Physics Reports vol.318

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

1

PATTERN FORMATION AND COMPETITION IN NONLINEAR OPTICS

F. Tito ARECCHI 
, Stefano BOCCALETTI, PierLuigi RAMAZZA Istituto Nazionale di Ottica, Largo E. Fermi, 6, 150125, Florence, Italy
Department of Physics, University of Florence, Florence, Italy  Dept. of Physics and Applied Mathematics, Universidad de Navarra, Irunlarrea s/n, Pamplona, Spain

AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO

Physics Reports 318 (1999) 1}83

Pattern formation and competition in nonlinear optics F. Tito Arecchi 
, Stefano Boccaletti, PierLuigi Ramazza * Istituto Nazionale di Ottica, Largo E. Fermi, 6, I50125, Florence, Italy
Department of Physics, University of Florence, Florence, Italy Dept. of Physics and Applied Mathematics, Universidad de Navarra, Irunlarrea s/n, Pamplona, Spain Received January 1999; editor: I. Procaccia Contents 1. Introduction 1.1. Optical patterns 1.2. Aspect ratios 1.3. Classi"cation of laser systems depending on damping rates 1.4. Outline of this review 2. Patterns in active optical systems 2.1. The theory of patterns in lasers 2.2. Experiments with lasers 2.3. Patterns in photorefractive systems 3. Patterns in passive optical systems 3.1. Filamentation in single-pass systems 3.2. Solitons in single-pass systems 3.3. Counterpropagating beams in a nonlinear medium 3.4. Nonlinear medium in an optical cavity 3.5. Nonlinear slice with optical feedback 3.6. Nonlocal interactions 4. Defects and phase singularities in optics 4.1. Phase singularities and topological defects in linear waves 4.2. Phase singularities in nonlinear waves

4 4 4 7 9 11 11 14 17 24 24 25 28 32 38 45 51

5. Open problems and conclusions 5.1. Localized structures in feedback systems 5.2. Control of patterns 5.3. Patterns in atom optics Acknowledgements Appendix A. A reminder of nonlinear optics A.1. Nonlinear susceptibility A.2. The two level approximation A.3. The s and s nonlinear optics A.4. The photorefractive (PR) e!ect Appendix B. Rescaling the Maxwell}Bloch equations to account for detuning and a large aspect ratio Appendix C. Multiple scale analysis of the bifurcation problem for the non lasing solution of the Maxwell}Bloch equations Appendix D. Symmetries and normal form equations References

51 53

* Corresponding author. 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 0 7 - 1

57 57 60 62 65 65 65 65 68 70

71

71 73 76

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

3

Abstract Pattern formation and competition occur in a nonlinear extended medium if dissipation allows for attracting sets, independently of initial and boundary conditions. This intrinsic patterning emerges from a reaction di!usion dynamics (Turing chemical patterns). In optics, the coupling of an electromagnetic "eld to a polarizable medium and the presence of losses induce a more general (di!raction-di!usion) mechanism of pattern formation. The presence of a coherent phase propagation may lead to a large set of unstable bands and hence to a richer variety with respect to the chemical case. A review of di!erent experimental situations is presented, including a discussion on suitable indicators which characterize the di!erent regimes. Vistas on perspective new phenomena and applications include an extension to atom optics.  1999 Elsevier Science B.V. All rights reserved. PACS: 05.45.#b Keywords: Nonlinear optics; Nonlinear dynamics; Pattern formation

4

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

1. Introduction 1.1. Optical patterns Pattern formation in extended media is the result of the interaction between a local nonlinear dynamics and space gradient terms which couple neighboring spatial regions. Furthermore, nonlocal terms may bring interactions from far away either in space or time. Some previous review papers are available on the matter [1,2]. A recent comprehensive review [3] is mainly devoted to #uid dynamic or chemical patterns, where the gradient terms result either from momentum transport of from di!usion processes. Optical patterns, on the other hand, are characterized by a wave transport, mainly pointing in one direction. Furthermore, use of laser sources and resonant media to enhance the nonlinearities restricts the time dependence to a quasi-monochromatic behavior. Therefore, the functional terms we have to deal with in optics are of the type f (x, y, z, t) e IX\SR ,

(1)

where the exponential term accounts for a plane wave moving along a direction z ((x, y) being the plane orthogonal to z) and the residual z and t dependence is slow, i.e.



Rf ;k" f ", Rz



Rf ;u" f " . Rt

(2)

This is currently called SVEA (slowly varying envelope approximation) and it is a sensible approximation even when the slow and fast scales di!er by a factor less than 10, as it occurs e.g. for femto-second pulses. Based upon this wave transport feature, optical patterns present all classes of relevant phenomena reported elsewhere, plus some ones which are speci"c of optics. It seems then appropriate to introduce a general classi"cation of optical patterns, within which it is easy to include also classes of phenomena observed in #uid and chemistry. For convenience, we have collected in Appendix A some introductory facts on nonlinear optics, together with the corresponding jargon. 1.2. Aspect ratios In the classi"cation we distinguish between the longitudinal space direction z and the transverse plane (x, y), since the boundary conditions are usually drastically di!erent for the two cases. We classify patterns as 0, 1, 2, or 3-dimensional depending on the functional space dependence of the envelope f. Notice that, at variance with condensed matter instabilities, a 0-dimensional dynamics (i.e. ruled by an ordinary di!erential equation for f ) still refers to a wave pattern which is mono-directional and mono-chromatic. Crucial parameters to estimate the dimensionality are the aspect ratios, which will be de"ned as follows. Let us con"ne the optical dynamics within a rectangular box of sides ¸ , ¸ , ¸ and take V W X ¸ <¸ , ¸ . X V W Optical patterns emerge from coupling Maxwell equation to the constitutive matter equations. The Maxwell "eld e(r, t) induces a polarization p(r, t) in the medium. This polarization acts as

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

5

a source for the "eld which then obeys the wave equation (for simplicity, we refer to scalar "elds, neglecting for the moment the polarizations) 䊐e"!kRp , R

(3)

where 䊐,R#R#R, k is a suitable parameter, and we are denoting the partial derivations as V W X R "R , etc. We expand the "eld as V Rx e(r, t)"E(x, y, z, t)e IX\SR .

(4)

If the longitudinal variations are mainly accounted for by the plane wave, then we can take the envelope E as slowly varying in t and z with respect to the variation rates u and k in the plane wave exponential (Eq. (2)). Furthermore, we shall de"ne P to be the projection of p on the plane wave. By neglecting second order envelope derivatives in z and t, it is easy to approximate the operator on E as






1 䊐P2ik R # R #R#R , X c R V W

(5)

where c is the light velocity. Eq. (5) is usually called the eikonal approximation of wave optics. Eq. (5) suggests that the comparison between transverse variations along x and y are ruled by the aspect ratios ¸ F " V, V j¸ X

¸ F" W , W j¸ X

(6)

where j,2n/k is the optical wavelength. The aspect ratios are called Fresnel numbers, and they have the following heuristic meaning. The geometric angle of view of an object of linear size ¸ from a distance ¸ is ¸ /¸ . Within this angle V X V X only details of minimal angular separation j/¸ can be resolved, due to di!raction. Thus, V the number of independent resolution elements along ¸ detectable at a distance ¸ is given by F . V X V The same holds for F . W So far we have referred to Eq. (5) in free space, considering p as an external perturbation, thus introducing a bare aspect ratio. In fact, the constitutive matter equations provide a generally nonlinear and nonlocal functional dependence p(E). To de"ne a dressed aspect ratio, we must consider the linear part of the polarization P"e s*E , 

(7)

where s is the linear susceptibility and the star convolution operator accounts for nonisotropic e!ects (tensor relations) as well as for nonlocalities in time and space (temporal and spatial dispersion). Furthermore we must consider the appropriate boundary conditions. Let us con"ne the dynamics within a volume bound by two parallel mirrors in the longitudinal direction and consider free lateral boundary conditions. We have the standard longitudinal and transverse modes well known in laser phenomenology.

6

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

Fig. 1. Temporal (a) and spatial (b) frequency space pictures of the active modes in the (i) (1#0), (ii) (1#1) and (iii) (1#2) dimensional cases.

In the active case, i.e. when the medium provides energy to the "eld, the frequency width *u of  the medium gain line can be compared with the so-called FSR (free spectral range) that is, with the separation *u ,c/2¸ of the adjacent ith and (i#1)th longitudinal modes. The ratio GG> X *u  (8) C" X *u GG> de"nes a longitudinal aspect ratio. If C (1 only one longitudinal resonance can be excited, and X the cavity "eld is uniform along z, whereas if C '1 many excited longitudinal modes give rise to X a short pulse whose spatial length is smaller than the cavity length ¸ . X In Fig. 1a we report the frequency position of the modes and the medium line for 0-, 1- and 2-dimensional cases, in Fig. 1b we report the corresponding wavenumbers. In fact, for the 1- and 2-dimensional cases, we do not consider di!erent wavenumbers, but we use a pseudo-spectral method consisting in Fourier expanding around the central wavenumber k , and  considering the residual spread as a slow time-space dependence in terms of evolution equations including the nonlinearities.

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

7

Table 1 Classi"cation of optical pattern forming systems depending on the longitudinal and transverse aspect ratios Dimension

X

>

Z

Systems

0

F (1 V

F (1 W

C (1 X

䢇

Single longitudinal and transverse mode laser

1

F (1 V

F (1 W

C '1 X

䢇

Single transverse mode, longitudinally multi-mode laser Temporal solitons in "bers

䢇

1

F '1 V

F (1 W

C (1 X

䢇

䢇

Linear arrays of semiconductor lasers Nonlinear optics of slab interferometers Single mode laser with delayed feedback Spatial solitons in planar waveguides

䢇 䢇

2

F '1 V

F (1 W

C (1 X

䢇

n.l.o. of planar waveguides

2

F "F '1 V W

F "F '1 V W

C (1 X

䢇

Transverse optical patterns in active media (lasers, PRO) and passive media (liquid crystals, resonant gases)

F "F '1 V W

F "F '1 V W

C (1 X

䢇

3

Co-propagating beam interaction as e.g. four wave mixing

䢇

Break-up and "lamentation of beams in n.l.o.

Note: n.l.o."Nonlinear optics, PRO"Photorefractive oscillator.

In Table 1 we list optical pattern forming systems of di!erent dimensions. It is understood that along a dimension there is no evolution whenever the corresponding aspect ratio is less than 1. 1.3. Classixcation of laser systems depending on damping rates It is well known that a discrete nonlinear dynamical system can undergo a chaotic motion, that is, at least one of its Liapunov exponents can be positive, only when the number of degrees of freedom (phase space dimension) is at least 3. We "nd it convenient to refer to dissipative systems, that is, systems with damping terms for which the phase-space volume is not conserved. In such systems the sum of all Liapunov exponents is negative, and initial conditions tend asymptotically to an attractor [4]. For dimensions N"1, the attractor is a "xed point, for N"2 a "xed point or a limit cycle, for N"3 it can be a "xed point (all the three Liapunov exponents negative), a limit cycle (two Liapunov exponents negative and one zero), or a torus (one Liapunov exponent negative and two zero), or even a chaotic attractor (one Liapunov exponent negative, one zero and one positive). An example of chaotic motion is o!ered by the Lorenz model of hydrodynamic instabilities [5], which corresponds to the following equations, where the parameter values have been chosen so as

8

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

to yield one positive Liapunov exponent: 8 x "!10x#10y, y "!y#28x!xz, z "! z#xy , 3

(9)

where now x, y, z are suitable variables, and dots denote temporal derivatives. Also the fundamental equations for "eld matter interaction can exhibit chaotic features. Indeed, if we couple Maxwell equations with Schroedinger equations for N atoms con"ned in a cavity, and expand the "eld in cavity modes, keeping only the "rst mode which goes unstable, its amplitude E is coupled with the collective variables P and D describing respectively the atomic polarization and the population inversion. The resulting equations are EQ "!kE#gP,

PQ "!c P#gED , , (10) DQ "!c (D!D )!4gPE . ,  For simplicity, we consider the cavity frequency at resonance with the atomic one, so that we can take E and P as real variables and we have three real equations. Here k, c , c are the loss rates for , , "eld, polarization and population, respectively, g is a coupling constant and D is the population  inversion which would be established by a pump mechanism in the atomic medium, in the absence of the coupling. While the "rst equation comes from the Maxwell equation, the two others imply the reduction of each atom to a two-level atom resonantly coupled with the "eld, that is, a description of each atom is an isospin space of spin 1/2. The last two equations are like Bloch equations which describe the spin precession in the presence of a magnetic "eld. For such a reason, Eqs. (10) are called Maxwell}Bloch equations. The presence of loss rates means that the three relevant degrees of freedom are in contact with a sea of other degrees of freedom. In principle, Eqs. (10) could be deduced from microscopic equations by statistical reduction techniques [6,365]. The similarity of Eqs. (10) with Eqs. (9) would suggest the easy appearance of chaotic instabilities in single mode, homogeneous-line lasers. However, time-scale considerations rule out the full dynamics of Eqs. (10) for most of the available lasers. Eqs. (9) have damping rates which lie within one order of magnitude of each other. On the contrary, in most lasers the three damping rates are widely di!erent from one another. The following classi"cation has been introduced [7]. Class A lasers (e.g. He}Ne, Ar, Kr, dye): c Kc
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

9

the addition of an overall feedback, besides that provided by the cavity mirrors [10,387]. All these con"gurations imply a third degree of freedom, besides the two ones intrinsic of a class B laser, thus making chaos possible. Lorenz chaos was later extensively investigated in far infrared lasers, corresponding to molecular (rotational) transitions with damping rates comparable with the cavity decay rate [11]. A few comprehensive reviews on the subject are available in Refs. [12,13]. In the forthcoming sections, Maxwell}Bloch equations will be generalized as follows: 1. introducing a detuning *u"u!u between the laser frequency u and the center u of the ? ? atomic line, thus E and P must be considered as complex quantities, and Eqs. (10) transform to a set of 5 real equations; 2. increasing the cavity aspect ratios, transforming the problem from a discrete to an extended one in the real space. One should carefully distinguish between the dimensions N of the phase space (N"number of dynamical degrees of freedom) and the dimension D of the real space within which the physics takes place. As one goes from D"0 to D"1, then the time derivative EQ in the equation for E will be replaced by R #cR . This extension was "rst considered in connection R X with the propagation of pulses in an excited two-level medium in presence of scattering losses compensating for the gain, so that soliton pulses (so called n pulses) resulted propagating at the light speed, the nonlinearity compensating for the dispersion [14]. In large aspect ratio cavities (D"2) the time derivative in the equation for E will be replaced by R #(ic/2k)  , where R ,

 ,R#R is the transverse Laplacian operator. This extension was "rst considered in , V W "lamentation problems [15]. In fact, the self focusing and self defocusing phenomena imply the whole 3D structure of the "eld, and hence require the use of the operator R #cR #(ic/2k)  , where the time derivative will be R X , dropped in case of stationary phenomena [16]. 1.4. Outline of this review As it has become customary, we call active optical devices those ones in which the medium is in an excited state and it can transfer energy to the light "eld, whereas we call passive optical devices those in which the medium is in its ground state. As a result, active optics can start from spontaneous emission processes, whereas passive optics always requires an incident "eld. Laser are prototypical examples of active optics. In the case of photorefractive oscillators, which are based on two or four wave mixing (see Appendix A) one or more input "elds will have the role of pumps. As we will see in this review, evidence of reliable patterns in lasers has been made di$cult by two reasons: (i) laser cavities have usually small transverse aspect ratios, (ii) the time scales of the dynamics are so fast that only averaged patterns can be visualized. Both limitations are not present in a photorefractive oscillator which has then become a very useful testbench to explore active pattern formation. Let us detail what we mean by the title of this report. `Pattern formationa refers to the fact that in an extended nonlinear medium, above a suitable threshold, any uniform amplitude distribution becomes unstable and the space}time distribution of the amplitude splits into correlated domains. The "rst symmetry is ruled by the boundary conditions. We call a pure pattern an eigenstate of the

10

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

propagation problem associated with the linear part of the dynamics, in the absence of nonlinearities. Nonlinearities imply the interaction of many eigenstates, or pure patterns. This may induce many scenarios of `Pattern competitiona, which, in order of increasing complexity, are respectively: 1. winner-takes-all dynamics: one pure pattern prevails on the others; 2. cooperation: many pure patterns coexist (e.g. hexagons as coexistence of three roll sets at 1203 with each other, mutually coupled by a quadratic nonlinearity); 3. locking of space and time phases of the wavevectors corresponding to di!erent hexagon families; 4. time alternation of patterns, which "ll the whole available region, inducing time chaotic phenomena and yet keeping a spatial coherence; 5. segregation of di!erent domains in each of which a di!erent stationary pattern is present; 6. space}time chaos with a limited correlation length and correlation time. Examples of the six cases are discussed in the following. This report is organized as follows. In Section 2 we report the theoretical and experimental results on pattern formation and competition in nonlinear active optics. In Section 2.1 we show how Maxwell}Bloch equations can be generalized to account for spatial dependence, and how the two homogeneous stationary solutions (lasing solution and nonlasing solution) bifurcate to patterned states which can be described by suitable amplitude equations. In Section 2.2 we review the most signi"cant results on patterns in laser systems, in both the low and high dimensional cases. In Section 2.3 we report one of the "rst evidences of spatial dependent chaotic regimes in a photorefractive oscillator, and we show that the main features of its dynamics can be captured by simple normal forms equations accounting for the symmetry requirements. In Section 3 we review the case of passive nonlinear optics, mainly referring to Kerr media. We consider various experimental situations, namely, "lamentation in single-pass systems (Section 3.1), formation of solitons in single-pass systems (Section 3.2), counterpropagating beams in a nonlinear medium (Section 3.3), patterned states originated by a nonlinear medium con"ned within an optical cavity (Section 3.4), patterns in a nonlinear slice with optical feedback (Section 3.5) and the e!ects of nonlocal interactions in the selection of the pattern shape and of the relevant pattern size (Section 3.6). In Section 4 we de"ne what is a phase singularity, or defect, or optical vortex, and we discuss how the presence of these defects is related to the dynamics of the pattern forming system. In Section 4.1, we summarize the main properties of phase singularities and topological defects in linear waves, and we report their scaling laws. In Section 4.2, we analyze the case of phase singularities in nonlinear waves, and we report the experimental evidence of the dynamical transition from a boundary dominated regime to a bulk dominated regime, in terms of the defect statistical properties. In Section 5 we discuss the perspectives of this area of investigation, referring to some challenging cases, namely, the formation and evolution of localized structures in nonlinear optics (Section 5.1), the problem of stabilizing unstable patterns within space}time chaotic states (Section 5.2), and the pattern formation in atom optics (Section 5.3), where the di!ractive properties are associated with the atom Schroedinger "eld, and the coherence requirements have only recently been reached by the evidence of atomic BEC (Bose}Einstein condensation) [17,388,389], as well as of the atom laser [18].

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

11

2. Patterns in active optical systems 2.1. The theory of patterns in lasers The theory of pattern formation in the transverse pro"le of a laser cavity is quite recent. If one considers the dynamics of the electromagnetic "eld in a cavity with #at end mirrors (Fabry-Perot con"guration) housing an active medium done by two level atoms, it is described by the Maxwell}Bloch equations [19] (see Appendix B), which read EQ !ia E"!pE#pP,

PQ #(1#iX)P"(r!N)E,

NQ #bN"(EHP#EPH) .  (11)

The above equations rule the behavior of the con"ned electromagnetic complex "eld E in the transverse plane ( ,(R/Rx)#(R/Ry); (x, y) being the plane transverse to the direction z of the cavity axis). They are written in the complex Lorenz notation [20], and derive directly from the Maxwell equation for the "eld E and from the Bloch equations for the complex atomic polarization P and the real population inversion N, under the assumption that E and P have a preferred plane-wave dependence in the z direction, and a slow residual dependence upon the transverse variables x and y. In Eq. (11), p and b are respectively the decay rates of E and N scaled to the decay rate of the polarization, r accounts for the pumping process, a is a real parameter which will be later speci"ed, and X represents the detuning, i.e. the di!erence between atomic and cavity frequencies. The Maxwell}Bloch equations possess two homogeneous stationary solutions: E"P"N"0 (nonlasing solution) and E"PO0, NO0 (lasing solution). The destabilizations of both solutions, leading to di!erent pattern formation, are described by suitable amplitude equations. For the nonlasing solution, Refs. [21,22] report the application of weakly nonlinear analysis to Eq. (11), and show that its bifurcation leads to the complex Swift}Hohenberg equation [23] for class A and C lasers, and to a complex Swift}Hohenberg equation coupled to a mean #ow for class B lasers. As for the bifurcation of the lasing solution, amplitude equations have been derived for class B lasers by linear analysis and symmetry techniques Ref. [24], showing that patterns are here ruled by a complex Swift}Hohenberg equation coupled to a Kuramoto}Shivasinsky equation [25,366]. The derivation of the amplitude equations for the case of the nonlasing solution is reported in Appendix C. Let us begin with the nonlasing solution. It is straightforward to show that such a solution becomes unstable for pumping coe$cients r larger than a critical value (X!ak)  , r "1#  (1#p)

(12)

which is called lasing threshold. In Eq. (12) k is a critical value for the transverse wavenumber k.  Furthermore k "0 as far as X(0, while k"X/a in the case X'0. Therefore, the nature of the   bifurcation depends on the sign of the detuning. In other words, if X(0, the bifurcation occurs at a homogeneous state (k "0), while for X'0 the bifurcation takes place at a preferred wavenum ber k O0. The physical reason for this di!erence is that the emergence of a patterned state at kO0 

12

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

would imply an o!-axis propagation of the beam, which introduces a k-dependent phase retardation. If we are in presence of a positive detuning, the o!-axis propagation may compensate for it, and therefore a patterned state with a suitable k (that for which the phase retardation compensates for the detuning) locates at the center of the atomic gain line, thus having the minimum value for the instability threshold. In the opposite case (negative detuning), the phase retardation can never compensate for the detuning, so that the state with minimal threshold is that at k"0. Refs. [21,22] consider small detuning values, and expand all relevant variables of Eq. (11) in powers of a small parameter e, by writing X"eX , (E, P, N)"(E , P , N )#e(E , P , N )#e(E , P , N )#2 . (13)           Furthermore, the spatial scaling are easily derived from the breadth of the band of the unstable modes above threshold. The resulting scaling is X"(r!1)x,

>"(r!1)y .

(14)

If one further assumes r"1#e, it results X"(ex, >"(ey. As for the time scales, Refs. [21,22] consider ¹ "et and ¹ "et as two di!erent secular time scales. Plugging all those   expressions into the Maxwell}Bloch equations, and identifying the coe$cients of each power of e, one gets for the "rst order "eld amplitude E "t the equation (for the derivation see Appendix C)  p p Rt (X#a )t#ia t!iXpt! "t"t , (15) (p#1) "p(r!1)t! (1#p) b Rt which is the Swift}Hohenberg equation. The derivation process (Appendix C) was carried out under the hypothesis that both p and b are of the order of one, since they have not been expanded as function of the smallness parameter e. Therefore, one can say that Eq. (15) describes what happens in class C lasers. For class A lasers, instead, p is small, and a derivation similar to what reported in Appendix C leads to a Swift}Hohenberg equation where the coe$cient 1/(1#p) in Eq. (15) is expanded in series of p. Finally, for class B lasers (bP0), the same procedure with small b and at order four in e leads to two coupled equations, namely, p Rt (X#a )t#ia t!iXpt!pmt, (p#1) "p(r!1)t! (1#p) Rt Rm "!bm#"t" , (16) Rt that is, a Swift}Hohenberg equation and a real mean #ow m. Similar equations for the bifurcation of the nonlasing solution have been obtained through other techniques [26]. In particular, several di!erent partial di!erential equations have been derived from the Maxwell}Bloch equations through center manifold techniques [27]. Ref. [28], even though formally less detailed than the approach summarized in Appendix C, is heuristically appealing insofar as it starts from the simple naive Ginzburg}Landau (GL) equation derivable for a class A laser (pP0) with the inclusion of the di!ractive Laplacian term ia E. Noticing that such GL equation would be structurally unstable, Ref. [28] releases the strong

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

13

Fig. 2. From Ref. [26]. Stability analysis of the non lasing solution: perturbation growth exponents vs. perturbation wavenumber for three di!erent laser models. Squares: classical complex Ginzburg}Landau equation, circles: Swift}Hohenberg equation, triangles: full Maxwell}Bloch description.

adiabatic elimination hypothesis by expanding the polarization in terms of a small parameter gJp. The "rst contribution amounts to correcting the Laplacian term with a di!usive constribution d E, where, however, d depends on the sign of the detuning, and would provide an   unphysical antidi!usion in the case of a negative detuning. Introducing the successive higher order terms in g, Ref. [26] arrives to the Swift}Hohenberg equation. Fig. 2 shows the stability analysis of the nonlasing solution for the Ginzburg}Landau equation, for the Swift}Hohenberg equation and for the fully Maxwell}Bloch equations. In Ref. [29], a Ginzburg}Landau equation was found near the lasing threshold, when considering the interaction of an electromagnetic "eld with matter in a laser cavity without the assumption of a "xed direction of the transverse electric "eld. Pattern formation and evolution in a single longitudinal mode two-level and Raman laser with #at end mirrors, with uniform transverse pumping was investigated numerically [30], and analyzed at and beyond threshold [31]. Other theoretical analyses of transverse pattern forming instabilities in lasers are given in Refs. [32}38]. In particular, the roles of symmetries and symmetry breaking mechanisms have been investigated, with the aim of extracting simple normal form equations for the dynamics of the "rst order transverse modes arising from the primary bifurcation of the nonlasing state [39}42]. Further theoretical analysis has regarded the study of tilted and standing waves in class A lasers [43], and that of polarization e!ects in the transverse mode dynamics [44], leading to the so called vector complex Ginzburg}Landau equation [45]. Also the case of a laser with injected signal has been the object of theoretical investigation [46]. Recently, the problem of boundary-driven selection of patterns has been studied in class B lasers with reference to Eqs. (16) [47]. Furthermore, analytic treatments of Maxwell}Bloch equations for fully 3D lasers have been provided [48]. Finally, analysis of a coherently optically pumped three level laser has pointed out the possibility of spiral formation by a mechanism similar to that occurring in excitable media, but here uniquely due to nonlinear di!ractive interactions [49]. A completely di!erent scenario characterizes the bifurcation of the homogeneous lasing solution [24,50]. In this case, when performing a linear stability analysis of a lasing homogeneous solution

14

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

E of Maxwell}Bloch equations in Fourier space and for class B lasers, the dynamics is governed  by a real "eld u and a complex amplitude A. In Ref. [24], it is shown that the bifurcation is described by






a 1#X "E" K1! 1#

u!i(Ae SR!AHe\ SR)#O["A", ( u)] , 4 "E " "E "  

(17)

where u is the frequency of the bifurcation (uK(2pb"E "/(1#X)).  Linear properties and symmetry arguments lead to equations for u and A of the form c R u,!k u!  u#c ( u)#c "A" , R    2 R A"kA!il(q# )A!a(q# )A![b "A"#b ( u)]A#b u ) A , R     

(18)

where c , c , c , b , b , b are suitable parameters, kKpXa/2, k ;q , q is the well de"ned           wavenumber of the bifurcation, k, l and a are parameters directly derived from linear stability analysis arguments. The "rst of Eq. (18) is the Kuramoto}Shivasinsky equation, while the second is the Swift} Hohenberg equation. A similar scenario has been observed in recent hydrodynamical experiments [51,52]. The above theoretical predictions have been experimentally tested for a CO laser with  large transverse aperture, thus allowing a large number of possible transverse modes [24], with an experimental setup similar to that already used in Ref. [53]. 2.2. Experiments with lasers Even though the theory provides accurate amplitude equations, no global solutions of them are available in 2D, thus no detailed comparison can be drawn with the existing experiments on pattern formation and competition in laser systems. Only recently, evidence of vortices, shocks, domains of tilted waves and cross-roll patterns on a photorefractive oscillator has been compared with the corresponding numerical solutions of the complex Swift}Hohenberg equation [54]. At this point, we should distinguish between two very di!erent experimental conditions. As we have pointed out in the previous subsection, patterns arise in the plane transverse to the cavity axis, that is, to the propagation direction of the light within the optical cavity. Therefore, there is an important role played by the transverse aspect ratio. Namely, when the transverse "nite size e!ects allows only few transverse modes to survive, we will generally speak of a low-dimensional situation. In this case, usually, boundary symmetries are crucial in selecting the pattern shape and in imposing suitable couplings in their dynamics. On the contrary, when the aspect ratio is su$ciently large to allow for a very large number of independent modes, the phase space is high dimensional, and the role of the boundary conditions will be irrelevant. A clear dynamical transition between these two regimes will be presented in Section 4.2. At this stage, this di!erence has been presented only qualitatively, so as the reader might be able to understand the jargon that we will use hereafter.

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

15

Fig. 3. From Ref. [57]. Transverse pattern arising from a cooperative frequency locking of two modes, each one possessing a cylindrical symmetry.

Even though patterns in lasers have been observed since their discovery [55], one of the "rst systematic experimental evidences of their competition in a laser was realized in Ref. [56]. In this case, the competition between the "rst orders transverse modes compatible with a cylindrical symmetry was studied in a helium}neon laser. Ref. [57] reports a stable spatial pattern emerging from a cooperative frequency locking of the "rst two modes. Such a pioneering work dealt with one of the most simple situations, wherein the transverse dynamics was ruled by the competition of two modes (each one of them possessing a cylindrical symmetry). The arising patterns are reported in Fig. 3. Similar investigations on a Na ring laser have been reported in Ref. [57]. Also in this case  boundary regulated patterns have been studied in a low-dimensional con"guration. Following previous experimental studies [58], and theoretical speculations on spontaneous symmetry breaking phenomena [59], the appearance of traveling waves in the transverse intensity pro"le of a CO laser was reported in Ref. [60]. These observations have con"rmed that the  spatiotemporal behavior of a laser in the low-dimensional regime crucially depends on symmetries, and that most of the dynamical features can be directly captured as consequences of spontaneous symmetry breaking mechanisms. The theoretical study of the transverse boundary e!ects on a positively detuned laser system is the subject of Ref. [61]. Further theoretical investigation on standing and travelling waves in homogeneously and inhomogeneously broadened lasers is contained in Ref. [62]. More recently, the attention has moved from boundary e!ects (small transverse sizes) to the high dimensional situation, where the large aspect ratio allows for many di!erent modes. Patterns in a highly pumped high dimensional CO laser has been the object of Refs. [24,63]. In these  conditions, the observed pattern (Fig. 4) is characterized by a high degree of complexity and by the absence of zeroes in the intensity pro"le. This last feature has supported the claim that, in the high dimensional case, patterns are generated by a di!erent bifurcation with respect to the low dimensional case. Namely, the experimental observations seem to support the hypothesis that patterns are here obtained as a modulation of the lasing state, as opposed to the previous cases, wherein the boundary symmetries induced a bifurcation of the nonlasing state toward suitable modes.

16

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

Fig. 4. From Ref. [24]. (a) Pattern of intensity observed on an infrared image plate. The pattern refers to the transverse intensity distribution in a highly pumped CO laser with large Fresnel number. (b1) Intensity distribution vs. transverse  coordinate, obtained through a measurement wherein a rotating mirror de#ects the beam onto a fast detector. The pattern is here characterized by a high degree of complexity and by the absence of zeroes in the intensity pro"le. (b2) Temporal #uctuations of the intensity on a point in the transverse section of the beam.

Fig. 5. From Ref. [64]. Experimental patterns (a)}(c) and pattern reconstructed in terms of superpositions of Hermite}Gauss modes (d)}(f). The experiments are carried out with a CO laser with large aspect ratio. 

In another experimental study of a CO laser with a large aspect ratio [64], many complicated  patterns have been observed in various experimental con"gurations (Fig. 5) and heuristically described in terms of superpositions of Hermite}Gauss modes. In the same years, after many theoretical [65] and experimental [66,67] studies on bidirectional ring lasers, it has been shown that the cavity resonances for transverse modes depend crucially on the number of mirrors housed within the cavity, thus implying that di!erent cavity con"gurations may lead to di!erent dynamical behaviors [68].

