j G G are related to charges of the generating solutions in a simple way, which we show in the following.
For 0-branes in D"4, the matrix Z is de"ned as follows [159,381]. The electric q and magnetic p charges of the ' ' ;(1) gauge group of the N"8, D"4 supergravity are combined into a 56-vector Z2"(p', q ), which transforms ' under G as ZPKZ. The dressed 0-brane charge ZM "< Z"(p ', q )2 is invariant under G but transforms under local ' S;(8). The central charge matrix Z , which is the complex antisymmetric representation of S;(8), that appears in the N"8, D"4 supersymmetry algebra is related to the `dresseda charges q and p ' as Z "(q #ip ')t' , where ' ' t' "!t' are the generators of SO(8).
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We now discuss the subsets of H that generate 0-branes with the most general charge con"gurations from the generating solutions. E D"4: The most general 0-brane carries 28#28 electric and magnetic charges of the ;(1) gauge group. The U-duality group is G"E with the maximal compact subgroup H"S;(8). The skew eigenvalues j (i"1,2, 4) are related to the charges Q ,Q $Q , G 0 * P ,P $P and q of the generating solution as 0 * j "Q #P , j "Q !P , 0 0 0 0 (537) j "Q #P #2iq, j "Q !P !2iq . * * * * The subset of H"S;(8) that leaves the generating solution unchanged is SO(4) ;SO(4) . The * 0 63!(6#6)"51 parameters of H/H "S;(8)/[SO(4) ;SO(4) ] are introduced to the generat * 0 ing solution. E D"5: The `dresseda 27 electric charges of the most general 0-brane transform as a 27 of the ;Sp(8) maximal compact subgroup of U-duality group E . The skew eigenvalues j (i"1,2, 4) G ,Q $Q and Q of the generating with a constraint j "0 are related to the charges Q G G 0 * solution as
E
E
E
E
j "Q#Q , j "Q!Q , j "!Q#Q , j "!Q!Q . (538) 0 0 * * The subset SO(4) ;SO(4) L;Sp(8) leaves this charge con"guration intact. The * 0 ;Sp(8)/[SO(4) ;SO(4) ] transformation introduces remaining 36!12"24 charge degrees of * 0 freedom into the generating solution. D"6: The most general 0-brane carries 16 electric charges, which transform as a 16 (spinor) of the SO(5, 5) U-duality group, whereas the `dresseda charges transform as (4, 4) under the maximal compact subgroup SO(5);SO(5). The skew eigenvalues j (i"1, 2) are related to the G charges Q ,Q $Q as 0 * j "Q , j "Q . (539) 0 * The subgroup SO(3) ;SO(3) of the maximal compact subgroup SO(5);SO(5) leaves the * 0 generating solution intact. The transformation [SO(5);SO(5)]/[SO(3) ;SO(3) ] introduces * 0 remaining 2(10!3)"14 charge degrees of freedom. D"7: The most general 0-brane carries 10 electric charges, which transform as a 10 under the S¸(5, R) U-duality group, whereas the `dresseda charges also transform as 10 under SO(5). The skew eigenvalues j (i"1, 2) are related to the charges Q ,Q $Q in the same way as G 0 * the D"6 case. The subgroup SO(2) ;SO(2) of the maximal compact subgroup SO(5) preserves * 0 the generating solution. 10!2"8 parameters of SO(5)/[SO(2) ;SO(2) ] are introduced into * 0 the generating solution. D"8: 6 electric charges of the general 0-brane transform as (3, 2) under the U-duality group S¸(3, R);S¸(2, R). There is no subgroup of the maximal compact subgroup SO(3);;(1) that leaves the generating solution intact. The SO(3);;(1) transformation induces 3#1"4 remaining charge degrees of freedom into the generating solution. D"9: 4 electric charges of the general 0-brane transform as (3, 1) under the U-duality group S¸(2, R);R>. The maximal compact subgroup ;(1) introduces an additional electric charge into the generating solution.
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Since the equations of motion and, especially, the Einstein frame metric are invariant under the U-duality, it is natural to expect that quantities derived from the metric, e.g. the ADM mass and the Bekenstein}Hawking entropy, are U-duality invariant. In fact, one can express the Bekenstein} Hawking entropy in a manifestly U-duality invariant form in terms of unique G invariants of D"11 supergravity on tori. Such manifestly U-duality invariant entropy depends only on `integer-valueda quantized bare charges [260]. In the following, we give the manifestly U-duality invariant form for the Bekenstein}Hawking entropy of black holes with general charge con"guration. E D"4: The quartic E invariant is given in terms of Z as [160] J "Z ZM !Z ZM " !(Z ZM )# (e ZM ZM !"ZM #$ZM %& !"#$%& !" #e !"#$%&Z Z Z Z ). (540) !" #$ %& In terms of the skew-eigenvalues j , J takes the form G J " "j "!2 "j ""j "#4(jM jM jM jM #j j j j ) . (541) G G H G HG By substituting j in (537) into the following E invariant entropy [405,413], one reproduces the G Bekenstein}Hawking entropy (203) of the generating solution: p S " (J . & 8
(542)
E D"5: The cubic E invariant has the form [156,157]: X#$Z , (543) J "! X Z X!"Z ! "# $ 2 $ which is expressed in terms of the real skew eigenvalues j as G J "2 j . (544) G G Here, X is the USp(8) symplectic invariant. The manifestly E invariant expression for the entropy of general solution is of the form [171]: (545) S "p( J , & which reproduces the entropy (276) of the generating solution if the expressions for j in (538) are G substituted. E 64D49: There is no non-trivial U-duality invariant in D56. This is consistent with the fact that the Bekenstein}Hawking entropy of the general BPS black holes in D56 is zero, which is the only U-duality invariant in D56. For near-extreme black holes, which has non-zero Bekenstein}Hawking entropy, the entropy can be expressed in a duality invariant form in terms of `dresseda electric charges and, therefore, has dependence on scalar asymptotic values [171]. The ADM mass M of the BPS solution is given by the largest eigenvalue max+"j ", of Z . The G " U-duality invariant form of the ADM mass can be expressed in terms of the U-duality invariant quantities Tr(>K) (m"1,2, [N/2]!p#1; > ,Z2 Z ) and corresponds to the largest root of " " " "
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a polynomial of degree [N/2]!p#1 in M with coe$cients involving Tr(>K). The BPS solution " preserves p/N of supersymmetry if p of the central charge eigenvalues have the same magnitude, i.e. "j ""2""j ". This depends on charge con"gurations of black holes. As for the generating N solutions, the number of identical eigenvalues "j " can be determined from (537), (538) and (539). In G the following, we discuss D"4 black holes as an example [171]. E p"4 case: The generating solution preserves 1/2 of supersymmetry when only one charge is non-zero. The U-duality invariant ADM mass is M"!Tr(> ). E p"3 case: An example is the case where (Q , Q , P "P )O0 with q"((Q #P )!Q . 0 0 * The U-duality invariant mass has the form M"!Tr(> )#( Tr(>)! (Tr > ). E p"2 case: An example is the case where only Q and Q are non-zero. The U-duality invariant mass M is the largest root of a cubic equation in M with coe$cients involving U-duality invariants Tr(>K) (m"1, 2, 3). E p"1 case: Examples are the case where only Q , Q and P are non-zero, or the case where all the "ve charges are non-zero and independent. The largest root of a quartic equation involving invariants Tr(>K) (m"1, 2, 4) corresponds to the ADM mass of the BPS solution. 6.3.3. p-Brane embedding of black holes We discuss the D"10, 11 p-brane embeddings of black holes in D(10. Starting from p-branes in D"10, 11, one obtains 0-branes in D(10 by wrapping all the constituent p-branes around the cycles of the internal manifold. The resulting black hole solution has the following generic form: ds "h "\(r)[!h\(r) f (r) dt#f \(r) dr#r dX ] , " "\ where 2m 2m sinh d ? f"1! H "1# ? r"\ r"\
(546)
is harmonic function associated with non-extremality parameter m [charge Q " ? (D!3)m sinh 2d ] and h(r)", H (r). The ADM mass and the Bekenstein}Hawking entropy ? ? ? are D!2 N , , ! k, M "2m (D!3) sinh d #D!2 " (Q#k#2 ? ? "+ D!3 2 ? ? (547) 1 , , S " (2m)"\ "\u cosh d &k"\ "\\, ((Q#k#k) , "\ & 4 ? ? ? ?
The fraction of supersymmetry preserved by N"8, D"4 BPS black holes can also be determined from the Killing spinor equations [140]. Generally, for a multi-charged black p-brane with the Page charges j "(p #1)m sinh 2d (a"1,2, N), ? ? , , p #2 N M "2m (p #1) sinh d #p #2 " (j#k#2 ! k ? ? "+ p #1 2 ? ? 1 , , S " (2m)N > N >u cosh d &kN > N >\, u ((j#k#k) , & 4 N> ? N> ? ? ? where k,(p #1)m is the rescaled non-extremality parameter.
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where k,(D!3)m is the rescaled non-extremality parameter and we neglected overall factor related to the gravitational constant, since we are interested only in the dependence on m, d and Q . ? ? As can be seen from (546), dimensional reduction of single-charged p-branes leads to black holes with singular horizon and zero horizon area in the BPS limit. To construct black holes with regular event horizon and non-zero horizon area in the BPS limit, one has to start from multi-charged p-branes in higher dimensions. This is achieved in D"4, 5 with N"4, 3, respectively, which can be seen from the BPS limit of entropy in (547). In fact, it is shown in [75] that the number N of distinct BPS black hole solutions to the equations of motion for the action (530) that have intersecting p-brane origins in D"11, 10 is N"4, 3 and 2 for D"4, 5 and D56, respectively. In the following, we discuss intersecting p-branes which give rise to regular BPS black holes in D"4, 5, as well as black holes with singular BPS limit. We concentrate on intersecting M-branes; intersecting p-branes in D"10 are related to intersecting M-branes through dimensional reduction and dualities. All the possible D"10, 11 BPS, intersecting p-branes that satisfy intersection rules are classi"ed in [75]. Also, there is a M-brane con"guration (473) interpreted as a M 2-brane within a M 5-brane (2L5), which preserves 1/2 of supersymmetry [388]. By including 2L5 con"gurations, one constructs new type of black holes [149,150] which are mixture of marginal and non-marginal bound states. Namely, the ADM mass and horizon area of p-branes that contain 2L5 have the forms M& (a#k#ck and A &kAY (k#(a#k) with a " G G G G & G (Q#P for each 2L5 constituent, where Q [P ] is charge of M 2-brane (M 5-brane) in the G G G G 2L5 constituent and c, c are appropriate constants. One can also add KK monopole to intersecting M-branes with overall transverse space dimensions higher than 3. We will not show the explicit intersecting p-brane solutions, since one can straightforwardly construct them applying harmonic superposition rules discussed in the previous section; explicit solutions can be found, for example, in [149,150,175,287,432,611]. 6.3.3.1. Four-dimensional black holes. Intersecting M-branes which reduce to D"4 black holes with 4 charges, i.e. (546) with N"4, should have 4 or 3 (with momentum along an isometry direction) M p-brane constituents and at least 3 overall transverse directions. Such con"gurations are (i) 2N2N5N5 for N"4, and (ii) 5N5N5, 2N5N5 and 2N2N5 for N"3. Additionally, one has the following D"11 con"gurations that reduce to D"4 black holes preserving 1/8 of supersymmetry: (i) 2N2N2#KK monopole, (ii) 2N5#boost#KK monople, (iii) (2L5)N(2L5)N(2L5), (iv) (2L5)N5#boost, (v) (2L5)N2#KK monopole, (vi) (2L5)#boost#KK monopole. Also, the boost#KK monopole con"guration reduces to D"4 black hole that carries KK electric and magnetic charges and preserves 1/4 of supersymmetry. 6.3.3.2. Five-dimensional black holes. Intersecting M-branes with 3 or 2 (with a boost along an isometry direction) M p-brane constituents and at least 4 overall transverse directions can be reduced to D"5 black holes with 3 charges. These are 2N2N2 and 2N5 with a momentum along an isometry direction. An additional M-brane con"guration that reduces to D"5 black hole with 3 charges is (2L5)N2, which preserves 1/4 of supersymmetry. 6.3.3.3. Black holes in D56. 0-branes in D56 can be supersymmetric with up to 2 constituent p-branes. Supersymmetric 2 intersecting M-branes are 5N5, 2N5 and 2N2, which are compacti"ed
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to 2-charged black holes in D"4, 5 and 7, respectively, after wrapping each constituent M p-brane around cycles of a compact manifold. One can compactify overall transverse directions of these D"5, 7 black holes to obtain black holes with 2 charges in D"4 and D46, respectively. One can also construct black holes in D49 and D45 by compactifying M 2-brane and M 5-brane with momentum along a longitudinal direction, respectively. Additionally, the M-brane con"guration (2N5)#boost reduces to D"6 black hole that preserves 1/4 of supersymmetry.
7. Entropy of black holes and perturbative string states One of challenging problems in quantum gravity for past decades is the issues related to black hole thermodynamics. It was early 1970s [38,63}65,126] when it was "rst noticed that the event horizon area A behaves much like entropy S of classical thermodynamics. Namely, it is observed [343,496] that the event horizon area tends to grow (dA50), resembling the second law of thermodynamics (dS50). Furthermore, Bardeen et al. [38] proved that the surface gravity i of a stationary black hole is constant over the event horizon, resembling the zeroth law of thermodynamics, which states that the temperature is uniform over a body in thermal equilibrium. They also realized the following relation between the ADM mass M of black holes and the event horizon area A: 1 i dA , (548) dM" 8pG , which resembles the thermodynamic relation between energy E and entropy S (the "rst law of thermodynamics): dE"¹dS ,
(549)
if one identi"es the energy E with the ADM mass M and the entropy S with the event horizon area A with some unknown constant of proportionally. Such analogy between horizon area and entropy met initially with skepticism, until Hawking discovered [345,346] that black hole is indeed thermal system which radiates (quasi-Planckian black body) thermal spectrum with temperature ¹ " i/2p, due to quantum e!ect. Since then, it is widely accepted that a black hole, as a thermal & system, is endowed with `thermodynamica entropy given by a quarter of the event horizon area in Planck units, the so-called Bekenstein}Hawking entropy [38,63}65,343,347,411,477,627]: S "A/4 G . (550) & , Puzzles on black hole entropy stem from the fact that entropy is a thermodynamic quantity, which arises from the fundamental microscopic dynamics of a large complicated system as
For the Kerr}Newmann black hole, this relation is generalized to 1 dM" i dA#U dQ#X dJ, 8pG , where U (X) is the potential (angular velocity) at the event horizon and Q [J] is a ;(1) charge (angular momentum).
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a universal macroscopic quantity which does not depend on the details of the underlying microscopic dynamics. So, if the correspondence between laws of black hole mechanics and thermodynamic laws is to be valid, the thermodynamic black hole entropy (550) should have a statistical interpretation in terms of the degeneracy of the corresponding microscopic degrees of freedom. Based upon our knowledge of statistical mechanics, one could guess several possible interpretations of the statistical origin of black hole entropy: (i) internal black hole states associated with a single black hole exterior [64,66,67,347], (ii) the number of di!erent ways the black holes can be formed [64,347], (iii) the number of horizon quantum states [587,594], (iv) missing information during the black hole evolution [302,348]. Another di$culty comes from the fact that contrary to ordinary thermodynamics, understanding of black hole thermodynamics requires the treatment of quantum e!ects, as we noted the crucial role that the Hawking e!ect plays in establishing black hole thermodynamics. Thus, the statistical interpretation of black hole entropy should necessarily entail quantum theory of gravity, of which we have only rudimentary understanding. The early attempts during the 1970s and 1980s were not successful in the sense that the most of approaches either (i) did not touch upon the statistical meaning of entropy, since the calculations were mostly based on thermodynamic relations (e.g. calculating entropy using Clausius's rule S"dM/¹ given that the black hole temperature ¹ is determined by the surface gravity method [38,626]), or (ii) is purely classical (e.g. in Gibbons}Hawking Euclidean (on-shell functional integral) method [294] involving grand partition function, the black hole entropy A/(4 G ) is , reproduced at tree-level of quantum gravity calculation). Another major was the ultraviolet quadratic divergence (related to the divergence of the number of energy levels a particle can occupy in the vicinity of black hole horizon) in black hole entropy when the Euclidean functional formulation of the partition function is evaluated for quantum "elds in the black hole background. To avoid such divergences, 't Hooft [593] introduced `cuto! a at a small distance e just above the event horizon in the path integral of real free scalar "eld , assuming that there are no states at the interval between the event horizon and the cuto! (the so-called brick-wall method). The black hole entropy in "eld theory based on brick-wall method in general depends on the cuto! distance e in the form S &A/4e, re#ecting the quadratic divergence. It was conjectured [594] by 't Hooft ( that such ultraviolet divergence of the statistical entropy might be related to Hawking's information paradox [348], i.e. a black hole is an in"nite sink of information. In [588], it is shown that the divergence associated with the Euclidean functional formulation of the partition function for canonical quantum gravity (of point-like particles) is related to the renormalization of the gravitational coupling G . When the contributions to entropy (obtained , from the partition function) from the pure gravity and matter "elds are added, the entropy takes a suggestive form
c A 1 # S" e 4 G ,
This is closely related to the fact that quantum gravity in point-like particle "eld theory is non-renormalizable, as we will discuss below. 't Hooft proposed that the entropy of black hole is nothing but the entropy of particles which are in thermal equilibrium with black hole background [592,593,595].
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similar to (550) but the bare gravitational constant G is renormalized. The explicit calculation , of quantum corrections of quantum gravity shows [202,394,395] that the renormalized gravitational constant G takes the same form. Since superstring theory is a promising candidate for , a "nite theory of quantum gravity, the contradiction encountered in the point-like particle "eld theory has to be resolved [583,585,588,589]. It is indeed shown in [186,588] that theory of superstring propagating in a black hole background gives rise to a xnite expression for black hole entropy in the calculation of partition function through Euclidean path integral, with the xnite renormalized gravitational coupling G : the genus zero contribution gives rise to the classical result , (550) with a bare Newton's constant G and the higher genus terms contribute to "nite corrections , to G . , This can be seen intuitively by considering microscopic states near the event horizon [584]. For point-like particles, due to the arbitrarily small longitudinal Lorentz contraction near the event horizon, an arbitrarily large number of particles can be packed close to the event horizon, giving rise to a divergent entropy. However, the Lorentz contraction of strings along the longitudinal direction is eventually halted to a "nite extent and, therefore, only "nite number of strings can be packed near the horizon, leading to "nite entropy. Just as only contribution to the "rst-quantized path integral of the point-like particle "eld theory is from the set of paths that encircle or touch the black hole event horizon, only string graphs which contribute to the entropy through the partition functions are those that are somehow entangled with the event horizon. From the point of view of an external observer, this kind of closed strings, which are partially hidden behind the event horizon, look like open strings frozen to the horizon. Thus, the black hole entropy can somehow be interpreted as being associated with oscillation degrees of freedom of #uctuating open strings whose ends are attached to the black hole horizon [588]. This section is organized as follows. In Section 7.1, we discuss connection between black holes and perturbative string states. Identi"cation of black holes with string states makes it possible to do explicit calculations of statistical entropy of black holes, based on the conjecture that microscopic origin of entropy is from degenerate string states with mass given by the corresponding ADM mass of the black hole. In Section 7.2, we discuss Sen's original calculation of statistical entropy of the BPS static black hole, which was compared to Bekenstein}Hawking entropy evaluated at the stretched horizon. In Section 7.3, we generalize Sen's result to near-extreme rotating black holes. In Section 7.4, we discuss the level matching of black holes to macroscopic string states at the core. This justi"es our working hypothesis that black holes are perturbative string states. In Section 7.5, we summarize Tseytlin's method of chiral null model for calculating statistical entropy of the BPS black holes that carries magnetic charges as well as perturbative NS-NS electric charges.
7.1. Black holes as string states It is not a new idea that elementary particles might behave like black holes [342,526,594]. A particle whose mass exceeds the Plank mass and therefore whose wavelength is less than its Schwarzschield radius exhibits an event horizon, a characteristic property of black holes. Since typical massive excitations of strings have mass of the order of the Planck mass, one would expect that massive string states become black holes when gravitational coupling (or string coupling) is
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large enough. It has been shown that black holes with given charges and angular momenta behave like string states with the corresponding quantum numbers. Before we discuss identi"cation of black holes with string states, we summarize [308] some aspects of perturbative spectrum, moduli space and T-duality of heterotic string on a torus. Heterotic string [322] is a theory of closed string whose left- and right-moving modes are respectively described by bosonic and super string theories. So, the critical dimensions, in which the conformal anomali is absent, for each mode are di!erent: D"26 for the left-movers and D"10 for the right-movers. In compactifying the extra 16 coordinates of the left-movers on ¹, one obtains a rank 16 non-Abelian gauge group which is associated with the even-self-dual lattice of the type E ;E or Spin(32)/Z . Thus, the massless bosonic modes of heterotic string in D"10 at a generic point of moduli space are ;(1) gauge "elds, as well as graviton, 2-form "eld and dilaton in the NS-NS sector ground state. We compactify the extra d spatial coordinates XI (k"1,2, d) on ¹B to obtain a theory in D"10!d. The toroidal compacti"cation is de"ned by the periodic identi"cation of each internal coordinate, i.e. XI&XI#2pmI, where mI is the integer-valued string `winding modea. Since the holonomy of a torus is trivial, all of supersymmetry is preserved in compacti"cation, i.e. N"4 for the compacti"cation on ¹. The ¹B part of the heterotic string worldsheet action in #at background, including the coupling to gauge "elds A? and a 2-form potential B , is I IJ 1 dz[(G #B )RXIRM XJ#A RXIRM X? S" IJ IJ I? 2p
(551) # (G #B )RX?RM X@]#(ferminionic terms) , ?@ ?@ where k, l"1,2, d (a"1,2, 16) correspond to the coordinates of ¹B (¹ of left-movers), and the complex worldsheet coordinate and derivative are de"ned as 1 1 (q#ip), R" (R !iR ) . z" N (2 O (2
(552)
The internal coordinates X? live on the weight lattice of E ;E or Spin(32)/Z . The background "elds G , B and A , which parameterize the moduli space O(d#16, d, Z)/[O(d#16, Z) IJ IJ I? ;O(d, Z)] of ¹B;¹, can be organized into the `background matrixa of the form:
(G#B#A)A ) A ) GH G( , (553) 0 (G#B) '( where B [G] is the antisymmetric (symmetric) part of E. (For the relations between the background "elds in (551) and E, see the next footnote.) Here, the indices i, j [I, J], associated with ¹B [¹], run from 1 to d (from 1 to 16). In particular, the components E "(B#G) of E are related to the '( '( Cartan matrix C of E ;E or Spin(32)/Z as '( (554) E "C (I'J), E "C , E "0 (I(J) . '( '( '( '' '' The Narain lattice [480,481] KB>, which de"nes ¹B;¹ is spanned by basis vectors a,(a , a ) of the form: G ' a "(e , A)E ) , a "(0, E ) , (555) G G G ) ' ' E,B#G"
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where the vectors +E? " I"1,2, 16, and +eI " i"1,2, d, are de"ned through ' G E ) E "2G , e ) e "2G , e ) E "0 . (556) ' ( '( G H GH G ' The zero modes (p , p ) of the right- and left-moving momenta form an even self-dual lattice 0 * CB B>"CB B C . Quantized momentum zero modes (p , p ) are embedded into 0 * CB> B> as
p pG 0 " 0 pH p * *
0
p( *
,
(557)
where p "[nR#mR(B!G)]aH, p "[nR#mR(B#G)]aH, m,n3RB> . 0 * Here, aH is the basis vector of the lattice dual to KB>:
(558)
aGH"(eGH, 0), a'H"(!A'eGH, E'H) , (559) G where E'H [eGH] are dual to E [e ], i.e. B eIeHH"dH [ E?E(H"d( ]. ' G I G I G ? ' ? ' The heterotic string with the action (551) has an O(d#16, d, Z) ¹-duality symmetry. This group is a subgroup of the following O(d#16, d#16, Z) transformation that preserves the triangular form of E in (553): EPE "(aE#b)(cE#d)\,
a b c
d
3O(d#16, d#16, Z) ,
(560)
and (p , p ) in CB B> transforms as a vector. T-duality is proven [307] to be exact to all orders in 0 * string coupling. The mass of perturbative states for heterotic string on a torus is [322] 1 1 +(p )#2N !1," +(p )#2N !2, , (561) M" 0 * 8j * 8j 0 where N are left- and right-moving oscillator numbers, j is the vacuum expectation value of * 0 the dilaton (or string coupling). We now identify string states with black holes. The mass of the BPS purely electric black holes in heterotic string on ¹, which preserves of the N"4 supersymmetry, is [337,544,558,560,564]: 1 m" a?(M#¸) a@ , (562) ?@ 16j where a is the charge lattice vector on an even, self-dual, Lorentzian lattice K with the O(6, 22) metric ¸ and the subscript (0) denotes asymptotic values. Here, M is the moduli matrix of ¹ de"ned in (121). Under the T-duality, the moduli matrix and the charge lattice transform
By contracting with eGH and E'H, one obtains the background "elds in (551) from E. For example, I ? G?@"2G (E'H)?(E(H)@ and BIJ"2B (eGH)I(eHH)J, etc. '( GH
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as [560] MPXMX2, KP¸X¸K, X3O(6, 22)
(563)
and the BPS mass (562) is invariant. With a choice of the asymptotic values j"1 and M"I , the mass takes a simple form: #¸) a@"(a ), a? ,(I $¸) a@ . (564) m" a?(I ?@ 0 0 * ?@ The string momentum (winding) zero modes are identi"ed with the quantized electric charges of KK (2-form) ;(1) gauge "elds, i.e. a "p . Then, m"M, provided N " [248]: the BPS black 0 * 0 * 0 holes are identi"ed with the ground states of the right movers. With a further inspection of (561), one "nds N in terms of a [248]: * (565) N !1"((a )!(a ))"a2¸a , * * 0 leading to a2¸a5!2. So, the various BPS black holes in the heterotic string on a torus are identi"ed [248] as string states with the corresponding value of N [or a2¸a]. * Non-extreme black holes are identi"ed with string states with the right movers excited as well. Identi"cation of black holes in other dimensions and in type-II theories with string states is proceeded similarly as above. For type-II string theories, the mass of perturbative string state is 1 1 M" +(p )#2N !1," +(p )#2N !1, . (566) 0 * 8j 0 8j * Since the type-II strings have supersymmetry in both the right- and left-moving sectors, perturbative string states can (i) preserve supersymmetry in both sectors (N "N "1/2), leading to 0 * short supermultiplet; (ii) preserve supersymmetry in one sector, only (N 'N "1/2), leading 0 * * 0 to intermediate supermultiplet; (iii) break supersymmetry in both sectors (N , N '1/2), leading to 0 * long supermultiplet. Further study of equivalence of string states and extreme black holes, including spins of string states and dipole moments of rotating black holes, is carried out in [237,249].
7.2. BPS, purely electric black holes and perturbative string states In the previous section, we showed that BPS electric black holes in the string low energy e!ective actions are identi"ed with perturbative string states. Thus, it is natural to infer that the microscopic degeneracy of black holes originates from the degenerate string states in a corresponding level. In general, non-extreme black holes are also identi"ed with perturbative string states. However, non-extreme solutions are plagued with (unknown gravitational) quantum corrections and, therefore, the ADM mass cannot be trusted. In fact, the number of states in non-extreme black hole grows with the ADM mass M like &e+ [594], whereas the string state level density grows with the string state mass M as &e+ . Thus, if one is to identify string states with black hole states, one is forced to identify M with M [517,583]. In [583], Susskind attributes the discrepancy to the mass renormalization due to unknown quantum corrections. (See also Section 8.2, where it is discussed that the Bekenstein}Hawking entropy of non-extreme black holes has to be evaluated
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at the speci"c string coupling at the black hole and microscopic con"guration (D-branes and fundamental string) transition point.) As "rst pointed out by Vafa [583], the BPS solutions do not receive quantum corrections [485] due to renormalization theorem of supersymmetry. Such class of solutions are, therefore, suitable for testing the hypothesis that the statistical origin of black hole entropy is from the degenerate string states with mass given by the ADM mass of black hole. So, one can calculate the `statisticala entropy by taking logarithm of the string level density. This yields the "nite non-zero entropy &(N . However, the `thermala entropy of the BPS purely * electric black holes in heterotic string is zero. In [564], Sen circumvented with problem by postulating that the `thermala entropy of the BPS black hole is not the event horizon area, but the area of a surface close to the event horizon, a so-called `stretcheda horizon [586,587,596]. Although the BPS electric black hole solutions are free of quantum corrections, they receive (classical) stringy a corrections due to the singularity at the event horizon. This leads to the shift of the event horizon by the amount of an order of a . Originally, the stretched horizon is de"ned [587] as the surface where the local Unruh temperature for an observer, who is stationary in the Schwarzschield coordinate, is of the order of the Hagedorn temperature [331]. Namely, it is a surface where the string interactions become signi"cant. In [419,475,582], it is observed that the transverse size of strings diverges logarithmically and "ll up a region at the stretched horizon, melting to form a single string. Thereby, information in the string states is stored and thermalized with black hole environment in the region near the stretched horizon [452,454,475,582,584], and black hole states are in one-to-one correspondence with single string states. So, the statistical entropy is due to degenerate strings states in equilibrium with the black hole background at the stretched horizon [588]. In this section, we summarize [495,564] to illustrate this idea. The electric black hole considered in [564] is a special case of the general solution [178] discussed in Section 4.2.1. But for the purpose of illustrating the idea of perturbative string state and black hole correspondence, we follow Sen's parameterization of solution in terms of left-moving and right-moving electric charges, rather than in terms of KK and 2-form electric charges. 7.2.1. Black hole solution In Sen's notation, the most general non-rotating, electric black hole solution in the heterotic string on ¹, in the Einstein-frame, is [564] D r(r!2m) dt# dr#D (dh#sinh du) , g dxI dxJ"! IJ r(r!2m) D
(567)
where D,r[r#2mr(cosh a cosh b!1)#m(cosh a!cosh b)].
It is argued in [350] that the Bekenstein}Hawking formula for entropy, i.e. (entropy)Jhorizon area), ceases to hold for extreme case and entropy of an extreme black hole is always zero, displaying discontinuity in going from non-extreme to extreme case. This is attributed [350] as being due to di!erence in Euclidean topologies for the two cases. The cure for this discontinuity is proposed in [291], where it is suggested that one has to extremize after quantization, rather than quantizing after extremization.
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The ADM mass and the electric charges are 1 m(1#cosh a cosh b) , M " & 2G , m g nG sinh a cosh b for 14i422 , Q (2 QG" m g pG\ sinh b cosh a for 234i428 , (2 Q
(568)
where n [p] is an arbitrary 22 [6] component unit vector. The left- and right-handed charges are de"ned as nG 1 !¸) QH" * g m sinh a cosh b , QG , (I GH * 2 (2 Q nG 1 #¸) QH" 0 g m sinh b cosh a , QG , (I GH 0 2 (2 Q
(569)
and the 28-component left- and right-handed unit vectors n and n are similarly de"ned. The * 0 solution (567) is in the frame where the O(6, 22) invariant metric ¸ (127) is diagonal. This parameterization of black hole solution has a convenient form in which only left (right) handed charges are non-zero when b"0 (a"0) with all the parameters "nite. The solution has 2 horizons at r"r "2m, 0. The event horizon area is > \
A" dh du(g g " >"8pm(cosh a#cosh b) . FF PP PP
(570)
The surface gravity at the event horizon is 1 i" lim (gPPR (!g " " . P RR F 2m(cosh a#cosh b) P P>
(571)
7.2.2. Extreme limit and string states The extreme limit is de"ned as a limit where the inner and outer horizons coincide, i.e. mP0. In taking the `non-extremality parametera m to zero, one has to let one (or both) of the boost parameters a and b go to in"nity so that the electric charges (569) do not vanish. Since we are interested in the BPS solutions, we let the ADM mass depend only on the right-handed electric charge. This is achieved by taking the limit bPR and mP0 such that m( "me@ remains as a "nite non-zero constant, while a remaining "nite. In this limit, the ADM mass and the electric charges are 1 m( cosh a , " .1 2G , 1 QG " g nG m( sinh a, * (2 Q * M
1 QG " g nG m( cosh a , 0 (2 Q 0
(572)
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thereby the ADM mass depends on the right-handed electric charge, only: 1 Q , M " .1 8g 0 Q where G "2. In this limit, the solution has the form (567) with m"0 and , D"r(r#2m( r cosh a#m( ) .
(573)
(574)
The event horizon area (570) is zero in the BPS limit. However, the string states are degenerate. One can circumvent such problem by calculating entropy at the `stretched horizona right above the event horizon. To "nd a location of the stretched horizon, one considers a region close to the event horizon in the `string framea metric: r dS,g dxI dxJK! g dt#g dr#gr(dh#sin h du) Q Q IJ m( Q "!r dtM #dr #r (dh#sin h du) ,
(575)
where r ,g r and tM ,t/m( . Note, in the frame (tM , r , h, ), all the dependence on the other parameters Q has disappeared. One can show that the other background "elds also become independent of the parameters near the event horizon, if one performs a suitable O(6, 22) transformation. Thus, the location of stretched horizon, i.e. the location where higher-order stringy corrections become important, is unambiguously estimated to be located at r "C, a distance of order 1 (in unit of string scale) from the event horizon. In terms of the original coordinate, the stretched horizon is located at r"C/g ,g. Q The stretched horizon area, calculated from (567) with m"0 and (574), is AK4pgm( "4pm( C/g , Q where only the term leading order in g is kept, and therefore the thermal entropy is
(576)
A p m( C S , " . & 4G 2 g Q , To compare this expression with the statistical entropy, one expresses S in terms of electric & charges by using the relation
Q m( "4 M ! * & 8g Q derived from (572):
Q 2pC M ! * . (577) S " & 8g & g Q Now we compare the thermal entropy (577) with the degeneracy of string states. Since string states identi"ed with BPS black holes have the right movers in ground state (N "), the string 0 state degeneracy is from the left movers with N given in terms of electric charges as (details are *
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along the same line as in Section 7.1):
Q 4 (578) N K M ! * , .1 8g * g Q Q for large Q. So, the statistical entropy associated with the degeneracy of string states is * 8p Q M ! * . (579) S ,ln d(N )K4p(N K 12 2 * * g & 8g Q Q This entropy expression has the same dependence on M and Q as the thermal entropy (577) .1 * calculated at the stretched horizon, and the two expressions agree if one chooses C"4 in (577). Note, it is crucial that the constant C does not depend on parameters of black holes; otherwise, the dependence of S (579) on Q and M cannot be trusted because of the unknown dependence of 12 2 * & C on these parameters.
7.3. Near-extreme black holes as string states In the previous section, we saw that thermal entropy of the BPS, non-rotating, electric black holes agrees (up to numerical factor of order one) with statistical entropy associated with the degeneracy of string states, if it is evaluated at the `stretched horizona. However, the rotating black hole case is problematic for the following reasons. Since the electric, rotating black hole (285) in the BPS limit with all the angular momenta non-zero has naked singularity, thermal quantities cannot be de"ned. The BPS limit with a horizon is possible in D56 with at most 1 non-zero angular momentum [366]. Even for this case, not only the event horizon is singular (i.e. the event horizon and the singularity coincide) and has zero surface area, but also the area of the stretched horizon (which is assumed to be independent of parameters of the black hole) is independent of angular momenta. We surmise that this is due to the unknown dependence of the location of the stretched horizon on physical parameters, unlike the non-rotating black hole case. The determination of the stretched horizon location may require understanding of a corrections with rotating black hole as the target space con"guration, which is di$cult to estimate at this point. We propose [184] an alternative way to circumvent the problems of the BPS electric black holes. Instead of de"ning the thermal entropy of the BPS black holes at the stretched horizon, we propose to calculate the thermal entropy of near extreme black holes at the event horizon. Then, the thermal entropy of near-extreme black holes takes suggestive form which can be interpreted in terms of string state degeneracy. We attempt statistical interpretation of such thermal entropy expression by using the conformal "eld theory of p-model with the near-extreme solution as a target space con"guration and with angular momenta identi"ed with [(D!1)/2] ;(1) left-moving world-sheet currents. 7.3.1. Thermal entropy of near-BPS black holes The proper way of taking the near-BPS limit of rotating black holes is to take the limit in such a way that the angular momentum contribution to the thermal entropy is not negligible compared with the contribution of the other terms, while ensuring the regular horizon so that thermal quantities can be evaluated at the event horizon. This is achieved as follows. First of all, the
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near-BPS limit is de"ned as the limit in which the non-extremal parameter m'0 is very small and the boost parameters d are very large such that the combinations meBG (i"1, 2) remain as "nite, G non-zero constants. Then, as long as l are non-zero, J (287) do not vanish. Second, the requirement G G of the regular event horizon restricts the range of the parameters of the solution [478], e.g. m5"l " for D"4 and m5("l "#"l ") for D"5. For an arbitrary D, we write such a constraint generally as . QQ<J 2 "\
Third, for the thermal entropy to be macroscopically non-negligible, the electric charges have to be very large, i.e. Q <m"O(1). This is required also for the statistical entropy, since the system has to be very large so that statistical quantities approach the exact values. Fourth, while keeping the angular momenta small so that the regular horizon is always ensured as m+0, one has to make sure that angular momenta contribution to entropy is not macroscopically negligible compared with the other terms. This is achieved by taking the limit <(QQ . J 2 "\
In such a near-BPS limit, the thermal entropy (288) takes the form [184]:
4 2 "\
J . S "2p QQ(2m) "\! G (D!3) (D!3) G
(580)
7.3.2. Microscopic interpretation In this section, we calculate the statistical entropy of near-extreme, rotating black holes by counting the degenerate string states with the speci"c angular momenta. In principle, to calculate the statistical entropy of rotating black holes, one has to extract the degenerate string states (in a given level) with the speci"c values of angular momenta. This was "rst attempted in [517]. Note, string states in a given level consist of states with di!erent angular momenta (with the maximum angular momentum determined by the level). Alternatively, one can use the level density formula that has contribution from all the possible angular momenta in a given level and employ the technique of conformal "eld theory to extract the speci"c contribution of states with given angular momenta. The main point is that the ;(1) charges of the left-moving worldsheet currents are interpreted as target space spins of string states. States with non-zero spins are obtained by applying the a$ne ;(1) current operators to spin zero states. The procedures described in the following paragraphs are also applicable to D-brane interpretation of entropy of rotating black holes [95,98,471]. As in the BPS case, one identi"es the KK electric charges Q [the 2-form electric charges Q] G G with the (internal) momentum zero modes [string winding modes]. Then, mass of the perturbative string states takes the form:
1 4 4 M "(Q#Q)# N ! "(Q!Q)# (N !1) , 0 2 a * a
(581)
where each circle in the torus has self-dual radius R"(a . From the second equality in (581), i.e. the Virasoro constraint, one has the following relation between N and N : * 0 N "a Q Q #N # . (582) * 0
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Note, for the statistical interpretation to be valid, the electric charge (quantized in unit 1/(a ) has to be very large, i.e. Q <1/(a . In the near extreme limit (m+0), the BPS mass (287) takes the form M +(Q#Q)#O(m) . (583) & By identifying the ADM mass (583) with the mass of string state (581), one "nds that the right movers are barely excited: N +#O(m). So, N is negligible compared to N and N +a QQ 0 0 * * to a good approximation: N +a QQ#1#O(m)+a QQ
1 ln d(N , N )+2p c N "4p(N , * 0 * 6 *
(584)
with the right-mover contributions neglected, just like BPS black holes. Here, c "26!2"24 since we are considering left moving bosonic string modes, only. Note, this level density contains contribution from all the spin states in the level (N , N ). * 0 To extract the contribution by the states with particular spins, we employ conformal "eld theory technique. Recall that the p-model with the target space con"guration [104,105] given by a rotating black hole is described by the WZNW model [289,437,484,632] with the ;(1) "\ a$ne Lie algebra (i.e. Cartan sub-algebra of the O(D!1) rotational group), or the conformal "eld theory with the ;(1) "\ group manifold. The eigenvalues of the left-moving ;(1) worldsheet currents j "iR HG (i"1,2, [(D!1)/2]) are G X interpreted as the [(D!1)/2] spins of string states. A general state in this WZNW model is labeled by charges of the a$ne Lie algebra as well as by the oscillator numbers. The conformal "eld can be expressed as U 2 "\ with ;(1) charges J 2 "\
( "\
U 2 "\ " eG(G&GU , (585) ( G where U is a conformal "eld without ;(1) charges, or without target space spins. Thus, the left-moving conformal dimensions hM 's, i.e. the eigenvalues of the left-moving Virasoro generator ¸ , of U 2 "\ and U are related as: ( 1 "\
hM U( 2 "\ " J#hM U . (586) G 2 G This implies that the total number N of left moving oscillations of spinless states is reduced by the * amount "\ J relative to the total number N of left moving oscillations of states with the G G * speci"c spins J 2 : "\
1 "\
N PN "N ! J . (587) * * * 2 G G Note, the level density for spinless states in a given level (N , N ) di!ers from the level density * 0 d(N , N ) of all the states in the level (N , N ) by a numerical factor, which can be neglected in the * 0 * 0
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large (N , N ) limit if one takes logarithm of the level densities. So, one can use the formula (584) as * 0 the logarithm of the level density of spinless states to a good approximation. Then, the statistical at the level entropy associated with degenerate string states with particular spins J 2 "\
(N , N ) is * 0 1 "\
J , (588) S ,log d(N , N )+4p(N "4p N ! G * 2 * 0 * G in the limit
1 "\
N < J. * 2 G G For the angular momenta contribution to be statistically non-negligible, "\
J<1 G G has to be satis"ed. In the near-extreme limit, N +a QQ. So, in terms of electric charges and angular momenta, * the statistical entropy takes the form: (N *
"\
. S "2p 4a Q Q !2 J G G This qualitatively agrees with the thermal entropy (580).
(589)
7.4. Black holes and fundamental strings In the previous sections, we calculated statistical entropy of black holes by assuming that perturbative string states are black holes. Based on this assumption, we equated the mass of string states with the ADM mass of black holes, and identi"ed the left and right moving momentum zero modes of the string states with the left and right handed electric charges of black holes. This "xes N (which determines the microscopic degeneracy of states) in terms of the macroscopic para* 0 meters of black holes, making it possible to calculate the statistical entropy of black holes. In this section, we justify [110,189] such identi"cation of perturbative string states with microscopic black hole states. The starting point is the fundamental string in 54D410. Here, the fundamental string is de"ned as a 1-brane solution of the combined action S#S with macroN scopic string (described by S ) as its electric charge source. When the fundamental string is N compacti"ed on S along its longitudinal direction, it asymptotically approaches black hole, as rPR, with its core having milder singularity (than black hole in (D!1) dimensions) of a D-dimensional string source. With this identi"cation, microscopic degrees of freedom of the asymptotic black hole in (D!1) dimensions is interpreted as being due to oscillating macroscopic string at its core. And electric charges and angular momenta of the black hole are determined by momentum and winding modes of the core string along its longitudinal direction and the frequency of string oscillation in the rotational planes.
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7.4.1. Fundamental string in D dimensions We consider the following worldsheet action of macroscopic string moving in a background of the string massless modes:
1 1 dp (cc?@R X+R X,GK #e?@R X+R X,BK ! (cUK R . S" ? @ +, ? @ +, 2 N 4pa
(590)
The conformal invariance of S leads to the equations of motion for the massless background "elds N [606,607]. These equations of motion can be reproduced [111] by the Euler}Lagrange equations of the e!ective action:
1 1 d"x (!GK e\UK R K #4R UK R+UK ! HK HK +,. , (591) S" % + 12 +,. 2i " where the D-dimensional gravitational constant i is related to the Newton's constant G as " " G "i /8p. " " When the target space is #at, i.e. GK "g and BK "0, one can exactly solve the p-model +, +, +, (590) to construct perturbative string states. We concentrate on compacti"cation of the p-model on S of radius R in #at background. The string states in this model are characterized by string winding number n and quantized momentum zero mode m/R along the S-direction. The right- and left-moving momenta along the S-direction are nR m p " ! , 0 2R 2a
m nR p " # . * 2R 2a
(592)
For BPS states in heterotic string, whose supersymmetry is generated by right-moving worldsheet current, all the right movers are in the ground state (N ") and the total number of left-moving 0 oscillations is determined by the Virasoro constraint to be: N "1#a (p!p)"1!mn . (593) * 0 * The mass of BPS states depend p , only: 0 M "4p . (594) 0 From now on, we concentrate on BPS fundamental string solution [187,189}191,380] to this theory. The background "elds of such `straighta fundamental string solution have the form [190]: BK "(eUK !1) , GK dx+ dx,"!eUK du dv#dx ) dx, ST +, Q i " e\UK "1# , Q" , (595) r"\ pa (D!4)X "\ where u,x!x"\ and v"x#x"\ are the lightcone coordinates along the string worldsheet, and xK (m"1,2, (D!2)) are the transverse coordinates. In deriving this solution, we chose the static gauge for X+:
XI"mI, XK"constant , G is related to the quantities in (590) as G "a eUK . " "
(596)
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where mI"(q, p) is the worldsheet coordinate. This fundamental string solution preserves 1/2 of the spacetime supersymmetry [190]. When the fundamental string is compacti"ed along its longitudinal direction on S of radius R, i.e. x"\"x"\#2pR, one obtains point-like solution in D!1 dimensions with its charge proportional to the winding number n along the S-direction. We now obtain solution that also has an arbitrary left-moving oscillation, which is a source for microscopic degeneracy. The zero-modes of the left-moving oscillation induce momentum m/R in the S-direction. Applying the general prescription of solution generating transformation discussed in [279,280,282,617,618] to the straight fundamental string solution (595), one obtains the following left-moving oscillating fundamental string solution GK dx+ dx,"!eUK (du dv!¹(v, x) dv)#dx ) dx , +, 1 Q BK " (eUK !1), e\UK "1# , ST 2 r"\
(597)
where ¹(v, x) is a solution to Rx ¹(v, x)"0. The general form of ¹(v, x) that can be matched onto the string source at the core is ¹(v, x)"f (v) ) x#p(v)r\"> ,
(598)
where the "rst term corresponds to oscillating string source and the second term corresponds to a momentum without oscillations. One can bring (597) to a manifestly asymptotically #at form by applying the coordinate transformations
u"u !2FQ ) x #2FQ ) F!
v"v ,
TY
FQ dv,
x"x !F ,
(599)
where Q,R/Rv, f (v)"!2F$ and FQ "FQ ) FQ . In this new coordinates, (597) takes the form (with primes suppressed) GK dx+ dx,"!eUK du dv#[eUK p(v)r\">!(eUK !1)FQ ] dv#2(eUK !1)FQ ) dx dv#dx ) dx , +, BK "(eUK !1), BK "FQ (eUK !1) , (600) ST TG G Q e\UK "1# . "x!F""\ This solution has the ADM mass p and the ADM momentum per unit length p G , and the "+ "+ D-momentum #ow along the string p"\ + given by "+ p + Q#QFQ #p QFQ G !QFQ !p "+ "(D!4)X"\ . (601) 2i p"\ + !QF Q !p !QF Q G !Q#QF Q #p " "+
7.4.2. Level matching condition In Section 7.4.1, we constructed oscillating fundamental string solution by solving the equations of motion (following from S#S ) of the target space background "elds and imposing solution N generating transformations. Note, all of such solutions do not correspond to the underlying string states. To ensure that these solutions match onto the perturbative string states at the core, one has
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to additionally solve the equations of motion for string coordinates X+ and the Virasoro constraints. From the Virasoro constraints (or the level matching conditions), one extracts the relations between the macroscopic quantities of the oscillating fundamental string (600) and the microscopic quantities of the perturbative string states. This allows to interpret the entropy of the target space solution in terms of the perturbative string state degrees of freedom. We start by choosing the following static gauge for the string coordinates X+ in the coordinate frame corresponding to the solution in (597): ;";(p>, p\),
<"<(p>, p\),
XK"0 ,
(602)
where p!"q$p are the light-cone worldsheet coordinates. Also, we choose the conformal gauge for strings, i.e. c "diag(!1, 1). From the Virasoro constraints ¹ "0"¹ , one has the ?@ >> \\ following form of string coordinates (602): ;"(2Rn#a)p\,
<"2Rnp> ,
(603)
where a,(1/p)p0LFQ is the zero mode of FQ . And the constant Q in (597) is expressed in terms of the perturbative string state quantities as ni " Q" . (604) pa (D!4)X "\ More information on matching of the spacetime solution onto states of the core string source is extracted by taking the #at spacetime limit i P0, in which for example the Virasoro relations " (592), (593) and(561) are valid. For this purpose, we go to the frame represented by (600), where the metric is manifestly asymptotically #at, by applying the transformations (599). In this new frame (denoted by primes), X+ take the form:
FQ , X "F(< ) , (605) 4Y manifestly showing that the core string is oscillating with pro"le F(< ). In the #at spacetime limit, i.e. i P0 or 1UK 2P0, a perturbative string state has the momentum p+ (conjugate to X+, " obtained from S ) and the winding vector n+ given in the coordinates (X , X , X "\) by N n+"(0, 0, n), p+"(2a )\(2nR#a, 0,!a) . (606) < "2Rnp>,
; "(2Rn#a)p\#
This expression for p+ agrees with the i P0 limit of the ADM momentum p + (601) of the target " "+ space solution (600) with Q given by (604). This con"rms that oscillating fundamental strings are matched onto perturbative states of the core macroscopic string. Since the momentum zero mode of a perturbative string state along the (compacti"ed) X"\direction is m/R and N "!nm ((593) in the large N limit), one can read o! the expression for * * pI in (606) to express m and N in terms of the macroscopic quantities of the fundamental string * solution: m"!Ra/2a , N "nRa/2a . (607) * Thus, we see that the oscillating fundamental string solution (600) with p(v)"0 is matched onto the perturbative string state with n, m and N given in (607). These are a subclass of solutions (600) that * can be matched onto perturbative states of the core source string.
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7.4.3. Black holes as string states When the longitudinal direction of the fundamental string (600) is compacti"ed, one has point-like object in D!1 dimensions. Such a solution approaches Sen's BPS electric black hole [562] as rPR. This allows one to relate the macroscopic quantities, which are de"ned at spatial in"nity, of black holes to the microscopic quantities of perturbative string states. Note, such point-like solutions in D!1 dimensions asymptotically approach only a subset of Sen's black holes that can be matched onto perturbative string states. So, for example, the angular momenta of such solutions follow the Regge bound of perturbative string states, whereas Sen's rotating black holes [562], in general, take arbitrary values of angular momenta, which do not satisfy the Regge bound. In the following, we discuss the D"5 case for the purpose of illustrating basic ideas. The generalization to an arbitrary D is straightforward; one starts from D-dimensional solution (600) with more general pro"le function F(v). One compacti"es the longitudinal direction of the D"5 fundamental string (600) with p(v)"0 on S of radius R to obtain a point-like solution in D"4. To make the resulting D"4 point-like solution approach a `rotatinga black hole asymptotically, one chooses the following form of F that describes rotation in the (x, x)-plane with amplitude A and angular frequency u: F"A(e( cos ut#e( sin ut) , (608) where e( is a unit vector in the xG-direction. G Since the D"4 point-like solution depends on the compacti"ed coordinate x and the time coordinate t through v"x#t, the compacti"cation on x (i.e. taking average over x so that only the zero modes of "elds are kept) is equivalent to taking the time-average. By taking the time-average of the leading order terms of the "elds at large r, one can read o! the following ADM mass M , angular momentum J, the right- and left-handed electric charges & Q ,($Q#Q)/(2, and the right- and left-handed magnetic moments k of the rotating 0 * 0 * black hole that the point-like solution approaches asymptotically: Q(1#Au) M " , & 4
QAu J" , 4
Q Q (Au!1), Q "! (1#Au) , Q "! 0 * 2(2 2(2
(609)
k "0, k "!(2J , * 0 with choice of unit in which G "1. With this choice of normalization, Q in (604) becomes " Q"4nR/a , and p and Q are related as Q "2(2p . Furthermore, the periodicity of * 0 * 0 * 0 * 0 x requires u"l/(nR) for some integer l. With this identi"cation, one has M "2p , as one would expect from the fact that the target & 0 space solution with the ADM mass M is matched onto the string source with the right moving & momentum p . This proves the assumption that the perturbative string states are black holes. 0 The generalization to a (D!1)-dimensional rotational black hole with [(D!2)/2] angular momenta involves F representing independent rotations in [(D!2)/2] mutually orthogonal planes.
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Furthermore, J in (609), which reduced to the form J"Al/a , satis"es the Regge bound J4 " 2#a (p!p)""Al/a , with J forming the Regge trajectory when l"1. Finally, the 0 * gyromagnetic ratios g , de"ned by k "g Q J/2M , are * 0 * 0 * 0 * 0 & g "0, *
g "2 . 0
(610)
This result is consistent with the fact that the BPS black holes correspond to the perturbative string states with only left-mover excited, since the gyromagnetic ratios are related to the left- and right-moving angular momenta J as * 0 g
J "2 0 * . * 0 J #J 0 *
Moreover, the right-moving gyromagnetic ratio is 2 as expected from the fact that the underlying states are fundamental. 7.5. Dyonic black holes and chiral null model So far, we discussed statistical interpretation for entropy of purely electric black holes in terms of microscopic degrees of freedom of perturbative string states. The BPS electric black holes have a nice virtue of being free of quantum corrections, thereby the ADM mass can be trusted. However, due to singularity at the horizon, the horizon gets shifted by &(a through the a -corrections. Thus, entropy is known only up to the order of a . Furthermore, such solutions are not black holes in the conventional sense, since the event horizon coincides with the singularity and has zero area. It is the construction of dyonic solutions [178] in the heterotic string on ¹ that triggered renewed interests in black hole entropy and made the precise calculation of the statistical entropy possible. Such dyonic solutions not only do not receive quantum corrections, but also are free of classical a -corrections [173] since they are described by exact conformal p-model and the event horizon is free of singularity. Such dyonic solutions contain as a subset the Reissner}NordstroK m solution and have non-zero event horizon area. Since the event horizon is regular, the a corrections are under control at the event horizon. And as in the pure electric case, the dilaton is "nite at the event horizon, implying that the string loop corrections are under control. Being free of plagues (i.e. a -corrections of purely electric solutions and the string loop corrections of the purely magnetic solutions) su!ered by the previously known solutions in string theories, the dyonic solution [178] is suitable for studying statistical origin of black hole entropy. Such observation was "rst made in [441] (see also Refs. [363,364]), where it is proposed that the microscopic degrees of freedom of the dyonic black holes are due to the hair associated with the oscillations in the internal dimensions. The "rst attempt to explain the statistical origin of the BPS black holes with non-zero event horizon area is based upon a special class of string worldsheet p-model called `chiral null modela. In this approach, the BPS black holes are embedded as background "elds of the chiral null model and the throat region conformal model is studied for understanding microscopic degeneracy of string states. Remarkably, the throat region conformal theory approximates to the
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WZNW model of perturbative string theory with string tension rescaled by magnetic charges of the black holes. So, the degeneracy of string states carrying magnetic charges is obtained by applying level density formula of perturbative string states. In this section, we summarize the results of [174,609,612], which study chiral null model interpretation of black hole entropy. 7.5.1. Chiral null model String theory is a promising candidate for consistent quantum gravity theory, being free from ultra-violet divergences, which plagued quantum gravity of point-like particles. So, it is useful to study classical string solutions to address problems in quantum gravity. However, it is almost hopeless to obtain exact classical solutions to the equations of motion following from e!ective "eld theory of string massless states, since the e!ective action consists of in"nite series of terms of all derivatives multiplied by powers of a , which are also ambiguous due to the freedom of choosing di!erent renormalization schemes (or "eld rede"nitions). So, the only exact classical string solutions that one can study are those that do not have a -corrections. In fact, there exist classes of string p-models whose background "elds do not receive a -corrections in a special renormalization scheme. One starts from a string p-model which is shown to be conformal to all orders in a and looks for classical solutions to the leading order (in a ) e!ective "eld theory which can be embedded as target space background "eld con"gurations of the p-model, or vice versa. This approach of studying classical solutions of string theory is based upon a remarkable relationship between the conformally invariant string p-model and the extremum of the e!ective action. To the leading order in string coupling, the string "eld equations are obtained by the conformal (Weyl) invariance condition of p-model, which are equivalent to the stationary conditions of the e!ective action due to the proportionality between the Weyl anomality coe$cients and derivatives (with respect to "elds) of the e!ective action. Given a conformal p-model, one obtains a string solution not modi"ed by a -corrections. The general bosonic p-model describing string propagation in background of massless "elds G , B and U is +, +, 1 dz[(G #B )(X)RX+RM X,#a RU(X)] . (611) I" +, +, pa
The chiral null model is a special case of (611) with the Lagrangian: ¸"F(x)Ru[RM v#K(x, u)RM u#2A (x, u)RM xG]#(G #B )(x)RxGRM xH#RU(x) , (612) G GH GH where X+ are splitted into `light-conea coordinates u, v and `transversea coordinates xG, i.e. X+"(u, v, xG). Note, F does not depend on u. There exists a special renormalization scheme in which the p-model (612) is conformal to all orders in a , provided (i) the transverse p-model ¸ "(G #B )(x)RxGRM xH#R (x), where , GH GH
,U! ln F, is conformal and (ii) F, K, A and U satisfy conformal invariance conditions, which G In di!erent approach [332}335] based upon the Rindler geometry in throat region, black hole in the weak coupling limit is described by closed strings in the background of black hole carrying non-perturbative charges. The e!ect of these non-perturbative charges on closed strings is to rescale the string tension, as in the case of chiral null model approach.
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for the case ¸ has at least (4, 0) supersymmetry take the form [174]: , ! F\#RG R F\"0, ! K#RG R K#R AG"0 , G G S G (613) i.e. (e\(FGH)!e\(F HGIH"0 , ! K FGH#R FGH"0, G G GI G where F "R A !R A , H "3R B and the covariant derivative K , (CK ) is de"ned in terms GH G H H G GHI G HI
of the generalized connection CK G ,CG #HG with torsion. HI HI HI The chiral null model has one null Killing vector which generates shifts of v: the action is invariant under an a$ne symmetry vPv "v#h(q#p). The associated null Killing vector R/Rv gives rise to the conserved current J "F(x)Ru on the string worldsheet. A balance between the T metric and the antisymmetric tensor (G "B ) implies that the conserved current J is chiral, SG SG T which is a crucial condition for the conformal invariance [369,371]. The action (613) is also invariant under the following subgroup of coordinate transformations on v combined with a gauge transformation of K and A : G vPv!2g(x, u), KPK#2R g, A PA #R g . (614) S G G G Unless K, A and U do not depend on u, one can choose a gauge in which K"0 by applying (614). G When the "elds are independent of u, the chiral null model turns out to be self-dual. Namely, a leading-order duality transformation along any non-null direction in the (u, v)-plane (say along the u-direction, where v"v( #au with a constant) leads to a p-model of the same form with duality transformed background "elds given by (615) F "(K#a)\, K "F\, A "A , U "U! ln[F(K#a)] . G G When background "elds are independent of u, the conformal invariance conditions (613) take the form of the Laplace equations in the transverse space. Background "eld solutions to the conformal invariance conditions are then parameterized by harmonic functions in the transverse space. Since the equations are linear, one can superpose harmonic functions to generate multi-center solutions. One can further generalize background "elds to depend on u in such a way that the conformal invariance conditions (613) are still satis"ed. Such changing of background "elds is viewed as `marginal deformationsa [371] of the conformal "eld theory. In particular, adding zero modes of A has the e!ect of adding a Taub-NUT charge, angular momenta or extra electric/magnetic G charges to the original solutions. The chiral null model (612) generalizes K-model (plane fronted wave solution) and F-model (a generalization of the fundamental string solution). First, in the limit F"1, (612) describes a class of plane fronted wave backgrounds which have a covariantly constant null vector R/Rv (K-model). For this case, one can add another vector coupling 2A M (x, u)RxGRM u to the Lagrangian while still G preserving conformal invariance but breaking chiral structure. Second, when K"0 with background "elds independent of u, (612) reduces to the F-model, which has two null Killing vectors R/Ru and R/Rv associated with a$ne symmetries u "u#f (q!p) and v "v#h(q#p). Since the
This follows [371] from the standard leading order conformal invariance conditions RK #2DK DK U"0 of \+, \+ \, the general p-model (611). Here, the Ricci tensor RK and covariant derivative DK are de"ned in terms of the \+, \+ generalized connection CK . "C. $H. with torsion H . !+, +, +, +,.
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coupling to u and v is chiral (G "B ), the associated 2 conserved currents JM "FRM v and J "FRu ST ST S T are chiral. The F model and the K model are related by a duality transformation (615) along u with a"0. The dimensional reduction of chiral null model leads to various charged (under the KK or 2-form gauge "eld) black hole and string solutions, which can also carry angular momenta or the Taub-NUT charge. The chiral coupling leads to a no-force condition (a characteristic of BPS solutions) on the solutions, which allows the construction of multi-centered solutions. The balance between G and B in chiral theories manifests in lower dimensional solutions as the "xed ratio +, +, of mass and charge, i.e. the BPS condition. In the following, we discuss black hole solutions that satisfy the conformal invariance conditions (613) and therefore are exact to all orders in a . For the purpose of obtaining general D"4, 5 black holes, we split the transverse coordinates xG into non-compact ones xQ and compact ones yL, i.e. xG"(xQ, yL), where s"1,2, D!1 and n"1,2, 9!D. We decompose the 8-dimensional transverse space into the direct product M;¹ of some 4-space M and ¹. The chiral null model Lagrangian (612) is accordingly expressed as the sum of two terms associated with each space. Here, M has SO(3) [SO(4)] symmetry for the D"4 [D"5] black holes and, therefore, M is parameterized by (xQ, y) [xK] with background "elds depending on the coordinates only through r"(xQxQ [r"(xKxK]. We consider the case where M has the torsion related to dilaton
in the speci"c way HKLI"!(2/(G)eKLIJR , so that the last conformal invariance condition J in (613) simpli"es to a Laplace-form (e\(FGH )"0, where FGH ,FGH#夹FGH with G > > 夹FGH"(1/2(G)eGHIJF . IJ 7.5.1.1. General four-dimensional, static, BPS black hole. We consider the case where 4-dimensional transverse part of the metric has the form G "f (x)g where f"0, , (g) and g is GH GH GH a hyper-KaK hler metric with a translational isometry in the x-direction. The D"6 part (u, v, x,2, x) of (612) then takes the special form: ¸"F(x)Ru[RM v#K(x)RM u#2A(x)(RM x#a (x)RM xQ)]#R ln F(x)#¸ , Q , ¸ "f (x)k(x)(Rx#a (x)RxQ)(RM x#a (x)RM xQ)#f (x)k\(x)RxQRM xQ , Q Q #b (x)(RxRM xQ!RM xRxQ)#R (x) , Q
(616)
where xQ"(x, x, x) are non-compact coordinates and compact coordinates are x"y and u"y . Here, we chose A in (612) to take the form A "A and A "Aa so that the D"4 metric G Q Q has no Taub-NUT term. The Lagrangian (616) is invariant under T-duality transformations in the x-direction (P P and qP!q): fPk\, kPf \, a b , AP( f k)\A , Q Q
(617)
and in the u-direction (Q Q ): FPK\, KPF\, U(F)PU(K\) .