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

17

Another important subject of recent investigation is the polarization dynamics in the low-order transverse modes of a CO laser in a Fabry-Perot con"guration [69]. While polarization e!ects in  single mode lasers had been modeled in optically pumped far infrared lasers [70,71], "ber lasers [72], edge-emitting semiconductor lasers [73] and vertical cavity semiconductor lasers [74,75], the corresponding experiments [72,76,77] were often sensitive to anisotropies. In Ref. [68] the polarization properties of the lowest order transverse modes of a CO laser have been thoroughly  investigated. Finally, we summarize the main results on semiconductor lasers. They represent the most convenient source of high-power optical coherent radiation, therefore the dynamics of a single laser and of an array of them has been object of a huge body of investigations. One of the "rst theoretical approaches to spatiotemporal behavior of broad-area semiconductor lasers discovered the presence of "lamentation phenomena in the transverse pro"le [78,79]. This pioneering work stimulated many other theoretical analyses, which have considered the e!ects of spatial inhomogeneities in the active medium on the onset of a spatiotemporal dynamics [80,81]. On the other side, spatiotemporal phenomena were studied in large arrays of spatially coherent semiconductor lasers. Following the idea of chaos synchronization [82], it has been numerically shown that a large array of semiconductor lasers can cluster in many subsets of synchronized chaotic systems, and the disintegration of such clusters leads to the appearance of space}time chaos [83], which is signaled by a spatial symmetry breaking mechanism. Also phase locking phenomena were studied in such a kind of system [84,85]. The dynamics of large arrays of lasers was modeled under various laboratory conditions [86,87]. 2.3. Patterns in photorefractive systems In the late 1980s spatial e!ects in optical oscillators di!erent from lasers have received serious consideration for the reasons listed in Section 1.4. We report here the investigation of a PRO (photorefractive oscillator) consisting of a photorefractive crystal pumped by a laser and emitting within a ring cavity [88]. Control of the number of transverse modes which can oscillate (that is, the aspect ratio) is performed by varying the aperture of a cavity pupil. The setup is shown in Fig. 6. The photorefractive crystal is a 5;5;10 mm BSO (bismuth silicon oxide) to which a dc electric "eld is applied. The medium is pumped by a cw Argon laser with intensity around 1 mW/cm. The pump beam shines the rear part of the crystal forming an angle h with the cavity axis, thus inducing an intensity grating by interference with the cavity "eld. The cavity is made by four high-re#ectivity dielectric mirrors plus a lens L (500 mm focal length), whose role is to enhance the cavity mode stability by providing a near-confocal con"guration. A pinhole is inserted within the optical path between two confocal lenses L of short focal length. The displacements of the pinhole along the optical axis yield a continous change of the ratio between the aperture and the spot size. As a consequence, a di!erent number of transverse modes is inhibited. The e!ective Fresnel number F is the ratio of the area of the di!racting aperture that limits the system (pupil) to that of the fundamental Gaussian-mode spot in the plane where the aperture is placed. Changing the position of the pupil along the optical axis means varying F in the range from 0 to approximately 100. The corresponding number of transverse modes that can oscillate is proportional to F, so that it ranges from 0 to 10 000.

18

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

Fig. 6. From Ref. [88]. Experimental setup for the photorefractive oscillator. VD is a videocamera recording the wavefront pattern, PM is a photomultiplier measuring the time evolution at a "xed point selected by the "ber OF. BSO is the photorefractive crystal, M are mirrors delimiting the ring cavity, A is a pupil controlling the Fresnel number, L and L are lenses of suitable focal length so as to make the cavity quasi confocal, BS is a beam splitter, V is an applied voltage on the BSO.

From the above de"nition, F is the ratio of the area of the pinhole (a) to that of the fundamental mode spot in the pinhole plane (w(z )):  N a . (19) F" w(z )  N On the other hand, the ABCD matrix method for propagation of Gaussian beams provides w (z) at  each z position. The overall size of the Gaussian mode of order n is w K(nw . This implies that  L the highest allowed mode (of order n "n and size w "aK(n w ) will be such that 

 L n w (20) FK "n . w  Eq. (20) tells us that the Fresnel number gives the maximum order of the transverse modes that can oscillate. The experimental intensity patterns observed by increasing Fresnel numbers F are shown in Fig. 7, together with the spatial correlation functions. The low F regime (F44) is characterized by a time alternation between pure cavity modes (Fig. 7a), with a spatial correlation length m covering the whole transverse size D of the beam. On the contrary, for large F (FK15), the signal is a complicated pattern obtained by the superposition of many modes irregularly evolving in space and time (Fig. 7c). It is characterized by a short correlation length (m/D(0.1). The transition between these two regimes is marked by a continuous variation of the ratio m/D. An intermediate situation is shown in Fig. 7b. Fig. 8 shows an example of alternating pure mode con"gurations (low F regime). Since here m/D&O(1), the temporal evolution of the pattern is coherent in space, so that studying the global features of the dynamics is fully equivalent to analysing the local temporal behavior of the intensity in an arbitrary point on the wavefront. Let us interpret the above results. The cylindrical geometry of the cavity constrains the symmetry of the output "eld to be O(2). However, the pumping process breaks the O(2) symmetry by

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

19

Fig. 7. From Ref. [88]. Intensity distribution of the wavefront (left column) and spatial autocorrelation function (right column) for increasing Fresnel numbers. (a) F"5, one single mode at a time is present, and the ratio between the correlation length m and the transverse size of the system D is of the order of one. (b) F"20, m/DK0.25. (c) F"70, m/DK0.1.

introducing a privileged plane, de"ned by the propagation vectors of pump and signal "elds. The oscillator generates "eld patterns varying in time. Two di!erent dynamical regimes arise by changing the size of the cavity pupil. For large pupils, the "eld displays a complex pattern made of a large number of solutions of the free propagation problem (the so called cavity modes). For small pupils, the "eld is made of a single mode at any time, but a small number of modes (from two to about ten) can alternate. The alternation is an ordered sequence of quasi-stationary modes. Depending on some control parameter (tiny adjustments of F or of the pump), the time of persistence of each mode is either regular (periodic alternation, PA) or irregular (chaotic alternation, CA). Apart from the short switching time from one mode to another, the amount of mode mixing is here negligible. This kind of dynamics is not peculiar of the system of Ref. [88]. Indeed, a phenomenon similar to CA, called chaotic itinerancy, was reported theoretically for a one-dimensional laser [89], and for an array of coupled lasers [90]. Later, the phenomenon was found also in globally coupled

20

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

Fig. 8. From Ref. [88]. An example of periodic alternation obtained for low Fresnel number. The pictures show the intensity patterns of the pure modes in their order of consecutive appearance in a cycle of periodic alternation obtained for F"5.

iteration maps [91] and in nonequilibrium neural networks [92]. While chaotic itinerancy implies erratic jumps among the available quasi-stationary states, here CA keeps the ordering sequence. By increasing the Fresnel number F, a new regime, called spatio-temporal chaos (STC) is observed, where a large number of modes coexist. From a statistical point of view, this regime has been characterized by Hohenberg and Shraiman [93] in the following way. Suppose a generic "eld u(r, t) be ruled by a PDE including nonlinear and gradient terms, and take the "eld of deviations away from the local time average du(r, t)"u(r, t)!1u(r, t)2 ,

(21)

(122 denotes time average). Under very broad assumptions, the leading part of the correlation function is an exponential, C(r, r)"1du(r, t)du(r, t)2Ke\P\PYK .

(22)

Whenever the correlation length m exceeds the system size ¸(m'¸), low dimensional chaos is observed. This means that the system can be chaotic in time, but its evolution is coherent in space, that is, the system is single mode, in a suitable mode expansion, and the corresponding chaotic attractor is low dimensional. On the contrary, in the limit m;¸, any local chaotic signal cannot be con"ned within a low dimensional space. In this case, a new outstanding feature appears. The transition between these two dynamical regimes (low and high F) was observed in Ref. [94], and will be reported in Section 4.2. Herewith we summarize the theoretical description of the low

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

21

Fresnel number regime, where boundary e!ects are dominant, and therefore the dynamical equations can be retrieved by looking at the symmetry properties imposed by the boundary conditions. Periodic and chaotic alternation is a dynamics proper of systems with imperfect O(2) symmetry [95]. Let us consider a situation involving three transverse modes competing nonlinearly according to the symmetry constraints. The considered modes are a central one, with complex amplitude z , and two higher order ones  (rotating and counter rotating along an azimuthal coordinate h) with respective complex amplitudes z and z and angular momenta $1. The cavity "eld can be expressed as   (23) E"f (r)(z e F#z e\ F)e SR#f (r)z e SR ,      where f and f are the space distributions of the modes. The optical frequencies u and u are in     general di!erent. The slow time dependence due to the dynamics is accounted for in the amplitudes z (t) (i"0, 1, 2). G The zero intensity situation is described by z "z "z "0, the central mode by z "z "0      and an azimuthal standing wave by z "0, z "z .    The time sequence of these three con"gurations is the simplest case experimentally observed in Ref. [88], and model equations having the above sets of z values as "xed points can be built for reproducing such a behavior. The experimental observations tell us that all quasi-stationary points persist only for a "nite time, thus each of the "xed points will be considered to have at least an unstable direction. These general rules are compounded now with the symmetry requirements. The cylindrical geometry of the cavity imposes the following constraints on mode amplitudes [96] H : (z , z , z )P(e Fz , e\ Fz , z ) ,       (24) K : (z , z , z )P(z , z , z ) ,       (H being the rotation operation, and K the re#ection operator around a privileged plane). If one considers the modes as born from Hopf bifurcations, then there is an additional time symmetry (25) B : (z , z , z )P(e @z , e @z , e @z ) .       The normal form for the nonlinear interaction among the three modes, assuming it to be invariant under the above symmetries is [97,98] (dots denote time derivatives) z "j z #(a("z "#"z ")#b"z ")z ,        z "j z #(c"z "#d"z "#e"z ")z #ez , (26)         z "j z #(d"z "#c"z "#e"z ")z #ez ,         (j , j , a, b, c, d, e being complex coe$cients and e"o e PC being a symmetry breaking parameter).   C The parameter e is reminiscent of the breaking of the cylindrical symmetry induced by the pumping procedure which privileges a de"ned plane, thus breaking the rotational invariance. The analysis of Eq. (26) is reported in Appendix D. The numerical solutions of Eq. (26) (Fig. 9) shows and example  Both Ref. [97] as well as the application of the symmetry arguments for the patterns of a CO laser [98] refer to a two  mode interaction and are concerned with the steady solutions whereas this is a three mode case.

22

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

Fig. 9. From Ref. [95]. (o , o ) projection of the solution of Eqs. (26) for aP"3, bP"1, cP"!1, dP"!2, eP"!1,   jP "0.5, jP "!1.5, o "0.5, u "0, cG!dG"1. P is marginally stable, and any initial condition generates a periodic   C C orbit. The closer a periodic orbit is to the three "xed points (O, C, SW), the larger the period. The imaginary parts of the coe$cients j , b, a and e only contribute to the dynamical evolution of u and u .   

of periodic alternation, in the case of Fig. 9, (Re b/Re e)'(2Re a/Re (c#d)) (see Appendix D). For any initial condition a periodic solution exists passing through it, and this suggests the existence of integrals of motion. There is an heteroclinic solution connecting the three "xed points on the axes, and therefore the period of a periodic solution is larger the closer it is to the heteroclinic solution. As already discussed, the experimental evidence of PA is, however, largely limited, whereas the natural evolution of the dynamics seems to lead to a chaotic alternation among the available con"gurations. This feature can be explained by invoking a frequency degeneracy u "u .   A resonance between the three states gives rise to an additional symmetry. Precisely, the time symmetry now becomes [96] B : (z , z , z )P(e @z , e @z , e @z ) .       and additional terms survive in the normal form equation, which becomes

(27)

z "j z #(a("z "#"z ")#b"z ")z #fz z zH ,           (28) z "j z #(c"z "#d"z "#e"z ")z #ez #gzzH ,           z "j z #(d"z "#c"z "#e"z ")z #ez #gzzH ,           ( f and g being complex coe$cients). The additional terms due to resonance act as forcing terms with frequency 2u !(u #u ). They induce dramatic changes in the structure of the solutions.    Indeed, if the additional terms are `turned ona when the other parameters are close to an heteroclinic solution, Melnikov like arguments [99] suggest that some of the periodic solutions can

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

23

Fig. 10. From Ref. [95]. (o , o ) projection of the solution of Eqs. (28) for the same parameters as in the caption of Fig. 9,   plus f "!0.01, gP"0.02, f G"0 and gG"0. The result is a chaotic alternation between modes O, C and SW. P

Fig. 11. From Ref. [54]. Vortices separated by shocks. Numerically (a)}(c) and experimentally (d)}(e) obtained "eld distributions: near "eld amplitude (a), phase (b), far "eld amplitude (c). (d) and (e) are experimental images of the near "eld and far "eld plane. Experiments are carried out with a photorefractive oscillator.

disappear, some can bifurcate to solutions of di!erent periodicity, and even chaotic behavior can be expected. Fig. 10 reports the integration of Eq. (28), and highlights the presence of a chaotic solution getting close to the pure modes (CA). It is crucial to notice that both PA and CA are structurally stable, in so far as they persist over wide ranges of parameter values.

24

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

In summary, by symmetry arguments, one can construct a model for PA and CA in a dynamical system with a broken O(2) symmetry. Even though the model has been referred to a particular system, a similar behavior can be expected in other systems with the same symmetry properties. Recently, rather than limiting to a low dimensional dynamics described by a small set of ordinary di!erential equations, a numerical solution of a Swift}Hohenberg equation has shown qualitative agreement with experiments on a photorefractive oscillator [54]. Fig. 11 reports numerical as well as experimental evidence of vortices separated by shocks for an aspect ratio F"10.

3. Patterns in passive optical systems 3.1. Filamentation in single-pass systems A laser beam propagating through a nonlinear optical medium, perhaps the simpler system one can think of in nonlinear optics, can undergo spatial instabilities leading to spot formation. This phenomenon has been already described in the early times of nonlinear optics development [16,100,101]. Consider a medium the optical response of which is described by a Kerr nonlinearity. Its polarization has a s term, P"sEEE. In the paraxial approximation the slowly varying envelope A(z, q"t!z/v) of an optical wave propagating through the medium obeys the following stationary equation iv

RA "k  A"a"A"A , , Rz

(29)

where v is the light velocity in the medium,  ,(R/Rx)#(R/Ry) denotes the transverse , laplacian, and a gives the sign and the strength of the nonlinearity. Eq. (29) admits the stationary solution: A (z, q)"A (q) exp (ia"A (q)"z/v).    Let us introduce a spatially dependent perturbation:

(30)

(31) A(z, x , q)"(A (q)#f (z, q)e qx,) exp(ia"A (q)"z/v)  ,  with x ,(x, y). By substituting this expression in the wave equation (29) we obtain, at "rst order , in f iv

Rf !kqf#a"A "f#"A "f H"0 .   Rz

(32)

The eigenvalues of the associated system of linear equation governing the evolution of f, f H along z are q j "$i (kq!2aI k .  ! c

(33)

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

25

Fig. 12. Growth rate j of the transverse perturbations at spatial frequency q for the "lamentation in a focusing Kerr O medium (see Eq. (33)). Lower curve: aI "1; upper curve: aI "2.   Fig. 13. From Ref. [148]. Filamentation of a wide radially symmetric beam passing through a photorefractive SBN crystal, for increasing voltage < applied to the crystal. (a) <"0, (b) <"600 V, (c) <"1200 V, (d) <"1500 V. The small horizontal modulation of the beam observed in (a) is due to striations in the crystal.

The spatial growth of the perturbation f requires the existence of a real, positive j, which is possible only for a'0, i.e. for focusing media. In this case, a whole band of wavenumbers q is unstable (Fig. 12). The maximum growth rate occurs for a q " I k 

(34)

this corresponds to a wave that at any value of z is exactly phase-matched with the medium nonlinear polarization. This instability is referred to as "lamentation because eventually light "laments of various diameters, small compared to the initial beam size, propagate through the medium. Recent investigations have been devoted to "lamentation in non-Kerr media [102}104,148], e.g. in photorefractives and in liquid crystals. Fig. 13 shows the "lamentation of the light beam passing a photorefractive crystal. Though the instability mechanism is still based on the self-focusing properties of these materials, other phenomena (beam bending, longitudinal beam undulations) related to the anisotropic nature of the materials considered are also observed in these cases (see Fig. 13). 3.2. Solitons in single-pass systems A phenomenon strictly related to "lamentation in self-focusing media is beam self-trapping, i.e., the propagation of light beams that conserve a limited transverse size by a balance between the

26

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

Fig. 14. From Ref. [107]. Intensity and phase as functions of coordinate x for bright (a), dark (b) and gray (c) solitons.

tendency to spread due to di!raction, and the focusing properties of the nonlinear medium. Such beams are usually called solitons, even though in many cases they do not satisfy all the properties required by the mathematical de"nition of a soliton [105]. The possibility of a beam self-trapping has been pointed out already in the early days of nonlinear optics [15,16,101,106]. Since then, soliton physics has been investigated extensively from a theoretical as well as from an experimental point of view. The richness of results and the rami"cations of this matter has rendered prohibitive a satisfactory coverage of this subject in the present review. We will therefore limit to survey rapidly some of the most important ideas and experimental results about solitons, addressing the interested reader to speci"c review articles fully devoted to this topic [107]. For the very special case in which propagation in a Kerr medium is considered and the transverse geometry is limited to one dimension x, it is possible to give an analytic form for the electric "eld distribution. The evolution is described in this case by Eq. (29), the general solution of which in the one-dimensional case and for a focusing medium is [105,107}111] A e TV\ T\X , (35) A(x, z)"  cosh A (x!vz)  where A is the soliton amplitude and v is its velocity (all variables are suitably rescaled to a non  dimensional form). For v"0 the fundamental bright soliton (Fig. 14a) is de"ned as A e X . A(x, z)"  cosh A x  In the defocusing case, the general localized solution reads [105,107,110}112]

(36)

(37) A(x, z)"A (b tanh h#ia)e X ,  where h"A b(x!aA z). The parameters a and b satisfy a#b"1, hence a single parameter   u can be introduced such that a"sin u, b"cos u. The value A sin u has the meaning of soliton 

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

27

velocity along x. For uO0, Eq. (37) represents a gray soliton shown in Fig. 14c. For u"0, the "eld distribution (37) reduces to (38) A(x, z)"A tanh(A x)e X .   That is referred to as the fundamental dark soliton (Fig. 14b). Linear stability analysis [107,108,112,113] shows that these solitons are stable in the (1#1) geometry (one transverse coordinate plus one longitudinal coordinate). Observation of bright solitons in this geometry has been reported in nonlinear liquids [114}117] and in planar waveguides [118], while dark solitons have been observed in photorefractives [119]. In the (2#1) dimensional case, both the solitons described by Eqs. (36) and (38) are unstable. Instability of bright solitons is a generic feature in (2#1) dimensional Kerr media [120}122]. It has been shown in general [113] that a soliton propagating in a (D#1) dimensional geometry through a focusing medium described by a nonlinearity of the type *nJIN is stable for p(2/D. Hence, in a Kerr medium, bright solitons are stable in (1#1), but unstable in (2#1) dimensions. From a physical point of view, the dependence of the soliton stability upon the transverse dimensionality D is due to the fact that, while spreading by di!raction acts in a way independent of D, focusing in a Kerr medium depends on intensity, i.e. on the ratio P/gD, where P is the total beam power, and g is its size. Bright solitons in (2#1) dimensions can nevertheless exist, and have been described in many circumstances. Saturation of the nonlinearity has been proposed [123,124] as a "rst mechanism able to stabilize these beams. Early experimental observations of bright solitons in sodium vapours [125] have con"rmed the validity of this suggestion. More recently, bright solitons have been observed in photorefractive crystals both operating in a regime in which the nonlinearity is local and saturable [126}133], and in a regime in which it is intensity independent but nonlocal [134}136]. In this latter case, solitons exist only in the transient regime. Finally, bright solitons have been shown to exist in media described by a s nonlinearity, both in regime of second harmonic generation [137,138], and in regime of parametric ampli"cation [139,367}374]. The limitation and stability derived for the Kerr medium do not apply in this case, in which instead there is a mutual trapping between the three optical waves at di!erent frequencies coupled via the s in the crystal. A mutual guiding is also at the base of the existence of vectorial solitons, formed by a pair of solitons that can be either one bright and one dark, or having di!erent polarizations [140}142]. As for the dark solitons in (2#1) geometry, the planar ones described by Eq. (38) are known to be unstable with respect to long wavelength transverse modulations [143]. This mechanism, resulting in a modulation of the initial dark stripe and successive creation of pairs of vortices of opposite polarity [144,145], has been observed both in atomic vapours [146] and in photorefractives [147,148] (Fig. 15). However, dark stripes can be stabilized by the medium di!usion, or be observed even if unstable, when the transverse size of the beam is small compared to the typical scale of the instability. Several observations of stable dark stripes have indeed been reported in absorbing liquids, [149,150] in semiconductors [151,152] and in photorefractives [153]. Dark solitons with circular symmetry in (2#1) dimensions have been considered for long time stable in defocusing media [121,122], even though recent results [107] show that some instabilities may occur also in this case. A particularly interesting structure is the vortex soliton, consisting of

28

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

Fig. 15. From Ref. [146]. Transverse instability of a dark stripe soliton in a Rb cell. The vapor concentration increases from vanishingly small (a) to the order of 10 cm\ (f). The inset shows the intensity distribution along a line through the center of the upper vortex in (f).

a "eld distribution characterized by a dip reaching the zero in its center, and a helical structure of the phase distribution. These vortices appear to be unstable in focusing media, in which they break resulting in a pair of bright solitons [154}156]. On the contrary, vortices have been shown to constitute stable solitons in defocusing media [157,158]. Several experimental con"rmations of this fact have been given, using as materials absorbing liquids [159], vapours [160,161] (see Fig. 16) or photorefractive crystals [162,163]. 3.3. Counterpropagating beams in a nonlinear medium Let us consider a pair of optical beams of wavenumber k and amplitudes E , E counterN N propagating inside a Kerr medium of length l (Fig. 17a). For low values of the wave intensities, a stationary, plane standing wave is found in the medium. Perturbations to this situation are taken into account by introducing the small amplitude probe beams E , E propagating at a small angle   h with respect to E , and E , E counterpropagating to E , E (Fig. 17b). In the limit of undepleted N    

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

29

Fig. 16. From Ref. [161]. Vortex soliton formation in a Rb cell. Experimental beam pro"les recorded at the input window of the cell (a); and at the output window in the regime of linear propagation (b), and nonlinear propagation regime (c); interference pattern at the output of the cell in the nonlinear propagation regime (d).

Fig. 17. Two pump beams E and E interact with a nonlinear Kerr medium of length ¸ (a). Spontaneous coherent N N emission can appear in a direction making an angle h with respect to the pump beam axis (b).

pumps E , E , the stationary evolution of E is governed by [164}166] N N  RE "!im[(1#a)EYH#qEHe IFX#(1#a)E e IFX#qE ] ,     Rz

(39)

where m is proportional to E E (m is negative in the focusing case and positive for a defocusing N N medium), q""E "/"E ", a (04a41) weights the wavelength-scale refractive index gratings due N N to the counterpropagating "elds. In media in which the Kerr excitation is very mobile, e.g. by di!usion, these short scale gratings are washed out, resulting in a"0. Each term at the r.h.s. of Eq. (39) represent a four-wave mixing (4WM) contribution to the evolution of E ; similar equations hold for E , E and E . The "rst term is associated with the     backward 4WM gain, i.e. with absorption of one photon in each pump and emission of photons in beams E , E . The second term describes forward 4WM, due to the annihilation of two photons of  

30

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

Fig. 18. From Ref. [165]. Instability threshold curves "m"¸ vs. h. (a) focusing case, a"0; (b) defocusing case, a"0; (c) focusing case, a"1; (d) defocusing case, a"1. In all plots the solid curves are associated to equal pump beams ("E """E "), the dashed curves to "E ""0.85"E ". The horizontal lines are the thresholds predicted by a model in which N N N N also the backward 4WM is accounted for [169].

the beam E and emission of photons in the beams E , E . The third term describes the parametric N   process in which one photon is absorbed in beams E , E and one photon is emitted in beams N  E , E . The last term describes nonlinear self modulation of E . Eq (39), together with the N   analogous equations for E , E and E , forms a set of four linear di!erential equations that must be    supplemented with the boundary conditions E (0)"E (0)"0, E (¸)"E (¸)"0. Solutions of     this boundary-value problem [164}168] gives rise to the marginal stability curves of Fig. 18. Several important features are visible in these curves. The horizontal lines at "m" ¸"n/2 (a"0) and "m" ¸"n/4 (a"1) correspond to the instability without any spatial structure predicted in [169] for the case in which only the h-independent backward 4WM is retained. We notice that the inclusion of h-dependent 4WM terms leads both to a selectivity in h, and to a lowering of the instability threshold, with the exception of the defocusing case for a"0. At the physical origin of these phenomena lies the fact that the inclusion of h-dependent 4WM processes results in an increasing of net gain for some h component, with respect to the case in which these processes are neglected. This gain enhancement is e!ective, however, only if the various 4WM processes are phase-matched along the medium length ¸. This constraint leads to the selection of transverse wavenumbers q,kh" O(2k/¸) [164}168]. In the self focusing case, phase-matching is

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

31

Fig. 19. From Ref. [170]. Shape of one transmitted beam in the far "eld of a sodium cell pumped by counterpropagating beams. The medium nonlinearity is of focusing type in these experimental conditions. The picture (a) corresponds to a situation where the pump beams are well aligned at counterpropagation, while they make a small angle (K5;10\ rad.) in picture (b).

achieved through a balance between di!raction and nonlinear self modulation of each probe wave. This balance is not possible in a defocusing material, since the e!ects of these two mechanisms here add instead of subtracting. For this reason, as shown in Fig. 18b, the h-dependent threshold is higher than the h-dependent one for defocusing media in the case a"0. If aO0, however, cross phase modulation among the counterpropagating waves can balance for the loss of e$ciency of the o!-axis gain [166,167], so that again instabilities at a "nite value of h is predicted. From an experimental point of view, instabilities of counterpropagating waves have been observed in atomic vapors [170}172] and in photorefractive crystals [173}180]. In some circumstances, it has been theoretically discussed and experimentally demonstrated that instabilities similar to the ones here discussed occur also in copropagating beam geometries [168,171,181]. Fig. 19 displays far-"eld hexagons observed in sodium vapors pumped with counterpropagating beams of light at frequency u'u , u being the frequency of the D resonance line. In these    conditions the nonlinearity is of focusing type. The symmetry of the experimentally observed patterns is often hexagonal, though also rolls, squares and other less generic structures have been reported. For the Kerr medium hexagons have been theoretically and numerically shown to be generic close to the instability threshold in the focusing case [182}184], while rolls or rhombs occur in the focusing case. We notice that neither atomic vapours nor photorefractives are adequately described by a Kerr-like nonlinearity [164,185]; this does not seem to pose a serious limitation to the predictions of the above reported model in comparison with the experiments. The reason lies in the fact that on

32

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

one side what is important is the ability of the material to support four-wave mixing interactions [167,185}187]. On the other side, the geometrical constraint given by the longitudinal boundary conditions results in a scale selection mechanism that is largely independent of the details of the nonlinearity. It is a reasonable guess that the speci"c kind of nonlinear material in#uences more strongly the pattern symmetry selection rules. This topic however has not been investigated in detail. Fig. 20 reports the patterns observed in the far and near "eld for counterpropagating beams in a photorefractive SBN crystal to which an external electric "eld is applied. The anisotropic nature of the photorefractive nonlinearity accounts for selection of rolls rather than hexagons for low values of the applied "eld. 3.4. Nonlinear medium in an optical cavity Optical resonators containing a passive nonlinear medium have been extensively investigated earlier in the context of optical bistability [188], and more recently as systems capable to display pattern formation. Two typical geometries for these systems, namely, a ring resonator partially "lled with the medium and a Fabry-Perot resonator entirely "lled with the medium, are shown in Fig. 21. Let us consider the case in which the cavity is uniformly "lled with a Kerr medium, and driven by a coherent external "eld E [189}194]. If only one longitudinal mode of the cavity is excited, it is possible to neglect the z dependence of the "eld envelope E. This approximation is sometimes referred to as the mean "eld limit [190]. In these conditions, the evolution of the cavity "eld is ruled by: RE "!E#E #ig("E"!h)E#ia  E , , Rq

(40)

where q represents a scaled time t/t , t being the mean cavity lifetime for the photons;   a,1/4nF, F being the Fresnel number of the cavity of transverse size b; h is the detuning between "eld and cavity frequencies; g"$1 determines whether the nonlinearity is of focusing or defocusing type. Eq. (40) is known to give rise to optical bistability in the transverse uniform limit. The relation between input and output optical intensities in this case reads I "I (1#(h!I )) ,  

(41)

where I ""E ", I ""E " in the steady state. It follows from Eq. (41) that the cavity is bistable for   h'(3, independently of the sign of the nonlinearity. By introducing the variable transformation E"E (1#A) 

(42)

Eq. (40) can be rewritten as [187,194] RA "![1#ig(h!I )]A#ia  A#igI (A#AH#A#2"A"#A"A") .  ,  Rq

(43)

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

33

Fig. 20. From Ref. [180]. Far "eld (left) and near "eld (right) patterns observed in a photorefractive SBN crystal pumped by counterpropagating beams, for increasing values of the voltage applied to the crystal (reported in volts in the upper corner of each "gure).

Notice that no linearity approximation has been used in passing from Eqs. (41) to (43), hence the two equations are fully equivalent. Eq. (43) shows that a perturbation (not necessarily small) to the uniform steady solution can undergo gain via the four-wave mixing (4WM) term igI AH. This  mixing describes annihilation of two photons of the steady uniform solution, and creation of one

34

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

Fig. 21. Passive optical cavities with nonlinear medium and injected signal: (a) ring con"guration; (b) Fabry-Perot con"guration.

photon in mode A and one in mode AH. Since these modes can be spatially dependent in the transverse plane, the 4WM term can lead to the destabilization of the uniform solution and eventually to the formation of transverse structures. Linear stability analysis of the system formed by Eq. (43) and its complex conjugate is performed by introducing the ansatz (A, AH)JeHRe qr .

(44)

The boundary between stable and unstable perturbations is given by the curve j"0 in the hyperplane (I , aq, h, g). By "xing the values of h and g, we obtain marginal stability curves  in the (aq, I ) plane of the kind of the ones shown in Fig. 22 [194]. Let us consider in more  detail the focusing case g"#1. The minimum of the marginal stability curve for h "xed has coordinates I "1, aq"2I !h"2!h . (45)    Eq. (45) determines the most unstable transverse mode, i.e. the one that will "rst bifurcate when increasing the input pump intensity I . The selection of the critical wavenumber q arises from  a balance between the di!ractive phase modulation aq, the nonlinear phase modulation 2I , and   the cavity detuning h. Exact compensation among these three e!ects, expressed by Eq. (45), results in a perfect phase-matching of the four-wave mixing interaction that enhances the perturbation, and hence in an optimum gain. Notice that physical values of q are positive, so that Eq. (18) is  meaningful only for h(2.

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

35

Fig. 22. From Ref. [194]. Domains in the (aq, I ) plane in which the transversally homogeneous stationary solution of  Eq. (43) is unstable to the growth of inhomogeneous perturbations. (a) focusing case, h"1 (right curve), h"5 (left curve); (b) defocusing case, h"5.

The bistability boundary for the uniform solution, h"(3, correspond to tangency of the marginal stability curve to the vertical axis. For h((3, the steady input-output characteristic of the cavity is monostable, and an instability with respect to a "nite transverse q whose critical value is given by Eq. (45) exists. Fig. 23a shows the regions of stability of the transverse homogeneous solution. For (3(h(2 the system is bistable in the transverse homogeneous case,

36

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

Fig. 23. From Ref. [194]. Transverse homogeneous stationary solutions of Eq. (41). (a): focusing case, h"1; (b) focusing case, h"5; (c) defocusing case, h"5. The dotted portions of the curves are unstable with respect to homogeneous or modulated perturbations.

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

37

Fig. 24. From Ref. [195]. Examples of the change in the patterns numerically observed in a model of a cavity "lled with a purely absorptive medium, for increasing input intensity from (a) to (c).

but instabilities with respect to a "nite wavenumber exist for both the lower and the upper stability branches. For h'2 the bistable uniform solution is stable with respect to any q in its lower branch, but unstable with respect to "nite transverse wavenumbers in its upper branch (Fig. 23b). In the self defocusing case, the marginal stability curves are as the one plotted in Fig. 22b. The coordinates of the minimum are I "1, 

aq"h!2I "h!2 .  

(46)

A physically meaningful minimum of the curve exists only for h'2, i.e. fully in the region of bistability de"ned by h'(3. Fig. 23c shows the stability of the uniform solution in this case. Since the upper branch is now stable with respect to any perturbation, it is to be expected that perturbations in the instability region of the lower branch lead the system to switch to the upper uniform solution. Hence, no pattern forming instabilities occur in the defocusing case. The model here discussed, in which the nonlinearity is of Kerr type, is also valid for a cavity "lled with two-level atoms in the limit of large detuning between input "eld and atomic line and fast atomic relaxation times [190]. For this model, both nonlinear stability analysis [193,194] and numerical simulations predict the bifurcation of subcritical hexagons close to threshold. This is a general feature related to the existence of quadratic nonlinearities in Eq. (43). Other mean "eld models have considered the situations in which either the medium nonlinearity is purely absorptive [195] (see Fig. 24), or polarization e!ects are taken into account. In these cases, the occurrence of rolls and negative hexagons has been predicted. It has also been shown that polarization e!ects lead to the occurrence of transverse instabilities for a defocusing Kerr nonlinearity, at variance with the scalar model. In mean "eld models, considering a cavity "lled with two-level atoms instead of a Kerr medium, temporal and spatiotemporal instabilities have been predicted as well as the purely spatial ones here discussed. Similar results have been found when the mean "eld limit approximation is not adopted, though the material response is assumed to be fast with respect to the "eld dynamics, and the system evolution is described by an in"nite-dimensional map [196}200]. This last case was actually historically the "rst in which spatial instabilities in nonlinear interferometers were described. In the region of nascent bistability for the homogeneous wave transmitted through the cavity, the amplitude equation for the optical "eld is a real Swift}Hohenberg equation [201,202].

38

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

In a model of ring cavity partially "lled with two level atoms, relaxation of the mean "eld approximation as well as the assumption of instantaneous material response has led to the prediction of multiconical emission [203}205], i.e. the instability of many di!erent transverse wavenumbers having approximately the same threshold value. Despite the relevant amount of theoretical and numerical studies about nonlinear passive cavities, the experimental results in this "eld are not abundant. Experiments involving the use of Fabry-Perot resonators "lled with sodium vapors have demonstrated the occurrence of both time oscillations and spatial pattern formation in the output beam [206}211]. Though, a direct comparison between the experimental results and the theoretical predictions is di$cult. Among the reasons of these di$culties are the low values of the atomic nonlinearities, that imposes the use of a narrow laser beam and consequently a low Fresnel number system, the fast response of the nonlinearity, that introduces serious di$culties for an experimental detection resolved both in space and time, the mismatch between the two-level description of the atoms and their real behaviour [212], the motion of atoms, resulting in di!usion processes. 3.5. Nonlinear slice with optical feedback In the previous paragraphs we dealt with optical systems in which medium nonlinearity and wave propagation take place in the same physical location. Here we report investigations about setups in which the action of these two mechanisms occurs at distinct regions in space. As a consequence of this spatial separation, an easier identi"cation of the speci"c role played by nonlinearity and propagation on pattern formation is possible. Let us consider the system shown in Fig. 25 [213}216], formed by a thin slice of Kerr material and a mirror. A plane wave of intensity I ""E " is sent through the material, propagates up to the   mirror and is re#ected back onto the Kerr slice, which induces a phase retardation on the plane wave, given by u(r , t)"aI (r , t), where I is the total intensity impinging on the material, a gives , R ,  the sign and strength of the Kerr nonlinearity, and r denotes the coordinates transverse to wave , propagation. If we neglect the small scale interference gratings due to the superposition of the forward and backward waved, which is a reasonable assumption if the di!usion length of the

Fig. 25. Kerr medium with a purely di!ractive feedback.

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

39

material is much larger than the optical wavelength, I is given by the sum of I with the intensity   I of the backward feedback "eld. 
Since we are interested in pattern formation, i.e. in instabilities involving "nite wavenumbers, the role of the spatially uniform intensity I in determining the evolution of the phase u can be  neglected, because I just give rise to a uniform additive variation of u. The evolution of the phase  retardation induced by the medium on the incoming plane wave is then ruled by [214}216] u(r , t) Ru(r , t) , "! , #D  u(r , t)#aI (r , t) , , , 
, q Rt

(47)

where q and D are, respectively, the local relaxation time and di!usion constant of the medium. In the simple geometry here considered, the feedback "eld distribution is due to di!ractive propagation of the "eld E(r , z"0)"E e PP, exiting the Kerr slice due to the propagation of the incoming ,  wave through the slice. In the usual paraxial approximation, E(r , z) evolves following , i RE((r , z, t) , "  E(r , z, t) , (48) 2k , , Rz  where k ,2n/j is the optical wavenumber. The formal solution of this equation is   (49) E(r , z, t)"e I,XE(r , z, t) , , substituting this expression into Eq. (47) and calling ¸ the propagation length, the equation governing the phase evolution becomes u(r ) Ru(r ) , "! , #D  u(r )#aI "e I,Be Pr," . , ,  q Rt

(50)

For small phase perturbations u;1, Eq. (50) can be linearized, giving Ru u O"! O#aI "e\ OI* (1#iu )" ,  O Rt q

(51)

where the evolution of a single spatial Fourier component at spatial frequency q has been singled out. Eq. (51) highlights the speci"c role of di!raction and nonlinearity in destabilizing the plane wave solution u(r )"0. Indeed if di!raction is absent (¸"0), the feedback term in Eq. (51) , reduces to I "I "1#iu "KI (52) 
 O  at "rst order in u . Thus no feedback is e!ective at this order, and the plane wave solution is stable. O Indeed, in such a case, the feedback beam is phase but not intensity modulated. This is ine!ective on the phase evolution, since the Kerr nonlinearity responds to intensity. When free propagation over a length ¸ is introduced, the Fourier component at frequency q is dephased by a factor e\ IOB with respect to the continuous component at q"0, represented by the factor `1a in Eq. (52). The feedback intensity reads now I "I "1#iu e\ OI*" . 
 O

(53)

40

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

For q¸/2k "(n/2)#2Kn or q¸/2k "n#2Kn, K integer, the feedback intensity is respec   tively I "I (1#2u ), I "I (1!2u ) at "rst order in u . The phase modulation has been in 
 O 
 O O this case completely converted into intensity modulation by the di!ractive propagation, hence an e$cient feedback action on the phase via the Kerr nonlinearity is possible. For generic values of the parameters, the linear growth rate j for a mode at spatial frequency O q is




q¸ 1 j "! !Dq#2aI sin  O 2k q  the curve j "0 gives the marginal stability in the (q, aI ) plane [216]: O  1#lq  qaI "  2sin(q¸/2k ) 

(54)

(55)

where the material di!usion length l ,(Dq has been introduced.  In Fig. 26 we report the marginal stability curves for two di!erent values of l . The upper and  lower half planes correspond respectively to focusing and defocusing nonlinearity (a'0, a(0). The extrema of each "nger-like branch correspond to q¸/2k K(n/2)#2Kn, q¸/2k K   n#2Kn for the focusing and defocusing case respectively in the limit l<j¸. The extrema   corresponding to successive values of the integer K lie, in the same limit, on the curve qaI "  (1#lq)/2. Di!usion is seen to lift the degeneracy of the threshold condition for the successive  bands, leading to lower threshold values for the ones occurring at low spatial frequencies. The predictions of this simple model have been experimentally veri"ed in systems using various kinds of nonlinear materials, among which liquid crystals are the one that "t better the Kerr-like description of the nonlinearity. Liquid crystals have been used both in simple layers [217}223], and in a hybrid electrooptical device named Liquid Crystal Light Valve (LCLV) [214,224}251]. This device is still adequately described by an optical Kerr model for a broad range of experimental parameters, and by the use of an auxiliary electric "eld it provides nonlinear susceptibilities much larger than a simple liquid crystal layer, thus allowing low intensity operation and the consequent possibility of investigating large Fresnel number systems. The typical patterns observed close to the instability threshold are hexagonal (Fig. 27) [217,220,223,234,245}248], due to the superposition of three Fourier modes oriented at 1203 one with respect to the other. These hexagons bifurcate subcritically [241], in agreement with the predictions of a nonlinear stability analysis. Simple roll solutions appear instead to be always unstable [252]. The scaling of the selected wavenumber q with the free propagation length ¸ has also been veri"ed [217,234]. Competition and cooperative phenomena among patterns with closeby threshold values have been predicted. In particular, it has been shown how spatial mode locking among wavevectors belonging to di!erent bands in Fourier space can result in singledomain, multiscale patterns of the kind shown in Fig. 28. The transverse boundary conditions have been shown to play an important role on symmetry selection when the system size is of the order of few unstable wavelength [218,222,231,236,253]. In this case, polygons di!erent from hexagons are observed. For values of the input intensity far above the instability threshold, broadening of the excited spectral bands leads to appearance and motion of defects in the hexagonal structure, and "nally to a space}time chaotic situation (Fig. 27) [216,223,248].

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

41

Fig. 26. Marginal stability curves for the Kerr medium with purely di!ractive feedback. The upper half plane refers to a focusing medium, the lower one to a defocusing one. The curves are obtained for ¸"30 cm, j,2n/k "514 nm.  l "20 lm (solid line), l "50 lm (dashed line).   Fig. 27. From Ref. [248]. Large aspect ratio structures observed in a LCLV with di!ractive feedback. Top: hexagonal structures, bottom: space}time chaotic state observed for higher values of pumping intensity.

The use of photorefractive crystals instead of liquid crystals or LCLVs as nonlinear media has led to the observation of both patterns with hexagonal symmetry [178,254], as well as squeezes hexagons and squares [255]. The reason for this symmetry breaking seems to be due to the anisotropic nonlinear response of photorefractives. Investigations of photorefractives [254] and organic materials [256] have clari"ed that the instability mechanism here discussed is not limited to the case in which the medium nonlinearity is purely dispersive, but basically holds also for absorptive or absorptive-dispersive nonlinearities. Other studies have been devoted to the use of atomic vapors as the nonlinear medium [213,257}269]. The nonlinearity in this case can be either focusing or defocusing, depending on the experimental conditions, and is poorly described by the Kerr model [261,265]. Furthermore, important polarization e!ects come into play when atomic vapors are used [260,261,265,269]. In particular, transitions between stable square or rolls and stable hexagons have been predicted [260] and experimentally observed [269] when varying the polarization of the light incident on a cell "lled with sodium vapors. In other experimental conditions, for "xed input polarization and varying pump intensity, a variety of di!erent structures have been reported [261,264}266,269], including polygons, positive

42

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

Fig. 28. From Ref. [237]. Near "eld (left) and far "eld (right) images of multiscale frequency locked patterns in a LCLV with di!ractive feedback close to threshold.

and negative hexagons, turbulent states and multipetal patterns in#uenced by the symmetry and size of the input beam. Most of these structures have been reproduced in simulations using suitable models for the sodium nonlinear response [259,260,265,266] (see Fig. 29). In the case of atomic vapors, the occurrence of simultaneous instability of several Fourier bands [257], as well as the instability of the basic hexagonal patterns with respect to its own harmonics or subharmonics spatial frequencies [268] have been predicted. The mechanism governing the instabilities here discussed can be conceptually split in two steps. First, a conversion of phase #uctuations into intensity #uctuations is needed. Second, intensity #uctuations must give rise to phase #uctuations, in order to close the feedback loop. Up to now we considered the situation in which the "rst process take place via optical free propagation, however, this is not the only possibility. An easy way to convert phases into amplitudes is by means of an interferometric process. Let us consider the setup shown in Fig. 30. The Kerr medium is operating only in re#ection (in practice, this is actually the case if a LCLV is used). The light fed back to the

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

43

Fig. 29. From Ref. [265]. Experimental (top) and corresponding simulated (bottom) patterns observed in a laser beam transmitted through a Na cell for di!erent experimental parameters.

Fig. 30. Setup for a Kerr medium operating in re#ection with di!ractive and interferential feedback. The linear operator M can account for di!raction, imaging, or nonlocal operators.

medium is a superposition of the wave re#ected by the medium itself, and a reference wave re#ected by mirror A. A generic linear operator M is included in the loop. Historically, a similar device was introduced with the aim of correcting the phase distorsions of the input beam, and a very rich fenomenology of pattern forming instabilities was early reported using this system [214,224}227]. If the arm containing the mirror A is blocked and the operator M represents simply di!raction, this setup is fully equivalent to the basic one of Fig. 25. Another limit case is that in which the arm containing A is unblocked, and M represents a one to one imaging of the front face of the medium onto its rear face. The evolution of the phase #uctuations u(r , t) follows , u(r , t) Ru(r , t) , "! , #D  u(r , t)#aI [1#m cos(u(r , t)!u )] , (56) , ,  ,  q Rt

44

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

Fig. 31. (a) Graphical solution of Eq. (57), showing the onset of optical bistability in the Kerr medium with interferential feedback. (b) A typical uniform steady state characteristic u vs. I for the same problem. 

where m is a modulation factor depending on the medium and mirror re#ectivities, and u is the  phase di!erence between the in-axis waves travelling in the two arms of the interferometer. Eq. (56) admits the uniform stationary solution u/aqI "1#m cos(u!u ) . (57)   This displays bistability (or multistability) for high enough values of I (Fig. 31). A linear stability  analysis of this solution shows that the mode with maximum temporal growth rate is the one at zero spatial frequency. This is to be expected, since di!usion is the only space-dependent phenomenon that comes into play. Hence, as I is varied, jumps between di!erent uniform stable solutions  of Eq. (57) are expected, but no pattern formation takes place. A completely di!erent situation occurs if both interference and di!raction are at work [227,228,232,251]. The evolution equation of the phase #uctuations in this case is [251] u(r ) Ru(r ) , "! , #D  u(r )#aI "e I,*(A#Be Pr,\P" , , ,  q Rt

(58)

where A and B are the normalized amplitudes of the waves travelling the two arms of the interferometer. Fig. 32 shows the marginal stability curves on the (q, I ) plane in the focusing case 

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

45

Fig. 32. Marginal stability curves for the Kerr medium with di!ractive plus interferential feedback (see Eq. (58)), for increasing values of the modulation parameter m,2AB/(A#B). (a) m"0, (b) m"0.42, (c) m"0.68.

(a'0), for di!erent values of the modulation parameter m,2AB/(A#B). It can be seen that for m"0 the pure di!ractive behavior is recovered (Fig. 32a). For increasing values of m, the curves are progressively distorted, till touching the q"0 axis. In these conditions the instability at a "nite q is accompanied by instability of the uniform background state with respect to uniform perturbations. Bistability of the uniform solutions and bifurcations at "nite q thus occur together, resulting in a complicate dynamical behaviour about which is not captured by the linear stability analysis. In Fig. 33 we show some of the spatial structures that have been numerically predicted for this regime [229,215]. From an experimental point of view, though a rich variety of results have been reported in pioneering experiments [214,224}227], systematic observations in the regime in which both di!raction and interference are present are rather recent [232,250]. Among the main results of these studies there is the observation of localized structures [250]. 3.6. Nonlocal interactions The introduction of nonlocal interactions in a system formed by a nonlinear material slice with optical feedback leads to qualitatively di!erent scenarios of pattern forming instabilities as compared to the ones above discussed. Nonlocality is introduced via rotation [213,225}230,234,235,237,238,240}243,270], translation [219,226,244}246,262,263] or magni"cation [213,226] in the optical feedback loop. In some cases, the e!ects of a global feedback have also been considered [239].

46

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

Fig. 33. From Ref. [229]. Examples of patterns numerically predicted for a Kerr medium with di!ractive plus interferential feedback, for various values of free propagation length, and relative phase between the beams coming from the two interferometer arms.

In the system with purely interferential feedback the discrete transport due to rotation or translation can destabilize the uniform steady states both in the monostable and in the bistable regimes. This occurs if the local feedback of these states, given by the term aI [1#m cos( ! )]   in Eq. (56) is negative and strong enough. Suppose that, in these circumstances, a translation of an amount *x is introduced in the feedback loop. Perturbations of spatial period 2*x can be destabilized since the feedback on these waves is now positive, resulting from the spatial shift by n of the original negative feedback in the absence of translation. This instability mechanism is the spatial counterpart of the well known temporal oscillatory behavior in an electronic ampli"er with delayed negative feedback. The same mechanism operates if rotation rather than translation is introduced in the feedback loop, resulting in the bifurcation of spatial Fourier bands limited only by the material di!usion. The occurrence of this kind of instabilities have been experimentally veri"ed in systems based on LCLV [226,230,240,241]. For increasing pump parameters, secondary instabilities and the "nal occurrence of space}time chaotic regimes [230,240,241] (Fig. 34) have been reported. The introduction of nonlocal interactions has important consequences also in systems with purely di!ractive feedback. In this case an e$cient conversion of phase into amplitude #uctuations via di!raction over a length ¸ is required in order to have a strong feedback, leading to the scale selection rules

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

47

Fig. 34. From Ref. [241]. Patterns observed in a LLV with purely interferential feedback, and a rotation of L in the  feedback loop. The control parameter is the voltage < applied to the LCLV, that controls the sensitivity of this Kerr like  device. < increases from (a) to (j). 

Fig. 35. From Ref. [242]. Marginal stability curves for the LCLV with purely di!ractive feedback and a rotation of D,2n/N in the feedback loop, evaluated for ¸"75 cm and l "15.5 lm, j"633 nm. 

q(¸/2k )K(n/2) or q(¸/2k )K3n/2 for focusing and defocusing materials respectively. It is not   expected that these rules are drastically changed by the discrete transport introduced, though it is possible to convert an e$cient negative feedback into an e$cient positive one via a spatial shift or rotation, just the same way we discussed for systems with interferential feedback. For these reasons, in presence of transport, a defocusing medium can be unstable with respect to spatial frequencies that are proper of a focusing medium in conditions of local feedback, and vice versa [234,238,242,244,245]. In Fig. 35 we plot the marginal stability curves for a defocusing medium in a di!ractive feedback loop, when a rotation of an angle D"2n/N, N integer is introduced. The bistability band at q(¸/2k )K3n/2 (band II) is the one that exists also for local feedback. The one  at q(¸/2k )Kn/2 (band I), that would exist only for a focusing medium if the feedback was local, 

48

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

Fig. 36. From Ref. [242]. Crystalline and quasi crystalline patterns observed in the LCLV with purely di!ractive feedback and rotation of D,2n/N in the feedback loop. Left: near "eld, right: far "eld. The integer N is reported close to each picture.

for N even has always a minimum lower than the band II. For N odd, still band I exists, but its instability threshold is lower than that of band II only for rather high values of N. The in#uence of the rotation on the selection of the symmetry of the patterns is even stronger. Instead of the hexagons formed in the case of local feedback, it is now possible to observe structures displaying crystalline or quasicrystalline symmetries [242,270] close to the instability threshold (Fig. 36). Competition between band I and II has been studied for "xed N"7, by increasing the

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

49

Fig. 37. From Ref. [243]. Transition from patterns on band II to patterns on band I, observed for increasing the input intensity (from (a) to (c)) in a LCLV with di!ractive feedback rotated of D"2n/7. (a)}(c): near "eld, (d) and (e): far "eld. Notice the coexistence of patterns at di!erent scales in di!erent spatial domains in (b).

input pump intensity I [243]. For low values of I , only band II bifurcates, and quasicrystalline   patterns are observed (Fig. 37a). High values of I lead to saturation of the gain for band II and  domination of patterns belonging to band I (Fig. 37b). At intermediate values of I , irregular time  alternation between structures belonging to the two bands occur. These features can be inferred from Fig. 38, showing the time series of the quantity g(t)"S (t)/[S (t)#S (t)] ,   

(59)

where S (t) denotes the total power in band j at the time t. In this time alternation regime, H coexistence of patterns belonging to the two bands occur via a spatial segregation of scales resulting in formation of domains, in each one of which only patterns at one scale is present (Fig. 37b). If the nonlocality is of translational type, a "rst e!ect expected is the occurrence of drifting instabilities [244}246,262,263,271]. Consider for simplicity a 1-dimensional system in which a discrete transport *x is introduced in the feedback loop. If patterns exists at a spatial frequency q, a phase shift "q*x arise between the material excitation and the optical intensity fed back to V the medium [263]. To compensate this phase shift, a drift of the pattern is expected, leading to a wave varying as cos(qx!ut) instead of cos qx, since in the time dependent case the medium steady state excitation exhibits a phase shift K t, i.e. uKq*x/q. The occurrence of drifting R instabilities and the agreement of the drift frequency with the prediction of the above heuristics

50

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

Fig. 38. From Ref. [243]. Temporal evolution of the normalized power spectrum g(t) (see Eq. (59)) for the LCLV with di!ractive feedback rotated of an angle D"2n/7, for di!erent values of the above threshold parameter e. Lower curve: e"1, middle curve: e"2.1, and upper curve: e"4.1.

Fig. 39. From Ref. [245]. Near "eld (above) and far "eld (below) patterns observed in the LCLV with purely di!ractive feedback translated by an amount *x. The input intensity I and free propagation length ¸ are kept "xed  (I "72 lW/cm, ¸"26 cm). (a,a): *x"0; (b,b): *x"50 lm; (c,c): *x"180 lm; (d,d): *x"220 lm; (e,e):  *x"400 lm.

have been veri"ed in experiments based both on atomic vapours [263], and on the LC [219] or LCLVs [244,245]. As in the case of rotation, translation induces important di!erences in the pattern selection rules with respect to the case of local feedback. In particular, series of transitions among di!erent structures for increasing transport length *x have been reported both for systems using focusing or defocusing materials [245,246]. In Fig. 39 we report the near and far "eld patterns observed in a LCLV (defocusing medium) in which the e!ective feedback is translated of an increasing amount of *x. All these structures are fairly accounted for by the linear stability analysis. There is a striking similarity between some of these patterns and similar ones observed in hydrodynamical systems

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

51

subjected either to an increase of the `pump parametera, or to an inclination of the gravity "eld. Clearly transport processes play a central role in all these phenomena, though a real understanding of the similarities and di!erences among pattern forming instabilities in these di!erent systems has not yet been achieved.

4. Defects and phase singularities in optics 4.1. Phase singularities and topological defects in linear waves A topological defect (or phase singularity) is a point of the space where the circulation of the phase gradient around any closed path surrounding it is equal to $2mn. The integer m is called topological charge. For the wave equation it has been demonstrated that only $1 charges are stable [272]. Let us recall the Berry's de"nition of a defect [272]: `Singularities, when considered in the modern way as geometric rather than algebraic structures are morphologies, that is form rather than matter; and waves are morphologies too (it is not matter, but form that moves with a wave). Therefore singularities of waves represent a double abstraction } forms of forms, as it were } and so it comes as something of a surprise to learn that they represent observable phenomena in a very direct waya. The topological defects represent one of the most important features of wavefronts, insofar as they correspond to singularities of the phase function U(r, t). The nature of these singularities is determined by the fact that the "eld E is a smooth single valued function of r and t. Single valuedness implies that U may change by 2mn along a circuit C in space}time. When m is not zero and C is shrunk to a very small loop, then C encloses a singularity, because U is varying in"nitely fast. The smoothness of E now entails "E""0. The vanishing of "E" requires to satisfy simultaneously two conditions (RE"IE"0). As a consequence, phase singularities are lines in space or points in the plane. Consider a random complex "eld E(r) de"ned in the r"(x, y) plane. If it arises from the interference of a large number of independent components, RE and IE are two independent random functions with Gaussian statistics. The zeroes of the functions RE(x, y) and IE(x, y) determine a number of curves in the (x, y) plane, and the intersections of curves of one family with those of the other give a set of points where "E(x, y)""0. It is relevant to address the problem of the propagation of such zeroes along the direction z. According to the wave equation, these points are converted into lines. Those last ones, in general, do not intersect in three dimensional space. Moreover, a given line cannot appear singly at same plane z"const, nor it can disappear singly. Topological defects must appear or be annihilated in pairs. All topological dimensional arguments do not depend, however, on the nature of the interfering "elds. The unbalance N !N (di!erence between the numbers of zeroes with positive and negative > \ charges) is conserved along the propagation. On the average, in a cross section of a random "eld (called speckle "eld) N "N because the beam is statistically homogeneous. > \

52

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

Refs. [273,375] derives the expression of the total number of zeroes (N"N #N ) in an area > \ S of the transverse (x, y) plane:



NS"



dx dyd(E (x, y))d(E (x, y)) " R(E , E )/R(x, y)"    

.

(60)

Here E"E #iE . The angle brackets denote an ensemble average process over a set of random   "elds. Each defect of the "eld gives a contribution unity to the right hand side of Eq. (60). Moreover, positive and negative defects correspond to positive and negative signs of the Jacobian G"R(E , E )/R(x, y). Eq. (60) can be rewritten as (denoting the gradients as RE /Rx"E , etc.)    V 1 N" dx dy dE dE dE dE dE dE = (E , E , E , E , E , E )d(E )d(E )"G"   V V W W    V W V W   S

  

" dE 2dE = (0, 0, E , E , E , E )"G" V W  V W V W

(61)

(= being the joint probability of the quantities E , E and their gradients at the given point). The    Gaussian assumption makes it possible to factor out = in terms of correlations of the complex  "eld. Precisely, the Van Cittert}Zernicke theorem tells us that the correlation is



1EH(r )E(r )2"I j(h)e IFP\P dh ,  

(62)

where j(h) is the normalized angular spectrum and h"(h , h ). Furthermore V W RE(r )  EH(r ) "ik1h 2"ik j(h)h dh . (63)  Rx G G PP G 1h 2 and 1h 20 can be made equal to zero by a suitable rotation of the z-axis. This corresponds to V W choosing a z axis in the direction of the center of gravity of the angular distribution. This way, the complex gradients RE/Rx and RE/Ry are independent of the "eld E(r) itself at each point (x, y). The correlation matrix













1 REH(r ) RE(r )   1h h 2" " j(h)h h dh (64) G I G I Rx Rx k G I PP can be transformed to principal axes by a rotation of axes in the x, y plane. All three complex quantities E, RE/Rx and RE/Ry are mutually independent at the given point. In this framework, with 1h2"0 and with 1h h 2 diagonal, = becomes: G I  = (E , E , E , E , E , E )    V W V W 1 (E #E ) (E #E ) V exp ! W W exp[!(E#E)/I] " exp ! V (65)   nk1h21h2I k1h2I k1h2I V W V W and Eq. (61) gives



k N" (1h21h2) . W 2n V

 



(66)

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

53

Now, the radius of the correlation of the speckle "eld (the transverse speckle size), is proportional to j/*h (*h being the angular divergence of the beam). Therefore Eq. (66) shows that the density of dislocations coincides with the number of speckles per unit area. This was veri"ed in a series of experiments [273,375] where the "eld was produced by transmitting a laser beam through a distorting phase plate. The structure of the speckle "eld wavefront was investigated by interference with a plane reference wave directed at a certain angle. The fringe separation identi"es the tilting angle between speckle and reference "elds. Bending of fringes corresponds to the curvature of the wavefront, while termination or birth of a fringe is a signature of a negative or positive phase singularity respectively. From Eq. (66), N scales like the square a of the diaphragm diameter on the phase plate, that is linearly in the Fresnel number F (see Eq. (19) for de"nition). The total number of dislocations NS,Na scales like F, since S is proportional to F. 4.2. Phase singularities in nonlinear waves In nonlinear physics, the dynamical role of a topological defect emerges by mediating the transition between two di!erent types of symmetry. More generally, one can state that the appearance of a defect marks a symmetry breaking. When the defect is a space structure (as grain boundaries or point defects in crystals), it is called `structural defecta. In wave patterns, defects appear in space}time, and they are called `topological defecta [274]. An important point here is that not all phase singularities should be identi"ed by defects, but only those ones localized at the edge between two patterns with di!erent symmetries. Defects and their role in mediating turbulence have been widely investigated in hydrodynamic systems with large aspect ratios as #uid thermal convection [275,276,376], in nematic liquid crystals [277,377,378], in surface waves [278], and in analytic treatments and numerical simulations [279,379] of partial di!erential equations in 2#1 space}time dimensions. The presence and the role of defects in nonlinear optics has been discussed theoretically in Refs. [28]. The nucleation and the evolution of phase dislocations in a laser beam interacting with a photorefractive medium has also been studied in Refs. [280}282]. The appearance of defects has been largely investigated, both theoretically and experimentally, in many other optical systems, such as a self-defocusing Kerr nonlinear medium [283], a nonlinear Fabry-Perot resonator [284], a class A laser [43], a saturable self-focusing medium [285]. In this last case, three dimensional bright spatial solitons have been observed as a result of a modulation instability of an optical defect [286]. The evolution of the optical defects and their propagation in a three dimensional self-defocusing medium has been studied in Ref. [287]. Motion and propagation of defects have been investigated in various experimental conditions [288,289]. The transformation of the topological charge during free-space propagation of light beams carrying phase singularities, and the conservation of the associated total angular momentum during the propagation have been studied in Ref. [290]. The possibility for a phase singularity to produce particle trapping, due to the absorption of the angular momentum has been shown in Refs. [291,292]. Finally, the motion of optical phase singularities has been theoretically and experimentally compared to that of a #uidodynamical system [293,294].

54

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

Fig. 40. From Ref. [295]. (a) Single optical vortex in a doughnut mode, with a spiral wave around, which, in the course of the time changes its sense of rotation (left, clockwise; right, anticlockwise). (b) Reconstruction of the instantaneous phase portrait for an optical vortex, perspective (left) and equi-phase (right) plots. The phase has been experimentally reconstructed via the algorithm of Ref. [296].

We here summarize the "rst experimental evidence of the existence of defects in nonlinear optics, provided in Ref. [295], with an experimental setup similar to that discussed in Section 2.3 [88]. Since a defect implies a phase singularity, in order to detect it experimentally, one needs to perform a phase measurement. In the case of an optical "eld, a phase measurement can be provided by heterodyning against an external reference, that is by beating the signal with a reference beam onto a CCD videocamera. Ref. [296] suggests a suitable algorithm, by whom the instantaneous surfaces of phase can be reconstructed. Fig. 40 shows that the phase surface of a doughnut mode is a helix of pitch 2n around the core (vortex). When many vortices are present, in order to count each vortex, a tilted reference beam is sent to the CCD, so that the video signal is now given by I(x, y)"A#B#2AB cos(Kx#U(x, y)) ,

(67)

where A and B are the amplitude of reference and signal "eld, K the fringe frequency due to tilting, x the coordinate normal to the fringes and U the local phase. A phase singularity appears as a dislocation in the system of fringes, so that the topological charge can be visually evaluated. This way, one can measure the mean number of defect, their mean distance, and the mean value of the unbalance (di!erence between positively and negatively charged defects) as a function of some extensive parameter (in Ref. [295] it was the Fresnel number). In Ref. [94], the defect statistic was measured in order to mark a transition between patterns ruled by the boundary constraints and patterns ruled by the bulk properties of the active medium.