(618)
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When A"0, the Lagrangian has remarkable manifest invariance under D"4 S-duality, under which (618) transforms as u x and FPf \, KPk\, fPF\, kPK\ ,
(619)
and under the D"6 string}string duality (GPe\UG, dBPe\U夹dB, UP!U) between the heterotic string on ¹ and the type-II string on K3: FPf \, KPK,
fPF\, kPk .
(620)
Note, the invariance of (616) under the T-duality is manifest only when all the charges associated with 4 harmonic functions F, K, f and k are non-zero. The self-dual case F"K\"f \"k and a "b corresponds to the D"4 Reissner}NordstroK m solution. As expected, the combined Q Q transformation of T-duality and the string}string duality yields the D"4 S-duality (619). One can obtain D"4 black hole solution which is exact to all orders in a by solving (613) with (616) and all the background "elds depending on non-compact transverse coordinates xQ, only. Solutions for background "elds are expressed in terms of harmonic functions f, k, F and K, which satisfy (linear) Laplace equations. Particularly, A is given in terms of harmonic functions by A"q k\#q f k (q " const). If one further assumes the asymptotic #atness condition (i.e. : q "!q . kP1, fP1, AP0 as r"(xQx PR), coe$cients in A are restricted such that q " Q The solutions for background "elds are: Q P Q F\"1# , K"1# , f"1# , r r r
P k\"1# , r
a dxQ"P (1!cos h)du, b dxQ"P (1!cos h)du , Q Q
(621)
r#P q r#(P #P ) , eU"Fe(" , A" ) r#P r#Q r where q,2q (P !P ). Since the resulting (conformal invariance condition) equations are of Laplace-type, one can superpose harmonic functions to obtain multi-center generalization of the above. The D"4 spherically symmetric solution in (200) is obtained by applying the standard KK procedure with all the background "eld in (612) properly identi"ed with those in (611) and setting u"y , v"2t, x "y . 7.5.1.2. General xve-dimensional, rotating, BPS black hole. We consider the case where M-part is (locally) SO(4)-invariant. The D"6 part of chiral null model Lagrangian is again given by (616) with A "(A , 0) and the transverse part ¸ replaced by G K , ¸ "f (x)RxKRM xK#B (x)RxKRM xK#R (x) , , KL
(622)
where m, n,2"1,2, 4, and the "elds are given in terms of a harmonic function f (x) (Rf"0) R "!e R f. By solving the conformal by " ln f, G "fd and H "!2(GGNJe KL KL KLI KLIN J KLIJ J
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invariance conditions (613) with above AnsaK tze, one obtains Q P f"1# , F\"1# , r r
Q K"1# , r
(623)
r#P c c e "Ff" , A " sin h, A " cos h , P P r#Q r r U
where r,(xKxK and c is related to the angular momenta as J "J "(p/4G )c. Note, the , conformal invariance condition (e\(FKL)"0 is solved by imposing the #at-space anti-selfK > duality condition F "0 on the "eld strength of the potential A , and as a result the 2 angular >KL K momenta are the same. By superposing harmonic functions, one obtains the multi-center generalization of the above. The dimensional reduction along u leads to D"5, rotating, BPS black hole with charge con"guration (Q "Q, Q "Q, P), where P is a magnetic charge of the NS-NS 3-form "eld strength (or an electric charge of its Hodge-dual). The Einstein-frame metric is
c ds"!j(dt#A dxK)#j\ dxK dx "!j dt# (sinh du #cos h du ) # K K r #j\[dr#r(dh#sin h du#cosh du)] r j"(F\Kf )\ " . [(r#Q )(r#Q )(r#P)]
(624)
7.5.2. Level matching condition To calculate statistical entropy of black holes, one has to relate macroscopic quantities of black holes to microscopic quantities of perturbative string states through `level matching conditiona [189]. Strictly speaking, level matching process is possible for electric solutions, only, since perturbative string states do not carry magnetic charges and string momentum [winding] modes are matched onto `electrica charges of KK [2-form "eld] ;(1) "elds. Furthermore, magnetic solutions can be supported without source at the core (cf. magnetic solutions are regular everywhere including the core), since they are topological in character. However, it turns out [174] that the dyonic solution found in [178], which is a `bound statea of fundamental string and solitonic 5-brane, still needs a source for its support and satis"es the same form of level matching condition as the fundamental string. The crucial point in the level matching of such dyonic solutions onto the perturbative string spectrum is that as in the purely magnetic case "elds are perfectly regular near the horizon (or the throat region), making it possible to describe the solutions at the throat region in terms of WZNW conformal model [289,437,484,632]. Since the solution is regular and the dilaton is "nite near the event horizon (implying that the classical a and the string loop corrections are under control), such e!ective WZNW model near the horizon can be trusted. For large magnetic charges (or large level), the theory e!ectively looks like a free p-model for perturbative string theory with the string tension rescaled by magnetic charges. Namely, for large magnetic charges, the dyonic
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solutions are matched onto perturbative string states with string tension rescaled by magnetic charges. To match background "eld solutions onto the macroscopic string source at the core, one considers the combined action of string p-model and the e!ective "eld theory. Among the equations of motion of the combined action, the relevant parts are the Einstein equations for target space metric, equations of motion for XI and the Virasoro conditions. Requiring that all the solutions are supported by sources, one obtains the level matching condition:
1 p0 E(u), du E(u)"0, (E(u),[F(x)K(u, x)] "0) . V 2pR
(625)
This condition is satis"ed without modi"cation even when solutions carry magnetic charges. 7.5.3. Throat region conformal model and magnetic renormalization of string tension 7.5.3.1. Four-dimensional dyonic solutions. The p-model (616) of dyonic black hole (193) with charge con"guration (P, Q, P, Q),(P , Q , P , Q ) takes the following form near the hor izon (rP0) [174,609,612]:
1 1 dp ¸ " dp(e\XRuRM v#Q Q\RuRM u) I" P pa pa
P P # dp(RzRM z#Ry RM y #RuRM u#RhRM h!2 cos hRy RM u) . pa
(626)
This is the S¸(2, R);S;(2) WZNW model with the level i"(4/a )P P . (Since the level has to be an integer, one has the quantization condition 4P P /a 3Z.) Here, the coordinates are de"ned as z,ln(Q /r)PR, u ,(Q\Q P P )\ u, v ,(Q Q\P P )\ v, and y ,P\y #u. For large P (or i), the transverse (o, y , u, h) part of (626) looks like a free theory of perturbative string with the string tension ¹"1/(2pa ) renormalized by P : 1 P P P P 1 P " " , a R a a a ,
(627)
where R "(a is the radius of the internal coordinate associated with P . 7.5.3.2. Five-dimensional dyonic solution. The p-model with the target space con"guration given by the 3-charged, BPS, non-rotating black hole, i.e. (263) with J"0, takes the following form in the limit rP0 [609,612]:
1 1 dp ¸ " dp(e\XRuRM v#Q Q\RuRM u) I" P pa pa #
P dp(RzRM z#Ru RM u #Ru RM u #RhRM h!2 cos hRu RM u ) , 4pa
(628)
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where z,ln(Q /r)PR. This is the S¸(2, R);S;(2) WZNW model with the level i,(1/a )P. In the limit of large P or large i, the transverse part (o, u , u , h) of (628) reduces to free perturbative string theory with the renormalized string tension: 1 P 1 P " . 4a a a ,
(629)
7.5.4. Marginal deformation The degeneracy of micro-states responsible for statistical entropy is traced to the degrees of freedom associated with oscillations or marginal deformations around the classical solutions. The marginal deformations lead to a family of all the possible solutions (obeying conformal and BPS conditions) with the same values of electric/magnetic charges but di!erent short distance structures that depend on a choice of oscillation pro"le function. Thus, one has to consider the region near the horizon (at r"0) to determine the microscopic degrees of freedom, since in this region degeneracy of solutions is lifted. General chiral null model action which represents deformation from the classical BPS solutions in Section 7.5.1 and preserves the BPS and conformal invariance properties is given by (612) with K and A "(A , A ) having an additional dependence on u, where u,z!t with the longitudinal G K ? direction z satisfying the periodicity condition z,z#2pR. Here, A &q (u)/r and A "q (u)/r K K ? ? (r,x x ) are respectively `deformationsa in the non-compact xK and the compact y? directions. K K On the other hand, the perturbation K(u, x)"h (u)xK#k(u)/r does not contribute to the K degeneracy, since h (u)xK drops out and k(u) has zero mean value. K The perturbations K, A and A represent various `left-movinga waves propagating along the ? K string and are invisible far away from the core. The mean values q (u) and q(u) are related to the ? K oscillation numbers of the macroscopic string at the core as p pa NK" q (u), N?" q(u) , (630) * * 16G K 16G ? , , thereby contributing to the microscopic degrees of freedom. These marginal deformations do not contribute to the microscopic degeneracy of black holes with the same order of magnitude [612]. This can be inferred from the fact that the classical BPS black holes are solutions of both heterotic and type-II string. Namely, although the thermal entropy is the same whether one embeds the solutions within heterotic or type-II string, one faces the discrepancy in factor of 2 in the statistical entropies within the two theories, if one takes the degeneracy contributions of all the oscillators to be of the same order of magnitude. In fact, as can be seen from the conformal invariance condition (e\(FGH )"0, the perturbations A in the G > ? compact directions y are decoupled from the non-trivial non-compact parts of the solution. On ? the other hand, the perturbations A in the non-compact directions x are non-trivially coupled to K K the magnetic harmonic functions, with the net e!ect being the rescaling of q (u) terms by the K The chiral null model corresponding to rotating black hole (624) also approaches the S¸(2, R);S;(2) WZW model in the throat limit. The requirements that the level i"P/a is an integer and u "u #2cP\Q\u (u ,u #u ) has the period 4p lead to the quantization of P and J: P"a i and J"iwl with k, l3Z (w" string winding number). The regularity of the underlying conformal model requires that J(imw, i.e. the `Regge bounda.
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magnetic charges. (This is related to the scaling of the string tension in the transverse directions by the magnetic charge(s).) So, the marginal deformation contributions from the compact directions, which are di!erent for the two theories, are suppressed relative to those of non-compact directions by the factor of the inverse of magnetic charge(s), thereby negligible for a large magnetic charge(s) or the large level i. Only the marginal deformations from the non-compact directions and the compact direction associated with non-zero magnetic charge(s), which are common for both theories, have the leading contribution to the degeneracy. Only these 4 string coordinates get their tension e!ectively rescaled by the magnetic charge(s). Furthermore, the marginal deformations on the original black hole solutions have to be only left-moving (i.e. depend only on u, not on v) so that marginally deformed p-models are conformaly invariant. This is related to the `chirala condition on the p-model; only left-moving deformations lead to supersymmetric action. Thus, for large magnetic charge(s), the statistical entropy calculations within heterotic and type-II strings agree. As pointed out, the marginal deformations A (u, x)"q (u)/r in the compact directions ? ? y contribute to the microscopic degeneracy to sub-leading order (suppressed by the inverse of ? magnetic charge(s)), which can be neglected for large value of magnetic charge(s). But the zero modes q of the Fourier expansions of q (u)"q #q (u) (q (u) denoting the oscillating parts) ? ? ? ? ? produce additional left-handed electric charges [174] of D"6 strings. Namely, the internal marginal deformation A (u)"q (u)/r on the p-model associated with D"4 4-charged BPS black ? ? hole (193) leads to 5-charged BPS black hole solution (621) with the zero mode q corresponding to ? an additional charge parameter q. The mean oscillation values q (u) of the marginal deformations A (u, x)&q (u)/r K K K (q (u)"q #q (u) with q and q (u) respectively denoting the zero modes and oscillating parts) K K K K K in the non-compact directions x contribute to N to the leading order. Meanwhile, the zero modes K * q have an interpretation as angular momenta [610] of black holes. Namely, the rotational K marginal deformation corresponding to S;(2) Cartan current deformation A (u, x) dxK" K (c(u)/r)(sin h du #cos h du ) on the p-model action of the D"5 3-charged non-rotating solution leads to the rotating solution (624) [609,610]. To calculate the statistical entropy associated with degeneracy of solutions (i.e. all the possible marginal deformations of the original classical solution), one has to determine N of the macro* scopic string at the core. For this purpose, we write the marginally deformed p-model action (612) in the throat region in the form:
1 dp(marginal deformation terms) I "I# P pa
1 dp [e\XRuRM v#E(u)RuRM u]#other terms , " pa
(631)
where I stands for the throat limit WZNW models (626) and (628) of the (undeformed) classical solutions. First, for the D"4 dyon, the marginal deformation (612) gives rise to the following expression for E(u): E(u)"Q Q\!(P P )\Q\q(u)!Q\q(u) L ? "(P P )\Q\[Q Q P P !q(u)!P P q(u)] . L ?
(632)
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Note, since the string tension a (627) of the transverse parts is rescaled by the magnetic charges, , the coe$cient in front of the term q(u) is rescaled by (P P )\. Applying the level matching L condition E(u)"0 (625), one "nds that q(u)"P P (Q Q !q ), where q denote the zero modes L ? ? of the oscillations q (u) in the compact directions. Thus, the statistical entropy is [174] ? p (P P (Q Q !q ) , (633) S +2p(N " ? * 2G , in agreement with the thermal entropy. Second, we consider D"5 solutions. For the non-rotating solutions, the marginal deformation (612) leads to E(u)"Q Q\#Q\k(u)!P\Q\q (u)#O(P\) . (634) K Applying the level matching condition E(u)"0, one "nds that q (u)"Q Q P for large P, which K reproduces the thermal entropy [609,612]: p (Q Q P . (635) S +2p(N " * 2G , For the rotating solution, one introduces the marginal deformation A in a non-compact K direction. Then, one has E(u)"Q Q\!P\Q\c(u)"P\Q\[Q Q P!c(u)] , (636) where c(u)"c#c(u) with c and c(u) respectively denoting zero and oscillating modes of c(u). From the level matching condition E(u)"0, one has c(u)"Q Q P!c, reproducing entropy of the rotating black hole [609,610] in the limit of P +P and Q large: p (Q Q P!c . (637) S +2p(N " * 2G , 8. D-branes and entropy of black holes Past year or so has been an active period for investigation on microscopic origin of black hole entropy. The construction of general class of BPS black hole solution in heterotic string on ¹ [178] motivated renewed interest [441] in the study of black hole entropy within perturbative string theory. The explicit calculation of statistical entropy of BPS solution in [178] by the method of WZNW model in the throat region of black hole reproduced the Bekenstein}Hawking entropy. Realization [498] that D-branes in open string theory can carry R-R charges motivated the explicit D-brane calculation of statistical entropy of non-rotating BPS solution in D"5 with 3 charges [580]. This is generalized to rotating black hole [98] in D"5, near extreme black hole [109,368] in D"5 and near extreme rotating black hole [95] in D"5. Meanwhile, the WZNW model approach was generalized to the case of non-rotating BPS D"5 black hole [609] and rotating BPS D"5 black hole [612]. D-brane approach was soon extended to D"4 cases: non-rotating BPS case in [392,471], near extreme case in [361] and extreme rotating case in [361]. Later, it is shown [365] that the microscopic counting argument in string theory can be extended even to
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non-extreme black holes as well, provided the entropy is evaluated at the proper transition point of black hole and D-brane (or perturbative string) descriptions. In this section, we review the recent works on D-brane interpretation of black hole entropy (for other reviews, see Refs. [468,360]). With realization [498] that R-R charges, which were previously known to be decoupled from string states, can couple to D-branes [193], it became possible to do conformal "eld theory of extended objects (p-branes) within string theories and to perform counting [90,91,566,567,619,620] of string states that carries R-R charges as well as NS-NS charges. To apply D-brane techniques to the calculation of microscopic degeneracy of black holes, one has to map non-perturbative NS-NS charges of the generating black hole solutions of heterotic string on tori to R-R charges by applying subsets of U-duality transformations. In D-brane picture of black holes [109], the microscopic degrees of freedom are carried by oscillating open strings which are attached to D-branes. Whereas the e!ect of magnetic charges in the chiral null model and Rindler geometry approaches is to rescale the string tension, the e!ect of R-R charges on open strings in the D-brane description is to alter the central charge (i.e. the bosonic and fermionic degrees of freedom) of open strings from the free open string theory value. In the D-brane picture of [109] (see also Ref. [196]), the number of degrees of freedom of open strings is increased relative to the free open string value because of an additional factor (proportional to the product of D-brane charges) related to all the possible ways of attaching the ends of open strings to di!erent D-branes. So, the net calculation results of statistical entropy in both descriptions are the same. This chapter is organized as follows. In Section 8.1, we summarize the basic facts on D-branes necessary in understanding D-brane description of black holes. In Section 8.2, we discuss the D-brane embeddings of black holes. The D-brane counting arguments for the statistical entropy of black holes are discussed in Section 8.3. 8.1. Introduction to D-branes We discuss basic facts on D-branes necessary in understanding D-brane description of black holes. Comprehensive account of the subject is found, for example, in [193,391,445,498,501,504], which we follow closely. The basic knowledge on string theories is referred to [316,535]. Each end of open strings can satisfy two types of boundary conditions. Namely, from the boundary term (1/2pa ) M dp dXIR X , where RM is the boundary of the worldsheet M swept by L I . an open string and dXI[R X ] is the variation (the derivative) of bosonic coordinates XI parallel to L I (normal to) RM, in the variation (with respect to XI) of the worldsheet action, one sees that the ends of the string either can have zero normal derivatives R XI"0 , L called Neumann boundary condition, or have "xed position in target spacetime XI"constant ,
(638)
(639)
called Dirichlet boundary condition. In order for the T-duality to be an exact symmetry of the string theory, open string has to satisfy both the Neumann and Dirichlet boundary conditions [193]. Under the T-duality of open string theory with the coordinate XG"XG (z)#XG (z ) compacti"ed on S of radius R , R PR "1/R and 0 * G G G G
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XGP>G"XG (z)!XG (z ). So, the Neumann and the Dirichlet boundary conditions get inter0 * changed under T-duality:
Rz Rz Rz Rz RXG! RM XG" R>G# RM >G"R >G , R XG" O L Rq Rq Rq Rq
(640)
where q is the worldsheet time coordinate, which is tangent to RM. Starting from the D"10 open string with the Neumann boundary conditions and with the coordinates XG (i"p#1,2, 9) compacti"ed on circles of radii RG, one obtains open string theory with the ends of the dual coordinates >G con"ned to the p-dimensional hyperplane in the R P0 limit (or the decompacti"caG tion limit, i.e. 1/R PR, of the dual theory). Such p-dimensional hyperplane is called D-brane G [498]. A further T-duality in the direction tangent (orthogonal) to a Dp-brane results in a D(p!1)-brane (D(p#1)-brane). D-brane is a dynamical surface [498] with the states of open strings (attached to the D-brane) interpreted as excitations of #uctuating D-brane. The massless bosonic excitation mode in the open string spectrum is the photon with the vertex operator < "A R X+, where R is the + R R derivative tangent to RM. So, the bosonic low energy e!ective action is that of N"1, D"10 Yang-Mills theory [637]:
1 dx Tr[F FIJ] . IJ 2g
(641)
In the T-dual theory, the vertex operator < of the photon is decomposed into < " N A (XI)R XI and < " u (XI)R XG, corresponding respectively to ;(1) gauge boson GN G N I I R P and scalars on the p-brane worldvolume. The scalars u are regarded as the collective coordinates G for transverse motions of the p-brane. The bosonic low energy e!ective action of the T-dual theory is, therefore, obtained by compactifying (641) down to p#1 dimensions [637]. In this action, the worldvolume scalars u have potential term <" Tr[u , u ]. G H G H G Note, open strings can have non-dynamic degrees of freedom called Chan-Paton factors (i, j) at both ends of strings [139,316,501,504,637]. The indices (i, j), which label the state at each end of the string, run over the representation of the symmetry group G. When the state "K, ij2 describes a massless vector, (i, j) run over the adjoint representation of G. Each vertex operator of an open string state carries antihermitian matrices j? (a"1,2, dim G) representing the algebra of G and GH j? describe the Chan-Paton degrees of freedom of the open string states. The global symmetry G of GH the worldsheet amplitude manifests as a gauge symmetry in target space. For the oriented open strings, G is ;(N) and each end of the open string is respectively in complex and complex conjugate representations of ;(N). When open string states are invariant under the worldsheet parity transformation X (pPp!p or zP!z ), i.e. the exchange of two ends plus reversal of the orientation of an open string, the open string is called unoriented. For this case, the representations R and RM on both ends of the string are equivalent. For unoriented open strings, the transformation property of the Chan-Paton matrix j under the worldsheet parity X GH determines G [474,535]. Under the worldsheet parity symmetry, the open string state j "K, ij2 GH The tachyon is removed by the GSO projection of supersymmetric theory.
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transforms to Xj "K, ij2"j "K, ij2, j "Mj2M\ . GH GH
(642)
If M is symmetric, i.e. M"M2"I , then the photon j aI "k2 survives the projection under the , GH \ gauged worldsheet parity and the Chan-Paton factor is antisymmetric (j2"!j), giving rise to SO(N) gauge group. If M is antisymmetric, i.e.
0
M"!M2"i
!I ,
I , , 0
then the gauge group is ;Sp(N), i.e. j"!Mj2M. When the Chan-Paton factors are present at the ends of open string, a Wilson line, say, for the ;(N) oriented open string theory with the coordinate X compacti"ed on S of radius R given by A "diag(h ,2, h )/(2pR)"!iK\R K , ,
(643)
where K"diag(e 6F p0,2, e 6F, p0), receives non-trivial phase factor diag(e\ F,2, e\ F,) under the transformation XPX#2pR. As a result, the momentum number of an open string along S can have a fractional value. So, in the T-dual theory the winding number takes on fractional values [497], meaning that two ends of open strings can live on N di!erent hyperplanes (D-branes) located at >"h R "2pa A (k"1,2, N): I II
>(p)!>(0)"
p
dp R >"(2p#h !h )R , N H G
(644)
where R "a /R is the radius of the circle in the dual theory. Note, without the Chan-Paton factors taken into account, both ends of open strings of the dual theory are con"ned to the same hyperplane up to the integer multiple of periodicity 2pR of the dual coordinate >. The similar argument can be made for unoriented open string theories. With SO(N) Chan-Paton symmetry, the Wilson line can be brought to the form: diag(h ,!h ,2, h ,!h ) . , ,
(645)
Note, in the dual coordinate >K, worldsheet parity reversal symmetry (zP!z ) of the original theory is translated into the product of worldsheet and spacetime parity operations. Since unoriented strings are invariant under the worldsheet parity, the T-dual spacetime is a torus modded by spacetime parity symmetry Z . The "xed planes >K"0, pR under spacetime parity symmetry are called orientifolds [193]. Away from the orientifold plane, the physics is that of oriented open strings, with a string away from the orientifold "xed plane being related to the string at the image point. Open strings can be attached to the orientifolds, but the orientifolds do not correspond to the dynamic surface since the projection X"#1 [355,356,509] removes the open string states corresponding to collective motion of D-branes away from the orientifold plane. In this T-dual theory, there are N/2 D-branes on the segment 04>K(pR and the remaining N/2 are at the image points under Z . Open strings can stretch between pairs of Z re#ection planes, as well as between di!erent planes on one side.
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With a single coordinate X compacti"ed on S of radius R, the mass spectrum of the dual open string is
M"
[2pn#(h !h )]R 1 G H # (N!1) . 2pa a
(646)
Thus, the massless states arise in the ground state (N"1) with no winding mode (n"0) and both ends of the string attached to the same hyperplane (h "h ): G H aI "k, ii2, <"A R XI , \ I R (647) a "k, ii2, <"uR X"uR > , \ R L respectively corresponding to D-brane worldvolume photon and scalar. For the case of oriented open string theories, when all the N hyperplanes (located at h R ) do not coincide (i.e. I h Oh , ∀i, j), ;(N) is broken down to ;(1),, corresponding to N massless ;(1) gauge "elds at each G H hyperplane located at h R (k"1,2, N). When m hyperplanes (m4N) coincide, say I h "2"h , the additional massless ;(1) gauge "elds (associated with open strings originally K stretched between these m hyperplanes) contribute to the restoration of the symmetry to ;(m);;(1),\K [497,637]. Furthermore, m massless scalars (interpreted as positions [193,637] of m distinct hyperplanes) are promoted to m;m matrix of m massless scalars when these m hyperplanes coincide [637]. For unoriented open strings with SO(N) symmetry, the generic gauge group with all the N/2 D-branes distinct is ;(1), . When m D-branes coincide, the symmetry is enhanced to ;(m);;(1), \K as in the oriented case. But when m D-branes are located at an orientifold plane, the symmetry is enhanced to SO(2m);;(1), \K, due to additional massless ;(1) gauge bosons arising from open strings that originally stretched between pairs of Z image branes. In the language of e!ective "eld theory, this symmetry enhancement or reversely symmetry breaking to Abelian group is interpreted as Higgs mechanism with scalars associated with location and separation of D-branes interpreted as Higgs "elds. The fermionic spectrum of open strings is divided into subsectors according to the boundary conditions that fermions tQ I (04p4p, !R(q(R) satisfy at one end of string. There are two types of boundary conditions on the fermion: R: tI(0, q)"tI I(0, q) tI(p, q)"tI I(p, q) , NS: tI(0, q)"!tI I(0, q) tI(p, q)"tI I(p, q) .
(648)
De"ning tI(2p!p, q),tI I(p, q), one sees that the Ramond (R) [Neveu-Schwarz (NS)] boundary condition becomes the periodic (anti-periodic) boundary condition on the rede"ned fermion t(p, q)(04p42p, !R(q(R), leading to integer (half-integer) modded Fourier series decomposition. In the NS sector, the ground state consists of 8 transverse polarizations tI "k2 of massless \ open string photon A . In the R sector, the ground state is degenerate, transforming as 32 spinor I representation of SO(8). The Virasoro conditions pick out 2 irreducible representations 8 and 8 . Q A Here, the subscript s[c] means eigenstates "s , s , s , s , s 2 (s , s "$) with even (odd) number of G The same argument can be applied for the case of ;Sp(N) symmetry.
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eigenvalues ! of S "iS and S "SG G>, where SIJ"! t I tJ are the fermionic part of G P \P P the D"10 Lorentz generators. These 2 representations are physically equivalent for open strings. The GSO projection picks out 8 and, therefore, the ground state of the open string theory is 8 8 , Q T Q forming a vector multiplet of D"10, N"1 theory. Including the Chan-Paton factors, the gauge group G of the N"1, D"10 super-Yang-Mills theory is ;(N) [SO(N) or ;Sp(N)] for an oriented [an unoriented] open string theory. For an open string theory with 9!p coordinates compacti"ed, the massless spectrum of Dp-brane worldvolume theory of dualized open string is described by the D"10, N"1 supersymmetric gauge theory compacti"ed to D"p#1. We brie#y discuss some aspects of type-II closed string relevant for understanding Dp-branes of open string theory. For type-II string, i.e. the closed string theory with supersymmetry on both leftand right-moving modes, the two choices of the GSO projections in the R sector are not equivalent. So, there are two types of type-II theories de"ned according to the possible inequivalent choices of the GSO projections on the left- and the right-moving modes. The massless sectors of these two type-II theories are Type IIA: (8 8 ) (8 8 ) , T Q T A (649) Type IIB: (8 8 ) (8 8 ) . T Q T Q The massless modes 8 8 in the NS-NS sector of the both theories are the same: dilaton, gravitino T T and the 2-form "eld. In the R-R sector, the massless modes 8 ;8 [8 ;8 ] of type-IIA [type-IIB] Q A Q Q theory are 1- and 3-form potentials (0-, 2- and self-dual 4-form potentials) [275]. The massless modes in NS-R and R-NS sectors contain 2 spinors and 2 gravitinos of the same (opposite) chirality for the type-IIB (type-IIA) theory. When an oriented type-II theory with a coordinate compacti"ed on S is T-dualized [193,216], the chirality of the right-movers gets reversed. So, when odd (even) number of coordinates in type-IIA/B theory are T-dualized, one ends up with type-IIB/A (type-IIA/B) theory. The e!ect of `odda T-duality, which exchanges type-IIA and type-IIB theories, on massless R-R "elds is to add (remove) the indices (of (p#1)-form potential) corresponding to the T-dualized coordinates, if those indices are absent (present) in the (p#1)-form potential. For example, the T-duality on B [B ] along xM (kOoOl) produces B [B ]. IJ IM IJM I A worldsheet parity symmetry X in a closed string, de"ned as pP!p or zPz , interchanges left- and right-moving oscillators. The unoriented closed string is de"ned by projecting only even parity states, i.e. X"t2"#"t2, as in the open string case. When type-II string is coupled to open superstring (type-I string), the orientation projection of type-I string picks up only one linear combination of 2 gravitinos in type-II theory, resulting in an N"1 theory. The only possible consistent coupling of type-I and closed superstring theories is between (unoriented) SO(32) type-I theory [108,503] and unoriented N"1 type-II theory. But in the T-dual theory, type-II theories without D-branes are invariant under N"2 supersymmetry, with orientation projection relating a gravitino state to the state of the image gravitino. The chiralities of these 2 gravitinos are the same
T-duality on the type-II theory leads to 16 D-branes on a ¹ /Z orbifold. (The restriction to 16, i.e. SO(32) gauge \N symmetry, comes from the conservation of R-R charges.) However, in non-compact space, one can have a consistent theory with an arbitrary number of D-branes.
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(opposite) if even (odd) number of directions are T-dualized. In the presence of D-branes, only one linear combination of supercharges in the T-dual type-II theory is conserved, resulting in a theory with 1/2 of supersymmetry broken (N"1 theory), i.e. the BPS state [498]. For this case, the leftand right-moving supersymmetry parameters (of the T-dual type-II theory) are constrained by the relation [326,501,504] e "C2CNe . (650) 0 * The conserved charges carried by D-branes are charges of the antisymmetric tensors in the R-R sector. The worldvolume of a D p-brane naturally couples to a (p#1)-form potential in the R-R sector, with the relevant space-time and D p-brane actions given by
1 G 夹G #ik N> N 2 N>
C , (651) N> NU where k"2p(4pa )\N is the D p-brane charge [498]. The worldvolume action is given by the N following Dirac}Born}Infeld type action [269,445] describing interaction of the world-brane ;(1) vector "eld and scalar "elds with the background "elds [445]: !¹
N
dN>m e\P det (G #B #2pa F ) , ?@ ?@ ?@
(652)
where ¹ is the D p-brane tension [199,311], and G and B are the pull-back of the spacetime N ?@ ?@ "elds to the brane. In the amplitudes of parallel D-brane interactions, terms involving exchange of the closed string NS-NS states and the closed string R-R states cancel [498], a reminiscence of no-force condition of BPS states. Furthermore, the D-brane tension, which measures the coupling of the closed string states to D-branes, has the g\ behavior [445], a property of R-R p-branes [367,381,601]. The Q "eld strengths which couple to D p- and D(6!p)-branes are Hodge-dual to each other, and the corresponding conserved R-R charges are subject to the Dirac quantization condition [482,498,591] k k "2pn. \N N 8.2. D-brane as black holes In Section 8.1, we observed that D p-branes have all the right properties of the R-R p-branes in the e!ective "eld theories. As solutions of the e!ective "eld theories, which are compacti"ed from the D"10 string e!ective actions, black holes can be embedded in D"10 as bound states [224,448,637] of D p-branes. Here, p takes the even (odd) integer values for the type-IIA [type-IIB] theory. Such p-branes of the e!ective "eld theories correspond to the string background "eld con"gurations [104,105] (with the p-brane worldvolume action being the source of (p#1)-form charges) to the leading order in string length scale l "(a and describe the long range "elds away Q from D p-branes. As long as the spacetime curvature of the soliton solutions (at the event horizon in string frame) is small compared to the string scale 1/l, the e!ective "eld theory solutions can be Q trusted, since the higher-order corrections to the spacetime metric is negligible. The e!ective "eld theory metric description of solitons in superstring theories is valid only for length scales larger than a string. The D-brane picture of black holes, or more generally black p-branes, is as follows.
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The string-frame ADM mass M of p-branes carrying NS-NS electric charge [187,380], NS-NS magnetic charge [577] and R-R charge [381] behaves as &1,&1/g and &1/g (in the unit where Q Q l &1), respectively. Since the gravitational constant G is proportional to g, the gravitational Q , Q "eld strength (JG M) of the NS-NS electric charged and the R-R charged p-branes vanishes as , g P0. Namely, in the limit g P0, strings live in the #at spacetime background. Note, in the limit Q Q g P0, the description of R-R charged con"gurations in terms of black p-branes is not valid, since Q the size of the p-brane horizon (A&g) is smaller than D-brane size; the black hole is surrounded Q by a halo which is large compared to its Schwarzschield radius. In the limit g <1, one can Q integrate out massive string states (with their masses increasing in Planck units, de"ned as l"1, as g increases) to obtain string e!ective "eld theories. (So, one can trust black hole solutions in the Q e!ective "eld theories in the strong string coupling limit.) In this limit, the massive string states form black holes and these degenerate massive states (whose mass is identi"ed with black hole mass) are degenerate black hole microscopic states, which are origin of statistical entropy. To summarize, the weak string coupling description of R-R charged con"guration is the perturbative D-branes in #at background, and in the strong string coupling limit the horizon size (&G M) , becomes larger than the string scale (with string states undergoing gravitational collapse inside the horizon), thereby, the black p-brane description emerges. The transition point of the two descriptions occurs at the point where the horizon size r is of the order of the string length scale l [365]. Q This occurs when g N &1 or g Q&1, where N is the string excitation level and Q is an R-R Q Q charge. This implies that g is very small for large N or Q. (Note, however that the e!ective string Q coupling of these bound states is g&g N or g Q, which is of order 1.) At this transition point, Q Q Q the mass of a string state M&N/l becomes comparable to black hole mass: M &r/G . Q Q , The essence of the D-brane description of black hole entropy is that the number of degenerate BPS states is a topological invariant which is independent of (continuous) moduli "elds, including g [549}551]. Furthermore, mass of the BPS states is not renormalized [485]. It is argued Q [195,496,469] that even for near BPS D-brane states the D-brane counting results can be extrapolated invariantly to strong coupling limit. It is also shown that D-brane approach reproduces entropy of non-supersymmetric extreme black holes [188,361] and even non-extreme case [365] as well. So, the statistical entropy of p-branes can be calculated by counting the number of degenerate perturbative (g P0) string states in D p-brane con"guration. Q Each D p-brane carries one unit of R-R charge and the g PR limit of Q D p-branes is black Q N p-brane carrying R-R (p#1)-form charge Q . R-R p-branes have a p-volume tension behaving as N &g\ [601]. So, although these are non-perturbative, R-R p-branes have singularities, except for Q p"3. The transverse (longitudinal) directions of R-R p-branes correspond to open string coordinates with Dirichlet (Neumann) boundary condition. The same T-duality rules of D-branes hold for R-R p-branes: T-duality on the transverse (longitudinal) directions of p-branes produces (p#1)-[(p!1)-] branes. When longitudinal directions are compacti"ed, single-charged p-branes become black holes having singular horizon with zero surface area and diverging dilaton at the horizon. This is due to
The gravitational constant in D"d is GB "G/< , where < is the volume of the (10!d)-dimensional , , \B \B internal space and G"8pga &gl. , Q Q Q The Planck length is de"ned as l "m\"( G /c) . ,
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the brane tension which makes the volume parallel (perpendicular) to the brane shrinks (expands) as one gets close to the brane [109]. Black holes having regular horizon with non-zero area in the BPS limit are constructed from bound states of p-branes (with a momentum) to balance the tension to stabilize the volume internal to all the constituent p-branes. Such regular black holes are obtained with the minimum of 4 [3] p-brane charges for the D"4 [D"5] black holes. The basic constituent of black holes is the R-R p-brane in D"10 [367]: ds "g dxI dxJ"f \ (!dt#dx#2#dx) IJ N N #f (dx #2#dx) , N N> (653) e\P"f N\ , A 2 "!( f \!1) , N N N where f "1#Q c/r\N (r,(x #2#x ) ). Here, c is related to the basic (p#1)-form N N N N> N potential charge and can be estimated by comparing the ADM mass of (653) to the mass of D-brane state carrying one unit of the (p#1)-form potential charge. The Killing spinors of this solution are constrained by [367]: e "C2CNe , e "!C2CNe , (654) * * 0 0 where e denotes the left/right handed chiral spinor (Ce "e ). One can construct solutions * 0 * 0 * 0 for bound states of p-branes by applying the intersection rules. (See Section 6.2.2 for details on intersection rules.) In particular, the dilaton is the product of individual factors associated with those of the constituent p-branes: e\P"f N \ 2f NI I\ . N N To add a momentum along an isometry direction xG, one oscillates p-brane so that it carries traveling waves along the xG-direction. (See Section 7.4.1 for the detailed discussion on the construction of such solutions.) At a long distance region, the solutions approach the form where all the oscillation pro"le functions are (time or phase) averaged over. So, the long-distance region p-brane solution carrying a momentum along the xG-direction is obtained by just imposing SO(1, 1) boost among the coordinates (x, xG), with the net e!ect on the metric being the following substitution: !dt#dxP!dt#dx#k(dt!dx ) , (655) G G G where k"cN/r\N with N interpreted as a momentum along the xG-direction. The momentum N along the xG-direction adds one more constraint on the Killing spinor: e "CCGe , e "CCGe . (656) 0 0 * * In general, the intersecting n p-branes preserve at least 1/2L of supersymmetry; since a single p-brane breaks 1/2 of supersymmetry with one spinor constraint (654), as one increases the number of constituents more supersymmetry get broken. One obtains BPS con"gurations if spinor constraints of constituent p-branes are compatible with non-zero spinor e . The intersecting D p0 * and D p -branes preserve 1/2 of supersymmetry i! p"p mod 4 [224]. All the supersymmetries are broken when the dimension of the relative transverse space is neither 4 nor 8. When there is
It is shown [431] that the stringy BPS black holes with non-zero horizon area are not possible for D56. This can be seen from the explicit solutions discussed in Section 4.5.
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a momentum in the xG-direction, the additional Killing spinor constraint (656) breaks 1/2 of the remaining supersymmetry. Black holes in lower dimensions are obtained by compactifying (intersecting) p-branes in D"10. In the language of p-branes, this compacti"cation procedure corresponds to wrapping p-branes along the cycles of compact manifold. Since the compacti"ed space is very small, the con"guration looks point-like (0-brane) in lower dimensions. In the following subsections, we discuss various D p-brane embeddings of D"4, 5 black holes having the regular BPS limit with non-zero horizon area. 8.2.1. Five-dimensional black hole We discuss D"5 type-IIB black hole originated from intersecting Q D1-branes (along x) and Q D5-branes (along x,2, x) with a momentum P #owing in the common string direction [109], i.e. the x-direction. The 1- and 5-brane charges are electric and magnetic charges of the R-R 2-form "eld, and the momentum corresponds to the KK electric charge associated with the metric component G. To obtain a black hole in D"5, one wraps Q D1-branes around S (along x) of radius R and wrap Q D5-branes around ¹"¹;S. Here, ¹ has coordinates (x ,2, x ) and volume <. The momentum (of open string) P"N/R #ows around S. The resulting D"5 solution has the form [109,182]:
r \ 1! dr#r dX , r
r ds"!f \ (r) 1! dt#f (r) r
(657)
where f (r) is given by
r sinh d f (r)" 1# r
r sinh d 1# r
(658)
R
(659)
r sinh d N . 1# r
The three charges carried by the black hole are:
r Q " sinh 2d , 2g Q
The ADM energy E, entropy S, and the Hawking temperature ¹ of (657) are & R
(660)
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From the asymptotic values of G and G (i"5,2, 8), one obtains the following expressions GG for pressures in the x- and xG-directions:
1 R
(661)
So, in the xG-directions, which are parallel to 5-brane but perpendicular to 1-brane, shrinking e!ect of 5-brane and expanding e!ect of 1-brane compete and become balanced when d "d . In the x-direction, which are parallel to both 5- and 1-branes, the shrinking e!ects of 5- and 1-branes are compensated by momentum in the x-direction. The 6 parameters r , d , d , d , < and R of the solution (657) can be traded with the numbers N of 1-branes, anti-1-branes, 5-branes, anti-5-branes, right-moving momentum, and left-moving momentum, respectively given by [362]
(662)
rR< N " e\BN . * 4g Q
These parameters are related to the charges in (659) as Q "N !N , Q "N !N and N"N !N . In terms of the new parameters, the ADM energy and the Bekenstein}Hawking 0 * entropy take simple and suggestive forms [362]: R< 1 R E" (N #N )# (N #N )# (N #N ) , * g R 0 g Q Q
(663)
S"2p((N #(N )((N #(N )((N #(N ) , * 0 where
<"
N N , N N
R"
gN N Q 0 * . N N
(664)
8.2.2. Four-dimensional black hole There are various ways in which one can construct D"4 black holes having regular BPS-limit with non-zero horizon area. The criteria for such construction are that supersymmetries are preserved and all the contributions of D-brane tensions from constituents compensate for each other so that internal space is stable against the shrinking e!ect near the horizon.
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The "rst type of con"guration is obtained by "rst T-dualizing the type-IIB D-brane bound state discussed in Section 8.2.1 along x, resulting in bound state of Q D 2-branes (along x, x) and Q D 6-branes (along x,2, x) with the momentum N #owing along the x-direction. This con"guration has a zero horizon area in the BPS limit, since the radii along the directions x, x shrink as one approaches the horizon, due to the tensions of D 2- and D 6-branes and the momentum along the x direction. This is compensated by adding solitonic 5-brane along x,2, x and with magnetic charge of the NS-NS 2-form "eld B , where k"0,2, 3. Since the spinor constraint of I the additional solitonic 5-brane does not further reduce the Killing spinor degrees of freedom, the solution still preserves 1/8 of supersymmetry. D"4 black hole is obtained by compactifying this p-brane bound state on ¹"¹;SY;S [361,471]. Here, ¹ with the coordinates (x,2, x) has the volume < and SY [S] with the coordinate x [x] has the radius R [R ]. Namely, the Q D 6-branes wrap around ¹, Q D 2-branes wrap around SY;S, solitonic 5-branes wrap around ¹;S and the momentum #ows along S. The resulting D"4 solution has the form [178,180,181,361]:
r ds"!f \ (r) 1! dt#f (r) r
r sinh d f (r)" 1# r
r sinh d 1# r
r \ dr#r(dh#sin h d ) , 1! r
r sinh d 1# r
(665)
r sinh d N . 1# r
The electric/magnetic charges carried by the black hole are r < Q " sinh 2d , Q "r R sinh 2d , g Q r r
(666)
The pressures along the directions x, xG (i"5,2, 8) and x are r
(667)
r
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One can trade the 8 parameters r , d , d , d , d , <, R , and R with the numbers of right- (left-) N moving momentum modes, (anti-) 2-branes, (anti-) 5-branes, and (anti-) 6-branes: r
(668)
r r N " eB, N " e\B . 2g 2g Q Q In terms of these parameters, the ADM mass and entropy take forms [361]: 1 R R
(669)
N N N N gN N , R " , R R" Q 0 * . (670) N N N N gN N Q We discuss couple of other ways [392] to construct D"4 black holes. The "rst con"guration is a type-IIB con"guration where Q parallel D 1-branes (along x) and Q parallel D 5-branes (along x,2, x) intersect in the x-direction, along which momentum P"N/¸ #ows, and magnetic monopole in the subspace (t, r, h, , x). The D"4 black hole is obtained by "rst wrapping Q D 5-branes around 4-cycles of K3 surface (with the coordinates x,2, x), resulting in a D-string bound state along with Q D 1-branes in D"6. This bound state in D"6 has momentum P along x and the KK magnetic charge associated with the metric component g . One further compac( ti"es the coordinates (x, x) on ¹ to obtain D"4 black hole. Entropy of this solution is <"
A "2p(Q Q N . S" 4G , The second con"guration is a type-IIA solution obtained by T-dualizing the above con"guration along x. Namely, this is a bound state of Q D 0-branes and Q D 4-branes (along x,2, x) with open strings wound around the x-direction (with the winding number =) and the KK monopole in the subspace (t, r, h, , x). Similarly as in the "rst case, to have a D"4 black hole, one "rst wrappes Q D 4-branes around 4-cycles of K3 surface (with the coordinates x,2, x), resulting in a D particle bound state in D"6, and then compacti"es the coordinates (x, x) on ¹. Such a solution has entropy A "2p(Q Q = . S" 4G ,
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8.3. D-brane counting argument In this section, we discuss D-brane interpretation of black hole entropy. The number of degenerate BPS states is calculated in the weak string coupling regime (g +0), in which the Q D-brane picture, rather than black p-brane description, of charge con"gurations is applicable. Since mass of R-R charge carrier behaves as &1/g , D-branes become in"nitely massive in the perturQ bative region. So, the leading contribution to the degeneracy of states is from perturbative states of open strings, which are attached to D-branes. So, perturbative D-brane con"gurations are described by conformal "eld theory of open strings in the target space manifold determined by D-branes and the internal space. In the following, we discuss conformal "eld theories of D-branes which correspond to speci"c D"4, 5 black holes and calculate the number of degenerate perturbative open string states. More intuitive picture of D-brane argument is discussed in the subsequent subsection. 8.3.1. Conformal model for D-brane conxgurations We saw that the worldvolume theory of massless modes in a bound state of N D p-branes is described by D"10 ;(N) super-Yang-Mills theory compacti"ed to D"p#1 [637]. Namely, dynamics of collective coordinates [107] of N parallel D p-branes is described by a supersymmetric ;(N) gauge theory. The BPS states of D-brane con"guration correspond to the supersymmetric vacuum of the corresponding super-Yang-Mills theory. The ;(N) group is decomposed into a ;(1) group, describing the center of mass motion of D-branes, and an S;(N) group [637]. It is the supersymmetric S;(N) vacuum that contains states with a mass gap, which are relevant for degeneracy of D-brane bound states. When there is a mass gap, the number of ground states in the S;(N) super-Yang-Mills theory is the same as the degeneracy of the D-brane bound states. Here, the ground states are identi"ed with the cohomology elements of the S;(N) instanton moduli space M,, implying that degeneracy of D-brane bound states is given by the number of cohomology elements of M, [619,620]. D-brane bound states are e!ectively described by the p-model on the instanton moduli space M,. Generally, the p-model with target space manifold given by hyperKaK hler manifold of dimension 4k has the central charge c"6k. For example, the moduli space M, of S;(N) instantons on K3 with the instanton I number k has the dimension 4[N(k!N)#1]. It has been checked [90,91,566,567,619,620] that calculation of the degeneracy of D-brane bound states based on this idea yields the results which are consistent with conjectured string dualities that relate perturbative string states to D-brane bound states. In particular, P of BPS perturbative string states at the level N "1#P is * dualized to the intersection number of D-brane bound states [619]. One of setbacks in study of D-brane bound states is that a bound state of m D p-branes is marginally stable, i.e. there is no energy barrier against decay into m D p-branes (each carrying the unit R-R (p#1)-form charge). Such a problem is circumvented [566] by compactifying an extra one direction, say y, on S so that states in the multiplet carry momentum or winding number n along S. If the pair (m, n) is relatively prime, the corresponding states in the D"9!p theory, i.e.
In particular, the singular points in the moduli space [21,26,635], at which gauge symmetry is enhanced, correspond to the case where D-branes coincide.
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D"10 string bound states compacti"ed on ¹N;S, become absolutely stable against decay. Such worldvolume theory is described by the N"1 ;(m) gauge theory compacti"ed to D"p#1 with states characterized by n units of ;(1)L;(m) electric #ux along y. The non-trivial information on degeneracy of the BPS D-brane bound states is contained in the supersymmetric ground states of the S;(m) part of theory on the base manifold ¹N;R, where R is labeled by the time coordinate. It is shown [619] that for the type-IIA bound state of m 0-branes and 1 4-brane (or the bound state of m 1-branes and 1 5-brane in the T-dual theory) the instanton moduli space M of the corresponding ;(m);;(1) super-Yang-Mills theory is (¹)K/S , where S is the permutation A K K group on m objects. The degeneracy of the cohomology in this space is in one-to-one correspondence with the partition function of left-movers of the superstring. To put it another way, the ground states of this gauge theory are related to the degenerate string states at level m. The quantum numbers (F #F , F !F ) of the cohomology of (¹)K/S , where F [F ] is the [anti-] holomor0 * 0 K * 0 * phic degrees of the homology shifted by half the complex dimension, is mapped to the light-cone helicities of the left oscillator states of type-II superstring [623]. The generalization to D-brane bound states wrapped around K3 surface is carried out in [91,620]. The bound state of N D 4-branes wrapped around K3 is described by a ;(N) gauge theory on K3;R with R parameterized by the time coordinate. For this D-brane con"guration, it is shown [91] that the quantum corrections induce the e!ective D 0-brane charge M"k!N. So, the momentum square P"P!P is P/2"NM"N(k!N). If N and k are relatively prime, 0 * there is a mass gap and the moduli space M, is smooth with a discrete spectrum, with cohomology I of M, being that of N(k!N)#1 unordered points on K3. When N"2 and k is odd, the moduli I space is the symmetric product of K3's: M"(K3)I\/S . The Euler characteristic d(2k!3) of I I\ M is the same as the degeneracy d(N )"d(P#1) of string states at level N , or the degeneracy * I * of D-brane bound states with charges (N,M). Here, d(n) is de"ned in terms of the Dedekind eta function g(q) as g(q)\"q\ d(n)qL. L 8.3.1.1. Applications to statistical entropy of black holes. We consider a type-IIB D"5 black hole carrying electric charges Q and Q of ;(1) "elds, respectively, originated from R-R 2-form $ & potential AK 00 and NS-NS 2-form potential BK ,1 [95,98,580]. Here, the D"5 ;(1) "eld strength ++ ++ which is associated with R-R 2-form potential in D"10 is Hodge-dual to the "eld strength of R-R in D"5. From the D"10 perspective, Q (Q ) is magnetic (electric) charge of 2-form "eld AK 00 $ & I I AK 00 (BK ,1 ), which couples to the worldvolume of R-R 5-brane [NS-NS string]. The D"5 ;(1) ++ ++ "eld originated from AK 00 , therefore, carries electric charge. The internal index m in the D"5 ++ NS-NS ;(1) "eld BK ,1 is along S, upon which an extra coordinate of the D"6 type-IIB theory is IK compacti"ed. So, this black hole is regarded as a bound state of 5-brane with R-R charge Q and $ string with N-N charge Q with the 5-brane wrapped around a holomorphic 4-cycle of K3 and & partially around S, and the string wound round S. T-duality along S leads to a type-IIA solution corresponding to 4-brane with 5-form charge Q and momentum #owing along S. $ The non-rotating BPS black hole with the above charge con"guration has spacetime of the Reissner}Nordstrom black hole [182,580,609]:
r r \ dt# 1! dr#r dX , ds"! 1! r r
(671)
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where r "(8Q Q/p) . The scalars are constant everywhere and expressed in terms of the & $ (integer-valued) electric charges Q [260]. The Bekenstein}Hawking entropy is & $ A Q Q & $. S " "2p (672) & 4 2
For D"5 rotating solutions, the regular BPS limit is possible only for the case where 2 angular momenta have the same magnitude, i.e. "J """J ""J. The solution is [98,182,610] k ku sin h ku cos h dt! d # d
ds"! 1! r (r!k) (r!k)
k \ dr#r(dh#sin h d #cos h d ) , # 1! r
(673)
with Q "k/j, Q "!(p/2(2)kj and J "!J "(p/4)ku. The Bekenstein}Hawking entropy & $ is
Q Q A & $!J . S " "2p & 4 2
(674)
We discuss the D-brane interpretation of the entropies (672) and (674). In terms of D-brane language, the above black holes are the bound state of Q D 5-branes wrapped around K3;S and $ a fundamental string wound around S with the winding mode Q . The product Q ) Q is the & $ $ intersection number of D 5-branes in the K3 homology. We consider the case where K3 is small compared to the size of S. For this case, the solution looks like a fundamental string in the 5-brane background. The theory is e!ectively described by the conformal "eld theory on S;R with the target space manifold given by the symmetric product of K3 surfaces [580]: (K3)B /$>
. (675) M" S $ / >
This conformal "eld theory has central charge c"6(Q#1), since the real dimension of this $ manifold is 2(Q#2). $ First, we consider the non-rotating black hole (671) with the thermal entropy (672) [580]. The statistical entropy of the BPS solutions is given in terms of degeneracy d(n, c) of conformal states with the left-moving oscillators at level ¸ "n (n<1) and the right-moving oscillators in ground state:
S "ln d(n, c)&2p
nc . 6
(676)
For the solution (671), n"Q and, therefore, the statistical entropy (676) reproduces the thermal & entropy (672): (677) S &2p(Q (Q#1) . & $ Next, we consider the thermal entropy (674) of the rotating black hole (673) [98]. The rotation group SO(4) of the D"6 lightcone-frame theory is identi"ed with the S;(2) ;S;(2) symmetry of * 0
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the N"4 superconformal algebra. The charges (F , F ) of ;(1) ;;(1) LS;(2) ;S;(2) are * 0 * 0 * 0 interpreted as spins of string states [98,623], and are related to angular momenta of the black hole (673) as J "(F #F ), J "(F !F ) . (678) * 0 * 0 Note, the ;(1) current J can be bosonized as J "((c/3)R and a conformal state U * carrying * * * $ ;(1) charge F is obtained by applying an operator exp(iF /(c/3) to the state U without ;(1) * * * * charge. The net e!ect is to shift the left-moving oscillator level n of string states carrying the ;(1) * charge F with respect to the level n for string states without the ;(1) charge F : n "n!3F/2c. * * * * For a large value of n , one can safely use the string level density formula d , which includes L A contributions of string states with all the possible spin values, in calculating the statistical entropy. Here, c"6(Q#1) and n"Q , as in the non-rotating case. Substituting all of these values into & $ (676), one obtains the following statistical entropy in agreement with the thermal entropy (674): S &2p
1 n c 1 &2p Q Q#1 ! ("J "#"J ") , & 2 $ 4 6
(679)
where J "!J "J. 8.3.2. Another description We discuss more intuitive D-brane picture of black holes based upon the arguments in [109]. In this picture, one counts all the possible oscillator contributions of open strings attached to D-branes. D-brane con"gurations, which are the weak string coupling limit of black holes, are described by the p-model of open strings with the target space background determined by D-branes, upon which the ends of the open strings live. The statistical entropy of black holes is due to degenerate states in "xed oscillator levels N * 0 (or a "xed mass) of open strings attached to D-branes:
c ((N #(N ) , S "ln d(N , N ; c)&2p * 0 * 0 6
(680)
in the limit N <1. The central charge c has an additive contribution of 1 () for each bosonic * 0 (fermionic) coordinate. N are determined by the NS-NS electric charges (i.e. momentum and winding modes in the * 0 compacti"ed space) from the mass formula of open string states, which is derived from the Virasoro condition ¸ !a"0. Here, a is the zero point energy (or the normal ordering constant of oscillator modes). The contribution to the zero point energy a is additive with the contribution of ! ( ) for each bosonic coordinate with the periodic (anti-periodic) boundary condition, i.e. the integer (half-integer) moding of oscillator, and for a fermionic coordinate there is an extra minus sign.
For detailed discussions on this point, see Section 7.3.2.
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Note, the Virasoro operator ¸ , say of the bosonic coordinates, is de"ned in terms of the oscillator modes as ¸ "aI a # aI a , where k is a D"10 vector index. Here, aI "pI is L \L LI I the center-of-mass frame momentum of an open string and N,(a /4) aI a is the oscillator L \L LI number operator. For string theory compacti"ed to D"d (d(10), we divide D"10 momentum pI into the D"d part pI and the internal part pK, i.e. (pI)"(pI , pK). Since mass in D"d is de"ned as M"!p pI , one obtains the following mass formula from the Virasoro condition ¸ !a"0: I 4 4 (681) M"(p )# (N #a )"(p )# (N #a ) , * 0 0 * a 0 a * where the subscripts L and R denote left- and right-moving sectors, and p are momenta in the * 0 internal directions. Note, this expression for mass has all the contributions from bosonic and fermionic coordinates. In particular, for the compacti"cation on S of radius R, the left- and right-moving momenta in the S-direction is p "n/RGmR/a , where n and m are respectively * 0 momentum and winding modes around S. Mass of the BPS states is M "(p ). So, for the BPS states, the right-movers are in ground .1 0 state and only left-movers are excited with their total oscillation numbers determined (from the mass formula (681)) by the left- and right-moving momenta in the internal space: a N "!a , N " [(p )!(p )]!a . * * 0 0 * 4 0 ), the right movers, as well as the left movers, are excited: .1 a N "!a #k, N " [(p )!(p )]!a #k (k3Z) . 0 0 * 4 0 * *
(682)
For non-BPS states (M'M
(683)
Since N are "xed by NS-NS electric charges, the main problem of calculating the statistical * 0 entropy in D-brane picture is to determine the value of c, i.e. the total degrees of freedom of bosonic and fermionic coordinates of open strings. There are two types of open strings. The "rst type is open strings that stretch between the same type of D-brane, called the (p, p) type. The second type is open strings that stretch between di!erent types of D-branes, called the (p, p ) type, pOp . For the second type, (p, p ) and (p , p) are not equivalent, since open strings are oriented. Open strings of the (p, p) type can be ignored in the calculation of the open string state degeneracy, since the excited states of open strings which stretch between D-branes of the same type become very heavy as the relative separation between a pair of D-branes gets large. We brie#y discuss some aspects of open string states of the (p, p ) type. The D"10 spacetime coordinates XI for this type of open strings are divided into 3 classes. The "rst (second) type, called the DD (NN) type, corresponds to coordinates for which both ends of strings satisfy the Dirichlet (Neumann) boundary conditions (i.e. coordinates which are transverse (longitudinal) to both branes). For these cases, the bosonic string coordinates are integer modded and, thereby, have the
The fermionic coordinates have the similar expressions. In the presence of both bosonic and fermionic coordinates, the Virasoro operator is sum of the 2 contributions, i.e. ¸ "¸#¸ .
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zero point energy contribution of ! for each bosonic coordinate. The third type, called DN or ND type, corresponds to coordinates for which each end of open strings satis"es di!erent boundary conditions (i.e. coordinates which are transverse to one brane and longitudinal to the other brane). For this type, the bosonic string coordinates are half-integer modded and, therefore, have the zero point energy contribution of for each bosonic coordinate. The fermionic coordi nates have the same moding (as the bosonic coordinates) in the R sector and the opposite in the NS sector. So, the total zero point energy is always 0 for the R sector. For the NS sector, the zero point energy depends on the total number l of the ND and DN coordinates, which is always even:
1 1 1 1 1 l #l # "! # . (8!l) ! ! 24 48 24 48 2 8
(684)
Only when l is a multiple of 4, D-brane con"gurations are supersymmetric and degeneracy between the NS and R sectors is possible. When l"4, N"2 supersymmetry is preserved in D"4 and the zero point energy is zero in both the R and NS sectors. An example is the intersecting D string and D 5-brane, corresponding to the D"5 black hole that we consider in the following. For this type of open strings, degrees of freedom of the fermionic and bosonic coordinates are determined as follows. In the NS sector, among 4 fermionic coordinates s in the ND directions only 2 of them survive the GSO projection CGCGCGCGs"s, where i ,2, i correspond to the ND directions. In the R sector, also only 2 of 4 periodic transverse fermions (in the NN and DD directions) survive the GSO projection. Furthermore, there are Q Q ways to attach open strings between D p- and D p -branes. Since open N NY strings are oriented, we have the same numbers of fermionic and bosonic degrees of freedom satisfying the above constraints in the (p, p ) and the (p , p) sectors. So, there are 4Q Q bosonic and N NY 4Q Q fermionic degrees freedom for each (p, p ) and (p , p) types. Thus, the central charge of open N NY strings of (p, p ) and (p , p) types is (685) c"4Q Q (1#)"6Q Q . N NY N NY We comment on di!erent interpretation of the entropy expression (680). Let us consider a BPS con"guration (N "0 case) with the total momentum number N along S of radius R. When the 0 number of the bosonic (fermionic) degrees of freedom are N (N ), the central charge is $ c"N #N . Then, the statistical entropy (680) becomes the following entropy formula describ $ ing the N bosonic and N fermionic species with energy E"N/R in a 1-dimensional space of $ length ¸"2pR [200,471]: S"(p(2N #N )E¸/6 . (686) $ Note, the entropy formula (680) is derived assuming that N <1, i.e. the NS electric charges are * 0 much bigger than the number Q of D p-branes. When Q and the NS electric charges are of the N N same order in magnitude, the string level density formula fails to reproduce the Bekenstein} Hawking entropy [472]. This stems from the fact that (with Q of the order of N) the notion of N extensivity (of entropy, energy, etc.) fails when radius R of S, around which D-branes are wrapped,
These two types are di!erent since open strings are oriented.