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

55

In a system far from equilibrium, indeed, two types of patterns can arise. When the symmetries are imposed by the bulk parameters of the active medium, the pattern is called `spontaneousa. Instead, when the symmetries are dominated by the boundary, either by the geometry of the system or by an external driving force, the pattern is called `forceda. A prototype of spontaneous pattern is the Turing instability [297]. An example of forced patterns is given by `dispersivea patterns, where the linearized dynamics of the system provides a dispersion relation f (u, k)"0. In those systems, when u is real, an external forcing frequency u makes the choice of the length scale k\ by constrain of dispersion. An example is the case of capillary patterns in #uid layers submitted to a periodic vertical force (Faraday instability) [298,380]. Experimental evidence of the transition from a dispersive length to a dissipative one was given for a liquid-vapor interface close to the critical point [299]. When u is imaginary, then f (u, k)"0 provides an interval of possible unstable k values, and selection of one (or a few) particular k is provided by the boundary constraints [300]. In both cases, `dispersivea patterns are dominated by an external in#uence. The evidence of boundary independent patterns in nonlinear optics was given by Ref. [94]. In that reference it was reported the transition from dispersive patterns, dominated by the geometric parameters, to dissipative patterns whose scale length is imposed by the bulk properties of the medium. The second patterns are not usual in optics, insofar as the patterns properties depend in general upon the Fresnel number F"a/j¸, which accounts for competition between geometric acceptance and di!raction phenomena. A "rst attempt to overcome such a limitation was given theoretically in Ref. [189]. In the experiment of Ref. [94] a fundamental geometric parameter is the spot size of the central mode constrained by the quasi-confocal con"guration to be [301]



w " 

j¸ . n

(68)

If the mirror size a is larger than w (that is, the Fresnel number is larger than 1), the cavity houses  higher order modes, made of regular arrangements of bright spots (in cylindrical geometry they are Gauss}Laguerre functions) separated by w DK  . (F

(69)

The overall spot size of a transverse mode of order n scales as (nw , so that n"F represents the  largest order mode allowed by the boundary conditions (that is, "lling all the aperture area). It is worth to remind that patterns built by superposition of Gauss}Laguerre functions have an average separation 1D2 of zeroes approximately equal to the average separation D of bright peaks [301]. Exploring very large F regimes, a plot of 1D2 versus F (Fig. 41a) shows that Eq. (69) is not valid everywhere, but it is restricted up to a critical value F , above which D is almost independent of F.  Similarly, the total number N of phase singularities scales as F or F, respectively below and above F . This transition in the scaling law properties is reported in Fig. 41b.  The di!erence in scaling laws for 1D2 and N is a signature of the transition of the system from a dynamical regime to another. This transition has the following root. Assume that the

56

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

Fig. 41. From Ref. [94]. (a) Mean nearest neighbor separation 1D2 (the scale is in lm) between phase singularities and (b) average total number 1N2 of phase singularities as functions of the Fresnel number F of the cavity. Dashed lines refer to best "ts for the boundary dependent scaling laws 1D2&F\ and 1N2&F. Solid lines are best "ts for the boundary independent scaling laws 1D2&F and 1N2&F. The transition between the two dynamical regime occurs at F&11.

photorefractive crystal is a collection of uncorrelated optical domains, each one having a transverse size limited by a correlation length l intrinsic of the crystal excitations. Then the medium gain will  have un upper cuto! at a transverse wavenumber 1/l and the ampli"cation of spatial details will be  e!ective only up to that frequency. This implies that, for F such that D"w /(F "l , a transition     from a boundary to a bulk dominated regime occurs. In the former regime, the separation between phase singularities is given by Eq. (69), in the latter it is independent of F. Fig. 41 reports the two regimes, and yields a value F K11 corresponding to l &170 lm, for w &600 lm and    ¸"200 cm. The reduction of the boundary in#uence is also marked by the reduction of the topological charge imbalance. This is because a regular "eld should have a balance between topological charges of di!erent sign, while any imbalance means that two phase singularities of opposite

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

57

sign created close to the boundary have been divided so that one has remained within the boundary. Therefore, a boundary layer of area a ) 1D2 can be de"ned, which contains N &a/1D2  defects (N ;N when 1D2;a). Only within such layer a topological imbalance occurs. Other  independent evidences of such a dynamical transition are o!ered in Ref. [94].

5. Open problems and conclusions We devote this conclusive section to a brief outline of three problem areas in optical patterns. The "rst one (Section 5.1) deals with localized structures in devices made of nonlinear slices in a feedback con"guration. How easy is to create and destroy small isolated spots of light intensity over a uniform dark background (or the complementary, that is, dark spots on a bright background) furthermore being able to change their positions? This might be the clue for a new information storage system, to be used in a next generation of computers. The second problem (Section 5.2) is how costly, in terms of probes and e!ectors, is to control chaotic patterns. For a discrete dynamical system, a single probe on the PoincareH section can signal chaotic instabilities and suggest how to correct them by small perturbations [302]. It is not even necessary to have access to the whole PoincareH section, but one single coordinate can be compared with its value at a previous period, provided the correction is continuous [303]. An adaptive version of these controls [304,381] provides higher order corrections. But in the case of space}time chaos, there may be defects, or large discontinuity points, whereby any perturbative approach fails. Furthermore, pattern control is a "eld problem, formally in"nite dimensional. Is there any suitable tool to reduce the number of probes to a manageable one? Finally, a new blossoming "eld is that of atom optics [305]. In this case, the propagating "eld is represented by the atomic Schroedinger t(r, t) describing the space}time amplitude probability of localizing the atom. In the case of an atomic beam pointing toward a direction z, we can extract a main plane wave e IX\SR and account for the variations of the transverse (x, y) plane through the Laplacian term of the Schroedinger equation. The structure of this equation in such a case includes an eikonal operator. We should expect all phenomena of optics occurring in such a case, and a few of them have already been observed [305]. Can we foresee a spontaneous patterning of a uniform Schroedinger wavefront due to some interaction of Kerr type with a light "eld? Such a matter would have been highly speculative a few years ago, because coherence constraints were di$cult to satisfy with thermal beams having a De Broglie length much smaller than the interatomic separation. After the beautiful evidence of Bose}Einstein condensation [17] and the make up of the atom laser [18], these coherence problems have been overcome, thus the set up presented in Section 5.3 may "nd a laboratory implementation within a short time. 5.1. Localized structures in feedback systems Localization of single light (or dark) spot in feedback systems constitute a counterpart of soliton formation in free propagation in nonlinear media. Though the existence of this kind of structures and their relation with modulational instabilities in optical cavities "lled with a nonlinear medium have been theoretically explored in the 80's in some pioneering works [197,306}314] full attention to this matter is rather recent.

58

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

Fig. 42. From Ref. [315]. The continuous-dotted line represents the homogeneous stationary solution of Eq. (70) in the monostable regime (a"0.1, c"!0.001). This solution is stable in the continuous region, unstable with respect to modulated perturbations in the dotted region. The circles indicate the minimum and maximum values of the numerically evaluated pattern solutions. Localized structures exist in regions B, B. Fig. 43. From Ref. [315]. Same as Fig. 41, but for c"0.025. The uniform stationary state is now bistable.

The localized structures (LS) that have been studied can be roughly classi"ed in two categories. The "rst one occurs in systems displaying a spatial instability of the basic uniform state of a "nite wavenumber, close to or in the bistability regime for the uniform state itself. The second one is associated with bistability between uniform states, without the presence of bifurcations to delocalized patterned states. As an example of the "rst type, consider pattern formation in a bistable cavity close to nascent optical bistability [201,202,315]. The model consists of a cavity "lled with two-level atoms, with an injected "eld. Assuming the validity of the mean "eld approximation, in the limit of weak dispersion the cavity electric "eld evolves according to 4 RE "4y#E(c!E)!4* E! E . 3 Rt

(70)

Here, E and c are the deviations of the electric "eld and cooperation parameter (see Appendix A) from their values at the critical point corresponding to nascent bistability, y is the injected "eld, and * is the detuning between the atom and "eld frequencies. The input-output characteristics y vs E describing the uniform solutions of (70) is monostable for c(0, bistable for c'0. In both cases, subcritical bifurcations to patterned states exist for a given range of y values. These are due to the instability mechanism described in Section 3.2, though an expansion close to the critical point has led to a formally di!erent model equation. Figs. 42 and 43 display the uniform solution for c(0, c'0 in the plane E!y, together with the maximum and minimum values of the "eld E in the regime in which bifurcation to hexagons occur. In both cases, localized spots of the kind shown in Fig. 44 can be created by addressing a narrow light pulse to speci"c spatial locations. Similar patterns have been predicted in the simulation of a bistable cavity "lled with a Kerr medium [194,316], and in systems with purely absorptive nonlinearity [317,318]. In other studies the mean "eld approximation as well as the instantaneous response of the medium have been relaxed, and still localization of patterns has been observed [319].

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

59

Fig. 44. From Ref. [315]. The two dimensional localized structures observed in simulation of the model described by Eq. (70) for D"0.1, c"0.025, y"!0.0005.

Due to the potential interest of these kinds of structures as pixels in information processing systems, investigations have been devoted to the existence of these spots in a Fabry-Perot cavity "lled with semiconductors [320,321]. It has been shown that the localized structures are robust with respect to the perturbations introduced by charge carriers di!usion and dynamics. Finally, the existence of stable localized structures has been predicted also for cavities "lled with a medium displaying optical nonlinearities [322]. In all the cases reported above, the shape of a localized structure looks as shown in Fig. 45. The plot shows also the "eld distribution of a LS as compared to that of the delocalized hexagonal pattern that can be excited for the same parameter values. In its central point, the LS "eld distribution "ts nicely the hexagons distribution, thus stressing the role of the modulational instability in the LS formation. Each spot can be interpreted as a single, elementary building block of the underlying set of hexagons. The peak is connected to the low uniform state by a shallow trough (see the oscillation at bottom left of Fig. 45). It has been argued that these peripheral undulations have a central role in determining the stability of the structure observed, introducing a sort of `pinninga that prevents the system to switch uniformly to the upper patterned or lower uniform state [113,309,323,324]. From an experimental point of view, LS of this kind have been reported in Fabry-Perot cavities containing a LC cell [325,326] and in the Kerr slice with feedback [327,250] experiment (Fig. 46). The experimental observation agrees with the theoretical predictions for these systems [251,326].

60

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

Fig. 45. From Ref. [315]. Plot in the complex E plane of the stable localized structures observed in simulations of a model of a cavity "lled with purely absorptive medium. Also plotted is the coexistent hexagonal solution. The localized state solutions spiral out, and back to the plane wave "xed point (bottom left). The inset shows the spatial pro"le of the localized structure. Fig. 46. From Ref. [250]. Localized structures observed in a LCLV with di!ractive plus interferential feedback. (a) near "eld image, (b) intensity pro"le.

A di!erent kind of LS occurs when a system displays bistability between uniform states, without the presence of bifurcations to patterned states. This is the case of nonliner interferometers or lasers in which the role of di!raction is negligible, due e.g. to the choice of a self-impinging geometry or to the presence of strong di!usion [113,328}331]. A typical example of this class of structures arises in a class A laser with saturable absorber, for which the electric "eld obeys the equation [330]: RE cE (1!d) "aE#b E# # E. Rt 1#"E" 1#e"E"

(71)

This equation supports stable LS like the one shown in Fig. 47. One immediately notices that the lundulations at the periphery of the peak are absent in this case. The stability of these structures is attributed to nonvariational e!ects leading to a coupling of the spot amplitude and phase, resulting in a negative feedback for perturbations around these stationary solutions [330}332]. LS of this second kind have been observed in cavities with nonlinear gain and/or absorption [333}336], using dyes or photorefractives as active media. It has been shown that structures of various sizes can exist, depending both on the material di!usion and the level of pumping [336]. 5.2. Control of patterns Another research line which is currently attracting a lot of interest in the optical community is the possibility of controlling desired or regular patterns within space}time chaotic regimes.

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

61

Fig. 47. From Ref. [330]. Two dimensional localized structure numerically found in a model of laser with saturable absorber (Eq. (71)).

The idea of control of chaos was initially introduced in the case of non extended chaotic systems. It consists in stabilizing a desired temporal dynamics within a chaotic regime by means of tiny, judiciously chosen perturbations. Such perturbations can a!ect either a control parameter of the system, or a state variable of the system. Since a chaotic dynamics can be seen as a continuous motion in the phase space visiting closely an in"nite set of unstable periodic behaviors (the so-called unstable periodic orbits or UPO) [337], then control of chaos allows one to stabilize each UPO and then to use the same chaotic system to produce an in"nite set of periodic motions. The "rst method for the control of chaos is reported in Ref. [302], and it consists in a tiny perturbation of a control parameter each time the chaotic trajectory intersects the PoincareH section of the #ow. However, the time lapse for a natural passage of the #ow at the right point of the PoincareH section suitable for control may be very large. To minimize such a waiting time, a technique of targeting has been also introduced [338]. Another technique to constrain a nonlinear system x(t) to follow a prescribed goal dynamics u(t) is based upon the addition to the equation of motion dx/dt"F(x) of a term U(t) choosen in such a way that "x(t)!u(t)"P0 as tPR. Refs. [339,382,383] considers U(t)"(du/dt)!F(u(t)). In other papers the e!ects of periodic [340,341,384] and stochastic [342] perturbations are shown to produce outstanding changes in the dynamics, which however are quite di$cult to predict and in general are not goal oriented. A further method [303] has been proposed, based upon the continuous application of a delayed feedback term in order to force the dynamical evolution of the system toward the desired periodic dynamics whenever the system gets close to such a periodic behavior. On the other hand, many experimental systems have been studied with the aim of establishing control over chaos. Experimental chaos control and higher order periodic orbit stabilization have been successfully demonstrated for a thermal convection loop [343], a yttrium iron garnet oscillator [341], a diode resonator [344], an optical multimode chaotic solid-state laser [345], a Belouzov}Zabotinsky chemical reaction [346,385], a CO laser with modulation of losses [347]. In most cases stabiliz ation of UPOs was achieved by the technique of occasional proportional feedback (OPF) introduced in Ref. [344]. Even though a discrete hyperchaotic (with more than one positive Liapunov exponent) dynamics has been controlled [348] and targetted [349], extension of the above methods from a return map for a discrete system to a continuous dynamics for an extended system was still an open problem.

62

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

A recent advancement is constituted by an adaptive technique, initially introduced for chaos recognition [350,386], and then applied to chaos control [304], targeting [351], and synchronization [352], as well as to "ltering noise from a chaotic time series [353]. Such a technique has been shown to be e!ective in controlling defects and space-like structures emerging in delayed dynamical systems [354], that is, large dimensional systems displaying many aspects in common with space}time chaos. Space}time chaos control would indeed consist in slightly perturbing an extended system in order to stabilize some of the unstable patterns embedded in the turbulent regime. Some preliminary attempts have been done [355], by trying to extend the method of Ref. [303] to a space extended system. Another proposal [356] suggests to use a spatial modulation of the input pump "eld in an optical pattern forming system to stabilize a series of unstable homogeneous solutions, such as squares, hexagons and honeycombs. A further method has been proposed, based on a spatial "lter with delayed feedback, and able to stabilize and steer the weakly turbulent output of a spatially extended system [357]. The method was shown to be e!ective in the case of the generalized complex Swift}Hohenberg equation introduced in Refs. [21,358] as the generic model for pattern formation in the transverse section of semiconductor lasers. Many experimental implementations of the above methods are still in progress, however a general question is in order, namely, extension of the control techniques from discrete dynamical systems to patterns has been shown to be e!ective only when one forces an apriori known pattern, as e.g. by a Fourier mask; then, is the number of independent controller (probes and e!ectors) increasing with the size of the system (that is, with the aspect ratio) in a linear way or with a larger power? In what circumstances does this increase of complication pay for, that is, what is the trade o! for pattern control? 5.3. Patterns in atom optics The dual version of the Young double-slit experiment [359] consists of a supercooled atomic beam, with a de Broglie wavelength comparable with the optical wavelength, crossing the standing wave of a laser beam detuned by D"u !u with respect to the atomic transition frequency   u so that a local dephasing d (r) is induced on the t-function at each transverse coordinate  r depending on the local "eld intensity "E(r)" l d "(n(r)!1) . v

(72)

Here l is the size of the laser beam along the atomic path (interaction length), v is the speed of the atomic beam, m n(r)K1! ;(r)

k

(73)

is the pseudo refraction index for the propagating wavefunction, where the energy shift ;(r) is given in terms of the atomic-dipole moment k and the local intensity n" f (r)" (n being the average photon

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

63

Fig. 48. From Ref. [360]. Interaction between the wavefunction of an atomic beam and the laser standing wave propagating along x and con"ned within a narrow range l along z. The forward-propagating wavefunction undergoes a dephasing as it crosses the laser region. The transmitted wavefunction is further dephased along the free propagation path, and it seems to the incident one in the laser volume, giving rise to a phase to amplitude conversion which modi"es the laser intensity distribution and hence the induced dephasing on W. (a) Forward and backward beams in a counterpropagating con"guration; (b) feedback con"guration. Fig. 49. From Ref. [360]. Two level atom (transition frequency u ) interacting with a detuned laser "eld (optical  frequency u ). r,(x, y) are the transverse coordinate and z the longitudinal one, l is the longitudinal size of the  interaction region. The laser "eld is a standing wave made of two counterpropagatig waves perpendicular to z and with a pattern f (r) in the transverse direction.

number and f (r) the normalized "eld distribution) as n u " f (r)"k . (74) ;(r)"  D 2e <  A?T Here, starting from Eqs. (72)}(74), the novel mechanism to be considered is the following (see Fig. 48): 1. the dephased W-"eld is propagated along a closed loop (this is feasible, since total re#ection atomic mirrors are nowadays available) and hence di!raction provides a phase-to-amplitude conversion, whereby d (r) induces a local modi"cation of "W(r)". 2. The atomic probability density "W" modi"es the stationary "eld pattern f (r). Thus, beyond a threshold controlled by the density of the atomic beam and by the frequency position, the uniform transverse phase of the W wavefront spontaneously destabilizes toward a pattern. The symmetry of the pattern can be studied in terms of the symmetry changes induced on the laser "eld. In fact, the energy shift ;(r) appears as a spatially inhomogeneous optical potential [359] to be introduced into the Schroedinger equation for the wavefunction W(r, z, t). In the case of the forward beam (see Fig. 49) we can write the wavefunction as t"W(r, z, t)e SR\IX ,

(75)

64

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

where

k

u" 2m

(76)

is the dispersion relation for a free particle. It is important not to confuse the plane-wave frequency u, corresponding to the translational degrees of freedom of the atom, with the frequency u corre sponding to the internal degree of freedom (see Fig. 49 and Eq. (74)). The two level optical transition has been treated separately using the formalism of Ref. [359]. We consider the potential ; localized over a z interval so small that the main z dependence in Eq. (75) is included in the exponential factor. The evolution equation for the wavefunction is z

 i R t" t#;trect , R l 2m

(77)

where rect (z/l)"1 in the z interval of width l, where the laser is active and vanishes identically elsewhere. #R, where  is the transverse Laplacian and taking for W a slow Expanding "

X , NCPN z dependence, so that RW RW ;k , Rz Rz

(78)

Eq. (77) reduces to






R

i z R #v #i  W"! ;Wrect . Rz 2m ,

l Rt

(79)

Eq. (79) is formally equal to the equation describing transverse pattern formation in an optical beam propagating along z and with di!ractive e!ects in the transverse direction. Therefore, it should be expected the possibility of occurrence of all phenomena described for patterns in optical "elds. By inspection, Eq. (79) is a legitimate Schroedinger equation, without ad hoc approximations, as e.g., Hartree}Fock type, corresponding to consider ; proportional to "W". A detailed analysis of Eq. (79) is contained in Ref. [360]. Ref. [360] establishes the longitudinal and transverse coherence constraints which the atomic W has to satisfy in order to assure successful interference between the input and feedback wavefronts. These conditions are very restrictive for thermal atomic beams, where the de Broglie wavelength is comparable with the optical wavelength. They have become feasible nowadays that Bose}Einstein condensates have a coherence length of macroscopic size. Based on these considerations, renewed interest [361] has arised to the problem of longitudinal patterning in atomic beams, originally considered in Ref. [362].

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

65

Acknowledgements F.T.A. and P.L.R. acknowledge "nancial support from the EEC Contract no. FMRXCT960010. S.B. acknowledges "nancial support from the EEC Contract no. ERBFMBICT983466. Work partly supported also by the Coordinated Project `Nonlinear dynamics in optical systemsa of the Italian CNR, and by the 1998 Italy}Spain Integrated Action.

Appendix A. A reminder of nonlinear optics We recall some general notions of nonlinear optics, which can be found in many standard textbooks. We refer, e.g., to the recent book by Boyd [363]. A.1. Nonlinear susceptibility The dipole moment per unit volume, or polarization P(t), of a material system depends upon the strength of the applied optical "eld. Taking for simplicity scalar relations, the suitable generalization of the linear relation P(t)"e sE(t)  of conventional optics is (in SI units)

(80)

P(t)"e (sE(t)#sE(t)#sE(t)#2) . (81)  More generally, s would be a second-rank tensor, s a third-rank tensor, etc. As well known, s is a nondimensional quantity, of the order of unity for condensed matter; s (m/V) and s (m/V) are of the order respectively of the reciprocal of the atomic "eld E and  of its square. If we call a K0.5;10\ m the Bohr radius of the hydrogen atom, the order of  magnitude of E is E &(1/4pe ) (e/a)&5;10 V/m, e"1.6;10\C being the electron charge.     Therefore s s& &2;10\ m/V E  and similarly s s& &4;10\ (m/V) . E 

(82)

(83)

A.2. The two level approximation In the case of light resonant with an atomic transition between a ground level "g2 and an excited level "e2, the contributions from other energy levels are negligible at all orders of perturbation. The problem of the quantum atoms interacting with a classical electromagnetic "eld has a simple

66

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

solution which is an oscillatory energy exchange between the atoms and the "eld, at the Rabi frequency kE , X"

(84)

where k,1e"ed"g2 is the matrix element of the atomic dipole operator ed between upper and lower states. For allowed transitions, d is of the order of a Bohr radius, thus k&10\ C m and X&10 s\ for E"1 V/cm. This coherent Rabi oscillation is eventually cut o! by coupling the atoms to the environment, which acts as a thermal bath, providing two damping rates: the "rst, denoted c , due to decay of any coherent in-phase superposition of the atomic wavefunctions of , upper and lower states, due e.g. to elastic collisions, and the second, denoted c corresponding to , the decay of the energy stored in the atomic medium, due, e.g., to spontaneous emission. In dilute gases, the two rates are equal; in condensed matter the phase decay is much faster than the energy decay (elastic collisions occur more frequently than spontaneous emission processes). Numerically, for the D transition in Na vapours at the pressure of a few millibar, c &c &10 s\; for dye  , , molecules in liquid solution c &10 s\ whereas c &10 s\; for a partly forbidden transition, , , where the spontaneous lifetime is very long as for the Nd> ions in a glass matrix, c &10 s\ , whereas c &10 s\. , The e!ect of the damping rates is to quench the coherent Rabi oscillation. Rather than a whole sinusoidal waveform in time, whenever (c,c <X we have just a piece of straight line which is , equivalent to a "rst order perturbation theory, as if the atom was a harmonic oscillator rather than a two-level system. This will provide the standard linear relation (80), where now s can be evaluated by Fermi golden rule. We present as numerical example the calculation for the D line of  Na atoms at a density o of 10 m\ corresponding to a pressure of a few millibar. Fermi golden rule gives the transition rate w per unit time in terms of the Rabi frequency and of the resonant density in frequency (dn/du)"1/c of the "nal states (c"10 s\ being the width of the D line)  2po X (s\ m\) . (85) w" c As the energy exchange rate w u is equated to the standard electromagnetic rate uPE" ue sE, we obtain   2pok . (86) s"  c e  Numerically, we evaluate s&10, whence a dramatic enhancement with respect to the standard  s of a nonresonant dilute gas (s&10\). As the ratio X/c c becomes important, one should introduce the correction , , e sE e sE  P" "  , (87) 1#X/(c c ) 1#E/E  , , where E is the E value for which the Rabi frequency kE / is equal to the mean damping rate   (c c . , ,

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

67

Fig. 50. (a) Resonant two level interaction. Upper: absorption; middle: emission; lower: emission with "rst nonlinear correction. (b) Nonresonant two level interaction (scattering). On the left: energy levels with photon exchanges (wavy lines). "g2 represents the ground state, "e2 the excited state. On the right: matter propagators (solid lines) and light propagators (wavy lines). In case (b) the lifetime of the excited level is very short (of the order of the reciprocal of the o! set frequency /*=) so that the interaction diagram can be simpli"ed to the photon exchange.

An expansion of the denominator of Eq. (87) will provide a generalized P(E) where only odd terms (s, s, s, etc.) are present. Thus, in the resonant interaction E replaces what in general  would be E . For the sake of numerical evaluation, in the case of Na vapours  (88) E " (c c "10 V/m ,  I , , thus s&1/E "10\ (m/V), rather than 4;10\ as in the nonresonant case. This shows  again the dramatic enhancement of nonlinear e!ects in resonant media. As we move away from resonance (Fig. 50b) the strength of the interaction is reduced by the square ratio of the Rabi frequency to the o!-set frequency *=/ . Thus, the scattering process depicted in Fig. 50b corresponds to a s reduced with respect to the resonant value s by    X  . (89) s "s    *=/






The reduction factor corresponds to the passage from the one-vertex diagram of Fig. 50a (1st order perturbation theory) to the two-vertex diagram of Fig. 50b (2nd order perturbation theory). For the above example, take X"10 s\ and *=/ "10 s\; thus s &0.1.  

68

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

The nonlinear relation (87) is the basis of optical bistability. If we shine an external "eld E on  a Fabry-Perot cavity "lled with a medium, the transmission properties of the cavity will be provided by the boundary conditions combined with the medium refractive index. In the case of tuning between the input "eld frequency and the cavity resonance, the input intensity I yields an  in-cavity intensity I through the Airy's equation [363]  ¹I  , I "  (1!R(1!al))

(90)

where ¹ and R are the transmission and re#ection of the mirrors, l is the length of the intra-cavity absorbing medium and a its absorption per unit length. This equation can be simpli"ed introducing the so-called cooperation number c"Ral/(1!R). The equation can be rewritten as I 1  . I "  ¹ (1#c)

(91)

If we now account for the nonlinear polarization, the input-output relation E vs. E can have two  branches, depending on the s value, that is, on the density of resonant atoms "lling the cavity. Indeed, from Eq. (87) the absorption factor is intensity dependent as a  , a" 1#I /I  

(92)

where I J"E ". The relation (91) between I and I can be rewritten as    






c  I "¹I 1# ,   1#2I /I  

(93)

where c is the cooperation number corresponding to the unperturbed a . It is easy to show that   the output}input relation I vs. I is monotonic for c (5 (monostable) and two-branched    (bistable) for c '5.  A.3. The s and s nonlinear optics In the case of a two-level medium, we have seen that the nonresonant interaction can be represented by a simpli"ed one-vertex diagram, displaying only the light propagators. Indeed, the lifetime of the excited state /*= as given by Heisemberg principle is very short, and we can consider its role as that of a catalyst which eventually leaves the medium in the unperturbed "g2 state. In the case of s processes (Fig. 51a) we must conserve the energy of the incident and outgoing quanta. On the left we present a di!erence frequency generation or down-conversion process, on the right a sum frequency generation or up-conversion process. If the impinging and outgoing "elds are all light "elds, we speak of parametric processes. In particular for X"u/2 we have the degenerate parametric down-conversion (left) and the second harmonic generation (right). But X may be the quantum of a wave"eld, which is di!erent from

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

69

Fig. 51. Interaction diagrams for (a) s or three-quanta and (b) s or four-quanta processes. Table 2 From Ref. [363]. Typical strength and response time of optical nonlinearities depending on the physical mechanisms lying at the basis of the nonlinear response Mechanism

n  (cm/W)

s  (esu)

Response time (s)

Electronic polarization Molecular orientation Electrostriction Saturated atomic absorption Thermal e!ects Photorefractive e!ect

10\ 10\ 10\ 10\ 10\ (large)

10\ 10\ 10\ 10\ 10\ (large)

10\ 10\ 10\ 10\ 10\ (intensity-dependent)

The photorefractive e!ect often leads to a very stron nonlinear response. These responses usually cannot be described in terms of a s (or an n ) nonlinear susceptibility, because the nonlinear polarization does not depend on the applied "eld  strength in the same manner as the other mechanisms listed.

a light "eld. If it is a molecular vibration, we will speak of a Raman process, if it is a sound wave, we will speak of a Brillouin process. In the left case, the energy of an incident photon is splitted into a lower frequency photon plus a quantum of a material excitation (Stokes process). In the right case, the outgoing photon energy is increased by the material quantum (anti-Stokes process). The s nonlinearity (Fig. 51b) gives rise to an intensity dependent refractive index. Indeed, neglecting for the time being the tensor nature of the sQ, the induced polarization can be written as P "(s#s"E")E , e  corresponding to a refractive index

(94)

n"n #n I , (95)   where IJ"E" is the light intensity. Such a change in the refractive index is called optical Kerr e!ect. Some of the physical processes that can produce a Kerr e!ect are listed in Table 2, together with the characteristic time scale for the nonlinear response to develop.

70

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

A.4. The photorefractive (PR) ewect The photorefractive e!ect is the change in refractive index of a medium resulting from the optically induced redistribution of electrons and holes. Suppose we shine on a PR crystal two input optical "elds of equal frequency and amplitudes A
(96)

where E is an inner "eld, function of the material properties. The dephasing between the

impinging "elds and the ripple "eld E acts as a nonlinear transfer of energy from the pump to the  signal wave. The two intensities I and I are respectively ampli"ed and attenuated as   dI II "$C   , dz I #I  

(97)

where C depends upon the PR properties. As it appears from this skeleton description, the PR behavior is in general more complex than a Kerr medium, and it is approximated by a s process only in special domains of operations. We can generalize the two beam coupling to a four wave mixing, as shown in the con"gurations of the passive phase-conjugate mirrors (Fig. 53) and the double phase-conjugate mirrors (Fig. 54). This latter con"guration has the remarkable property that one of the output waves can be an ampli"ed phase-conjugate wave, even though the two input waves A and A are mutually   incoherent, so that no gratings are formed by their interference. The nonlinear interaction leads to the generation of the output waves A (phase-conjugate of A ) and A (phase-conjugate of A );     however A is incoherent with A and A with A , whence the creation of the grating shown in the     "gure.

Fig. 52. From Ref. [363]. Typical geometry for studying two-bam coupling in a photorefractive crystal. Fig. 53. From Ref. [363]. Geometry of the linear passive phase-conjugate mirror. Only the A wave is applied externally;  this wave excites the oscillation of the waves A and A , which act as pump waves for the four-wave mixing process that   generates the conjugate wave A . 

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

71

Fig. 54. From Ref. [363]. Geometry of the double phase-conjugate mirror. Waves A and A are applied externally and   need not to be phase-coherent. The generated wave A is the phase conjugate of A , and the generated wave A is the    phase conjugate of A . 

Appendix B. Rescaling the Maxwell}Bloch equations to account for detuning and a large aspect ratio For a single mode, the Maxwell}Bloch equations have been reported in Section 1.3. The stationary solutions are c k DM " , , g

EM "E





D !1 , DM

(98)

where E"c c /(4g) is called the saturation intensity and  , , g PM " DM EM . (99) c , Let us now replace the inversion D with N"D !D (o!set with respect to the pump value).  Furthermore, we introduce dimensionless variables E D D P e" , d" , r" , p" , (100) E DM DM DM E g/c   , and rescale the time with respect to the polarization decay time c . The equations then are , rewritten as [13,19] e "p(p!e), p "!p#(r!N)e, NQ "!bN#pe .