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is smaller than the size of black holes ( fat black hole limit), since the wavelength (&1/¹, where ¹ is the e!ective temperature of the left movers) of a typical quantum exceeds the size ¸"2pR of the system [472]. The remedy to this problem proposed in [472] is as follows. One considers a bound state of Q D 5-branes and Q D 1-branes which, respectively, wrap around M ;S and S. Here, M is a 4-dimensional compact manifold and S has radius R. One neglects the directions associated with M and therefore the D 5-branes are regarded as strings wrapped around S. Instead of taking Q N D p-branes as Q numbers of strings wrapped around S once, one regards it as a single string N wrapped Q times around S, thereby having length 2pQ R [467]. So, a bound state of D 1- and N N D 5-branes is described by a single string of the length 2pQ Q R wrapped Q Q times around S. This is interpreted as a single species with the energy E"N/R"N /R in a 1-dimensional space of length ¸ "2pR (thereby simulating a spectrum of fractional charges), where N ,Q Q N and R ,Q Q R. Note, both the oscillator level and the radius of S are multiplied by Q Q . In this description of D-brane bound states, the size ¸ of systems is always bigger than the wavelength of a typical quantum, thereby the extensivity condition is always satis"ed. Furthermore, since the (e!ective) oscillator level N "Q Q N is always much bigger than the number of D-branes (which is 1) even when Q +N+Q , one can still use the statistical entropy expression of the type (680). The statistical entropy is then S"(2pE¸ "2p(N , where ¸ "Q Q ¸"(N #N )¸ and $ N "Q Q N. Note, this is the same as (686) when expressed in terms of the original variables, but with this new description the approximation (of string level density) leading to the statistical entropy is valid. Near-extreme black holes correspond to D-brane con"gurations with a small amount of right moving oscillations excited, i.e. N given by (683) with a small integer n. As long as the string * 0 coupling is very small and the density of strings (J1/R) is low, the interaction between the left- and the right-movers is negligible [368]. In this limit, entropy contributions from the left- and the right-movers are additive, leading to [362,368] S "(p(2N #N )E ¸/6#(p(2N #N )E ¸/6 , $ * $ 0
(687)
where ¸"2pR and E "N /R with N very small. This expression correctly reproduces * 0 * 0 0 the Bekenstein}Hawking entropy of near extreme black holes. We comment that the above Dbrane interpretation of near extreme black holes is valid when only one of the constituents of the D-brane bound state has energy contribution much smaller than the others. For example, the above calculation is done in the limit where R is larger than the size < of the other internal manifold and, thereby, black holes are e!ectively described by (oscillating) strings in spacetime 1 dimension higher. In this limit, the momentum modes (of open strings) are much lighter than the branes (cf. (663)); the leading order contribution to the black hole entropy is from open string modes. For other limits where one of branes has much smaller energy than the other constituents, one can apply the similar argument in calculating statistical entropy of near extreme black holes [362] and the result is consistent with the conjectured U-duality. (In these cases, the leading contribution to the degeneracy is from D-branes rather than from open strings.) For the case where more than one constituents are light or all the constituents have energies of the same order of magnitude, the statistical interpretation of near extreme black holes is not known yet.
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Generalization to the rotating black hole case [95,98,361] is along the same line described in Section 7.3.2. Note, only the fermionic coordinates of the (p, p ) type can carry angular momentum under the spatial rotational group. These are 4 periodic transverse fermions (from the NN and DD directions) in the R sector, which are spinors under the Lorentz group of the external spacetime. When the GSO projection is imposed, only the positive chirality representation of the external Lorentz group survives. We concentrate on the (1, 5) type, which is related to the D"5 (D"4) black holes under consideration in this section (through T-duality). For this case, the D"10 Lorentz group SO(1, 9) is decomposed as: SO(1, 9)MSO(1, 1) SO(4) SO(4) , (688) # ' where the "rst (third) factor acts on the D 1-brane worldsheet (the space internal to D 5-brane) and the second factor on the space external to the con"guration. Worldsheet spinors are decomposed into the 2-dimensional positive chirality representation 2> (with R-type quantization in ND directions of the NS sector) under the internal rotational group SO(4) (thereby a boson under the ' spacetime SO(1, 5) Lorentz group) and the 2-dimensional representation 2> (with NS-type quantiz> ation in ND directions of the R sector) which is positive-chiral under the external group SO(1, 1);SO(4) LSO(1, 5) (thereby a spacetime fermion). # Here, the spatial rotation group SO(4) (external to the D-brane con"guration) is isomorphic to # S;(2) ;S;(2) , which is identi"ed with the symmetry of (4, 4) superconformal theories. This is 0 * related to the fact that worldsheet fermions, which carry angular momenta, manifest themselves as spacetime fermions. The ;(1) LS;(2) charges F are related to angular momenta J of the * 0 * 0 0 * spatial rotational group SO(4) as in (678). # The e!ect of angular momenta is to reduce the total oscillation numbers [98]: 3F 3F N "N ! *, N "N ! 0 , 0 0 * * 2c 2c
(689)
where N (N ) are the total oscillation numbers of states that do not carry the ;(1) charges * 0 * 0 * 0 (have the ;(1) charges F ). To obtain the statistical entropy of black holes, one plug this * 0 * 0 expression for N into (680). * 0 For non-extreme or near-extreme black holes, which do not saturate the BPS bound, the right moving as well as the left moving supersymmetries are preserved, and thereby F are both 0 * non-zero. The statistical entropy of near-extreme black holes is:
c S &2p ((N #(N ) , * 0 6
(690)
where N are given in (689). * 0 However, for D"5 BPS black holes, F "0 (leading to J "J ":J) since only the left-moving 0 supersymmetry survives (i.e. (0, 4) superconformal theory). So, for BPS black holes, N "N "0 0 0 and N "N !(6/c)J, leading to the statistical entropy [361]: * *
c c S &2p (N "2p N !J . * 6 6 *
(691)
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For the D"5 rotating black hole with charge con"guration described in the above, c"6Q Q and N "N. So, the statistical entropy (691) correctly reproduces the Bekenstein}Hawking * entropy: S "2p(NQ Q !J. To obtain D"4 rotating black hole with regular extreme limit, one "rst applies T-duality along the common string direction of the intersecting D 1- and D 5-branes with open strings wound around the common string direction. The resulting con"guration is bound state of D 2- and D 6-branes with momentum #owing along one of the common intersection directions. To have non-zero horizon area in the BPS limit, one adds solitonic 5-brane with charge Q . Here, D 6-branes, D 2-branes and solitonic 5-brane, respectively, wrap around ¹"¹;S ;S, S ;S and ¹;S, and momentum #ows along S. Since the right moving supersymmetry breaks in the presence of solitonic 5-branes, the ;(1) charge F in the right moving sector vanishes (i.e. 0 J "J "J). Particularly, D"4 extreme rotating black hole corresponds to the minimum energy con"guration which is regular with non-zero angular momentum, which happens when N "0 * [361]. So, the left movers are constrained to carry angular momentum, only. In this case, N "N 0 0 and N "6J/c. By plugging these into (683), one obtains * a 6 N " J! [p!p] , * 0 c 4 0 leading to the statistical entropy:
c a c S &2p (N "2p J! [p!p] . 0 * 6 24 0
(692)
For the D-brane bound state corresponding to the D"4 black hole described above, c"6Q Q Q (since there are 4Q Q Q species of bosons and fermions) and (a /4)[p!p]"N is the total * 0 momentum mode of open strings around S. So, the statistical entropy S "2p(J#NQ Q Q agrees with the Bekenstein}Hawking entropy. For further reading, see Refs. [15,16,28,31,53,78,153,172,176,256,402,417,433,502,508,616].
Acknowledgements I would like to thank M. Cvetic\ for suggesting me to write this review paper, for discussion during the initial stage of the work and for proof-reading part of the review. Part of the work was done while the author was at University of Pennsylvania. The work is supported by DOE grant DE-FG02-90ER40542.
The extra factor of Q comes from the fact that whenever D 2-branes (which intersect the solitonic 5-brane along S) cross the solitonic 5-brane they can break up and ends separate in di!erent positions in ¹. Thus, Q D 2-branes break up into Q Q D 2-branes.
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J. Main/Physics Reports 316 (1999) 233}338
USE OF HARMONIC INVERSION TECHNIQUES IN SEMICLASSICAL QUANTIZATION AND ANALYSIS OF QUANTUM SPECTRA
JoK rg MAIN Institut fu( r Theoretische Physik I, Ruhr-Universita( t Bochum, D-44780 Bochum, Germany
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
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Use of harmonic inversion techniques in semiclassical quantization and analysis of quantum spectra JoK rg Main Institut fu( r Theoretische Physik I, Ruhr-Universita( t Bochum, D-44780 Bochum, Germany Received June 1998; editor: J. Eichler
Contents 1. Introduction 1.1. Motivation of semiclassical concepts 1.2. Objective of this work 1.3. Outline 2. High precision analysis of quantum spectra 2.1. Fourier transform recurrence spectra 2.2. Circumventing the uncertainty principle 2.3. Precision check of the periodic orbit theory 2.4. Ghost orbits and uniform semiclassical approximations 2.5. Symmetry breaking 2.6. expansion of the periodic orbit sum 3. Periodic orbit quantization by harmonic inversion 3.1. A mathematical model: Riemann's zeta function
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3.2. Periodic orbit quantization 3.3. The three-disk scattering system 3.4. Systems with mixed regular-chaotic dynamics 3.5. Harmonic inversion of cross-correlated periodic orbit sums 3.6. expansion for the periodic orbit quantization by harmonic inversion 3.7. The circle billiard 3.8. Semiclassical calculation of transition matrix elements for atoms in external "elds 4. Conclusion Acknowledgements Appendices References
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Abstract Harmonic inversion is introduced as a powerful tool for both the analysis of quantum spectra and semiclassical periodic orbit quantization. The method allows one to circumvent the uncertainty principle of the conventional Fourier transform and to extract dynamical information from quantum spectra which has been unattainable before, such as bifurcations of orbits, the uncovering of hidden ghost orbits in complex phase space, and the direct observation of symmetry breaking e!ects. The method also solves the fundamental convergence problems in semiclassical periodic orbit theories } for both the Berry}Tabor formula and Gutzwiller's trace formula } and can therefore be applied as a novel technique for periodic orbit quantization, i.e., to calculate semiclassical eigenenergies from a "nite set of classical periodic orbits. The advantage of periodic orbit quantization by harmonic inversion is the universality and wide applicability of the method, which will be demonstrated in this work for various open and bound systems with underlying regular, chaotic, and even mixed classical dynamics. The e$ciency of the method is increased, i.e., the number of orbits required for periodic orbit quantization is reduced, when the harmonic inversion technique is generalized to the analysis of 0370-1573/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 8 ) 0 0 1 3 1 - 8
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cross-correlated periodic orbit sums. The method provides not only the eigenenergies and resonances of systems but also allows the semiclassical calculation of diagonal matrix elements and, e.g., for atoms in external "elds, individual non-diagonal transition strengths. Furthermore, it is possible to include higher-order terms of the expanded periodic orbit sum to obtain semiclassical spectra beyond the Gutzwiller and Berry}Tabor approximation. 1999 Elsevier Science B.V. All rights reserved. PACS: 05.45.#b; 03.65.Sq Keywords: Spectral analysis; Periodic orbit quantization
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1. Introduction 1.1. Motivation of semiclassical concepts Since the development of quantum mechanics in the early decades of this century quantum mechanical methods and computational techniques have become a powerful tool for accurate numerical calculations in atomic and molecular systems. The excellent agreement between experimental measurements and quantum calculations has silenced any serious critics on the fundamental concepts of quantum mechanics, and there are nowadays no doubts that quantum mechanics is the correct theory for microscopic systems. Nevertheless, there has been an increasing interest in semiclassical theories during recent years. The reasons for the resurgence of semiclassics are the following. Firstly, the quantum mechanical methods for solving multidimensional, nonintegrable systems generically imply intense numerical calculations, e.g., the diagonalization of a large, but truncated Hamiltonian in a suitably chosen basis. Such calculations provide little insight into the underlying dynamics of the system. By contrast, semiclassical methods open the way to a deeper understanding of the system, and therefore can serve for the interpretation of experimental or numerically calculated quantum mechanical data in physical terms. Secondly, the relation between the quantum mechanics of microscopic systems and the classical mechanics of the macroscopic world is of fundamental interest and importance for a deeper understanding of nature. This relation was evident in the early days of quantum mechanics, when semiclassical techniques provided the only quantization rules, i.e., the WKB quantization of one-dimensional systems or the generalization to the Einstein}Brillouin}Keller (EBK) quantization [1}3] for systems with n degrees of freedom. However, the EBK torus quantization is limited to integrable or at least near-integrable systems. In non-integrable systems the KAM tori are destroyed [4,5], a complete set of classical constants of motion does not exist any more, and therefore the eigenstates of the quantized system cannot be characterized by a complete set of quantum numbers. The `breakdowna of the semiclassical quantization rules for non-regular, i.e., chaotic systems was already recognized by Einstein in 1917 [1]. The failure of the `olda quantum mechanics to describe more complicated systems such as the helium atom [6], and, at the same time, the development and success of the `moderna wave mechanics are the reasons for little interest in semiclassical theories for several decades. The connection between wave mechanics and classical dynamics especially for chaotic systems remained an open question during that period. 1.1.1. Basic semiclassical theories 1.1.1.1. Chaotic systems: Gutzwiller1s trace formula. The problem was reconsidered by Gutzwiller around 1970 [7,8]. Although a semiclassical quantization of individual eigenstates is, in principle, impossible for chaotic systems, Gutzwiller derived a semiclassical formula for the density of states as a whole. Starting from the exact quantum expression given as the trace of the Green's operator he replaced the exact Green's function with its semiclassical approximation. Applying stationary phase approximations he could "nally write the semiclassical density of states as the sum of a smooth part, i.e., the Weyl term, and an oscillating sum over all periodic orbits of the corresponding classical system. For this reason, Gutzwiller's theory is also commonly known as
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periodic orbit theory. Gutzwiller's semiclassical trace formula is valid for isolated and unstable periodic orbits, i.e., for fully chaotic systems with a complete hyperbolic dynamics. Examples of hyperbolic systems are n-disk repellers, as models for hyperbolic scattering [9}15], the stadium billiard [16}18], the anisotropic Kepler problem [8], and the hydrogen atom in magnetic "elds at very high energies far above the ionization threshold [19}21]. Gutzwiller's trace formula is exact only in exceptional cases, e.g., for the geodesic #ow on a surface with constant negative curvature [8]. In general, the semiclassical periodic orbit sum is just the leading order term of an in"nite series in powers of the Planck constant, . Methods to derive the higher-order contributions of the
expansion are presented in [22}24]. 1.1.1.2. Integrable systems: the Berry}¹abor formula. For integrable systems a semiclassical trace formula was derived by Berry and Tabor [25,26]. The Berry}Tabor formula describes the density of states in terms of the periodic orbits of the system and is therefore the analogue of Gutzwiller's trace formula for integrable systems. The equation is known to be formally equivalent to the EBK torus quantization. A generalization of the Berry}Tabor formula to near-integrable systems is given in [27,28]. 1.1.1.3. Mixed systems: uniform semiclassical approximations. Physical systems are usually neither integrable nor exhibit complete hyperbolic dynamics. Generic systems are of mixed type, characterized by the coexistence of regular torus structures and stochastic regions in the classical PoincareH surface of section. A typical example is the hydrogen atom in a magnetic "eld [29}31], which undergoes a transition from near-integrable at low energies to complete hyperbolic dynamics at high excitation. In mixed systems the classical trajectories, i.e., the periodic orbits, undergo bifurcations. At bifurcations periodic orbits change their structure and stability properties, new orbits are born, or orbits vanish. A systematic classi"cation of the various types of bifurcations is possible by application of normal form theory [32}35], where the phase space structure around the bifurcation point is analyzed with the help of classical perturbation theory in local coordinates de"ned parallel and perpendicular to the periodic orbit. At bifurcation points periodic orbits are neither isolated nor belong to a regular torus in phase space. As a consequence the periodic orbit amplitudes diverge in both semiclassical expressions, Gutzwiller's trace formula and the Berry}Tabor formula, i.e., both formulae are not valid near bifurcations. The correct semiclassical solutions must simultaneously account for all periodic orbits which participate at the bifurcation, including `ghosta orbits [36,37] in the complex generalization of phase space, which can be important near bifurcations. Such solutions are called uniform semiclassical approximations and can be constructed by application of catastrophe theory [38,39] in terms of catastrophe di!raction integrals. Uniform semiclassical approximations have been derived in [40,41] for the simplest and generic types of bifurcations and in [37,42}44] for nongeneric bifurcations with higher codimension and corank. With Gutzwiller's trace formula for isolated periodic orbits, the Berry}Tabor formula for regular tori, and the uniform semiclassical approximations for orbits near bifurcations we have, in principle, the basic equations for the semiclassical investigation of all systems with regular, chaotic, and mixed regular-chaotic dynamics. There are, however, fundamental problems in practical applications of these equations for the calculation of semiclassical spectra, and the development of techniques to overcome these problems has been the objective of intense research during recent years.
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1.1.2. Convergence problems of the semiclassical trace formulae The most serious problem of Gutzwiller's trace formula is that for chaotic systems the periodic orbit sum does not converge in the physically interesting energy region, i.e., on and below the real axis, where the eigenstates of bound systems and the resonances of open systems are located, respectively. For chaotic systems the sum of the absolute values of the periodic orbit terms diverges exponentially because of the exponential proliferation of the number of orbits with increasing periods. The convergence problems are similar for the quantization of regular systems with the Berry}Tabor formula, however, the divergence is algebraic instead of exponential in this case. Because of the technical problems encountered in trying to extract eigenvalues directly from the periodic orbit sum, Gutzwiller's trace formula has been used in many applications mainly for the interpretation of experimental or theoretical spectra of chaotic systems. For example, the periodic orbit theory served as basis for the development of scaled-energy spectroscopy, a method where the classical dynamics of a system is "xed within long-range quantum spectra [45,46]. The Fourier transforms of the scaled spectra exhibit peaks at positions given by the classical action of the periodic orbits. The action spectra can therefore be interpreted directly in terms of the periodic orbits of the system, and the technique of scaled-energy spectroscopy is now well established, e.g., for investigations of atoms in external "elds [47}56]. The resolution of the Fourier transform action spectra is restricted by the uncertainty principle of the Fourier transform, i.e., the method allows the identi"cation of usually short orbits in the low-dense part of the action spectra. A fully resolved action spectrum would require an in"nite length of the original scaled quantum spectrum. Although periodic orbit theory has been very successful in the interpretation of quantum spectra, the extraction of individual eigenstates and resonances directly from periodic orbit quantization remains of fundamental interest and importance. As mentioned above the semiclassical trace formulae diverge at the physical interesting regions. They are convergent, however, at complex energies with positive imaginary part above a certain value, viz. the entropy barrier. Thus the problem of revealing the semiclassical eigenenergies and resonances is closely related to "nding an analytic continuation of the trace formulae to the region at and below the real axis. Several re"nements have been introduced in recent years in order to transform the periodic orbit sum in the physical domain of interest to a conditionally convergent series by, e.g., using symbolic dynamics and the cycle expansion [9,57,58], the Riemann}Siegel look-alike formula and pseudo-orbit expansions [59}61], surface of section techniques [62,63], and heat-kernel regularization [64}66]. These techniques are mostly designed for systems with special properties, e.g., the cycle expansion requires the existence and knowledge of a symbolic code and is most successful for open systems, while the Riemann}Siegel look-alike formula and the heat-kernel regularization are restricted to bound systems. Until now there is no universal method, which allows periodic orbit quantization of a large variety of bound and open systems with an underlying regular, mixed, or chaotic classical dynamics. 1.2. Objective of this work The main objective of this work is the development of novel methods for (a) the analysis of quantum spectra and (b) periodic orbit quantization. The conventional action or recurrence spectra obtained by Fourier transformation of "nite range quantum spectra su!er from the fundamental resolution problem because of the uncertainty principle of the Fourier transformation. The broadening of recurrence peaks usually prevents the detailed analysis of structures near
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bifurcations of classical orbits. We will present a method which allows, e.g., the detailed analysis of bifurcations and symmetry breaking and the study of higher order corrections to the periodic orbit sum. For periodic orbit quantization the aim is the development of a universal method for periodic orbit quantization which does not depend on special properties of the system, such as the existence of a symbolic code or the applicability of a functional equation. As will be shown both problems can be solved by applying methods for high resolution spectral analysis [67]. The computational techniques for high-resolution spectral analysis have recently been signi"cantly improved [68}70], and we will demonstrate that the state of the art methods are a powerful tool for the semiclassical quantization and analysis of dynamical systems. 1.2.1. High precision analysis of quantum spectra Within the semiclassical approximation of Gutzwiller's trace formula or the Berry}Tabor formula the density of states of a scaled quantum spectrum is a superposition of sinusoidal modulations as a function of the energy or an appropriate scaling parameter. The frequencies of the modulations are determined by the periods, i.e., the classical action of orbits. The amplitudes and phases of the oscillations depend on the stability properties of the periodic orbits and the Maslov indices. To extract information about the classical dynamics from the quantum spectrum it is therefore quite natural to Fourier transform the spectrum from `energya domain to `timea (or, for scaled spectra, `actiona) domain [45}47,71}77]. The periodic orbits appear as peaks at positions given by the periods of orbits, and the peak heights exhibit the amplitudes of periodic orbit contributions in the semiclassical trace formulae. The comparison with classical calculations allows the interpretation of quantum spectra in terms of the periodic orbits of the corresponding classical system. However, the analyzed experimental or theoretical quantum spectra are usually of "nite length, and thus the sinusoidal modulations are truncated when Fourier transformed. The truncation implies a broadening of peaks in the time domain spectra because of the uncertainty principle of the Fourier transform. The widths of the recurrence peaks are determined by the total length of the original spectrum. The uncertainty principle prevents until now precision tests of the semiclassical theories because neither the peak positions (periods of orbits) nor the amplitudes can be obtained from the time domain spectra to high accuracy. Furthermore, the uncertainty principle implies an overlapping of broadened recurrence peaks when the separation of neighboring periods is less than the peak widths. In this case individual periodic orbit contributions cannot be revealed any more. This is especially a disadvantage, e.g., when following, in quantum spectra of systems with mixed regular-chaotic dynamics, the bifurcation tree of periodic orbits. All orbits involved in a bifurcation, including complex `ghosta orbits [36,37,44], have nearly the same period close to the bifurcation point. The details of the bifurcations, especially of catastrophes with higher codimension or corank [38,39], cannot be resolved with the help of the conventional Fourier transform. The same is true for the e!ects of symmetry breaking, e.g., breaking of the cylindrical symmetry of the hydrogen atom in crossed magnetic and electric "elds [54] or the `temporal symmetry breakinga [55,56] of atoms in oscillating "elds. The symmetry breaking should result in a splitting of peaks in the Fourier transform recurrence spectra. Until now this phenomenon could only be observed indirectly via constructive and destructive interference e!ects of orbits with broken symmetry [54,55]. To overcome the resolution problem in the conventional recurrence spectra and to achieve a high-resolution analysis of quantum spectra it is "rst of all necessary to mention that the
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uncertainty principle of the Fourier transformation is not a fundamental restriction to the resolution of the time domain recurrence spectra, and is not comparable to the fundamental Heisenberg uncertainty principle in quantum mechanics. This can be illustrated for the simple example of a sinusoidal function with only one frequency. The knowledge of the signal at only three points is su$cient to recover, although not uniquely, the three unknown parameters, viz. the frequency, amplitude and phase of the modulation. Similarly, a superposition of N sinusoidal functions can be recovered, in principle, from a set of at least 3N data points (t , f ). If the data points G G are exact the spectral parameters +u , A , , can also be exactly determined from a set of 3N I I I nonlinear equations , f " A sin(u t ! ) , (1.1) G I IG I I i.e., there is no uncertainty principle. The main di!erence between this procedure and the Fourier transformation is that we use the linear superposition of sinusoidal functions as an ansatz for the functional form of our data, and the frequencies, amplitudes, and phases are adjusted to obtain the best approximation to the data points. By contrast, the Fourier transform is not based on any constraints. In case of a discrete Fourier transform (DFT) the frequencies are chosen on a (usually equidistant) grid, and the amplitudes are determined from a linear set of equations, which can be solved numerically very e$ciently, e.g., by fast Fourier transform (FFT). As mentioned above the high precision spectral analysis requires the numerical solution of a nonlinear set of equations, and the availability of e$cient algorithms has been the bottleneck for practical applications in the past. Several techniques have been developed [67], however, most of them are restricted } for reasons of storage requirements, computational time, and stability of the algorithm } to signals with quite low number of frequencies. By contrast, the number of frequencies (periodic orbits) in quantum spectra of chaotic systems is in"nite or at least very large, and applied methods for high precision spectral analysis must be able to handle signals with large numbers of frequencies. This requirement is ful"lled by the method of harmonic inversion by ,lter-diagonalization, which was recently developed by Wall and Neuhauser [68] and signi"cantly improved by Mandelshtam and Taylor [69,70]. The decisive step is to recast the nonlinear set of equations as a generalized eigenvalue problem. Using an appropriate basis set the generalized eigenvalue equation is solved with the "lter-diagonalization method, which means that the frequencies of the signal can be determined within small frequency windows and only small matrices must be diagonalized numerically even when the signal is composed of a very large number of sinusoidal oscillations. We will introduce harmonic inversion as a powerful tool for the high precision analysis of quantum spectra, which allows to circumvent the uncertainty principle of the conventional Fourier transform analysis and to extract previously unattainable information from the spectra. In particular, the following items are investigated: E High precision check of semiclassical theories: the analysis of spectra allows a direct quantitative comparison of the quantum and classical periodic orbit quantities to many signi"cant digits. E Uncovering of periodic orbit bifurcations in quantum spectra, and the veri"cation of ghost orbits and uniform semiclassical approximations. E Direct observation of symmetry breaking e!ects.
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E Quantitative interpretation of the di!erences between quantum and semiclassical spectra in terms of the expansion of the periodic orbit sum. Results will be presented for various physical systems, e.g., the hydrogen atom in external "elds, the circle billiard, and the three disk scattering problem. 1.2.2. Periodic orbit quantization The analysis of quantum spectra by harmonic inversion and, e.g., the observation of `ghosta orbits, symmetry breaking e!ects, or higher-order corrections to the periodic orbit contributions provides a deeper understanding of the relation between quantum mechanics and the underlying classical dynamics of the system. However, the inverse procedure, i.e., the calculation of semiclassical eigenenergies and resonances directly from classical input data is of at least the same importance and even more challenging. As mentioned above, the periodic orbit sums su!er from fundamental convergence problems in the physically interesting domain and much e!ort has been undertaken in recent years to overcome these problems [9,57}66]. Although many of the re"nements which have been introduced are very e$cient for a speci"c model system or a class of systems, they all su!er from the disadvantage of non-universality. The cycle expansion technique [9,57,58] requires a completely hyperbolic dynamics and the existence of a symbolic code. The method is most e$cient only for open systems, e.g., for three-disk or n-disk pinball scattering [9}15]. By contrast, the Riemann}Siegel look-alike formula and pseudo orbit expansion of Berry and Keating [59}61] can only be applied for bound systems. The same is true for surface of section techniques [62,63] and heat-kernel regularization [64}66]. We will introduce high resolution spectral analysis, and in particular harmonic inversion by "lter-diagonalization as a novel and universal method for periodic orbit quantization. Universality means that the method can be applied to open and bound systems with regular and chaotic classical dynamics as well. Formally, the semiclassical density of states (more precisely the semiclassical response function) can be written as the Fourier transform of the periodic orbit recurrence signal C(s)" A d(s!s ) , (1.2) with A and s the periodic orbit amplitudes and periods (actions), respectively. If all orbits up to in"nite length are considered the Fourier transform of C(s) is equivalent to the non-convergent periodic orbit sum. For chaotic systems a numerical search for all periodic orbits is impossible and, furthermore, does not make sense because the periodic orbit sum does not converge anyway. On the other hand, the truncation of the Fourier integral at "nite maximum period s yields
a smoothed spectrum only [78]. However, the low resolution property of the spectrum can be interpreted as a consequence of the uncertainty principle of the Fourier transform. We can now argue in an analogous way as in the previous section that the uncertainty principle can be circumvented with the help of high-resolution spectral analysis, and propose the following procedure for periodic orbit quantization by harmonic inversion. Let us assume that the periodic orbits with periods 0(s(s are available and the semiclassi cal recurrence function C(s) has been constructed. This signal can now be harmonically inverted and thus adjusted to the functional form of the quantum recurrence function C (s), which is a superposition of sinusoidal oscillations with frequencies given by the quantum mechanical
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eigenvalues of the system. The frequencies obtained by harmonic inversion of C(s) should therefore be the semiclassical approximations to the exact eigenenergies. The universality and wide applicability of this novel quantization scheme follows from the fact that the periodic orbit recurrence signal C(s) can be obtained for a large variety of systems because the calculation does not depend on special properties of the system, such as boundness, ergodicity, or the existence of a symbolic dynamics. For systems with underlying regular or chaotic classical dynamics the amplitudes A of periodic orbit contributions are directly obtained from the Berry}Tabor formula [25,26] and Gutzwiller's trace formula [8], respectively. As mentioned above, the harmonic inversion technique requires the knowledge of the signal C(s) up to a "nite maximum period s . The e$ciency of the quantization method strongly
depends on the signal length which is required to obtain a certain number of eigenenergies. In chaotic systems periodic orbits proliferate exponentially with increasing period and usually the orbits must be searched numerically. It is therefore highly desirable to use the shortest possible signal for periodic orbit quantization. The e$ciency of the method can be improved if not just the single signal C(s) is harmonically inverted but additional classical information obtained from a set of smooth and linearly independent observables is used to construct a semiclassical crosscorrelated periodic orbit signal. The cross-correlation function can be analyzed with a generalized harmonic inversion technique and allows calculating semiclassical eigenenergies from a signi"cantly reduced set of orbits or alternatively to improve the accuracy of spectra obtained with the same set of orbits. However, the semiclassical eigenenergies deviate } apart from a few exceptions, e.g., the geodesic #ow on a surface with constant negative curvature [8] } from the exact quantum mechanical eigenvalues. The reason is that Gutzwiller's trace formula and the Berry}Tabor formula are only the leading-order terms of the expansion of periodic orbit contributions [22}24]. It will be shown how the higher-order corrections of the periodic orbit sum can be used within the harmonic inversion procedure to improve, order by order, the semiclassical accuracy of eigenenergies, i.e., to obtain eigenvalues beyond the Gutzwiller and Berry}Tabor approximation. 1.3. Outline The paper is organized as follows. In Section 2 the high precision analysis of quantum spectra is discussed. After general remarks on Fourier transform recurrence spectra in Section 2.1, we introduce in Section 2.2 harmonic inversion as a tool to circumvent the uncertainty principle of the conventional Fourier transformation [79]. In Section 2.3 a precision check of the periodic orbit theory is demonstrated by way of example of the hydrogen atom in a magnetic "eld. For the hydrogen atom in external "elds we furthermore illustrate that dynamical information which has been unattainable before can be extracted from the quantum spectra. In Section 2.4 we investigate in detail the quantum manifestations of bifurcations of orbits related to a hyperbolic umbilic catastrophe [44,80] and a butter#y catastrophe [37,81], and in Section 2.5 we directly uncover e!ects of symmetry breaking in the quantum spectra. In Section 2.6 we analyze by way of example of the circle billiard and the three disk system the di!erence between the exact quantum and the semiclassical spectra. It is shown that the deviations between the spectra can be quantitatively interpreted in terms of the next order contributions of the expanded periodic orbit sum.
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In Section 3 we propose harmonic inversion as a novel technique for periodic orbit quantization [82,83]. The method is introduced in Section 3.1 for a mathematical model, viz. the calculation of zeros of Riemann's zeta function. This model shows a formal analogy with the semiclassical trace formulae, and is chosen for the reasons that no extensive numerical periodic orbit search is necessary and the results can be directly compared to the exact zeros of the Riemann zeta function. In Section 3.2 the method is derived for the periodic orbit quantization of physical systems and is demonstrated in Section 3.3 for the example of the three disk scattering system with fully chaotic (hyperbolic) classical dynamics [82,83], and in Section 3.4 for the hydrogen atom in a magnetic "eld as a prototype example of a system with mixed regular-chaotic dynamics [84]. In Section 3.5, the e$ciency of the method is improved by a generalization of the technique to the harmonic inversion of cross-correlated periodic orbit sums, which allows to signi"cantly reduce the number of orbits required for the semiclassical quantization [85], and in Section 3.6 we derive the concept for periodic orbit quantization beyond the Gutzwiller and Berry}Tabor approximation by harmonic inversion of the expansion of the periodic orbit sum [86]. The methods of Sections 3.5 and 3.6 are illustrated in Section 3.7 for the example of the circle billiard [85}88]. Finally, in Section 3.8 we demonstrate the semiclassical calculation of individual transition matrix elements for atoms in external "elds [89]. Section 4 concludes with a summary and outlines possible future applications, e.g., the analysis of experimental spectra and the periodic orbit quantization of systems without scaling properties. Computational details of harmonic inversion by "lter-diagonalization are presented in Appendix A, and the calculation and asymptotic behavior of some catastrophe di!raction integrals is discussed in Appendix C.
2. High precision analysis of quantum spectra Semiclassical periodic orbit theory [7,8] and closed orbit theory [90}93] have become the key for the interpretation of quantum spectra of classically chaotic systems. The semiclassical spectra at least in low resolution are given as the sum of a smooth background and a superposition of modulations whose amplitudes, frequencies, and phases are solely determined by the closed or periodic orbits of the classical system. For the interpretation of quantum spectra in terms of classical orbits it is therefore most natural to obtain the recurrence spectrum by Fourier transforming the energy spectrum to the time domain. Each closed or periodic orbit should show up as a sharp d-peak at the recurrence time (period), provided, "rst, the classical recurrence times do not change within the whole range of the spectrum and, second, the Fourier integral is calculated along an in"nite energy range. Both conditions are usually not ful"lled. However, the problem regarding the energy dependence of recurrence times can be solved in systems possessing a classical scaling property by application of scaling techniques. The second condition is never ful"lled in practice, i.e., the length of quantum spectra is always restricted either by experimental limitations or, in theoretical calculations, by the growing dimension of the Hamiltonian matrix which has to be diagonalized numerically. The given length of a quantum spectrum determines the resolution of the quantum recurrence spectrum due to the uncertainty principle, *E ) *¹& , when the conventional Fourier transform is used. Only those closed or periodic orbits can be clearly identi"ed quantum mechanically which appear as isolated non-overlapping peaks in the quantum
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recurrence spectra. This is especially not the case for orbits which undergo bifurcations at energies close to the bifurcation point. As will be shown the resolution of quantum recurrence spectra can be signi"cantly improved beyond the limitations of the uncertainty principle of the Fourier transform by methods of high resolution spectral analysis. In the following, we "rst review the conventional analysis of quantum spectra by Fourier transformation and discuss on the example of the hydrogen atom in a magnetic "eld the achievements and limitations of the Fourier transform recurrence spectra. In Section 2.2 we review methods of high-resolution spectral analysis which can serve to circumvent the uncertainty principle of the Fourier transform. In Section 2.3 the harmonic inversion technique is applied to calculate high-resolution recurrence spectra beyond the limitations of the uncertainty principle of the Fourier transformation from experimental or theoretical quantum spectra of ,nite length. The method allows to reveal information about the dynamics of the system which is completely hidden in the Fourier transform recurrence spectra. In particular, it allows to identify real orbits with nearly degenerate periods, to detect complex `ghosta orbits which are of importance in the vicinity of bifurcations [36,37,44], and to investigate higher order corrections of the periodic orbit contributions [22}24]. 2.1. Fourier transform recurrence spectra According to periodic orbit theory [7,8] the semiclassical density of states can be written as the sum of a smooth background . (E) and oscillatory modulations induced by the periodic orbits, 1 . (E)". (E)# Re A (E)e P1# , (2.1) p P with A (E) the amplitudes of the modulations and the classical actions S (E) of a primitive periodic orbit (po) determining the frequencies of the oscillations. Linearizing the action around E"E yields (2.2) S(E)+S(E )#dS/dE" (E!E )"S(E )#¹ (E!E ) , # with ¹ the time period of the orbit at energy E . When Eq. (2.2) is inserted in Eq. (2.1) the semiclassical density of states is locally given as a superposition of sinusoidal oscillations, and it might appear that the problem of identifying the amplitudes A and time periods ¹ that contribute to the quantum spectrum . (E) can be solved by Fourier transforming . (E) to the time domain, i.e., each periodic orbit should show up as a d-peak at the recurrence time t"¹ . However, fully resolved recurrence spectra can be obtained only if the two following conditions are ful"lled. First, the Fourier integral is calculated along an in"nite energy range, and, second, the classical recurrence times do not change within the whole range of the spectrum. The "rst condition is usually not ful"lled when quantum spectra are obtained from an experimental measurement or a numerical quantum calculation. In that case the Fourier transformation is restricted to a "nite energy range and the resolution of the time domain spectrum is limited by the uncertainty principle of the Fourier transform. It is the main objective of this section to introduce high-resolution methods for the spectral analysis which allow to circumvent the uncertainty principle of the Fourier transform and to obtain fully resolved recurrence spectra from the analysis of quantum spectra with "nite length. However, the second condition is usually not ful"lled either. Both the
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amplitudes and recurrence times of periodic orbits are in general non-trivial functions of the energy E, and, even worse, the whole phase space structure changes with energy when periodic orbits undergo bifurcations. For the interpretation of quantum spectra in terms of periodic orbit quantities it is therefore in general not appropriate to analyze the frequencies of long ranged spectra . (E). The problems due to the energy dependence of periodic orbit quantities have been solved by the development and application of scaling techniques. 2.1.1. Scaling techniques Many systems possess a classical scaling property in that the classical dynamics does not depend on an external scaling parameter w and the action of trajectories varies linearly with w. Examples are systems with homogeneous potentials, billiard systems, or the hydrogen atom in external "elds. In billiard systems the shapes of periodic orbits are solely determined by the geometry of the borders, and the classical action depends on the length ¸ of the periodic orbit, S " k¸ , with k the wave number. For a particle with mass m moving in a billiard system it is therefore most appropriate to take the wave number as the scaling parameter, i.e., w"k"(2mE/ . For a system with a homogeneous potential <(q) with <(cq)"c?<(q) only the size of periodic orbits but not their shape changes with varying energy E. Introducing a scaling parameter as a power of the energy, w"E ?> , the classical action of a periodic orbit is obtained as S (E)"(E/E ) ?> S (E ), with E being a reference energy, i.e., the action depends linearly on a scaling parameter de"ned as w"(E/E ) ?> . For example the Coulomb potential is a homogeneous potential with a"!1 and the bound Coulomb spectrum at negative energies (E "!1/2) is transformed by the scaling procedure to a simple spectrum with equidistant lines at w"n with n"1,2,2 the principal quantum number. For atoms in external magnetic and electric "elds the shape of periodic orbits changes if the "eld strengths are "xed and only the energy is varied. However, these systems possess a scaling property if both the energy and the "eld strengths are simultaneously varied. Details for the hydrogen atom in a magnetic "eld will be given below. The scaling parameter plays the role of an inverse e!ective Planck constant, w" \. In theoretical investigations it is even possible to apply scaling techniques to general systems with non-homogeneous potentials if the Planck constant is formally used as a variable parameter. Non-homogeneous potentials are important, e.g., in molecular dynamics, and the genaralized scaling technique has been applied in [94] to analyze quantum spectra of the HO molecule. When the scaling technique is applied to a quantum system the scaling parameter w is quantized, i.e., bound systems exhibit sharp lines at real eigenvalues w"w and open systems have resonances I related to complex poles w"w of the scaled Green's function G>(w). By varying the scaled energy I a direct comparison of the quantum recurrence spectra with the bifurcation diagram of the underlying classical system is possible [45,47]. The semiclassical approximation to the scaled spectrum is given by 1 . (w)". (w)# Re A e PQU , p P
(2.3)
with
s "
1 pJ dqJ " w
p dq
(2.4)
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the scaled action of a primitive periodic orbit. The vectors qJ and pJ are the coordinates and momenta of the scaled Hamiltonian. In contrast to Eq. (2.1) the periodic orbit sum Eq. (2.3) is a superposition of sinusoidal oscillations as a function of the scaling parameter w. Therefore, the scaled spectra . (w) can be Fourier-transformed along arbitrarily long ranges of w to generate Fourier transform recurrence spectra of in principle arbitrarily high resolution, i.e., yielding sharp d-peaks at the positions of the scaled action s"s of periodic orbits. The high-resolution analysis of quantum spectra in the following sections is possible only in conjunction with the application of the scaling technique. The periodic orbit amplitudes A in Gutzwiller's trace formula Eq. (2.3) for scaled systems are given by s e\ Pp I , (2.5) A " ("det(MP !I)" with M and k the monodromy matrix and Maslov index of the primitive periodic orbits, respectively, and r the repetition number of orbits. The monodromy matrix M is the stability matrix restricted to deviations perpendicular to a periodic orbit after period time ¹. We here discuss systems with two degrees of freedom. If dq(0) is a small deviation perpendicular to the orbit in coordinate space at time t"0 and dp(0) an initial deviation in momentum space, the corresponding deviations at time t"¹ are related to the monodromy matrix [93,95]:
dq(¹)
dq(0)
m m dq(0) . (2.6) dp(¹) dp(0) m m dp(0) To compute M, one considers an initial deviation solely in coordinate space to obtain the matrix elements m and m , and an initial deviation solely in momentum space to obtain m and m . In practice, a linearized system of di!erential equations obtained by di!erentiating Hamilton's equations of motion with respect to the phase space coordinates is numerically integrated. The Maslov index k increases by one every time the trajectory passes a conjugate point or a caustic. Therefore the amplitudes A are complex numbers containing phase information determined by the Maslov indices of orbits. The classical actions s are usually real numbers, although they can be complex in general. Non-real actions s indicate `ghosta orbits [36,37,44] which exist in the complex continuation of the classical phase space. As mentioned above the problem of identifying A and s that contribute to the quantum spectrum . (w) can in principle be solved by Fourier transforming . (w) to the action domain, "M
"
1 U C(s)" (2.7) [. (w)!. (w)]e\ QU dw . 2p U If the quantum spectrum . (w) has in"nite length and the periodic orbits (i.e., the actions s ) are real the Fourier transform of Eq. (2.3) indeed results in a fully resolved recurrence spectrum
1 1 > C(s)" [. (w)!. (w)]e\ QU dw" (A d(s!rs )#AH d(s#rs )) . (2.8) 2p 2p \ P The periodic orbits are identi"ed as d-peaks in the recurrence spectrum at positions s"$s (and their repetitions at s"$rs ) and the recurrence strengths are given as the amplitudes
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A (Eq. (2.5)) of Gutzwiller's trace formula. However, for "nite range spectra the d-functions in Eq. (2.8) must be replaced apart from a phase factor with d(s)Psin((w !w )s/2)/ps . Unfortunately, the recurrence peaks are broadened by the uncertainty principle of the Fourier transform and furthermore the function sin((w !w )s/2)/ps has side-peaks which are not related to recurrences of periodic orbits but are solely an undesirable e!ect of the sharp cut of the Fourier integral at w and w . In complicated recurrence spectra it can be di$cult or impossible to separate the physical recurrences and the unphysical side-peaks. The occurrence of side-peaks can be avoided by multiplying the spectra with a window function h(w), i.e.
1 U (2.9) h(w)[. (w)!. (w)]e\ QU dw , C(s)" 2p U where h(w) is equal to one at the center of the spectrum and decreases smoothly close to zero at the borders w and w of the Fourier integral. An example is a Gaussian window (w!w ) h(w)"exp ! 2p
centered at w "(w #w )/2 and with su$ciently chosen width p+(w !w )/6. The Gaussian window suppresses unphysical side-peaks in the recurrence spectra, however, the decrease of the side-peaks is paid by an additional broadening of the central peak. Various other types of window functions h(w) have been used as a compromise between the least uncertainty of the central recurrence peak and the optimal suppression of side-peaks. However, the fundamental problem of the uncertainty principle of the Fourier transform cannot be solved with any window function h(w). As a "rst example for the analysis of quantum spectra we introduce the hydrogen atom in a magnetic "eld (for reviews see [29}31]) given by the Hamiltonian [in atomic units, magnetic "eld strength c"B/(2.35;10 T), angular momentum ¸ "0] X (2.10) H" p!(1/r)#c(x#y) . As mentioned above, this system possesses a scaling property. Introducing scaled coordinates, rJ "c r, and momenta, pJ "c\ p, the classical dynamics of the scaled Hamiltonian HI " pI !(1/rJ )#(xJ #yJ )"Ec\ (2.11) does not depend on two parameters, the energy E and the magnetic "eld strength c, but solely on the scaled energy EI "Ec\ .
(2.12)
The classical action of the trajectories scales as S "s c\ "s w . (2.13) Based on the scaling relations of the classical Hamiltonian the experimental technique of scaled energy spectroscopy was developed [45}47]. Experimental spectra on the hydrogen atom at constant scaled energy have been measured by varying the magnetic "eld strength linearly on
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a scale w,c\ , adjusting simultaneously the energy E (via the wave length of the exciting laser light) so that the scaled energy E"EI c\ is kept constant at a given value. The spectra have been Fourier-transformed in the experimentally accessible range 364c\ 450. Experimental recurrence spectra at scaled energies !0.34EI 40 are presented in Fig. 1a. The overlay exhibits a well-structured system of clustered branches of resonances in the scaled energy-action plane. For comparison, Fig. 1b presents the energy-action spectrum of the closed classical orbits, i.e. the scaled action of orbits as a function of the energy. As can be seen the clustered branches of resonances in the experimental recurrence spectra (Fig. 1a) well resemble the classical bifurcation tree of closed orbits in Fig. 1b, although a one to one comparison between recurrence peaks and closed orbits is limited by the "nite resolution of the quantum recurrence spectra. Closed orbits bifurcating from the kth repetition of the orbit parallel to the magnetic "eld axis are called `vibratorsa <J , and orbits bifurcating from the kth repetition of the perpendicular orbit are called I `rotatorsa RJ in Fig. 1. Orbits X are created mainly in tangent bifurcations. The graphs of some I I closed orbits are given in Fig. 2. A detailed comparison of the peak heights of resonances in the experimental recurrence spectra with the semiclassical amplitudes obtained from closed orbit theory can be found in [47].
Fig. 1. (a) Experimental recurrence spectra of the hydrogen atom in a magnetic "eld in overlay form. Even-parity, m"0 "nal state. (b) Scaled action of closed classical orbits through origin (from [45]).
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Fig. 2. Closed classical orbits of the hydrogen atom in a magnetic "eld in (o,z) projection: (a) vibrator, (b) rotator, and (c) `exotica orbits (from [46]).
Scaled energy spectroscopy has become a well-established method to investigate the dynamics of hydrogenic and non-hydrogenic atoms in external "elds. Recurrence spectra of helium in a magnetic "eld are presented in [48,49]. Recurrence peaks of the helium atom which cannot be identi"ed by hydrogenic closed orbits have been explained by scattering of the highly excited electron at the ionic core [49,75,76]. Rubidium has been studied in crossed electric and magnetic "elds in [50] and unidenti"ed recurrence peaks have been interpreted in terms of classical core scattering [77]. The Stark e!ect on lithium has been investigated by the MIT group [51}53]. For atoms in an electric "eld F the scaling parameter is w,F\ , and EF\ is the scaled energy. Experimental recurrence spectra of lithium in an electric "eld and the corresponding closed orbits are presented in Fig. 3. Strong recurrence peaks occur close to the bifurcations of the parallel orbit and its repetitions marked by open circles in Fig. 3. The new orbits created in bifurcations have almost the same action as the corresponding return of the parallel orbit, and, similar as in Fig. 1 for the hydrogen atom in a magnetic "eld, the small splittings of recurrence peaks are not resolved in the experimental recurrence spectra. It is important to note that the "nite resolution of the experimental recurrence spectra in Figs. 1 and 3 is not caused by the "nite bandwidth of the exciting laser but results as discussed above from the "nite length of the Fourier-transformed scaled spectra and the uncertainty principle of the Fourier transform. This can be illustrated by analyzing a numerically computed spectrum instead
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Fig. 3. Experimental recurrence spectra of lithium in an electric "eld in overlay form. The curves mark the scaled action of the parallel orbit and its repetitions as a function of scaled energy. The open circles mark the bifurcation points of closed orbits, whose shapes are shown along the bottom (from [52]).
of an experimentally measured spectrum. We study the hydrogen atom in a magnetic "eld at constant scaled energy EI "!0.1. At this energy the classical dynamics is completely chaotic and all periodic orbits are unstable. We calculated 9715 states in the region w(140 by numerical diagonalization of the Hamiltonian matrix in a complete basis set. For details of the quantum calculations see, e.g., [96]. In Fig. 4 the quantum density of states is analyzed by the conventional Fourier transform. To get rid of unphysical side-peaks the spectrum was multiplied with a Gaussian function h(w) with width p chosen in accordance with the total length of the quantum spectrum. Fig. 4 clearly exhibits recurrence peaks which can be related to classical periodic orbits. However, the widths of the peaks are approximately *s/2p"0.03, and it is impossible to determine the periods of the classical orbits to higher accuracy than about 0.01 from the Fourier analysis of the quantum spectrum. Recurrence peaks of orbits with similar periods overlap, as can be clearly seen around s/2p"2.1, and at least guessed, e.g., at s/2p"1.1, or s/2p"2.6. A precise determination of the amplitudes is impossible especially for the overlapping peaks. Furthermore, the Fourier transform does not allow to distinguish between real and ghost orbits. In the following, we will demonstrate that the quality of recurrence spectra can be signi"cantly improved by application of state of the art methods for high-resolution spectral analysis.
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Fig. 4. Recurrence spectrum (Fourier transform) for the density of states of the hydrogen atom in a magnetic "eld at scaled energy EI "Ec\ "!0.1.
2.2. Circumventing the uncertainty principle Instead of using the standard Fourier analysis, to extract the amplitudes and actions we propose to apply methods for high-resolution spectral analysis. The fundamental di!erence between the Fourier transform and high-resolution methods is the following. Assume a complex signal C(t) given on an equidistant grid of points t "t #n*t as a superposition of exponential functions, i.e. L C(t ),c " d e\ RLSI, n"1,2,2,N . L L I I In the discrete Fourier transform (DFT) N real frequencies 2pk u" , k"1,2,2,N I N*t
(2.14)
(2.15)
are "xed and evenly spaced. The N complex amplitudes d of the Fourier transform are determined I by solving a linear set of equations, yielding 1 , (2.16) d " c e RLSI, k"1,2,2,N . L I N L The sums in Eq. (2.16) can be calculated numerically very e$ciently, e.g., by the fast Fourier transform (FFT) algorithm. The resolution *u of the Fourier transform is controlled by the total length N*t of the signal, *u"2p/N*t ,
(2.17)
which is the spacing between the grid points in the frequency domain. Only those spectral features that are separated from each other by more than *w can be resolved. This is referred to as the `uncertainty principlea of the Fourier transform. By contrast, both the amplitudes d and the I frequencies u in Eq. (2.14) are free parameters when methods for high-resolution spectral analysis I are applied. Because the frequencies u are free adjusting parameters they are allowed to appear I
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very close to each other and therefore the resolution is practically in"nite. Using the data points of the signal C(t) at t"t the set of in general complex parameters +d ,u , are given as the solution of L I I a nonlinear set of equations. Unfortunately, the nonlinear set of equations does not have a solution in closed form similar to Eq. (2.16) for the discrete Fourier transform. In fact, the numerical calculation of the parameters +d ,u , from the nonlinear set of equations is the central and I I nontrivial problem of all methods for high-resolution spectral analysis. The numerical harmonic inversion of a given signal like Eq. (2.14) is a fundamental problem in physics, electrical engineering and many other diverse "elds. The problem has already been addressed (in a slightly di!erent form) in the 18th century by Baron de Prony, who converted the nonlinear set of equations Eq. (2.14) to a linear algebra problem. There are several approaches related to the Prony method used for a high-resolution spectral analysis of short time signals, such as the modern versions of the Prony method, MUSIC (MUltiple SIgnal Classi"cation) and ESPRIT (Estimation of Signal Parameters via Rotational Invariance Technique) [67,97,98]. As opposed to the Fourier transform these methods are not related to a linear (unitary) transformation of the signal and are highly nonlinear by their nature. However, the common feature present in most versions of these methods is converting the nonlinear "tting problem to a linear algebraic one. Note also that while ESPRIT uses exclusively linear algebra, the Prony method or MUSIC require some additional search in the frequency domain which makes them less e$cient. To our best knowledge none of these nonlinear methods is able to handle a signal Eq. (2.14) that contains `too manya frequencies as they lead to unfeasibly large and typically ill-conditioned linear algebra problems [70]. This is especially the case for the analysis of quantum spectra because such spectra cannot be treated as `short signalsa and contain a high number of frequencies given by (see below) the number of periodic orbits of the underlying classical system. Decisive progress in the numerical techniques for harmonic inversion has recently been achieved by Wall and Neuhauser [68]. Their method is conceptually based on the original "lter-diagonalization method of Neuhauser [99] designed to obtain the eigenspectrum of a Hamiltonian operator in any selected small energy range. The main idea is to associate the signal C(t) (Eq. (2.14)) with an autocorrelation function of a suitable dynamical system, (2.18) C(t)"(U ,e\ XK RU ) , where the brackets de"ne a complex symmetric inner product (i.e., no complex conjugation). Eq. (2.18) establishes an equivalence between the problem of extracting information from the signal C(t) with the one of diagonalizing the evolution operator exp(!iXK ) of the underlying dynamical system. The frequencies u of the signal C(t) are the eigenvalues of the operator XK , i.e. I XK "U )"u "U ) , (2.19) I I I and the amplitudes d are formally given as I d "(U ,U ) . (2.20) I I After introducing an appropriate basis set, the operator exp(!iXK ) can be diagonalized using the method of ,lter diagonalization [68}70]. Operationally, this is done by solving a small generalized eigenvalue problem whose eigenvalues yield the frequencies in a chosen window. Typically, the numerical handling of matrices with approximate dimension 100;100 is su$cient even when the number of frequencies in the signal C(s) is about 10 000 or more. The knowledge of the operator
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XK itself is not required as for a properly chosen basis the matrix elements of XK can be expressed only in terms of the signal C(t). The advantage of the "lter-diagonalization technique is its numerical stability with respect to both the length and complexity (the number and density of the contributing frequencies) of the signal. Details of the method of harmonic inversion by "lter-diagonalization are given in [70] and in Appendix A.1. We now want to apply harmonic inversion as a tool for the high precision analysis of quantum spectra. In quantum calculations the bound state spectrum is given as a sum of d-functions,
. (w)" d(w!w ) . (2.21) I I Instead of using the Fourier transform (Eq. (2.7)) we want to adjust . (w) to the functional form of Gutzwiller's semiclassical trace formula Eq. (2.3), which can be written as 1 . (w)". (w)# +A e PQU#AH e\ PQHU, . (2.22) 2p P The #uctuating part of the semiclassical density of states Eq. (2.22) has exactly the functional form of the signal C(t) in Eq. (2.14) with t replaced by the scaling parameter w. The amplitudes and frequencies +d ,u , in Eq. (2.14) are the amplitudes and scaled actions +A /2p,!rs , of the I I periodic orbit contributions and their conjugate pairs +AH /2p, rsH ,. In order to obtain . (w) on an evenly spaced grid the spectrum is regularized by convoluting it with a narrow Gaussian function having the width p;1/s , where s is the scaled action of the longest orbit of interest.
The regularized density of states reads 1 . (w)" d (w!w )" e\U\UI N , (2.23) N N I (2pp I I and is the starting point for the harmonic inversion procedure. The step width for the discretization of . (w) is typically chosen as *w+p/3. The convolution of . (w) with a Gaussian function does N not e!ect the frequencies, i.e., the values obtained for the scaled actions s of the periodic orbits, but just results in a small damping of the amplitudes A PAN "A e\QN . For open systems the density of states is given by
(2.24)
1 1 . (w)"! Im , (2.25) p w!w I I with complex resonances w . If the minimum of the resonance widths C "!2 Im w is larger than I I I the step width *w, there is no need to convolute . (w) with a Gaussian function and the density of states can directly be analyzed by harmonic inversion in the same way as for bound systems. 2.3. Precision check of the periodic orbit theory As a "rst application of harmonic inversion for the high-resolution analysis of quantum spectra we investigate the hydrogen atom in a magnetic "eld given by the Hamiltonian Eq. (2.11) and
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compare the results of the harmonic inversion method to the conventional Fourier transform presented in Fig. 4. We analyze the density of states Eq. (2.23) with p"0.0015 at constant scaled energy EI "!0.1. From the discussion of the harmonic inversion technique and especially Eq. (2.19) it follows that the frequencies in . (w), i.e., the actions of periodic orbits, are not N obtained from a continuous frequency spectrum (or a spectrum de"ned on an equidistant grid of frequencies) but are given as discrete and in general complex eigenvalues. Actions with imaginary part signi"cantly below zero indicate `ghosta orbit contributions to the quantum spectrum. The actions s /2p and absolute values of the amplitudes "A " are given in the "rst three columns of Table 1. The last two columns present the classical actions s /2p of the periodic orbits and the absolute values of the semiclassical amplitudes "A ""s /2p("det(M !I)" ,
(2.26)
Table 1 Hydrogen atom in a magnetic "eld at scaled energy EI "!0.1. s and "A ": Actions and absolute values of amplitudes obtained by harmonic inversion of the quantum spectrum. s and "A ": Actions and absolute values of amplitudes obtained by periodic orbit theory Re s /2p
Im s /2p
"A "
s /2p
0.67746349 1.09456040 1.11461451 1.35492782 1.56500143 1.69802779 1.79106306 1.87643070 1.93352314 1.99328138 2.03199050 2.07515409 2.10239191 2.13389568 2.15304246 2.17505591 2.18921873 2.20287712 2.30616383 2.40955313 2.58910596 2.60055850 2.62069577 2.70985681 2.76685370 2.84036264 2.87476026 2.95857109
0.00000000 0.00003414 !0.00000723 0.00000089 0.00000054 !0.00000017 0.00000075 0.00000051 0.00000177 0.00000352 !0.00017651 0.00002170 0.00003926 0.00007623 0.00035849 !0.00021399 0.00034303 !0.00045692 !0.00000277 0.00000007 !0.00002565 !0.00002093 !0.00001446 !0.00000436 !0.00008167 !0.01311947 !0.00973465 !0.00000045
0.29925998 1.07896385 0.51757197 0.28203634 0.42062528 0.39860976 0.34636062 0.34352848 0.31816675 0.32173573 0.31129071 0.31251061 0.30768786 0.33987795 0.34510366 0.24582893 0.79129103 0.35778573 0.47937694 0.35374810 0.57871391 0.94842381 0.39565368 0.09049970 0.70094383 0.00065741 0.00390219 0.31520915
0.67746283 1.09457049 1.11457036 1.35492566 1.56499821 1.69802585 1.79106067 1.87642962 1.93352213 1.99328294 2.03194819 2.07517790 2.10234679 2.13380984 2.15300805 2.17556880 2.18914098 2.20439817 2.30615714 2.40954609 2.58913149 2.60051951 2.62066666 2.70985132
"A "
0.29929657 1.14785331 0.51991351 0.28236677 0.42053218 0.39857924 0.34672899 0.34375997 0.31888913 0.32238331 0.30888967 0.31598033 0.30947859 0.31999751 0.31929721 0.33461616 0.84694565 0.36428325 0.47918792 0.35398381 0.57621849 0.94785988 0.39496450 0.09040051 ghost orbit ghost orbit ghost orbit 2.95857293 0.31514488
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with s the action of the primitive periodic orbit, i.e., the "rst recurrence of the orbit. A graphical comparison between the Fourier transform, the high-resolution quantum recurrence spectrum, and the semiclassical recurrence spectrum is presented in Fig. 5. The crosses in Fig. 5b are the (complex) actions obtained by the harmonic inversion of the quantum mechanical density of states, and the squares are the actions of the (real) classical periodic orbits. The crosses and squares are in excellent agreement with a few exceptions, e.g., around s/2p+2.2 and s/2p+2.8 which will be discussed later. The amplitudes of the periodic orbit contributions are illustrated in Fig. 5a. The solid sticks are the amplitudes obtained by harmonic inversion of the quantum spectrum and the dashed sticks (hardly visible under solid sticks) present the corresponding semiclassical results. For comparison, the conventional Fourier transform recurrence spectrum is drawn as a solid line. To visualize more clearly the improvement of the high-resolution recurrence spectrum compared to the conventional Fourier transform a small part of the recurrence spectrum (Fig. 5a) around s/2p"2.6 is enlarged in Fig. 6. The smooth line is the absolute value of the conventional Fourier transform. Its shape suggests the existence of at least three periodic orbits but obviously the recurrence spectrum is not completely resolved. The results of the high-resolution spectral analysis are presented as sticks and crosses at the positions de"ned by the scaled actions s with peak heights "A ". Note that the positions of the peaks are considerably shifted with respect to the maxima of the conventional Fourier transform. To compare the quantum recurrence spectrum with Gutzwiller's periodic orbit theory the semiclassical results are presented as dashed sticks and squares in Fig. 6. For illustration the shapes of periodic orbits are also shown (in semiparabolic coordinates k"(r#z, l"(r!z). For these three orbits the agreement between the semiclassical and the highresolution quantum recurrence spectrum is nearly perfect, deviations are within the stick widths.
Fig. 5. (a) Recurrence spectrum for the density of states of the hydrogen atom in a magnetic "eld at scaled energy EI "Ec\ "!0.1. Smooth line: conventional Fourier transform. Solid stick spectrum: high-resolution quantum recurrence spectrum. Dashed sticks (hardly visible under solid sticks): recurrence spectrum from semiclassical periodic orbit theory. (b) Complex actions. Crosses and squares are the quantum and classical results, respectively.
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Fig. 6. Recurrence spectrum for the density of states of the hydrogen atom in a magnetic "eld at scaled energy EI "Ec\ "!0.1. Smooth line: conventional Fourier transform. Solid stick spectrum and crosses: high resolution quantum recurrence spectrum. Dashed sticks (hardly visible under solid sticks) and squares: recurrence spectrum from semiclassical periodic orbit theory. The recurrence peaks are identi"ed by periodic orbits drawn in semiparabolical coordinates k"(r#z) , l"(r!z) (from [79]).
Table 2 Hydrogen atom in a magnetic "eld at scaled energy EI "!0.1. Relative deviations between the quantum mechanical and classical actions and amplitudes of the three recurrence peaks around s/2p+2.6 s /2p
"s !s "/s
"A !A "/"A "
2.589131 2.600520 2.620667
9.86;10\ 1.50;10\ 1.11;10\
0.00433 0.00060 0.00174
The relative deviations between the quantum mechanical and classical actions and amplitudes are given in Table 2. The excellent agreement between the classical periodic orbit data and the quantum mechanical results obtained by harmonic inversion of the density of states may appear to be in contradiction to the fact that Gutzwiller's trace formula is an approximation, i.e., only the leading term of the semiclassical expansion of the periodic orbit sum. The small deviations given in Table 2 are certainly due to numerical limitations and do not indicate e!ects of higher-order corrections. The reason is that the higher-order contributions of the periodic orbits do not have the functional form of Eq. (2.22) as a linear superposition of exponential functions of the scaling parameter w. As the harmonic inversion procedure adjusts the quantum mechanical density of states . (w) to the ansatz Eq. (2.22) with free parameters s and A , the exact parameters s and A of the classical periodic orbits should provide the optimal adjustment to the quantum spectrum within the lowest-order approximation. However, the harmonic inversion technique can also be used to reveal the higher-order contributions of the periodic orbits in the quantum spectra, as will be demonstrated in Section 2.6.