(101)

If furthermore we introduce a normalized detuning X"(u !u )/c between the cavity line   , u and the atomic line u , as well as a di!ractive operator, we obtain Eqs. (11), but now "eld and  polarization are complex quantities. Appendix C. Multiple scale analysis of the bifurcation problem for the non lasing solution of the Maxwell}Bloch equations Expanding all relevant variables of the Maxwell}Bloch equations in powers of a smallness parameter e, by writing X"eX , (E, P, N)"(E , P , N )#e(E , P , N )#e(E , P , N )#2          

(102)

72

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

and deriving the spatial scaling from the breadth of the band of the unstable modes above threshold X"(r!1)x,

>"(r!1)y ,

(103)

one can assume r"1#e. It results X"(ex, >"(ey. As for the time scales, Refs. [21,22] consider ¹ "et and ¹ "et as two di!erent secular time scales. Plugging all those expressions   into the Maxwell}Bloch equations, and identifying the coe$cients of each power of e, one gets, at order zero 0"!pE #pP ,  

P "(1!N )E , bN "(EHP #E PH) ,         

(104)

which gives E "P "N "0. At order one, the relations are    0"!pE #pP ,  

P "E ,  

bN "0 , 

(105)

implying N "0 and E "P "t, t being a complex variable. Finally, at order two, one gets the    following equations: RE  !ia E "!pE #pP ,    R¹ 

RP  #P #iX P "E ,     R¹ 

bN "(EHP #E PH) ,       (106)

which yield, in terms of the new variable t Rt pE !pP "! #ia t,   R¹ 

Rt !E #P "! !iX t,    R¹ 

bN ""t" . 

(107)

In order for the "rst two equations of (107) to be compatible, one must require a condition on t, which in fact corresponds to the dynamical equation sought for. At the actual order, this condition reads Rt "ia t!iX pt . (p#1)  R¹ 

(108)

Finally, choosing P "!(R/R¹ #iX )t"t and N ""t", the next      @ order (order three) yields RE RE  #  !ia E "!pE #pP ,    R¹ R¹   RP RP  #  #P #iX P "E #E !N E !N E ,          R¹ R¹   RN RN 1 # #bN " (EHP #EHP #E PH#E PH) ,        2   R¹ R¹  

(109)

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

73

which can be rewritten as Rt pE !pP "! ,   R¹  R 1 Rt (110) ! ! #iX P #t! "t"t , !E #P "!     R¹ b R¹   R ia 1 "t"# (t tH!tH t) . bN "! !  2(1#p) b R¹  The new solvability condition represents the behavior of t as a function of ¹ . If one uses the  relation










R R #iX P " #iX    R¹ R¹   one gets








Rt 1 ! !iX t " (X #a )t ,  R¹ (1#p)  

p p Rt "pt! "t"t! (X #a )t . (p#1) b (1#p)  R¹ 

(111)

(112)

The "nal equation for t is obtained by writing Rt/Rt"e Rt/R¹ #e Rt/R¹ . Reintroducing the   original variables x"X/(e, y">/(e, X"eX , r!1"e and rede"ning et"t, the "nal equa tion reads (p#1)

Rt p p "p(r!1)t! (X#a )t#ia t!iXpt! "t"t . Rt (1#p) b

(113)

Appendix D. Symmetries and normal form equations We analyze the normal form equations arising for symmetry requirements in the case of three transverse modes: a central one, with complex amplitude z , and two higher order ones (rotating  and counter rotating along an azimuthal coordinate h) with respective complex amplitudes z and  z and angular momenta $1. The cavity "eld can be expressed as  (114) E"f (r)(z e F#z e\ F)e SR#f (r)z e SR ,      where f and f are the space distributions of the modes. The optical frequencies u and u are in     general di!erent. The slow time dependence due to the dynamics is accounted for in the amplitudes z (t) (i"0, 1, 2). G The zero intensity situation is described by z "z "z "0, the central mode by z "z "0      and an azimuthal standing wave by z "0, z "z .    The cylindrical geometry of the cavity imposes the following constraints on mode amplitudes [96] H : (z , z , z )P(e Fz , e\ Fz , z ) ,       K : (z , z , z )P(z , z , z ) ,      

(115)

74

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

(H being the rotation operation, and K the re#ection operator around a privileged plane). If one considers the modes as born from Hopf bifurcations, then there is an additional time symmetry B : (z , z , z )P(e @z , e @z , e @z ) .      

(116)

The normal form for the nonlinear interaction among the three modes, assuming it to be invariant under the above symmetries is [97,98] (dots denote time derivatives) z "j z #(a("z "#"z ")#b"z ")z ,        z "j z #(c"z "#d"z "#e"z ")z #ez ,         z "j z #(d"z "#c"z "#e"z ")z #ez ,        

(117)

(j , j , a, b, c, d, e being complex coe$cients and e"o e PC being a symmetry breaking parameter).   C The parameter e is reminiscent of the breaking of the cylindrical symmetry induced by the pumping procedure which privileges a de"ned plane, thus breaking the rotational invariance. Putting z "o e PG, and operating a change in the variables (o "A cos(a/2), o "A sin(a/2) and   G G d"u !u ) Eqs. (117) are rewritten as   AQ "(jP #(cP!(1/2)(cP!dP)sin(a))A)A#(o sin(a)cos(d)cos(u )#ePo)A ,  C C  a"!(cP!dP)sin(2a)A#2o (cos(u )cos(d)cos(a)#sin(u )sin(d)) , C C C  a a sin(u !d)!tg sin(u #d) , dQ "!(cG!dG)Acos(a)#o cotan C C C 2 2 o "(jP #aPA#bPo)o ,    





u "jG #aGA#bGo ,    a a #dG sin u "jG #A cG cos   2 2















(118)

(119)

#eGo . 

The solutions of Eqs. (118) and (119) reproduce the experimental behavior for certain parameter values that it is possible to derive explicitly. First of all, it is to notice that Eqs. (118) constitute a closed four dimensional system. The laboratory experiment shows that any initial condition close to a central mode evolves in time toward a zero intensity state. Therefore, in the phase-space of the solutions of Eqs. (118) and (119), the zero con"guration will be stable in the o direction. A "rst condition for the correspond ence with the experiments is thus: jP (0 and bP'0. The fact that jP (0 is supported by the linear   stability analysis contained in Ref. [364]. Furthermore, if o "0, there are "xed points at a"p/2, d"0, p, de"ned by  A"!2(jP $o cos(u ))/(cP#dP) .  C C

(120)

These solutions are standing waves, and they are part of the `skeletona of pure modes in the dynamical model. Stability of these solutions in the (a, d) directions can be inspected looking at the eigenvalues of the Jacobian of the "rst three Eqs. (118). The experimentally observed dynamics again shows that

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

75

states close to zero are followed by a standing wave. This implies that the zero con"guration must be unstable in the o "o "o direction, and this occurs if !2jP G2o cos(u )'0. If the    C C eigenvalues of the Jacobian have negative real parts, the two standing waves will be stable in the a and d directions also. The time average of the intensity for these states can be calculated as 1EEH2, with E given by E"e P\B o(e F\B #e\ F\B ) f (r)e SR .

(121)

The pattern looks like a set of two bright spots either parallel (d"0) or perpendicular (d"p) to the privileged plane (the plane with respect to whom the symmetry is broken). The conditions for switching to the central mode is now derived. This can be obtained by imposing the standing wave to be unstable in the o and o directions, in order that initial states   close to them (with small components of o ) evolve towards a central mode con"guration.  In the subspace (o "o , d"0), the dynamics is ruled by   o "(jP #aPA#bPo)o ,     AQ "(jP #o cos(u )#(cP#dP)A#ePo)A .   C C 

(122)

In the plane (o , A), three "xed points are present (O, S= and C), de"ned by (0, 0),  (0, (!2(jP #o cos(u ))/(cP#dP)) and ((!jP /bP, 0) respectively. If the following conditions are  C C  satis"ed: aP (jP #o cos(u ))'0 , jP !2 C C  cP#dP  eP ! jP #(jP #o cos(u ))(0 ,  C C bP 

(123)

then S= is unstable in the o direction, whereas C is stable in the A direction. Moreover, since  bP'0, also C is unstable in the o direction. In order for both conditions to be satis"ed at the same  time, D,aPeP!bP(cP#dP)/2 must be negative. This implies that the (o , A) plane includes a fourth  "xed point P de"ned by






1 cP#dP jP !aP(jP #o cos(u )) , o"   C C  D 2 1 A" (!ePjP #bP (jP #o cos(u ))) .   C C D P undergoes a Hopf bifurcation if




o"! 



cP#dP A . 2bP

(124)

(125)

A periodic orbit which emerges from P and which gets close to the other "xed points, will su!er a critical slowing down that gives rise to the PA phenomenon among the O, S= and C con"gurations.

76

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

References [1] N.B. Abraham, W.J. Firth, J. Opt. Soc. Am. B 7 (1990) 951. [2] L.A. Lugiato (Ed.), Nonlinear Optical Structures, Patterns and Chaos (special issue), Chaos, Solitons and Fractals 4 (8/9) (1994). [3] M.C. Cross, P.C. Hohenberg, Rev. Mod. Phys. 65 (1993) 851. * [4] H.G. Schuster, Deterministic Chaos: An Introduction, Physik-Verlag, Frankfurt, 1984. [5] E.N. Lorenz, J. Atmos. Sci. 20 (1963) 130. * [6] H. Haken, Synergetics: An Introduction, Springer, Berlin, 1977. [7] F.T. Arecchi, G.L. Lippi, G.P. Puccioni, J.R. Tredicce, Opt. Commun. 51 (1984) 308. [8] F.T. Arecchi, R. Meucci, G.P. Puccioni, J.R. Tredicce, Phys. Rev. Lett. 49 (1982) 1217. [9] G.L. Lippi, J.R. Tredicce, N.B. Abraham, F.T. Arecchi, Opt. Commun. 53 (1985) 129. [10] F.T. Arecchi, W. Gadomski, R. Meucci, Phys. Rev. A 34 (1986) 1617. [11] C.O. Weiss, J. Brock, Phys. Rev. Lett. 57 (1986) 2804. [12] F.T. Arecchi, R.G. Harrison (Eds.), Instabilities and Chaos in Quantum Optics, Springer, Berlin, 1987. [13] Y.I. Khanin, Principles of Laser Dynamics, North-Holland, Amsterdam, 1995. [14] F.T. Arecchi, R. Bonifacio, IEEE J. Quantum Electron. 1 (1965) 1691. [15] R.Y. Chiao, E. Garmire, C.H. Townes, Phys. Rev. Lett. 13 (1964) 479. [16] S.A. Akhmanov, R.V. Khokhlov, A.P. Sukhorukov, in: F.T. Arecchi, E. Schulz-DuBois (Eds.), Laser Handbook, vol. 2, North-Holland, Amsterdam, 1972, p. 1151. [17] M.H. Anderson, J.H. Ensher, M.R. Mattews, C.E. Wieman, E.A. Cornell, Science 69 (1995) 198. [18] M.-O. Mewes, M.R. Andrews, D.M. Kurn, D.S. Durfee, C.G. Townsend, W. Ketterle, Phys. Rev. Lett. 78 (1997) 582. [19] L.A. Lugiato, C. Oldano, L.M. Narducci, J. Opt. Soc. Am. B 5 (1988) 879. [20] A.C. Newell, J.V. Moloney, Nonlinear Optics, Addison-Wesley, Redwood City, CA, 1992. [21] J. Lega, J.V. Moloney, A.C. Newell, Phys. Rev. Lett. 73 (1994) 2978. ** [22] J. Lega, J.V. Moloney, A.C. Newell, Physica D 83 (1995) 478. [23] J. Swift, P.C. Hohenberg, Phys. Rev. A 15 (1977) 319. ** [24] G. Huyet, M.C. Martinoni, J.R. Tredicce, S. Rica, Phys. Rev. Lett. 75 (1995) 4027. * [25] Y. Kuramoto, T. Tsuzuki, Prog. Theor. Phys. 55 (1976) 356. ** [26] K. Staliunas, Phys. Rev. A 48 (1993) 1573. [27] G.L. Oppo, G. D'Alessandro, W.J. Firth, Phys. Rev. A 44 (1991) 4712. [28] P. Coullet, L. Gil, F. Rocca, Opt. Commun. 73 (1989) 403. * [29] L. Gil, Phys. Rev. Lett. 70 (1993) 162. [30] P.K. Jakobsen, J. Lega, Q. Feng, M. Staley, J.V. Moloney, A.C. Newell, Phys. Rev. A 49 (1994) 4189. [31] J. Lega, P.K. Jakobsen, J.V. Moloney, A.C. Newell, Phys. Rev. A 49 (1994) 4201. [32] L.M. Narducci, J.R. Tredicce, L.A. Lugiato, N.B. Abraham, D.K. Bandy, Phys. Rev. A 33 (1986) 1842. [33] L.A. Lugiato, G.L. Oppo, M.A. Pernigo, J.R. Tredicce, L.M. Narducci, D.K. Bandy, Opt. Commun. 68 (1988) 63. [34] L.A. Lugiato, F. Prati, L.M. Narducci, P. Ru, J.R. Tredicce, D.K. Bandy, Phys. Rev. A 37 (1988) 3847. [35] L.A. Lugiato, G.L. Oppo, J.R. Tredicce, L.M. Narducci, M.A. Pernigo, J. Opt. Soc. Am. B 7 (1990) 1019. [36] H.G. Solari, R. Gilmore, J. Opt. Soc. Am. B 7 (1990) 828. * [37] L. Gil, K. Emilson, G.L. Oppo, Phys. Rev. A 45 (1992) 567. [38] P.K. Jakobsen, J.V. Moloney, A.C. Newell, R.E. Indik, Phys. Rev. A 45 (1992) 8129. [39] E.J. D'Angelo, E. Izaguirre, G.B. Mindlin, G. Huyet, L. Gil, J.R. Tredicce, Phys. Rev. Lett. 68 (1992) 3702. [40] G. D'Alessandro, G.L. Oppo, Opt. Commun. 88 (1992) 130. [41] E.J. D'Angelo, C. Green, J.R. Tredicce, N.B. Abraham, S. Balle, Z. Chen, G.L. Oppo, Physica D 61 (1992) 6. [42] R. Lopez-Ruiz, G.B. Mindlin, C. Perez-Garcia, J.R. Tredicce, Phys. Rev. A 49 (1994) 4916. [43] K. Staliunas, C.O. Weiss, Physica D 81 (1995) 79. [44] J. Martin-Regalado, S. Balle, M. San Miguel, Opt. Lett. 22 (1997) 460. [45] M. San Miguel, Phys. Rev. Lett. 75 (1995) 425. ** [46] S. Longhi, Phys. Rev. A 56 (1997) 2397. [47] J. Aranson, D. Hochheiser, J.V. Moloney, Phys. Rev. A 55 (1997) 3173.

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83 [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96]

77

N.L. Komarova, B.A. Malomed, J.V. Moloney, A.C. Newell, Phys. Rev. A 56 (1997) 803. D. Yu, W. Lu, R.G. Harrison, Phys. Rev. Lett. 77 (1996) 5051. G. Huyet, S. Rica, Physica D 96 (1996) 215. B.J. Gluckmann, P. Marcq, J. Bridger, J.P. Gollub, Phys. Rev. Lett. 71 (1993) 2034. L. Ning, Y. Hu, R. Hecke, G. Ahlers, Phys. Rev. Lett. 71 (1993) 2216. D. Dangoise, D. Hennequin, C. Lepers, E. Lovergneaux, P. Glorieux, Phys. Rev. A 46 (1992) 5955. K. Staliunas, G. Slekys, C.O. Weiss, Phys. Rev. Lett. 79 (1997) 2658. W.W. Rigrod, Appl. Phys. Lett. 2 (1963) 51. Chr. Tamm, Phys. Rev. A 38 (1988) R5960. W. Klische, C.O. Weiss, B. Wellegehausen, Phys. Rev. A 39 (1989) R919. J.R. Tredicce, E. Quel, C. Green, A. Ghazzawi, L.M. Narducci, M. Pernigo, L.A. Lugiato, Phys. Rev. Lett. 62 (1989) 1274. P. Chossat, M. Golubitsky, B.L. Key"tz, Dyn. Stab. Sysy. 1 (1987) 255. C. Green, G.B. Mindlin, E.J. D'Angelo, H.G. Solari, J.R. Tredicce, Phys. Rev. Lett. 65 (1990) 3124. G.K. Harkness, W.J. Firth, J.B. Geddes, J.V. Moloney, E.M. Wright, Phys. Rev. A 50 (1994) 4310. G. Huyet, C. Mathis, H. Grassi, J.R. Tredicce, N.B. Abraham, Opt. Commun. 111 (1994) 488. G. Huyet, J.R. Tredicce, Physica D 96 (1996) 209. E. Louvergneaux, D. Hennequin, D. Dangoise, P. Glorieux, Phys. Rev. A 53 (1996) 4435. N.V. Kravstov, E.G. Lariontsev, Quantum Electron. 24 (1994) 841. G.L. Lippi, J.R. Tredicce, N.B. Abraham, F.T. Arecchi, Opt. Commun. 53 (1985) 129. L.M. Ho!er, G.L. Lippi, N.B. Abraham, P. Mandel, Opt. Commun. 66 (1988) 219. C. Mathis, C. Taggiasco, L. Bertarelli, I. Boscolo, J.R. Tredicce, Physica D 96 (1996) 242. C. Taggiasco, R. Meucci, M. Cio"ni, N.B. Abraham, Opt. Commun. 133 (1997) 507. C. Serrat, A. Kul'miniskii, R. Vilaseca, R. Corbalan, Opt. Lett. 20 (1995) 1353. R. Corbalan, R. Vilaseca, M. Arjona, J. Pujol, E. Roldan, J.G. de Valcarel, Phys. Rev. A 48 (1993) 1483. S. Bielawski, D. Derozier, P. Glorieux, Phys. Rev. A 46 (1992) 2811. Y. Ozeki, C.L. Tang, Appl. Phys. Lett. 58 (1991) 2214. M. San Miguel, Q. Feng, J. Moloney, Phys. Rev. A 52 (1995) 1728. J. Martin-Regalado, M. San Miguel, N.B. Abraham, F. Prati, Opt. Lett. 21 (1996) 351. A. Sapia, P. Spano, B. Daino, Appl. Phys. Lett. 50 (1987) 57. C.J. Chang-Hasnaim, M. Orestein, A. Von Lehmen, L.T. Florez, J.P. Harbison, N.G. Sto!el, Appl. Phys. Lett. 57 (1990) 218. H. Adachihara, O. Hess, E. Abraham, P. Ru, J.V. Moloney, J. Opt. Soc. Am. B 10 (1993) 658. O. Hess, S.W. Koch, J.V. Moloney, IEEE J. Quantum Electron. 31 (1995) 35. O. Hess, T. Kuhn, Phys. Rev. A 54 (1996) 3347. O. Hess, T. Kuhn, Phys. Rev. A 54 (1996) 3360. L.M. Pecora, T.L. Carroll, Phys. Rev. Lett. 64 (1990) 821. H.G. Winful, L. Rahman, Phys. Rev. Lett. 65 (1990) 1575. S.S. Wang, H.G. Winful, Appl. Phys. Lett. 52 (1988) 1774. H.G. Winful, S.S. Wang, Appl. Phys. Lett. 52 (1988) 1894. H. Adachihara, O. Hess, R. Indik, J.V. Moloney, J. Opt. Soc. Am. B 10 (1993) 496. R. Li, T. Erneux, Phys. Rev. A 49 (1994) 1301. F.T. Arecchi, G. Giacomelli, P.L. Ramazza, S. Residori, Phys. Rev. Lett. 65 (1990) 2531. * K. Ikeda, K. Matsumoto, K. Otsuka, Progr. Theor. Phys. (Suppl.) 99 (1989) 295. ** K. Otsuka, Phys. Rev. Lett. 65 (1990) 329. K. Kaneko, Physica D 54 (1991) 5. I. Tsuda, Neural Networks 5 (1992) 313. P.C. Hohenberg, B.I. Shraiman, Physica D 37 (1989) 109. * F.T. Arecchi, S. Boccaletti, P.L. Ramazza, S. Residori, Phys. Rev. Lett. 70 (1993) 2277. * F.T. Arecchi, S. Boccaletti, G.B. Mindlin, C. Perez-Garcia, Phys. Rev. Lett. 69 (1992) 3723. M. Golubitsky, I. Stewart, D. Schae!er, Singularities and Groups in Bifurcation Theory, Appl. Math. Sciences, vol. 69, Springer, New York, 1983.

78

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

[97] G. Dangelmayr, E. Knobloch, Nonlinearity 4 (1991) 399. [98] E.J. D'Angelo, E. Izaguirre, G.B. Mindlin, G. Huyet, L. Gil, J.R. Tredicce, Phys. Rev. Lett. 68 (1992) 3702. [99] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Texts in Appl. Math. vol. 2, Springer, New York, 1990. [100] V.I. Bespalov, V.I. Talanov, JETP Lett. 3 (1966) 307. ** [101] O. Svelto, in: E. Wolf (Ed.), Progress in Optics, North-Holland, Amsterdam, 1974. [102] A.A. Zozulya, M. Sa!man, D.Z. Anderson, Phys. Rev. Lett. 73 (1994) 818. [103] A.A. Zozulya, D.Z. Anderson, A. Mamaev, M. Sa!man, Phys. Rev. A 57 (1998) 522. [104] E. Braun, L.P. Facheux, A. Libchaber, D.W. McLaughlin, D.J. Muraki, M.J. Shelley, Europhys. Lett. 23 (1993) 239. [105] A.C. Newell, Solitons in Mathematics and Physics, SIAM, Philadelphia, 1987. [106] S.A. Akhmanov, A.P. Sukhorukov, R.V. Kholkhov, Sov. Phys. JETP 23 (1966) 1025. * [107] Y.S. Kivshar, B. Luther-Davies, Phys. Rep. 298 (1998) 81. * [108] V.E. Zakharov, A.B. Shabat, JETP 34 (1972) 62. * [109] M.J. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur, Phys. Rev. Lett. 31 (1973) 125. [110] D.J. Kaup, A.C. Newell, Proc. R. Soc. London A 361 (1978) 413. [111] E.A. Kustnezov, A.M. Rubenchik, V.E. Zakharov, Phys. Rep. 142 (1986) 103. [112] V.E. Zakharov, A.B. Shabat, JETP 37 (1973) 823. [113] N.N. Rosanov, in: Progress in Optics, vol. 35, Elsevier, Amsterdam, 1996. * [114] A. Barthelemy, S. Maneuf, C. Froehly, Opt. Commun. 55 (1985) 201. [115] S. Maneuf, R. Desailly, C. Froehly, Opt. Commun. 65 (1988) 193. [116] S. Maneuf, F. Reynaud, Opt. Commun. 66 (1988) 325. [117] F. Reynaud, A. Barthelemy, Europhys. Lett. 12 (1990) 401. [118] J.S. Aitchison, A.M. Weiner, Y. Silberberg, M.K. Oliver, J.L. Jackel, D.E. Leaird, E.M. Vogel, P.W.E. Smith, Opt. Lett. 15 (1990) 471. [119] M.D. Iturbe Castillo, J.J. Sanchez-Mondragon, S.I. Stepanov, M.B. Klein, B.A. Wechsler, Opt. Commun. 118 (1995) 515. [120] P.L. Kelley, Phys. Rev. Lett. 15 (1965) 1005. [121] A.W. Snyder, D.J. Mitchell, L. Poladian, F. Ladouceur, Opt. Lett. 16 (1991) 21. [122] D.J. Mitchell, A.W. Snyder, J. Opt. Soc. Am. B 10 (1993) 1572. [123] J.H. Marburger, E. Dawes, Phys. Rev. Lett. 21 (1968) 556. [124] E. Dawes, J.H. Marburger, Phys. Rev. 179 (1969) 862. [125] J.E. Bjorkholm, A. Ashkin, Phys. Rev. Lett. 32 (1974) 129. [126] M.D. Iturbe Castillo, P.A. Marquez Aguilar, J.J. Sanchez-Mondragon, S. Stepanov, V. Vyslouk, Appl. Phys. Lett. 64 (1994) 408. [127] M. Segev, G.C. Valley, B. Crosignani, P. Di Porto, A. Yariv, Phys. Rev. Lett. 73 (1994) 3211. [128] D.N. Christodoulides, M.I. Carvalho, J. Opt. Soc. Am. B 12 (1995) 1628. [129] G. Duree, G. Salamo, M. Segeev, A. Yariv, B. Crosignani, P. Di Porto, E. Sharp, Opt. Lett. 19 (1994) 1195. [130] M. Mitchell, Z. Chemg, M. Shih, M. Segeev, Phys. Rev. Lett. 77 (1996) 490. [131] D.N. Christodoulides, T.H. Coskun, M. Mitchell, M. Segeev, Phys. Rev. Lett. 78 (1997) 646. [132] M. Shih, M. Segeev, G. Salamo, Phys. Rev. Lett. 78 (1997) 2551. [133] G.C. Valley, M. Segeev, B. Crosignani, A. Yariv, M.M. Fejer, M.C. Bashaw, Phys. Rev. A 50 (1994) R4457. [134] M. Segev, B. Crosignani, A. Yariv, B. Fisher, Phys. Rev. Lett. 68 (1992) 923. [135] G.C. Duree Jr., J.L. Schultz, G.J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E.J. Sharp, R.R. Neurgaonkar, Phys. Rev. Lett. 71 (1993) 533. [136] M. Segev, B. Crosignani, P. Di Porto, A. Yariv, G. Duree, G. Salamo, E. Sharp, Opt. Lett. 19 (1994) 1296. [137] W.E. Torruellas, Z. Wang, D.J. Hagan, E.W. VanStryland, G.I. Stegeman, L. Torner, C.R. Menyuk, Phys. Rev. Lett. 74 (1995) 5036. [138] L. Torner, C.R. Menyuk, W.E. Torruellas, G.I. Stegeman, Opt. Lett. 20 (1995) 13. [139] K. Hayata, M. Koshiba, Phys. Rev. Lett. 71 (1993) 3275. [140] M. Haelterman, A.P. Sheppard, Phys. Rev. E 49 (1994) 3389. [141] M. Haelterman, A.P. Sheppard, Phys. Rev. E 49 (1994) 4512.

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83 [142] [143] [144] [145] [146] [147] [148] [149] [150] [151] [152] [153] [154] [155] [156] [157] [158] [159] [160] [161] [162] [163] [164] [165] [166] [167] [168] [169] [170] [171] [172] [173] [174] [175] [176] [177] [178] [179] [180] [181] [182] [183] [184] [185] [186] [187] [188] [189] [190]

79

A.P. Sheppard, M. Haelterman, Opt. Lett. 19 (1994) 859. E.A. Kuznetsov, S.K. Turitsyn, JETP 67 (1988) 1583. C.T. Law, G.A. Swartzlander Jr., Opt. Lett. 18 (1993) 586. G.S. McDonald, K.S. Syed, W.J. Firth, Opt. Commun. 95 (1993) 281. * V. Tikhonenko, J. Christou, B. Luther-Davies, Y.S. Kivshar, Opt. Lett. 21 (1996) 1129. A.V. Mamaev, M. Sa!man, A.A. Zozulya, Phys. Rev. Lett. 76 (1996) 2262. A.V. Mamaev, M. Sa!man, D.Z. Anderson, A.A. Zozulya, Phys. Rev. A 54 (1996) 870. * D.R. Andersen, D.E. Hooton, G.A. Swartzlander Jr., A.E. Kaplan, Opt. Lett. 15 (1990) 783. G.A. Swartzlander Jr., D.R. Andersen, J.J. Regan, H. Yin, A.E. Kaplan, Phys. Rev. Lett. 66 (1991) 1583. G.R. Allan, S.R. Skinner, D.R. Andersen, A.L. Smirl, Opt. Lett. 16 (1991) 156. S.R. Skinner, G.R. Allan, D.R. Andersen, A.L. Smirl, IEEE J. Quantum Electron. 27 (1991) 2211. M. Taya, M.C. Bashaw, M.M. Fejer, M. Segev, G.C. Valley, Phys. Rev. A 52 (1995) 3095. V. Tikhonenko, J. Christou, V. Luther-Daves, J. Opt. Soc. Am. B 12 (1995) 2046. V. Tikhonenko, J. Christou, B. Luther-Daves, Phys. Rev. Lett. 76 (1996) 2698. A.V. Mamaev, M. Sa!man, A.A. Zozulya, Phys. Rev. Lett. 77 (1996) 4544. A.W. Snyder, L. Poladian, D.J. Mitchell, Opt. Lett. 17 (1992) 789. G.S. McDonald, K.S. Syed, W.J. Firth, Opt. Commun. 94 (1992) 469. G.A. Swartzlander Jr., C.T. Law, Phys. Rev. Lett. 69 (1992) 2503. B. Luther-Daves, R. Powles, V. Tikhonenko, Opt. Lett. 19 (1994) 1816. V. Tikhonenko, N.N. Akhmediev, Opt. Commun. 126 (1996) 108. G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, A. Yariv, Phys. Rev. Lett. 74 (1995) 1978. Z. Chen, M. Segev, D.W. Wilson, R.E. Muller, P.D. Maker, Phys. Rev. Lett. 78 (1997) 2948. G. Grynberg, Opt. Commun. 66 (1988) 321. G. Grynberg, J. Paye, Europhys. Lett. 8 (1989) 29. *** W.J. Firth, C. Pare, Opt. Lett. 13 (1988) 1990. W.J. Firth, A. Fitzgerald, C. Pare, J. Opt. Soc. Am. B 7 (1990) 1087. ** G.G. Luther, C.J. McKinstrie, J. Opt. Soc. Am. B 7 (1990) 1125. A. Yariv, D.M. Pepper, Opt. Lett. 1 (1977) 16. G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J.R.R. Leite, D. Bloch, S. Le Boiteux, M. Ducloy, Opt. Commun. 67 (1988) 363. * J. Pender, L. Hesselink, J. Opt. Soc. Am. B 7 (1990) 1361. A. Petrossian, M. Pinard, A. Maitre, J.-Y. Courtois, G. Grynberg, Europhys. Lett. 18 (1992) 689. A.D. Novikov, V.V. Obukhovskii, S.G. Odulov, B.I. Sturman, JETP Lett. 44 (1987) 538. T. Honda, Opt. Lett. 18 (1993) 598. T. Honda, H. Matsumoto, M. Sedlatschek, C. Denz, T. Tschudi, Opt. Commun. 133 (1998) 293. N.V. Kuktharev, T. Kukhtareva, H.J. Caul"eld, P.P. Banerjee, H.-L. Yu, L. Hesselink, Opt. Eng. 34 (1995) 2261. T. Honda, Opt. Lett. 20 (1995) 851. T. Honda, P.P. Banerjee, Opt. Lett. 21 (1996) 779. P.P. Banerjee, H.-L. Yu, D.A. Gregory, N. Kukhtarev, H.J. Caul"eld, Opt. Lett. 20 (1995) 10. A.V. Mamaev, M. Sa!man, Europhys. Lett. 34 (1996) 669. G. Agrawal, J. Opt. Soc. Am. B 7 (1990) 1072. J.Y. Courtois, G. Grynberg, Opt. Commun. 87 (1992) 186. R. Chang, W.J. Firth, R. Indik, J.V. Moloney, E.M. Wright, Opt. Commun. 88 (1992) 167. * J.B. Geddes, R.A. Indik, J.V. Moloney, W.J. Firth, Phys. Rev. A 50 (1994) 3471. M. Sa!man, D. Montgomery, A.A. Zozulya, K. Kuroda, D.Z. Anderson, Phys. Rev. A 48 (1993) 3209. M. Sa!man, A.A. Zozulya, D.Z. Anderson, J. Opt. Soc. Am. B 11 (1994) 1409. B. Sturman, A. Chernykh, J. Opt. Soc. Am. B 12 (1995) 1384. H.M. Gibbs, P. Mandel, N. Peyghambarian, S.D. Smith (Eds.), Optical Bistability III, Springer, Berlin, 1986. L.A. Lugiato, L. Lefever, Phys. Rev. Lett. 58 (1987 2209. ** L.A. Lugiato, C. Oldano, Phys. Rev. A 37 (1988) 3896. **

80 [191] [192] [193] [194] [195] [196] [197] [198] [199] [200] [201] [202] [203] [204] [205] [206] [207] [208] [209] [210] [211] [212] [213] [214] [215] [216] [217] [218] [219] [220] [221] [222] [223] [224] [225] [226] [227] [228] [229] [230] [231] [232] [233] [234] [235] [236] [237] [238] [239]