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The quantum mechanical high-resolution recurrence spectrum is not in good agreement with the classical calculations around s/2p+2.2 (see crosses and squares in Fig. 5b). This period is close to the scaled action s /2p"(!2EI )\ "2.2361 of the parallel orbit, which undergoes an in"nite series of bifurcations with increasing energy [100]. If orbits are too close to bifurcations they can no longer be treated as isolated periodic orbits, which in turn is the basic assumption for Gutzwiller's semiclassical trace formula Eq. (2.3). The disagreement between the crosses and squares in Fig. 5 around s/2p+2.2 thus indicates the breakdown of the semiclassical ansatz Eq. (2.3) for nonisolated orbits. The structures around s/2p+2.8 can also not be explained by real classical periodic orbits. They are related to uniform semiclassical approximations and complex ghost orbits as will be discussed in the next section. 2.4. Ghost orbits and uniform semiclassical approximations Gutzwiller's periodic orbit theory Eq. (2.3) is valid for isolated periodic orbits where the determinant det(MP !I), i.e., the denominator in Eq. (2.5) is non-zero. However, the periodic orbit amplitudes A diverge close to the bifurcation points of orbits where MP !I has a vanishing P determinant. To remove the unphysical singularity from the semiclassical expressions all periodic orbits which participate at the bifurcation must be considered simultaneously in a uniform semiclassical approximation [40,41]. The uniform solutions can be constructed with the help of catastrophe theory [38,39], and have been studied, e.g., for the kicked top [36,42,43,101] and the hydrogen atom in a magnetic "eld [37,44]. In the vicinity of bifurcations `ghosta orbits, i.e., periodic orbits in the complex continuation of phase space can be very important. In general, the ghost orbits have real or complex actions, s . As can be shown from the asymptotic expansions of the uniform semiclassical approximations [37,44] those ghosts with positive imaginary part of the action, Im s '0, are of physical relevance. They contribute as A e 0QUe\' QU to Gutzwiller's periodic orbit sum Eq. (2.3), i.e., the modulations of the ghost orbits are exponentially damped with increasing scaling parameter w. Here we will only present a brief derivation of uniform semiclassical approximations. For details we refer the reader to the literature [36,37,42}44,101]. Our main interest is to demonstrate how bifurcations and ghost orbits can directly be uncovered in quantum spectra with the help of the harmonic inversion technique and the high-resolution recurrence spectra. Note that a detailed investigation of these phenomena is in general impossible with the Fourier transform because the orbits participating at the bifurcation have nearly the same period and thus cannot be resolved in the conventional recurrence spectra. We will discuss two di!erent types of bifurcations by way of example of the hydrogen atom in a magnetic "eld: The hyperbolic umbilic and the butter#y catastrophe. In the following section we will adopt the symbolic code of [102] for the nomenclature of periodic orbits. Introducing scaled semiparabolic coordinates k"c (r#z and l"c (r!z the scaled Hamiltonian of the hydrogen atom in a magnetic "eld reads h"(p#p)!EI (k#l)#(kl#kl)"2 . J I
(2.27)
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Fig. 7. Schematic view of the four-disk scattering problem (from [102]).
As pointed out in [102] the e!ective potential <(k,l)"!EI (k#l)#(kl#kl)/8 is bounded for EI '0 by four hyperbolas. When the smooth potential is replaced with hard walls periodic orbits can be assigned with the same ternary symbolic code as orbits of the four-disk scattering problem (see Fig. 7). In a "rst step a ternary alphabet +0AC, is introduced. The symbol 0 labels scattering to the opposite disk, and the symbols C and A label scattering to the neighboring disk in clockwise or anticlockwise direction, respectively. The orbit in Fig. 7 is coded 0C0A. The ternary code can be made more e$cient with respect to the exchange symmetry of the C and A symbol if it is rede"ned in the following way. The C's and A's are replaced with the sympol #, if consecutive letters ignoring the 0's are equal and they are replaced with the symbol !, if consecutive letters di!er. With this rede"nition the orbit shown in Fig. 7 is coded 0!0!, or because of its periodicity even simpler 0!. For the hydrogen atom in a magnetic "eld there is a one to one correspondence between the periodic orbits and the ternary symbolic code at energies EI 'EI "#0.3287 [19,21]. Below the critical energy orbits undergo bifurcations and the code is not unique. However, nearly all unstable periodic orbits can be uniquely assigned with this code even at negative energies. 2.4.1. The hyperbolic umbilic catastrophe As a "rst example we investigate the structure at s /2p"2.767 in the recurrence spectrum of the hydrogen atom in a magnetic "eld at constant scaled energy EI "Ec\ "!0.1 (see Fig. 5 and Table 1). At this energy no periodic orbit with an action close to s /2p"2.767 does exist. However, the strong recurrence peak in Fig. 5a indicates a near bifurcation. This bifurcation and the corresponding semiclassical approximation have been studied in detail in [44,80]. Four periodic orbits are created through two nearby bifurcations around the scaled energy EI +!0.096
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Fig. 8. Di!erence *s between the classical action of the four periodic orbits involved in the bifurcations. Dashed lines: analytical "ts related to the hyperbolic umbilic catastrophe. Inset: graphs of periodic orbits 0!#!! and its time reversal 0!!!# (solid line), 00#! (dashed line), and ###!!! (dash-dotted line) drawn in semiparabolical coordinates k"(r#z) , l"(r!z) (from [44]).
where we search for both real and complex `ghosta orbits. For the nomenclature of the real orbits we adopt the symbolic code of [102] as explained above. At scaled energy EI "!0.09689, the two orbits 00#! and ###!!! are born in a tangent bifurcation. At energies EI (EI , a prebifurcation ghost orbit and its complex conjugate exist in the complex continuation of the phase space. Orbit 00## is born unstable, and turns stable at the slightly higher energy EI "!0.09451. This is the bifurcation point of two additional orbits, 0!#!! and its time reversal 0!!!#, which also have ghost orbits as predecessors. The graphs of the real orbits at energy E"0 are shown as insets in Fig. 8, and the classical periodic orbit parameters are presented as solid lines in Figs. 8 and 9. Fig. 8 shows the di!erence in scaled action between the orbits. The action of orbit 0!#!! (or its time reversal 0!!!#), which is real also for its prebifurcation ghost orbits, has been taken as the reference action. The uniform semiclassical approximation for the four orbits involved in the bifurcation can be expressed in terms of the di!raction integral of a hyperbolic umbilic catastrophe
W(x, y)"
>
\
dp
>
\
dq e UN O_V W
(2.28)
with U(p, q; x, y)"p#q#y(p#q)#x(p#q) .
(2.29)
For our convenience the function U(p, q; x, y) slightly di!ers from the standard polynomial of the hyperbolic umbilic catastrophe given in [39] but the di!raction integral Eq. (2.28) can be easily
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transformed to the standard representation. The four stationary points of the integral Eq. (2.28) are readily obtained from the condition U"0 as p "!q "$(!x/3NU(p , q ; x, y)"0
(2.30)
and (2.31) p "q "!y$(y!V NU(p , q x, y)"y(y!x)G4(y!V) . The function U(p , q ; x, y) must now be adapted to the classical action of the four periodic orbits, i.e., *S"w*s+U(p , q ; x, y), which is well ful"lled for x"aw (EI !EI ), y"bw , (2.32) and constants a"!5.415, b"0.09665, as can be seen from the dashed lines in Fig. 8. Note that the agreement holds for both the real and complex ghost orbits. The next step to obtain the uniform solution is to calculate the di!raction integral Eq. (2.28) within the stationary-phase approximation. For EI 'EI there are four real stationary points (p ,q ) (see Eqs. (2.30) and (2.31)), and after expanding U(p, q; x, y) around the stationary points up to second order in p and q, the di!raction integral becomes the sum of Fresnel integrals, viz. W(x, y)V&
2p
# (!3x > \
p e W W\V8 W\V !p
((4y!3x)G2y(4y!3x
.
(2.33)
The terms of Eq. (2.33) can now be compared to the standard periodic orbit contributions Eq. (2.26) of Gutzwiller's trace formula. In our example the "rst term is related to the orbit 0!#!! (with a multiplicity factor of 2 for its time reversal 0!!!#), and the other two terms are related to the orbits 00#! and ###!!! for the upper and lower sign, respectively. The phase shift in the numerators describe the di!erences of the action *S and of the Maslov index *k"G1 relative to the reference orbit 0!#!!. The denominators are, up to a factor cw , with c"0.1034, the square root of "det(M!I)", with M the stability matrix. Fig. 9 presents the comparison for the determinants obtained from classical periodic orbit calculations (solid lines) and from Eqs. (2.32) and (2.33) (dashed lines). The agreement is very good for both the real and complex ghost orbits, similar to the agreement found for *s in Fig. 8. The constant c introduced above determines the normalization of the uniform semiclassical approximation for the hyperbolic umbilic bifurcation which is "nally obtained as [44,80] (2.34) (EI , w)"(c/p)s w W(aw (EI !EI ), bw )e QU\p I , with s and k denoting the orbital action and Maslov index of the reference orbit 0!#!!, and the constants a,b, and c as given above. The comparison between the amplitudes Eq. (2.26) of the conventional semiclassical trace formula for isolated periodic orbits and the uniform approximation Eq. (2.34) for the hyperbolic umbilic catastrophe is presented in Fig. 10 at the magnetic "eld strengths c"10\, c"10\, and c"10\. For graphical purposes we suppress the highly oscillatory part resulting from the function exp[i(S / !(p/2)k )] by plotting the absolute value of A(EI ,w) instead of the real part. The dashed line in Fig. 10 is the superposition of the isolated periodic orbit contributions from the A
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Fig. 9. Same as Fig. 8 but for the determinant det(M!I) of the periodic orbits (from [44]).
Fig. 10. Semiclassical amplitudes (absolute values) for magnetic "eld strength (a) c"10\, (b) c"10\, and (c) c"10\ in units of the time period ¹ . Dashed line: amplitudes of the standard semiclassical trace formula. Dash-dotted line: ghost orbit contribution. Solid line: uniform approximation of the hyperbolic umbilic catastrophe (from [44]).
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four orbits involved in the bifurcations. The modulations of the amplitude are caused by the constructive and destructive interference of the real orbits at energies EI 'EI and are most pronounced at low magnetic "eld strength (see Fig. 10c). The amplitude diverges at the two bifurcation points. For the calculation of the uniform approximation Eq. (2.34) we numerically evaluated the catastrophe di!raction integral Eq. (2.28) using a more simple and direct technique as described in [103]. Details of our method which is based on Taylor series expansions are given in Appendix C.1. The solid line in Fig. 10 is the uniform approximation Eq. (2.34). It does not diverge at the bifurcation points but decreases exponentially at energies EI (EI . At these energies no real orbits exist and the amplitude in the standard formulation would be zero when only real orbits are considered. However, the exponential tail of the uniform approximation Eq. (2.34) is well reproduced by a ghost orbit [37,36] with positive imaginary part of the complex action. As can be shown, the asymptotic expansion of the di!raction integral Eq. (2.28) has, for x<0, exactly the form of Eq. (2.26) but with complex action s and determinant det(M!I). The ghost orbit contribution is shown as dash-dotted line in Fig. 10. To verify the hyperbolic umbilic catastrophe in the quantum spectrum we analyze all three, the exact quantum spectrum, the uniform semiclassical approximation Eq. (2.34), and Gutzwiller's periodic orbit formula for isolated periodic orbits by means of the harmonic inversion technique at scaled energy EI "!0.1, which is slightly below the two bifurcation points. The part of the complex action plane which is of interest for the hyperbolic umbilic catastrophe is presented in Fig. 11. The two solid peaks mark the positions s /2p and the absolute values of amplitudes "A " obtained from I I the quantum spectrum. As mentioned above, at this energy only one classical ghost orbit is of physical relevance and marked as dash-dotted peak in Fig. 11. The position of that peak is in good agreement with the quantum result but the amplitude is enhanced. This enhancement is expected for isolated periodic orbit contributions near bifurcations which become singular exactly at the bifurcation points. The harmonic inversion analysis of the uniform approximation Eq. (2.34) at constant scaled energy EI "!0.1 in the same range 0(w(140 is presented as dashed peaks in Fig. 11. The two peaks agree well with the quantum results for both the complex actions and
Fig. 11. High-resolution recurrence spectra at scaled energy EI "!0.1. Solid peaks: part of the quantum recurrence spectra. Dash-dotted peak: classical ghost orbit contribution. Dashed peaks: uniform approximation of the hyperbolic umbilic catastrophe (from [44]).
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amplitudes. The enhancement of the ghost orbit peak and the additional non-classical peak observed in the quantum spectrum are therefore clearly identi"ed as artifacts of the bifurcation, i.e., the hyperbolic umbilic catastrophe. 2.4.2. The butteryy catastrophe: uncovering the `hiddena ghost orbits We now investigate the butter#y catastrophe, which is of importance, e.g., in photoabsorption spectra of the hydrogen atom in a magnetic "eld. In contrast to the density of states the photoabsorption spectra for dipole transitions from a low-lying initial state to highly excited "nal states can be measured experimentally [45}47]. The semiclassical photoabsorption cross-section is obtained by closed orbit theory [90}93] as the superposition of a smooth background and sinusoidal modulations induced by closed orbits starting at and returning back to the nucleus. Although the derivation of the semiclassical oscillator strength for dipole transitions (closed orbit theory) di!ers from the derivation of the semiclassical density of states (periodic orbit theory) the "nal results have a very similar structure and therefore spectra can be analyzed in the same way by conventional Fourier transform or the high-resolution harmonic inversion technique. In closed orbit theory the semiclassical oscillator strength is given by f"f #f
(2.35)
with f the oscillator strength of the "eld free hydrogen atom at energy E"0 and 2(2p) (sin 0 0 f "2(E !E )w\ Im G I D I " I ("mJ I ;Y (0 )Y (0 )exp+i(s w#m¹I !pk#p), (2.36) K G I K D I I I I the #uctuating part of the oscillator strength. In Eq. (2.36) E and E are the energies of the "nal and initial state, m is the magnetic quantum number, s , ¹I "c¹ , and k are the scaled action, scaled I I I I time, and the Maslov index of closed orbit k, 0 and 0 are the starting and returning angle of the orbit with respect to the magnetic "eld axis, and mJ is an element of the scaled monodromy matrix. The angular functions Y (0) depend on the initial state and polarization of light (see K Appendix B). For more details see [47,91,93]. The #uctuating terms Eq. (2.36) of the photoabsorption cross section are sinusoidal functions of the scaling parameter w"c\ despite a factor of w\ . To obtain the same functional form as Eq. (2.22) for the density of states, which is required for harmonic inversion, we multiply the oscillator strength f by w for both the semiclassical and quantum mechanical photoabsorption spectra. In analogy to Gutzwiller's trace formula Eq. (2.36) for photoabsorption spectra is valid only for isolated orbits and diverges at bifurcations of orbits, where mJ is zero. Near bifurcations the closed orbit contributions in Eq. (2.36) must be replaced with uniform semiclassical approximations, which have been studied in detail for the fold, cusp, and butter#y catastrophe in [37,81]. Here we restrict the discussion to the butter#y catastrophe, which is especially interesting because of the existence of a `hiddena ghost orbit, which can be uncovered in the photoabsorption spectrum of the hydrogen atom in a magnetic "eld with the help of the harmonic inversion technique. As an example we investigate real and ghost orbits related to the period doubling of the perpendicular orbit R . (For the closed orbits we adopt the nomenclature of [45,46], see also Fig. 2.) This closed
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orbit bifurcation is more complicated because various orbits with similar periods undergo two di!erent elementary types of bifurcations at nearly the same energy. The structure of bifurcations and the appearance of ghost orbits can clearly be seen in the energy dependence of the starting angles 0 in Fig. 12a. Two orbits R and R are born in a saddle node bifurcation at EI "!0.31735345, 0 "1.3465. Below the bifurcation energy we "nd an associated ghost orbit and its complex conjugate. Orbit R is real only in a very short energy interval (*EI +0.001), and is then involved in the next bifurcation at EI "!0.31618537, 0 "p/2. This is the period doubling bifurcation of the perpendicular orbit R , which exists at all energies (0 "p/2 in Fig. 12a). The real orbit R separates from R at energies below the bifurcation point, i.e., a real orbit vanishes with increasing energy. Consequently, associated ghost orbits are expected at energies above the bifurcation, i.e. EI 'EI , and indeed such `postbifurcationa ghosts have been found. Its complex starting angles are also shown in Fig. 12a. The energy dependence of scaled actions, or, more precisely, the di!erence *s with respect to the action of the period doubled perpendicular orbit R ,
Fig. 12. (a) Real and imaginary part of starting angle 0 for closed orbits related to the bifurcating scenario of the (period G doubled) perpendicular orbit R . (b) Di!erence *s/2p between the classical action of the (period doubled) perpendicular orbit R and real and ghost orbits bifurcating from it. (c) Monodromy matrix element mJ of the perpendicular orbit R and orbits bifurcating from it. Dashed lines: Analytical "ts (see text) (from [37]).
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is presented in Fig. 12b (solid lines), and the graph for the monodromy matrix element mJ is given in Fig. 12c. It can be seen that the actions and the monodromy matrix elements of the ghost orbits related to the saddle node bifurcation of R become complex at EI (EI , while these parameters remain real for the postbifurcation ghosts at EI 'EI . The two bifurcations are so closely adjacent that formulae for the saddle node bifurcation and the period doubling are no reasonable approximation to *s(EI ) and mJ (EI ) in the neighborhood of the bifurcations. However, both functions can be "tted well by the more complicated formulae [37] *s"k(pJ (EI !EI )#+1$[pJ (EI !EI )#1] ,)
(2.37)
and mJ "!M I (EI !EI ) (orbit R ) 4M I mJ "4M I (EI !EI )# [1$(pJ (EI !EI )#1](R, R, and ghosts) pJ
(2.38)
with k"3.768;10\, pJ "763.6, and M I "13.52 (see dashed lines in Fig. 12b and Fig. 12c). Note that Eqs. (2.37) and (2.38) describe the complete scenario for the real and the ghost orbits including both the saddle node and period doubling bifurcations. We also mention that orbits with angles 0 O0 have to be counted twice because they correspond to di!erent orbits when traversed in either direction, and therefore a total number of ,ve closed orbits, including ghosts, is considered here in the bifurcation scenario around the period doubling of the perpendicular orbit. The bunch of trajectories forming the butter#y is given by the Hamilton}Jacobi equations (with p "RS/Rm and p "RS/Rg) K E 3dp#2cpp#p[(m!m )p !gp ]"0, E K E K E K
p#p"4 , K E
(2.39)
where m and g are rotated semiparabolic coordinates
0 0 0!0 , m"k cos #l sin "(2r cos 2 2 2
(2.40)
0 0 0!0 , g"l cos !k sin "(2r sin 2 2 2
(2.41)
so that the m and g axes are now parallel and perpendicular to the returning orbit. The parameters c, d, and m in Eq. (2.39) will be speci"ed later. With p /p +a*0 we obtain E K g(m)"3ad(*0 )#2ac(*0 )#a(*0 )(m!m ) .
(2.42)
The butter#y is illustrated in Fig. 13. Depending on the number of real solutions *0 of Eq. (2.42) there exist one, three, or "ve orbits returning to each point (m,g). The di!erent regions are separated by caustics.
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Fig. 13. (a) Butter#y catastrophe of returning orbits (in rotated semiparabolic coordinates) related to the bifurcation of the perpendicular orbit at EI "!0.31618537. (b) Magni"cation of the marked region close to the nucleus. There are one, three, or "ve orbits returning to each point (m, g) in coordinate space (from [37]).
The uniform semiclassical approximations for closed orbits near bifurcations can in general be written as
p p f "2(E!E ) (sin 0 sin 0 Y (0 )Y (0 ) Im A exp i SI ! kI# K K K 2 I 4
,
(2.43)
where the complex amplitude A is de"ned implicitly by the di!raction integral [37,81]
p exp(i[*SH I(r, 0)!p *kH I]) cos((8r!p) . d0"A; I(r), 2p (2r) ("det(R(k, l)/R(q, 0 ))H I" H
(2.44)
To "nd a uniform semiclassical approximation for the butter#y catastrophe we have to solve the Hamilton}Jacobi equations Eq. (2.39) at least in the vicinity of the central returning orbit and to insert the action S(r, 0) into Eq. (2.44). For the classical action we obtain *SH(r, 0)"$(8r#m (0!0 )# c(0!0 )! d(0!0 ) ,
(2.45)
and for the determinant in the denominator of Eq. (2.44) we "nd $a(8r in the limit r<0 and 0+0 . Summing up in Eq. (2.44) the contributions of the incoming and the outgoing orbit (with
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Maslov indices *kH as for the cusp) we obtain the integral I(r) and the amplitude A for the butter#y catastrophe (t,0!0 ): p 3 I(r)"2 r\ "a"\ exp !i cos (8r! p 4 4
> exp(i[(m /4)t!(c/16)t!(d/64)t]) dt \ p NA"2 p"a"\ exp !i d\ W(!d\ m ,!cd\ ) , 4 ;
(2.46)
where
(2.47)
> exp(!i[xt#yt#t]) dt (2.48) \ is an analytic function in both variables x and y. Its numerical calculation and asymptotic properties are discussed in Appendix C.2. The uniform result for the oscillatory part of the transition strength now reads W(x, y),
f "2(E!E ) (sin 0 sin 0 Y (0 )Y (0 )2 p"a"\ d\ K K p ;Im exp i S ! k W(!d\ m ,!cd\ ) . K 2
(2.49)
It is very illustrative to study the asymptotic behavior of the uniform approximation Eq. (2.49) as we obtain, on the one hand, the relation between the parameters a, c, d, and m and the actions and the monodromy matrix elements of closed classical orbits, and, on the other hand, the role of complex ghost orbits related to this type of bifurcation is revealed. In the following we discuss both limits m <0, i.e. scaled energy EI <EI , and m ;0, i.e. EI ;EI . 2.4.2.1. Asymptotic behavior at scaled energy EI <EI . Applying Eq. (C.10) from Appendix C.2 to the W-function in the uniform approximation Eq. (2.49), we obtain the asymptotic formula for m <0 f "2(E!E ) (sin 0 sin 0 Y (0 )Y (0 )2(2p) "am "\ K K \ p p c (1#((3d/c)m #1) ; sin S ! k# # 1# K 2 4 3dm c 3d 2 3d p p m # 1# 1# m ! (k#1)# . (2.50) ; sin S # K 9d c 3 c 2 4
Comparing with the solutions for isolated closed orbits Eq. (2.36) we can identify the contributions of three real closed orbits. The classical action of orbit 1 is S , its Maslov index is k and the K monodromy matrix element m is given by m"!am ,!c\ M I (EI !EI ) , (2.51) where the parameter M I can be determined by closed orbit calculations (see Eqs. (2.37) and (2.38)). Orbits 2 and 3 are symmetric with respect to the z"0 plane and have the same orbital parameters,
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i.e. Maslov index k#1 and the classical action and the monodromy matrix element
3d 2 c 3d m # 1# m #1 S "S #*S"S # K K 9d c 3 c
,
(2.52)
c (1#((3d/c)m #1) . (2.53) m "4am 1# 3dm In the example of the bifurcation of orbit R at scaled energy EI "!0.31618537, orbit 1 is the orbit R perpendicular to the magnetic "eld axis, while orbits 2 and 3 can be identi"ed with R traversed in both directions (0 "p!0 ). With the help of Eqs. (2.51), (2.52) and (2.53) and classical scaling properties of the action and the monodromy matrix the parameters a, c, d, and m in the uniform approximation Eq. (2.49) can now be expressed completely in terms of closed orbit parameters k, pJ , and M I (see Eq. (2.37) and Eq. (2.38)), "a"\ d\ "3 c k (pJ /MI ) , d\ m "3 k c\ pJ (EI !EI ) , cd\ "(9k) c\ ,
(2.54)
and the uniform approximation for the butter#y catastrophe "nally reads f "2(E!E ) (sin 0 sin 0 Y (0 )Y (0 ) K K
p ;pc 3 k (32pJ /M I ) Im exp i S ! k K 2
;W(!3 k c\ pJ (EI !EI ),!(9k) c\ ) .
(2.55)
In the classical analysis complex ghost orbits were discovered both below and above the bifurcation energy. At EI 'EI they have the property that the classical action and the monodromy matrix remain real, although coordinates and momenta in phase space are complex. These ghost orbits do not appear in the asymptotic expansion Eq. (2.50) of the uniform approximation Eq. (2.49), and therefore, in analogy with the cusp catastrophe (see [37]), they do not have a physical meaning. The situation is di!erent at energy EI (EI where a `hidden ghosta with physical meaning will be revealed in the following. 2.4.2.2. Asymptotic behavior and **hidden ghost++ at scaled energy EI ;EI . At scaled energies below the bifurcation point we can apply the asymptotic formula Eq. (C.12) from Appendix C.2 to the W-function in the uniform approximation Eq. (2.49) and obtain
p p f "2(E!E ) (sin 0 sin 0 Y (0 )Y (0 )2(2p) "am "\ sin S ! (k#1)# K 2 K K 4
\ c 1# (1!i(!(3d/c)m !1) 3dm c 3d 2 3d m # 1#i ! m !1 ;exp i S # K 9d c 3 c #Im
p p ! k# 2 4
.
(2.56)
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The "rst term in Eq. (2.56) can be identi"ed as a real orbit with the same classical action and monodromy matrix element as in Eq. (2.50), but with a Maslov index increased by one. The second term in Eq. (2.56) is a ghost orbit contribution resulting from a superposition of two closed orbits with complex action and monodromy matrix element,
2 3d c 3d m # 1#i ! m !1 S "S # K 9d c 3 c
,
(2.57)
c m "4am 1# (1!i(!(3d/c)m !1) , (2.58) 3dm traversed in both directions. The positive imaginary part of the classical action results in an exponential damping of the ghost orbit contribution to the oscillator strength amplitude with decreasing energy similar to the situation at a fold catastrophe. In contrast to the fold catastrophe (see [37]) and to the hyperbolic umbilic catastrophe discussed in Section 2.4.1, the ghost orbit related to a butter#y catastrophe is always accompanied by a real orbit with almost the same classical action. Because the contribution of the real orbit is not exponentially damped its amplitude at energies where the asymptotic formula Eq. (2.56) is valid is much stronger than the ghost contribution. Therefore we call the ghost orbit in Eq. (2.56) a `hidden ghosta [37,81]. Note that classically the complex conjugate of the hidden ghost orbit also exists. The negative imaginary part of its classical action would result in an unphysical exponential increase of amplitude with decreasing energy, and consequently the complex conjugate ghost orbit does not appear in the asymptotic formula Eq. (2.56). In [37] it was admitted that it might be rather di$cult to "nd evidence for hidden ghost orbits in the Fourier transform of experimental or theoretical scaled energy spectra. However, with the harmonic inversion technique we are now able to uncover the hidden ghosts in high-resolution quantum recurrence spectra. 2.4.2.3. ;ncovering the hidden ghost orbit. To uncover the hidden ghost orbit related to the period doubling bifurcation of the perpendicular orbit in photoabsorption spectra of the hydrogen atom in a magnetic "eld we calculated the quantum spectrum at constant scaled energy EI "!0.35, which is su$ciently far below the two bifurcation energies around EI "!0.317 so that the real orbit R and the prebifurcation ghost orbit of R are approximately isolated orbits. We calculated 2823 transitions from the initial state "2p02 to "nal states with magnetic quantum number m"0 in the region w"c\ (100. The scaled photoabsorption spectrum was analyzed by conventional Fourier transform and by the high-resolution harmonic inversion technique. The interesting part of the recurrence spectrum in the region 1.86(s/2p(1.93 is presented in Fig. 14. The conventional Fourier transform (smooth line) has a maximum at s/2p"1.895 which is roughly twice the period of the perpendicular orbit but does not give any hint on the existence of a ghost orbit. The key points of the harmonic inversion analysis are that the resolution of the recurrence spectrum is not restricted by the uncertainty principle of the Fourier transform and that the method supplies complex frequencies s of the analyzed quantum spectrum f (w), which can be interpreted as complex actions s of ghost orbits. The high-resolution spectral analysis uncovers one real and two complex actions s around s/2p"1.9, which are marked as crosses in Fig. 14. They can be compared to actions of real and complex classical orbits. We "nd two closed orbits, i.e., the period doubling of the perpendicular orbit with scaled action s /2p"1.896011 and a ghost orbit with
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Fig. 14. (a) Recurrence spectrum for the photoabsorption cross section of the hydrogen atom in a magnetic "eld at scaled energy EI "Ec\ "!0.35. Transition "2p02P"mLX"0>2. Smooth line: conventional Fourier transform. Solid stick spectrum and crosses: high resolution quantum recurrence spectrum. Dashed sticks and squares: recurrence spectrum from semiclassical closed orbit theory. The two strongest recurrence peaks are identi"ed by a real and complex ghost orbit which are presented as insets. The solid and dashed lines in the insets are the real and imaginary part in semiparabolical coordinates k"(r#z) ,l"(r!z) . (b) Complex actions. Crosses and squares are the quantum and classical results, respectively (from [79]).
scaled action s /2p"1.894401!i0.006372, which are marked as squares in Fig. 14. The shapes of the real and complex closed orbits are presented as insets in Fig. 14 (in semiparabolic coordinates k,l). The real part of the ghost orbit (solid line) is similar to the shape of orbit R which is created as a real orbit at much higher energy EI "!0.317. The actions and also the amplitudes in the quantum and classical recurrence spectrum (see crosses and squares in Fig. 14) agree very well for the two closed orbit recurrences. The deviation between the imaginary parts of the actions of the quantum and classical ghost orbit can be explained by the fact that the orbits are only approximately isolated at energy EI "!0.35. The correct semiclassical formula is the uniform semiclassical approximation Eq. (2.55) of the butter#y catastrophe. This interpretation is supported by the occurrence of a third complex resonance with large imaginary part of the action and small amplitude in the quantum recurrence spectrum (Fig. 14), which has no classical analogue. The non-classical peak is similar to the occurrence of a non-classical peak in Fig. 11 for the hyperbolic umbilic catastrophe and is clearly an artifact of the near bifurcation. The uncovering of the hidden ghost orbit, which obviously is impossible with the conventional Fourier transform (solid line in Fig. 14) demonstrates that harmonic inversion is a very powerful tool for the analysis of quantum spectra, which can reveal structures and information from the spectra that have been unattainable before. 2.5. Symmetry breaking Methods for high-resolution spectral analysis are also helpful for a direct observation of symmetry breaking e!ects in quantum spectra. In many cases physical systems possess symmetries,
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e.g., a cylindrical symmetry around a "xed axis. In such situations the closed or periodic orbits of the classical system appear in continuous families. All members of the family have the same stability parameters and periods, i.e., they are observed as one peak in the Fourier transform recurrence spectrum of the quantum mechanical density of states or the transition spectrum. The dynamics is profoundly changed when the symmetry is broken by a (weak) external perturbation. The behavior of the dynamics and the corresponding semiclassical theory is described in [104]. In general, out of a continuous family of orbits only two closed or periodic orbits survive the symmetry breaking. Recently, quantum manifestations of symmetry breaking have been observed experimentally for atoms in external "elds: The cylindrical symmetry of the hydrogen atom in a magnetic "eld was broken in crossed magnetic and electric "elds [54], and a `temporal symmetry breakinga was studied on lithium atoms in an oscillating electric "eld [55,56]. However, the expected splitting of recurrence peaks cannot be observed directly because of the "nite resolution of the Fourier transform. This can be seen in Fig. 15 for the hydrogen atom in crossed "elds. Fig. 15 presents segments of the experimental recurrence spectra at constant scaled energy EI "!0.15 and scaled electric "eld strengths 04f"Fc\ 40.055. F is the electric "eld strength in atomic units (F "5.14;10 V/cm). For comparison, Fig. 15 shows the theoretical recurrence spectra obtained from semiclassical closed orbit theory [54]. Spectra have been Fourier transformed in the range 34.04w"c\ 461.7. The recurrence structure corresponds to the classical orbit in the plane perpendicular to the magnetic "eld axis drawn in Fig. 16. At vanishing electric "eld trajectories starting at the origin are exactly closed at each return to the nucleus (thin line in Fig. 16). For non-zero electric "eld, only two orbits with slightly di!erent periods (shown
Fig. 15. Segments of (a) experimental and (b) theoretical scaled action spectra for the hydrogen atom in crossed magnetic and electric "elds at constant scaled energy EI "!0.15 (from [54]).
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Fig. 16. (a) Classical trajectories for the hydrogen atom in crossed "elds in the plane perpendicular to the magnetic "eld axis. Scaled energy EI "!0.15 and "eld strength f"0.012. (b) Close-up of the returning part of the trajectories as they approach the nucleus (from [54]).
with heavy lines in Fig. 16) return exactly to the origin. A closeup of the returning part of the trajectories as they approach the nucleus at scaled energy EI "!0.15 and "eld strength f"0.012 is given in Fig. 16. The two closed orbits return to the nucleus diagonally; all others follow near-parabolic paths near the nucleus. In Fig. 15, no splitting of the recurrence peak can be observed due to the "nite resolution of the Fourier transform spectra. However, the amplitude of the peak changes with increasing electric "eld strength, and indeed in [54] the symmetry breaking was identi"ed indirectly by the constructive and destructive interference of the two orbits resulting in a Bessel function type modulation of amplitudes of recurrence peaks as a function of the strength of the symmetry breaking perturbation. We now want to apply harmonic inversion to directly uncover the splitting of recurrence peaks when symmetries are broken. As in [54] we investigate the hydrogen atom in crossed magnetic and electric "elds. SchroK dinger's equation for hydrogen in crossed "elds reads (2.59) [ p!(1/r)#c¸ #c(x#y)#Fx]W"EW , X with c and F being the magnetic and electric "eld strength (in atomic units), and E the energy. The eigenvalue problem Eq. (2.59) was solved numerically for "xed external "elds c and F in [96,105]. To study the e!ects of symmetry breaking on the quantum spectra we want to use the scaling properties of the classical system and to analyze spectra at constant scaled energy EI "Ec\ and scaled electric "eld f"Fc\ . Eigenvalues are obtained for the scaling parameter w"c\ . Introducing dilated coordinates rM "c r, Eq. (2.59) reads w[ pN ]W#w[¸ ]W#[(xN #yN )#2fxN !2EI !(2/rN )]W"0 . (2.60) X The peculiarity of Eq. (2.60) is that it is a quadratic (instead of linear) eigenvalue equation for the scaling parameter w, which cannot be solved straightforwardly with standard diagonalization routines for linear eigenvalue problems. To solve the quadratic SchroK dinger equation Eq. (2.60) we use the following technique. We write w"j"j #*j ,
(2.61)
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with j "xed and assume that *j is small. Now the scaling parameter w in the paramagnetic term of Eq. (2.60) can be approximated by w"(j+(j #(1/2(j )*j . (2.62) Replacing w in Eq. (2.60) with this approximation yields a generalized linear eigenvalue problem for *j,
1 2 1 j pN #(j ¸ # (xN #yN )#2fxN !2EI ! W"!*j pN # ¸ W, (2.63) X 4 rN 2(j X which can be solved numerically with the spectral transformation Lanczos method (STLM) [106]. The numerical details for the diagonalization of the linearized SchroK dinger equation Eq. (2.63) in a complete set of Sturmian-type basis functions are similar to the diagonalization of Eq. (2.59) at constant "eld strengths as described in [96]. It is convenient to choose identical parameters for both j and the center of the spectral transformation, so that the STLM method provides eigenvalues in the local neighborhood of j+j , i.e., *j is small. However, the obtained eigenvalues are still not very precise because of the approximation Eq. (2.62). In a second step we therefore apply perturbation theory and use the eigenvectors "W2 of Eq. (2.63) to solve the quadratic equation Eq. (2.60) separately for each eigenvalue, w , in the corresponding one-dimensional L subspace, i.e.
2 1 w 1W " pN "W 2#w 1W "¸ "W 2# W (xN #yN )#2fxN !2EI ! W "0 . L4 L L L L L X L rN L
(2.64)
By this procedure the accuracy of the eigenvalues is signi"cantly improved. An alternative method for the exact solution of the quadratic eigenvalue equation (2.60) is described in [107]. We now investigate the symmetry breaking in the hydrogen atom at constant scaled energy EI "!0.5. Without electric "eld (f"0) the two shortest closed orbit recurrences are the perpendicular orbit at s/2p"0.872 and the parallel orbit at s/2p"1. When the cylindrical symmetry is broken by a crossed electric "eld, all three-dimensional orbits should split into two nearby peaks. The only exception is the orbit parallel to the magnetic "eld axis, which does not appear as a continuous family of closed orbits. To verify the symmetry breaking in quantum spectra we calculated the photoabsorption spectrum (transitions from the initial state "2p02 to the "nal states with even z-parity) up to w"50 for the crossed "eld atom at constant scaled energy EI "!0.5 and scaled "eld strength f"0.02. The interesting part of the resulting recurrence spectrum obtained by both the conventional Fourier transform and the high-resolution harmonic inversion technique are presented in Fig. 17. The conventional Fourier transform (smooth line) shows two peaks around s/2p"0.87 (the perpendicular orbit) and s/2p"1 (the parallel orbit). However, none of the peaks is split. The high-resolution recurrence spectrum obtained by harmonic inversion is drawn as solid sticks and crosses in Fig. 17 and clearly exhibits a splitting of the recurrence peak of the perpendicular orbit at s/2p+0.87. For comparison the semiclassical recurrence spectrum is presented as dashed sticks and squares. In the plane perpendicular to the magnetic "eld axis two closed orbits have been found with slightly di!erent classical actions s /2p"0.864 and s /2p"0.881. The shapes of these orbits are illustrated as insets in Fig. 17. As can be seen the semiclassical and the high-resolution quantum recurrence spectrum are in excellent agreement,
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Fig. 17. Recurrence spectrum for the photoabsorption cross section of the hydrogen atom in crossed magnetic and electric "elds at scaled energy EI "Ec\ "!0.5 and "eld strength f"Fc\ "0.02. Transition "2p02P"n "#12. X Smooth line: conventional Fourier transform. Solid stick spectrum and crosses: high resolution quantum recurrence spectrum. Dashed sticks and squares: recurrence spectrum from semiclassical closed orbit theory. The two recurrence peaks around s/2p"0.87 are identi"ed by closed orbits in the (x,y) plane presented as insets in the "gure (from [79]).
i.e., the symmetry breaking in crossed "elds has been directly uncovered by harmonic inversion of the quantum mechanical photoabsorption spectrum. 2.6. expansion of the periodic orbit sum In the previous sections we have introduced harmonic inversion as a powerful method for the high-resolution analysis of quantum spectra, which allows a direct and quantitative comparison of the quantum spectra with semiclassical theories. However, the excellent agreement to many signi"cant digits for both the periods and amplitudes of quantum mechanical recurrence peaks with the semiclassical periodic orbit contributions (see Section 2.3) may be surprising for the following reason. Periodic orbit theory is exact only for a special class of systems, e.g., the geodesic motion on a surface with constant negative curvature [8]. In general, the semiclassical periodic orbit sum is only the leading order contribution of an in"nite series in powers of the Planck constant [22}24]. Therefore, it might be expected that only the high-resolution analysis of semiclassical spectra yields perfect agreement with periodic orbit theory while the harmonic inversion of quantum spectra should show small but noticeable deviations from the semiclassical recurrence spectra. As will be shown in the following, the absence of such deviations is related to the functional form of the expansion of the periodic orbit sum and special properties of the harmonic inversion method. In scaling systems, where the classical action of periodic orbits scales as S / "s / "s w , the scaling parameter plays the role of an e!ective Planck constant, i.e. w, \ ,
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and the expansion of the periodic orbit sum can therefore be written as a power series in w\. The #uctuating part of the semiclassical response function is given by 1 ALe QU . (2.65) g(w)" g (w)" L wL L L The AL are the complex amplitudes of the nth-order periodic orbit contributions including phase information from the Maslov indices. When quantum spectra are analyzed in the semiclassical regime, i.e., at su$ciently high scaling parameter w, the higher-order correction terms (n51) of the series Eq. (2.65) are certainly small compared to the zeroth-order terms. However, the reason why the higher-order contributions are not uncovered by the harmonic inversion method is that only the zeroth-order (n"0) terms in Eq. (2.65) ful"ll the ansatz Eq. (2.14) required for harmonic inversion, i.e., they are exponential functions with constant frequencies and amplitudes. The higher order terms (n51) have amplitudes decreasing &w\L with increasing scaling parameter, and thus do not ful"ll the ansatz Eq. (2.14). Therefore, only the zeroth-order amplitudes A can be obtained as converged parameters from the high resolution harmonic inversion analysis of the quantum spectra. The terms with n51 have similar properties as weak `noisea [70] and are separated by the harmonic inversion method from the `truea signal. In other words, the zerothorder approximation of the periodic orbit sum Eq. (2.65) with amplitudes A given by Gutzwil ler's trace formula is the best "t to the quantum spectra within the given ansatz as a linear superposition of exponential functions of the scaling parameter w, and the corresponding parameters are obtained from the harmonic inversion procedure. However, the higher-order terms of the expansion Eq. (2.65) can be revealed by harmonic inversion as will be demonstrated in the following. The periodic orbit terms AL can be obtained provided that the quantum spectrum and the (n!1)th-order eigenvalues w are given. We can I L\ then calculate the di!erence between the exact quantum mechanical and the (n!1)th-order response function L\ 1 AHe QU . g (w)! g (w)" g (w)" H H wH H HL HL The leading-order terms in Eq. (2.66) are &w\L, i.e., multiplication with wL yields
(2.66)
1 L\ wL g (w)! g (w) " ALe QU#O . (2.67) H w H In Eq. (2.67) we have restored the functional form Eq. (2.14), i.e., a linear superposition of exponential functions of w. The harmonic inversion of the function Eq. (2.67) will now provide the periods s and the nth-order amplitudes AL of the expansion Eq. (2.65). We will illustrate the method on two di!erent examples, i.e., the circle billiard and the three-disk scattering problem. 2.6.1. The circle billiard The circle billiard is an integrable and even separable bound system, and has been chosen here mainly for the sake of simplicity, since all the relevant physical quantities, i.e., the quantum and semiclassical eigenenergies, and the periodic orbits can easily be obtained. In polar coordinates
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(o, ) and after separation of the -motion SchroK dinger's equation reads
m
1 R R o ! W (o)"EW (o) , ! K 2M o Ro Ro o K
(2.68)
with M the mass of the particle and m the angular momentum quantum number. The wave functions must ful"ll the boundary condition W (R)"0 with R the radius of the circle billiard. K De"ning E" k/2M and after substitution of z,ko SchroK dinger's equation Eq. (2.68) is transformed into the di!erential equation for the Bessel functions [108] zJ (z)#zJ (z)#(z!m)J (z)"0 . (2.69) K K K The quantum mechanical eigenvalues k are obtained from the boundary condition J (k R)"0 L K K L K as zeros of the Bessel functions. We calculated numerically the "rst 32 469 eigenenergies of the circle billiard in the region k R(510. Note that states with mO0 are twofold degenerate. L K The semiclassical eigenenergies are obtained from an EBK torus quantization or, after separation of the -motion, even more simply from the one-dimensional WKB quantization of the radial motion in the centrifugal potential, i.e.
0
0 3 (2.70) p do"2 k (1!(m/ko) do"2p n# , M 4 K I K I with n"0,1,2,2 the radial quantum number. The r.h.s. of Eq. (2.70) takes into account the correct boundary conditions of the semiclassical wave functions at the classical turning points. The zeroth-order semiclassical eigenenergies are "nally obtained from the quantization condition 2
kR(1!(m/kR)!"m"arccos
"m" 3 "p n# . kR 4
(2.71)
The semiclassical spectrum has been calculated with the help of Eq. (2.71) in the same region k R(510 as the exact quantum spectrum. L K For the comparison of the spectra with periodic orbit theory we need to calculate the periodic orbits and their physical quantities. For the circle billiard all quantities are obtained analytically. In the following we choose R"1. The periodic orbits of the circle billiard are those orbits for which the angle between two bounces is a rational multiple of 2p, i.e., the periods l are obtained from the condition l "2m sin c ,
(2.72)
with c,pm /m , ( m "1,2,2 the number of turns of the orbit around the origin, and m "2m , 2m #1,2 the ( ( ( number of re#ections at the boundary of the circle. Periodic orbits with m O2m can be traversed ( in two directions and thus have multiplicity 2. Because the classical dynamics of the circle billiard is
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regular the Berry}Tabor formula [25] must be applied instead of Gutzwiller's trace formula for the calculation of the semiclassical density of states. We obtain 1 1 g (k)" g(k)" ALe lI , L kL (k L (k L 1
(2.73)
which basically di!ers from Eq. (2.65) by a factor of k\ on the l.h.s. of Eq. (2.73). [Note that for billiard systems the scaling parameter is the absolute value of the wave vector, w,k"" p"/ , and the action is proportional to the length of the orbit, S " kl .] The zeroth-order periodic orbit amplitudes obtained from the Berry}Tabor formula read
p l e\ p I>p , (2.74) A" 2 m with k "3m the Maslov index. For the calculation of the "rst-order periodic orbit amplitudes A in Eq. (2.73) we adopt the method of Alonso and Gaspard [23]. After a lengthy calculation we "nally obtain 1 A" (pm 2
e\ p I\p .
(2.75)
A detailed derivation of Eqs. (2.74) and (2.75) will be given elsewhere [87,88]. When the quantum mechanical density of states, or, more precisely, the spectrum k\ . (k) is analyzed by the harmonic inversion method the periodic orbit quantities l and A are obtained to very high precision [87]. However, we are now interested in the expansion of the periodic orbit sum and want to verify the "rst-order corrections A directly in the quantum spectrum. We therefore analyze the di!erence spectrum *. (k)". (k)!. (k) between the quantum and the semiclassical density of states. A small part of this spectrum at k+200 is presented in Fig. 18. The absolute values of the peak heights mark the multiplicities of the states. To restore the functional form which is required for the harmonic inversion procedure the di!erence spectrum
Fig. 18. Part of the di!erence spectrum *. (k)". (k)!. (k) between the quantum and the semiclassical density of states for the circle billiard with radius R"1.
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Fig. 19. First-order correction terms of the semiclassical periodic orbit sum for the circle billiard. Solid line: semiclassical theory. Crosses: harmonic inversion analysis of the di!erence spectrum k [. (k)!. (k)]. The periodic orbits are marked by the numbers (m ,m ). (
was multiplied by (k, and the resulting signal (k*. (k) was analyzed in the region 100(k(500. The results are presented in Fig. 19. For a direct comparison with Eq. (2.75) we transform for each periodic orbit the obtained period l and the "rst-order amplitude A into the quantities c"arcsin(l /2m ) and f(c),(2/(pm )"A" . These quantities are plotted as crosses in Fig. 19. The periodic orbits are marked by the numbers (m ,m ). The solid line ( f (c)"(5!2 sin c)/3 sin c is the result of Eq. (2.75). As can be seen the theoretical curve and the crosses obtained by harmonic inversion of the di!erence between the quantum and semiclassical density of states are in excellent agreement. The semiclassical accuracy of the zeroth-order eigenenergies of the circle billiard has been discussed in [109,110] in terms of, e.g., the average error in units of the mean level spacing. The analysis presented here provides a more physical interpretation of the deviations between the quantum and semiclassical eigenenergies in terms of the higher-order corrections of the expanded periodic orbit sum Eq. (2.65). The direct application of the series Eq. (2.65) for the semiclassical quantization beyond the Gutzwiller and Berry}Tabor approximation will be discussed in Section 3.6. 2.6.2. The three-disk scattering system As a second example for the investigation of the expansion of the periodic orbit sum we now consider a billiard system consisting of three identical hard disks with unit radius R"1, displaced from each other by the same distance, d. The classical, semiclassical, and quantum dynamics of this scattering system has been studied by Gaspard and Rice [111]. In recent years the system has served as a prototype model for periodic orbit quantization by cycle expansion techniques [9}11,14,15]. If the disks are separated by a distance d'2.0481419 there is a one-to-one identity between the periodic orbits and a symbolic code, whereas for d(2.04821419 pruning of orbits sets in [112]. The geometry of the three-disk scattering system is shown in Fig. 20. The symbolic code
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Fig. 20. The scattering geometry for the three-disk system. (a) The three disks with 12, 123, and 121313232 cycles indicated. (b) The fundamental domain, i.e., a wedge consisting of a section of a disk, two segments of symmetry axis acting as straight mirror walls, and an escape gap. The above cycles restricted to the fundamental domain are now the 0,1, and 100 cycle (from [9]).
of a periodic orbit is obtained by numbering of the disks, `1a, `2a, and `3a, and by bookkeeping the re#ections of the orbit at the three-disks. E.g., 121313232 is the symbolic code of the primitive orbit with cycle length n"9 drawn with a dashed line in Fig. 20. The bar, which is often omitted, indicates the periodicity of the orbit. The three-disk scattering system is invariant under the symmetry operations of the group C , i.e., three re#ections at symmetry lines and two rotations by T 2p/3 and 4p/3. Periodic orbits have p symmetry if they are invariant under re#ections and C ,C T symmetry if they are invariant under rotations. After symmetry decomposition the periodic orbits in the fundamental domain (see Fig. 20b) can be classi"ed by a binary symbolic code of symbols `0a and `1a, where each `0a represents a change between clockwise and anticlockwise scattering in the original three-disk system [9]. The symbolic code of the orbit 121313232 in Fig. 20a restricted to the fundamental domain is 100100100 or because of its periodicity even simpler 100. The quantum resonances are also classi"ed by symmetries. Resonances with A (A ) symmetry are symmetric (antisymmetric) under re#ections at a symmetry line, and resonances with E symmetry are invariant (up to a complex phase factor c with c"1) under rotations by 2p/3 and 4p/3. The resonances in the E subspace are twofold degenerate. In the following we analyze resonances with A symmetry. The three-disk scattering system does not have any bound states. However, the #uctuating part of the density of states can be written as 1 1 . (k)"! Im k!k p L L
(2.76)
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Fig. 21. Spectra of the three disk scattering system (A subspace) with radius R"1 and distance d"6 between the disks. (a) Fluctuating part of the quantum mechanical density of states. (b) Di!erence spectrum k[. (k)!. (k)] between the quantum and semiclassical density of states. (c) Quantum (#) and semiclassical (;) A resonances of the Green's function.
with k the complex resonances (poles) of the Green's function. For distance d"6, the quantum L mechanical A resonances have been calculated by Wirzba [14,15,113] in the region 0(Re k(250, and are plotted with the # symbols in Fig. 21c. The corresponding #uctuating part of the quantum density of states is shown in Fig. 21a. This spectrum can be analyzed by harmonic inversion to extract the lengths (periods) of the orbits and the zeroth-order amplitudes. However, as in the previous section for the circle billiard we here want to investigate the di!erence between the quantum and semiclassical spectrum. The semiclassical A resonances have been calculated by Wirzba using the 12th order in the curvature expansion of the Gutzwiller}Voros zeta-function [14,15,113] and are plotted by the crosses (;) in Fig. 21c. The di!erences between the quantum and semiclassical resonances are usually much smaller than the size of the symbols in Fig. 21b and are visible only at small values of Re k. However, the deviations become clearly pronounced in Fig. 21b, which presents the di!erence between the quantum and zeroth-order semiclassical density of states
1 1 . k[. (k)!. (k)]"! Im Ae lI#O k p
(2.77)
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Table 3 Periodic orbit quantities for the three disk scattering system with R"1, d"6. P: symbolic code; ¸ and A/A: . periods and ratio of zeroth- and "rst-order amplitudes obtained by harmonic inversion of spectra (hi) and by the classical calculations (cl) of Alonso and Gaspard in [23] P
¸ .
¸ .
"A/A"
"A/A"
0 1 01 001 011 0001 0011 0111 00001 00011 00101 00111 01011 01111
4.000498 4.268069 8.316597 12.321782 12.580837 16.322308 16.585261 16.849133 20.322343 20.585725 20.638284 20.853593 20.897413 21.117009
4.000000 4.267949 8.316529 12.321747 12.580808 16.322276 16.585243 16.849072 20.322330 20.585690 20.638238 20.853572 20.897369 21.116994
0.3135 0.5554 1.0172 1.3465 1.5558 1.6584 1.8514 2.1172 1.9961 2.1682 2.4061 2.4387 2.6127 2.6764
0.31250 0.56216 1.01990 1.35493 1.55617 1.67009 2.12219
2.45127
Eq. (2.77) implies that by harmonic inversion of the spectrum in Fig. 21b we can extract the "rst-order amplitudes A of the expansion of the periodic orbit sum. In Table 3 we present the ratio of the zeroth- and "rst-order amplitudes, "A/A" obtained by harmonic inversion of the spectra in Fig. 21a and Fig. 21b for all periodic orbits up to cycle length 5 in the symmetry reduced symbolic code [9]. These quantities have been calculated by Alonso and Gaspard [23] and we present their results "A/A" in Table 3 for comparison. In [23] the periodic orbits have not been symmetry reduced and the set of orbits chosen by Alonso and Gaspard is complete only up to cycle length 3 in the symmetry reduced symbolic code. However, Table 3 clearly illustrates a very good agreement between the amplitudes obtained by the harmonic inversion analysis of spectra and the results of [23]. We have here obtained information about the "rst-order terms of the expansion of the periodic orbit sum from the analysis of the quantum and the lowest (zeroth)-order semiclassical resonances. In general, the quantum spectrum and its nth-order semiclassical approximation are required to extract information about the (n#1)th-order terms of the expansion series Eq. (2.65). 3. Periodic orbit quantization by harmonic inversion Since the development of periodic orbit theory by Gutzwiller [7,8] it has become a fundamental question as to how individual semiclassical eigenenergies and resonances can be obtained from periodic orbit quantization for classically chaotic systems. A major problem is the exponential proliferation of the number of periodic orbits with increasing period, resulting in a divergence of
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Gutzwiller's trace formula at real energies and below the real axis, where the poles of the Green's function are located. The periodic orbit sum is a Dirichlet series g(w)" A e QLU , L L
(3.1)
where the parameters A and s are the amplitudes and periods (actions) of the periodic orbit L L contributions. In most applications, Eq. (3.1) is absolutely convergent only in the region Im w'c'0 with c the entropy barrier of the system, while the poles of g(w), i.e., the bound states and resonances, are located on and below the real axis, Im w40. Thus, to extract individual eigenstates, the semiclassical trace formula Eq. (3.1) has to be analytically continued to the region of the quantum poles. Up to now no general procedure is known for the analytic continuation of a non-convergent Dirichlet series of the type of Eq. (3.1). All existing techniques are restricted to special situations. For bound and ergodic systems the semiclassical eigenenergies can be extracted with the help of a functional equation and the mean staircase function (Weyl term), resulting in a Riemann}Siegel look-alike formula [59}61]. Alternative semiclassical quantization conditions based on a semiclassical representation of the spectral staircase [65,66] and derived from a quantum version of a classical PoincareH map [63] are also restricted to bound and ergodic systems. For systems with a symbolic dynamics the periodic orbit sum Eq. (3.1) can be reformulated as an in"nite Euler product, which can be expanded in terms of the cycle length of the symbolic code. If the contributions of longer orbits are shadowed by the contributions of short orbits the cycle expansion technique can remarkably improve the convergence properties of the series and allows to extract the bound states and resonances of bound and open systems, respectively [9}11,58]. A combination of the cycle expansion technique with a functional equation for bound systems has been studied by Tanner et al. [114]. However, the existence of a simple symbolic code is restricted to very few systems, and cycle expansion techniques cannot be applied, e.g., to the general class of systems with mixed regular-chaotic classical dynamics. In this section we present a general technique for the analytic continuation and the extraction of poles of a non-convergent series of the type of Eq. (3.1). The method is based on harmonic inversion by "lter diagonalization. The advantage of the method is that it does not depend on special properties of the system such as ergodicity or the existence of a symbolic dynamics for periodic orbits. It does not even require the knowledge of the mean staircase function, i.e., the Weyl term in dynamical systems. The only assumption we have to make is that the analytic continuation of the Dirichlet series g(w) (Eq. (3.1)) is a linear combination of poles (w!w )\, which is exactly the I functional form of, e.g., a quantum mechanical response function with real and complex parameters w representing the bound states and resonances of the system, respectively. To demonstrate the I general applicability and accuracy of our method we will apply it to three systems with completely di!erent properties, "rst the zeros of the Riemann zeta function [115,116], as a mathematical model for a bound system, second the three-disk scattering system as a physical example of an open system with classically chaotic dynamics, and third the circle billiard as an integrable system. As pointed out by Berry [117] the density of zeros of Riemann's zeta function can be written, in formal analogy with Gutzwiller's semiclassical trace formula, as a non-convergent series, where the `periodic orbitsa are the prime numbers. A special property of this system is the existence of a functional equation which allows the calculation of Riemann zeros via the Riemann}Siegel
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formula [115}118]. An analogous functional equation for quantum systems with an underlying chaotic (ergodic) classical dynamics has served as the basis for the development of a semiclassical quantization rule for bound ergodic systems [59}61]. The Riemann zeta function has also served as a mathematical model to study the statistical properties of level distributions [118}120]. We will demonstrate in Section 3.1 that harmonic inversion can reveal the Riemann zeros with extremely high accuracy and with just prime numbers as input data. The most important advantage of our method is, however, its wide applicability, i.e., it can be generalized in a straightforward way to non-ergodic bound or open systems, and the procedure for periodic orbit quantization by harmonic inversion will be discussed in Section 3.2. As an example of periodic orbit quantization of a physical system we investigate in Section 3.3 the three disk scattering problem, which is an open and non-ergodic system. Its classical dynamics is purely hyperbolic, and the periodic orbits can be classi"ed by a complete binary symbolic code. This system has served as the prototype for the development of cycle expansion techniques [9}11]. When applying the harmonic inversion technique to the three disk scattering system we will highlight the general applicability of our method by not having to make use of its symbolic dynamics in any way. The power of the method will be illustrated in Section 3.4 on a challenging physical system which has not been solved previously with any semiclassical quantization technique. The hydrogen atom in a magnetic "eld shows a transition from near-integrable to chaotic classical dynamics with increasing excitation energy. We apply periodic orbit quantization by harmonic inversion to the hydrogen atom in the mixed regular-chaotic regime. An important question is the e$ciency of methods for periodic orbit quantization, i.e., the number of periodic orbits required for the calculation of a certain number of poles of the response function g(w). It is evident that methods invoking special properties of a given system may be remarkably e$cient. E.g., the Riemann}Siegel-type formulae [59}61] require the periodic orbits up to a maximum period which is by about a factor of four shorter compared to the required signal length for the harmonic inversion technique discussed in Section 3.2. We therefore propose in Section 3.5 an extension of the harmonic inversion method to the harmonic inversion of crosscorrelated periodic orbit sums. The method uses additional semiclassical information obtained from a set of linearly independent smooth observables and allows to signi"cantly reduce the number of periodic orbits and thus to improve the e$ciency of periodic orbit quantization by harmonic inversion. Periodic orbit theory yields exact eigenenergies only in exceptional cases, e.g., for the geodesic motion on the constant negative curvature surface [8]. As already discussed in Section 2.6, Gutzwiller's periodic orbit sum is, in general, just the leading order term of an in"nite series in powers of the Planck constant. Methods for the calculation of the higher order periodic orbit contributions were developed in [22}24]. In Section 3.6 we demonstrate how the higher-order
corrections of the periodic orbit sum can be used to improve the accuracy of the semiclassical eigenenergies, i.e., to obtain eigenenergies beyond the standard semiclassical approximation of periodic orbit theory. Both methods, the harmonic inversion of cross-correlated periodic orbit sums, which allows to signi"cantly reduce the required number of periodic orbits for semiclassical quantization, and the calculation of eigenenergies beyond the lowest order approximation will be illustrated in Section 3.7 by way of example of the circle billiard. Furthermore, for this system we will calculate
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semiclassically the diagonal matrix elements of various operators. The circle billiard is an integrable system and therefore the Berry}Tabor formula for integrable systems [25,26] is valid in this case rather than Gutzwiller's trace formula. However, periodic orbit quantization by harmonic inversion can be applied for both regular and chaotic systems as well which demonstrates the universality and wide applicability of the method. Finally, in Section 3.8 we will calculate semiclassically the photoabsorption spectra of atoms in external "elds. Applying a combination of closed orbit theory [90}93], the cross-correlation approach of Section 3.5, and the harmonic inversion method we obtain individual non-diagonal transition matrix elements between low-lying initial states and strongly perturbed Rydberg states of the magnetized hydrogen atom. 3.1. A mathematical model: Riemann1s zeta function Our goal is to introduce our method for periodic orbit quantization by harmonic inversion using, as an example, the well de"ned problem of calculating zeros of the Riemann zeta function. There are essentially two advantages of studying the zeta function instead of a `reala physical bound system. First, the Riemann analogue of Gutzwiller's trace formula is exact, as is the case for systems with constant negative curvature [8,66], whereas the semiclassical trace formula for systems with plane geometry is correct only to "rst order in . This allows a direct check on the precision of the method. Second, no extensive periodic orbit search is necessary for the calculation of Riemann zeros, as the only input data are just prime numbers. It is not our intention to introduce yet another method for computing Riemann zeros, which, as an objective in its own right, can be accomplished more e$ciently by speci"c procedures. Rather, in our context the Riemann zeta function serves primarily as a mathematical model to illustrate the power of our technique when applied to bound systems. 3.1.1. General remarks Before discussing the harmonic inversion method we start with recapitulating a few brief remarks on Riemann's zeta function necessary for our purposes. The hypothesis of Riemann is that all the non-trivial zeros of the analytic continuation of the function f(z)" n\X" (1!p\X)\ (Re z'1, p: primes) L N
(3.2)
have real part , so that the values w"w , de"ned by I f(!iw )"0 , I
(3.3)
are all real or purely imaginary [115,116]. The Riemann staircase function for the zeros along the line z"!iw, de"ned as N(w)" H(w!w ) , I I
(3.4)
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i.e. the number N(w) of zeros with w (w, can be split [115}117] into a smooth part, I w 1 1 1 NM (w)" arg C # iw ! ln p#1 2p 4 2 p
7 1 7 w w ln !1 # # ! #O(w\) , " 8 48pw 5760pw 2p 2p
(3.5)
and a #uctuating part,
1 1 (3.6) N (w)"! lim Im ln f !i(w#ig) . 2 p E Substituting the product formula Eq. (3.2) (assuming that it can be used when Re z") into Eq. (3.6) and expanding the logarithms yields 1 1 N (w)"! Im e UK N . mpK p N K Therefore, the density of zeros along the line z"!iw can formally be written as dN 1 . (w)" "! Im g(w) dw p
(3.7)
(3.8)
with the response function g(w) given by the series ln(p) e UK N , (3.9) g(w)"i pK N K which converges only for Im w'. Obviously, Eq. (3.9) is of the same type as the response function Eq. (3.1), with the entropy barrier c", i.e., Eq. (3.9) does not converge on the real axis, where the Riemann zeros are located. The mathematical analogy between the above equation and Gutzwiller's periodic orbit sum 1 (E)+! Im A e 1 , (3.10) p with A the amplitudes and S the classical actions (including phase information) of the periodic orbit contributions, was already pointed out by Berry [117,59]. For the Riemann zeta function the primitive periodic orbits have to be identi"ed with the primes p, and the integer m formally counts the `repetitionsa of orbits. The `amplitudesa and `actionsa are then given by
.
A "i (ln(p)/pK ) , (3.11) NK S "mw ln(p) . (3.12) NK Both Eq. (3.9) for the Riemann zeros and } for most classically chaotic physical systems } the periodic orbit sum Eq. (3.10) do not converge. In particular, zeros of the zeta function, or
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semiclassical eigenstates, cannot be obtained directly using these expressions. The problem is to "nd the analytic continuation of these equations to the region where the Riemann zeros or, for physical systems, the eigenenergies and resonances, are located. Eq. (3.9) is the starting point for our introduction and discussion of the harmonic inversion technique for the example of the Riemann zeta function. The generalization of the method to periodic orbit quantization (Eq. (3.10)) in Section 3.2 will be straightforward. Although Eq. (3.9) is the starting point for the harmonic inversion method, for completeness we quote the Riemann}Siegel formula, which is the most e$cient approach to computing Riemann zeros. For the Riemann zeta function it follows from a functional equation [115] that the function Z(w)"exp+!i[arg C(#iw)!w ln p],f(!iw)
(3.13)
is real, and even for real w. The asymptotic representation of Z(w) for large w, ' (U p cos+pNM (w)!w ln n, Z(w)"!2 n L ! (!1)' (U p
2p cos(2p(t!t!1/16)) #2 , w cos(2pt)
(3.14)
with t"(w/2p!Int[(w/2p] is known as the Riemann}Siegel formula and has been employed (with several more correction terms) in e!ective methods for computing Riemann zeros [118]. Note that the principal sum in Eq. (3.14) has discontinuities at integer positions of (w/2p, and therefore the Riemann zeros obtained from the principal sum are correct only to about 1}15% of the mean spacing between the zeros. The higher order corrections to the principal Riemann}Siegel sum remove, one by one, the discontinuities in successive derivatives of Z(w) at the truncation points and are thus essential to obtaining accurate numerical results. An alternative method for improving the asymptotic representation of Z(w) by smoothing the cuto!s with an error function and adding higher-order correction terms is presented in [61]. An analogue of the functional equation for bound and ergodic dynamical systems has been used as the starting point to develop a `rule for quantizing chaosa via a `Riemann}Siegel look-alike formulaa [59}61]. This method is very e$cient as it requires the least number of periodic orbits, but unfortunately it is restricted to ergodic systems on principle reasons, and cannot be generalized either to systems with regular or mixed classical dynamics or to open systems. By contrast, the method of harmonic inversion does not have these restrictions. We will demonstrate that Riemann zeros can be obtained directly from the `ingredientsa of the non-convergent response function Eq. (3.9), i.e., the set of values A and NK S , thus avoiding the use of the functional equation, the Riemann}Siegel formula, the mean NK staircase function Eq. (3.5), or any other special property of the zeta function. The comparison of results in Section 3.1.3 will show that the accuracy of our method goes far beyond the Riemann}Siegel formula Eq. (3.14) without higher order correction terms. The main goal of this section is to demonstrate that because of the formal equivalence between Eqs. (3.9) and (3.10) our method can then be applied to periodic orbit quantization of dynamical systems in Section 3.2 without any modi"cation.
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3.1.2. The ansatz for the Riemann zeros To "nd the analytic continuation of Eq. (3.9) in the region Im w( we essentially wish to "t g(w) to its exact functional form, d I , (3.15) g (w)" w!w #i0 I I arising from the de"nition of the Riemann staircase Eq. (3.4). The `multiplicitiesa d in Eq. (3.15) I are formally "tting parameters, which here should all be equal to 1. It is hard to directly adjust the non-convergent (on the real axis) series g(w) to the form of g (w). The "rst step towards the solution of the problem is to carry out the adjustment for the Fourier components of the response function,
ln(p) 1 > d(s!m ln(p)) , (3.16) g(w)e\ QUdw"i C(s)" pK 2p \ N K which after certain regularizations (see below) is a well-behaved function of s. Due to the formal analogy with the results of periodic orbit theory (see Eqs. (3.11) and (3.12)), C(s) can be interpreted as the recurrence function for the Riemann zeta function, with the recurrence positions s "S /w"m ln(p) and recurrence strengths of periodic orbit returns A "i ln(p)p\K . The NK NK NK exact functional form which now should be used to adjust C(s) is given by
1 > g (w)e\ QUdw"!i d e\ UIQ . (3.17) C (s)" I 2p \ I C (s) is a superposition of sinusoidal functions with frequencies w given by the Riemann zeros and I amplitudes d "1. (It is convenient to use the word `frequenciesa for w referring to the sinusoidal I I form of C(s). We will also use the word `polesa in the context of the response function g(w).) Fitting a signal C(s) to the functional form of Eq. (3.17) with, in general, both complex frequencies w and amplitudes d is known as harmonic inversion, and has already been introduced I I in Section 2.2 for the high-resolution analysis of quantum spectra. The harmonic inversion analysis is especially non-trivial if the number of frequencies in the signal C(s) is large, e.g., more than a thousand. It is additionally complicated by the fact that the conventional way to perform the spectral analysis by studying the Fourier spectrum of C(s) will bring us back to analyzing the non-convergent response function g(w) de"ned in Eq. (3.9). Until recently the known techniques of spectral analysis [67] would not be applicable in the present case, and it is the "lter-diagonalization method [68}70] which has turned the harmonic inversion concept into a general and powerful computational tool. The signal C(s) as de"ned by Eq. (3.16) is not yet suitable for the spectral analysis. The next step is to regularize C(s) by convoluting it with a Gaussian function to obtain the smoothed signal,
i ln(p) > 1 e\Q\K N N (3.18) C(s )e\Q\QY N ds " C (s)" N pK (2pp N K (2pp \ that has to be adjusted to the functional form of the corresponding convolution of C (s). The latter is readily obtained by substituting d in Eq. (3.17) by the damped amplitudes, I (3.19) d PdN"d e\UIN . I I I
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The regularization Eq. (3.18) can also be interpreted as a cut of an in"nite number of high frequencies in the signal which is of fundamental importance for numerically stable harmonic inversion. Note that the convolution with the Gaussian function is no approximation, and the obtained frequencies w and amplitudes d corrected by Eq. (3.19) are still exact, i.e., do not depend I I on p. The convolution is therefore not related to the Gaussian smoothing devised for Riemann zeros in [121] and for quantum mechanics in [122], which provides low resolution spectra only. The next step is to analyze the signal Eq. (3.18) by harmonic inversion. The concept of harmonic inversion by "lter-diagonalization has already been explained in Section 2.2 and the technical details are given in Appendix A.1. Note that even though the derivation of Eq. (3.18) assumed that the zeros w are on the real axis, the analytic properties of C (s) imply that its representation by I N Eq. (3.18) includes not only the non-trivial real zeros, but also all the trivial ones, w "!i(2k#), k"1,2,2, which are purely imaginary. The general harmonic inversion proceI dure does not require the frequencies to be real. Both the real and imaginary zeros w will be I obtained as the eigenvalues of the non-Hermitian generalized eigenvalue problem, Eq. (A.7) in Appendix A.1. 3.1.3. Numerical results For a numerical demonstration we construct the signal C (s) using Eq. (3.18) in the region N s(ln(10)"13.82 from the "rst 78498 prime numbers and with a Gaussian smoothing width p"0.0003. Parts of the signal are presented in Fig. 22. Up to s+8 the Gaussian approximations to the d-functions do essentially not overlap (see Fig. 22a), whereas for s<8 the mean spacing *s between successive d-functions becomes much less than the Gaussian width p"0.0003 and the signal #uctuates around the mean CM (s)"ieQ (see Fig. 22b). From this signal we were able to calculate about 2600 Riemann zeros to at least 12-digit precision. For the small generalized eigenvalue problem (see Eq. (A.7) in Appendix A.1) we used matrices with dimension J(100. Some Riemann zeros w , the corresponding amplitudes d , and the estimated errors e (see Eq. (A.12) I I in Appendix A.1) are given in Table 4 and Table 5. The pole of the zeta function yields the smooth background CM (s)"ieQ of the signal C(s) (see the dashed line in Fig. 22). Within the numerical error the Riemann zeros are real and the amplitudes are consistent with d "1 for non-degenerate I
Fig. 22. `Recurrencea function !iC (s) for the Riemann zeros which has been analyzed by harmonic inversion. (a) N Range 04s46, (b) short range around s"13. The d-functions have been convoluted by a Gaussian function with width p"0.0003. Dashed line: Smooth background CM (s)"ieQ resulting from the pole of the zeta function at w"i/2 (from [83]).
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Table 4 Non-trivial zeros w , multiplicities d , and error estimate e for the Riemann zeta function I I k
Re w I
Im w I
Re d
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
14.13472514 21.02203964 25.01085758 30.42487613 32.93506159 37.58617816 40.91871901 43.32707328 48.00515088 49.77383248 52.97032148 56.44624770 59.34704400 60.83177852 65.11254406 67.07981053 69.54640171 72.06715767 75.70469070 77.14484007 79.33737502 82.91038085 84.73549298 87.42527461 88.80911121 92.49189927 94.65134404 95.87063426 98.83119422 101.31785101
4.05E!12 !2.23E!12 1.66E!11 !6.88E!11 7.62E!11 1.46E!10 !3.14E!10 1.67E!11 4.35E!11 7.02E!11 1.92E!10 !1.30E!10 5.40E!11 !3.94E!10 !4.98E!09 !2.05E!10 2.51E!11 !5.74E!10 3.93E!10 !2.70E!12 !3.58E!11 1.56E!10 3.34E!10 1.20E!09 !9.42E!10 !4.11E!09 !7.11E!09 8.06E!09 !1.78E!11 4.22E!11
1.00000011 1.00000014 0.99999975 0.99999981 1.00000020 1.00000034 0.99999856 1.00000008 0.99999975 1.00000254 1.00000122 0.99999993 0.99999954 1.00000014 0.99998010 0.99999892 0.99999951 0.99999974 1.00000082 0.99999979 1.00000086 0.99999912 0.99999940 0.99999866 1.00000101 0.99999761 1.00000520 0.99999001 0.99999936 0.99999969
I
Im d I
e
!5.07E!08 1.62E!07 !2.64E!07 !1.65E!07 5.94E!08 5.13E!07 1.60E!06 3.29E!07 !1.35E!07 !4.59E!07 7.31E!07 4.51E!07 2.34E!06 1.11E!06 !8.30E!06 !8.04E!07 9.45E!07 8.63E!06 1.07E!06 1.25E!06 3.03E!07 !8.58E!07 !7.09E!07 1.39E!06 1.49E!06 !1.93E!06 !1.27E!06 !1.15E!05 5.70E!07 !4.73E!07
3.90E!13 9.80E!13 5.20E!12 1.90E!12 7.10E!13 1.00E!12 4.90E!11 1.90E!12 1.40E!12 1.10E!10 6.00E!11 5.50E!12 2.30E!10 3.00E!11 2.70E!08 5.30E!11 6.80E!12 4.30E!10 3.20E!11 1.30E!11 9.20E!12 1.60E!11 2.50E!11 6.70E!11 4.80E!11 1.50E!10 6.80E!10 5.20E!09 3.10E!12 4.50E!12
zeros. To fully appreciate the accuracy of our harmonic inversion technique we note that zeros obtained from the principal sum of the Riemann}Siegel formula Eq. (3.14) deviate by about 1}15% of the mean spacing from the exact zeros. Including the "rst correction term in Eq. (3.14) the approximations to the "rst "ve zeros read w "14.137, w "21.024, w "25.018, w "30.428, and w "32.933, which still signi"cantly deviates from the exact values (see Table 4). Considering even higher-order correction terms the results will certainly converge to the exact zeros. However, the generalization of such higher-order corrections to ergodic dynamical systems is a non-trivial task and requires, e.g., the knowledge of the terms in the Weyl series, i.e., the mean staircase function after the constant [61,123]. The perfect agreement of our results for the w with the exact I Riemann zeros to full numerical precision is remarkable and clearly demonstrates that harmonic inversion by "lter diagonalization is a very powerful and accurate technique for the analytic continuation and the extraction of poles of a non-convergent series such as Eq. (3.1).
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Table 5 Non-trivial zeros w , multiplicities d , and error estimate e for the Riemann zeta function I I k
Re w I
Im w I
Re d
2532 2533 2534 2535 2536 2537 2538 2539 2540 2541 2542 2543 2544 2545 2546 2547 2548 2549 2550 2551 2552 2553 2554 2555 2556 2557 2558 2559 2560 2561
3063.43508648 3065.28655558 3066.32025039 3067.07132023 3068.01350133 3068.98426618 3069.78290477 3070.54262154 3072.00099337 3073.18523777 3074.52349428 3075.03387288 3075.83347924 3077.42747330 3078.28622690 3078.89737915 3079.87139464 3080.85638233 3082.16316375 3083.36135798 3084.83845150 3085.37726898 3085.96552225 3087.01881535 3088.08343703 3089.22230894 3090.28219490 3091.15446969 3092.68766704 3093.18544571
!1.64E!09 1.15E!09 !1.66E!10 !3.68E!09 !1.51E!09 !5.92E!09 !4.40E!09 !7.71E!10 !6.44E!11 9.17E!11 6.73E!09 !1.22E!08 !3.13E!09 5.76E!10 1.34E!08 1.61E!09 1.70E!09 8.67E!10 !5.88E!10 8.43E!10 2.72E!09 !1.37E!08 6.39E!09 3.46E!11 !3.89E!10 !3.31E!10 2.97E!10 1.10E!09 2.25E!09 !2.33E!09
0.99999901 1.00000107 1.00000231 1.00000334 1.00000291 1.00000205 1.00000237 1.00000169 0.99999908 0.99999942 1.00000391 1.00000117 1.00000013 1.00000561 1.00001283 1.00000487 1.00000275 1.00000159 1.00000013 0.99999923 1.00000057 0.99999576 0.99999667 0.99999845 0.99999931 1.00000017 1.00000069 1.00000052 1.00000033 1.00000168
I
Im d I
e
1.34E!06 !3.63E!07 1.00E!06 2.73E!07 1.46E!06 2.94E!06 2.51E!06 9.57E!07 2.17E!07 !1.07E!06 !6.51E!07 !5.69E!06 !3.86E!06 4.69E!06 1.10E!06 !8.04E!06 !2.32E!06 !4.12E!07 8.44E!07 3.45E!07 !2.86E!06 !2.88E!06 1.50E!06 !3.63E!07 !8.44E!07 !9.21E!07 !7.17E!07 !6.59E!07 1.45E!06 !1.50E!07
5.50E!11 2.40E!11 1.20E!10 2.20E!10 2.10E!10 2.60E!10 2.40E!10 7.90E!11 2.00E!11 3.00E!11 3.50E!10 7.30E!10 3.10E!10 1.10E!09 3.80E!09 2.10E!09 3.00E!10 5.90E!11 1.70E!11 1.50E!11 1.80E!10 5.50E!10 2.80E!10 5.20E!11 2.40E!11 1.80E!11 2.10E!11 1.50E!11 5.20E!11 6.40E!11
A few w have been obtained (see Table 6) which are de"nitely not located on the real axis. I Except for the "rst at w"i/2 they can be identi"ed with the trivial real zeros of the zeta function at z"!2n, n"1,2,2 In contrast to the non-trivial zeros with real w , the numerical accuracy for I the trivial zeros decreases rapidly with increasing n. The trivial zeros w "!i(2n#) are the L analogue of resonances in open physical systems with widths increasing with n. The fact that the trivial Riemann zeros are obtained emphasizes the universality of our method and demonstrates that periodic orbit quantization by harmonic inversion can be applied not only to closed but to open systems as well. The decrease of the numerical accuracy for very broad resonances is a natural numerical consequence of the harmonic inversion procedure [69,70]. The value w"i/2 in Table 6 is special because in this case the amplitude is negative, i.e., d "!1. Writing the zeta function I
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Table 6 Trivial zeros and pole of the Riemann zeta function Re w I
Im w I
Re d
0.00000000 !0.00000060 !0.00129915 !0.09761173
0.50000000 !2.49999941 !4.49987911 !6.53286064
!1.00000002 0.99992487 1.00069939 1.07141445
I
Im d I
e
!4.26E!08 !3.66E!05 !3.25E!03 !1.49E!01
1.80E!14 1.80E!07 4.40E!05 1.70E!03
in the form [116]
1 !iw "C (w!w )BIA(w, w ) , (3.20) I I 2 I where C is a constant and A a regularizing function which ensures convergence of the product, integer values d are the multiplicities of zeros. Therefore, it is reasonable to relate negative integer I values with the multiplicities of poles. In fact, f(z) has a simple pole at z"!iw"1 consistent with w"i/2 in Table 6. f
3.1.4. Required signal length We have calculated Riemann zeros by harmonic inversion of the signal C (s) (Eq. (3.18)) which N uses prime numbers as input. The question arises what are the requirements on the signal C (s), in N particular what is the required signal length. In other words, how many Riemann zeros (or semiclassical eigenenergies) can be converged for a given set of prime numbers (or periodic orbits, respectively)? The answer can be directly obtained from the requirements on the harmonic inversion technique. In general, the required signal length s for harmonic inversion is related to
the average density of frequencies .N (w) by [70] (3.21) s +4p.N (w) ,
i.e., s is about two times the Heisenberg length
(3.22) s ,2p.N (w) . & From Eq. (3.21) the required number of primes (or periodic orbits) can be directly estimated as +C primes p " ln p(s , or +C periodic orbits " s (s ,. For the special example of the
Riemann zeta function the required number of primes to have a given number of Riemann zeros converged can be estimated analytically. With the average density of Riemann zeros derived from Eq. (3.5),
.N (w)"dNM /dw"(1/2p) ln(w/2p) ,
(3.23)
we obtain s "ln(p )"2 ln(w/2p)Np "(w/2p) .
(3.24)
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Fig. 23. Estimated number of converged zeros of the Riemann zeta function, which can be obtained by harmonic inversion for given number of primes p (from [83]).
The number of primes with p(p can be estimated from the prime number theorem
p (w/2p) p(p )& " . (3.25)
ln(p ) 2 ln(w/2p)
On the other hand, the number of Riemann zeros as a function of w is given by Eq. (3.5). The estimated number of Riemann zeros which can be obtained by harmonic inversion from a given set of primes is presented in Fig. 23. For example, about 80 zeros (w(200) can be extracted from the short signal C (s) with s "ln(1000)"6.91 (168 prime numbers) in agreement with the estimates N
given above. Obviously, in the special case of the Riemann zeta function the e$ciency of our method cannot compete with that of the Riemann}Siegel formula method Eq. (3.14) where the number of terms is given by n "Int [(w/2p] and, e.g., "ve terms in Eq. (3.14) would be
su$cient to calculate good approximations to the Riemann zeros in the region w(200. Our primary intention is to introduce harmonic inversion by way of example of the zeros of the Riemann zeta function as a universal tool for periodic orbit quantization, and not to use it as an alternative method for solving the problem of "nding most e$ciently zeros of the Riemann zeta function. For the semiclassical quantization of bound and ergodic systems a functional equation can be invoked to derive a Riemann}Siegel look-alike quantization condition [59}61]. In this case the required number of periodic orbits can be estimated from the condition , (3.26) s +p.N (w)"s &
which di!ers by a factor of 4 from the required signal length Eq. (3.21) for harmonic inversion. Obviously, the limitation Eq. (3.21) is unfavorable for periodic orbit quantization because of the exponential proliferation of the number of orbits in chaotic systems, and methods for a signi"cant reduction of the signal length s are highly desirable. In fact, this can be achieved by an extension
of the harmonic inversion technique to cross-correlation signals. We will return to this problem in Section 3.5. 3.2. Periodic orbit quantization As mentioned in Section 3.1 the basic equation Eq. (3.9) used for the calculation of Riemann zeros has the same mathematical form as Gutzwiller's semiclassical trace formula. Both series,
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Eq. (3.9) and the periodic orbit sum Eq. (3.10), su!er from similar convergence problems in that they are absolutely convergent only in the complex half-plane outside the region where the Riemann zeros, or quantum eigenvalues, respectively, are located. As a consequence, in a direct summation of periodic orbit contributions smoothing techniques must be applied resulting in low-resolution spectra for the density of states [78,122]. To extract individual eigenstates the semiclassical trace formula has to be analytically continued to the region of the quantum poles, and this was the subject of intense research during recent years. For strongly chaotic bound systems a semiclassical quantization condition based on a semiclassical representation of the spectral staircase N(E)" H(E!E ) (3.27) L L has been developed by Aurich et al. in [65,66]. They suggest to replace the spectral staircase N(E) with a smooth semiclassical approximation N (E), and to evaluate the semiclassical eigenenergies from the quantization condition cos+pN (E),"0 . (3.28) For the even-parity states of the hyperbola billiard the exact spectral staircase N>(E) and the semiclassical approximation N>(E) evaluated from 101265 periodic orbits are shown in Fig. 24.The function cos+pN>(E), obtained from the smooth spectral staircase N>(E) in Fig. 24 is presented in Fig. 25. According to the quantization condition Eq. (3.28) the zeros of this function are the semiclassical energies, and are in good agreement with the true quantum mechanical energies marked by triangles in Fig. 25. The quantization condition Eq. (3.28) was also successfully
Fig. 24. The spectral staircase N>(E) and its semiclassical approximation N>(E) for the hyperbolic billiard. N>(E) was calculated using 101265 periodic orbits (from [65]).
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Fig. 25. The function cos+pN>(E), for the hyperbolic billiard. N>(E) was evaluated as in Fig. 24. The triangles mark the positions of the true quantum mechanical energies (from [65]).
applied to the motion of a particle on various Riemann surfaces with constant negative curvature, e.g., Artin's billiard or the Hadamard}Gutzwiller model [65,66]. Another technique for bound and ergodic systems is to apply an approximate functional equation and generalize the Riemann}Siegel formula Eq. (3.14) to dynamical zeta functions [59}61]. The Riemann}Siegel look-alike formula has been applied, e.g., for the semiclassical quantization of the hyperbola billiard [123]. These quantization techniques cannot be applied to open systems. However, if a symbolic dynamics for the system exists, i.e., if the periodic orbits can be classi"ed with the help of a complete symbolic code, the dynamical zeta function, given as an in"nite Euler product over entries from classical periodic orbits, can be expanded in terms of the cycle length of the orbits [9,58]. The cycle expansion series is rapidly convergent if the contributions of long orbits are approximately shadowed by contributions of short orbits. The cycle expansion technique has been applied, e.g., to the three-disk scattering system [9}11] (see also Section 3.3), the three-body Coulomb system [6,124], and to the hydrogen atom in a magnetic "eld [20]. It turns out that the cycle expansion of dynamical zeta functions converges very slowly for bound systems. Therefore, a combination of the cycle expansion method with a functional equation has been developed by Tanner et al. [114,125]. Applying a functional equation to the dynamical zeta function they conclude that the zeros of the real expression D(E)"e\ pNM #Z(E)#e pNM #ZH(E) ,
(3.29)
with N M (E) the mean spectral staircase and Z(E) the cycle expanded dynamical zeta function should be semiclassical approximations to the eigenvalues. The semiclassical functional determinant D(E) is illustrated in Fig. 26 for the anisotropic Kepler problem [8] and in Fig. 27 for the closed three-disk billiard, i.e., the system shown in Fig. 20 with touching disks. The vertical bars in Figs. 26 and 27 mark the exact eigenvalues.
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Fig. 26. The cycle-expanded functional determinant for the mp"0> subspace of the anisotropic Kepler problem. The quantum eigenvalues are marked with vertical bars on the real axis. (a) All orbits up to length 4 (8 in number) included. (b) All orbits up to length 8 (71) included (from [114]). Fig. 27. The cycle-expanded functional determinant for the closed three disk billiard (disk radius R"1). The vertical bars mark the exact quantum eigenvalues. (a) All orbits up to length 2 (3 in number) included. (b) All orbits up to length 3 (5) included (from [114]).
The existence of a complete symbolic dynamics is more the exception than the rule, and therefore cycle expansion techniques cannot be applied, in particular, for systems with mixed regular-chaotic classical dynamics. In this section, we apply the same technique that we used for the calculation of Riemann zeros to the calculation of semiclassical eigenenergies and resonances of physical systems by harmonic inversion of Gutzwiller's periodic orbit sum for the propagator. The method only requires the knowledge of all orbits up to a su$ciently long but "nite period and does neither rely on an approximate semiclassical functional equation, nor on the existence of a symbolic code for the orbits. The universality of the method will therefore allow the investigation of a large variety of systems with an underlying chaotic, regular, or even mixed classical dynamics. The derivation of an expression for the recurrence function to be harmonically inverted is analogous to that in Section 3.1.2. 3.2.1. Semiclassical density of states Following Gutzwiller [7,8] the semiclassical response function for chaotic systems is given by g(E)"g(E)# A e 1 ,
(3.30)
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where g(E) is a smooth function and the S and A are the classical actions and weights (including phase information given by the Maslov index) of periodic orbit contributions. Eq. (3.30) is also valid for integrable [25,26] and near-integrable [27,28] systems but with di!erent expressions for the amplitudes A . It should also be possible to include complex `ghosta orbits [36,37] and uniform semiclassical approximations [40,44] close to bifurcations of periodic orbits in the semiclassical response function Eq. (3.30). The eigenenergies and resonances are the poles of the response function but, unfortunately, its semiclassical approximation Eq. (3.30) does not converge in the region of the poles, whence the problem is the analytic continuation of g(E) to this region. In the following we make the (weak) assumption that the classical system has a scaling property (see Section 2.1), i.e., the shape of periodic orbits does not depend on the scaling parameter, w, and the classical action scales as S "ws . (3.31) Examples of scaling systems are billiards [9,16], Hamiltonians with homogeneous potentials [126,127], Coulomb systems [6], or the hydrogen atom in external magnetic and electric "elds [47,20]. Eq. (3.31) can even be applied for non-scaling, e.g., molecular systems if a generalized scaling parameter w, \ is introduced as a new dynamical variable [94]. Quantization yields bound states or resonances, w , for the scaling parameter. In scaling systems the semiclassical I response function g(w) can be Fourier transformed easily to obtain the semiclassical trace of the propagator
1 > g(w)e\ QU dw" A d(s!s ) . (3.32) C(s)" 2p \ The signal C(s) has d-peaks at the positions of the classical periods (scaled actions) s"s of periodic orbits and with peak heights (recurrence strengths) A , i.e., C(s) is Gutzwiller's periodic orbit recurrence function. Consider now the quantum mechanical counterparts of g(w) and C(w) taken as the sums over the poles w of the Green's function, I d I , (3.33) g (w)" w!w #i0 I I 1 > C (s)" (3.34) g (w)e\ QU dw"!i d e\ UIQ , I 2p \ I with d being the multiplicities of resonances, i.e., d "1 for non-degenerate states. In analogy with I I the calculation of Riemann zeros from Eq. (3.18) the frequencies, w , and amplitudes, d , can be I I extracted by harmonic inversion of the signal C(s) after convoluting it with a Gaussian function, i.e.,
1 A eQ\Q N . (3.35) C(s)" N (2pp By adjusting C(s) to the functional form of Eq. (3.34), the frequencies, w , can be interpreted as the N I semiclassical approximation to the poles of the Green's function in Eq. (3.33). Note that the harmonic inversion method described in Appendix A.1 allows studying signals with complex
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frequencies w as well. For open systems the complex frequencies can be interpreted as semiclassical I resonances. Note also that the w in general di!er from the exact quantum eigenvalues because I Gutzwiller's trace formula Eq. (3.30) is an approximation, correct only to the lowest order in . Therefore the diagonalization of small matrices (Eq. (A.7) in Appendix A.1) does not imply that the results of the periodic orbit quantization are more `quantuma in any sense than those obtained, e.g., from a cycle expansion [9,58]. However, the harmonic inversion technique also allows the calculation of higher-order corrections to the periodic orbit sum, and we will return to this problem in Section 3.6. 3.2.2. Semiclassical matrix elements The procedure described above can be generalized in a straightforward manner to the calculation of semiclassical diagonal matrix elements 1t "AK "t 2 of a smooth Hermitian operator AK . In I I this case we start from the quantum mechanical trace formula 1t "AK "t 2 I I , (3.36) g (w)"tr G>AK " w!w #i0 I I which has the same functional form as Eq. (3.33), but with d "1t "AK "t 2 instead of d "1. For the I I I I quantum response function g (w) (Eq. (3.36)) a semiclassical approximation has been derived in [128,129], which has the same form as Gutzwiller's trace formula Eq. (3.30) but with amplitudes A e\ p I N A "!i (3.37) ("det(M !I)" where M is the monodromy matrix and k the Maslov index of the periodic orbit, and QN (3.38) A " A(q(s), p(s)) ds N is the classical integral of the observable A over one period s of the primitive periodic orbit. Note N that q(s) and p(s) are functions of the classical action instead of time for scaling systems [130]. Gutzwiller's trace formula for the density of states is obtained with AK being the identity operator, i.e., A "s . When the semiclassical signal C(s) (Eq. (3.32)) with amplitudes A given by N N Eqs. (3.37) and (3.38) is analyzed with the method of harmonic inversion the frequencies and amplitudes obtained are the semiclassical approximations to the eigenvalues w and matrix I elements d "1t "AK "t 2, respectively. I I I
3.3. The three-disk scattering system Let us consider the three-disk scattering system which has served as a model for periodic orbit quantization by cycle expansion techniques [9}15]. For a brief introduction of the system, its symbolic dynamics and symmetries we refer the reader to Section 2.6.2. The starting point for the cycle expansion technique is to rewrite Gutzwiller's trace formula as a dynamical zeta function [131] 1/m" (1!t ) , N N
(3.39)
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with p indicating the primitive periodic orbits, 1 e QNU\p IN , (3.40) t, N (K N k the Maslov index, and K the largest eigenvalue of the monodromy matrix. [The dynamical zeta N N function, Eqs. (3.39) and (3.40) is based on an approximation. By contrast, the Gutzwiller}Voros zeta function [132,133] is exactly equivalent to Gutzwiller's trace formula. See [14,15] for more details.] The semiclassical eigenvalues w are obtained as zeros of the dynamical zeta function Eq. (3.39). However, the convergence problems of the in"nite product Eq. (3.39) are similar to the convergence problems of Gutzwiller's periodic orbit sum. The basic observation for the three-disk system with su$ciently large distance d was that the periodic orbit quantities of long orbits can be approximated by the periodic orbit quantities of short orbits, i.e., for two orbits with symbolic codes p and p we have s +s #s , N N NN K +K K , (3.41) NN N N k "k #k , N N NN and thus t
+t t . (3.42) NN N N This implies that in the cycle expansion of the dynamical zeta function Eq. (3.39) the contributions of long periodic orbits are shadowed by the contributions of short orbits. E.g., for the three-disk scattering system the cycle expansion up to cycle length n"3 reads 1/m"1!t !t ![t !t t ]![t !t t #t !t t ]!2 . (3.43) The terms t and t are the fundamental contributions, while the terms in brackets are the curvature corrections, ordered by cycle length, and can rapidly decrease with increasing cycle length. With the cycle expansion Eq. (3.43) the dynamical zeta function Eq. (3.39) is analytically continued to the physically important region below the real axis, where the resonances are located. For d"6 semiclassical resonances were calculated by application of the cycle expansion technique including all (symmetry reduced) periodic orbits up to cycle length n"13 [11,14,15]. We now demonstrate the usefulness of the harmonic inversion technique for the semiclassical quantization of the three disk scattering system. In contrast to the cycle expansion we will not make use of the symbolic code of the periodic orbits and the approximate relations Eq. (3.42) between the periodic orbit quantities. As in the previously discussed examples for billiard systems the scaled action s is given by the length ¸ of orbits (s"¸) and the quantized parameter is the absolute value of the wave vector k""k""(2mE/ . For the three-disk system the periodic orbit signal C(¸) reads ¸ e\ Pp IN N d(¸!r¸ ) , (3.44) C(¸)"!i N N ("det(MP !I)" N N P N with ¸ , M , and k the geometrical length, monodromy matrix, and Maslov index of the N N N symmetry reduced primitive periodic orbit p, respectively. The weight factors N depend on the N
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Table 7 Weight factors N for the symmetry decomposition of the three-disk N scattering system C T
A
A
E
e C , C p T
1 1 1
1 1 !1
2 !1 0
Fig. 28. Three-disk scattering system (A subspace) with R"1, d"6. (a) Periodic orbit recurrence function, C(¸). The signal has been convoluted with a Gaussian function of width p"0.0015. (b) Semiclassical resonances (from [82]).
irreducible subspace (A , A , and E), where resonances are calculated, and the symmetries of the orbits. Periodic orbits without symmetries, with rotational symmetry, and with symmetry under re#ection are characterized by e, C , C, and p , respectively. The weight factors N are the same as T N for the cycle expansion, and are given in Table 7. For details of the symmetry decomposition see [134]. In the following we calculate resonances in the irreducible subspace A , i.e., N "1 in N Eq. (3.44) for all orbits. We "rst apply harmonic inversion to the case R : d"1 : 6 studied before. Fig. 28a shows the periodic orbit recurrence function, i.e., the trace of the semiclassical propagator C(¸). The groups with oscillating sign belong to periodic orbits with adjacent cycle lengths. To obtain a smooth function on an equidistant grid, which is required for the harmonic inversion method, the d-functions in Eq. (3.44) have been convoluted with a Gaussian function of width p"0.0015. As explained in Section 3.1.2 this does not change the underlying spectrum. The results of the harmonic inversion analysis of this signal are presented in Fig. 28b. The crosses in Fig. 28b represent semiclassical poles, for which the amplitudes d are very close to 1, mostly within one I percent. Because the amplitudes converge much slower than the frequencies these resonance positions can be assumed to be very accurate within the semiclassical approximation. For some broad resonances marked by diamonds in Fig. 28b the d deviate strongly from 1, within 5 to I maximal 50%. It is not clear whether these strong deviations are due to numerical e!ects, such as
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convergence problems caused by too short a signal, or if they are a consequence of the semiclassical approximation. A direct comparison between the semiclassical resonances k obtained by harmonic inversion of the periodic orbit sum, and results of Wirzba [14,15,113], i.e., the cycle expansion resonances k calculated up to 12th order in the curvature expansion of the Gutzwiller}Voros zeta function, and the exact quantum resonances k is given in Table 8 for resonances Table 8 Semiclassical and exact quantum mechanical resonances for the three disk scattering problem (A subspace) with R"1, d"6. Resonances k have been obtained by harmonic inversion of the periodic orbit sum, resonances k (cycle expansion) and the exact quantum resonances k have been calculated by Wirzba Re k
Im k
Re k
Im k
Re k
Im k
0.7583139 2.2742786 3.7878768 4.1456898 5.2960678 5.6814976 6.7936365 7.2240580 8.2763906 8.7792134 9.7476329 10.3442257 11.2134778 11.9134496 12.6775319 13.4826489 14.1424136 15.0473050 15.6114431 16.6025598 17.0875557 18.1465009 18.5733904 19.6808375 20.0679755 21.2080634 21.5736413 22.7296581 23.0872357 24.2458779 24.6079881 25.7560497 26.1353684 27.2592433 27.6694386 28.7553526 29.2098235
!0.1228222 !0.1330587 !0.1541274 !0.6585397 !0.1867873 !0.5713721 !0.2299221 !0.4954243 !0.2770805 !0.4302561 !0.3208170 !0.3781988 !0.3599639 !0.3357346 !0.3961154 !0.2969478 !0.4300604 !0.2578357 !0.4603838 !0.2188731 !0.4826796 !0.1842319 !0.4913642 !0.1575927 !0.4814959 !0.1403086 !0.4643351 !0.1323433 !0.4363434 !0.1331136 !0.4038977 !0.1415265 !0.3694453 !0.1556272 !0.3353429 !0.1727506 !0.3041527
0.7583139 2.2742786 3.7878768 4.1474774 5.2960678 5.6820274 6.7936365 7.2242175 8.2763906 8.7791917 9.7476329 10.3442254 11.2134779 11.9134496 12.6775320 13.4826489 14.1424117 15.0473050 15.6113293 16.6025599 17.0876372 18.1465009 18.5731865 19.6808375 20.0685648 21.2080634 21.5736471 22.7296581 23.0872405 24.2458779 24.6079949 25.7560497 26.1353709 27.2592433 27.6694384 28.7553526 29.2098230
!0.1228222 !0.1330587 !0.1541274 !0.6604761 !0.1867873 !0.5715543 !0.2299221 !0.4954066 !0.2770805 !0.4302718 !0.3208170 !0.3781988 !0.3599639 !0.3357346 !0.3961159 !0.2969477 !0.4300584 !0.2578357 !0.4604377 !0.2188731 !0.4827914 !0.1842319 !0.4914087 !0.1575927 !0.4842307 !0.1403086 !0.4643077 !0.1323433 !0.4363396 !0.1331136 !0.4038962 !0.1415265 !0.3694420 !0.1556272 !0.3353414 !0.1727506 !0.3041533
0.6979958 2.2396014 3.7626868 4.1316606 5.2756666 5.6694976 6.7760661 7.2152706 8.2611376 8.7724709 9.7345075 10.3381881 11.2021099 11.9075971 12.6675941 13.4769269 14.1337039 15.0416935 15.6037211 16.5970551 17.0809957 18.1411380 18.5673580 19.6756560 20.0633947 21.2030727 21.5689872 22.7248396 23.0829816 24.2411967 24.6040788 25.7514701 26.1317670 27.2547515 27.6661128 28.7509680 29.2067213
!0.0750137 !0.1187664 !0.1475455 !0.6170418 !0.1832203 !0.5534079 !0.2275078 !0.4856243 !0.2749083 !0.4241019 !0.3188052 !0.3737056 !0.3582265 !0.3322326 !0.3946675 !0.2941108 !0.4288264 !0.2555072 !0.4592793 !0.2170025 !0.4815364 !0.1827959 !0.4899438 !0.1565422 !0.4825586 !0.1395793 !0.4625282 !0.1318732 !0.4345679 !0.1328421 !0.4022114 !0.1413856 !0.3679001 !0.1555416 !0.3339924 !0.1726604 !0.3030272
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Table 9 Same as Table 8 but for resonances with 120(Re k(132 Re k
Im k
Re k
Im k
Re k
Im k
120.0966075 120.3615795 120.8941439 121.2644970 121.6158735 121.9157839 122.3933221 122.7533968 123.1345930 123.4680089 123.8916697 124.2423970 124.6520739 125.0191421 125.3924617 125.7306095 126.1681278 126.5700003 126.8986333 127.2175968 127.6830865 128.1211609 128.4113722 128.7033407 129.1973295 129.6731970 129.9292721 130.1879622 130.7109808 131.2271782 131.4488921 131.6713958
!0.1313240 !0.4242281 !0.5145734 !0.4017485 !0.1451572 !0.3979469 !0.5441615 !0.3809547 !0.1656546 !0.3672776 !0.5717434 !0.3593830 !0.1905948 !0.3358345 !0.5950322 !0.3386874 !0.2172657 !0.3071799 !0.6105834 !0.3201029 !0.2434140 !0.2838964 !0.6157741 !0.3044266 !0.2678886 !0.2671784 !0.6091832 !0.2922354 !0.2904524 !0.2573647 !0.5905439 !0.2842971
120.0966093 120.3624892 120.8959933 121.2643338 121.6158734 121.9158422 122.3962246 122.7533176 123.1345929 123.4679489 123.8969484 124.2423480 124.6520738 125.0191284 125.4000681 125.7305801 126.1681277 126.5700090 126.9067485 127.2175825 127.6830866 128.1211675 128.4171380 128.7033374 129.1973310 129.6731961 129.9304559 130.1879664 130.7109850 131.2271733 131.4451877 131.6714040
!0.1313207 !0.4246464 !0.5143028 !0.4018042 !0.1451572 !0.3978279 !0.5427722 !0.3809838 !0.1656547 !0.3672571 !0.5675655 !0.3593825 !0.1905950 !0.3358584 !0.5858492 !0.3386762 !0.2172664 !0.3071895 !0.5956839 !0.3200895 !0.2434159 !0.2838934 !0.5966404 !0.3044145 !0.2678924 !0.2671733 !0.5891942 !0.2922273 !0.2904576 !0.2573650 !0.5734808 !0.2842951
120.0956703 120.3617516 120.8949118 121.2633401 121.6149357 121.9151103 122.3951716 122.7523307 123.1336504 123.4672237 123.8959339 124.2413639 124.6511239 125.0184111 125.3990978 125.7295973 126.1671733 126.5692997 126.9058213 127.2166018 127.6821330 128.1204656 128.4162452 128.7023607 129.1963819 129.6725028 129.9295889 130.1869953 130.7100422 131.2264935 131.4443391 131.6704389
!0.1313419 !0.4244457 !0.5143053 !0.4017384 !0.1452113 !0.3976081 !0.5427575 !0.3809219 !0.1657315 !0.3670296 !0.5675273 !0.3593262 !0.1906812 !0.3356413 !0.5857818 !0.3386268 !0.2173496 !0.3069994 !0.5955889 !0.3200477 !0.2434894 !0.2837384 !0.5965218 !0.3043815 !0.2679547 !0.2670549 !0.5890492 !0.2922052 !0.2905099 !0.2572819 !0.5733029 !0.2842886
with Re k(30 and in Table 9 for 120(Re k(132. Apart from a few resonances with large imaginary parts the di!erences "k !k" between the semiclassical resonances obtained by harmonic inversion and cycle expansion are by several orders of magnitude smaller than the semiclassical error, i.e., the di!erences "k !k ". To compare the e$ciency of the harmonic inversion and the cycle expansion method we calculated the A resonances of the three disk system with distance d"6 from a short signal C(¸) with ¸424, which includes the recurrences of all periodic orbits with cycle length n45. The resonances obtained by harmonic inversion of the short signal are presented as squares in Fig. 29. The resonances of the two bands closest to the real axis qualitatively agree with the correct
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Fig. 29. Three-disk scattering system (A subspace) with R"1, d"6. Semiclassical resonances obtained by harmonic inversion of the periodic orbit recurrence function C(¸) with ¸424 (squares) and ¸450 (crosses).
Fig. 30. Three-disk scattering system (A subspace) with R"1, d"2.5. (a) Periodic orbit recurrence function, C(¸). The signal has been convoluted with a Gaussian function of width p"0.0003. (b) Semiclassical resonances (from [82]).
semiclassical resonances (crosses in Fig. 29). The accuracy is similar to the accuracy obtained by the cycle expansion up to third order in the curvature expansion [15], which includes all periodic orbits with cycle length n44. The reason for the somewhat higher e$ciency of the cycle expansion compared to harmonic inversion is probably that the basic requirement Eq. (3.42) for the cycle expansion is a very good approximation at the large distance d"6 between the disks. We now study the three-disk scattering system with a short distance ratio d/R"2.5. The signal C(¸) is constructed from 356 periodic orbits with geometrical length ¸47.5 (see Fig. 30a). For large ¸ groups of orbits with the same cycle length of the symbolic code strongly overlap and cannot be recognized in Fig. 30a. Note that the signal contains complete sets of orbits up to topological length (cycle length) n"9 only. The resonances obtained by harmonic inversion of the signal C(¸) are presented in Fig. 30b. A comparison between the semiclassical resonances k obtained by harmonic inversion, the cycle expansion resonances k calculated up to ninth order
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Table 10 Semiclassical and exact quantum mechanical resonances for the three disk scattering problem (A subspace) with R"1, d"2.5. Resonances k have been obtained by harmonic inversion of the periodic orbit sum, resonances k (cycle expansion) and the exact quantum resonances k have been calculated by Wirzba Re k
Im k
Re k
Im k
Re k
Im k
4.5811788 7.1436576 13.0000861 17.5688087 18.9210923 26.6126422 27.8887512 30.3885091 32.0961916 33.7905462 36.5066867 39.8138808 42.6585345 44.3271565 45.1114372 48.8432026 51.9171049 53.3788306 60.5907039 62.2000118 64.8371533 65.6804780 67.8612092 69.3238832 71.1163719 74.8566924 77.3118119 78.5807841 80.3395505 83.9450649 85.8041790 88.5979364 93.2377427 94.4188748 97.5331975
!0.0899874 !0.8112421 !0.6516382 !0.6848798 !0.7860616 !1.8066664 !0.5431988 !0.1134324 !0.6218604 !1.9974557 !0.3845559 !0.3582253 !0.3514612 !0.3596874 !0.4173349 !0.5925064 !0.6951700 !0.1062924 !0.8405420 !0.2161506 !2.1940372 !0.2791339 !0.2865945 !0.2988268 !0.5635882 !0.3058255 !0.3223758 !0.9179862 !0.3052621 !0.4910890 !0.3988778 !0.6089526 !0.1707857 !0.5644403 !0.4015302
4.5811768 7.1441170 13.0000510 17.5699350 18.9266454 26.5430240 27.8881874 30.3884554 32.0966982 33.9721550 36.5066384 39.8139242 42.6556455 44.2457223 45.0602989 48.8417782 51.9146021 53.3766455 60.6204084 62.2004010 64.1528609 65.6804671 67.8688350 69.3443564 71.0822786 74.8552672 77.3193218 78.9236428 80.4173794 83.8740934 85.8000989 88.4703014 93.0234303 94.4284207 97.5576754
!0.0899911 !0.8107256 !0.6516077 !0.6845598 !0.7836853 !1.8607912 !0.5431833 !0.1134504 !0.6223710 !2.0780146 !0.3846400 !0.3580251 !0.3492807 !0.4072838 !0.3453843 !0.5913485 !0.6791551 !0.1005604 !0.8126455 !0.2137105 !1.6713984 !0.2737728 !0.2881904 !0.3123706 !0.5381921 !0.3022484 !0.3129301 !0.9433778 !0.3670163 !0.5035157 !0.4147646 !0.6739426 !0.1231810 !0.5013008 !0.4167467
4.4692836 7.0917132 12.9503201 17.5042296 18.9254527 26.5316680 27.8577850 30.3528872 32.0693735 33.9323425 36.4822805 39.7859698 42.6312420 44.2295828 45.0400958 48.8203073 51.8983106 53.3584390 60.6075221 62.1832903 64.1457569 65.6638670 67.8518132 69.3334558 71.0672666 74.8405304 77.3088118 78.9042646 80.4002169 83.8631132 85.7918872 88.4561398 93.0113364 94.4351896 97.5478240
!0.0015711 !0.7207876 !0.6282388 !0.6352583 !0.7662881 !1.8273425 !0.5499256 !0.1056653 !0.6077432 !2.0379172 !0.3839172 !0.3508678 !0.3403609 !0.4015643 !0.3421851 !0.5853389 !0.6721309 !0.1005951 !0.8098089 !0.2111191 !1.6638733 !0.2725830 !0.2865564 !0.3092481 !0.5353411 !0.2994128 !0.3107104 !0.9416087 !0.3628903 !0.5005356 !0.4152898 !0.6778186 !0.1220480 !0.5267138 !0.4181066
in the curvature expansion of the Gutzwiller}Voros zeta function, and the exact quantum resonances k is given in Table 10. The cycle expansion and exact quantum calculations have been performed by Wirzba [113]. The results of both semiclassical methods are in good agreement, although a detailed comparison reveals that for some resonances towards the end of Table 10 the values of the cycle expansion are somewhat closer to the exact quantum mechanical results than those values obtained by harmonic inversion.
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3.4. Systems with mixed regular-chaotic dynamics In Section 3.1 we have applied harmonic inversion for the calculation of zeros of Riemann's zeta function as a mathematical model of a strongly chaotic bound system, and in Section 3.3 we have used harmonic inversion for the periodic orbit quantization of the three-disk scattering system. Both systems have been solved with other especially designed methods, i.e., the Riemann zeta function with the help of the Riemann}Siegel formula and the three-disk system by application of cycle expansion techniques. However, none of the special methods, which were designed to overcome the convergence problems of the semiclassical trace formula (see, e.g., [9,61,63,65]), has succeeded so far in correctly describing generic dynamical systems with mixed regular-chaotic phase spaces. In this section we want to demonstrate the universality of periodic orbit quantization by harmonic inversion by investigating generic dynamical systems. It will be the objective to contribute to solving the long-standing problem of semiclassical quantization of non-integrable systems in the mixed regular-chaotic regime. A "rst step towards the periodic orbit quantization of mixed systems has been done by Wintgen [78] on the hydrogen atom in a magnetic "eld. From Gutzwiller's truncated periodic orbit sum he obtained the smoothed part of the density of states presented in Fig. 31 at scaled energy EI "!0.2 which is in the mixed regular-chaotic regime. The resolution of the smoothed spectra depends on the cuto! value of the periodic orbit sum which is s /2p"1.33 and s /2p"3.00 in Fig. 31a and
Fig. 31b, respectively. Nine approximate eigenvalues have been estimated from the low-resolution truncated periodic orbit sum in Fig. 31b (for comparison the exact quantum eigenvalues are
Fig. 31. Smoothed #uctuating part of the density of states for the hydrogen atom in a magnetic "eld at scaled energy EI "!0.2. The quantum results (thick lines) are smoothed over the "rst 100 eigenstates. Semiclassical results (thin lines) are obtained by including (a) two orbits (three contributions including repetitions) with scaled actions s/2p(1.33, and (b) 13 orbits (19 including repetitions) with s/2p(3. The lowest quantum eigenvalues are marked as vertical bars (from [78]).
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marked by vertical bars in Fig. 31), but neither the accuracy nor the number of semiclassical eigenvalues could be improved by increasing the number of periodic orbits because of the non-convergence property of Gutzwiller's trace formula. The results of [78] therefore clearly demonstrate that methods to overcome the convergence problems of the semiclassical periodic orbit sum are of crucial importance to obtain highly resolved semiclassical spectra in the mixed regime. Note also that Gutzwiller's trace formula is not valid for non-isolated periodic orbits on invariant tori in the regular part of the phase space. This fact has not been considered in [78]. The limiting cases of mixed systems are strongly chaotic and integrable systems. As has been proven by Gutzwiller [8], for systems with complete chaotic (hyperbolic) classical dynamics the density of states can be expressed as an in"nite sum over all (isolated) periodic orbits. On the other extreme of complete integrability, it is well known that the semiclassical energy values can be obtained by EBK torus quantization [1]. This requires the knowledge of all the constants of motion, which are not normally given in explicit form, and therefore practical EBK quantization based on the direct or indirect numerical construction of the constants of motion turns out to be a formidable task [135]. As an alternative, EBK quantization was recast as a sum over all periodic orbits of a given topology on respective tori by Berry and Tabor [25,26]. The Berry}Tabor formula circumvents the numerical construction of the constants of motion but usually su!ers from the convergence problems of the in"nite periodic orbit sum. The extension of the Berry}Tabor formula into the near-integrable (KAM) regime was outlined by Ozorio de Almeida [41] and elaborated, at di!erent levels of re"nement, by Tomsovic et al. [27] and Ullmo et al. [28]. These authors noted that in the near-integrable regime, according to the PoincareH }Birkho! theorem, two periodic orbits survive the destruction of a rational torus with similar actions, one stable and one hyperbolic unstable, and worked out the ensuing modi"cations of the Berry}Tabor formula. In this section we go one step further by noting that, with increasing perturbation, the stable orbit turns into an inverse hyperbolic one representing, together with its unstable companion with similar action, a remnant torus. We include the contributions of these pairs of inverse hyperbolic and hyperbolic orbits in the Berry}Tabor formula and demonstrate for a system with mixed regular-chaotic dynamics that this procedure yields excellent results even in the deep mixed regular-chaotic regime. As in [78] we choose the hydrogen atom in a magnetic "eld, which is a real physical system and has served extensively as a prototype for the investigation of `quantum chaosa [29}31]. The fundamental obstacle bedeviling the semiclassical quantization of systems with mixed regular-chaotic dynamics is that the periodic orbits are neither su$ciently isolated, as is required for Gutzwiller's trace formula [8], nor are they part of invariant tori, as is necessary for the Berry}Tabor formula [25,26]. However, as will become clear below, it is the Berry}Tabor formula which lends itself in a natural way for an extension of periodic orbit quantization to mixed systems. As previously, we consider scaling systems where the classical action S scales as S"sw with w" \ the scaling parameter and s the scaled action. For scaling systems with two degrees of freedom, which we will focus on, the Berry}Tabor formula for the #uctuating part of the level density reads w sM 1 . (w)" Re e QMU\p EM\p , M "g " p M #
(3.45)
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with M"(M , M ) pairs of integers specifying the individual periodic orbits on the tori (numbers of rotations per period, M /M rational), and sM and gM the scaled action and Maslov index of the periodic orbit M. The function g in Eq. (3.45) is obtained by inverting the Hamiltonian, expressed # in terms of the actions (I , I ) of the corresponding torus, with respect to I , viz. H(I , I "g (I ))"E [136]. The calculation of g from the actions (I , I ) can be rather laborious # # even for integrable and near-integrable systems, and, by de"nition, becomes impossible for mixed systems in the chaotic part of the phase space. Here we will adopt the method of [27,28] and calculate g , for given M"(k , k ), with (k , k ) coprime integers specifying the primitive periodic # orbit, directly from the parameters of the two periodic orbits (stable (s) and hyperbolic unstable (h)) that survive the destruction of the rational torus M, viz.,
1 1 \ 2 # , g " # pk*s (det(M !I) (!det(M !I)
(3.46)
with *s"(s !s ) (3.47) the di!erence of the scaled actions, and M and M the monodromy matrices of the two orbits. The action sM in Eq. (3.45) is to replaced with the mean action sN "(s #s ) . (3.48) Eq. (3.46) is an approximation which becomes exact in the limit of an integrable system. It is a characteristic feature of systems with mixed regular-chaotic dynamics that with increasing non-integrability the stable orbits turn into inverse hyperbolic unstable orbits in the chaotic part of the phase space. These orbits, although embedded in the fully chaotic part of phase space, are remnants of broken tori. It is therefore natural to assume that Eqs. (3.45) and (3.46) can even be applied when these pairs of inverse hyperbolic and hyperbolic orbits are taken into account, i.e., more deeply in the mixed regular-chaotic regime. It should be noted that the di!erence *s between the actions of the two orbits is normally still small, and it is therefore more appropriate to start from the Berry}Tabor formula for semiclassical quantization in that regime than from Gutzwiller's trace formula, which assumes well-isolated periodic orbits. It is also important to note that the Berry}Tabor formula does not require an extensive numerical periodic orbit search. The periodic orbit parameters s/M and g are smooth # functions of the rotation number M /M , and can be obtained for arbitrary periodic orbits with coprime integers (M , M ) by interpolation between `simplea rational numbers M /M . 3.4.1. Hydrogen atom in a magnetic xeld We now demonstrate the high quality of the extension of Eqs. (3.45) and (3.46) to pairs of inverse hyperbolic and hyperbolic periodic orbits for a physical system that undergoes a transition from regularity to chaos, namely the hydrogen atom in a magnetic "eld. This is a scaling system, with w"c\ " \ the scaling parameter and c"B/(2.35;10 T) the magnetic "eld strength in atomic units. The classical Hamiltonian and the scaling procedure have been discussed in Section 2.1. At low scaled energies EI "Ec\ (!0.6, a PoincareH surface of section analysis [30] of the classical Hamiltonian (Eq. (2.11)) exhibits two di!erent torus structures related to a `rotatora
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and `vibratora type motion. The separatrix between these tori is destroyed at a scaled energy of EI +!0.6, and the chaotic region around the separatrix grows with increasing energy. At EI "!0.127 the classical phase space becomes completely chaotic. We investigate the system at scaled energy EI "!0.4, where about 40% of the classical phase space volume is chaotic (see inset in Fig. 32), i.e. well in the region of mixed dynamics. We use eight pairs of periodic orbits to describe the rotator type motion in both the regular and chaotic region. The results for the periodic orbit parameters s/2pM and g are presented as solid lines in Fig. 32. The squares on the solid # lines mark parameters obtained by pairs of stable and unstable periodic orbits in the regular region of the phase space. The diamonds mark parameters obtained by pairs of two unstable (inverse hyperbolic and hyperbolic) periodic orbits in the chaotic region of phase space. The cuto! is related to the winding angle "1.278 of the "xed point of the rotator type motion, i.e., the orbit perpendicular to the magnetic "eld axis, (M /M ) "p/ "2.458. The solid lines have been obtained by spline interpolation of the data points. In the same way the periodic orbit parameters for the vibrator type motion have been obtained from 11 pairs of periodic orbits (see the dashed lines in Fig. 32). The cuto! at M /M "p/ "1.158 is related to the winding angle "2.714 of the "xed point of the vibrator type motion, i.e., the orbit parallel to the "eld axis. With the data of Fig. 32 we have all the ingredients at hand to calculate the semiclassical density of states . (w) in Eq. (3.45). The periodic orbit sum includes for both the rotator and vibrator type motion the orbits with M /M '(M /M ) . For each orbit the action and the function g is # obtained from the spline interpolations. The Maslov indices are gM"4M !M for the rotator
Fig. 32. (a) Action s/2pM and (b) second derivative g as a function of the frequency ratio M /M for the rotator (solid # lines) and vibrator (dashed lines) type motion of the hydrogen atom in a magnetic "eld at scaled energy EI "!0.4. Inset: PoincareH surface of section in semiparabolical coordinates (k,p ;l"0) (from [84]). I
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and gM"4M #2M !1 for the vibrator type orbits. However, the problem is to extract the semiclassical eigenenergies from Eq. (3.45) because the periodic orbit sum does not converge. We now adopt the method discussed in Section 3.2 and adjust the semiclassical recurrence signal, i.e., the Fourier transform of the weighted density of states w\ . (w) (Eq. (3.45)) C(s)" AMd(s!sM) ,
(3.49)
M
with the amplitudes being determined exclusively by periodic orbit quantities, sM AM" e\ p EM , M "g " # to the functional form of its quantum mechanical analogue
(3.50)
C (s)"!i d e\ UIQ , (3.51) I I where the w are the quantum eigenvalues of the scaling parameter, and the d are the multiI I plicities of the eigenvalues (d "1 for non-degenerate states). The frequencies obtained from this I procedure are interpreted as the semiclassical eigenvalues w . The technique used to adjust I Eq. (3.49) to the functional form of Eq. (3.51) is harmonic inversion [82,83]. For the hydrogen atom in a magnetic "eld part of the semiclassical recurrence signal C(s) at scaled energy EI "!0.4 is presented in Fig. 33. The solid and dashed peaks mark the recurrencies
Fig. 33. Semiclassical recurrence signal C(s) for the hydrogen atom in a magnetic "eld at scaled energy EI "!0.4. Solid and dashed sticks: signal from the rotator and vibrator type motion, respectively (from [84]).
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of the rotator and vibrator-type orbits, respectively. Note that C(s) can be easily calculated even for long periods s with the help of the spline interpolation functions in Fig. 32. By contrast, the construction of the recurrence signal for Gutzwiller's trace formula usually requires an exponentially increasing e!ort for the numerical periodic orbit search with growing period. We have analyzed C(s) by the harmonic inversion technique in the region 0(s/2p(200. The resulting semiclassical spectrum of the lowest 106 states with eigenvalues w(20 is shown in the upper part of Fig. 34a. For graphical purposes the spectrum is presented as a function of the squared scaling parameter w, which is equivalent to unfolding the spectrum to constant mean level spacing. For comparison the lower part of Fig. 34a shows the exact quantum spectrum. The semiclassical and quantum spectrum are seen to be in excellent agreement, and deviations are less than the stick widths for nearly all states. The distribution P(d) of the semiclassical error with d"(w !w )/*w the error in units of the mean level spacing, *w "1.937/w, is presented in Fig. 34b. For most levels the semiclassical error is less than 4% of the mean level spacing, which is typical for a system with two degrees of freedom [110]. The accuracy of the results presented in Fig. 34 seems to be surprising for two reasons. First, we have not exploited the mean staircase function N M (w), i.e., the number of eigenvalues w with I w (w, which is a basic requirement of some other semiclassical quantization techniques for I bound chaotic systems [65,61]. Second, as mentioned before, Eq. (3.46) has been derived for near-integrable systems, and is only an approximation, in particular, for mixed systems. We have not taken into account any more re"ned extensions of the Berry}Tabor formula Eq. (3.45) as
Fig. 34. (a) Semiclassical and quantum mechanical spectrum of the hydrogen atom in a magnetic "eld at scaled energy EI "!0.4. (b) Distribution P(d) of the semiclassical error in units of the mean level spacing, d"(w !w )/*w for the lowest 106 eigenvalues (from [84]).