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83 L.A. Lugiato, W. Kaige, N.B. Abraham, Phys. Rev. A 49 (1994) 2049. M. Aguado, R.F. Rodriguez, M. San Miguel, Phys. Rev. A 39 (1989) 5686. W.J. Firth, A.J. Scroggie, G.S. McDonald, L.A. Lugiato, Phys. Rev. A 46 (1992) R3609. * A.J. Scroggie, W.J. Firth, G.S. McDonald, M. Tlidi, R. Lefever, L.A. Lugiato, Chaos Solitons and Fractals 4 (1994) 1323. W.J. Firth, A.J. Scroggie, Europhys. Lett. 26 (1994) 521. * J.V. Moloney, H.M. Gibbs, Phys. Rev. Lett. 48 (1982) 1607. ** D.W. McLaughlin, J.V. Moloney, A.C. Newell, Phys. Rev. Lett. 51 (1983) 75. ** D.W. McLaughlin, J.V. Moloney, A.C. Newell, Phys. Rev. Lett. 54 (1985) 681. ** A. Ouarzeddini, H. Adachihara, J.V. Moloney, Phys. Rev. A 38 (1988) 2005. J.V. Moloney, H. Adachihara, R. Indik, C. Lizarraga, R. Northcutt, D.W. McLaughlin, A.C. Newell, J. Opt. Soc. Am. B 7 (1990) 1039. P. Mandel, M. Georgiou, T. Erneux, Phys. Rev. A 47 (1993) 4277. ** M. Tlidi, M. Georgiou, P. Mandel, Phys. Rev. A 48 (1993) 4605. A.S. Patrascu, C. Nath, M. Le Berre, E. Ressayre, A. Tallet, Opt. Commun. 91 (1992) 433. M. Le Berre, A.S. Patrascu, E. Ressayre, A. Tallet, Opt. Commun. 123 (1996) 810. * M. Le Berre, D. Leduc, E. Ressayre, A. Tallet, Phys. Rev. A 54 (1992) 3428. J. Nalik, L.M. Ho!er, G.L. Lippi, Ch. Vorgerd, W. Lange, Phys. Rev. A 45 (1992) R4237. P. La Penna, G. Giusfredi, Phys. Rev. A 48 (1993) 2299. G.L. Lippi, T. Ackermann, L.M. Ho!er, A. Gahl, W. Lange, Phys. Rev. A 48 (1993) R4043. G.L. Lippi, T. Ackemann, L.M. Ho!er, W. Lange, Chaos Solitons and Fractals 4 (1994) 1409. G.L. Lippi, T. Ackemann, L.M. Ho!er, W. Lange, Chaos Solitons and Fractals 4 (1994) 1433. A. Gahl, T. Ackermann, W. Grosse-Nobis, G.L. Lippi, L.M. Ho!er, M. Moller, W. Lange, Phys. Rev. A 57 (1998) 4026. * T. Ackermann, T. Scholz, Ch. Vorgerd, J. Nalik, L.M. Ho!er, G.L. Lippi, Opt. Commun. 147 (1998) 411. G. Giusfredi, J.F. Valley, R. Pon, G. Khitrova, H.M. Gibbs, J. Opt. Soc. Am. B 5 (1988) 1181. S. Akhmanov, M.A. Vorontsov, V.Y. Ivanov, JETP Lett. 47 (1988) 707. *** W.J. Firth, J. Mod. Opt. 37 (1990) 151. G. D'Alessandro, W.J. Firth, Phys. Rev. Lett. 66 (1991) 2597. *** R. Mcdonald, H.J. Eichler, Opt. Commun. 89 (1992) 289. * R. Mcdonald, H. Danlewski, Opt. Commun. 113 (1994) 111. R. Mcdonald, H. Danlewski, Opt. Lett. 20 (1995) 441. E. Ciaramella, M. Tamburrini, E. Santamato, Appl. Phys. Lett. 63 (1993) 1604. M. Tamburrini, M. Bonavita, S. Wabnitz, E. Santamato, Opt. Lett. 18 (1993) 855. E. Ciaramella, M. Tamburrini, E. Santamato, Phys. Rev. A 50 (1994) R10. F. Castaldo, D. Paparo, E. Santamato, Opt. Commun. 143 (1997) 57. M.A. Vorontsov, M.E. Kirakosyan, A.V. Larichev, Sov. J. Quantum Electron. 21 (1991) 105. S.A. Akhmanov, M.A. Vorontsov, A.V. Larichev, Sov. J. Quantum Electron. 20 (1990) 325. S.A. Akhmanov, M.A. Vorontsov, V.Yu. Ivanov, A.V. Larichev, N.I. Zheleznykh, J. Opt. Soc. Am. B 9 (1992) 78. * M.A. Vorontsov, Quantum Electron. 23 (1993) 269. M.A. Vorontsov, A.Yu. Karpov, Opt. Lett. 20 (1995) 2466. M.A. Vorontsov, A.Yu. Karpov, J. Opt. Soc. Am. B 14 (1997) 34. A.V. Larichev, I.P. Nikolaev, A.L. Chulichkov, Opt. Lett. 21 (1996) 1180. M.A. Vorontsov, M. Zhao, D.Z. Anderson, Opt. Commun. 134 (1997) 191. M.A. Vorontsov, B.A. Samson, Phys. Rev. A 57 (1998) 3040. * M.A. Vorontsov, A.Yu. Karpov, J. Mod. Opt. 44 (1997) 439. E. Pampaloni, S. Residori, F.T. Arecchi, Europhys. Lett. 24 (1993) 647. * G. D'Alessandro, E. Pampaloni, P.L. Ramazza, S. Residori, F.T. Arecchi, Phys. Rev. A 52 (1995) 4176. * E. Pampaloni, P.L. Ramazza, S. Residori, F.T. Arecchi, Europhys. Lett. 25 (1994) 587. E. Pampaloni, S. Residori, S. Soria, F.T. Arecchi, Phys. Rev. Lett. 78 (1997) 1042. P.L. Ramazza, E. Pampaloni, S. Residori, F.T. Arecchi, Physica D 96 (1996) 259. A.V. Larichev, F.T. Arecchi, Opt. Commun. 113 (1994) 53.

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83 [240] [241] [242] [243] [244] [245] [246] [247] [248] [249] [250] [251] [252] [253] [254] [255] [256] [257] [258] [259] [260] [261] [262] [263] [264] [265] [266] [267] [268] [269] [270] [271] [272] [273] [274] [275] [276] [277] [278] [279] [280] [281] [282] [283] [284] [285] [286] [287] [288] [289] [290]

81

F.T. Arecchi, A.V. Larichev, P.L. Ramazza, S. Residori, J.C. Ricklin, M.A. Vorontsov, Opt. Commun. 117 (1995) 492. P.L. Ramazza, S. Residori, E. Pampaloni, A.V. Larichev, Phys. Rev. A 53 (1996) 400. E. Pampaloni, P.L. Ramazza, S. Residori, F.T. Arecchi, Phys. Rev. Lett. 74 (1995) 258. * S. Residori, P.L. Ramazza, E. Pampaloni, S. Boccaletti, F.T. Arecchi, Phys. Rev. Lett. 76 (1996) 1063. * P.L. Ramazza, P. Bigazzi, E. Pampaloni, S. Residori, F.T. Arecchi, Phys. Rev. E 52 (1995) 5524. P.L. Ramazza, S. Boccaletti, A. Giaquinta, E. Pampaloni, S. Soria, F.T. Arecchi, Phys. Rev. A 54 (1996) 3472. * P.L. Ramazza, S. Boccaletti, F.T. Arecchi, Opt. Commun. 136 (1997) 267. B. Thuring, R. Neubecker, T. Tschudi, Opt. Commun. 102 (1993) 111. R. Neubecker, B. Thuering, T. Tschudi, Chaos Solitons and Fractals 4 (1994) 1307. B. Thuring, A. Schreiber, M. Kreuzer, T. Tschudi, Physica D 96 (1996) 282. A. Schreiber, B. Thuring, M. Kreuzer, T. Tschudi, Opt. Commun. 136 (1997) 415. * R. Neubecker, G.L. Oppo, B. Thuering, T. Tschudi, Phys. Rev. A 52 (1995) 791. * G. D'Alessandro, W.J. Firth, Phys. Rev. A 46 (1992) 537. * F. Papo!, G. D'Alessandro, G.L. Oppo, W.J. Firth, Phys. Rev. A 48 (1993) 634. * T. Honda, H. Matsumoto, Opt. Lett. 20 (1995) 1755. T. Honda, H. Matsumoto, M. Sedlatschek, C. Denz, T. Tschudi, Opt. Commun. 133 (1997) 293. J. Gluckstad, M. Sa!man, Opt. Lett. 20 (1995) 531. M. Le Berre, E. Ressayre, A. Tallet, Phys. Rev. A 43 (1991) 6345. M. Le Berre, E. Ressayre, A. Tallet, J.-J. Zondy, J. Opt. Soc. Am. B 7 (1990) 1346. M. Le Berre, D. Leduc, E. Ressayre, A. Tallet, A. Maitre, Opt. Commun. 118 (1995) 447. A.J. Scroggie, W.J. Firth, Phys. Rev. A 53 (1996) 2752. G. Grynberg, A. Maitre, A. Petrossian, Phys. Rev. Lett. 72 (1994) 2379. * G. Grynberg, Opt. Commun. 109 (1994) 483. A. Petrossian, L. Dambly, G. Grynberg, Europhys. Lett. 29 (1995) 209. T. Ackemann, W. Lange, Phys. Rev. A 50 (1994) R4468. T. Ackemann, Yu.A. Logvin, A. Heuer, W. Lange, Phys. Rev. Lett. 75 (1995) 3450. * W. Lange, Yu.A. Logvin, T. Ackemann, Physica D 96 (1996) 230. Yu.A. Logvin, T. Ackemann, W. Lange, Europhys. Lett. 38 (1997) 583. Yu.A. Logvin, T. Ackemann, W. Lange, Phys. Rev. A 55 (1997) 4538. A. Aumann, E. Buthe, Yu.A. Logvin, T. Ackemann, W. Lange, Phys. Rev. A 56 (1997) R1709. B.Y. Rubinstein, L.M. Pismen, Phys. Rev. A 56 (1997) 4264. M. Haeltherman, G. Vitrant, J. Opt. Soc. Am. B 9 (1992) 1563. * M. Berry, in: R. Balian et al. (Eds.), Physics of Defects, North-Holland, Amsterdam, 1981, pp. 456}543. * N.B. Baranova, B.Ya. Zel'dovich, Sov. Phys. JETP 53 (1981) 925. P. Coullet, C. Elphick, L. Gil, J. Lega, Phys. Rev. Lett. 59 (1987) 884. S. Ciliberto, P. Coullet, J. Lega, E. Pampaloni, C. Perez-Garcia, Phys. Rev. Lett. 65 (1990) 2370. * G. Ahlers, R.P. Behringer, Prog. Theor. Phys. Suppl. 64 (1979) 186. R. Ribotta, A. Joets, in: J.E. Wesfried, S. Zaleski (Eds.), Cellular Structures and Instabilities, Springer, Berlin, 1984. J.P. Gollub, R. Ramshankar, in: S. Orszag, L. Sirovich (Eds.), New Perspectives in Turbulence, Springer, Berlin, 1990. K. Kawasaki, Prog. Theor. Phys. Suppl. 79 (1984) 161. G. Indebetouw, D. Korwan, J. Mod. Opt. 41 (1994) 941. A.V. Ilyenkov, A.I. Khiznyak, L.V. Kremiskaya, M.S. Soskin, M.V. Vasnetsov, Appl. Phys. B 62 (1996) 465. A.V. Mamaev, M. Sa!man, A.A. Zozulya, Phys. Rev. A 56 (1997) R1713. G.A. Swartzlander Jr., C.T. Law, Phys. Rev. Lett. 69 (1992) 2503. R. Neubecker, M. Kreuzer, T. Tschudi, Opt. Commun. 96 (1993) 117. V. Tikhonenko, J. Christou, B. Luther-Daves, J. Opt. Soc. Am. B 22 (1995) 2046. V. Tikhonenko, J. Christou, B. Luther-Daves, Phys. Rev. Lett. 76 (1996) 2698. G.S. McDonald, K.S. Syed, W.J. Firth, Opt. Commun. 94 (1992) 469. * J. Christou, V. Tikhonenko, Y.S. Kivshar, B. Luther-Daves, Opt. Lett. 21 (1996) 1649. D. Rozas, C.T. Law, G.A. Swartzlander Jr., J. Opt. Soc. Am. B 14 (1997) 3054. M.S. Soskin, V.N. Gorshov, M.V. Vasnetsov, J.T. Malos, N.R. Heckenberg, Phys. Rev. A 56 (1997) 4064. *

82 [291] [292] [293] [294] [295] [296] [297] [298] [299] [300] [301] [302] [303] [304] [305] [306] [307] [308] [309] [310] [311] [312] [313] [314] [315] [316] [317] [318] [319] [320] [321] [322] [323] [324] [325] [326] [327] [328] [329] [330] [331] [332] [333] [334] [335] [336] [337] [338] [339] [340] [341]

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83 H. He, M.E.J. Friese, N.R. Heckeneberg, H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75 (1995) 826. K.T. Gahagan, G.A. Swartzlander Jr., Opt. Lett. 21 (1996) 827. M. Vaupel, K. Staliunas, C.O. Weiss, Phys. Rev. A 54 (1996) 880. D. Rozas, Z.S. Sacks, G.A. Swartzlander Jr., Phys. Rev. Lett. 79 (1997) 3399. F.T. Arecchi, G. Giacomelli, P.L. Ramazza, S. Residori, Phys. Rev. Lett. 67 (1991) 3749. * M. Takeda, M. Ina, S. Kobayashi, J. Opt. Soc. Am. 72 (1982) 156. A.M. Turing, Phil. Trans. R. Soc. London B 237 (1952) 37. ** A.B. Ezerskii, P.I. Korotin, M.I. Rabinovich, JETP Lett. 41 (1985) 157. S. Fauve, K. Kumar, C. Laroche, D. Beysens, Y. Garrabos, Phys. Rev. Lett. 68 (1992) 3160. P. Manneville, Dissipative Structures and Weak Turbulence, Academic Press, San Diego, CA, 1990. * H. Kogelnik, in: A.K. Levine (Ed.), Lasers: A Series of Advances, Dekker, New York, 1966. E. Ott, C. Grebogi, J.A. Yorke, Phys. Rev. Lett. 64 1990 1196. * K. Pyragas, Phys. Lett. A 170 (1992) 421. S. Boccaletti, F.T. Arecchi, Europhys. Lett. 31 (1995) 127. T. Pfau, C. Kurtsiefer, C.S. Adams, M. Sigel, J. Mlynek, Phys. Rev. Lett. 71 (1993) 3427 and references therein. N.N. Rosanov, V.E. Semenov, Opt. Spectrosc. 48 (1980) 59. N.N. Rosanov, G.V. Khodova, Opt. Spectrosc. 65 (1988) 449. N.N. Rosanov, G.V. Khodova, Opt. Spectrosc. 65 (1988) 828. N.N. Rosanov, G.V. Khodova, J. Opt. Soc. Am. 7 (1990) 1057. * K. Tai, J.V. Moloney, H.M. Gibbs, Opt. Lett. 7 (1982) 429. J.V. Moloney, IEEE J. Quantum Electron. QE21 (1985) 1393. W.J. Firth, I. Galbraith, IEEE J. Quantum Electron. QE21 (1985) 1399. H. Haug, S.W. Koch, IEEE J. Quantum Electron. QE21 (1985) 1385. G.S. McDonald, W.J. Firth, J. Opt. Soc. Am. 7 (1990) 1328. M. Tlidi, P. Mandel, R. Lefever, Phys. Rev. Lett. 73 (1994) 640. * W.J. Firth, A. Lord, J. Mod. Opt. 43 (1996) 1071. W.J. Firth, A.J. Scroggie, Phys. Rev. Lett. 76 (1996) 1623. * M. Brambilla, L.A. Lugiato, M. Stefani, Europhys. Lett. 34 (1996) 109. M. Le Berre, A.S. Patrascu, E. Ressayre, A. Tallet, Phys. Rev. A 56 (1997) 3150. M. Brambilla, L.A. Lugiato, F. Prati, L. Spinelli, W.J. Firth, Phys. Rev. Lett. 79 (1997) 2042. D. Michaelis, U. Peschel, F. Lederer, Phys. Rev. A 56 (1997) R3366. C. Etrich, U. Peschel, F. Lederer, Phys. Rev. Lett. 79 (1997) 2454. P. Coullet, C. Elpick, D. Repaux, Phys. Rev. Lett. 58 (1987) 431. N.N. Rosanov, G.V. Khodova, Opt. Spectrosc. 72 (1992) 788. R. Neubecker, T. Tschudi, J. Mod. Opt. 41 (1994) 885. M. Kreuzer, H. Gottschling, R. Neubecker, T. Tschudi, Appl. Phys. B 59 (1994) 581. A.N. Rakhmanov, Opt. Spectrosc. 74 (1993) 701. N.N. Rosanov, A.V. Fedorov, S.V. Fedorov, G.V. Khodova, JETP 80 (1995) 199. N.N. Rosanov, A.V. Fedorov, S.V. Fedorov, G.V. Khodova, Physica D 96 (1996) 272. H.R. Brand, R.J. Deissler, Physica A 204 (1994) 87. H.R. Brand, R.J. Deissler, Physica A 216 (1995) 288. D. Thual, S. Fauve, J de Physique (Paris) 49 (1988) 1829. * B. Fischer, O. Werner, M. Horowitz, A. Lewis, Appl. Phys. Lett. 58 (1991) 2729. M. Sa!man, D. Montgomery, D.Z. Anderson, Opt. Lett. 19 (1994) 518. V.B. Taranenko, K. Staliunas, C.O. Weiss, Phys. Rev. A 56 (1997) 1582. K. Staliunas, V.B. Taranenko, G. Slekysm, R. Viselga, C.O. Weiss, Phys. Rev. A 57 (1998) 599. D. Auerbach, P. Cvitanovic, J.-P. Eckmann, G. Gunaratne, I. Procaccia, Phys. Rev. Lett. 58 (1987) 2387. T. Shinbrot, C. Grebogi, E. Ott, J.A. Yorke, Nature 363 (1993) 411. B.B. Plapp, A.W. Huebler, Phys. Rev. Lett. 65 (1990) 2302. R. Lima, M. Pettini, Phys. Rev. A 41 (1990) 726. A. Azevedo, S.M. Rezende, Phys. Rev. Lett. 66 (1991) 1342.

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83 [342] [343] [344] [345] [346] [347] [348] [349] [350] [351] [352] [353] [354] [355] [356] [357] [358] [359] [360] [361] [362] [363] [364] [365] [366] [367] [368] [369] [370] [371] [372] [373] [374] [375] [376] [377] [378] [379] [380] [381] [382] [383] [384] [385] [386] [387] [388] [389]

83

S. Fahy, D.R. Hamann, Phys. Rev. Lett. 69 (1992) 761. J. Singer, Y.-Z. Wang, H.H. Bau, Phys. Rev. Lett. 66 (1991) 1123. E.R. Hunt, Phys. Rev. Lett. 67 (1991) 1953. R. Roy, T.W. Murphy Jr., T.D. Maier, Z. Gills, E.R. Hunt, Phys. Rev. Lett. 68 (1992) 1259. * B. Peng, V. Petrov, K. Showalter, J. Phys. Chem. 95 (1991) 4957. R. Meucci, W. Gadomski, M. Cio"ni, F.T. Arecchi, Phys. Rev. E 49 (1994) R2528. F.J. Romeiras, C. Grebogi, E. Ott, W.P. Dayawansa, Physica D 58 (1992) 165. E.J. Kostelich, C. Grebogi, E. Ott, J.A. Yorke, Phys. Rev. E 47 (1993) 305. F.T. Arecchi, G. Basti, S. Boccaletti, A.L. Perrone, Europhys. Lett. 26 (1994) 327. S. Boccaletti, A. Farini, E.J. Kostelich, F.T. Arecchi, Phys. Rev. E 55 (1997) R4845. S. Boccaletti, A. Farini, F.T. Arecchi, Phys. Rev. E 55 (1997) 4979. S. Boccaletti, A. Giaquinta, F.T. Arecchi, Phys. Rev. E 55 (1997) 5393. S. Boccaletti, D. Maza, H. Mancini, R. Genesio, F.T. Arecchi, Phys. Rev. Lett. 79 (1997) 5246. * W. Lu, D. Yu, R.G. Harrison, Phys. Rev. Lett. 76 (1996) 3316. R. Martin, A.J. Scroggie, G.L. Oppo, W.J. Firth, Phys. Rev. Lett. 77 (1996) 4007. D. Hochheiser, J.V. Moloney, J. Lega, Phys. Rev. A 55 (1997) R4011. J. Lega, J.V. Moloney, A.C. Newell, Physica D 83 (1995) 478. S. Haroche, M. Brun, J.M. Raimond, Physica D 51 (1991) 355. F.T. Arecchi, Il Nuovo Cimento B 110 (1995) 625. M. Sa!man, Filamentation instability of a matter wave, Presented at the 1st EuroConf. on Trends in Optical Nonlinear Dynamics: Physical Problems and Applications, Pueblo Alcantilado, Alicante, Spain, May 21}23, 1998. W. Zhang, D.F. Wells, B.C. Sanders, Phys. Rev. Lett. 72 (1994) 60. R.W. Boyd, Nonlinear Optics, Academic Press, San Diego, 1992. F.T. Arecchi, S. Boccaletti, G.P. Puccioni, P.L. Ramazza, S. Residori, Chaos 4 (1994) 491. H. Haken, Advanced Synergetics, Springer, Berlin, 1983. G.I. Shivasinsky, Acta Astronaut. 4 (1977) 1177. A.V. Buryak, Y.S. Kivshar, Phys. Lett. A 197 (1995) 407. A.V. Buryak, Y.S. Kivshar, V.V. Steblina, Phys. Rev. A 52 (1995) 1670. R.A. Fuerst, D.-M. Baboiu, B. Lawrence, W.E. Torruellas, G.I. Stegeman, S. Trillo, S. Wabnitz, Phys. Rev. Lett. 78 (1997) 2756. R. Schiek, Y. Baek, G.I. Stegeman, Phys. Rev. E 53 (1996) 1138. L. Torner, D. Mazilu, D. Mihalache, Phys. Rev. Lett. 77 (1996) 2455. D. Mihalache, D. Mazilu, L.-C. Crasovan, L. Torner, Opt. Commun. 137 (1997) 113. T. Peschel, U. Peschel, F. Lederer, B.A. Malomed, Phys. Rev. E 55 (1997) 4730. W.J. Firth, D.V. Skryabin, Phys. Rev. Lett. 79 (1997) 2450. N.B. Baranova, A.V. Mamaev, N.F. Pilipetsky, V.V. Shkunov, B.Ya. Zel'dovich, J. Opt. Soc. Am. 73 (1983) 525. A. Pocheau, V. Croquette, P. Le Gal, Phys. Rev. Lett. 65 (1985) 1094. I. Rehberg, S. Rasenat, V. Steinberg, Phys. Rev. Lett. 62 (1989) 756. G. Goren, I. Procaccia, S. Rasenat, V. Steinberg, Phys. Rev. Lett. 63 (1989) 1237. E. Bodenschatz, W. Pesch, L. Kramer, Physica D 32 (1988) 135. N. Tu"llaro, R. Ramshankar, J.P. Gollub, Phys. Rev. Lett. 62 (1989) 422. S. Boccaletti, F.T. Arecchi, Physica D 96 (1996) 9. E.A. Jackson, A.W. Huebler, Physica D 44 (1990) 407. E.A. Jackson, Phys. Rev. A 44 (1991) 4839. Y. Braiman, I. Goldhirsch, Phys. Rev. Lett. 66 (1991) 2545. V. Petrov, V. Gaspar, J. Masere, K. Showalter, Nature 361 (1993) 240. F.T. Arecchi, G. Basti, S. Boccaletti, A.L. Perrone, Int. J. Bifurcation Chaos 4 (1994) 1275. F.T. Arecchi, W. Gadomski, A. Lapucci, H. Mancini, R. Meucci, J.A. Roversi, J. Opt. Soc. Am. B 5 (1988) 1153. C.C. Bradley, C.A. Sacket, J.J. Tollet, R.H. Hulet, Phys. Rev. Lett. 75 (1995) 1687. K.B. Davies, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, W. Ketterle, Phys. Rev. Lett. 75 (1995) 3969.

F.A. Escobedo, J.J. de Pablo / Physics Reports 318 (1999) 85}112

MOLECULAR SIMULATION OF POLYMERIC NETWORKS AND GELS: PHASE BEHAVIOR AND SWELLING

Fernando A. ESCOBEDO , Juan J. DE PABLO
School of Chemical Engineering, Cornell University, Ithaca, NY 14853-5201, USA
Department of Chemical Engineering, University of Wisconsin-Madison, Madison, WI 53706-1691, USA

AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO

85

Physics Reports 318 (1999) 85}112

Molecular simulation of polymeric networks and gels: phase behavior and swelling Fernando A. Escobedo *, Juan J. de Pablo
School of Chemical Engineering, Cornell University, Ithaca, NY 14853-5201, USA
Department of Chemical Engineering, University of Wisconsin-Madison, Madison, WI 53706-1691, USA Received December 1998: editor; M.L. Klein Contents 1. 2. 3. 4.

Introduction Polymeric networks Gel swelling: Theoretical considerations Gel swelling: Simulation methods 4.1. Simulations in the bond #uctuation model 4.2. Continuum space Monte Carlo simulations 5. Results of simulations of isotropic systems

88 89 91 92 93 93 94

6. Fluid}#uid phase transitions in #exible and rigid gels 6.1. Gel-multicomponent solvent systems 7. Semi#exible and mesogenic gels 8. Future challenges 8.1. Combining simulation and theory 8.2. Ionic systems 9. Closing remarks Acknowledgements References

97 100 104 106 106 107 108 108 108

Abstract Polymer gels are commonly used in industrial, analytical, and domestic applications; their uses are likely to continue expanding as gels with novel chemical and structural characteristics are developed. These applications often rely on the precise control of the adsorption behavior of a gel. Development of useful gels, however, has been hampered by a lack of molecular-level understanding of the physics underlying phase transitions in such materials. In this report, we review recent molecular simulation work related to the study of fundamental aspects of network elasticity and of phase transitions in polymeric gels. In particular, simulations of simpli"ed (coarse-grained) molecular models are described which provide insights into the general behavior of gels, as opposed to studies concerned with the properties of speci"c materials. Methodological aspects unique to the simulation of di!erent properties of polymeric gels are emphasized. We also pay special attention to the role of entropic factors (such as network topology, backbone sti!ness, chain length asymmetry), over that of energetic interactions (such as hydrofobic interactions or ionic forces) on the

* Corresponding author. Tel.: #1-607-255-8243; fax: #1-607-255-9166. E-mail address: [email protected] (F.A. Escobedo) 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 1 2 - 5

F.A. Escobedo, J.J. de Pablo / Physics Reports 318 (1999) 85}112

87

onset and characteristics of phase transitions in gels. In spite of the important advances made over the last years in methodology and computer hardware, many challenges remain if phase transitions for more realistic gel models are to be simulated.  1999 Elsevier Science B.V. All rights reserved. PACS: 31.15.Qg; 61.25.Hq; 61.43.Bn Keywords: Monte Carlo; Crosslinked polymers; Swelling

88

F.A. Escobedo, J.J. de Pablo / Physics Reports 318 (1999) 85}112

1. Introduction Polymer gels have important applications as absorbent materials and separation agents in various industries. They are found in industrial and domestic applications such as in ion exchangers, coatings, diapers, etc., and in analytical equipment (e.g., in gel permeation chromatography); they are also being tested for novel medical applications such as drug delivery devices [1,2] (e.g., for eye drops, skin care products, insulin controlled delivery, etc.). The key for development and use of novel gels resides in our ability to control the absorption and release of substances from the gel, e.g., the degree of gel swelling. An improved understanding and control of gel swelling will translate into novel, improved materials for separation media, molecular sieves, drug delivery systems, actuators, and sensors [3]. A crosslinked polymer can be viewed as a network consisting of #exible chains (strands) connected to multifunctional nodes (crosslinking sites). When the network is swollen by a solvent, the resulting gel can increase its volume, sometimes to several times its original (dried) size. When the properties of a gel are monitored as a function of an external controllable variable (e.g., temperature or pH), the degree of swelling usually varies in a continuous manner; under certain circumstances, however, the change in gel volume becomes discontinuous. These transitions are known as volume phase transitions and are of great interest for technological applications [3]. A volume phase transition can be used as a `switcha to turn on and o! the absorbing power of a gel; many interesting applications of polymeric gels are based on this property [1]. Volume phase transitions in a gel can be driven by changes in di!erent variables such as temperature, pressure, ionic concentration, pH, stress, electric "elds, and composition of the solvent. The chemistry of the components determines the dominant interaction force in the system (e.g., van der Waals, hydrophobic}hydrophilic, hydrogen-bonding, or ionic interactions) and the speci"c mechanism of the phase transition. Because di!erent mechanisms can compete in di!erent regions of thermodynamic space, the resulting phase diagram can be complex and di$cult to predict and interpret. Although studies of gel swelling have been con"ned to conditions where the solvent is subcritical, use of supercritical solvents, for example, can provide new opportunities for development of novel separation technologies. Supercritical #uids are used in extraction processes due to the ease with which their solvent quality can be tuned by small changes in pressure and temperature; it is therefore of interest to explore the possible advantages of separation processes comprising both a supercritical solvent and a polymeric gel. The topology of the network is believed to in#uence signi"cantly its volumetric properties. The degree of crosslinking, strand length distribution, fraction of active strands, distribution of defects (free ends, loops, etc.), rigidity of the strands, and the presence of entanglements can all alter the swelling pattern of a gel and, ultimately, the existence of a volume phase transition. These topological features determine the character and magnitude of the entropic (elastic) forces associated with the conformation of network strands. The complex interplay of entropic forces and di!erent types of enthalpic forces has led researchers to adopt a predominantly empirical approach for development of novel, useful polymer gels for speci"c applications. Much progress could be made towards understanding the relative importance of these forces if perfectly regular networks could be made. Synthesis of a highly regular, reproducible polymer network topology with a desirable distribution of active strands, however, remains an important challenge for synthetic chemistry.

F.A. Escobedo, J.J. de Pablo / Physics Reports 318 (1999) 85}112

89

Theoretical work has been helpful in correlating and explaining some experimentally observed features of phase diagrams for polymeric gels. This has been an active area of research since the early work of Flory and Rehner [4] (see, for example, [5}8]). The complexity of swelling phenomena, however, has precluded the development of a general, quantitative theory. Most of the proposed theories are, at best, of semi-quantitative value; they rely on the use of several adjustable parameters to match experimental data, and provide reliable extrapolations over a limited range of conditions. Molecular simulations provide an important complement to experimental work and analytical theories. In a simulation, the molecular architecture is speci"ed unambiguously; by studying a succession of carefully chosen molecular models, analytical theories could be tested and, more importantly, a more complete understanding of swelling phenomena could be achieved by decoupling di!erent entropic and enthalpic e!ects in a systematic manner. Simulations can also be used to explore the properties of gels with novel architecture and chemistry. Unfortunately, molecular simulation studies of swelling and phase equilibria for gels have been scarce. Conventional simulation methodologies are not e!ective at dealing with the topological complexity, the large size, and long relaxation times encountered in these systems. New methods designed speci"cally for these systems are being developed by several groups to make progress on this front [9,10]. The main goal of this review is to describe some of the advances in methodology and application of molecular simulation techniques to study swelling phenomena and #uid}#uid phase transitions in gels. Because this "eld is still in its infancy, the applications to be described pertain to highly simpli"ed molecular models of polymeric networks and gels; however, emphasis will be given to qualitative physical aspects elucidated by simulation studies. We restrict ourselves to gels with chemical crosslinking bonds, that is, gels in which the network junctions are "xed and do not dissociate by changes in the solvent or the environment as in physical gels (e.g., gelatin). In connection to phase transitions in chemically-bonded gels, we also discuss some recent simulation results for phase equilibria of #uids within rigid gel structures.