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discussed, e.g., in [27,28]. The answer to the second point is that the splitting of scaled actions of the periodic orbit pairs used in Fig. 32 does not exceed *s"0.022, and therefore for states with w(20 the phase shift between the two periodic orbit contributions is w*s"0.44, at most. For small phase shifts the extension of the Berry}Tabor formula to near-integrable systems results in a damping of the amplitudes of the periodic orbit recurrence signal in Fig. 33 but seems not to e!ect the frequencies, i.e., the semiclassical eigenvalues w obtained by the harmonic inversion of the function I C(s). To summarize, in this section we have presented a solution to the fundamental problem of semiclassical quantization of non-integrable systems in the mixed regular-chaotic regime. We have demonstrated the excellent quality of our procedure for the hydrogen atom in a magnetic "eld at a scaled energy EI "!0.4, where about 40% of the phase space volume is chaotic. The lowest 106 semiclassical and quantum eigenenergies have been shown to agree within a few percent of the mean level spacings. The same method can be applied straightforwardly to other systems with mixed dynamics. 3.5. Harmonic inversion of cross-correlated periodic orbit sums In the previous sections we have introduced harmonic inversion of semiclassical signals as a powerful and universal technique for the problem of periodic orbit quantization in that it does not depend on special properties of the system such as being bound and ergodic, or the existence of a symbolic dynamics. The method only requires the knowledge of periodic orbits and their physical quantities up to a certain maximum period, which depends on the average local density of states. Unfortunately, this method is not free of the general drawback of most semiclassical approaches, which su!er from a rapid proliferation of periodic orbits with their period, which in turn requires an enormous number of orbits to be taken into account. As discussed in Section 3.1.4 the required signal length for harmonic inversion is at least two times the Heisenberg period. This is by a factor of four longer than the signal length required for the application of the Riemann}Siegel look-alike formula [59}61]. In chaotic systems, where the number of periodic orbits grows exponentially with the period, the reduction of the required signal length by a factor of four implies that, e.g., instead of one million periodic orbits a reduced set of about 32 orbits is su$cient for the semiclassical quantization. This example clearly illustrates that a shortening of the required signal length is highly desirable. In this section we want to introduce harmonic inversion of cross-correlated periodic orbit sums as a method to reduce the required amount of periodic orbit data [85]. The idea is that the informational content of a D;D cross-correlated time signal is increased roughly by a factor of D as compared to a 1;1 signal. The cross-correlated signal is constructed by introducing a set of D smooth and linearly independent operators. Numerically, the harmonic inversion of the crosscorrelated periodic orbit sum is based on an extension of the "lter-diagonalization method to the case of time cross-correlation functions [68,137,138]. This extended method provides highly resolved spectra even in situations of nearly degenerate states, as well as the diagonal matrix elements for the set of operators chosen. The power of the method will be demonstrated in Section 3.7 for the circle billiard, as an example of a completely integrable system. Consider a quantum Hamiltonian HK whose eigenvalues are w and eigenstates "n2. [As preL viously, we consider scaling systems with w the scaling parameter, which is not necessarily the
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energy, and with linear scaling of the classical action, S "ws .] We introduce a cross-correlated response function (a, b"1,2,2,D) b b ?L @L , (3.52) g (w)" ?@ w!w #i0 L L where b and b are the diagonal matrix elements of two operators AK andK A , respectively, i.e. ?L @L ? @ b "1n"AK "n2 . (3.53) ?L ? Later we have to "nd a semiclassical approximation to Eq. (3.52). In this context, it is important to note that g (w) can only be written as a trace formula ?@ g (w)"tr+AK GK >(w)AK , ?@ ? @ with the Green function GK >(w)"1/(w!HK #i0) , if either AK or AK commutes with HK . The weighted density of states is given by ? @ (3.54) . (w)"!(1/p) Im g (w) . ?@ ?@ Let us assume that the semiclassical approximation g (w) to the quantum expression Eq. (3.52) is ?@ given. The general procedure of harmonic inversion as described in Section 3.2 would then be to adjust the Fourier transform of g (w) to the functional form of the quantum expression ?@ 1 > (3.55) C (s)" g (w)e\ QU dw"!i b b e\ ULQ . ?@ ?@ ?L @L 2p \ L The conventional harmonic inversion problem is formulated as a nonlinear "t of the signal C(s) by the sum of sinusoidal terms (see Section 2.2 and Appendix A.1),
C(s)" d e\ ULQ , L L with the set of, in general, complex variational parameters +w ,d "!ib b ,. As already disL L ?L @L cussed in Section 3.1.4, simple information theoretical considerations then yield an estimate for the required signal length, s &4p.N (w), for poles w 4w which can be extracted by this method. L
When a periodic orbit approximation of the quantum signal C(s) is used, this estimate results sometimes in a very unfavorable scaling because of a rapid (exponential for chaotic systems) proliferation of periodic orbits with increasing period. Let us consider now a generalized harmonic inversion problem, which assumes that the whole s-dependent D;D signal C (s) is adjusted ?@ simultaneously to the form of Eq. (3.55), with b and w being the variational parameters. The ?L L advantage of using the cross-correlation approach [68,137,138] is based on the simple argument that the total amount of independent information contained in the D;D signal is D(D#1) multiplied by the length of the signal, while the total number of unknowns (here b and w ) is ?L L (D#1) times the total number of poles w . Therefore, the informational content of the D;D signal L per unknown parameter is increased (compared to the case of Eq. (3.32)) by a factor of D. [Of course, this scaling holds only approximately and for su$ciently small numbers D of operators
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AK chosen.] Thus, we have the result that, provided we are able to obtain a periodic orbit ? approximation for C (s), with this procedure we can extract more information from the same set of ?@ periodic orbits. We now have to "nd a semiclassical approximation for C (s) in Eq. (3.55). The problem has been ?@ solved in the literature only for special cases, i.e., if one or both operators AK and AK are the identity ? @ or, somewhat more general, if at least one of the operators commutes with the Hamiltonian HK . For the identity operator AK "I the element C (s) is the Fourier transform of Gutzwiller's trace formula [7,8] for chaotic systems, and of the Berry}Tabor formula [25,26] for regular systems, i.e. C (s)" A d(s!s ) , (3.56) where s are the periods of the orbits and A the amplitudes (recurrence strengths) of the periodic orbit contributions including phase information. For AK "I and an arbitrary smooth operator AK the elements C (s) are obtained from a semiclassical approximation to the generalized trace ? ? formula g (w)"tr+GK >(w)AK , , (3.57) ? ? which has been investigated in detail in [128,129]. The result is that the amplitudes A in Eq. (3.56) have to be multiplied by the classical average of the observable A along the periodic ? orbit,
1 Q A (q(s), p(s)) ds , " ? ? s with A (q, p) the Wigner transform of the operator AK , i.e., the signal C (s) reads ? ? ? a
(3.58)
C (s)" a A d(s!s ) . (3.59) ? ? If at least one of the operators AK and AK commutes with HK , Eq. (3.52) can still be written as a trace ? @ formula, g (w)"tr+AK GK >(w)AK ,"tr+GK >(w)(AK AK ), , ?@ ? @ ? @ and Eq. (3.59) can be applied to the product AK AK . However, we do not want to restrict the ? @ operators to those commuting with HK , which obviously would be a severe restriction especially for chaotic systems, and the problem is now to "nd a semiclassical approximation to Eq. (3.55) for the general case of two arbitrary smooth operators AK and AK . A reasonable assumption is that the ? @ amplitudes A in Eq. (3.56) have to be multiplied by the product of the classical averages, a a , of these two observables, i.e. ? @ C (s)" a a A d(s!s ) . (3.60) ?@ ? @ Although no rigorous mathematical proof of Eq. (3.60) will be given here, we have strong numerical evidence, from the high-resolution analysis of quantum spectra, part of which will be
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given below, that the conjecture of Eq. (3.60) is correct. Details on semiclassical non-trace type formulae like Eq. (3.60) are given in [139]. Eq. (3.60) is the starting point for the following application of harmonic inversion of cross-correlation functions. Note that all quantities in Eq. (3.60) are obtained from the classical periodic orbits. The idea of periodic orbit quantization by harmonic inversion of cross-correlated periodic orbit sums is to "t the semiclassical functions C (s) given in a "nite range 0(s(s to the functional ?@
form of the quantum expression Eq. (3.55). As for the harmonic inversion of a one-dimensional signal (see Section 3.2) the frequencies of the harmonic inversion analysis are then identi"ed with the semiclassical eigenvalues w . The amplitudes b are identi"ed with the semiclassical approxiL ?L mations to the diagonal matrix elements 1n"AK "n2. Here we only give a brief description how the ? harmonic inversion method is extended to cross-correlation functions. The details of the numerical procedure of solving the generalized harmonic inversion problem Eq. (3.55) are presented in [68,137,138] and in Appendix A.2. As for the harmonic inversion of a single function the idea is to recast the nonlinear "t problem as a linear algebraic problem [68]. This is done by associating the signal C (s) (to be inverted) with a time cross-correlation function between an initial state U and ?@ ? a "nal state U , @ (3.61) C (s)"(U ,e\ XK QU ) , ? ?@ @ where the "ctitious quantum dynamical system is described by an e!ective Hamiltonian XK . The latter is de"ned implicitly by relating its spectrum to the set of unknown spectral parameters w and L b . Diagonalization of XK would yield the desired w and b . This is done by introducing an ?L L ?L appropriate basis set in which the matrix elements of XK are available only in terms of the known signals C (s). The Hamiltonian XK is assumed to be complex symmetric even in the case of a bound ?@ system. This makes the harmonic inversion stable with respect to `noisea due to the imperfections of the semiclassical approximation. The most e$cient numerical and practical implementation of the harmonic inversion method with all relevant formulae can be found in [137,138] and Appendix A.2. The method of harmonic inversion of cross-correlated periodic orbit sums will be applied in Section 3.7 to the circle billiard. As will be shown, for a given number of periodic orbits the accuracy of semiclassical spectra can be signi"cantly improved with the help of the crosscorrelation approach, or, alternatively, spectra with similar accuracy can be obtained from a periodic orbit cross-correlation signal with signi"cantly reduced signal length. 3.6. expansion for the periodic orbit quantization by harmonic inversion Semiclassical spectra can be obtained for both regular and chaotic systems in terms of the periodic orbits of the system. For chaotic dynamics the semiclassical trace formula was derived by Gutzwiller [7,8], and for integrable systems the Berry}Tabor formula [25,26] is well known to be precisely equivalent to the EBK torus quantization [1}3]. However, as already has been discussed in Section 2.6, the semiclassical trace formulae are exact only in exceptional cases, e.g., the geodesic motion on the constant negative curvature surface. In general, they are just the leading-order terms of an in"nite series in powers of the Planck constant and the accuracy of semiclassical quantization is still an object of intense investigation [109,110,140]. Methods for the calculation of the higher-order periodic orbit contributions were developed in [22}24]. In Section 2.6 we have
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demonstrated how the periodic orbit quantities of the expanded trace formula can be extracted from the quantum and semiclassical spectra. It is an even more fundamental problem to obtain semiclassical eigenenergies beyond the Gutzwiller and Berry}Tabor approximation directly from the expanded periodic orbit sum. Note that the expansion of the periodic orbit sum does not solve the general problem of the construction of the analytic continuation of the trace formula, which is already a fundamental problem when only the leading-order terms of the expansion is considered. Up to now the expansion for periodic orbit quantization is restricted to systems with known symbolic dynamics, like the three-disk scattering problem, where cycle expansion techniques can be applied [23,24], and semiclassical eigenenergies beyond the Gutzwiller and Berry}Tabor approximation cannot be calculated, e.g., for bound systems with the help of Riemann}Siegel-type formulae [60,61] or surface of section techniques [62,63]. In this section we extend the method of periodic orbit quantization by harmonic inversion to the analysis of the
expansion of the periodic orbit sum. The accuracy of semiclassical eigenvalues can be improved by one to several orders of magnitude, as will be shown in Section 3.7 by way of example of the circle billiard. As in Section 2.6 we consider systems with a scaling property, i.e., where the classical action scales as S "ws , and the scaling parameter w, \ plays the role of an inverse e!ective Planck constant. The expansion of the periodic orbit sum is given (see Eq. (2.65)) as a power series in w\, 1 ALe QU . (3.62) g(w)" g (w)" L wL L L The complex amplitudes AL of the nth-order periodic orbit contributions include the phase information from the Maslov indices. For periodic orbit quantization the zeroth-order contributions A are usually considered only. The Fourier transform of the principal periodic orbit sum
1 > C (s)" g (w)e\ QU dw" Ad(s!s ) 2p \
(3.63)
is adjusted by application of the harmonic inversion technique (see Section 3.2) to the functional form of the exact quantum expression
1 > d I C(s)" e\ UQ dw"!i d e\ UIQ , I 2p w!w #i0 \ I I I
(3.64)
with +w , d , the eigenvalues and multiplicities. The frequencies w obtained by harmonic I I I inversion of Eq. (3.63) are the zeroth-order approximation to the semiclassical eigenvalues. We will now demonstrate how the higher-order correction terms to the semiclassical eigenvalues can be extracted from the periodic orbit sum Eq. (3.62). We "rst remark that the asymptotic expansion Eq. (3.62) of the semiclassical response function su!ers, for n51, from the singularities at w"0, and it is therefore not appropriate to harmonically invert the Fourier transform of Eq. (3.62), although the Fourier transform formally exists. This means that the method of periodic orbit quantization by harmonic inversion cannot straightforwardly be extended to the expansion of the periodic orbit sum. Instead we will calculate the correction terms to the semiclassical eigenvalues separately, order by order, as described in the following.
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Let us assume that the (n!1)th-order approximations w to the semiclassical eigenvalues I L\ are already obtained and the w are to be calculated. The di!erence between the two subsequent I L approximations to the quantum mechanical response function reads [86]
d d *w d I I I L I ! + , g (w)" L #i0 (w!wN #i0) w!w #i0 w!w I L\ I L I L I I
(3.65)
with wN "(w #w )/2 and *w "w !w . Integration of Eq. (3.65) and multiplicaI L I L I L\ I L I L I L\ tion by wL yields
!d wL*w I I L , G (w)"wL g (w) dw" L L w!wN #i0 I L I
(3.66)
which has the functional form of a quantum mechanical response function but with residues proportional to the nth-order corrections *w to the semiclassical eigenvalues. The semiclassical I L approximation to Eq. (3.66) is obtained from the term g (w) in the periodic orbit sum Eq. (3.62) by L integration and multiplication by wL, i.e.
1 1 . G (w)"wL g (w) dw"!i ALe UQ#O L L w s
(3.67)
We can now Fourier transform both Eqs. (3.66) and (3.67), and obtain (n51)
1 > C (s), G (w)e\ UQ dw"i d (w )L*w e\ UIQ L L I I I L 2p \ I 1 "!i ALd(s!s ) . s
(3.68)
(3.69)
Eqs. (3.68) and (3.69) are the main result of this section. They imply that the expansion of the semiclassical eigenvalues can be obtained, order by order, by harmonic inversion (h.i.) of the periodic orbit signal in Eq. (3.69) to the functional form of Eq. (3.68). The frequencies of the periodic orbit signal Eq. (3.69) are the semiclassical eigenvalues w . Note that the accuracy I of the semiclassical eigenvalues does not necessarily increase with increasing order n. We indicate this in Eq. (3.68) by omitting the index n at the eigenvalues w . The corrections *w to the I I L eigenvalues are obtained from the amplitudes, d (w )L*w , of the periodic orbit signal. I I I L The method requires as input the periodic orbits of the classical system up to a maximum period (scaled action), s , determined by the average density of states [82,83]. The amplitudes A are
obtained from Gutzwiller's trace formula [7,8] and the Berry}Tabor formula [25,26] for chaotic and regular systems, respectively. For the next order correction A explicit formulae were derived y Gaspard and Alonso for chaotic systems with smooth potentials [22] and in [23,24] for billiards. With appropriate modi"cations [87,88] the formulae can be used for regular systems as well. As an example we investigate the expansion of the periodic orbit sum for the circle billiard in the next section.
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3.7. The circle billiard In Section 3.5 (see also [85]) we have introduced harmonic inversion of cross-correlated periodic orbit sums as a method to signi"cantly reduce the required number of periodic orbits for semiclassical quantization, and in Section 3.6 (see also [86]) we have discussed the expansion of the periodic orbit sum and the calculation of semiclassical eigenenergies beyond the Gutzwiller [8] and Berry}Tabor [25,26] approximation. We now demonstrate both methods by way of example of the circle billiard. The circle billiard is a regular system and has been chosen here for the following reasons. 1. The nearest-neighbor level statistics of integrable systems is a Poisson distribution, with a high probability for nearly degenerate states. The conventional method for periodic orbit quantization by harmonic inversion requires very long signals to resolve the nearly degenerate states. We will demonstrate the power of harmonic inversion of cross-correlated periodic orbit sums by fully resolving those nearly degenerate states with a signi"cantly reduced set of orbits. 2. All relevant physical quantities, i.e., the quantum and semiclassical eigenenergies, the matrix elements of operators, the periodic orbits and their zeroth- and "rst-order amplitudes of the
expanded periodic orbit sum, and the periodic orbit averages of classical observables can easily be obtained. 3. The semiclassical quantization of the circle billiard as an example of an integrable system demonstrates the universality and wide applicability of periodic orbit quantization by harmonic inversion, i.e., the method is not restricted to systems with hyperbolic dynamics like, e.g., pinball scattering. The circle billiard has already been introduced in Section 2.6.1. The exact quantum mechanical eigenvalues E" k/2M are given as zeros of Bessel functions J (kR)"0, where m is the angular K momentum quantum number and R, the radius of the circle. In the following we choose R"1. The semiclassical eigenvalues are obtained from the EBK quantization condition, Eq. (2.71), kR(1!(m/kR)!"m" arccos("m"/kR)"p(n#) , and the expansion of the periodic orbit sum reads (see Eq. (2.73)) 1 1 g(k)" ALe lI , g (k)" L kL (k L (k L 1
with (see Eqs. (2.74) and (2.75))
p l e\ p I>p , A" 2 m 1 5!2 sinc e\ p I\p . A" (pm 3sin c 2 The angle c is de"ned as c,pm /m , with m "1,2,2 the number of turns of the periodic orbit ( ( around the origin, and m "2m ,2m #1,2 the number of re#ections at the boundary of the ( (
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circle. l "2m sin c and k "3m are the geometrical length and Maslov index of the orbits, respectively. 3.7.1. Harmonic inversion of the cross-correlated periodic orbit sum We now calculate the semiclassical eigenenergies of the circle billiard by harmonic inversion of the cross-correlated periodic orbit sum Eq. (3.60) with A "A (Eq. (2.74)) the amplitudes of the Berry}Tabor formula [25,26], i.e., the lowest-order approximation. To construct the periodic orbit cross-correlation signal C (l) we choose three di!erent operators, ?@ AK "I the identity, AK "r the distance from the origin, and AK "(¸/k) the square of the scaled angular momentum. For these operators the classical weights a (Eq. (3.58)) are obtained as ? a "1 , a "(1#(cos c/tan c)arsinh tan c) , a "cos c . (3.70) Once all the ingredients of Eq. (3.60) for the circle billiard are available, the 3;3 periodic orbit cross-correlation signal C (l) can easily be constructed and inverted by the generalized "lter?@ diagonalization method. Results obtained from the periodic orbits with maximum length s "100 are presented in Fig. 35. Fig. 35a is part of the density of states, . (k), Fig. 35b and
Fig. 35c is the density of states weighted with the diagonal matrix elements of the operators AK "r and AK "¸, respectively. The squares are the results from the harmonic inversion of the periodic orbit cross-correlation signals. For comparison the crosses mark the matrix elements obtained by exact quantum calculations at positions k# ) obtained from the EBK quantization condition Eq. (2.71). In this section, we do not compare with the exact zeros of the Bessel functions because Eq. (3.60) is correct only to "rst order in and thus the harmonic inversion of C (s) cannot provide ?@ the exact quantum mechanical eigenvalues. The calculation of eigenenergies beyond the Berry}Tabor approximation will be discussed in Section 3.7.2. However, the perfect agreement between the eigenvalues k&' obtained by harmonic inversion and the EBK eigenvalues k# ) is remarkable, and this is even true for nearly degenerate states marked by arrows in Fig. 35a. The eigenvalues of some nearly degenerate states are presented in Table 11. It is important to emphasize that these states with level splittings of, e.g., *k"6;10\ cannot be resolved by the originally proposed method of periodic orbit quantization by harmonic inversion (see Section 3.2) with a periodic orbit signal length s "100. To resolve the two levels at k+11.049 (see Table 11)
a signal length of at least s +500 is required if a single periodic orbit function C(s) is used
instead of a cross-correlation function. The method presented in Section 3.5 can therefore be used
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Fig. 35. Density of states weighted with the diagonal matrix elements of the operators (a) AK "I, (b) AK "r, (c) AK "¸ for the circle billiard with radius R"1. Crosses: EBK eigenvalues and quantum matrix elements. Squares: eigenvalues and matrix elements obtained by harmonic inversion of cross-correlated periodic orbit sums. Three nearly degenerate states are marked by arrows (from [85]).
to signi"cantly reduce the required signal length and thus the required number of periodic orbits for periodic orbit quantization by harmonic inversion. As such the part of the spectrum shown in Fig. 35 can even be resolved, apart from the splittings of the nearly degenerate states marked by the arrows, from a short cross-correlation signal with s "30, which is about the Heisenberg period
s "2p.N (k), i.e. half of the signal length required for the harmonic inversion of a 1;1 signal. With & "ve operators and a 5;5 cross-correlation signal highly excited states around k"130 have even been obtained with a signal length s +0.7s [87,88], which is close to the signal length
& s +0.5s required for the Riemann}Siegel-type quantization [59}61]. The reduction of the
& signal length is especially important if the periodic orbit parameters are not given analytically, as in our example of the circle billiard, but must be obtained from a numerical periodic orbit search. How small can s get as one uses more and more operators in the method? It might be that half of
the Heisenberg period is a fundamental barrier for bound systems with chaotic dynamics in analogy to the Riemann}Siegel formula [61] while for regular systems an even further reduction of the signal length should in principle be possible. However, further investigations are necessary to clarify this point.
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Table 11 Nearly degenerate semiclassical states of the circle billiard. k# ): results from EBK-quantization. k&': eigenvalues obtained by harmonic inversion of cross-correlated periodic orbit sums. States are labeled by the radial and angular momentum quantum numbers (n,m) k# )
k&'
4 7
11.048664 11.049268
11.048569 11.049239
3 0
1 9
13.314197 13.315852
13.314216 13.315869
3 1
2 7
14.787105 14.805435
14.787036 14.805345
1 5
11 1
19.599795 19.609451
19.599863 19.608981
1 6
15 2
24.252501 24.264873
24.252721 24.264887
n
m
1 0
3.7.2. Periodic orbit quantization beyond the Berry}¹abor approximation The semiclassical eigenvalues obtained by harmonic inversion of a cross-correlated or a su$ciently long single signal are in excellent agreement with the results of the EBK torus quantization, Eq. (2.71). However, they deviate from the exact quantum mechanical eigenenergies, i.e., the zeros of the Bessel functions because the Berry}Tabor formula [25,26] is only the lowest-order approximation of the periodic orbit sum. We now demonstrate the expansion of the periodic orbit sum and apply the technique discussed in Section 3.6 to the circle billiard. The "rst-order corrections to the semiclassical eigenvalues, *k"k!k are obtained by harmonic inversion of the periodic orbit signal C (l) (see Eq. (3.69)), 1 i d k[k!k]e\ IHl , Ad(l!l )" C (l)"!i H H H H l H
(3.71)
with d the multiplicities of states. The signal C (l) in Eq. (3.71) can be inverted as a single function H as has been done in [86], where the accuracy of the eigenenergies was improved by one to several orders of magnitude, apart from the nearly degenerate states marked by arrows in Fig. 35. Here we go one step further and use Eq. (3.71) as part of a 3;3 cross-correlation signal. For the two other diagonal components of the cross-correlation matrix we use the identity operator, AK "I, and the distance from the origin, AK "r. By applying the cross-correlation technique of Section 3.5 we obtain both the zeroth- and "rst-order expansion of the eigenenergies and the diagonal matrix elements of the chosen operators simultaneously from one and the same harmonic inversion procedure. Furthermore, we can now even resolve the nearly degenerate states. The spectrum of the integrated di!erences of the density of states *. (k) dk obtained by harmonic inversion of the 3;3 cross-correlation matrix with signal length l "150 is shown in Fig. 36. The squares mark the
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Fig. 36. Integrated di!erence of the density of states, *. (k) dk, for the circle billiard with radius R"1. Crosses: *. (k)". (k)!. # )(k). Squares: . (k)". (k)!. (k) obtained from the expansion of the periodic orbit signal.
spectrum for *. (k)". (k)!. (k) obtained from the harmonic inversion of the signal C (s). For comparison, the crosses present the same spectrum but for the di!erence *. (k)" . (k)!. # )(k) between the exact quantum mechanical and the EBK-spectrum. The deviations between the peak heights exhibit the contributions of terms of the expansion series beyond the "rst-order approximation. The peak heights of the levels in Fig. 36 (solid lines and crosses) are, up to a multiplicity factor for the degenerate states, the shifts *k between the zeroth- and "rst-order semiclassical approximations to the eigenvalues k. The zeroth- and "rst-order eigenvalues, k and k"k#*k are presented in Table 12 for the 40 lowest eigenstates. The zeroth-order eigenvalues, k, agree within the numerical accuracy with the results of the torus quantization, Eq. (2.71) (see eigenvalues k# ) in Table 12). However, the semiclassical eigenvalues deviate signi"cantly, especially for states with low radial quantum numbers n, from the exact quantum mechanical eigenvalues k in Table 12. By contrast, the semiclassical error of the "rst-order eigenvalues, k, is by orders of magnitude reduced compared to the lowest-order approximation. An appropriate measure for the accuracy of semiclassical eigenvalues is the deviation from the exact quantum eigenvalues in units of the average level spacings, 1*k2 "1/.N (k). Fig. 37 presents the semiclassical error in units of the average level spacings 1*k2 +4/k for the zeroth-order (diamonds) and "rst-order (crosses) approximations to the eigenvalues. In the zeroth-order approximation the semiclassical error for the low lying states is about 3}10% of the mean level spacing. This error is reduced in the "rst-order approximation by at least one order of magnitude for the least semiclassical states with radial quantum number n"0. The accuracy of states with n51 is improved by two or more orders of magnitude. Finally, we want to note that the small splittings between the nearly degenerate states are extremely sensitive to the higher-order corrections. E.g., in the zeroth-order approximation the splitting between the two states around k+11.05 is *k"6;10\. In the "rst-order approximation the splitting between the same states is *k"0.0242, which is very close to the exact splitting *k"0.0217. The accuracy obtained here for the "rst order approximations to the nearly degenerate states goes beyond the results presented in [86], and is achieved by the combined application of the methods introduced in Sections 3.5 and 3.6, i.e., the harmonic inversion of cross-correlation functions and the analysis of the expanded periodic orbit sum, respectively.
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Table 12 The 40 lowest eigenstates of the circle billiard with radius R"1. n,m: radial and angular momentum quantum numbers; k# ): results from EBK-quantization; k and k: zeroth and "rst order semiclassical eigenvalues obtained by harmonic inversion of the periodic orbit signal; k: exact eigenvalues, i.e., zeros of the Bessel functions J (kR)"0 K n
m
k# )
k
k
k
0 0 0 1 0 1 0 1 2 0 1 0 2 1 0 2 3 0 1 2 3 0 1 2 0 3 1 4 0 2 1 3 4 0 2 1 3 0 4 5
0 1 2 0 3 1 4 2 0 5 3 6 1 4 7 2 0 8 5 3 1 9 6 4 10 2 7 0 11 5 8 3 1 12 6 9 4 13 2 0
2.356194 3.794440 5.100386 5.497787 6.345186 6.997002 7.553060 8.400144 8.639380 8.735670 9.744628 9.899671 10.160928 11.048664 11.049268 11.608251 11.780972 12.187316 12.322723 13.004166 13.314197 13.315852 13.573465 14.361846 14.436391 14.787105 14.805435 14.922565 15.550089 15.689703 16.021889 16.215041 16.462981 16.657857 16.993489 17.225257 17.607830 17.760424 17.952638 18.064158
2.356230 3.794440 5.100382 5.497816 6.345182 6.997006 7.553055 8.400145 8.639394 8.735672 9.744627 9.899660 10.160949 11.048635 11.049228 11.608254 11.780993 12.187302 12.322721 13.004167 13.314192 13.315782 13.573464 14.361846 14.436375 14.787091 14.805457 14.922572 15.550084 15.689701 16.021888 16.215047 16.462982 16.657846 16.993486 17.225252 17.607831 17.760386 17.952662 18.064201
2.409288 3.834267 5.138118 5.520550 6.382709 7.015881 7.590990 8.417503 8.653878 8.774213 9.761274 9.938954 10.173568 11.063791 11.087943 11.619919 11.791599 12.228037 12.338847 13.015272 13.323418 13.356645 13.589544 14.372606 14.478531 14.795970 14.821595 14.930938 15.593060 15.700239 16.038034 16.223499 16.470648 16.701442 17.003884 17.241482 17.615994 17.804708 17.959859 18.071125
2.404826 3.831706 5.135622 5.520078 6.380162 7.015587 7.588342 8.417244 8.653728 8.771484 9.761023 9.936110 10.173468 11.064709 11.086370 11.619841 11.791534 12.225092 12.338604 13.015201 13.323692 13.354300 13.589290 14.372537 14.475501 14.795952 14.821269 14.930918 15.589848 15.700174 16.037774 16.223466 16.470630 16.698250 17.003820 17.241220 17.615966 17.801435 17.959819 18.071064
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Fig. 37. Semiclassical error "k!k" (diamonds) and "k!k" (crosses) in units of the average level spacing 1*k2 +4/k.
3.8. Semiclassical calculation of transition matrix elements for atoms in external xelds The interpretation of photoabsorption spectra of atoms in external "elds is a fundamental problem of atomic physics. Although the `exacta quantum mechanics accurately describes the energies and transition strengths of individual levels it has completely failed to present a simple physical picture of the long-ranged modulations which have been observed in early low-resolution spectra of barium atoms in a magnetic "eld [141] and later in the Fourier transform recurrence spectra of the magnetized hydrogen atom [142,71]. However, the long-ranged modulations of the quantum photoabsorption spectra can be naturally interpreted in terms of the periods of classical closed orbits starting at and returning back to the nucleus where the initial state is localized. The link between the quantum spectra and classical trajectories is given by closed orbit theory [90}93] which describes the photoabsorption cross section as the sum of a smooth part and the superposition of sinusoidal modulations. The frequencies, amplitudes, and phases of the modulations are directly obtained from the quantities of the closed orbits. When the photoabsorption spectra are Fourier transformed or analyzed with a high resolution method (see Section 2.1) the sinusoidal modulations result in sharp peaks in the Fourier transform recurrence spectra, and closed orbit theory has been most successful to explain quantum mechanical recurrence spectra qualitatively and even quantitatively in terms of the closed orbits of the underlying classical system [46}48,51]. However, up to now practical applications of closed orbit theory have always been restricted to the semiclassical calculation of low-resolution photoabsorption spectra for the following two reasons. First, the closed orbit sum requires, in principle, the knowledge of all orbits up to in"nite length, which are usually not available from a numerical closed orbit search, and second, the in"nite closed orbit sum su!ers from fundamental convergence problems [90}93]. It is therefore commonly accepted that the calculation of individual transition matrix elements 1 "D"t 2 of the dipole operator D, which describe the transition strengths from the initial state " 2 to "nal states "t 2, is a problem beyond the applicability of the semiclassical closed orbit theory, i.e., is the domain of quantum mechanical methods. In this section we disprove this common believe and demonstrate that individual eigenenergies and transition matrix elements can be directly extracted from the quantities of the classical closed
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orbits. To that end, we slightly generalize closed orbit theory to the semiclassical calculation of cross-correlated recurrence functions. We then adopt the cross-correlation approach introduced in Section 3.5 to harmonically invert the cross-correlated recurrence signal and to extract the semiclassical eigenenergies and transition matrix elements. Results will be presented for the photoexcitation of the hydrogen atom in a magnetic "eld. The oscillator strength f for the photoexcitation of atoms in external "elds can be written as f (E)"!(2/p)(E!E )Im 1 "DG>D" 2 , #
(3.72)
where " 2 is the initial state at energy E , D is the dipole operator, and G> the retarded Green's # function of the atomic system. The basic steps for the derivation of closed orbit theory are to replace the quantum mechanical Green's function in Eq. (3.72) with its semiclassical Van Vleck}Gutzwiller approximation and to carry out the overlap integrals with the initial state " 2. Here we go one step further by introducing a cross-correlation matrix g "1 "DG>D" 2 ??Y ? # ?Y
(3.73)
with " 2, a"1,2,2,¸, a set of independent initial states. As will be shown below the use of ? cross-correlation matrices can considerably improve the convergence properties of the semiclassical procedure. In the following, we will concentrate on the hydrogen atom in a magnetic "eld with c"B/(2.35;10 T) the magnetic "eld strength in atomic units. As discussed in Section 2.1 the system has a scaling property, i.e., the shape of periodic orbits does not depend on the scaling parameter, w"c\ " \, and the classical action scales as S"sw with s the scaled action. As, e.g., in [47] we consider scaled photoabsorption spectra at constant scaled energy EI "Ec\ as a function of the scaling parameter w. We choose dipole transitions between states with magnetic quantum number m"0. Note that the following ideas can be applied in an analogous way to atoms in electric "elds. Following the derivation of [91,93] the semiclassical approximation to the #uctuating part of g in Eq. (3.73) reads ??Y !(2p) (sin 0 sin 0 Y (0 )Y (0 )e QU\p I>p , g (w)"w\ ? ?Y ??Y ("m "
(3.74)
with s and k the scaled action and Maslov index of the closed orbit (co), m an element of the monodromy matrix, and 0 and 0 the initial and "nal angle of the trajectory with respect to the magnetic "eld axis. The angular functions Y (0) depend on the states " 2 and the dipole operator ? ? D and are given as a linear superposition of Legendre polynomials, Y (0)" B P (cos 0) with J? J ? usually only few nonzero coe$cients B with low l. Explicit formulae for the Jcalculation of the J? coe$cients can be found in [91,93] and in Appendix B. The problem is now to extract the semiclassical eigenenergies and transition matrix elements from Eq. (3.74) because the closed orbit sum does not converge. The Fourier transformation of w g (w) yields the cross-correlated ??Y recurrence signals C (s)" A d(s!s ) , ??Y ??Y
(3.75)
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with the amplitudes (3.76) A "(!(2p) /("m ")(sin 0 sin 0 Y (0 )Y (0 )e \p I>p ? ?Y ??Y being determined exclusively by closed orbit quantities. The corresponding quantum mechanical cross-correlated recurrence functions, i.e., the Fourier transforms of w g (w) read ??Y C (s)"!i b b e\ UIQ , (3.77) ??Y ?I ?YI I with w the eigenvalues of the scaling parameter, and I b "w 1 "D"t 2 (3.78) ?I I ? I proportional to the transition matrix element for the transition from the initial state " 2 to the ? "nal state "t 2. I The method to adjust Eq. (3.75) to the functional form of Eq. (3.77) for a set of initial states " 2, a"1,2,2,¸, is the harmonic inversion of cross-correlation functions as discussed in Sec? tion 3.5 and Appendix A.2. We now demonstrate the method of harmonic inversion of the cross-correlated closed orbit recurrence functions Eq. (3.75) for the example of the hydrogen atom in a magnetic "eld at constant scaled energy EI "!0.7. This energy was also chosen for detailed experimental investigations on the helium atom [48]. We investigate dipole transitions from the initial state " 2""2p02 with light polarized parallel to the magnetic "eld axis to "nal states with magnetic quantum number m"0. For this transition the angular function in Eq. (3.76) reads (see Appendix B) Y (0)"(2p)\ 2e\(4 cos 0!1). For the construction of a 2;2 cross-correlated recurrence signal we use for simplicity as a second transition formally an outgoing s-wave, i.e., D" 2J> , and, thus, Y (0)"const. A numerical closed orbit search yields 1395 primitive closed orbits (2397 orbits including repetitions) with scaled action s/2p(100. With the closed orbit quantities at hand it is straightforward to calculate the cross-correlated recurrence functions in Eq. (3.75). The real and imaginary parts of the complex functions C (s), C (s), and C (s) with s/2p(50 are presented in Figs. 38 and 39, respectively. Note that for symmetry reasons C (s)"C (s). We have inverted the 2;2 cross-correlated recurrence functions in the region 0(s/2p(100. The resulting semiclassical photoabsorption spectrum is compared with the exact quantum spectrum in Fig. 40a for the region 16(w(21 and in Fig. 40b for the region 34(w(40. The upper and lower parts in Fig. 40 show the exact quantum spectrum and the semiclassical spectrum, respectively. Note that the region of the spectrum presented in Fig. 40b belongs well to the experimentally accessible regime with laboratory "eld strengths B"6.0 T to B"3.7 T. The overall agreement between the quantum and semiclassical spectrum is impressive, even though a line by line comparison still reveals small di!erences for a few matrix elements. It is important to note that the high quality of the semiclassical spectrum could only be achieved by our application of the cross-correlation approach. For example, the two nearly degenerate states at w"36.969 and w"36.982 cannot be resolved and the very weak transition at w"38.894 with 12p0"D"t 2"0.028 is not detected with a single (1;1) recurrence signal of the same length. However, these hardly visible details are indeed present in the semiclassical spectrum in Fig. 40b obtained from the harmonic inversion of the 2;2 cross-correlated recurrence functions.
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Fig. 38. Real parts of the cross-correlated recurrence functions for the hydrogen atom in a magnetic "eld at constant scaled energy EI "!0.7 (from [89]).
To summarize, we have demonstrated that closed orbit theory is not restricted to describe long-ranged modulations in quantum mechanical photoabsorption spectra of atoms in external "elds but can well be applied to extract individual eigenenergies and transition matrix elements from the closed orbit quantities. This is achieved by the high-resolution spectral analysis (harmonic inversion) of cross-correlated closed orbit recurrence signals. For the hydrogen atom in a magnetic "eld we have obtained individual transition matrix elements between low lying and highly excited Rydberg states solely from the classical closed orbit data.
4. Conclusion In summary, we have shown that harmonic inversion is a powerful tool for the analysis of quantum spectra, and is the foundation for a novel and universal method for periodic orbit quantization of both regular and chaotic systems. The high-resolution analysis of "nite range quantum spectra allows to circumvent the restrictions imposed by the uncertainty principle of the conventional Fourier transformation. Therefore, physical phenomena can directly be revealed in the quantum spectra which previously were
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Fig. 39. Same as Fig. 38 but for the imaginary parts of the cross-correlated recurrence functions.
unattainable. Topical examples are the study of quantum manifestations of periodic orbit bifurcations and catastrophe theory, and the uncovering of symmetry breaking e!ects. The investigation of these phenomena provides a deeper understanding of the relation between quantum mechanics and the dynamics of the underlying classical system. The high-resolution technique is demonstrated in this work for numerically calculated quantum spectra of, e.g., the hydrogen atom in external "elds, three-disk pinball scattering, and the circle billiard. Theoretical spectra are especially suited for the high-resolution analysis because of the very high accuracy of most quantum computational methods. In principle, harmonic inversion may be applied to experimental spectra as well, e.g., to study atoms in external "elds measured with the technique of scaled-energy spectroscopy [45}55]. However, the exact requirements on the precision of experimental data to achieve high-resolution recurrence spectra beyond the limitations of the uncertainty principle are not yet known. It will certainly be a challenge for future experimental work to verify the quantum manifestations of bifurcations, which have been extracted here from theoretically computed spectra, using real systems in the laboratory. We have also introduced harmonic inversion as a new and general tool for semiclassical periodic orbit quantization. Here we brie#y recall the highlights of our technique. The method requires the complete set of periodic orbits up to a given maximum period as input but does not depend on special properties of the orbits, as, e.g., the existence of a symbolic code or a functional equation. The universality and wide applicability has been demonstrated by applying it to systems with
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Fig. 40. Quantum (upper part) and semiclassical (lower part) photoabsorption spectra of the hydrogen atom in a magnetic "eld at scaled energy EI "!0.7. Transition matrix elements 12p0"D"t 2 for dipole transitions with light polarized parallel to the magnetic "eld axis (from [89]).
completely di!erent properties, namely the zeros of the Riemann zeta function, the three disk scattering problem, and the circle billiard. These systems have been treated before by separate e$cient methods, which, however, are restricted to bound ergodic systems, systems with a complete symbolic dynamics, or integrable systems. The harmonic inversion technique allows to solve all these problems with one and the same method. The method has furthermore been successfully applied to the hydrogen atom in a magnetic "eld as a prototype example of a system with mixed regular-chaotic dynamics. The e$ciency of the method can be improved if additional semiclassical information obtained from a set of linearly independent observables is used to construct a crosscorrelated periodic orbit sum, which can then be inverted with a generalized harmonic inversion technique. The cross-correlated periodic orbit sum allows the calculation of semiclassical eigenenergies from a signi"cantly reduced set of orbits. Eigenenergies beyond the Gutzwiller and Berry}Tabor approximation are obtained by the harmonic inversion of the expansion of the periodic orbit sum. When applied, e.g., to the circle billiard the semiclassical accuracy is improved by at least one to several orders of magnitude. The combination of closed orbit theory with the cross-correlation approach and the harmonic inversion technique also allows the semiclassical calculation of individual quantum transition strengths for atoms in external "elds.
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Periodic orbit quantization by harmonic inversion has been applied in this work to systems with scaling properties, i.e., systems where the classical actions of periodic orbits depend linearly on a scaling parameter, w. However, the method can even be used for the semiclassical quantization of systems with non-homogeneous potentials such as the potential surfaces of molecules. The basic idea is to introduce a generalized scaling technique with the inverse Planck constant w,1/ as the new formal scaling parameter. The generalized scaling technique can be applied, e.g., to the analysis of the rovibrational dynamics of the HO molecule [94]. By varying the energy of the system, the harmonic inversion method yields the semiclassical eigenenergies in the (E,w) plane. For nonscaling systems the semiclassical spectra can then be compared along the line with the true physical Planck constant, w"1/ "1, with experimental measurements in the laboratory. Acknowledgements The author thanks H.S. Taylor and V.A. Mandelshtam for their collaboration and kind hospitality during his stay at the University of Southern California, where part of this work was initiated. Financial support from the Feodor Lynen program of the Alexander von Humboldt foundation is gratefully acknowledged. Stimulating discussions with M.V. Berry, E.B. Bogomolny, D. BooseH , J. Briggs, D. Delande, J.B. Delos, B. Eckhardt, S. Freund, M. Gutzwiller, F. Haake, M. Haggerty, E. Heller, C. Jung, J. Keating, D. Kleppner, M. KusH , C. Neumann, B. Mehlig, K. MuK ller, H. Schomerus, N. Spellmeyer, F. Steiner, G. Tanner, T. Uzer, K.H. Welge, A. Wirzba, and J. Zakrzewski are also gratefully acknowledged. The author thanks V.A. Mandelshtam for his program on "lter-diagonalization and B. Eckhardt and A. Wirzba for supplying him with numerical data of the three-disk system. Part of the present work has been done in collaboration with K. Weibert and K. Wilmesmeyer, whose Ph.D. thesis [88] and diploma thesis [80] are prepared under the author's guidance. The author is indebted to G. Wunner for his permanent support and his interest in the progress of this work. The work was supported by the Deutsche Forschungsgemeinschaft via a Habilitandenstipendium (Grant No. Ma 1639/3) and the Sonderforschungsbereich 237 `Unordnung und gro{e Fluktuationena. Appendix A. Harmonic inversion by 5lter-diagonalization In the following we give details about the numerical method of harmonic inversion by "lterdiagonalization. We begin with the harmonic inversion of a single function and then extend the method to the harmonic inversion of cross-correlation functions. A.1. Harmonic inversion of a single function The harmonic inversion problem can be formulated as a nonlinear "t (see, e.g., [67]) of the signal C(s) de"ned on an equidistant grid, c ,C(nq)" d e\ LOUI, n"0,1,2,2,N , L I I
(A.1)
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with the set of generally complex variational parameters +w ,d ,. (In this context the discrete I I Fourier transform scheme would correspond to a linear "t with N amplitudes d and "xed real I frequencies w "2pk/Nq, k"1,2,2,N. The latter implies the so-called `uncertainty principlea, I i.e., the resolution, de"ned by the Fourier grid spacing, *w, is inversely proportional to the length, s "Nq, of the signal C(s).) The `high-resolutiona property associated with Eq. (A.1) is due to the
fact that there is no restriction for the closeness of the frequencies w as they are variational I parameters. In [68] it was shown how this nonlinear "tting problem can be recast as a linear algebraic one using the "lter-diagonalization procedure. The essential idea is to associate the signal c with an autocorrelation function of a suitable dynamical system, L c "(U ,;K LU ) , (A.2) L where ( ) , ) ) de"nes a complex symmetric inner product (i.e., no complex conjugation). The evolution operator can be de"ned implicitly by ) (A.3) ;K ,e\ OXK " e\ OUI"B )(B " , I I I where the set of eigenvectors +B , is associated with an arbitrary orthonormal basis set and the I eigenvalues of ;K are u ,e\ OUI (or equivalently the eigenvalues of XK are w ). Inserting Eq. (A.3) I I into Eq. (A.2) we obtain Eq. (A.1) with d "(B ,U ) , (A.4) I I which also implicitly de"nes the `initial statea U . This construction establishes an equivalence between the problem of extracting spectral information from the signal with the one of diagonalizing the evolution operator ;K "e\ OXK (or the Hamiltonian XK ) of the "ctitious underlying dynamical system. The "lter-diagonalization method is then used for extracting the eigenvalues of the Hamiltonian XK in any chosen small energy window. Operationally, this is done by solving a small generalized eigenvalue problem whose eigenvalues yield the frequencies in a chosen window. The knowledge of the operator XK itself is not required, as for a properly chosen basis the matrix elements of XK can be expressed only in terms of c .The advantage of the "lter-diagonalization L procedure is its numerical stability with respect to both the length and complexity (the number and density of the contributing frequencies) of the signal. Here we apply the method of [69,70] which is an improvement of the "lter-diagonalization method of [68] in that it allows to signi"cantly reduce the required length of the signal by implementing a di!erent Fourier-type basis with an e$cient rectangular "lter. Such a basis is de"ned by choosing a small set of values u in the frequency H interval of interest, qw (u (qw , j"1,2,2,J, and the maximum order, M, of the Krylov
H
vectors, U "e\ LOXK U , used in the Fourier series, L + + W ,W(u )" e LPHU , e LPH\OXK U . (A.5) H H L L L It is convenient to introduce the notations, (A.6) ;N,;N(u ,u )"(W(u ),e\ NOXK W(u )) , HY HHY H HY H for the matrix elements of the operator e\ NOXK , and UN, for the corresponding small J;J complex symmetric matrix. As such U denotes the matrix representation of the operator ;K itself and U,
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the overlap matrix with elements (W(u ),W(u )), which is required as the vectors W(u ) are not H HY H generally orthonormal. Now using these de"nitions we can set up a generalized eigenvalue problem, UNB "e\ NOUIUB , I I
(A.7)
for the eigenvalues e\ NOUI of the operator e\ NOXK . The column vectors B with elements B de"ne the I HI eigenvectors B in terms of the basis functions W as I H ( B" B W , (A.8) I HI H H assuming that the W 's form a locally complete basis. H The matrix elements Eq. (A.6) can be expressed in terms of the signal c , the explicit knowledge of L the auxiliary objects XK , B or U is not needed. Indeed, insertion of Eq. (A.5) into Eq. (A.6), use of I the symmetry property, (W,;K U)"(;K W, U), and the de"nition of c , Eq. (A.2), gives after some L arithmetics
+ + ;N(u, u )"(e\ P!e\ PY)\ e\ P e LPYc !e\ PY e LPc L>N L>N L L + + , uOu , (A.9) !e +P e L\+\PYc #e +PY e L\+\Pc L>N L>N L+> L+> + ;N(u, u)" (M!"M!n"#1)e LPc . L>N L (Note that the evaluation of UN requires the knowledge of c for n"p,p#1,2,N"2M#p.) L The generalized eigenvalue problem Eq. (A.7) can be solved by a singular value decomposition of the matrix U, or more accurately by application of the QZ algorithm [143], which is implemented, e.g., in the NAG library. Each value of p yields a set of frequencies w and, due to I Eqs. (A.4), (A.5) and (A.8), amplitudes,
( + (A.10) d " B c e LPH . HI L I H L Note that Eq. (A.10) is a functional of the half-signal c , n"1,2,2,M. An even better expression L for the coe$cients d (see [70]) reads I
1 ( 1 ( d" B (W(u ),W(w )) , B ;(u ,w ) , (A.11) I HI H I HI H I M#1 M#1 H H with ;(u ,w ) de"ned by Eq. (A.9). Eq. (A.11) is a functional of the whole available signal H I c , n"0,1,2,2M, and therefore sometimes provides more precise results than Eq. (A.10). L The converged w and d should not depend on p. This condition allows us to identify spurious or I I non-converged frequencies by comparing the results with di!erent values of p (e.g., with p"1 and
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p"2). We can de"ne the simplest error estimate e as the di!erence between the frequencies w obtained from diagonalizations with p"1 and p"2, i.e. I e""wN!wN" . (A.12) I I A.2. Harmonic inversion of cross-correlation functions We now consider a cross-correlation signal, i.e., a D;D matrix of signals de"ned on an equidistant grid (a,a "1,2,2,D): (A.13) c (n),C (nq)" b b e\ LOUI, n"0,1,2,2,N . ??Y ??Y ? I ?Y I I (We choose q"1 for simplicity in what follows.) Each component of the signal c (n) contains the ??Y same set of frequencies w , and the amplitudes belonging to each frequency are correlated, i.e., I d "b b with only D (instead of D) independent parameters b . As for the harmonic ??Y I ? I ?Y I ? I inversion of a single function the nonlinear problem of adjusting the parameters +w , b , can be I ? I recast as a linear algebra one using the "lter-diagonalization procedure [68,137,138] The crosscorrelation signal Eq. (A.13) is associated with the cross-correlation function of a suitable dynamical system, c "(U ,;K LU ) , (A.14) ??Y L ? ?Y with the same complex symmetric inner product as in Eq. (A.2), and the evolution operator ;K de"ned implicitly by Eq. (A.3). The extension from Eq. (A.2) to Eq. (A.14) is that the autocorrelation function (U , ;K LU ) built of a single state U is replaced with the cross-correlation function (U , ;K LU ) built of a set of D di!erent states U . Inserting Eq. (A.3) into Eq. (A.14) we obtain ? ?Y ? Eq. (A.13) with b "(B , U ) , (A.15) ? I I ? which now implicitly de"nes the states U . After choosing a basis set in analogy to Eq. (A.5), ? + (A.16) W "W (u )" e LPH\XK U , ? ?H ? H L and introducing the notations (A.17) UN,;N ";N (u , u )"(W (u ) ,e\ NXK W (u )) , ?Y HY ?H ?YHY ??Y H HY ? H for the small matrix of the operator e\ NXK in the basis set Eq. (A.16), we can set up a generalized eigenvalue problem, UNB "e\ NUIUB , I I for the eigenvalues e\ NUI of the operator e\ NXK , which is formally identical with Eq. (A.7). The matrix elements Eq. (A.17) can be expressed in terms of the signal c . The following expression ??Y L
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for the matrix elements of UN is derived in complete analogy with [69,70] with the additional indices a,a being the only di!erence,
+ + ;N (u,u )"(e\ P!e\ PY)\ e\ P e LPYc (n#p)!e\ PY e LPc (n#p) ??Y ??Y ??Y L L + !e +P e L\+\PYc (n#p) ??Y L+> + #e +PY e L\+\Pc (n#p) , uOu , (A.18) ??Y L+> + ;N , (u, u)" (M!"M!n"#1)e LPc (n#p) . (A.19) ??Y ??Y L Given the cross-correlation signal c (n), the solution of the generalized eigenvalue problem ??Y Eq. (A.7) yields the eigenfrequencies w and the eigenvectors B . The latter can be used to compute I I the amplitudes,
( " 1!e\A (A.20) B ;(u ,w #ic) , b " ?YH I ??Y H I ?I 1!e\+>A H ?Y where the adjusting parameter c is chosen so that ;(u ,w #ic) is numerically stable [138]. One H I correct choice is c"!Im w for Im w (0 and c"0 for Im w '0. I I I Appendix B. Angular function Y (0 ) K In Section 2.4.2 we have presented Eq. (2.36) as the "nal result of closed orbit theory for the semiclassical photoabsorption spectrum of the hydrogen atom in a magnetic "eld. Here we de"ne explicitly the angular function Y (0) in Eq. (2.36). K The angular function Y (0) solely depends on the initial state t and the dipole operator D and is K a linear superposition of spherical harmonics: Y (0)" (!1)lBl >l (0, 0) . K K K l K The coe$cients Bl are de"ned by the overlap integrals K
Bl " dx(Dt )(x)(2/rJ l ((8r)>H l (0, u) K K >
(B.1)
(B.2)
(with J (x) the Bessel functions) and can be calculated analytically [93]. For excitations of the J ground state t ""1s02 with p-polarized light (i.e. dipole operator D"z) the explicit result is (B.3) Y (0)"!p\ 2e\ cos 0 ,
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and for t ""2p02, i.e., the initial state in many spectroscopic measurements on hydrogen [45}47] we obtain Y (0)"(2p)\ 2e\(4 cos 0!1) .
(B.4)
Appendix C. Catastrophe di4raction integrals Here we give some technical details about the numerical calculation and the asymptotic expansion of catastrophe di!raction integrals for the hyperbolic umbilic and the butter#y catastrophe. C.1. The hyperbolic umbilic catastrophe The catastrophe di!raction integral of the hyperbolic umbilic (Eq. (2.28)) reads
W(x, y)"
>
dp
\
>
\
dq e N>O>WN>O>VN>O .
(C.1)
By substituting p"s !y , q"s !y , we obtain W(x, y)"e W W\V U(x!y, 2y) ,
(C.2)
with
U(m, g)"
>
ds
\
>
\
ds e Q>Q>KQ>Q>EQQ .
(C.3)
The integral U(m, g) can be expanded into a Taylor series around g"0. Using
> > RLU(m,g) " ds ds iL(s s )Le Q>Q>KQ>Q "iL E RgL \ \
>
\
ds sLe Q>KQ
,
(C.4)
and solving the one-dimensional integrals [144]
>
1 (!m)I k#n#1 ds sLe Q>KQ" C [e L>\Ip #(!1)Le\ L>\Ip ] , k! 3 3 \ I
(C.5)
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we "nally obtain
1 (ig)L (!m)I k#n#1 C [e L>\Ip #(!1)Le\ L>\Ip ] , (C.6) U(m, g)" n! k! 3 9 L I which is a convergent series for all m and g. C.2. The butteryy catastrophe The uniform phase integral W(x, y) of the butter#y catastrophe is expanded in a two-parametric Taylor series around x"y"0:
W(x, y),
> exp[!i(xt#yt#t)] dt \
1 xLyK > tL>Kexp[!it] dt . " iL>K n! m! \ L K
(C.7)
With the substitution z"tL>K> we obtain [144]
>
2 tL>Kexp[!it] dt" exp[!iz L>K>] dz 2n#4m#1 \
2n#4m#1 2n#4m#1 1 p C , " exp !i 12 6 3 and "nally
p 1 W(x, y)" exp !i 12 3
1 2n#4m#1 C n! m! 6 L K
2 ; x exp !i p 3
L
5 y exp !i p 6
K
,
(C.8)
which is a convergent series for all x and y. The asymptotic behavior of W(x,y) for xP$R is obtained in a stationary-phase approximation to Eq. (C.7) with the stationary points t being de"ned by t (6t#4yt#2x)"0 . (a) xP!R: There are three real stationary points given by t "0 and t"!y#(y!x
(C.9)
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and we obtain
V \ > W(x, y) & exp[!ixt] dt#2 exp[!i(xt#yt#t)] \ > ; exp [!i(x#6yt#15t) t] dt \
p p p exp i # 4 !x !x#y!y(y!x ;exp+i[2(y!x) !y(y!x)!p], . (C.10) (b) xP#R: The only real solution of Eq. (C.9) is t "0 and W(x, y) is approximated as "
p p V > > exp !i . exp[!ixt] dt" W(x, y) & 4 x \
(C.11)
y the complex zeros In the case of y(0 and x& t "$(!y$i(x!y of Eq. (C.9) are situated close to the real axis. When one considers these complex zeros in a stationary phase approximation, Eq. (C.11) is modi"ed by an additional term exponentially damped for large x: V > W(x, y) &
p p p exp !i # 4 x x!y!iy(x!y ;exp+!2(x!y) ,exp+!i[y(y!x)!p], .
(C.12)
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J. Main / Physics Reports 316 (1999) 233}338 T. Ericcson, A. Ruhe, Math. Comp. 35 (1980) 1251. K. Spreckelsen, Diploma Thesis, Ruhr-UniversitaK t Bochum, 1996, unpublished. M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions, Dover, New York, 1965. T. Prosen, M. Robnik, J. Phys. A 26 (1993) L37. P.A. Boasman, Nonlinearity 7 (1994) 485. P. Gaspard, S.A. Rice, J. Chem. Phys. 90 (1989) 2225, 2242, 2255. K.T. Hansen, Nonlinearity 6 (1993) 753. A. Wirzba, private communication. G. Tanner, P. Scherer, E.B. Bogomolny, B. Eckhardt, D. Wintgen, Phys. Rev. Lett. 67 (1991) 2410. H.M. Edwards, Riemann's Zeta Function, Academic Press, New York, 1974. E.C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd ed., Oxford University Press, Oxford, 1986. M.V. Berry, in: T.H. Seligman, H. Nishioka (Eds.), Quantum Chaos and Statistical Nuclear Physics, Lecture Notes in Physics, vol. 263, Springer, Berlin, 1986, pp. 1}17. A.M. Odlyzko, The 10th zero of the Riemann zeta function and 70 million of its neighbours, AT&T Bell Laboratories, 1990. O. Bohigas, M.J. Giannoni, in: J.S. Dehesa, J.M.G. Gomez, A. Polls (Eds.), Mathematical and Computational Methods in Nuclear Physics, Lecture Notes in Physics, vol. 209, Springer, Berlin, 1984, pp. 1}99. E. Bogomolny, J. Keating, Nonlinearity 8 (1995) 1115. J. Delsarte, J. Anal. Math. (Jerusalem) 17 (1966) 419. R. Aurich, M. Sieber, F. Steiner, Phys. Rev. Lett. 61 (1988) 483. J.P. Keating, M. Sieber, Proc. R. Soc. London A 447 (1994) 413. G.S. Ezra, K. Richter, G. Tanner, D. Wintgen, J. Phys. B 24 (1991) L413. G. Tanner, D. Wintgen, Chaos 2 (1992) 53. C.C. Martens, R.L. Waterland, W.P. Reinhardt, J. Chem. Phys. 90 (1989) 2328. S. Tomsovic, J. Phys. A 24 (1991) L733. M. Wilkinson, J. Phys. A 21 (1988) 1173. B. Eckhardt, S. Fishman, K. MuK ller, D. Wintgen, Phys. Rev. A 45 (1992) 3531. D. BooseH , J. Main, B. Mehlig, K. MuK ller, Europhys. Lett. 32 (1995) 295. D. Ruelle, Statistical Mechanics, Thermodynamical Formalism, Addison-Wesley, Reading, MA, 1987. M. Gutzwiller, J. Phys. Chem. 92 (1988) 3154. A. Voros, J. Phys. A 21 (1988) 685. P. CvitanovicH , B. Eckhardt, Nonlinearity 6 (1993) 277. I.C. Percival, Adv. Chem. Phys. 36 (1977) 1. O. Bohigas, S. Tomsovic, D. Ullmo, Phys. Rep. 223 (1993) 43. E. Narevicius, D. Neuhauser, H.J. Korsch, N. Moiseyev, Chem. Phys. Lett. 276 (1997) 250. V.A. Mandelshtam, J. Chem. Phys. 108 (1998) 9999. J. Main, G. Wunner, Phys. Rev. E, submitted. H. Primack, U. Smilansky, J. Phys. A 31 (1998) 6253. W.R.S. Garton, F.S. Tomkins, Astrophys. J. 158 (1969) 839. A. Holle, G. Wiebusch, J. Main, B. Hager, H. Rottke, K.H. Welge, Phys. Rev. Lett. 56 (1986) 2594. J.H. Wilkinson, Linear Algebra and Appl. 28 (1979) 285. I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1965.