2. Polymeric networks The topology, entanglements, and ultimately, the dynamic and thermophysical properties of a polymer gel depend on how the crosslinking reaction is carried out. Variations in the concentration of the polymer (from a melt-like state to a dilute solution) and crosslinking agents (from stoichiometric proportions to large disparities between reactants) are particularly important in determining the properties of the "nal product. Modeling and prediction of crosslinking reactions are di$cult problems which often times require the use of numerous empirical "tting parameters to correlate experimental data for a given system. The physical and chemical properties of the base polymer (like solvent a$nity, molecular weight distribution, density, etc.) provide the reference point to characterize the network. Important microscopic parameters in a network are the average strand length, functionality of crosslinks (and a suitable description of the overall network connectivity), topological defects (e.g., free ends), and entanglements. These properties are often di$cult (or impossible) to measure experimentally; they can be indirectly evaluated from their e!ect on macroscopic properties of the network such as elastic modulus and equilibrium swelling.

90

F.A. Escobedo, J.J. de Pablo / Physics Reports 318 (1999) 85}112

The formation of polymeric networks has been simulated in several studies [11}18]. The early work of Eichinger and coworkers [11,12] has become the basis for development of commercial software to study the structure and elasticity of polymer networks [13]. In a typical implementation of these methods, realistic models of speci"c molecules are (randomly) placed in a (nonperiodic) simulation box; however, the crosslinking reactions are not conducted by a molecular dynamics simulation approach but instead by resorting to probabilistic algorithms (e.g., by introducing models for a reaction probability and steric penalty), whereby molecules are essentially static but a capture radius for polymer-crosslinker reactions is gradually incremented. An elaborated bookkeeping scheme is also employed to keep track of the evolution of the network topology in the system as the allowed linkages are formed. These techniques have been successfully used to investigate a wide variety of materials (see, for example, Ref. [12]). The dynamics of crosslinking has been simulated by Grest et al. [16], Duering et al. [17], and Trautenberg et al. [18]. In the former two studies [16,17], an approximate description of a polymer network was employed by resorting to a large, o!-lattice, bead-spring model of the polymer chains (with repulsive Lennard}Jones interactions between chain beads). These authors prepared their networks by crosslinking a homopolymer melt with polyfunctional crosslinkers attached to chain ends. The dynamical evolution of the reactants yields direct information on the kinetics of crosslinking. A similar method was employed by Trautenberg et al. [18] but for an athermal lattice model; they found that the kinetics of crosslinking exhibits a cross-over from reaction-controlled to di!usion-limited behavior. Several studies of the structural and dynamic properties of continuum-space polymeric networks have resorted to molecular dynamic simulations [14}17]. Gao and Weiner [14,15] simulated small-size, highly idealized representations of a polymer network to study segment orientation and chain stresses in stretched networks. Duering et al. [17] studied tetrafunctional networks formed by a dynamic crosslinking process; these networks were used to determine the e!ect of entanglements on the motion of the crosslinks and modulus of the network. Strand lengths ranged from 12 to 100 monomers (which translated into 30 000 to 50 000 total monomers within the simulation box). These authors used a cluster search algorithm to determine the number of active strands and crosslinks, and the `burninga method from percolation analysis to determine the gel fraction of chains with no free ends, and the fraction of elastically active beads. This microscopic characterization of the networks allowed Duering et al. [17] to readily compare their results to the predictions of rubber elasticity theories. The elastic modulus was obtained through the long-time limit of the time-dependent shear relaxation modulus within the Rouse and reptation models (the Rouse modes are computed from random walk paths constructed along the network such that the end segments of each walk were not crosslinked). They found that for long chains, the simulated moduli were consistent with the Edwards tube model [20] and about two times larger than predicted by the a$ne model (due to the contributions from entanglements). These simulation results are in good agreement with experiments on `modela end-linked polydimethylsiloxane networks by Patel et al. [22]. Everaers and Kremer [32] have conducted molecular dynamic simulations to study the elastic properties of model polymer networks with diamond lattice connectivity. The systems were prepared by interpenetrating networks to get the desired concentration of crosslinks. The usefulness of these model networks is that the e!ect of topology conservation can be isolated from other sources of quenched disorder. They implemented a method to measure the shear modulus in which

F.A. Escobedo, J.J. de Pablo / Physics Reports 318 (1999) 85}112

91

the samples are gradually strained (at constant volume) and the stress is obtained from the plateau value (after relaxation) of the measured normal tension. Using this approach, these authors were able to obtain exact predictions for the classical models of rubber elasticity and found that the treatment of network strands as independent entropic springs neglects important entropic e!ects in a deformed network; such additional contributions arise from the quenched topology of the network (such as entanglements). Trautenberg et al. [18] and Sommer et al. [19] analyzed their end-linked lattice networks using combinatorial algorithms (to extract the topological sol fraction, unreacted chain ends, network chains, and the cycle rank per network chain). Strand lengths varied from 11 to 83 segments and the total number of precursor chains in the system was 5000. Periodic boundary conditions were not used in these simulations. It was found that the simulated networks were closer to a model network than the experimental model of Patel et al. [22]. Dimensional analysis showed that for intermediate length-scales, the connectivity structure of the simulated networks was closer to a ternary tree than to a diamond lattice. Because the simulated structures were highly back-folded at large length-scales, these networks could be described as self-interpenetrating. Recently, Holzl et al. [21] have devised a novel method to simulate stress}strain relation in polymer networks; the method mimics a real simple extension experiment where both ends of a piece of rubber are clamped to mountings. The results of this study provide further support to the view that a$ne and homogeneous deformation theories for rubbers are not adequate. The structure and relaxation of end-linked polymer networks have also been studied with discontinuous molecular dynamics methods by Kenkare et al. [29]. These authors studied nearperfect tri- and tetra-functional networks with hard chain beads at liquid-like packing fractions. Because the system evolves on a collision-by-collision basis, e$cient book-keeping algorithms can be used to speed up the calculations, thereby allowing longer times to be simulated. The networks were also constructed by end-linking free chains; precursor chains ranged from 20 to 150 beads; the total number of beads in the system ranged from 21 000 to 45 000. The crosslinking process implemented by Kenkare et al. [29] di!ers from that of Duering et al. [17] in that the crosslinking molecules were not explicitly present and the concentration of chains was lower than that in a melt (e.g. a scenario closer to crosslinking in solution), thereby resulting in somewhat collapsed chains and fewer entanglements. Kenkare et al. found that the dynamics of crosslinks and chain inner segments were similar to those in the melt at short times but exhibited spatial localization at long times. Similar results to those of Duering et al. [17] were found regarding the chain length dependence of the elastic modulus and entanglement e!ects. A recent review that describes in more detail the methodology and results of simulation studies of elastomers and networks [9] has appeared. A review of theoretical treatments on the thermodynamics of crosslinked polymers is beyond the scope of this work; For a recent discussion of this topic, the reader is referred to the text by Erman and Mark [5].

3. Gel swelling: Theoretical considerations Several theories are available to describe the equilibrium swelling of gels (see, for example, [4}8]). A basic element of the Flory}Rehner [4] and related theories, is the so-called `additivity assumptiona, which states that mixing (*A ) and elastic contributions (*A ) to the free energy

  

92

F.A. Escobedo, J.J. de Pablo / Physics Reports 318 (1999) 85}112

of a gel are additive, i.e., A +A #A #*A #*A . (1)      

   It is not clear, however, under what conditions this assumption is valid; while some experimental results appear to support it [22], others have questioned its validity altogether [35]. Because elastic contributions to the gel free energy are typically small [19,26], it is important to measure them accurately. Density changes play a particularly important role in systems where di!erent packing arrangements arise as a result of anisotropic interactions (e.g., liquid-crystalline materials, hydrogenbonding #uids, etc.), systems comprising asymmetric or partially miscible solvents, and systems containing gaseous components. For all of these cases, a theory of gel swelling must take into account the e!ects of mixing, elasticity, and compressibility on the properties of the system. A simple theory representative of previous work on gels is that proposed by Marchetti et al. [36], which combines elements from the Sanchez}Lacombe lattice theory [39] for the thermodynamics of mixing of compressible polymer-solvent systems, and the Flory}Rehner theory for the elastic free energy of the network [4]. Engineering models that describe the swelling of hydrogels have been developed by several groups (e.g., by Prausnitz et al. [40]). For polyelectrolyte gels, an additional term representing the ionic contributions must be added to the right-hand side of Eq. (1). The interested reader should consult several monographs that review the modeling of hydrogel swelling [5,64]. Several improvements can be introduced into simple theories based on Eq. (1) to address their known de"ciencies; for example, a more accurate description of elastic e!ects can be implemented (e.g., based on more recent theories [6,8] or on detailed simulations of networks with various topologies). We have recently formulated a theory that incorporates density e!ects in calculations of athermal gel swelling [26]; it is based on a statistical mechanical treatment of the mixing of hard-core molecules, the additivity assumption, and simulated elastic pressures for model networks. This theory was shown to provide quantitative agreement with simulated results for gel swelling by pure and mixed solvents, and it could serve as the basis for development of theories capable of describing the behavior of more realistic gels; for example, by using athermal networks with more realistic topology to measure the elastic contributions, and by introducing additional perturbation terms to account for dispersion forces and ionic interactions. This theory was used to obtain the results shown in Figs. 4, 5, and 11.

4. Gel swelling: Simulation methods Molecular dynamics provides a general framework to treat molecular systems with any degree of topological complexity; however, the slow relaxation of polymeric gels constitutes a signi"cant barrier to study gel swelling by following the natural evolution of the system. Thermodynamic properties and phase behavior (e.g., equilibrium swelling) can be more advantageously simulated by Monte Carlo methods, which can avoid dynamical bottlenecks and therefore relax the system more e!ectively. Note, however, that conventional Monte Carlo moves are not useful for equilibration of either the internal network conformation or the volume of a gel. In this context, two

F.A. Escobedo, J.J. de Pablo / Physics Reports 318 (1999) 85}112

93

di!erent strategies have been recently pursued, one based on a suitable lattice model and the other on custom-made molecular moves for continuum space models. 4.1. Simulations in the bond yuctuation model Sommer et al. [18,19] employed a particular lattice model, the so-called `bond #uctuation modela of Carmesin and Kremer [23], in which rearrangements of multifunctional sites are readily performed. This is done by taking advantage of the `#uctuatinga nature of the bonds (which can assume a large but "xed number of discrete lengths and orientations). These authors simulated a "nite-size network with non-periodic boundary conditions to simplify the swelling experiment. A swelling simulation was performed by placing a network at the center of larger lattice volume; unoccupied lattice sites then di!use into the network without interaction (ideal athermal swelling). Note that solvent molecules were not considered explicitly in these calculations; in other words, empty sites are indistinguishable from sites occupied by solvent molecules. Trautenberg et al. [18] started by forming randomly crosslinked networks (as described in Section 2). Swelling simulations on these networks showed that they deform in a highly non-a$ne manner (in marked contrast to the assumptions of theoretical models). Small substructures swell much less than the whole sample; that is, the degree of swelling is larger as the size of a control volume is increased. This behavior can be interpreted in terms of an unfolding scenario involving substructures much larger than a single precursor chain. This behavior has also been rationalized with a scaling analysis based on the assumption that the network structure possesses fractal properties with certain correlation length [19]. An important conclusion of this work is that a signi"cant di!erence in mixing entropy is predicted between a network (and solvent) and the corresponding uncrosslinked system. In other words, the *A term in Eq. (1) should not be

  evaluated based on the properties of the uncrosslinked polymer plus solvent system. The use of an improper mixing free energy term could explain the deviations from the predictions of the additivity assumption that several authors have found in experimental studies [35]. 4.2. Continuum space Monte Carlo simulations On the one hand, lattice implementations permit simulation of much larger systems than continuum calculations; a large system size is necessary to recreate a representative sample of a network (including all relevant topologies). On the other hand, continuum space models provide a much more realistic description of density and local entropic e!ects, which are likely to be important for supercritical solvents, anisotropic solvents (liquid crystals), and for systems where any type of shape asymmetry is present. Specialized Monte Carlo moves have been developed for e$cient intramolecular equilibration of polymeric networks [24], and for simulation of gel swelling [24}27]; here, the simulation box emulates an in"nitely large system through the use of periodic boundary conditions. Chain molecules and network strands are modeled as a collection of sites connected by rigid bonds. Intramolecular rearrangements of polymeric sites are performed by means of extended continuum con"gurational moves [24]; volume changes can be carried out by means of cluster moves [26] or slab moves [25]. Because the solvent is explicitly present in these systems, simulation of gel swelling must be conducted in a Grand canonical type of ensemble or, more properly, an osmotic ensemble.

94

F.A. Escobedo, J.J. de Pablo / Physics Reports 318 (1999) 85}112

Fig. 1. Schematic representation of a computer gel-swelling experiment. The gel is assumed to be in contact with an in"nite bath of solvent molecules (at "xed pressure, temperature, and chemical potential) with which it can exchange `volumea and molecules until equilibrium is attained. Network sites are shown as black beads (adapted from [26]).

In an osmotic ensemble, the number of network sites is "xed, but the number of solvent molecules and volume are allowed to #uctuate in response to the imposed chemical potential (of the solvent) and pressure, respectively. The setup for the simulation of gel swelling can be represented by the system shown in Fig. 1. Details of these calculations have been provided in [26]. The properties of the pure solvent (PVT and chemical potential) can be obtained from standard isothermal-isobaric simulations [31].

5. Results of simulations of isotropic systems Simulation of the swelling of large, realistic continuum space representations of crosslinked polymer is still beyond the capabilities of current computational tools. Important insights into the qualitative aspects of swelling and volume phase transitions can be gained by examining simpli"ed network topologies. The e!ect of some topological defects could then be isolated and subsequently studied by generating systematic modi"cations of a basic template. For tetrafunctional networks, a convenient template is provided by a diamond-like connectivity of monodisperse strands. The diamond lattice can be viewed as the most regular structure in three dimensions with tetrafunctional crosslinks. As described in Section 2, this perfect topology has been employed in several simulation studies aimed at understanding the elastic and dynamic properties of gels [26,32]. Using networks with diamond-like connectivity (`DC networksa), we have studied the pressure}volume}temperature (PVT) behavior of pure networks. Fig. 2 shows a snapshot of a low density perfect network with monodisperse strand length, `N a of 10 sites. The pressure}density relation ship for athermal networks with di!erent strand lengths is shown in Fig. 3. Packing e!ects, similar to those found in hard-sphere chains, dominate the volumetric behavior of athermal networks at intermediate-to-high densities. At such conditions, intramolecular entropic e!ects arising from a particular architecture are e!ectively screened by neighboring sites in the #uid; as a consequence,

F.A. Escobedo, J.J. de Pablo / Physics Reports 318 (1999) 85}112

95

Fig. 2. Instantaneous con"guration of a dilute gel; for clarity, only the network sites are shown (1344 beads). The network has a perfect tetrafunctional architecture and strands of regular length (10 sites). Tetrafunctional sites, shown as darker beads, are placed in a regular diamond lattice arrangement for clarity (their positions are free to move during the course of a simulation). Fig. 3. Simulated pressure-density curves for hard-core uncrosslinked and crosslinked chain systems having di!erent connectivity ratios (l) as de"ned in the text; l"0 (monomer), l"0.75 (tetramer), and l"1 (Rmer) correspond to linear (uncrosslinked) chains, symbols for l"1.048, 1.143, and 1.333 correspond to DC networks with 10, 3, and 1 sites per strand, respectively (adapted from [26]).

a unique parameter, the so-called `connectivity ratioa, is su$cient to fully characterize the compressibility of homonuclear systems [26]: Total number of bonds in the system . l" Total number of sites

(2)

This ratio is always less than unity for linear or branched chains, but it can be greater than one for crosslinked molecules. At low densities, the physical behavior of uncrosslinked and crosslinked (athermal) polymers is quite di!erent. As oP0 the pressure of a system of disconnected chains also vanishes (and the compressibility factor curve approaches unity with a slope proportional to the system's second virial coe$cient). For the crosslinked systems, however, the network strands must be stretched to attain low concentrations. Since this reduces the network entropy, a retractive (elastic) pressure must be balanced by an external negative pressure. Negative pressures are necessary to attain large (isotropic) expansions of networks with attractive intramolecular interactions; results have been reported for a DC network whose sites interact

96

F.A. Escobedo, J.J. de Pablo / Physics Reports 318 (1999) 85}112

through square-well potential energy functions [28]. In this case, however, and for a su$ciently low temperature, the cohesive energy between strands precludes the system from stabilizing at intermediate densities. Upon reducing the pressure of a melt-like network (at constant temperature), the density of the system is gradually reduced up to a point (a network cannot vaporize, even at zero pressure). When the pressure is set to negative values, the network undergoes a transition from a dense, liquid-like state, to a low-density, vapor-like state. This phenomenon is illustrated in Fig. 4. While the physics underlying such "rst-order transitions are easy to rationalize [28], it is unclear, however, whether such transitions could be realized in real networks that have highly irregular topologies and interconnected substructures. Fig. 5 shows simulation results for the swelling of athermal networks by a monomeric hardsphere solvent as a function of reduced pressure (PH"Pp/e). All these systems have a uniform bead diameter. Three strand lengths are considered, namely N "10, 16 and 32. The solvent  concentration in the gel is given as a site fraction, which can be interpreted as a weight fraction. It is found that the e!ect of increasing the strand length of the network is to increase substantially the relative amount of solvent uptake. This is a consequence of the fact that a gel with long strands exerts a weaker contractive (elastic) pressure than a gel with shorter strands, for the same concentration (on a volume-basis) of polymer network in the gel. It is also observed that in the range of low to intermediate pressures (irrespective of the strand length), solvent uptake increases with pressure. Experimental evidence in support of this latter result has been reported in the past for subcritical gels [38] and, from simulation, for attractive gels as well [27]. The explanation of this pressure e!ect is the following; as the gel and the coexisting solvent are compressed (at a given temperature), their densities increase at the expense of a reduction of free volume. Gel swelling will

Fig. 4. Isotherms for pure DC networks with 10 sites per strand and sites interacting via a square-well potential energy function. The temperatures are given in reduced units (¹H"k ¹/e, where k is Boltzmann's constant and e is the depth of the square-well potential). The lines are predictions of a mean "eld theory described in [28]. Fig. 5. Simulated gel composition and density as a function of pressure and network strand length for athermal gels in equilibrium with a monomeric solvent. The gel composition is given as solvent site fraction, which is the ratio of number of solvent sites (in the gel) to the total number of sites in the gel. The gel density is given in packing fraction or volume fraction occupied by all hard beads in the system.

F.A. Escobedo, J.J. de Pablo / Physics Reports 318 (1999) 85}112

97

be higher at elevated pressures, because the di!erence in density (or free volume) between the pure solvent and the pure network becomes smaller. The denser the solvent, the more e!ective it is at screening intrachain forces. 6. Fluid}6uid phase transitions in 6exible and rigid gels The e!ects of con"nement on the phase behavior of complex #uids and their mixtures are of general interest, as they are manifest in numerous applications ranging from catalysis in zeolites to separation processes with membranes. In particular, the question of how vapor} liquid and liquid}liquid binodal curves (and critical points) are modi"ed within a gel matrix (in contact with the bulk system) remains largely unsolved. A number of experimental (see, for example, [41}44]) and theoretical studies [45}49] have examined this issue in con"ned rigid geometries. Simulation work has been particularly useful in elucidating the physics of #uid phase transitions in model rigid pores [50}53] and rigid porous matrices [54,58]. A #exible, polymeric gel matrix, however, adds yet another degree of complexity to the problem as its volume can change appreciably. Fig. 6 shows results of simulations [27] of the swelling of a fully #exible polymeric network in contact with a monomeric solvent held both at subcritical and supercritical pressures. The high #exibility of the network allows it to `mixa with the solvent, so that changes in solvent quality (density, in this case) translate into di!erent degrees of swelling. At subcritical pressures, even if the bulk solvent is a vapor, it cannot exhibit a vapor-like state within the gel because a collapsible

Fig. 6. Simulated and theoretical isobars for equilibrium swelling of Lennard}Jones gels. Simulation data was obtained through osmotic simulations of perfect tetrafunctional networks of 20 (diamonds) and 40 (squares and triangles) sites per strand in contact with a monomeric solvent. The pressure was held "xed at P "1.64 (supercritical) and P "0.66 as   indicated in the "gure (reduced values of pressure and temperatures are with respect to the critical properties of the pure solvent). Theoretical curves (full lines) are predictions of the Marchetti et al. theory [36,37], with parameters "tted to the simulation data (at ¹ "0.8) of the 20-site strand network. 

98

F.A. Escobedo, J.J. de Pablo / Physics Reports 318 (1999) 85}112

polymer network will shrink out any vapor-like pocket that may form; in other words, vapor and liquid solvent phases cannot coexist within a gel. If two phases are to coexist within such a gel, at either sub- or super-critical conditions, they will likely be liquid-like phases with di!erent solvent concentration; such description of coexisting phases is consistent with a "rst-order volume phase transition. At supercritical pressures, the gel tends to shrink as the coil-to-globule transition temperature [59] for the network's strands (in solvent) is approached (near the critical point); an incipient discontinuous volume phase transition appears to take place for the 40-site strand gel. The location of this transition is near the lower critical solution temperature of the solvent plus the `uncrosslinkeda (linear) polymer at the same pressure. It is interesting to examine the cross-over that occurs from a highly expandable polymer gel to a #uid-rigid matrix system when the network sti!ness is increased. If the network conformation is quenched in a swollen state, and the solvent is evacuated (e.g., by freeze-drying or by supercritical extraction), the resulting product is a porous organic matrix. System-speci"c energetic interactions aside, the network entropy plays a crucial role in the behavior of a gel; the strands give rise to conformational entropy and also provide a con"ned environment for the #uid (through excluded volume interactions). In a rigid gel, however, the conformational entropy of the strands is lost, and its associated mixing and elastic e!ects disappear. Recent simulation studies in a grand canonical ensemble have provided evidence that (inorganic) porous materials can host vapor}liquid phase transitions [54,55]. Page and Monson [54] studied the adsorption of a (truncated) Lennard}Jones #uid within a matrix of large spheres intended to provide an approximate description of an aerogel; Saravanan and Auerbach studied the adsorption of benzene in a zeolite (faujasite) described by a lattice model. In both cases, a clear vapor}liquid transition for the #uid could be identi"ed (at least for the choice of force "elds employed by these authors). Intriguingly, in their pioneering work Page and Monson found evidence of additional transitions in disordered matrices; these second transitions may be related to predrying and prewetting transitions. For a composite matrix with attractive interactions, Page and Monson found evidence of a small binodal curve at low densities (appearing next to the vapor}liquid binodal). For the second transition, however, these authors did not conduct calculations for the grand potential (whose equality for the coexisting phases provides a necessary condition for phase equilibrium). Fig. 7 shows their estimates of the second transition (based on the shape of the adsorption isotherms) and our calculations based on histogram reweighting analysis [56] (in which the equality of grand potentials is ful"lled by enforcing the so-called equal area criterion for the bimodal density distributions). True macroscopic estimates of the coexistence curves can only be obtained by averaging results from very many realization of the ("nite-size) solid matrix (a problem akin to modeling the topological complexity in crosslinked polymers). Indeed, the results of simulations are somewhat di!erent when calculations are repeated on a di!erent matrix; nonetheless, the main qualitatively features are preserved and it is likely that the second transition may be smoothed but not suppressed. The origin of these secondary transitions is related to the wide variations in pore sizes that occur within a disordered matrix, and the ability of the #uid to wet denser regions of the matrix; note that simulations for ordered large-sphere matrices (at the same void fraction of the disordered matrix) do not exhibit secondary transitions [57]. Secondary transitions have also been observed for purely repulsive (rigid) matrices [54]. These systems are interesting as they provide a limiting scenario for #uid}matrix interactions. In this case, a saturated liquid will not condense within the matrix unless the pressure is increased and/or the

F.A. Escobedo, J.J. de Pablo / Physics Reports 318 (1999) 85}112

99

Fig. 7. Secondary transition for a 12}6 Lennard}Jones #uid in a composite large-sphere matrix with attractive interactions (for a particular choice of interaction parameters, see [54]). The squares are coexistence estimates of Page and Monson [54] and the lines are results from histogram reweighting analysis of simulation conducted in this work. The temperature and density are given in reduced units ¹H"k ¹/e , oH"op /(1!g); where e and p EE EE EE EE are the Lennard}Jones parameters for the #uid}#uid interactions and g is an e!ective hard-core packing fraction of the matrix.

temperature of the #uid lowered (because #uid}matrix interactions are purely repulsive, temperature only a!ects #uid}#uid interactions). We have studied two types of matrices [61], (i) a large-sphere matrix similar to that of Page and Monson but with slightly smaller spheres (and thus having a slightly higher void fraction), and (ii) a small-sphere matrix obtained by `freezinga a #exible DC network structure. These two systems are illustrated in Fig. 8. The free volume accessible to small spherical solvent molecules is identical in systems (i) and (ii), and corresponds to a void fraction of 0.614; however, the pore geometry and distribution of pore sizes are signi"cantly di!erent in these two systems. Fig. 9 shows the entire phase diagram for the hard, repulsive matrix studied by Page and Monson [54] and also our results from histogram reweighting analysis; in all cases, the main vapor}liquid binodal is `squeezeda towards the low density region and the critical temperature reduced (relative to the pure #uid bulk value), so that the curves are enclosed by the bulk binodal curve. The critical temperature of the #uid con"ned in large-sphere matrices increases with the matrix void fraction. The critical point for the #uid within the polymer network-like matrix is lower than that within a large-sphere matrix (with identical free volume) and the second transition is less pronounced. Stronger "nite-size e!ects are associated with the DC network matrix than with the large-sphere matrix (e.g., results vary more appreciably for di!erent matrix realizations); consequently, it is di$cult to predict the shape of the binodal in a `macroscopica system (and ascertain the existence of the second transition). At the microscopic level, however, our simulations reveal a rich behavior, sometimes exhibiting multiple free energy minima as suggested by multimodal density distributions such as those shown in Fig. 10. Each peak in the histograms of Fig. 10 is indicative of a preferred `#uid con"gurationa, in which #uid molecules tend to concentrate in particular combinations of regions within the matrix.

100

F.A. Escobedo, J.J. de Pablo / Physics Reports 318 (1999) 85}112

Fig. 8. Schematic comparison of two di!erent rigid gel matrices used for #uid adsorption simulations. The matrix on the left is made-up of disordered large-spheres and corresponds to a realization of a hard-sphere #uid at a packing fraction of g"0.386 (based on the system adopted by Page and Monson [54]). The matrix on the right corresponds to a realization of an athermal DC network with 10 beads per strand (1344 beads) and g&0.12. A #uid molecule is modeled as a Lennard}Jones sphere whose collision diameter is roughly the same size as one of the spheres of the network-like matrix on the right. The volume of space available to #uid molecules is essentially the same in both matrices.

6.1. Gel-multicomponent solvent systems Gels are widely used to perform selective adsorption of species from a #uid stream (for example, in ion-exchangers). Selectivity is usually accomplished by a proper choice of the gel chemistry; network structure, however, could also be used to aid selectivity. In gel-permeation chromatography, for example, a dynamic segregation of species is accomplished by exploiting di!usion rate di!erences among species within the gel. Segregation at equilibrium conditions in polymeric gels, however, has not been exploited as a means of optimizing selectivity; we have studied segregation driven by conformational (entropic) asymmetries among components. Fig. 11 shows our simulation data for the equilibrium partitioning of n-mers (n"4 or 8) from a n-mer/monomer solution, between a bulk and a gel; the relative composition of n-mer/monomer is smaller inside the gel than outside it. Because only hard-core interactions are considered for this system, purely entropic forces are responsible for the observed partitioning. These results, although theoretical, clearly suggest that network architecture could be tailored to enhance entropic selectivity. Entropic selectivity does play a role in several separation processes involving membranes and gels. For example, a recent report describes a semi-commercial process for production of soy protein isolate

F.A. Escobedo, J.J. de Pablo / Physics Reports 318 (1999) 85}112

101

Fig. 9. Phase diagram for a Lennard}Jones #uid con"ned by a hard-sphere matrix. The enclosing dotted line is the binodal curve for the bulk #uid. The squares are coexistence densities calculated by Page and Monson [54] for the vapor}liquid transition ("rst peak to the left) and for the second transition (smaller peak at the center) in a large-sphere matrix; the e!ective void fraction in this system is estimated to be &0.5 (#uid}solid collision distance"8.06p ). The EE solid lines correspond to histogram-reweighting calculations for simulations of #uid adsorption in a large-sphere #uid with void fraction of &0.6 (#uid}solid collision distance"7.06p ). The dashed lines are the corresponding coexistence EE densities for the adsorption of the #uid in a polymer network-like matrix (Fig. 8). The units are the same as those described in the caption of Fig. 7.

employing gels as separation agents [65]. In real systems, however, it is di$cult to assess the relative importance of entropy vs. chemically driven interactions. A study of athermal and thermal systems of di!erent molecular architecture could provide a basic thermodynamic understanding of entropic partitioning. A convenient way of tuning gel swelling is to vary the composition of a mixture of solvents that di!er in quality. An example of this approach is provided by the swelling of (non-ionic) poly N-normalpropylacrylamide gels by a water}propanol mixture; in this case, water is a poor solvent and propanol is a good solvent. It has been reported that these systems can exhibit multiple volume phase transitions [60], and that the polymer backbone is probably solvated by a layer of solvent molecules. Solvent-explicit simulation of gel swelling can unambiguously test the preferential adsorption hypothesis [60] and examine the physics underlying di!erent volume phase transitions. Fig. 12 shows simulation results for a network, species `Ca, in contact with a solvent bath of species `Aa and `Ba. In this example, A and B are identical, fully miscible monomeric solvents, but A}C interactions are more favorable than B}C interactions (described by Lennard}Jones potentials). Simulations were conducted at constant pressure and for di!erent compositions of the solvent bath in contact with the gel. As expected, the gel shrinks when pure B is present, and it gradually swells as the composition of A in the bath increases; the change in gel swelling is more pronounced around the equimolar bath composition. Fig. 12(b) shows that the chemical mismatch between sites B and C causes a relative depletion of B and enrichment of A within the gel; e.g., for any composition of A in the bath, a higher composition of A is observed in

102

F.A. Escobedo, J.J. de Pablo / Physics Reports 318 (1999) 85}112

Fig. 10. Probability density distributions for the number density of #uid molecules (op ) adsorbed in a hard matrix. The EE top "gure corresponds to vapor}liquid coexistence conditions at di!erent temperatures (from ¹H"k ¹/e "0.94 to EE ¹H"1.0) in the large-sphere matrix; the bimodal shape of the distributions is characteristic of a "rst-order phase transition (similar results are obtained for the `seconda transition). The distributions in the bottom "gure correspond to the small-sphere network-like matrix (from Fig. 8) for di!erent temperatures and di!erent reduced chemical potentials (CP) near the expected vapor}liquid transition; at least four distinct peaks are clearly detectable (and reproducible) for a particular matrix realization. Multimodal density distributions can be associated with localized density inhomogeneities within the porous matrix. These features (for both the vapor}liquid and second transition) tend to be suppressed by averaging the results from di!erent matrix realizations.

Fig. 11. Simulation data (symbols) and theoretical predictions (full lines) of the composition of chain molecules in the solvent bath phase and in the coexisting gel phase. The network has a strand length of N "10 and the system's pressure  is "xed at PH"Pe/p"0.3. Two cases are considered for the solvent bath; one containing a mixture of monomers and octamers and the other containing a mixture of monomers and tetramers. The solvent concentration in the gel is provided on a network-free basis. Adapted from [26].