N. KonjevicH /Physics Reports 316 (1999) 339}401
PLASMA BROADENING AND SHIFTING OF NON-HYDROGENIC SPECTRAL LINES: PRESENT STATUS AND APPLICATIONS
N. KONJEVICD Faculty of Physics, 11001 Belgrade, P.O. Box 368, Yugoslavia Institute of Physics, 11081 Belgrade, P.O. Box 68, Yugoslavia
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
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Physics Reports 316 (1999) 339}401
Plasma broadening and shifting of non-hydrogenic spectral lines: present status and applications N. KonjevicH Faculty of Physics, 11001 Belgrade, P.O. Box 368, Yugoslavia Institute of Physics, 11081 Belgrade, P.O. Box 68, Yugoslavia Received November 1998; editor: T.F. Gallagher Contents 1. Introduction 2. Theory 2.1. Neutral atom lines 2.2. Ionic lines 3. Experimental determination of line shapes and shifts 4. Other broadening mechanisms 4.1. Natural broadening 4.2. Resonance broadening 4.3. Van der Waals broadening 4.4. Doppler broadening 4.5. Instrumental broadening 4.6. Self-absorption 5. Electron temperature diagnostics
341 342 344 345 346 350 350 350 351 351 352 353 356
6. Regularities in experimental Stark widths and shifts 6.1. Widths 6.2. Shifts 7. Typical experimental procedure 8. Analysis of experimental results 8.1. Neutral atom lines 8.2. Singly ionized atom lines 8.3. Multiply ionized atom lines 8.4. Studies along isoelectronic sequences 9. Conclusions Acknowledgements Appendix References
360 360 361 361 363 364 375 384 387 393 396 396 397
Abstract The present status of the experimental studies of plasma broadening and shifting of non-hydrogenic neutral atom and positive ion spectral lines is discussed with an emphasis to the plasma diagnostic applications. A short overview of the available theoretical results is followed by the description of experimental techniques for the line shape and shift measurements. The in#uence of other broadening mechanisms to the Stark width and shift determination is discussed and a typical experimental procedure described. To select higher accuracy experimental data for plasma diagnostic purposes the available experimental results are analyzed and the uncertainty of recommended ones is estimated. The procedure for application of selected lines for plasma electron density diagnostics is described. The in#uence of ion dynamics to the width and shift of visible helium lines in low electron density plasmas is examined. In order to test theories the selected data are compared with results of various Stark width and shift calculations. Neutral atom lines asymmetry and new techniques for ion-broadening parameter measurements are reviewed. The studies of line widths and shifts along isoelectronic sequences of multiply ionized atoms are also discussed. 1999 Elsevier Science B.V. All rights reserved. PACS: 52.70.Kz; 32.70.Jz Keywords: Plasma spectroscopy; Electron density diagnostics; Stark broadening
0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 8 ) 0 0 1 3 2 - X
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1. Introduction Plasma-broadened and shifted spectral line pro"les have been used for a number of years as a basis of an important non-interfering plasma diagnostic method. The numerous theoretical and experimental e!orts have been made to "nd solid and reliable basis for this application. This technique became, in some cases, the most sensitive and often the only possible plasma diagnostic tool. In the early 1960s a number of attempts were made to improve and to check experimentally existing theories of spectral line broadening by plasmas. Most of these early works were concerned with the Stark broadening of hydrogen lines. Owing to the large, linear Stark e!ect in hydrogen, these studies were very useful for plasma diagnostic purposes. However, it is not always convenient to seed plasma with hydrogen, and sometimes this is not possible. Furthermore, due to the large Stark e!ect, hydrogen lines or the lines of hydrogen-like ions are sometimes inconvenient for plasma diagnostic purposes, since they become so broad at high electron densities that, due to the interference with other neighboring lines, it is di$cult to determine their shape correctly. Thus, from the beginning of this "eld of research, there was an interest for the plasma broadening of isolated non-hydrogenic lines of neutral atoms and positive ions. Due to relatively small Stark broadening of non-hydrogenic lines, they can be used for plasma diagnostic purposes at high electron densities and, in particular, at high electron temperatures when hydrogen is fully ionized. Hereafter, the word `isolateda is used for the lines originating from isolated energy levels in the sense that levels are not degenerate and do not overlap each other. First semiclassical calculations of Stark broadening parameters for isolated non-hydrogenic atomic and singly charged ion lines were carried out by Griem and co-workers [1,2]. After some improvements of the theory (see e.g. [3]) the new comprehensive calculations of Stark-broadening parameters of neutral (helium through calcium and cesium) and singly ionized atom lines (lithium through calcium) were published in 1974 [3]. Later on the calculations for neutral atom lines were extended to some other elements heavier than calcium [4]. Using another semiclassical perturbation method by Sahal-Brechot [5], DimitrijevicH and Sahal-Brechot performed numerous calculations of Stark broadening parameters for the lines of neutral, singly and multiply ionized atoms (see list of publications in [6]). Within `Opacity projecta, using close-coupling approximation, Seaton [7] evaluated a number of Stark broadening data for multiply ionized atoms. Large number of data were produced also using simple semiempirical formula [8] and its modi"ed version [9}11]. A simpli"ed semiclassical formula (Eq. (526) in [3]) is also used extensively for data evaluation and a large set of results is given in [12] together with those from the modi"ed semiempirical formula [9]. Here, only calculations which provide data for a large number of elements and their ions are given. Parallel with the development of the theory of Stark broadening, numerous experiments were performed to provide plasma broadening data and to test theoretical predictions. In order to evaluate and select reliable experimental results, which can be used with con"dence for both, plasma diagnostic purposes and for the testing of theory, KonjevicH and Roberts [13], KonjevicH and Wiese [14], KonjevicH et al. [15,16] and KonjevicH and Wiese [17] carried out analysis of all available experiments for neutral and ionized atoms until the end of 1988. The basis for these experimental reviews and in a good part for this report, was the bibliography on atomic line shapes and shifts [18]. In order to select reliable data, the authors of these reviews [13}17] imposed certain criteria on each experimental paper and, therefore, not all experiments reviewed were included in the "nal
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results [13}17]. The accuracies of experimental data, coded with letters A ($15%), B ($30%), C ($50%) and D (larger than 50%) are estimated in [13}17] irrespective of the agreement with the theory which was used only as a check for experimental consistency within experiments and between di!erent experiments. Although the theory was used only as a consistency check, an important conclusion is derived from the comparison with experiments: the results of semiclassical calculations [3] may be used for plasma diagnostic purposes with an average estimated accuracy of $20% and $30% for a neutral atom and singly charged ion lines, respectively; see also [3,19]. Here, on the basis of comparison of selected data, we shall make an attempt to extend this analysis and to identify lines of various elements which may be used with higher accuracy for plasma diagnostic purposes. Furthermore, this search will be extended to the lines of higher ionization stages as well. When one intends to use Stark broadening data for plasma diagnostic purposes, problems usually not encountered in experiments set to measure Stark broadening parameters have to be solved. In these experiments, the amount of investigated atoms and ions whose lines are studied is usually controlled so that the distortion caused by the radiative transfer is avoided as much as possible. Furthermore, plasma sources for these experiments are selected in such a way that the in#uence of plasma inhomogeneity to the line shapes is negligible or such that it can be taken into account (e.g. axially symmetric plasmas). In the case when one uses line shapes and shifts to diagnose plasma source, changes or control of plasma conditions is usually not feasible and often (e.g. astrophysical plasmas) not possible at all. So the measurements of line parameters and estimation of the in#uence of other broadening mechanisms will be discussed in detail. Since line widths and shifts also depend upon electron temperature, ¹ , for accurate N plasma diagnostics, ¹ measurements are required and they too will be discussed. In addition, a part of this article is devoted to the regularities and similarities of Stark widths and shifts which arise from atomic energy structure of the line emitter. The knowledge of these regularities and similarities may improve the accuracy or, in some cases, when Stark broadening data are not available, make possible the determination of plasma electron density. Finally, at the end of the general part of this paper typical experimental procedure for electron density plasma diagnostics is also described. An important aim of this work is to summarize results of experimental studies of the Stark broadening and shifting of spectral lines and to discuss the comparison of these data with theory. For this purpose critical evaluation of the recent works (published after 1988) has been performed and the results are used together with earlier data [13}17] to test consistency among experiments and between experiments and theories. The estimated accuracies of selected data will be given and some experimental di$culties underlined. The results related to di!erent atoms and their ionization stages are described separately. In this part the recommended data for plasma diagnostic purposes and their estimated accuracies are given. The recent studies of neutral line asymmetries and line width and shift investigations along with isoelectronic sequences of multiply ionized atoms are also described.
2. Theory The general area of line broadening has been reviewed extensively, the most complete treatment being that of Griem [3]. Here we shall con"ne our discussion towards the application of theoretical results for plasma diagnostic purposes.
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According to Stark broadening theory the shapes and shifts of plasma-broadened isolated lines are mainly determined by electron impacts with the radiating atom or ion and a smaller contribution arises from the electric micro"elds generated by essentially static plasma ions. The quadratic Stark e!ect due to the quasistatic "eld of ions shifts the energy of the upper and lower level [20] by an amount which depends on the instantaneous local "eld strength. The distribution of "elds in the plasma smears out these shifts and the neutral atom line is unsymmetrically broadened. The parameter A (in earlier literature designated as a), tabulated by Griem [3] is a measure of the e!ect this has on the width in relation to the electron impact width. The in#uence of static "eld to the Stark splitting and shifting of ionic lines is even smaller than for neutral atom lines, see e.g. [20], so the ion contribution to the line shapes and shifts of ionic lines is usually considered to be negligible in comparison with electron impacts. By taking into account the linear dependence of the Stark broadening parameters upon N for ionic lines and very close to linear for neutral atom lines (proven in a number of experiments, see e.g. [3,13}17]), unknown electron density may be determined from the comparison of the measured Stark width, w , and/or shift, d , with results from another experiment where these parameters are
measured at known N and at similar electron temperature ¹ . This procedure has a limited range of application and instead, theoretical data w(N ,¹ ) and d(N ,¹ ) in conjunction with w and
d are used for N plasma determination. In practice, to perform N measurements it is necessary to
determine experimentally the line width and/or shift and to compare with theoretical data for the same ¹ . This, well established procedure, was preceded by numerous experimental and theoretical studies which have proven that experiment agrees with theory. The degree of the discrepancy between various theoretical approaches and experiments was the main task of many earlier publications and it will be discussed here in detail. Two large sets of theoretical data are available for plasma diagnostic purposes. One is published in [3] for neutral atom lines (He through Ca and Cs) and for singly charged ion lines (Li II through Ca II). The other set of data covers large number of neutrals, singly, and multiply ionized atoms (see [6] and references therein). Furthermore, numerous calculations of Stark widths and shifts for ionic lines are performed using simple approximate formulas (see [6,9}12] and references therein) in particular for the radiators without su$ciently complete set of atomic data necessary for semiclassical calculations. The Stark broadening data are usually published in the form of tables where widths and shifts are given for a single electron density and for several electron temperatures. The results for atomic lines are usually presented for N "10 cm\ while data for ionic lines are given at N "10 cm\. For neutral atom lines (Appendix IV in [3]) in addition to electron impact half-halfwidth w (in [6] are always given full halfwidths 2w ) and electron impact shift, d , the corresponding ion broadening parameter, A, is given as well. Namely in [3] impact approximation is used for evaluation of electron collisions contribution to the line width while for ionic part quasistatic approximation is used. In [6] impact approximation is used for both, electrons and ions and the results for ion contribution are given in the form of widths and shifts induced by collisions with di!erent ions present in plasma. Therefore, from the data in [6], the total width and shift is a sum of the electron and all ion impact widths. For neutral atom lines the latter approach [6] always gives symmetric Lorentzian pro"le which is not in agreement with experiments at higher electron densities, where asymmetric line pro"les are detected. The ionic spectral line pro"les in both sets of theoretical data, Appendix V in [3] and [6], are symmetrical and well described by the Lorentzian function which is in a good agreement with experimental results.
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2.1. Neutral atom lines The total theoretical full-width at half-maximum (FWHM) w and the total theoretical shift at the line maximum d may be calculated using the approximate formulas [3] obtained from a numerical "t to the widths of the resultant j (x) pro"les where x is the reduced frequency which 0 is given by x"(u!u !d )/w with u being the frequency, u the center frequency of the unperturbed line, d and w the electron impact shift and width, respectively. In this case, w and d may be calculated from the following equations: w (N ,¹ ) 2w (¹ ) [1#gA (¹ )]N 10\ , (1) , d (N ,¹ ) [d (¹ )$2.0g A (¹ )w (¹ )]N 10\ , (2) , where g"1.75(1!0.75R), with R being Debye shielding parameter, A (¹ )"A(¹ )N 10\ and , g "g/1.75. Due to the asymmetry of plasma-broadened atom lines, the shift at the half-width of the line is slightly di!erent from the one measured at the peak of the line pro"le and may be calculated from [3,21] d
(N ,¹ ) [d (¹ )$3.2g A (¹ )w (¹ )]N 10\ . , From Eqs. (1)}(3) it follows that
(3)
w (N ,¹ ) 2w (¹ ) [1#1.75;10\N A(¹ ) (1!0.068N ¹\ )]10\N , (3a) d (N ,¹ ) [d (¹ )$2.0;10\N A(¹ )w (¹ ) (1!0.068N ¹\ )]10\N , (3b) d (N ,¹ ) [d (¹ )$3.2;10\N A(¹ )w (¹ ) (1!0.068N ¹\ )]10\N . (3c) In the above equations, w (¹ ) and d (¹ ) are electron impact half half-width and shift, respectively, A(¹ ) is the ion broadening parameter, all given in [3] at N "10 cm\, and N and ¹ are the electron density (cm\) and temperature (K). The sign of the ion quadratic contribution to the shift in Eqs. (2) and (3) is equal to that of the low-velocity limit for d . There are certain restrictions on the applicability of Eqs. (1)}(3) and they are R"8.99;10\N ¹\ 40.8 , (4) 0.054A(¹ )N 10\40.5 , (5) where R (the Debye shielding parameter) is de"ned as a ratio of the mean inter-ion distance o to the Debye radius o . For values of ion broadening parameter larger than 0.5 the forbidden " component begins to signi"cantly overlap, and the linear Stark e!ect becomes important. Other considerations, such as Debye shielding a!ecting line shapes, widths and shifts are covered in detail in [3]. Simple estimation of the conditions when Debye shielding correction may be omitted and when the assumption of the isolated line approximation is ful"lled, may be derived from the following conditions: , Debye shielding: j <j ??Y , Isolated line: j <j ??Y
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where j is the wave number (cm\) associated with perturbed level a and nearest perturbing level ??Y is the wave number (cm\) associated with a to which dipole transition is allowed, j plasma frequency given by j
"[(e/pmc)N ] "2.995;10\N , (6) and j is the wave number (cm\) associated with the half-width of the spectral line and given in terms of the broadened line's total half-width w and central wavelength, j by j "w /j . (7) Since impact approximation is used for evaluation of both, electron and ion contributions in [6], the resulting pro"le has always Lorentzian shape and the evaluation of theoretical FWHM w and shift d in this case is very simple whenever all necessary data are available: L w (N ,¹ ) 2w (¹ )10\N # 2w (¹ )10\N , (8) G G G G L d (N ,¹ ) d (¹ )10\N $ d (¹ )10\N , (9) G G G G where 2w , 2w and d and d are electron and ion impact full half-widths and shifts respectively at G G N "10 cm\; N and ¹ are the ion concentration and temperature, respectively. The sign of G G G the ion contribution to the shift in Eq. (9) is speci"ed for all ionic species and they are given in data tables. One can "nd the restrictions of the applicability of Eqs. (8) and (9) in [6] and references therein. Apart from extensive set of Stark broadening data for light elements He through Ca and Cs [3] and some lines of other heavy elements Zn, Ge, Br, Rb, Cd, Sn, Pb and Hg [4] calculated using the same semiclassical code [3], large number of theoretical results evaluated from another semiclassical approach [5] are also available for: E
E
some lines of C, N, Mg and Si calculated by Sahal-Brechot without the contribution of Feshbach resonances, see [22]; contribution of resonances are taken into account in an extensive set of calculations for the lines of 79 multiplets of He, 61 Li, 19 Be, 62 Na, 270 Mg, 25 Al 51 K, 24 Rb, 3 Pd, 31 Se; for some lines of F and Br; see [6] and references therein. Recently, extensive sets of data for neutral Sr [23] and Ba [24] lines have become available.
In the case when w , d and A data are not available one can use simple formulas [25,26] for their estimation. 2.2. Ionic lines The theoretical full half-width w and shift d of singly ionized atom lines may be calculated from data in [3]: w (N ,¹ ) 2w (¹ )10\N , d (N ,¹ ) d (¹ )10\N ,
(10) (11)
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where w and d are electron impact half half-width and shift at N "10 cm\, respectively. Here, ion broadening is assumed to be negligible. For evaluation of the theoretical widths and shifts from [6] one can use Eqs. (4) and (5) so the ion contribution may be included as well. Here it should be noted that in [6] electron impact full half-widths (2w ) and shifts (d ), and ion impact full half-widths (2w ) and shifts (d ) are calculated G G usually for the perturber concentration 10 cm\ and thus in Eqs. (8) and (9), 10\ changes to 10\. If one neglects the ion contribution to the total line width, reduced Eqs. (8) and (9) become identical with Eqs. (10) and (11). The omission of ion contribution which is usually of the order of several percent for ion lines may be easily justi"ed for most of applications. Since detailed composition for majority of plasmas is not known the omission of the ion contribution sums in Eqs. (8) and (9) is the only way to determine N . Semiclassical data for singly charged ions Li through calcium are available in [3]. Data for a number of alkali-like ion lines Be II, Mg II, Ca II, Sr II Ba II and Si II and Ar II are available, see [6] and references therein. Extensive calculations of Stark broadening parameters are also performed by DimitrijevicH and Sahal Brechot for 29 multiplets of Li II, 30 Be II, 52 Mg II, 3 Fe II, 2 Ni II; see [6] and references therein. For Ba II lines see [24]. Furthermore data for 23 multiplets of A III, 10 Sc III, 10 Ti IV, 19 SI IV, 90 C IV, 5 O IV, 19 O V, 30 N V, 25 C V. 51 P V, 30 O VI, 21 S VI, 10 F VII, 20 Ne VIII, 8 Na IX, 7 Al XI, and 9 Si XII; data for certain lines of Ga II, Ga III, Si II, Br II, Hg II, N III, S IV and F V also exist, see [6] and references therein. Recently, data for Be III, B III [27] and P IV [28] have been published. Modi"ed semiempirical approach [9}11] has been used for the radiators where there is not a su$ciently complete atomic data set for reliable semiclassical calculations, see [6] and references therein. In case when theoretical data for electron impact widths and shifts are not available one can use for E E E
singly ionized atom lines: semiempirical formula [8] for estimation of w and d , multiply ionized atom lines: simpli"ed semiclassical formula (Eq. (526) in [3]) for w , singly and multiply ionized atoms: modi"ed semiempirical formulas [9}11] for w and d , and classical-path approximation approach [29] for w .
3. Experimental determination of line shapes and shifts In principle, line shape determinations are straightforward. For the ideal case of homogeneous steady-state plasmas, which are approximately realized in end-on observations of stabilized arcs, and in fast homogeneous pulsed sources, one simply scans over the pro"le with a spectrometer of su$cient resolution. To carry out the very fast spectral scanning of pulsed plasmas, optical multi-channel analyzers, fast rotating mirror systems and fast scanning Fabry}Perot interferometers have been developed. Side-on observations of stabilized arcs and plasma jets are more tedious. In this case one has to obtain the intensity distribution over the cross section of the radially symmetric plasma for many "xed wavelength positions over the line pro"le. Then, by applying an Abel inversion technique (see e.g. [30,31]), one obtains the radial intensity distribution at each wavelength and can thus construct a pro"le for the radial positions of interest. An advantage of this method is that one may obtain pro"les for a range of electron densities corresponding to the
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Fig. 1. Set of Abel-inverted pro"les for the Ar I 425.9 nm line: 15 di!erent radial positions starting from the axis of a wall stabilized arc [32].
di!erent radial positions, i.e., one may measure the functional dependence of line shape parameters on electron density and temperature from a single experimental run, see example in Fig. 1. In line shape determinations from pulsed sources with standard monochromators, one may advance the instrument stepwise in small wavelength increments from shot-to-shot over the range of the line pro"le. Good reproducibility of the pulsed plasma source and convenient monitoring setup are required for this operation. With modern waveform recorders (digital oscilloscopes) this technique may be considerably improved. Time evolution of the light intensity at di!erent wavelengths may be stored so the analysis of the data may be performed in various times of plasma evolution and decay. This technique is useful in particular for repetitive plasma sources where the signal averaging technique may be employed. One can achieve a similar goal by using a boxcar averager instead of an oscilloscope. In this case signals are averaged only in a preselected time of the plasma existence. Of course, the most advanced technique of spectral line recording is to use optical multi-channel analyzer (OMA) whose detector head is made of a large number of diodes or charged coupled detectors (CCD) which may be gated and used for time-resolved spectra recordings. Unfortunately, the "nite dimensions of diodes or CCD (typically 15 or 25 lm) frequently limit spectral resolution of the detection system, therefore OMA is mainly used for studies of broader lines. Line shift is measured by comparing the investigated line with unshifted reference line generated in low-pressure lamps (see Fig. 2). The main di$culty one may encounter in these experiments is to obtain a comparable signal from the main discharge and from a low-intensity reference source. This is di$cult in particular, if the line shapes from powerful pulsed plasma sources are studied. Comparable intensities to those from the reference source may be achieved by selecting the transmittivity of the partially re#ecting mirror (see Fig. 2A). Another way is to extend the observation time (see e.g. [34]) during reference line recording. If this technique is combined with wavelength positioning of the monochromator (best results achieved with the stepping motor) and automatic radiation chopper control (see Fig. 2B) with all events synchronized and controlled by
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Fig. 2. Shift measurement techniques: (A) simultaneously recorded spectra from plasma and reference source (RS) and (B) alternatively recording radiation from plasma and reference source. PRM and S are partially re#ecting mirror and monochromator's slit respectively. Examples of spectra recordings for (A) and (B) are taken from [33] and [34], respectively.
PC, one obtains a powerful apparatus for line shift measurements. An alternative to the extension of reference source observation time is an increase of sensitivity of the radiation detection system during reference spectra recordings. This can be achieved with an externally controlled detector power supply which is commercially available nowadays. The line shift measurements are most accurate if one can use the same unshifted line from the reference source for comparison. In this way uncertainty in wavelength of the unshifted line cancels out. The line shift may be determined at the maximum of the line pro"le or at its half-width. In the case of asymmetric line pro"le (e.g. neutral atom lines) these two shifts slightly di!er, see Eqs. (2) and (3). For homogeneous steady-state plasmas, shift measurements may be e$ciently done by the photographic method because of the ease of spectra recording on a photographic plate and measuring small wavelength di!erences. A special method applicable to pulsed sources is to use as a reference line the line emission from this same source at a very late stage of the discharge, when the electron density has decreased to a very small value [35]. Another technique for shift measurements is to use an auxiliary low current discharge within the plasma source to determine an unshifted line position which is later during pulsed plasma evolution used for line shift measurements (see e.g. [36]). Here, one has to mention techniques of width and shift measurements based on linear and nonlinear laser spectroscopy (see e.g. [37]). These techniques are, in particular, advantageous for
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the study of small widths and shifts typical for low-pressure gas discharges. Here we draw attention to the use of optogalvanic signal from the reference source low-pressure discharge, as a wavelength indicator of unshifted line position [38]. Recently, tunable lasers have been used in absorption and induced #uorescence experiments for electron density diagnostics, see e.g. [39}42]. For this application tunable semiconductor diode lasers are in particular very popular and widely used in numerous laboratories. They have typical bandwidth of several tenths of megahertz while diode lasers with the bandwidth of 0.1 MHz are commercially available. The simplest way of semiconductor laser tuning is to change diode temperature, while for "ne wavelength scanning diode current is being used. In a typical experiment the continuous frequency scanning laser beam is directed through plasma and by monitoring either the intensity of transmitted radiation or by recording the induced #uorescence intensity it is possible to determine plasma line shape and in conjunction with the wavelength reference source, the line shift. If all necessary theoretical data are available, the measured half-width and/or shift can be used to determine N from Eqs. (1)}(3). All precautions related to the in#uence of radiative transfer on the line shapes, see Section 4.6, may be applied to induced #uorescence experiments. In case of the absorption experiment the intensity of transmitted radiation, I , through a homogeneous plasma slab of thickness l is H given by
J (12) I (l)"I exp ! k dx , H H H where I is the intensity of incident laser radiation. In the case of homogeneous plasma, H determination of the absorption coe$cient k is rather simple. For inhomogeneous plasma one has H to determine local absorption coe$cient and this may not be an easy task. In a special case of cylindrical plasmas one has, like in emission spectroscopy, to apply the Abel inversion procedure to determine radial distribution of k . Once k is deduced it may be further used for determination of H H plasma line shape function from the following relationship (see e.g. [43]): H g g N j (13) k " A N 1! , H H 8p g g N where j is the central wavelength for the observed transition from energy level 1 to energy level 2, A is the probability for spontaneous emission from 2 to 1 and g and g are the degeneracies of the levels in question. The line shape function is normalized to one. So H by measuring absorption coe$cient k it is possible to determine the line shape pro"le irrespectH ive of the broadening mechanism. One should take great care in recording k so as to H avoid saturation e!ects which may be induced by laser radiation. Particularly in cases of low current, low-pressure discharges where the number of collisions per unit time interval is relatively small, so due to the narrow laser bandwidth, even a laser beam of a few milliwatts only, may produce considerable saturation. This would result in a broader line pro"le and consequently larger plasma electron densities or gas temperatures would be detected. To avoid saturation e!ects it is necessary to reduce the intensity of the laser probing beam (usually with neutral density "lters) until the recorded line width does not depend upon the intensity of incident laser radiation.
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4. Other broadening mechanisms The experimentally observed line shapes are usually the result of several broadening factors from which the Stark pro"le component has to be isolated and retrieved. While the experiments reporting plasma broadening data are usually designed so as to have Stark broadening as the dominant cause of the width, the conditions can rarely be made so ideal that all broadening factors become insigni"cant. For the line shifts, the situation is less complex insofar as van der Waals shifts are the only competing factor. When line shapes and shifts are used for plasma diagnostic purposes, in most of the cases, one cannot change plasma conditions so the contribution of other broadening mechanisms has to be carefully estimated and, if necessary, measured widths and shifts corrected. Here we shall list the other broadening mechanisms which are present in plasmas and may in#uence Stark broadening parameter measurements, see also [21]. All widths given below are FWHM (full-widths at half-maximum). 4.1. Natural broadening Natural broadening arises because even an unperturbed level has a "nite lifetime, q, due to spontaneous emission. For a transition between states n and m,
(14) w (cm)"j A # A /2pc , KYK LYL * KY LY where A is the transition probability between the state m and any `alloweda level m . Natural KYK broadening is largest when one of the two levels is dipole-coupled to the ground state. Even in this case, it is usually negligible (of the order of 10\ nm). However, it may be of some importance for low electron density plasmas generated in low-pressure gas discharges. 4.2. Resonance broadening Resonance broadening occurs for transitions involving a level that is dipole-coupled to the ground state. Ali and Griem [44] derived an equation for width evaluation, w (cm)"1.63;10\(g /g ) jj f N , 0 G I 0 0
(15)
where j is the wavelength (in cm) of the observed radiation. Here, N is the density of groundstate particles which are the same species as the emitter, and g and g are the statistical weights G I of the upper and lower state; j and f are the wavelength and f-value, respectively, of the 0 0 resonance transition from level `Ra. The level `Ra is the upper or lower level of the observed transition, which happens also to be the upper level of a resonance transition to the ground state. Not all transitions involve such a level. The shifts due to resonance broadening are negligible. For 4s[3/2]!4p[3/2] transition of neutral Ar, j"810.37 nm; j (3p S!4s[3/2])" 0 106.67 nm; g "1; g "3; f "0.061 at N "1;10 cm\, w "0.00212 nm. G I 0 0
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4.3. Van der Waals broadening Van der Waals broadening results from the dipole interaction of an excited atom with the induced dipole of a neutral ground-state atom of density N . This is a short-range C /r interaction. Griem's estimate [45] for the full-width at half-maximum w can be written as w (cm)"8.18;10\j(aN R) (¹ /k) N where R"R !R . 3 *
(16)
(17)
R is the di!erence of the squares of coordinate vectors (in a units) of the upper and lower level, and k is the atom-perturber reduced mass in a.m.u. (k"19.97 for excited Ar perturbed by Ar atoms). Values of aN , the mean atomic polarizability of the neutral perturber, are tabulated for di!erent elements by Allen [46]: for argon aN "16.54;10\ cm. If the required value of aN is not tabulated, it can be estimated either from the expression given by Allen [46] or by Griem [45]: aN "(9/2)a(3E /4E ) , (18) & #6! where E is the ionization potential of hydrogen (109737.32 cm) and E is the energy of the "rst & #6! excited level of the perturber. In the Coulomb approximation the values of R and R in Eq. (17) * 3 may be calculated from R"nH/2[5nH#1!3l (l #1)] , (19) H H H H H where the square of e!ective quantum number nH is H nH"E /(E !E ) . (20) H & '. H Here E is the ionization potential of the studied element and E is the energy of the upper or lower '. H level of the transition. van der Waals broadening causes a red wavelength shift which is two-third of the size of the width w . For 4s[3/2]}4p[3/2] transition of neutral Ar, and with Ar as perturber; j"810.37 nm; I"127 109 cm\; E "106 087 cm\ and E "93751 cm; k"19.97; aN "16.54;10\. At 3 * ¹ "7000 K and N "1;10 cm\, w "0.00356 nm. 4.4. Doppler broadening For a Doppler-broadened spectral line the intensity distribution is Gaussian whose full halfwidth w is given by (21) w "7.16;10\j((¹ /M) , " where ¹ and M are gas kinetic temperature (in K) and mass of radiating atom (in a.m.u.), respectively.
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4.5. Instrumental broadening The standard practice to determine the apparatus pro"le of a spectrometer is to scan over a line whose intrinsic width is very small compared to the apparatus width so that the latter determines its shape. Typically, such narrow lines are obtained from low-pressure discharges, like Geissler tubes or hollow cathode lamps. The instrumental line shapes obtained with good spectrometers are usually approximately Gaussian pro"les in particular central core which usually is being used for deconvolution procedure. The far wings of the instrumental pro"les often decay more slowly. Since the Stark pro"les of ionic lines are, to a very good approximation, of dispersion type (Lorentzian shape), the well-known Voigt pro"le analysis (see [47] and references therein) can be applied to derive the Stark width from the observed line pro"le. In the case of neutral atom lines which are asymmetric due to the ion broadening contribution, another deconvolution procedure [48] has to be applied. The use of the deconvolution procedure [47] to the asymmetric line pro"les as applied by some authors introduces systematic error in Stark width (Lorentzian part of the experimental pro"le) measurements. This is well illustrated in Fig. 3 where coe$cients K and K , % * K "w /w , (22) % % K K "w /w , (23) * * K needed for evaluation of Gaussian w and Lorentzian w , half-widths from the measured width % * w are given as a function of the ratio of w /w where w and w are measured line widths K
Fig. 3. An example of K and K dependence upon w /w ratio for a symmetric Voigt pro"le (thick solid line) [47] % * and for asymmetric pro"les for o /o "0.40 (see Eq. (4)) and various ion broadening parameters A ranging from 0.1 " to 0.5 [48].
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at one-tenth and one-half of the maximum intensity. The solid line in Fig. 3 is used for deconvolution of symmetric Voigt pro"les [47] while for asymmetric line pro"les whose asymmetry depends upon the ion broadening parameter, A, coe$cients K and K have di!erent values for the same % * w w ratio. So the application of deconvolution procedure for the Voigt pro"le to the asymmetric line may introduce an error in determination of Gaussian and Stark widths whose magnitude depends upon ion broadening parameter A and may be as large as 25%; see Fig. 2 in [48]. In some cases, Fabry}Perot interferometer is employed for line pro"le measurements. The determination of the apparatus function is similar to the procedure described above but the analysis and deconvolution procedure is more complex (see [49,50] and references therein). Generally speaking it is best if one can virtually eliminate the instrumental broadening contribution by using a high-resolution spectrometer whose instrumental width is of the order of one-tenth or less than the observed experimental pro"le. 4.6. Self-absorption Self-absorption will have the e!ect of distorting and especially of broadening lines and will therefore produce an apparently large width. If the self-absorption originates mostly from the cooler boundary layer of much lower electron density and if the spectral resolution is su$cient, the line center exhibits readily recognizable self-reversal. But more often, self-absorption (especially when originates within homogeneous plasma layers) only slightly distorts the shape of a dispersion pro"le, even if it is appreciable [13]. Thus, it is very di$cult to judge the amount of self-absorption from the observed shape of a line. In fact, good adherence of Stark broadened lines to dispersion shapes may have sometimes mislead authors to the conclusion that line shapes have not su!ered self-absorption. A variety of well-established techniques exists to determine the presence of self-absorption e!ects. Particularly straightforward is the method of checking line intensity ratios within multiplets which are expected to adhere to LS-coupling. The well-known ratios for LS-coupling line strengths [46,51] are taken as the basis to which the observed line intensity ratios are compared, and reduction in the observed intensity of the strongest line in the multiplet relative to a weaker one indicates that self-absorption is present. In the case where transition probabilities within a multiplet are available, one can use them to evaluate line strengths (see e.g. [52,53]) and to compare those with experimentally determined values. In this way one is not related to the LS-coupling scheme only. The compilations of selected atomic transition probabilities data are available [52}54]. Furthermore, one can "nd vast transition probability data in the TOP base (the complete package of the Opacity Project data with the database management system which is usually referred to as TOP base) [55,56]. For new sources of transition probabilities data see also introduction of [54]. A modi"cation of this technique for self-absorption check, in a Stark parameters measurement, is to vary (if possible) the concentration of the species under study and to observe variations of the intensity ratio (and of the line widths) with decreased concentration. If the ratio of the line intensities (or line widths) remains constant within certain lines in multiplet these are optically thin and may be used for plasma diagnostic purposes without any correction for self-absorption.
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Fig. 4. Self-absorption test: (A) with external concave mirror; (B) doubling the plasma length by moving an auxiliary electrode within discharge and (C) recorded line pro"les I (j) and I (j) with single and double plasma length, respectively.
Another popular technique to check for self absorption is to double the optical path length by placing a concave mirror at two focal lengths behind the plasma (see Fig. 4A) and to "nd out if the increase in signal intensity, except for re#ection and transmission losses, also doubles. In practice, one scans over the line pro"le with and without an external mirror and observes whether the pro"les are proportional everywhere to the same factor. Self-absorption is present if the proportionality factor is not maintained for the high-intensity region near the line center but instead becomes smaller. If the optical depth k l (k is absorption coe$cient and l the plasma H length) is not large, k l(1, one may correct the measured intensity to the limit of an optically thin H layer [57]. Recently, another technique for self-absorption check in axially homogeneous pulsed sources has been developed [58,59]. Within a linear pulsed discharge one auxiliary movable electrode is introduced between the cathode and anode (see Fig. 4B). By changing the position of the movable electrode one can change the observed plasma length without changing plasma impedance. In this
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way, plasma parameters remain constant while observation length is being changed. By measuring line shapes with two plasma lengths it is possible to determine k l. H In order to evaluate correction for self-absorption for the case described in Fig. 4A and B we start from the expression for the intensity of radiation I from the homogeneous plasma layer H (length l) in LTE: I "B [1!exp(!k l)] , (24) H H 2 H where B is the Planck's function and k the absorption coe$cient which is related to the emission H 2 H coe$cient e "k B . Depending upon the value of k l one can distinguish three cases: H H H 2 H 1. k l;1, a very low absorption when exp(!k l) 1!k l so Eq. (24) may be written as H H H I B k l e l . (25) H H 2 H H 2. k l<1, a very large absorption so Eq. (24) may be written as I "B , i.e. the intensity of H H H 2 plasma radiation is equal to the radiation of a black body and the line pro"le cannot be recovered. 3. k l(1 when it is still possible to reconstruct line pro"le to the optically thin case. The H correction factor is equal to [57] K "k l/[1!exp(!k l)] (26) H H H and may be derived by dividing Eqs. (25) and (24). The corrected line pro"le may be obtained by multiplying the recorded line pro"le with correction factor K (Eq. (26)). To determine K (see H H Fig. 4A), we introduce factor G which takes into account mirror re#ectivity and geometrical e!ects. Obviously, the length of observed plasma layer l should be much smaller than the focal length of focusing system. By opening and closing the optical shutter, two line pro"les are recorded (see Fig. 4C) and they can be used for determination of K from Eq. (26). If the H frequency of radiation shutter is high enough both pro"les may be obtained simultaneously. From the line recordings with two plasma lengths (Fig. 4C), one can determine ratio R , H R "I /I "1#G exp(!k l) . (27) H H For continuum radiation, k "0 and H R"I /I "1#G . (28) Sometimes, when continuum is weak the accuracy of R determination is low. R may be determined at the line wings where absorption is negligible. From Eqs. (27) and (28) k l"ln[(R!1)/(R !1)] H H so correction factor K (Eq. (26)) may be expressed as H K "ln[(R!1)/(R !1)]/(1![(R !1)/(R!1)]) . (30) H H H To obtain the line pro"le for the optically thin case it is necessary to multiply measured pro"le I (j) (or I (j)) with the correction factor K (see Fig. 4C). In a similar way one can derive correction H
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factor K for the case described schematically in Fig. 4B: H K "ln[2(R /R)!1]/2[(R /R)!1] . (31) H H H Still another technique exists for correction of self-absorbed line pro"les and it consists of (a) calculating the intensity that would be emitted by a black body of the plasma temperature at the wavelength of the line peak and (b) comparing this with the actually observed (absolute) intensity at the line peak. If the observed peak intensity is not close to the calculated black body limit this intensity ratio may, within a certain range ((0.6), be used to apply correction to the line pro"le. In conclusion of this part dealing with the line shape recordings in emission experiments, one can point out that the importance of self-absorption cannot be overemphasized. There are cases however, when one has to diagnose plasmas where self-absorption cannot be corrected to the optically thin conditions. Furthermore, Stark pro"les may be distorted by the presence of cooler boundary layers or spatial and temporal inhomogeneities in plasma. These inhomogeneities may arise, e.g. when materials are introduced into plasmas by evaporation from the electrodes or as dust from the walls. In these cases, one can consider measured line shape as a superposition of line pro"les with di!erent Stark widths. Since the lower electron density plasma regions, which are usually close to the boundaries, will predominantly contribute to the line center and thus raise it up. So the overall pro"le is likely to show a width which is smaller than the one from the central hotter plasma region. Although such distortion of the line pro"les may be treated using radiative transfer calculations, due to the lack of all necessary data (distributions of temperature and particle densities along observation path, population of energy levels, etc.) this would be a very di$cult task. The diagnostics of inhomogeneous optically thick plasmas were treated elsewhere (see e.g. [30,60] and references therein).
5. Electron temperature diagnostics As it has been already pointed out, Stark widths and shifts depend also upon electron temperature, ¹ (see Section 2). Therefore, before one compares experimental Stark broadening para meters with theoretical results or with another experiment it is necessary to determine ¹ . Although several publications are devoted to this subject (see [30,31,45,61,62] and references therein) short overview of spectroscopic methods for electron temperature measurements will be given here for completeness. Absolute and relative line intensities. For the case of optically thin line emission the total line intensity I is given by I"hl A N l , (32) where N is the density of atoms in the upper state, A is the absolute transition probability for spontaneous emission, l is the emitting length, and l is the emission frequency of the line. Since the line intensity for an optically thin plasma is proportional to the number density of the upper state, by equilibrium relations, one can obtain a temperature from the measurement of the intensity. In the case of absolute line intensities, one must also have knowledge of the absolute transition probability and the plasma emitting length, and an absolute spectral radiance
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calibration must be performed. This can be seen from the following equation for the case of Boltzmann equilibrium: I"hl g A l(N /Z)exp(!E /k¹ ) , (33) where N is the total number density in cm\ of the radiating species, E is the upper state energy, g is the upper state degeneracy and is equal to 2J #1 (J is the upper-state total angular momentum), and Z is the partition function. For relative line intensities of the same species and stage of ionization, besides the equilibrium assumption (usually local thermal equilibrium, LTE), only relative transition probabilities and relative spectral radiance calibration are needed to determine ¹ . Because this method of relative line intensities depends on the di!erence in upper-state energies, it is quite insensitive for lines within the same stage of ionization when the energy di!erence is small compared with k¹ , as demonstrated by Eq. (34): I /I "[(gA) j /(gA) j ] exp[(E !E )/k¹ ] , (34) where 1 and 2 denote two di!erent transitions within the same species and stage of ionization. Instead of using relative intensities of only two lines it is much more reliable to use relative intensities of several lines originating from atomic states of di!erent excitation energies of the same species. In this frequently applied method, the relative populations of several atomic (or ionic) states of di!erent excitation energies are obtained from relative line intensity measurements. If the plasma is in partial LTE the population adhere to a Boltzmann distribution uniquely characterized by their excitation temperature, and this temperature ¹ may thus be derived conveniently from a Boltzmann plot; see e.g. [61,62]: ln(IH j )/(g A )"b#a*E , (35) where IH is the relative line intensity, a"!1/k¹ is the slope of the Boltzmann plot and b"ln(N /Z ) is the constant for the lines of the same ionization state (N and Z (g) are population of the ground state and its partition function respectively). The accuracy of this method depends critically on the number of lines available and especially on the energy spread between the excited atomic states involved, *E , which should preferably be several times larger than k¹ . In favorable situations the temperature may be determined to 2}3%, while normally an uncertainty of about 5}10% must be expected. For lines of di!erent stages of ionization of the same species the method of relative line intensities can be very sensitive because the energy di!erence is usually quite large. But because of the equilibrium assumption, described by Saha equilibrium, this method also depends on an independent knowledge of the electron density. Therefore, it depends strongly on LTE, as shown in Eq. (36) for Saha}Boltzmann equilibrium: (gA) j N h I exp[(E !E #E !*E)/k¹ ] , " (36) 2(2pmk) (gA) j ¹ I where 1 and 2 denote two transitions within same species but consecutive stages of ionization, N is the electron density (cm\), ¹ is the temperature in K, E and E are the excitation energies, E is the ionization potential of the lower stage of ionization, and *E is the lowering of the ionization
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potential (see e.g. [45,62,63]). Eq. (36) for determination of ¹ requires electron density and N is usually measured independently in an experiment set for measuring Stark broadening parameters. If one needs ¹ to evaluate N from Stark broadening parameters this method, after determination of N , may be used as a sensitive cross-check for electron temperatures measured using other spectroscopic methods. The validity criteria for LTE and consequently, criteria for the application of the above spectroscopic methods for ¹ measurements are extensively discussed by Griem [45,62] and Drawin [31] for spatially homogeneous and inhomogeneous plasmas and for steady-state and transient plasmas. A simple formula derived for steady-state plasma (applicable also to slow time-varying plasmas) may be used to illustrate the importance of emitter character for estimation of LTE criteria [45]: N 57;10Zn\ (k¹ /E ) (cm\) , (37) & where Z!1 is the emitter charge, and n is the principal quantum number of the lower level of considered transition. To demonstrate the importance of the ful"lment of the LTE criterion Eq. (37), the results of electron temperature measurements in a low-pressure pulsed arc in helium with small admixture of oxygen (about 0.5%) [64] are given in Fig. 5A using the above-mentioned techniques: Boltzmann plots of O III and O IV lines and ratios of O IV/O III and O V/O IV lines. The plasma electron density, N , was determined from the width of the He II P line. The LTE criterion Eq. (37) was not ? ful"lled for any of the thermometric species, so none of the used techniques may be applied for ¹ diagnostics. It is therefore not surprising that measured temperatures are di!erent and always higher if ¹ is determined using a method which is further from the ful"llment of LTE criterion Eq. (37). In order to derive correct electron temperature, ¹ at a certain period of plasma decay, measured ¹ are plotted versus log(N /N ) where N is derived from the LTE criterion Eq. (37) and N is measured electron density (see Fig. 5B). The data points in Fig. 5B are "tted and the intersection of the best-"t curve with ¹ axis at log(N /N )"0 (LTE criterion ful"lled) is taken as a correct electron temperature ¹ . During plasma decay in this way the other ¹ were derived and are presented in Fig. 5A. The main purpose of the example in Fig. 5 is to illustrate once more the importance of the ful"lment of LTE criteria for the application of certain method for ¹ measurements. Although this condition is discussed in many textbooks of plasma spectroscopy, see e.g. [30,31,45,62], some authors still apply one of the above methods to measure ¹ assuming LTE, i.e. without proving this assumption. Such erroneous ¹ measurements introduce di$culties in comparison or evaluation of Stark broadening data. Relative line-to-continuum intensities. This method is normally used for pure gases. Knowledge of the line transition probability and the continuum Gaunt factors are necessary, and this method is not dependent on an electron density but is dependent on an equilibrium assumption. There are many examples of this method throughout the literature, see e.g. [45,62]. In the case of hydrogen, all theoretical dependencies are well known and have been calculated so the ratio is accurately known, see e.g. [45,62]. Optically thick lines. Up to this point optically thin plasma has been assumed for all diagnostic methods. In this example of temperature determination, there is a line (or lines) whose peak intensity is equal to the black body intensity at the plasma temperature. This peak intensity is
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Fig. 5. (A) The example of time resolved electron temperature measurements, ¹ , in a low-pressure pulsed arc in helium with 0.5% of oxygen using di!erent spectroscopic methods: Boltzmann plots of O III and O IV lines and the ratio of line intensities of O IV/O III and O V/O IV. The so-called `corrected temperaturea ¹ (see text) is also presented. Time behavior of discharge current I, and electron density N are also given. (B) The measured electron temperatures 5.5 ls after the beginning of discharge current versus log(N /N ) where N is derived from the LTE criterion (37) and N is the measured electron density. All results are taken from [64].
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independent of transition probabilities and is simply described by Planck's law. An absolute intensity calibration and the assumption of LTE are also necessary. The major problem of this method is that the black body radiation may arise from layers at di!erent temperatures, e.g. inhomogeneous plasmas or plasmas with boundary layers, and the resultant intensity does not correspond to homogeneous optically thick plasma that was assumed under investigation. An additional method that relies on the absorption of radiation to determine temperature is the line-reversal method. This technique utilizes a variable-intensity (calibrated), usually continuous, light source that backlights the emission from the plasma. When the temperature of the calibrated continuous source equals the temperature of the plasma, i.e. the total intensity of the plasma plus the back lighting source equals the intensity of the black body at the plasma temperature, the plasma spectral line goes from an emission line to an absorption line. Since, the reversal method depends only on the population ratio of the levels within an atomic species, if spectral lines arising from di!erent excitation potentials all give the same reversal temperature, this allows one to attribute Boltzmann equilibrium to these excited levels of this species. Many other methods of electron temperature measurements exist and are discussed in detail in [30,31,45,62]. In particular, we draw attention to the method based on Thomson scattering (see e.g. [30,65]) which may be used for simultaneous ¹ and N determination. 6. Regularities in experimental Stark widths and shifts As already pointed out, see e.g. [3], broadening and shifting of spectral lines in plasmas is determined by two factors, the plasma environment and the atomic structure of the emitting atom or ion. Since atomic structure exhibits a great many regularities and similarities one must expect that these had their way into width and shift parameters of plasma-broadened lines. So before one begins with the analysis of experimental results it is of interest to establish regularities and similarities of Stark widths and shifts which may help to make a preliminary test of experimental data. Several types of studies of Stark widths and shifts regularities have been carried out. For example, PuricH [66] has presented recently a detailed overview of the numerous studies of his and his co-workers, showing that Stark widths and shifts linearly depend in a log}log scale upon the inverse of the di!erence between upper level of the transition and ionization potential, *I\. In another study, Sarandaev and Salakhov [67] used regularities of energy level broadening and shifting to determine Stark width and shift dependence upon the e!ective quantum number of the upper level of transition. Again linear dependence in a log}log scale is found. Here, we shall use main results of regularity and similarity studies of Stark widths and shifts by Wiese and KonjevicH [68,69] based on examination of a large sample of experimental line broadening data. The results of these studies may be summarized as follows. 6.1. Widths 1. Line widths in multiplets are usually the same within a few per cent, and possibly better than 1%. Exceptions are cases where some perturbing levels are accidentally extremely close to the upper (or lower) levels of multiplet. 2. Line widths in supermultiplets are usually the same within about $15%.
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3. Line widths within a transition array stay normally within a range of about #20%. 4. For complex spectra, line widths show pronounced stepwise increases with increasing n (principal quantum number) and l (orbital quantum number) of the upper state. 5. For simple spectra, where the line widths within a spectral series can be observed, the widths show a smooth increase with n of the upper state so that interpolations or limited extrapolations along the series should produce additional reliable data. 6. For most of the transitions studied in homologous atoms and ions, clear systematic trends are discernible for analogous lines. Therefore, interpolations or extrapolations in homologous atoms and ions appear feasible. 7. For ions along the isoelectronic sequences, clear trends of stepwise decrease in the widths are seen in the experimental data. In addition, recent studies of analogous transitions along isoelectronic Li}like [70] and B-like [71,72] sequences proved this conclusion. 6.2. Shifts 1. Stark shifts in multiplets generally agree within $10%. 2. Shifts within supermultiplets and transition arrays vary moderately within about $25%, when the shifts are relatively large fraction of the half-width, i.e. approximately 50% of the width. When the shifts are signi"cantly smaller fractions of the widths, the variations become much larger. 3. Shifts within a spectral series show pronounced increases with increasing principal quantum number of the upper state. Thus, interpolations or limited extrapolations along a spectral series should yield additional reliable data. 4. For analogous transitions in homologous atoms and ions, systematic trends in the Stark shifts are clearly observed. Therefore, interpolations and extrapolations to other homologous species should provide additional reliable data. 5. Very little is known about shift trends for isoelectronic ions. Only few cases are available where data for more than two isoelectronic ions are reported (and regular decreasing trend along the sequences is usually observed). In summary, the available experimental data on Stark widths and shifts clearly exhibit regularities, as numerous examples have shown [68,69]. Nevertheless, pronounced irregularities are encountered [68,69], too, and appear to be readily explainable in terms of special circumstances in the atomic structure.
7. Typical experimental procedure Initially, it is necessary to perform detailed analysis of the emitted spectra and select lines to be used for electron density and temperature diagnostics. For this purpose photographic spectra recording is very convenient. Although the photographic technique may be used in conjunction with high-speed cameras for the study of transient sources, OMA has an advantage and it may be used with continuous plasma sources as well. In the process of line selection one has to bear in mind that the line shapes may be distorted by the blending with nearby impurity lines. If strong magnetic
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"elds are present, Zeeman splitting must be taken into account as well. So, for the diagnostic purposes it is best to select, well isolated lines belonging, if possible, to two consecutive ionization stages of an element (or elements). The consistency of electron density measurements from the lines of two ionization stages indicates that the lines are emitted from the same region of plasma. Furthermore, in order to perform self-absorption check it is useful to select more than one line from a multiplet. Finally, for the selected lines theoretical values should be consulted. If they are not available one has to perform calculations of Stark widths and/or shifts. Although non-hydrogenic spectral lines are usually narrow and for the line shape recording one does not require correction for the sensitivity of the recording system, calibration of the sensitivity for monochromator-radiation detector system is usually required for electron temperature determination. In the next step the line shapes are recorded. This procedure is usually preceded by instrumental pro"le determination. For these measurements and line shape recordings, in general the photographic technique is signi"cantly more uncertain than the photoelectric technique because of the complicated conversion process from "lm densities to radiation intensities which involves a logarithmic dependence between two quantities. On the other hand, photoelectric signals are, over a wide range, directly proportional to radiative intensities. For shift measurements, photographic technique may be, in some cases (e.g. continuous plasma sources) still very useful. As a rule it is better to select weaker lines for electron density diagnostics since these are less a!ected by self-absorption. Nevertheless, the self-absorption test has to be performed (see Section 4.6). If the line is optically thin or if it can be corrected to optically thin conditions, one proceeds with the line shape measurements. Here, care should be taken to determine correctly continuous background. For this purpose it is advisable to record line wings as far as "ve half-widths (not less than three) in both, red and blue direction. The next step is to perform an appropriate deconvolution procedure (see Section 4.5). The neutral atom lines are asymmetric (see Section 2.1) due to the contribution of ion broadening while ion lines usually have symmetric pro"le of Lorentzian or Voigt type whenever Doppler and/or instrumental broadening are not negligible. Sometimes ionic lines are also found to be asymmetric and this is usually an indication of the presence of an inhomogeneous plasma layer along the observation path. In this case the radiation from the cooler and hotter part of the plasma are superimposed. The pro"les from the lower electron density region (less shifted) are superimposed on the top of more shifted line pro"les. The overall pro"le is asymmetric and cannot be used for plasma diagnostic purposes. This e!ect usually occurs with strong lines and in particular with resonance lines. If the instrumental half-width w is measured, after the deconvolution of experimental pro"le, the ' half-width of the Gaussian part w may be used to determine Doppler width w from the following % " relation (sum of Gaussian pro"les): w "w #w . (38) % " ' The Doppler width w may be further used to determine the emitter temperature ¹ from Eq. (21). " With a Fabry}Perot interferometer used for line shape recordings the deconvolution procedure is di!erent, see e.g. [49,50]. Up to now the discussion of the deconvolution of experimental line pro"les is related to the determination of half-widths. It is, of course, much better if one "ts the whole experimental pro"le
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before deconvolution starts. Commercial software packages for line "tting and deconvolution of Voigt pro"les are available. Before the Lorentzian part of the measured pro"le (Stark width) is used for N determination, one has to estimate the possible in#uence of the resonance and van der Waals broadening (see Sections 4.2 and 4.3) and to subtract their contributions from the measured width. The experimental shift measurement is much simpler to use for N determination. The only possible correction is due to van der Waals broadening. Unfortunately, theoretical results for the Stark shifts have larger uncertainties than for widths. This is in particular for those Stark shifts which are much smaller than widths. Furthermore, shifts are sometimes very small (or may be equal to zero) and the experimental error is large. After determination of Stark half-widths and/or shifts one should check whether the results are in agreement with the above-described regularities and similarities (see Section 5). In the case of deviations from these rules (this is in particular case for the lines within multiplet) either line originates from the energy level with very close perturbing level (for energy levels and Grotrian diagrams see e.g. [73,74]) or some experimental di$culties (erroneous identi"cation of the line, interference with impurity lines, etc.) have to be overcome.
8. Analysis of experimental results The measured Stark widths w and/or shift d and electron temperature ¹ are further used in
conjunction with theoretical data for N plasma diagnostics. It is necessary to replace w and/or d in one of Eqs. (1)}(3), (8)}(11) with measured values w and d , introduce necessary theoretical
data w , d and A for neutral atom lines and solve the selected equation. Depending upon the set of theoretical data used for N evaluation di!erent results may be obtained. The general statement that semiclassical theory for a neutral atom and ion line widths agree with experiment within $20% and $30% respectively is derived on the basis of comparison of a large number of experimental data [3,13}17,62], with the theory [3]. The discrepancy may well exceed estimated uncertainty for some lines and this is another argument to use several lines (if possible from di!erent multiplets) for plasma diagnostic purposes. The situation is even more complex since various theoretical calculations, sometimes, di!er considerably from each other. Furthermore, there are cases when theory does not describe correctly the Stark width and shift temperature dependence, which introduces an additional error in comparison to experimental results measured in a large temperature range. All these di$culties will be discussed in more detail later in relation to the results for various elements and their ions. In order to establish a reliable database which will be recommended with a higher estimated accuracy for plasma diagnostic purposes, we have selected experimental data reported by at least three independent experiments. Only data with an estimated uncertainty up to $30% (A,$15%, and B,$30%, in [13}17]) were taken for this study. The critical reviews [13}17] were the sources of experimental data published until the end of 1988. The later publications were selected and analyzed in accordance with already established criteria [13}17]: 1. The plasma source must be well characterized, i.e. it must be homogeneous in the observation region, be in the steady state during observation time and must be reproducible.
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2. An independent and accurate determination of plasma electron density N must have been carried out. 3. A reasonably accurate determination of the electron temperature must have been carried out. 4. Other contributing line broadening mechanisms and pertinent experimental problems must have been discussed and taken into account. The order of appearance of the selected results for various elements will be as in the Periodic Table. The data tables are subdivided into three principal parts. In the "rst part, which comprises three columns each spectral line is identi"ed by transition array, multiplet designation, and wavelength. In the second part of the table, ranges of electron temperature are given and they are followed by the ratios of measured and theoretical values. When the theoretical data used for comparsion with experiment are presented in the form of tables (see e.g. [3]) the cubic splines numerical method is used to determine particular values, otherwise they are calculated for the experimental conditions. In cases of ionized atom lines where linear w(N ) dependence exists, one more column with normalized widths w /N (10e/cm) is given in data
tables. In the third part of the table the accuracy of the data are quoted and the literature source is identi"ed. Through the selected ratios of measured w , and theoretical Stark widths w , the linear best "t
formula w /w "A#B¹ is determined and given for studied lines or multiples. The estimated
uncertainty (Eu) of this formula is estimated from (i) the scatter of w /w around best-"t value, and
(ii) typical error in electron density diagnostics which is usually by far the largest error in Stark broadening parameter measurements. The uncertainty of theoretical results is not taken into account. So the best-"t formula shows the correction factor R "w /w at various ¹ , which
should be applied when the theory used for comparison is applied for electron density evaluation, see for example the He I 501.57 nm line. Furthermore, from the same best-"t formula, the average deviation from the theory (ADT) is derived as 2"(A#B¹)!1" ;100 . (39) ADT(%)" 2 "¹ !¹ " Due to much larger scatter of the Stark shift experimental data they will not be considered here for high accuracy diagnostic purposes. The only exception are the shifts of visible He I lines (see below). First, neutral atom lines will be discussed, singly ionized atom lines will follow while the lines of multiply charged ions are treated separately. It should be noticed that there is no physical reason to separate singly ionized atom lines from those belonging to multiply charged ions. But since the most comprehensive set of semiclassical data [3] deals with singly charged ions only it is convenient to discuss singly ionized atoms separately. The results for multiply charged ions will be compared with another semiclassical approach [5,6]. 8.1. Neutral atom lines As a consequence of the fact that quasistatic ion contribution to the Stark widths and shifts does not depend linearly upon electron density and temperature for neutral atom lines (see Eqs. (1)}(3)) only lines with available theoretical data are taken for analysis.
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Table 1 Numerical data for He-I lines measured at electron densities N'10 cm\ Transition
j (nm)
Temperature range (K)
w /w
Accuracy
Reference
2S}3P
501.57
30 000 24 000 22 700 38 000 17 400 20 000 19 000}43 000
0.72 0.86 0.86 0.99}0.97 1.01 & 0.88 0.91 & 0.96 0.88}0.88
B B B B A B B
[75] [76] [77] [78] [79] [80] [81]
2S}3P
388.87
30 000 26 000 20 000 31 000 & 35 000
0.87 1.11 0.96}1.08 0.86 & 0.86
B B B B
[75] [76] [80] [59]
2P}4S
471.32
30 000 20 000 22 700 19 000}43 000
0.85 1.02 0.91 0.90}0.90
B B B B
[75] [76] [77] [81]
Not taken for averaging.
Helium: He I. The results of He I Stark width data are summarized in Table 1 where the average ratios of measured to theoretical [3] widths are given. For this table only experimental results obtained at electron densities above several times 10 cm\ were used. Lower electron density experiments will be discussed in relation to ion-dynamic correction and these results are summarized in Table 2. Apart from experiments critically evaluated in [13,15,17] some new results [59,81,82] published after the end of 1988 are introduced as well. The experiment [59] will be used in relation to the study of the in#uence of ion dynamics to the shape and shift of He I lines (see Table 2). Here, in Table 1, only the results obtained in He}Ar mixture are presented since for them the application of static ion approximation is justi"ed in this case. The comparison of selected experimental values with semiclassical theoretical results [3] (see Table 1 and Fig. 6) is very impressive indicating that the estimated experimental uncertainties [13,15,17] and Table 1 were, most likely, too conservative. The best-"t values are as follows: 501.6 nm, w /w "0.897#8.69;10\¹ for 19000(¹(43 000 K .
Estimated uncertainty (Eu):$12% and average deviation from the theory (ADT); see Eq. (39): 7.6% . 388.9 nm, w /w "1.310!1.31;10\¹ for 26 000(¹(35 000 K .
Eu:$17%, ADT"9.0% . 471.3 nm, w /w "0.986!2.59;10\¹ for 19 000(¹(43 000 K .
Eu:$14%, ADT"9.4% .
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Table 2 Average ratios of the measured He linewidths w and shifts d at N (10 cm\ [21,34,59] to the theoretical results
using data from Griem (G) [3], Basalo, Cattani and Walder (BCW) [86] and DimitrijevicH and Sahal-Brechot (DSB) [87] respectively. Theoretical results marked DSB#G are calculated with electron impact half half-width, from [87] and ion-broadening parameter A from [3]. The data given in parantheses represent maximal scatter from the average value d /d
w /w j (nm)
Transition
G
BCW
DSB
DSB#G
G
BCW
DSB
DSB#G
501.6 667.8 706.5 388.9 587.6 471.3 318.8
2S}3P 2P}3D 2P}P 2S}3P 2P}3D 2P}4S 2S}4P
0.93(0.03) 0.98(0.04) 0.85(0.03) 0.85(0.03) 0.90(0.02) 0.86(0.01) 0.91(0.03)
0.97(0.04) 1.07(0.05) 0.95(0.03) 0.98(0.03) 1.17(0.02) 0.98(0.02) 1.02(0.05)
1.07(0.04) 1.07(0.06) 1.10(0.03) 0.99(0.06) 1.04(0.07) 1.19(0.02) 1.14(0.05)
1.11(0.06) 1.13(0.04) 1.13(0.05) 1.03(0.03) 1.05(0.02) 1.21(0.01) 1.19(0.04)
0.66(0.02) 0.70(0.04) 0.90(0.02) 0.79(0.04) 0.72(0.04) 0.96(0.02) 0.98(0.03)
0.62(0.02) 0.79(0.05) 0.94(0.02) 0.95(0.04) 0.46(0.03) 1.00(0.03) 0.99(0.02)
0.79(0.10) 0.91(0.06) 1.03(0.05) 0.93(0.18) 1.16(0.22) 1.04(0.04) 1.26(0.09)
0.58(0.07) 0.58(0.06) 0.67(0.08) 0.83(0.06) 0.84(0.03) 0.87(0.04)
Theory predicts a sign opposite from the experimental shift.
Fig. 6. Ratios of measured, w , and theoretical widths, w , versus electron temperature, ¹ , and the best "t for the He
I 501.57 nm line. The experimental data: 䊏 [76], 䢇 [77], 䉱 [78], 䉲 [79], 䉬 [80], # [81].
The way to apply the above formulas is: (i) calculate R "w /w for measured ¹ , (ii) using
experimental w and ¹ and theoretical data [3], from Eq. (1) evaluate N and (iii) determine
electron density as N "N /R . From experimental results in data tables (see e.g. Table 1) one can derive best-"t formulas for any other theoretical approach. It is interesting to note that for all three selected He I lines (see Table 1) the theory [3] predicts always larger widths for 7}12%. The Stark width temperature dependence for 501.6 nm and for 471.3 nm lines is well predicted by the theory [3].
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Recently, Perez et al. [81,83] have calibrated Stark broadening parameters for several He I lines. On the basis of their and all other experimental data the authors [81,83] derived empirical formulas relating measured width w , and shift d with the plasma parameters N and ¹ :
E 501.6 nm, range of application for N : (1}16.5);10 cm\ and ¹ : 19 000}43 000 K: ln w (0.1 nm)"!38.99($1.4)#1.08($0.02)ln N (cm\)!0.12($0.04)ln ¹ (K) ; (40)
E 667.8 nm, range for N : (1}10);10 cm\; ¹ : 19 000}43 000 K: ln w (0.1 nm)"!34.90($1.5)#1.040($0.014)ln N (cm\)!0.35($0.04)ln ¹ (K) ;
(41) E 471.3 nm, range for N : (0.6}13);10 cm\; ¹ : 19 000}43 000 K: ln w (0.1 nm)"!39.97($1.6)#1.05($0.02)ln N (cm\)#0.13($0.05)ln ¹ (K) ; (42)
E 728.1 nm, range for N : (1.5}14.5);10 cm\; ¹ : 16 000}25 000 K: w (0.1 nm)"8.96($0.14);10\N (cm\)#0.282($0.108) ; (43)
d (0.1 nm)"4.28($0.09);10\N (cm\) . (44)
The authors [81,83] recommend the above empirical formulas for plasma diagnostic purposes, with an accuracy of $15%. The comparison between Eqs. (40)}(43) with [3] shows some discrepancies which are best illustrated by the ratio of Stark widths calculated from Eqs. (40)}(43) w , and from [3], w : 501.6 nm, N "(1}10);10 cm\, 20 000 K, w /w "0.82}0.90 , 40 000 K, 0.80}0.86 , 667.8 nm, N "(1}10);10 cm\,
20 000 K, 1.11}1.10 ,
471.3 nm, N "(1}10);10 cm\,
20 000 K, 1.01}1.07 ,
40 000 K, 0.92}0.90 , 40 000 K, 1.05}1.10 ,
728.1 nm, N "(1.5}10);10 cm\, 20 000 K, 1.22}1.00 . The results of the above comparison, Eq. (40), with [3] and indirectly with our recommended data for the 501.6 nm line show an agreement within 0}10% depending upon ¹ and N . The discrepancy for the 471.3 nm is larger and goes up to 20% (again depending upon both ¹ and N ) from our recommended values, see best-"t formula for this line and w /w results in Table 1. For the other
two lines, 667.8 and 728.1 nm, the discrepancy with [3] is of similar type and has to be examined further. In another experiment [34], broadening and shifting of "ve He I (318.8, 471.3, 501.6, 667.8 and 706.5 nm) lines has been studied at relatively low electron densities in the range (2.5}5.9);10 cm\, electron temperatures 19 300}23 600 K and gas temperature 5000}12 500 K
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368
in helium}hydrogen plasma. This work was a continuation of a similar study dealing with broadening and shifting of the He I 388.9 and 587.6 nm lines [59,84]. At these relatively low electron density plasmas [34,59] and [21] performed earlier, in the presence of light ions as perturbers (e.g. H> or He>), the ion dynamic e!ects near the line center may occur. Namely these light ions behave more like electrons, resulting in impact line broadening. The method for evaluation of the in#uence of ion dynamic e!ects on the shape of non-hydrogenic atom lines is developed independently by Griem [3] and Barnard, Cooper and Smith (BCS) [85]. A summary of both theoretical approaches [3,85], is given in an experimental paper on neutral helium lines [21]. The results for neutral atom and ion lines reported by DimitrijevicH and Sahal-Brechot (DSB) (see e.g. [5,6]) were calculated using impact approximation only. The inclusion of ion dynamics in both equivalent theoretical calculations [3,85] also produces asymmetrical line pro"les but with larger widths and shifts than if quasistatic ion approximation is applied only. However, it should be pointed out that ion dynamics increase only the width and does not in#uence line asymmetry. As a matter of fact the more dynamical, the less asymmetric, and in the ion impact limit the pro"le is a symmetric Lorentzian. The shape of the lines remains the same. The line widths and shifts in the presence of ion dynamics may be calculated using simple formulas [21,59] which were derived from BCS [85]. When compared with Eqs. (1)}(3) these formulas contain, in addition to static ion broadening parameter A the dynamic ion broadening parameters = and D so one now has H H w (N ,¹ ) 2w (¹ )[1#g= A (¹ )]N 10\ , H ,
(45)
d (N ,¹ ) [d (¹ )$2.0g A (¹ )D w (¹ )]N 10\ , , H
(46)
d
(N ,¹ ) [d (¹ )$3.2g A (¹ )D w (¹ )]N 10\ . , H
(47)
The parameters = and D may be calculated from the following expressions: H H
1.36B\ /g, B((1.36/g) , =" H 1, B5(1.36/g) ,
(2.35B\ !3A R)2g , B(1 , D" H 1, B51 ,
(48)
(49)
where R is given by Eq. (4) and B"A (¹ )[0.0806w (¹ )/jN ](k/¹ ) , with atom-ion perturber reduced mass k in amu and gas temperature ¹ . In case when = "1 H and/or D "1 the in#uence of ions on the line shape is treated quasistatically. H Thus, for the calculations of the ion-dynamic line widths and shifts [85] one requires, as in Eqs. (1)}(3), electron impact half-half-width w electron impact shift d , and ion broadening parameter A (see Section 2). For the He I lines these three quantities were calculated by several authors [3,86,87]. The experimental results [21,34,59] were compared in [34,59] with the data
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369
obtained using quasistatic, Eqs. (1)}(3), and ion-dynamic approximation, Eqs. (45)}(47). The overall trend of the results appears to favor the application of ion-dynamic theories at low electron densities (see also [88]). In Table 2 [34], the average ratios of measured to theoretical widths and shifts with incorporated ion-dynamics, Eqs. (44)}(46), are given and they may be used in conjunction with all three sets of theoretical He I data [3,86,87] for N plasma diagnostic with an estimated accuracy (based on the scatter of average ratios, see [34]) in the range $9}12%. This is close to the accuracy of N determination from the shape of hydrogen Balmer lines [89]. In view of the studied in#uence of ion dynamical e!ect on the width of He I lines [21,34,59] the results of an earlier experimental study of a wall-stabilized helium arc at 1.4 atm [90] were reanalyzed in [91]. Originally in [90] the radial distributions of electron densities were derived from experimentally determined Stark widths using semiclassical data [45] and quasi-static treatment of ion broadening (Eq. (1)). The results [90] are given in Fig. 7A together with N derived from Saha equation. Using same experimental results [90] but with ion dynamics taken into account (Eq. (45)) the consistency between various N -radial distributions is considerably im proved (see Fig. 7B) [91] which is another illustration of the importance of ion dynamics at low electron densities. Carbon: C I. All three selected experiments [92}94] were performed in a wall-stabilized arc at atmospheric pressure. In addition to Stark widths in [93] the experimental results of ion broadening parameters A were also reported. The same C I line pro"les [93] were reanalyzed in [95] and slightly di!erent values for A were reported. If one excludes from comparison data [92] which show systematic discrepancy with those of the other two experiments [93,94] (see Table 3) the results of the comparison with the theory [3] may be well presented by the following equations: 538.0 nm, w /w "0.862#1.71;10\¹ for 9900(¹(11 600 K ,
Eu.:$10%, ADT"4.6% , 505.2 nm, w /w "0.919#6.71;10\¹ for 9900(¹(11 600 K ,
Eu.:$12%, ADT"8.9% , 493.2 nm, w /w "0.850!4.43;10\¹ for 9900(¹(11 600 K ,
Eu.:$11%, ADT"19.8% . One can extend the use of above equations to the other lines of multiplets. Nitrogen: N I. Wall-stabilized arc discharges were used as a plasma source in four out of "ve selected experiments [96,97,99,100]; a plasma jet was employed in the "fth one [98]. The width of the hydrogen H line in conjunction with theoretical results [45] was used for electron density @ diagnostics in two earlier experiments [96,99]. Since it was shown, see e.g. [89], that this procedure underestimates electron densities for 9}12% in the density range of interest, the results for N in [96,99] were corrected in accordance with theoretical results of Vidal et al. [101]. Thus, the average ratios of measured and theoretical Stark widths [96,99] in Table 4 are obtained for these new N values.
370
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Fig. 7. Radial distribution of electron density in a helium arc: (A) evaluated from measured widths in [90] using quasi-static ion approximation and (B) evaluated in [91] from the same set of data [90] using dynamic treatment of ions.
The best "t of ratios in Table 4 are as follows: Mult.(3), w /w "1.197!2.97;10\¹ for 11 270(¹(14 750 K ,
Eu:$14%, ADT"18.9% , Mult.(6), w /w "1.060!1.97;10\¹ for 10 500(¹(14 160 K ,
Eu.:$13%, ADT"18.3% ,
N. Konjevic& / Physics Reports 316 (1999) 339}401
371
Table 3 Numerical data for CI Transition
Multiplet (No)
j (nm)
Temperature range (K)
w /w
Accuracy
Reference
2p2s}2p(P)4p
P}P (11)
538.02
9000}11 600 11 600 9900 & 10 300
(1.10}0.91) 1.06 1.03 & 1.04
B> A B>
[92] [93] [94]
P}D (12)
505.21
9000}11 600 11 600 9670}10 300
(1.14}0.84) 1.00 0.99}0.98
A A A
[92] [93] [94]
P}S (13)
493.20
9000}11 600 11 600 9900 & 10 300
(0.81}0.63) 0.80 0.81 & 0.80
A A A
[92] [93] [94]
Reference
Not taken for averaging.
Table 4 Numerical data for N I Transition
Multiplet (No)
j (nm)
Temperature range (K)
w /w
Accuracy
2p3s}2p(P)3p
P}S (3)
746.83
13 000 11 270}14 750 13 500 13 000 11 270}14 750 13 000 11 270}14 750
0.78 0.89}0.79 0.76 0.76 0.89}0.79 0.76 0.89}0.79
B B B B B B B
[96] [97] [98] [96] [97] [96] [97]
10 500}12 350 11 700}14 160 12 100 10 600}12 450 11 700}14 160 11 700}14 160 12 100
0.82}0.75 0.83}0.78 0.85 0.87}0.81 0.83}0.78 0.83}0.78 0.88
B B A B B B A
[99] [97] [100] [99] [97] [97] [100]
10 650}12 200 11 700}14 640 13 500 11 050}14 500 12 100
0.91}0.78 0.88}0.85 0.78 0.95}0.85 0.93
B B B B A
[99] [97] [98] [97] [100]
744.23 742.36
2p3s}2p(P)4p
P}S (6)
415.15
413.76 414.34 P}S (9)
493.50
491.49
N. Konjevic& / Physics Reports 316 (1999) 339}401
372 Table 5 Numerical data for O I Transition
Multiplet (No)
j (nm)
Temperature range (K)
w /w
Accuracy
Reference
2p3s}2p(S)4p
S}P (5)
436.82
12 080 10 100}12 700 11 580 & 12 500 10 600 & 10 980
0.94 1.22}1.20 1.16 & 1.04 1.29 & 1.28
B B B A
[102] [92] [103] [94]
Mult.(9), w /w "1.154!2.29;10\¹ for 10 650(¹(14 640 K ,
Eu.:$16%, ADT"13.6% , Oxygen: O I. In all four selected papers [92,94,102,103], wall-stabilized atmospheric pressure electric arc was used as a plasma source and for electron density diagnostics the hydrogen H line @ was employed. The results for N in [102] are corrected in accordance with [101]. The best-"t formula and accuracy in Table 5 are as follows: 436.82 nm, w /w "2.076!7.95;10\¹ for 10 100(¹(11 580 K ,
Eu.$18%, ADT"21.4% , Fluorine: F I. If one takes into account temperature dependence of the ratios w /w in Table 6
the mutual agreement between four selected experiments [104}107] is very good: Mult.(2), w /w "1.281!3.16;10\¹ for 10000(¹(16 400 K
Eu.:$13%, ADT"13.6% Mult.(3), w /w "1.483!5.10;10\¹ for 10000(¹(14 000 K
Eu.:$14%, ADT"12.9%. Argon: Ar I. By far the largest number of papers [32,108}114] are devoted to the study of Stark broadening and shifting of Ar I lines. There are however, still large discrepancies between experiments (see Table 7). The electron densities in some earlier experiments [108,109,113] are corrected in accordance with [101]. The best "ts are as follows: 420.07 nm, w /w "0.891!1.15;10\¹ for 9700(¹(13 400 K
Eu.:$25%, ADT"24.2%, 427.72 nm, w /w "0.708#6.42;10\¹ for 9750(¹(12 700 K
Eu.:$23%, ADT"22.0%, 451.07 nm, w /w "0.383#1.63;10\¹ for 9750(¹(13 400 K
Eu.:$17%, ADT"42.8%, 425.94 nm, w /w "0.981!2.51;10\¹ for 8900(¹(12 700 K
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373
Table 6 Numerical data for F I Transition
Multiplet (No)
j (nm)
Temperature range (K)
w /w
Accuracy
Reference
2p3s}2p(P)3p
P}D (2)
685.9 690.98 690.25 685.60 683.43 670.83
14 000 10 000 10 000 10 000 10 000 16 400
0.73 0.98 0.97 0.97 0.98 0.83
B B> B> B> B> B
[104] [107] [107] [107] [107] [105]
P}S (3)
630.4 641.36
623.96
14 000 12 700 10 000 12 700 10 000 10 000
0.77 0.83 0.97 0.84 0.97 0.98
B> A B> A B> B>
[104] [106] [107] [106] [107] [107]
634.85
Average value for a multiplet.
Table 7 Numerical data for Ar I Transition
Multiplet (No)
j (nm)
Temperature range (K)
w /w
Accuracy
Reference
4s}5p
[1 1/2]}[2 1/2]
420.1
9750}12 700 9700}12 600 13 400
0.67}0.60 0.90}0.82 0.80
B B B
[108] [109] [110]
4s}5p
[1 1/2]}[1 1/2]
427.17
9750}12 700 11 900 11 900
0.73}0.68 0.85 0.87
B A A
[108] [111] [95]
4s }5p
[1/2]}[1/2]
451.07
9750}12 700 11 900 & 12 100 10 800}13 400 11 900
0.50}0.49 0.63 & 0.64 0.55}0.58 0.64
B A B A
[108] [111] [110] [95]
4s }5p
[1/2]}[1/2]
425.94
9750}12 700 11 900 9800}12 600 11 900 8900}11 700
0.62}0.59 0.77 0.73}0.60 0.80 0.81}0.70
B A B A A
[108] [111] [110] [95] [32]
4p}5d
[2 1/2]}[3 1/2]
603.21
9550}11 500 9100}11 450 7800
0.74}0.74 0.64}0.65 0.65
B B B
[112] [108] [113]
4p}5d
[1 1/2]}[2 1/2]
573.95
9550}11 500 9100}11 450 7800
0.70}0.81 0.84}0.63 0.80
B B B
[112] [108] [113]
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Eu.:$20%, ADT"29.0%, 603.21 nm, w /w "0.563#1.22;10\¹ for 7800(¹(11 500 K
Eu.:$16%, ADT"31.9%, 573.95 nm, w /w "0.999!2.46;10\¹ for 7800(¹(11 500 K
Eu.:$20%, ADT"23.8% . The authors of [114] o!ered an experimentally determined relation between measured FWHM, w , and N and ¹ for the Ar I 430.01 nm line:
ln N (cm\)"44.232#0.992 ln w (0.1 nm)!0.612 ln ¹ (K)
(50)
which may be used in a large range of N (1.1}11.0);10 cm\ and ¹ 9000}15 500 K for plasma electron density diagnostics with an accuracy $6.5%. Recently, the same laboratory strong asymmetry of the Ar I 430.01 nm line at high electron densities (N'10cm\) has been reported [115]. The asymmetry is explained by the presence of nearby forbidden 429.9 nm Ar I line [115]. So the use of Eq. (49) for determination of N above 10 cm\ may not be justi"ed. 8.1.1. Ion broadening parameters and asymmetries of neutral atom lines As it was pointed out already for neutral atom lines, electron-impact broadening produces a symmetrical, shifted pro"le of the Lorentzian type, while the ion contribution (primarily due to quadratic Stark e!ect) introduces asymmetry as well as additional contribution to the width and shift of the pro"le. Asymmetries in plasma broadened lines have been observed earlier in helium [21,116], carbon [117], nitrogen [118] and krypton [119]. In these experiments all or parts of the experimental and theoretical pro"les were compared to determine ion broadening parameter A. Recently detailed measurements of the plasma broadened line shapes has been carried out in a plasma of a wall-stabilized atmospheric pressure arc [93,100,111,120], the asymmetries were isolated and comparison with the theory [3] performed. An example of line shape and isolated asymmetry of the Ar I 415.86 nm line is shown in Fig. 8 [120]. In Refs. [93,100,111,120] Lorentzian pro"les were "tted to the experimental and theoretical pro"les. A functional relationship between the maximum of the Lorentzian theory deviation curve and A was obtained. This function was then used to determine A from the maximum deviation between the experimental points and the "tted Lorentzian. The asymmetry patterns, see example in Fig. 7 have a common shape with a minimum, maximum and zero crossing at the same point on a reduced wavelength scale, but they vary widely in their amplitudes. These "ndings are in a very good qualitative agreement with the quasistatic theory of ion broadening [3]. It is interesting to note that the asymmetry patterns may be successfully used as a test for plasma homogeneity [100]. The ion broadening parameters of several C I, N I and Ar I lines were determined from the measured line pro"les. In [95] an alternative approach was utilized: a computer code was developed which would "t the experimental pro"le with an asymmetrical theoretical pro"le calculated from the theory [3] by varying the width, shift, ion broadening parameter, and the background of the theoretical curve. In this way ion broadening parameters A were determined for several Ar I and C I lines.
N. Konjevic& / Physics Reports 316 (1999) 339}401
375
Fig. 8. (A) Scan of the central portion of the Ar I 415.86 nm line including the much weaker Ar I 416.42 nm line. The points are the actual spectral radiance measurements and the solid line is the least-squares "t synthetic spectrum. The points at the bottom of the "gure are the residual deviations obtained by substracting the "tted synthetic spectrum from the data points. (B) The normalized deviations for the Ar I 415.86 nm line (corresponding to the residual deviations shown in A). They are given as a percentage of the peak spectral radiance (see Eq. (5) in [120]) and are plotted versus the wavelength o!set from the line center in units of the full half-width (FWHM). Both (A) and (B) are taken from [120].
8.2. Singly ionized atom lines For the lines of singly ionized atoms the situation is somewhat similar to the one already described for neutral atom lines. The accuracy of experimental data is, however, not as well established yet and more measurements are desirable. Nevertheless, the average agreement with the
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semiclassical results [3] is in the range $30%. There are several exceptions with large discrepancies and one of them are the results for widths and shifts of singly ionized earth alkaline resonance lines (ns}np transitions of Be II, Mg II, Ca II, Sr II and Ba II) see Figs. 7}12 in [14], Fig. 1 in [16] and Figs. 1 and 2 in [17]. Like most of the resonance lines these transitions are very sensitive to self-absorption, which distorts their line shapes. Nearly optically thin conditions may be achieved by keeping the earth alkaline metals concentration in the plasma very small and the optical thickness must be accurately determined and monitored. The Stark widths and shifts for the resonance lines are very small. Therefore, high spectral resolution is required to minimize instrumental broadening and pro"le analysis is necessary to separate usually appreciable Doppler broadening contribution from the Stark pro"le (see Section 4). Also, the possible presence of other broadening mechanisms needs to be checked (see Section 4). As an example of the unsatisfactory data situation for earth alkaline ions results for Ca II lines will be summarized in Table 14 and will be accompanied with an appropriate discussion. Another case of large discrepancy between experiments and theories are Si II lines, see Tables 11 and 12. The normalized widths w /N in data tables are given in [0.1 nm/(10 e/cm)] units.
Nitrogen: N II. Large number of papers [121}126] are selected for Table 8. Unfortunately, there are a number of di$culties in data comparison. For multiplet 3 theoretical results [3] are only available up to 40 000 K so comparison above this value is not possible. Furthermore, both sets of compared experimental results (see Fig. 9 and Table 8) show variation of Stark widths within a multiplet. If the half-widths are related to the line strengths (see Fig. 9) it becomes evident that all lines were not optically thin. So for the best "t in Fig. 9 the weakest lines denoted by full squares and circles are taken into account.
Fig. 9. Same as Fig. 6 but for the N II lines from, mult.3. The experimental results: [121,122]. The line strength values, S, were taken from [54]. The data denoted with full squares and circles are taken for the best-"t line. The experimental results: 䊐 [121], * [122].
N. Konjevic& / Physics Reports 316 (1999) 339}401
377
Table 8 Numerical data for N II Transition
Multiplet (No)
j (nm)
Temperature range (K)
w /w
w /N
Accuracy
Reference
2p3s}2p(P)3p
P}D (3)
567.96
31 000 6500 54 000 31 000 6500 54 000 31 000 6500 54 000 31 000 6500 54 000
0.70 0.67
0.40 0.54 0.27 0.40 0.47 0.27 0.40 0.51 0.24 0.38 0.49 0.23
B B B B B B B B B B B B
[121] [122] [123] [121] [122] [123] [121] [122] [123] [121] [122] [123]
23 150 53 000 6500 23 150 31 000 6500 6500 23 150 6500 6500 6500
1.06 1.39 0.65 1.06 0.97 0.51 0.60 1.00 0.60 0.65 0.60
0.33 0.31 0.38 0.33 0.26 0.30 0.35 0.31 0.35 0.38 0.35
B B B B B B B B B B B
[124] [126] [122] [124] [121] [122] [122] [124] [122] [122] [122]
566.66
567.60
568.62
P}P (5)
463.05
461.39
464.31 462.14 460.15 460.72
2p3d}2p(P)4f
0.67 0.55 0.66 0.59 0.62 0.57
P}D (12)
399.50
23 150 31 000 6500 54 000
1.23 1.63 0.60 1.02
0.31 0.36 0.28 0.19
B B B B
[124] [121] [122] [123]
F}G (39)
404.13
23 150 28 300}32 300 53 000 31 000 6500 28 300}32 300 28 300}32 300
0.66 0.65}0.56
0.83 0.80}0.67 0.52 0.90 1.09 0.99}0.84 0.79}0.76
B B B B B B B
[124] [125] [126] [121] [122] [125] [125]
1.36}1.25 1.25 2.19
B B B
[125] [121] [122]
404.35 403.51 F}G (59)
453.04
28 300}32 300 31 000 6500
0.75 0.66 0.81}0.70 0.65}0.63
Temperature range 5000}8000 K.
Theoretical results for mult.5 and 12 are calculated using MSE [9]. Best "t for mult.5 shows a very strong temperature dependence. The scatter of data for mult.12 is rather large.
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The theoretical results for multiplets 39 and 59 are not available in [3] and the lack of energy level data prevented us to perform calculations. Here, we draw attention to the work of Hey and Blaha [127] where the in#uence of the change from LS to LK coupling to the widths of lines belonging to these two multiplets was studied. Mult.3, w /w "0.557#2.04;10\¹ for 6500(¹(31 000 K ,
Eu.:$15%, ADT"40.5% , Mult.5, w /w "0.512#1.79;10\¹ for 6500(¹(53 000 K ,
Eu.:$20%, ADT"21.0% , Mult.12, w /w "0.879#8.29;10\¹ for 6500(¹(54 000 K ,
Eu.:$30%, ADT"14.1% , Mult.39, w /w "0.671#1.11;10\¹ for 6500(¹(32 300 K ,
Eu.:$15%, ADT"32.6% . Oxygen: O II. Three papers [128}130] are selected for Table 9. Since theoretical data for multiplet 3 are not available in [3] w values in Table 9 were calculated using modi"ed semiempiri cal approach (MSE) [9]: Mult.3, w /w "1.24#2.17;10\¹ for 25 900(¹(54 000 K ,
Eu.:$25%, ADT"32.7% . Neon: Ne II. All three selected experiments [131}133] were performed in pulsed plasma sources. Due to the lack of theoretical data above 40 000 K [3] the results [132] were not used for the derivation of best "t for mult.2. The theoretical data for comparison with experiments, mult.7, were obtained from MSE [9]. Strong temperature dependence is detected for mult.2. Best "t for mult.7 is derived without w /w "1.77 [131] (see Table 10):
Mult.2, w /w "1.836!2.39;10\¹ for 27 000(¹(40 000 K ,
Eu.:$17%, ADT"8.2% , Mult.7, w /w "1.406!1.42;10\¹ for 28 000(¹(84 700 K ,
Eu.:$16%, ADT"32.6% . Silicon: Si II. The Stark widths for Si II exhibit large scatter (see e.g. [17] and references therein). This is, in particular, the case with the lines from multiplet no. 1. The experimental results for two strong lines, 386.2 and 385.6 nm, di!er more than a factor of two (see Table 11) where for the purpose of comparison all experimental data including accuracy C are given. To illustrate the discrepancies between various theoretical approaches the results of calculations are also given but without estimated uncertainties. We draw attention here to the recent shock tube experiment [134]
N. Konjevic& / Physics Reports 316 (1999) 339}401
379
Table 9 Numerical data for O II Transition
Multiplet (No)
j (nm)
Temperature range (K)
w /w
w /N
Accuracy
Reference
2p3s}2p(P)3p
P}S (3)
374.95 372.73 371.28
54 000 25 900 25 900 43 400
1.23 1.19 1.32 1.54
0.14 0.19 0.21 0.19
B B B B
[128] [129] [129] [130]
Table 10 Numerical data for Ne II Transition
Multiplet (No)
2p3s}2p(P)3p P}D (2)
j (nm)
Temperature range (K)
w /w
w /N
Accuracy
Reference
333.49
30 000}40 000 84 700 & 95 200 27 000 30 000}40 000 27 000 27 000 27 000
1.04}0.90 1.21 1.04}0.90 1.21 1.21 1.26
0.117}0.102 0.063 & 0.062 0.147 0.117}0.102 0.147 0.147 0.151
B> B B B> B B B
[131] [132] [133] [131] [133] [133] [133]
28 000 30 000}40 000 84 700 84 700
1.37 1.36}1.77 1.24 1.33
0.136 0.130}0.147 0.077 0.082
B B B B
[133] [131] [132] [132]
335.50 336.06 334.44 331.13 P}P (7)
332.37
337.83
Table 11 Numerical data for Si IIa Transition
Multiplet (No)
Temperature range (K)
w /N
Accuracy
Reference
3s3p}3s(S)4p
D}P (1)
8500}9700 8700}16 400 10 000 18 000 16 000}22 000 11 000}14 500
0.40}0.38 0.52}0.60 1.07 1.00 0.64}0.66 0.56
C C C C C> A
[136] [139] [140] [137] [141] [134]
10 000}20 000 10 000 18 000 10 000}20 000
1.15}0.98 0.50 0.44 0.56}0.42
[3] [140] [142] [143]
N. Konjevic& / Physics Reports 316 (1999) 339}401
380 Table 12 Numerical data for Si IIb Transition
Multiplet (No)
j (nm)
Temperature range (K)
w /w
w /N
Accuracy
Reference
3s4s}3s(S)4p
S}P (5)
634.71
8500}10 000 18 000 13 900 16 400 31 500 8500}10 000 18 000 13 900 16 400 31 500
0.47 1.06 1.01 0.87 0.72 0.49 1.10 1.03 0.92 0.76
1.10 2.14 2.12 1.89 1.33 1.15 2.22 2.16 1.86 1.40
B B B B B B B B B B
[136] [137] [138] [138] [138] [136] [137] [138] [138] [138]
637.14
where in addition to two strong lines, the width of the weak line 385.4 nm from the same multiplet 1 was also measured. To determine the half-width of a weak line, the authors [134] applied Biraud's deconvolution method [135]. Good mutual agreement for the widths of all three lines is found. The mutual agreement of experimental results for the lines of multiplet no. 5 is not better than for mult.1 (see Table 12), so no attempt was made to determine the best-"t curve through w /w data
points. Argon: Ar II. All six selected experiments [144}149] (see Table 13) were performed with continuous plasma sources: wall-stabilized arcs and plasma jets. The ratios of experimental and theoretical [3] results for mult.2 are given in Table 13 together with data for other multiplets. Large spread of Stark widths within a multiplet [149] with the widths in some cases proportional to the line strengths S (the strongest line has a largest width) is an indication of the presence of selfabsorption which may be, most probably, neglected for the weakest lines. Unfortunately the other two experiments [147,148] were performed using end-on spectroscopic observations in a wall-stabilized arc and they may be distorted by radiative transfer through the cooler plasma layers in the electrode region. Thus for mult.2 the best-"t ratio is not derived. As far as the relation width-line strength is concerned, similar reasoning may be applied for the results of mult.6 (see Fig. 10A), for the best-"t data from [144,145] and the weak line of [149] were taken into account only. All three selected experiments [144,145,149] have in common that spectroscopic plasma observations were performed side-on. Same type of arguments was applied to mult.7. One should note that data for mult.7 [149] show again large spread of widths with the weakest line in the multiplet having the largest width (see Table 13 and Fig. 10B). This may be due to the small signal-to-noise ratio for this line but for other lines (see Table 13) it is di$cult to o!er an explanation. The best "t is derived between data [145] and the weak line from [149] (see Fig. 10B). For multiplets 10 and 15, theoretical widths are calculated from MSE [9]. The best "ts d for mult.10 and 15 were derived using all data from [145,149]: Mult.6, w /w "0.853#3.43;10\¹ for 11 000(¹(22 000 K ,
Eu.:$15%, ADT"9.0% ,
N. Konjevic& / Physics Reports 316 (1999) 339}401
381
Table 13 Numerical data for Ar II Transition
Multiplet (No)
j (nm)
Temperature range (K)
w /w
w /N
Accuracy
Reference
3p3d}3p(P)4p
D}D (2)
401.39
12 800 12 400 22 000 22 000 22 000 22 000 22 000 22 000 22 000 22 000
0.87 1.04 1.23 1.17 1.06 1.11 1.00 1.16 1.08 0.97
0.20 0.24 0.25 0.24 0.21 0.22 0.19 0.24 0.22 0.19
B> B> B> B> B> B> B> B> B> B>
[147] [148] [149] [149] [149] [149] [149] [149] [149] [149]
11 800}13 000 13 000 11 000}14 000 12 800 12 400 22 000 12 800 22 000 22 000 12 400 22 000 12 800 13 000 12 400 22 000 22 000 22 000
0.87} 0.90 0.91 0.39}0.51 0.61 0.55 0.94 0.57 0.88 0.94 0.71 1.02 0.60 0.91 0.68 0.96 0.92 1.05
0.30}0.30 0.36 0.16}0.20 0.24 0.22 0.32 0.24 0.32 0.35 0.29 0.34 0.24 0.36 0.27 0.34 0.34 0.40
B B B B> B> B> B> B> B> B> B> B> B B> B> B> B>
[144] [145] [146] [147] [148] [149] [147] [149] [149] [148] [149] [147] [145] [148] [149] [149] [149]
13 000 12 800 22 000 13 000 11 000}14 000 12 800 12 400 22 000 22 000 22 000 13 000 22 000 13 000 12 800 12 400 22 000
1.02 0.60 1.01 1.02 0.62}0.56 0.67 0.79 1.02 0.96 1.15 1.02 1.18 1.02 0.57 0.73 1.07
0.32 0.20 0.29 0.32 0.22}0.19 0.23 0.26 0.30 0.28 0.32 0.32 0.34 0.32 0.19 0.24 0.32
B B B> B B B> B> B> B> B> B B> B B B> B>
[145] [147] [149] [145] [146] [147] [148] [149] [149] [149] [145] [149] [145] [147] [148] [149]
396.84 391.48 394.43 387.53 403.88 399.20 393.12 3p4s}3p(P)4p
P}P (6)
480.60
493.32 497.22 473.59 484.78
500.93 506.20 P}D (7)
434.81
442.60
443.02 426.65 433.12 437.97
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382 Table 13 (continued) Transition
Multiplet (No)
P}S (10)
j (nm)
Temperature range (K)
w /w
w /N
Accuracy
Reference
417.84 428.29
22 000 22 000
1.20 0.90
0.32 0.25
B> B>
[149] [149]
372.93
12 400 22 000 13 000 12 400 22 000 13 000 12 800 12 400 22 000
0.69 1.17 1.01 0.80 1.22 1.01 0.62 0.84 1.17
0.18 0.23 0.26 0.21 0.24 0.26 0.16 0.22 0.23
B> B> B B> B> B B> B> B>
[148] [149] [145] [148] [149] [145] [147] [148] [149]
13 000 12 800 12 400 22 000 22 000 13 000 12 800 12 400 22 000 12 800 12 400 22 000
1.00 0.68 0.78 1.24 1.34 1.00 0.66 0.75 1.27 0.79 0.80 1.30
0.38 0.26 0.30 0.36 0.39 0.38 0.25 0.29 0.37 0.30 0.31 0.38
B B> B> B> B> B B> B> B> B> B> B>
[145] [147] [148] [149] [149] [145] [147] [148] [149] [147] [148] [149]
385.06
392.86
P}P (15)
454.50
488.90 465.79
476.49
The average value for 10 900(¹(13 900 K.
Mult.7, w /w "1.089!5.28;10\¹ for 11 000(¹(22 000 K ,
Eu.:$20%, ADT"1.5% , Mult.10, w /w "0.756#1.98;10\¹ for 12 500(¹(22 000 K ,
Eu.:$17%, ADT"9.8% , Mult.15, w /w "0.592#3.16;10\¹ for 12 500(¹(22 000 K ,
Eu.:$18%, ADT"13.7% . Calcium: Ca II. All experimental and most of the theoretical results for the Stark widths of Ca II resonance lines are given together in Fig. 1 [17]. The experimental low temperature data points fall into two groups separated by more than a factor of two, see also selected experiments in Table 14. Comparisons with other experiments (see Fig. 1 [17]) are di$cult to interpret since they were carried out at signi"cantly higher temperatures and the temperature dependence of the widths is not clearly established. Interestingly, the Stark widths of Goldbach et al. [153] for the two lines are
N. Konjevic& / Physics Reports 316 (1999) 339}401
383
Fig. 10. Same as Fig. 6 but for the Ar II lines from multiplets 6 and 7, respectively. The line strength values, S, were taken from [53]. The experimental results: (A) multiplet 6 䢇 [144], 䊏 [145], 夹 [146], £ [147], # [148], 䉭 [149]. (B) multiplet 7 䊏 [145], ; [146], * [147], £ [148], 䉭 [149].
larger for their higher temperature point, while all theories predict decreasing widths with increasing temperature in this range. Similar increasing temperature trend of Mg II experimental Stark widths for the data of Goldbach et al. [156] and Roberts and Barnard [157] may be detected in Fig. 1 [16]. Thus it is very di$cult to recommend best set of experimental data, but it is important to note that the results by Goldbach et al. [153] have the smallest estimated uncertainty. In this situation and with the large scatter of results in Table 14 determination of the best "t through data was not performed.
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384 Table 14 Numerical data for Ca II Transition
Multiplet (No)
j (nm)
Temperature range (K)
w /w
w /N
Accuracy
Reference
4s}4p
S}P (1)
393.37
12 000 19 000 13 000 12 240 & 13 350 18 560 13 000 7450 12 240 & 13 350
0.71 0.69 0.78 0.41 & 0.50 0.75 0.78 0.67 0.38 & 0.44
0.20 0.17 0.22 0.11 & 0.14 0.19 0.22 0.21 0.11 & 0.12
B B B B> B B B> B>
[150] [151] [152] [153] [154] [152] [155] [153]
396.85
Table 15 Numerical data for Xe II Transition
j (nm)
Temperature range (K)
w /w
w /N
Accuracy
Reference
(P )6s[2]}(P )6p[1]
460.30
10 000 12 700}15 300 10 000 8000 13 000}15 300
1.11 2.13 0.89 1.07 1.59
0.44 0.68 0.48 0.67 0.69
B B B B B
[147] [158] [147] [159] [158]
10 000 11 000 11 100}15 300 8000 11 000 12 300}15 300 10 000 13 000}15 300
0.84 1.23 1.86 0.99 1.20 1.60 0.88 1.51
0.43 0.59 0.80 0.60 0.59 0.68 0.58 0.79
B B B B B B B B
[147] [160] [158] [159] [160] [158] [147] [158]
10 000 12 500}13 000 11 100}15 300 12 500}15 300 12 500}13 000 11 300}15 300
0.80 1.40 2.22 1.21 1.39 1.88
0.37 0.55 0.83 0.46 0.68 0.90
B B B B B B
[147] [160] [158] [158] [160] [158]
537.24
(P )6s[2]}(P )6p[2]
529.22
533.94
597.65 (P )6s[2]}(P )6p[3]
484.43
489.01 541.92
Xenon: Xe II. The results of four experiments [147,158}160] are compared with results of MSE [9] calculated in [161] using jK coupling scheme. Too large a scatter of data is evident from Table 15. New results are needed to clear the large discrepancy between various experiments. 8.3. Multiply ionized atom lines An intensive activity in this "eld is initiated by both, need of reliable data for theory testing and lack of data for high temperature, high electron density plasma diagnostics (primarily laser
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385
produced and astrophysical plasmas). So, after the publication of last critical review [17] a large number of experimental results became available. The shapes and shifts of many spectral lines were remeasured or numerous new transitions were investigated. For example, Stark widths of N III spectral lines multiplet 3sS}3pP were studied intensively for more than 20 years (see Table 17 and [16,17]). As for singly charged ions, the experimental results for multiply ionized atom lines are presented in tabular form. For each set of data the source of semiclassical theoretical results taken for comparison is given. Although ion-impact broadening data are available for a number of investigated lines only electron impact contribution is taken into account. The normalized line widths w /N in data tables are given in [0.1 nm/(10e/cm)] units.
Carbon: C I<. Three experiments [70,162,163] were selected (see Table 16). Semiclassical results for comparison were taken from [163]. The best-"t line is as follows: Mult.1, w /w "0.579#3.30;10\¹ for 38 000(¹(99 800 K ,
Eu.:$20%, ADT"19.4% . Nitrogen: N III. Data from an experiment with a gas-liner pinch discharge [71] and two experiments with low-pressure pulsed arcs [72,126] were used to set Table 17. The electron density N in [72] was determined from the width of the He II P line using empirical formula by Pittman ? and Fleurier [164]. On the basis of the recent results of a new experimental study by BuK scher et al. [165] the relationship between width of P and N has been slightly changed. In accordance with ? new empirical formula [165] the results for N in [72] were changed for about 8%. The theoretical results taken for comparison are from [72]. Without data [126], see Table 17, the best-"t result is as follows: Mult.1, w /w "0.589#6.82;10\¹ for 33 000(¹(96 300 K ,
Eu.:$18%, ADT"11.0% . Nitrogen: N I<. The semiclassical data [163] were taken for comparison with results of three experiments [123,126,163] (see Table 17). The best-"t formula is derived without results from [126]: Mult.1, w /w "0.325#9.86;10\¹ for 50 000(¹(82 300 K ,
Eu.:$25%, ADT"8.1% . Table 16 Numerical data for C IV Transition
Multiplet (No)
j (nm)
Temperature range (K)
w /w
w /N
Accuracy
Reference
3s}3p
S}P (1)
580.13
38 000 81 200 & 99 800 72 400}80 100 38 000 72 400 & 78 300
0.78 0.93 & 0.86 0.84}0.85 0.61 0.79 & 0.82
0.55 0.45 & 0.40 0.42}0.41 0.43 0.39 & 0.40
B B B B B
[162] [70] [163] [162] [163]
581.20
Transition
3s}(S)3p
2s3s}2s(S)3p
3s}3p
Ion
N III
N IV
NV
S}P (1)
S}P (1)
409.73
S}P (1)
462.00
460.37
348.50
348.30
347.87
410.34
j (nm)
Multiplet (No)
Table 17 Numerical data for N III, N IV, N V
50 000 145 000 173 000 217 000 253 000 277 000 72 400}82 300 50 000 72 400}82 300
50 000 72 400}82 300 54 000 72 400}82 300 54 000 50 000 72 400}80 100 54 000
50 000 87 000 & 96 300 33 000}44 300 50 000 33 000}44 300
Temperature range (K)
0.69 1.53 1.09 1.09 1.16 1.16 0.90}0.94 0.69 0.90}0.94
0.70 1.12}1.06 0.99 1.12}1.08 0.80 0.70 1.12}1.05 0.97
0.52 1.27 & 1.19 0.85}0.84 0.45 0.85}0.84
w /w
0.190 0.273 0.183 0.169 0.170 0.165 0.216}0.212 0.190 0.216}0.212
0.108 0.142}0.129 0.146 0.142}0.131 0.118 0.108 0.142}0.129 0.143
0.12 0.24 & 0.22 0.23}0.20 0.10 0.23}0.20
w /N
B B B B B B B B B
B B B B B B B B
B B B B B
Accuracy
[126] [166] [70] [70] [70] [70] [163] [126] [163]
[126] [163] [123] [163] [123] [126] [163] [123]
[126] [71] [72] [126] [72]
Reference
386 N. Konjevic& / Physics Reports 316 (1999) 339}401
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Nitrogen: N <. Four experiments [70,126,163,166] were selected for Table 17. The theoretical results were taken from [163]. Without results from [126,166] the best-"t formula is Mult.1, w /w "0.723#1.76;10\¹ for 50 000(¹(277 000 K ,
Eu.:$20%, ADT"10.0% . Oxygen: O I<. Three experiments [71,72,167] were selected (see Table 18). In [72], experimental N data were corrected for about 9% in accordance with a new empirical formula [166] for N determination from the width of the He II P line. The theoretical results for comparison were ? taken from [72]. For the best-"t formulas for mult.1 all data from Table 18 were taken into account. For mult.2 results from [167] were omitted. Mult.1, w /w "0.903#1.09;10\¹ for 42 500(¹(131 800 K
Eu.:$22%, ADT"2.4% , Mult.2, w /w "1.125!4.42;10\¹ for 42 500(¹(131 800 K
Eu.:$15%, ADT"8.6% . Oxygen: O
Eu.:$23%, ADT"3.6% . 8.4. Studies along isoelectronic sequences Study of plasma broadening and shifting of analogous spectral lines along isoelectronic sequences o!er an opportunity for testing theory under the conditions of gradual change of emitter's energy levels structure with gradual increase of ionic charge. These studies also enable determination of the Stark width and/or shift dependence upon spectroscopic charge number Z of the emitter. This is also of importance for the estimation of broadening parameters of ions with no available data. The greatest di$culty in this type of studies is the necessity to compare results measured at di!erent electron temperatures. Namely experimental data for low charge ions usually determined at relatively low temperatures have to be compared with results for higher charge ions measured at much higher temperatures. Since Stark broadening parameters depend upon both, electron density N and electron temperature ¹ , one has to perform temperature scaling whenever comparison of results measured at di!erent ¹ is performed. Since w(¹ ) and d(¹ ) dependencies are not yet "rmly established one has to be cautious and try to determine these dependencies. Experimentally this is a very di$cult task which is illustrated best by the fact that we are still lacking good tests of theories in a wider temperature range. Nevertheless, for the comparisons along isoelectronic sequences theoretical predictions of w(¹ ) and d(¹ ) are applied and therefore theory used for temperature scaling should be clearly stated. According to the "rst systematic studies of experimental Stark widths [68] and later shifts [69] both quantities gradually decrease with an increase of Z along the isoelectronic sequence. This
O VI
3s}3p
S}P (1)
P}D (2)
3p}(S)3d
383.42
381.13
340.36
341.17
307.16
306.34
S}P (1)
3s}(S)3p
O IV
j (nm)
Transition
Ion
Multiplet (No)
Table 18 Numerical data for O IV, O VI
145 000 96 300}203 000 65 500}79 700 65 500}79 700
42 500 76 600 62 600}131 800 42 500 62 600}131 800
42 500 98 600 & 119 500 62 600}131 800 42 500 62 600}131 800
Temperature range (K)
1.66 0.92}1.08 1.00}1.05 0.97}1.09
0.86 1.11 1.09}1.05 0.82 1.09}1.08
0.85 1.09 & 0.93 1.09}1.06 0.85 1.09}1.02
w /w
0.154 0.100}0.088 0.125}0.120 0.121}0.125
0.099 0.099 0.104}0.075 0.094 0.104}0.077
0.092 0.099 & 0.079 0.099}0.072 0.092 0.099}0.069
w /N
B B B B
B B B B B
B B B B B
Accuracy
[168] [70] [163] [163]
[167] [71] [72] [167] [72]
[167] [71] [72] [167] [72]
Reference
388 N. Konjevic& / Physics Reports 316 (1999) 339}401
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conclusion [68,69] is drawn on the basis of relatively small number of experimental data. Since then several systematic experimental studies along Li and B sequence are published [70}72], all of them indicating some di$culties in comparison between theory and experiment. Studies of the largest number of ions are performed along Li-isoelectronic sequence for 3sS}3pP transitions of C IV, N V, O VI, F VII and Ne VIII [70]. The experimental results were compared with theoretical obtained from simpli"ed semiclassical formula (SSF) (Eq. (526) in [3]), modi"ed semiempirical formula, (MSE) [9] and classical-path approximation approach (CPA) [29]. None of these three sets of theoretical data could explain the trend of measured Stark widths along the Li-sequence. In a recent experimental and theoretical work, Stark widths of the same 3s}3p doublets along the Li-isoelectronic sequence have been studied [163]. Stark widths for C IV, N V and O VI were remeasured and data extended to B III, another member of the same sequence. This experiment [163] is in good agreement with [70]. Both sets of experimental data and theoretical results are presented in Fig. 11. The best agreement is achieved with results [163] obtained using semiclassical-perturbation formalism [5,6]. In addition to the electron impact widths and shifts, ion widths and shifts due to collisions with ions of interest are calculated as well. It is interesting that the largest discrepancy between theory and experiment (see Fig. 11) is for B III
Fig. 11. Stark widths of lithium like spectral lines (in angular frequency units) as a function of log Z [163]. Experimental data are scaled linearly to a value of the electron density of 10 e/cm and to a value of the electron temperature 87 000 K (7.5 eV) using w (¹ ) dependence from theoretical data in [163]. Experimental results: 䉭 Glenzer et al. [70] and * BlagojevicH et al. [163]. Error #ags are calculated uncertainities including the error in determination of the full-width at half-maximum and of electron density measurements. Theoretical results: **, electron-impact half-widths [163]; - - -, simpli"ed semiclassical formula (SSF), after Griem (Eq. (526) in [3]); } } }, modi"ed semiempirical formula (MSE), after DimitrijevicH and KonjevicH [9].
390
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and Ne VIII ions with lowest and largest ionic charge number Z. The inclusion of ion broadening does not change this conclusion essentially [163]. The gradual change of the discrepancy between theory and experiments seems to indicate a future trend for the improvement of theory. In another systematic study, experimental Stark widths of 3sS}3pP, 3pP}3dD and 2s2p3sP}2s2p3pD transitions of N III, O IV, F V and Ne VI along the B-sequence have been reported [71]. For comparison, theoretical results of SSF [3], MSE [9] and CPA [29] were used. Again none of the theoretical results fully predicted the trend of Stark widths along the Bisoelectronic sequence; the discrepancy of experimental widths (always being larger than the theoretical predictions) is increasing with ionic charge Z. The results of SSC [3] are in the best agreement with experiment. A recent paper [72] reports results of an experimental and theoretical study of the same 3s}3p and 3p}3d doublet transitions of N III, O IV and F V along the B-sequence. For evaluation of Stark broadening parameters the same semiclassical perturbation formalism [5,6] is used (for details see [72]). In addition to electron impact half-widths and shifts, ion impact widths and shifts are calculated as well. For the calculation of O IV transitions, perturbing energy levels with di!erent parent terms were also taken into account. With the exception of a high-temperature result for the F V 3s}3p transition [71], theoretical results agree well with both experiments [71,72] (see Fig. 12). The inclusion of the energy levels with di!erent parent term improved the agreement for the Stark widths and explained the change of the sign of shift for analogous transitions of N III and O IV [72]. The estimated contribution of ion broadening did not exceed 5%, so, within the precision of the experiment, it was not possible to detect its presence with certainty. The results of the recent Stark width measurements [163,169] along the Be-sequence are summarized in Fig. 13. There is a large spread of N IV Stark widths [169] which is di$cult to explain. It seems that the F VI data [169] are too high but to draw conclusion new experimental data are needed. Finally, it is interesting to notice gradual increase of the discrepancy with lowering Z between semiclassical results [163] and experiment [163]. This behavior is already noticed for the Li-sequence and the B-sequence (see Figs. 11 and 12). The experimental results for the Stark shifts along the Li-sequence show an unexpected small increase with Z and a change of shift sign from red for N V to blue for O VI (see Fig. 14). In spite of the fact that measured Stark shifts along the Li-sequence are small and their uncertainties are very large, the change of the shift sign from red to blue is certain. Although the theory [5,6,163] was successful to explain the change of shift sign for 3s}3p doublets from N III to O IV along the B-sequence [72], in cases of 3s}3p doublets and triplets along the Li-sequence and the Be-sequence, respectively, the theory does not predict always the shift direction correctly. Here when we are discussing the comparison between experiment and theory for the Stark shifts, one has to draw attention once more to the large experimental and theoretical di$culties in measurements and evaluation of small line shifts. The "nal conclusion of the comparison between experimental Stark widths and shifts along isoelectronic sequences (see Figs. 11}14) are as follows: 1. The semiclassical calculations [5,6,163] reasonably well describe experimental Stark widths although certain improvements are still required in particular for lowest and highest Z ions. The existence of ion broadening contribution cannot be con"rmed with certainty. On the other hand, additional precision experimental data for the lines of high and low Z ions are required.
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Fig. 12. Same as in Fig. 11 but for boron-like spectral lines as a function of log Z [72]: A for 3s}3p and B for 3p}3d transitions. Experimental results: 䉭, Glenzer et al. [71] and *, BlagojevicH et al. [72]. Theoretical results: ** electron#ion impact width (data for O IV lines with transitions without di!erent parent term included); - - -., electron#ion impact width (data for O IV lines with transitions with di!erent parent term included); - - - , SSF after [3]; } } }, MSE after [9]; 22, classical-path approximation (CPA), after Hey and Breger [24] calculated in [71].
2. Although it is well known that evaluation of small Stark shifts is a great challenge, changes of the shift sign experimentally detected in both sequences should be possible to predict. Experimental Stark shifts for higher Z ions are almost completely missing in literature so data are urgently needed.
392
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Fig. 13. Same as in Fig. 11 but for berillium-like spectral lines as a function of log Z. Experimental results: #, PuricH et al. [126]; 䉭, Wrubel et al. [169] and *, BlagojevicH et al. [163]. Theoretical results: *, semiclassical electron impact widths [183]; } } }, SSF after [3]; - - -, MSE after [9].
3. The approximate formulas SSC [3] and MSE [9] and CPA [29] for Stark widths should be further improved to describe better width dependence upon e!ective charge of ion. The discrepancy between experiment [70,71] and simpli"ed theoretical approaches [3,9,29] initiated semiclassical perturbative (PR) [170] and non-perturbative (NP) [171] calculations. Unfortunately, the results [170,171] were reported only for the experimental conditions of certain experiments so they cannot be easily tested in a way presented in Figs. 11}13. In order to illustrate the possibility of various theoretical methods for the evaluation of multiply charged ion (MCI) Stark widths in Table 19, in addition to the results (in the form of ratios w /w ) of close-coupling (CC) calculations by Seaton [7], perturbative, PR, by Alexiou and Ratchenko [170] and non-perturbative results, NP, by Alexiou [171], data for semiclassical electron-impact widths (EI) by DimitrijevicH and Sahal-Brechot (DSB) [5,6,72,163] are also given. Table 19 is similar to the one in [172] but extended for the results DSB whenever they are available. The results of Table 19 suggest that the results of electron impact EI and non-perturbative NP calculations agree best with experiment. The only larger exception between EI and NP are results for Ne VIII lines where NP agrees better with experiment. Although Ne VIII lines are supposed to be the key test to prove superiority of NP over EI, we do not share the opinion expressed in [172] that it is time for closing in on Stark broadening for weakly coupled plasmas in favor of NP. There are several objections to this early conclusion: (i) NP theory [171] has not been tested yet on a wider
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Fig. 14. Stark shifts of the Li-like spectral lines (in angular frequency units) as a function of log Z [163]. Experimental data are scaled linearly to a value of the electron density of 10 e/cm and to a value of the electron temperature 87 000 K (7.5 eV) using w (¹ ) dependence from theoretical data in [163]. Experimental results: 䉫, Djeniz\ e et al. [162], *, BlagojevicH et al. [163]. Theory: 22, semiclassical electron impact widths [163]; **}, semiclassical electron#ion evaluated only for experimental conditions [163]; } } }, MSE after [10].
sample of experimental data; many results from the same experimental group available at the time of publication [172] were not tested (e.g. N III, C IV, F IV, F V, Ne II}Ne VI, Ar IV, etc.). (ii) Irrespective of what one would expect all conclusions about the agreement between NP and experiment were drawn for low-lying transitions (3s}3p, 2s}2p) while higher ones remain as an open question and last but not least, only two results of a single experiment for Ne VIII (see Table 19) were taken as decisive to make so important a conclusion in favor of NP. To our opinion a lot of experimental and theoretical work has to be involved before "nal conclusions can be drawn.
9. Conclusions This paper in its essence is divided into two parts. In the "rst part theoretical and experimental procedure used to determine plasma electron density were described in detail. Although some steps in the experimental procedure are known for many years they are frequently overlooked in a number of experiments (self-absorption, other broadening mechanisms, deconvolution of line pro"les, importance of temperature measurements, etc.), so they were summarized once more. Some new techniques of line shape and shift measurements were also discussed. In particular,
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Table 19 Theory versus experiment for the lines of multiply charged ions. Calculations: CC-close coupling. Seaton [7], PRperturbative, Alexiou and Ratchenko [170], NP-non-perturbative, Alexiou [171] and EI-electron impact, DimitrijevicH and Sahal-Breshot [5,6,72,163]. Experiment: Li-like Glenzer et al. [70], B III Glenzer and Kunze [173], Ne VII [174] and O IV Glenzer et al. [71] ¹
N
w /w
(eV)
(10 e/cm)
CC
PR
7.0 8.6 14.9 18.7 21.8 23.9 8.3 11.5 15.6 17.5 14.4 16.6 18.5 29.7 42.5 10.6
1.5 2.4 1.2 1.6 2.0 2.3 1.0 1.3 2.1 2.4 1.57 2.1 2.92 2.8 3.2 1.82
0.71 0.78
2s}2p
C IV C IV NV NV NV NV O VI O VI O VI O VI F VII F VII F VII Ne VIII Ne VIII B III
Be-like 2s3s}2s3p Singlet Triplet
Ne VII Ne VII
19.0 20.5
O O O O
4.7 7.5 8.5 10.3
Transition
¸i-like 3s}3p
B-like 3s}3p
Ion
IV IV IV IV
NP
EI
1}1.44 1.04}1.5 0.73}1.14 0.74}1.13 0.7}1.07 0.70}1.07 0.86}1.3 0.71}1.05 0.8}1.18 0.75}1.11 0.70}1.22 0.70}1.21 0.7}1.2 0.52}0.76 0.52}0.77 0.53}1.19
0.92}1.03 0.91}1.04 0.85}0.99 0.85}1 1.16}1.24 0.94}1.03 1}1.13 0.96}1.07 0.96}1.06 0.96}1.06 0.94}1.05 0.8}0.93 0.77}0.97 0.85}1.02
1.09 1.15 0.88 0.92 0.86 0.86 1.09 0.90 0.99 0.93 0.94 0.91 0.90 0.60 0.57 1.11
3.5 3.0
0.63}1.03 0.71}1.13
0.95}1.10 0.98}1.13
0.56 1.03 0.99 1.63
0.36}1.12 0.41}1.19 0.38}1.06 0.47}1.25
0.93}1 0.96}1.05 0.86}0.96 0.99}1.13
0.55 0.7 0.66
0.35 0.3 0.55
0.88 0.83 0.75 0.89
PR and NP results are presented in the form of two ratios which represent the lower and upper limit of the comparison with experiment.
attention is drawn to the regularities in Stark widths and shifts which may be used as a guideline in the analysis of experimental results. In the second part of this review available experimental results were analyzed in order to determine higher accuracy data which may be recommended for plasma diagnostic purposes. The selected data were compared with theory (see examples in Figs. 7, 9 and 10) best-"t formulas through data points derived and from Eq. (39) average deviation from theory determined. The main results of this analysis are summarized in the appendix and they are very simple to use, see
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example with He I lines. The estimated uncertainty of the best "t formula combined with an error of line half-width determination and to the smaller extent ¹ measurements would be the total error of N diagnostics. So data in the appendix o!er, in a number of cases, a possibility of N diagnostics with an accuracy better than $15%. To our opinion this attempt is a step forward in establishing standards among non-hydrogenic lines of various elements for plasma electron density diagnostics and in the same time results may be used for theory testing. Using the same methodology and experimental data presented in the tables, comparison with other theories may be easily performed. Due to the in#uence of ion dynamics to the line shapes and shifts plasma diagnostics from He I lines at relatively low electron densities (several times below 10 e/cm) is treated with special attention, see Eqs. (44)}(46) and Table 2. The results of comparison with semiclassical calculations for neutral atom lines [3] are very encouraging. For a number of He I, C I, N I, O I and F I lines the agreement is well within $20% and in several cases, He I and C I, better than $15%. In comparison with neutrals, for singly ionized atom lines the results of the test experiment versus semiclassical calculations [3] are rather meager. If one takes into account the unsettled situation for Ca II, Si II and Xe II (see the corresponding tables) it seems that a lot of theoretical and experimental e!ort has to be involved to improve the present situation. It is worth noting here that if one lacks results of sophisticated calculations, the modi"ed semiempirical formula [9] may be used successfully for the evaluation of Stark widths of singly charged ion lines, see the appendix. The results for multiply ionized atom lines are a pleasant surprise. The average deviation between semiclassical electron impact calculations [5,6] does not exceed 20% and in several cases is smaller than 10%. Apart from the search for reliable data for plasma diagnostic purposes, studies of neutral atom line asymmetry and new methods for ion-broadening parameter measurements are discussed. The studies of Stark width and shifts along isoelectronic sequences were of particular interest. They are used to demonstrate the applicability of various theoretical approaches for the evaluation of the Stark broadening parameters along a sequence. Finally, the results of various theoretical calculations are compared in Table 19. Although an evident progress is achieved, see results for NP [171] and EI [5,6] in Table 19 and the appendix, further testing of essentially di!erent theoretical approaches is required before one makes a decision about the future development of the theory. As a conclusion one can state that gradual progress in both, theory and experiment dealing with plasma broadening and shifting of spectral lines has been achieved. Nowadays spectral line shapes and shifts of many non-hydrogenic lines may be used with con"dence for plasma diagnostic purposes. In some cases isolated lines of non-hydrogenic atoms start to compete with other techniques as a high accuracy plasma diagnostic method. Still there is a lot of space for improvement and this has been illustrated by a number of examples. Apart from further sophistications some of the older theoretical calculations may be considerably improved by using more complete atomic data sets which are available nowadays. A number of old experimental data require to be remeasured, if possible with Stark width and shift temperature dependence "rmly established. New experimental data are always required especially for highly charged ions. For further development of new standards and for improvement of the proposed ones high accuracy results are more needed than large quantity of low accuracy data.
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Acknowledgements The author gratefully acknowledges the numerous discussions and comments from R. KonjevicH , J. Labat, S. DjurovicH , Z. PavlovicH and many other colleagues in Belgrade and Novi Sad. This work is supported by the Ministry of Science and Technology of the Republic of Serbia.
Appendix A The summarized results of experimental data and their comparison with theory: Eu } estimated uncertainty of the best "t formula, ADT } average deviation from the theory (see Eq. (39)) temperature range and in the last column theory used for comparison is identi"ed. Wherever the He I line widths and shifts are taken for plasma diagnostic purposes at electron densities several times below 10 e/cm, the ion dynamic e!ects should be taken into account (see Eqs. (44)}(46) and Table 2).
Ion
Line or Mult.
Eu (%)
ADT (%)
Temp. range (kK)
He I
501.57 388.86 471.32 538.02 505.21 493.20 Mult.3 Mult.6 Mult.9 436.82 Mult.2 Mult.3 420.07 427.72 451.07 425.94 603.21 573.95
12 17 14 10 12 11 14 13 16 18 13 14 25 23 17 20 16 20
7.5 9.0 9.4 4.6 8.9 19.8 18.9 18.3 13.6 21.4 13.6 12.9 24.2 22.0 42.8 29.0 31.9 23.8
19.0}43.0 26.0}35.0 19.0}43.0 9.9}11.6 9.9}11.6 9.9}11.6 11.3}14.8 10.5}14.2 10.6}14.6 10.1}11.6 10.0}16.4 10.0}14.0 9.7}13.4 9.8}12.7 9.8}13.4 8.9}12.7 7.8}11.5 7.8}11.5
[3] [3] [3] [3] [3] [3] [3] [3] [3] [3] [3] [3] [3] [3] [3] [3] [3] [3]
Mult.3 Mult.5 Mult.12 Mult.39
15 20 30 15
40.5 21.0 14.1 32.6
6.5}31.0 6.5}53.0 6.5}54.0 6.5}32.3
[3] [9] [9] [3]
CI
NI
OI FI Ar I
N II
Theory
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Ion
Line or Mult.
Eu (%)
ADT (%)
Temp. range (kK)
O II Ne II
Mult.3 Mult.2 Mult.7 Mult.6 Mult.7 Mult.10 Mult.15
25 17 16 15 20 17 18
32.7 8.2 32.6 9.0 1.5 9.8 13.7
25.9}54.0 27.0}40.0 28.0}84.7 11.0}22.0 11.0}22.0 12.5}22.0 12.5}22.0
Mult.1 Mult.1 Mult.1 Mult.1 Mult.1 Mult.2 Mult.1
20 18 25 20 22 15 23
19.4 11.0 8.1 10.0 2.4 8.6 3.6
38.0}99.8 33.0}96.3 50.0}82.3 50.0}277.0 42.5}131.8 42.5}131.8 65.5}203.0
Ar II
C IV N III N IV NV O IV O VI
397
Theory [9] [3] [9] [3] [3] [9] [9] [163] [72] [163] [163] [72] [72] [163]
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CONTENTS VOLUME 316 D. Youm. Black holes and solitons in strong theory
Nos. 1}3, p. 1
J. Main. Use of harmonic inversion techniques in semiclassical quantization and analysis of quantum spectra
Nos. 4}5, p. 233
N. KonjevicH . Plasma broadening and shifting of non-hydrogenic spectral lines: present status and applications
No. 6, p. 339
CONTENTS VOLUMES 311}315 I.V. Ostrovskii, O.A. Korotchcenko, T. Goto, H.G. Grimmeiss. Sonoluminescence and acoustically driven optical phenomena in solids and solid}gas interfaces R. Singh, B.M. Deb. Developments in excited-state density functional theory D. Prialnik, O. Regev (editors). Processes in astrophysical #uids. Conference held at Technion } Israel Institute of Technology, Haifa, January 1998, on the occasion of the 60th birthday of Giora Shaviv S.J. Sanders, A. Szanto de Toledo, C. Beck. Binary decay of light nuclear systems B. Wolle. Tokamak plasma diagnostics based on measured neutron signals F. Gel'mukhanov, H. Agren. Resonant X-ray Raman scattering J. Fineberg. M. Marder. Instability in dynamic fracture Y. Hatano. Interactions of vacuum ultraviolet photons with molecules. Formation and dissociation dynamics of molecular superexcited states J.J. Ladik. Polymers as solids: a quantum mechanical treatment D. Sornette. Earthquakes: from chemical alteration to mechanical rupture S. Schael. B physics at the Z-resonance D.H. Lyth, A. Riotto. Particle physics models of in#ation and the cosmological density perturbation R. Lai, A.J. Sievers. Nonlinear nanoscale localization of magnetic excitations in atomic lattices A.J. Majda, P.R. Kramer. Simpli"ed models for turbulent di!usion: theory, numerical modelling, and physical phenomena T. Piran. Gamma-ray bursts and the "reball model E.H. Lieb, J. Yngvason. Erratum. The physics and mathematics of the second law of thermodynamics (Physics Reports 310 (1999) 1}96) G. Zwicknagel, C. Toep!er, P.-G. Reinhard. Erratum. Stopping of heavy ions at strong coupling (Physics Reports 309 (1999) 117}208) F. Cooper, G.B. West (editors). Looking forward: frontiers in theoretical science. Symposium to honor the memory of Richard Slansky. Los Alamos NM, 20}21 May 1998 D. Bailin, A. Love. Orbifold compacti"cation of string theory W. Nakel, C.T. Whelan. Relativistic (e, 2e) processes
311, No. 1, p. 1 311, No. 2, p. 47 311, Nos. 3}5, p. 95 311, No. 6, p. 487 312, Nos. 1}2, p. 1 312, Nos. 3}6, p. 87 313, Nos. 1}2, p. 1 313, No. 313, No. 313, No. 313, No. 314, Nos. 314, No.
3, p. 109 4, p. 171 5, p. 237 6, p. 293 1}2, p. 1 3, p. 147
314, Nos. 4}5, p. 237 314, No. 6, p. 575 314, No. 6, p. 669 314, No. 6, p. 671 315, Nos. 1}3, p. 1 315, Nos. 4}5, p. 285 315, No. 6, p. 409