F.A. Escobedo, J.J. de Pablo / Physics Reports 318 (1999) 85}112

103

Fig. 12. Simulated swelling curves for a network (species C) in contact with a solvent bath of species A (good solvent) and B (poor solvent) at various bath concentrations. The solvent quality is tuned by introducing a chemical mismatch in the e parameter of the Lennard}Jones potential; in our case e "e "e "e "e "e but e "0.85e. The broken GH  !!  ! ! and full lines correspond to 20-site and 40-site strand networks, respectively. The conditions are "xed at P "0.43 and  ¹ "0.87 (reduced with respect to the solvent's critical properties). In (a) the weight fraction of (A#B) in the gel is  plotted as a function of the weight fraction of A in the solvent bath; in (b) the partitioning of A between phases is shown [e.g. the fraction A/(A#B) in the gel vs. that in the bath]. Fig. 12(c) shows the radial distribution functions for A sites (solid line) and B sites (dashed line) surrounding a C site. The results correspond to the N "40 network in contact with  a bath with 0.55 mole fraction of A (e.g., about 0.7 mole fraction of A in the solvent absorbed in the gel).

the (total) solvent absorbed in the gel. Fig. 12(c) shows that species A is not only more concentrated in the bulk solvent in the gel but also locally, near a network site (the radial distribution functions correspond to the N "40 network, at 0.55 mole fraction of A in the bath). Clearly, A-sites are  preferentially absorbed by C-sites; that is, A-sites tend to concentrate near a network site more strongly than B-sites, as evidenced by the di!erence in peak heights of the corresponding radial distribution functions [g(r)]. Note that each g(r) is normalized with respect to the bulk concentration of the relevant solvent species in the gel, therefore, the bulk concentration di!erence between A and B (within the gel) is factored out in the comparison of Fig. 12(c).

104

F.A. Escobedo, J.J. de Pablo / Physics Reports 318 (1999) 85}112

Although a "rst-order phase transition is not observed in Fig. 12 (it is, however, incipient), it may occur for longer strand lengths, or for di!erent chemical mismatch between components. A systematic study of di!erent molecular models could clarify the role of these factors on volume phase transitions.

7. Semi6exible and mesogenic gels If chain molecules or network strands in a gel are su$ciently sti!, a uniaxial deformation can induce a nematic phase transition [66,67]. The interest in strain-induced nematic elastomers lies in their potential use for electro-optics, integrated optics, and data storage devices [67]. An important question for design of these materials is: how should the mesogenic chains be incorporated into the gel in order to achieve an optimal optical e!ect?. Mesogenic groups can consist of free chains [68], side chains attached to the network [66], part of active strands, or a combination of the above. Furthermore, the system can be prepared under isotropic or anisotropic conditions. Fig. 13 illustrates some of the applications of systems that combine a #exible matrix and mesogenic chains and the architecture of systems that have been studied in the past.

Fig. 13. Applications and microscopic structure of systems that combine a #exible polymer matrix with mesogenic chains.

F.A. Escobedo, J.J. de Pablo / Physics Reports 318 (1999) 85}112

105

It has been shown that an isotropic-nematic transition can be achieved in a stretched gel, even if the mesogenic chains are not linked to the network and even if the solvent in equilibrium with the gel is isotropic. This transition is illustrated in Fig. 14(a); in this case, a fully #exible network is in contact with an isotropic bath of rod-like molecules. In spite of the tendency to demix that is

Fig. 14. Equilibrium swelling of uniaxially stretched gels (N "10) in equilibrium with a solvent bath of rigid 12-mers.  (a) Simulated composition (>"solvent site fraction in the gel) for a bath of 12-mers (diamonds) and for bath of monomers (squares). D is the relative deformation of the network with respect to the maximum possible uniaxial V elongation. The lateral pressure is "xed at PH"Pe/p"0.16. The leftmost points correspond to conditions of free swelling. (b) Snapshots of typical con"gurations of the polymer network (left boxes) and solvent within the gel (right boxes) of the stretched network-dodecamer system for D "0.606 (isotropic state) at the top, and D "0.68 (nematic V V state) at the bottom. For clarity, only the molecular backbones are depicted (by straight lines). Adapted from [68].

106

F.A. Escobedo, J.J. de Pablo / Physics Reports 318 (1999) 85}112

observed in athermal mixtures of #exible and rigid molecules [68], under proper conditions, a #exible gel can absorb large amounts of rod-like molecules. It appears that only at high degrees of swelling the mesogenic particles can compensate the entropic penalty associated with con"nement in a gel; further, the gain in entropy of mixing o!sets the loss of conformational entropy of the expanded network. As the gel is stretched, the concentration of nematogens remains almost constant until a sudden, pronounced drop is observed corresponding to the isotropic-nematic transition. In Fig. 14(b) we show two representative con"gurations of network and solvent molecules before and after the isotropic-nematic transition. This result emphasizes the importance of entropic forces in driving the isotropic-nematic phase transition. In its current form, however, it might not be directly applicable to practical applications because the nematic phase is stable for a highly stretched, shrunk gel (Fig. 14). A more useful material could be prepared, for example, if in the isotropic, highly swollen state, the mesogenic molecules were a$xed to the gel by end-chain crosslinking.

8. Future challenges 8.1. Combining simulation and theory One of the main challenges encountered in simulations of gel swelling is the large size of the system. A minimal representation of a polymer network will contain O(10) sites; larger networks are necessary if topological irregularities are to be accounted for. If we consider that a network can easily absorb ten times its own weight of solvent, it becomes clear that typical systems comprise O(10) interaction sites. Systems of these sizes pose a great challenge for conventional MC equilibration moves and, in particular, for volume-#uctuation moves, which require repositioning of all sites in the system. Equally important, polymer conformations relax slowly in dense media, thereby resulting in poor sampling of the network's elasticity. It is therefore important to complement full-blown solvent-explicit simulations with faster, albeit approximate calculations if the e!ect of topological defects and wide ranges of parametric space are to be explored. In this context, simulations of lattice systems provide an attractive means of approaching the problem, particularly if density e!ects are not important. Such an approach has been pioneered by Trautenberg et al. [18] and others [19,21], and it is likely to continue generating valuable results. A complementary approach entails the use of continuum space models in which the e!ect of solvent molecules is replaced by a mean-"eld continuum, i.e., by a solvation potential. Solvation potentials have been successfully used before in conjunction with integral-equation or density-functional theories for polymers [69,71]. In this approach, information pertaining to molecular structure is extracted from simulations of a single chain in a solvation potential (e.g. intramolecular interactions are treated exactly but intermolecular interactions are treated approximately). Note that a polymer network can be viewed as a single, giant molecule. The solvation potential = is de"ned through the probability distribution of the N-mer polymer chain `p a as , , p "Z\ exp[!b(; #= )] (3) ,  , where Z is the single chain partition function, and ; is the intramolecular energy of the network  (in vacuum). In general, = is a function of temperature, composition, and solvent density. The ,

F.A. Escobedo, J.J. de Pablo / Physics Reports 318 (1999) 85}112

107

calculation of = can be accomplished by resorting to theories capable of describing the micro, scopic structure of a #uid, such as the Polymer Reference Interaction Site Model (PRISM) [70], the Born}Green}Yvon integral equation theory [33], Scaled-particle theory [34], and density functional theory [71]. Simulation of gel swelling using a network in a solvation potential will be di!erent from other, `singlea chain simulations in that the ensemble employed is not canonical but osmotic. Because the solvent is absent, however, microscopic #uctuations can be sampled more e!ectively and ergodic (metastable) traps can be circumvented. 8.2. Ionic systems Polyelectrolyte gels have been the object of much attention for many decades because of their proclivity to exhibit volume phase transitions [62}64] and their abundance in nature. In dilute aqueous solution, a polyelectrolyte chain tends to expand because its ionized groups repel each other. If hydrophobic groups are present in the polyelectrolyte, their interaction gives rise to a net self-attraction which competes with the ionic repulsion. In these hydrophobic polyelectrolytes, the degree of ionization and the hydrophobicity can be tuned by changes of pH and chain chemistry; coil-to-globule transitions, for example, have been reported for these systems [72]. In recent years, several simulation studies have been conducted to study these structural transitions in single chains on a lattice [30,74], and in a continuum [75}79]. To some extent, the behavior of a large, single polyelectrolyte chain can be related to that of a polymer gel (e.g., the chain can be seen as a micro-size gel particle); the gel can undergo transitions from a collapsed (hydrophobic) state to an expanded (hydrophilic) state [73]. In spite of their great practical importance, however, only one simulation study has been reported concerning volume phase transitions for polyelectrolyte gels. Furthermore, that study was limited to highly simpli"ed, defect-free lattice networks in two dimensions [80]. This author did not simulate the solvent or counterions explicitly; an e!ective polymer}polymer interaction was described by a square-well potential (this interaction measures the relative preference for a site to sit next to a solvent vs. some other site). Although ionic groups within the network were not simulated, a hydrogen ion (or counterion) pressure was included which, as a "rst approximation, was assumed to exert a force inversely proportional to the gel volume. This simple model, however, was su$cient to give rise to "rst-order phase transitions in two-dimensional systems. There is a need for more elaborate simulation studies of titrating (e.g., weak polyelectrolyte) semi#exible gels in a three-dimensional continuum which should incorporate ionizable groups in the network, counterions, and approximate solvent e!ects through a solvation potential (see previous section). Such level of detail is necessary to generate realistic predictions of gel behavior. It has been shown [78] that, for linear polyelectrolytes, molecular structures and chain conformations treated within the Debye}HuK ckel approximation exhibit a di!erent density dependence than those obtained for Coulombic systems. For the latter systems, it was also found that both chain density and counterion condensation can dramatically shrink a polyelectrolyte chain [78]; these phenomena are likely to play an important role in volume phase transitions for polyelectrolyte gels. Likewise, the introduction of chain sti!ness (either intrinsic or solvent induced) can drastically change the swelling pattern of a gel because of the competition of di!erent persistence lengths [79].

108

F.A. Escobedo, J.J. de Pablo / Physics Reports 318 (1999) 85}112

9. Closing remarks In this review, we have focused on recent methodological advances, applications, and limitations associated with the simulation of phase transitions in polymeric networks and gels. Not withstanding the fact that most simulation studies so far have dealt with coarse-grained representations of a network, the work of di!erent research groups has already provided useful tests to theoretical models and unveiled the important role of entropy on the properties of polymeric gels. While the need for further methodological and theoretical advances is clear, the tools that are already available can, if applied to thoughtfully chosen simpli"ed models, generate a great deal of new, useful information. Such simulation studies are needed to provide a deeper level of understanding of the physics underlying volume phase transitions in gels, which are the cornerstone for countless known and prospective applications of both #exible and rigid gels. The information collected from simulations can aid scientists to test and develop theories to describe and correlate the properties of polymeric gels; it can also be useful to experimentalists to design gels with improved performance in absorption, release, and separation processes. The ultimate goal of this line of research is to establish a modeling and simulation methodology that can become a widely used auxiliary tool for synthetic chemists and biomedical engineers for the development of the next generation of organic gels. Acknowledgements The authors are grateful to the Dreyfus Foundation and to the National Science Foundation for "nancial support of this work. This review was completed while one of the authors (JJdP) was a visiting Professor at the Institut fuK r Polymere, ETH, ZuK rich; JJdP wishes to express his gratitude to the Institute and, in particular, to Prof. H.C. Ottinger for their hospitality. References [1] R. Dagani, Intelligent gels, Chem. Eng. News (9 June 1997) p. 26. [2] F.J. Doyle, C. Dorski, J.E. Harting, N.A. Peppas, Control and modeling of drug delivery devices for the treatment of diabetes, Proceedings of the 1995 American Control Conference, 1995; 6 vol. LXXII#4483 pp. 776}80. [3] F.J. Doyle, Responsive Gels: Volume Transitions, Vols. I and II, in Advances in Polymer Sciences 109}110, Edited by K. Dusek, Springer, Berlin, 1993. [4] P.J. Flory, Principles of Polymer Chemistry, Cornell University Press, New York, 1953. [5] B. Erman, J.E. Mark, Structures and Properties of Rubberlike Networks, Oxford, New York, 1997. [6] J.E. Mark, B. Erman, Rubberlike Elasticity. A Molecular Primer, Wiley, New York, 1988. [7] B. Erman, J.E. Mark, Use of the Fixman-Alben distribution function in the analysis of non-Gaussian rubber-like elasticity, J. Chem. Phys. 89 (1988) 3314. [8] E. Geissler, F. Horkay, A.M. Hecht, Structure and thermodynamics of #exible polymer gels, J. Chem. Phys. 100 (1994) 8418. E. Geissler, F. Horkay, A.M. Hecht, M. Zring, Elastic free energy in swollen polymer networks, J. Chem. Phys. 90 (1989) 1924. [9] K. Kremer, Numerical studies of polymer networks and gels, Comput. Mat. Sci. 10 (1998) 168. [10] J.J. de Pablo, F.A. Escobedo, Monte Carlo Methods for Polymeric Systems, Advan. Chem. Phys. 105 (1999) 337.

F.A. Escobedo, J.J. de Pablo / Physics Reports 318 (1999) 85}112

109

[11] Y.-K. Leung, B.E. Eichinger, Computer simulation of end-linked elastomers, Parts 1 and 2, J. Chem. Phys. 80 (1984) 3877. V. Galiatstos, B.E. Eichinger, Simulation of the Formation of Elatomers, Rubber Chemistry and Technology, 61 (1988) 205. [12] K.-J. Lee, B.E. Eichinger, Computer simulation of the structure and elasticity of polyurethane networks (parts 1 and 2), Polymer 31 (1990) 406. B.E. Eichinger, D.R. Rigby, M.H. Muir, Computational chemistry applied to materials design } Contact lenses, Comput. Polymer Sci. 5 (1995) 147. [13] O. Argiray, Computer simulation of the formation of polymer networks, Makromol. Chem., Macromol. Symp. 76 (1993) 211. [14] J. Gao, J.H. Weiner, Excluded volume e!ects in rubber elasticity, Macromol. 20 (1987) 2525. J. Gao, J.H. Weiner, Anisotropy e!ects on chain-chain interactions in stretched rubber, Macromol. 24 (1991) 1519. [15] J. Gao, J.H. Wiener, Monomer-level description of stress and birefringence relaxation in polymer melts, Macromol. 27 (1994) 1201. [16] G.S. Grest, K. Kremer, E.R. Duering, Kinetics and relaxation of end cross-linked polymer networks, Physica A 194 (1993) 330. [17] E.R. Duering, K. Kremer, G.S. Grest, Structure and relaxation of end-linked polymer networks, J. Chem. Phys. 101 (1994) 8169. [18] H.L. Trautenberg, J.-U. Sommer, D. Goritz, Kinetics of Network formation by end-linking- a Monte Carlo study, Macromol. Symp. 81 (1994) 153. H.L. Trautenberg, J.-U. Sommer, D. Goritz, Structure and swelling of end-linked model networks, J. Chem. Soc. Faraday Trans. 91 (1995) 2649. [19] J.-U. Sommer, T.A. Vilgis, G. Heinrich, Fractal properties and swelling behavior of polymer networks, J. Chem. Phys. 100 (1994) 9181. [20] S.F. Edwards, T.A. Vilgis, Rep. Progr. Phys. 51 (1988) 243. [21] T. Holzl, H.L. Trautenberg, D. Goritz, Monte carlo Simulations on Polymer Network deformation, Phys. Rev. Lett. 79 (1997) 2293. [22] S.K. Patel, S. Malone, C. Cohen, J.R. Gillmor, R.H. Colby, Elastic-modulus and equilibrium swelling of poly (dimethylsiloxane) networks, Macromol. 25 (1992) 5241. S.P. Malone, C. Vosburgh, C. Cohen, Validity of the swelling method for the determination of the interaction parameter, Polymer 34 (1993) 5149. [23] I. Carmesin, K. Kremer, The bond #uctuation method } A new e!ective algorithm for the dynamics of polymers in all spatial dimensions, Macromol. 21 (1988) 2819. H.P. Deutsch, K. Binder, Interdi!usion and self-di!usion in polymer mixtures } a Monte Carlo study, Chem. Phys. 94 (1991) 2294. [24] F.A. Escobedo, J.J. de Pablo, Monte Carlo simulation of branched and crosslinked polymers, J. Chem. Phys. 104 (1996) 4788. [25] F.A. Escobedo, J.J. de Pablo, A new method for generating volume changes in constant-pressure simulations of #exible molecules, Macromol. Chem. & Phys. Theory and Simulation 4 (1995) 691. [26] F.A. Escobedo, J.J. de Pablo, Simulation and theory of the swelling of athermal gels, J. Chem. Phys. 106 (1996) 793. [27] F.A. Escobedo, J.J. de Pablo, Simulation of swelling of model polymeric gels by subcritical and supercritical solvents, in press, J. Chem. Phys. (1998). [28] F.A. Escobedo, J.J. de Pablo, Phase behavior of model polymeric networks and gels, Molec. Phys. 90 (1997) 437. [29] N.R. Kenkare, S.W. Smith, C.K. Hall, S.A. Khan, Discontinuous molecular dynamics studies of end-linked polymer networks, Macromol. 31 (1998) 5861. [30] A.P. Sassi, S. Beltran, H.H. Hooper, H.W. Blanch, J. Prausnitz, R.A. Siegel, Monte Carlo simulation of hydrophobic weak polyelectrolytes } Titration properties and pH-induced structural transitions for polymers containing weak electrolytes, J. Chem. Phys. 97 (1992) 8767. [31] M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Clarendon Press, Oxford, 1987. [32] R. Everaers, K. Kremer, Topological interactions in model polymer networks, Phys. Rev. E 53 (1996) R37. Entanglement E!ects in Model Polymer networks with diamond Lattice Connectivity, Phys. Rev. E 53 (1996) R37.

110

F.A. Escobedo, J.J. de Pablo / Physics Reports 318 (1999) 85}112

[33] M.P. Taylor, J.E.G. Lipson, A Born}Greem}Yvon equation for #exible chain molecule #uids, part 1, J. Chem. Phys. 102 (1995) 2118. Ibid 102 (1995) 6272. [34] C.J. Grayce, The conformation of hard-sphere polymers in hard-sphere solution calculated by single-chain simulation in a many-body solvent in#uence functional, J. Chem. Phys. 106 (1996) 5171. H. Reiss, H.L. Frisch, J.L. Lebowitz, Statistical mechanics of rigid spheres, J. Chem. Phys. 31 (1959) 369. [35] N.A. Neuburger, B.E. Eichinger, Critical experimental test of the Flory-Rehner theory of swelling, Macromol. 21 (1988) 3060. Y. Zhao, B.E. Eichinger, Study of solvent e!ects of the dilation modulus of poly(dimethylsiloxane), Macromol. 25 (1992) 6988. [36] M. Marchetti, S. Prager, E.L. Cussler, Thermodynamic predictions of volume changes in temperature-sensitive gels. 1. Theory, Macromol. 23 (1990) 1760. [37] M. Marchetti, S. Prager, E.L. Cussler, Thermodynamic predictions of volume changes in temperature-sensitive gels. 2. Experiments, Macromol. 23 (1990) 3445. [38] K.K. Lee, E.L. Cussler, M. Marchetti, M. McHugh, Chem. Eng. Sci. 45 (1990) 766. [39] I.C. Sanchez, R.H. Lacombe, An elementary molecular theory of classical #uids. Pure #uids, J. Phys. Chem. 80 (1976) 2352. R.H. Lacombe, I.C. Sanchez, Statistical thermodynamics of #uid mixtures, J. Phys. Chem. 80 (1976) 2568. [40] J.P. Baker, D.R. Stephens, H.W. Blanch, J.M. Prausnitz, Swelling equilibria for acrylamide-based polyampholyte hydrogels, Macromolecules 25 (1992) 1955. [41] D.D. Awschalom, J. Warnock, M.W. Shafer, Liquid-"lm instabilities in con"ned geometries, Phys. Rev. Lett. 57 (1986) 1607. [42] M. Schoen, M. Thommes, G.H. Findenedd, Aspects of sorption and phase behavior of near-critical #uids con"ned to mesoporous media, J. Chem. Phys. 107 (1997) 3262. [43] A.P.Y. Wong, M.H.W. Chan, Liquid-vapor critical point of He in aerogel, Phys. Rev. Lett. 65 (1990) 2567. [44] A.P.Y. Wong, S.B. Kim, W.I. Goldburg, M.H.W. Chan, Phase separation, density #uctuations, and critical dynamics of N in aerogel, Phys. Rev. Lett. 70 (1993) 954.  J. Yoon, N. Mulders, M.H.M. Chan, He}He mixtures in 95% porous aerogel, J. Low Temp. Phys. 110 (1998) 585. [45] B.C. Freasier, S. Nordholm, A generalized van der Waals model for solvation forces between solute particles in a colloidal suspension, J. Chem. Phys. 79 (1983) 4431. B.C. Freasier, S. Nordholm, C.E. Woodward, Generalized van der Waals theory of hard sphere oscillatory structure, J. Chem. Phys. 90 (1989) 5657. [46] R. Evans, U.M.B. Marconi, Fluids in narrow pores: Adsorption, capillary condensation, critical points, J. Chem. Phys. 84 (1986) 2376. Phase equilibria of #uid interfaces and con"ned #uids. Non-local versus local density functional, P. Tarazona, U.M.B. Marconi, R. Evans, Molec. Phys. 60 (1987) 573. [47] B.K. Peterson, K.E. Gubbins, Phase transitions in a cylindrical pore. Grand canonical Monte Carlo, mean "eld theory and the Kelvin equation, Molec. Phys. 62 (1987) 215. [48] W.G. Madden, E.D. Glandt, Distribution functions for #uids in random media, J. Stat. Phys. 51 (1988) 537. W.G. Madden, J. Chem. Phys. 96 (1992) 5422. [49] J. Given, G.R. Stell, The replica Ornstein-Zernike equations and the structure of partly quenched media, Physica A 209 (1994) 495. E. Kierlik, M.L. Rosinberg, G. Tarjus, P.A. Monson, Phase diagrams of a #uid con"ned in a disordered porous material, J. Phys.: Condens. Matter 8 (1996) 9621. [50] B.K. Peterson, K.E. Gubbins, G.S. He!el"nger, U. Marini, B. Marconi, F. van Swol, Lennard}Jones #uids in cylindrical pores: nonlocal theory and computer simulation, J. Chem. Phys. 88 (1988) 6487. D. Nicholson, K.E. Gubbins, Separation of carbon dioxide-methane mixtures by adsorption: e!ects of geometry and energetics on selectivity, J. Chem. Phys. 104 (1996) 8126. [51] A.Z. Panagiotopoulos, Adsorption and capillary condensation of #uids on cylindrical pores by Monte Carlo simulation in the Gibbs ensemble, Mol. Phys. 62 (1987) 701. W.T. Gozdz, K.E. Gubbins, A.Z. Panagiotopoulos, Liquid-liquid phase transitions in pores, Mol. Phys. 84 (1995) 825.

F.A. Escobedo, J.J. de Pablo / Physics Reports 318 (1999) 85}112

111

[52] S.K. Kumar, H. Tang, I. Szleifer, Phase transition in thin "lms of symmetric binary polymer mixtures, Mol. Phys. 81 (1994) 867. H. Tang, I. Szleifer, S.K. Kumar, Critical temperature shifts in thin polymer blend "lms, J. Chem. Phys. 100 (1994) 5367. [53] J. Forsman, C.E. Woodward, Simulation of phase equilibria in planar slits, Mol. Phys. 90 (1997) 637. [54] K.S. Page, P.A. Monson, Monte Carlo calculations of phase diagrams for a #uid con"ned in a disordered porous material, Phys. Rev. E 54 (1996) 6557. [55] C. Saravanan, S.M. Auerbach, J. Chem. Phys. 109 (1998) 8755. [56] A.M. Ferrengber, R.H. Swendsen, Phys. Rev. Lett. 63 (1989) 1195. N.B. Wilding, Critical-point and coexistence-curve properties of the Lennard}Jones #uid, Phys. Rev. E 52 (1995) 602. [57] L. Sarkisov, K.S. Page, P.A. Monson, Molecular modeling of #uid phase equilibrium in disordered porous materials, to appear in Fund. Adsorption VI, F. Mennier, editor. [58] P.A. Gordon, E.D. Glandt, Liquid}liquid equilibrium for #uids con"ned within random porous materials, J. Chem. Phys. 105 (1996) 4257. [59] G. Luna-Barcenas, J.C. Meredith, I.C. Sanchez, K.P. Johnston, D.G. Gromov, J.J. de Pablo, Relationship between polymer chain conformation and phase boundaries in a supercritical #uid, J. Chem. Phys. 107 (1997) 10 782. D.G. Gromov, J.J. de Pablo, G. Luna-Barcenas, I.C. Sanchez, K.P. Johnston. Simulation of phase equilibria for polymer-supercritical solvent mixtures, J. Chem. Phys. 108 (1998) 4647. [60] H. Kawasaki, T. Nakamura, K. Miyamoto, M. Tokita, T. Komai, Multiple volume phase transition of nonionic thermosensitive gel, J. Chem. Phys. 103 (1995) 6241. [61] F.A. Escobedo, J.J. de Pablo, Phase transitions within rigid matrices, in preparation. [62] T.L. Hill, An Introduction to Statistical Thermodynamics, Dover Edition (reprint), 1986. [63] T. Tanaka, Collapse of gels and the critical endpoint, Phys. Rev. Lett. 40 (1978) 820. Y. Hirokawa, T. Tanaka, Volume phase transition in a nonionic gel, J. Chem. Phys. 81 (1984) 6379. [64] M. Shibayama, T. Tanaka, Volume phase transition and related phenomena of polymer gels, Adv. Polymer Sci. 109 (1993) 1. [65] K.L. Wang, J.H. Burban, E.L. Cussler, Hydrogels as separation agents, Adv. in Poly. Sci. 110 (1993) 67. [66] G.R. Mitchell, F.J. Davis, W. Guo, Strain-induced transitions in liquid-crystal elastomers, Phys. Rev. Lett. 71 (1993) 2947. W. Guo, G.R. Mitchell, A study of the phase-behavior of blends of side-chain liquid-crystal polymers, Polymer 35 (1994) 3706. W. Guo, F.J. Davis, G.R. Mitchell, Side-chain liquid-crystal copolymers and elastomers with a null coupling between the polymer backbone and the mesogenic groups, Polymer 35 (1994) 2952. [67] J. Kupfer, H. Finkelmann, Nematic liquid single crystal elastomers, Makromol. Chem. 12 (1991) 717. [68] F.A. Escobedo, J.J. de Pablo, Monte Carlo simulation of athermal mesogenic chains: Pure systems, mixtures, and constrained environments, J. Chem. Phys. 106 (1997) 9858. [69] C.J. Grayce, K.S. Schweizer, Solvation potentials for macromolecules, J. Chem. Phys. 100 (1994) 6846. C.J. Grayce, A. Yethiraj, K.S. Schweizer, Liquid-state theory of the density dependence conformation of nonpolar linear-polymers, J. Chem. Phys. 100 (1994) 6857. D. Gromov, J.J. de Pablo, Structure of binary polymer blends: Multiple time step hybrid Monte Carlo simulations and self-consistent integral-equation theory, J. Chem. Phys. 103 (1995) 8247. [70] K.S. Schweizer, J.G. Curro, Integral-equation theory of the structure and thermodynamics of polymer blends, J. Chem. Phys. 91 (1989) 5059. [71] D. Chandler, J.D. McCoy, S.J. Singer, Density functional theory of nonuniform polyatomic systems, J. Chem. Phys. 85 (1986) 5971 ibid. 85 (1986) 5977. S.K. Nath, J.D. McCoy, J.G. Curro, R.S. Saunders. Density functional theory of polymer-polymer phase separation behavior, J. Pol. Sci. B 33 (1995) 2307. [72] U.P. Strauss, Hydrophobic poly-electrolytes, Adv. Chem. Ser. 22 (1989) 317. Macromol. 15 (1982) 1567. [73] R.A. Siegel, B.A. Firestone, pH-Dependent equilibrium swelling properties of hydrophobic polyelectrolyte copolymer gels, Macromol. 21 (1988) 3254.

112

F.A. Escobedo, J.J. de Pablo / Physics Reports 318 (1999) 85}112

[74] H.H. Hooper, H.W. Blanch, J.M. Prausnitz, Con"gurational properties of partially ionized polyelectrolytes from Monte Carlo simulation, Macromol. 23 (1990) 4820. H.H. Hooper, S. Beltran, A.P. Sassi, H.W. Blanch, J.M. Prausnitz, Monte Carlo simulations of hydrophobic polyelectrolytes, J. Chem. Phys. 93 (1990) 2715. [75] B. Jonsson, C. Peterson, B. Soderberg, J. Phys. Chem. 99 (1995) 1251. M. Ullner, B. Jonsson, B. Soderberg, C. Peterson, A Monte Carlo study of titrating polyelectrolytes, J. Chem. Phys. 104 (1996) 3048. [76] C.E. Reed, W.F. Reed, Monte Carlo study of titration of linear polyelectrolytes, J. Chem. Phys. 96 (1992) 1609. [77] C. Jeppesen, K. Kremer, Single-chain collapse as a "rst-order transition: model for PEO in water, Europhys. Lett. 34 (1996) 563. [78] M.J. Stevens, K. Kremer, The nature of #exible linear polyelectrolytes in salt free solution: a molecular dynamics study, J. Chem. Phys. 103 (1995) 1669. M.J. Stevens, K. Kremer, Structure of slat-free linear polyelectrolytes in the Debye}Huckel approximation, J. Phys. II France 6 (1996) 1607. [79] U. Micka, K. Kremer, Persistence length of the Debye}Huckel model of weakly charged #exible polyelectrolyte chains, Phys. Rev. E 54 (1996) 2653. U. Micka, K. Kremer, The persistence length of polyelectrolyte chains, J. Phys.: Condens. Matter 8 (1996) 9463. [80] D.P. Aalberts, Microscopic simulation of phase transition in interacting ionic gels, J. Chem. Phys. 104 (1996) 4309. D.P. Aalberts, Monte Carlo study of polyelectrolyte gels, J. Pol. Sci. B 34 (1996) 1127.

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

U-DUALITY AND M-THEORY

N.A. OBERS , B. PIOLINE
Theory Division, CERN, CH-1211 Geneva 23, Switzerland
Centre de Physique TheH orique, Ecole Polytechnique, F-91128 Palaiseau, France

AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO

113

Physics Reports 318 (1999) 113}225

U-duality and M-theory N.A. Obers *, B. Pioline
 Theory Division, CERN, CH-1211 Geneva, 23, Switzerland
Centre de Physique The& orique, Ecole Polytechnique, F-91128 Palaiseau, France Received October 1998; editor: A. Schwimmer Contents 1. Introduction 1.1. Setting the scene 1.2. Sources and omissions 1.3. Outline 2. M-theory and BPS states 2.1. M-theory and type IIA string theory 2.2. M-theory superalgebra and BPS states 2.3. BPS solutions of 11D SUGRA 2.4. Reduction to type IIA BPS solutions 2.5. T-duality and type IIA/B string theory 3. T-duality and toroidal compacti"cation 3.1. Continuous symmetry of the e!ective action 3.2. Charge quantization and T-duality symmetry 3.3. Weyl and Borel generators 3.4. Weyl generators and Weyl re#ections 3.5. BPS spectrum and highest weights 3.6. Weyl-invariant e!ective action 3.7. Spectral #ow and Borel generators 3.8. D-branes and T-duality invariant mass 4. U-duality in toroidal compacti"cations of M-theory 4.1. Continuous R-symmetries of the superalgebra

116 116 118 120 120 120 123 125 127 131 133 135 136 137 138 140 141 142 143 146 146

4.2. Continuous symmetries of the e!ective action 4.3. Charge quantization and U-duality 4.4. Weyl and Borel generators 4.5. Type IIB BPS states and S-duality 4.6. Weyl generators and Weyl re#ections 4.7. BPS spectrum and highest weights 4.8. The particle alias #ux multiplet 4.9. T-duality decomposition and exotic states 4.10. The string alias momentum multiplet 4.11. Weyl-invariant e!ective action 4.12. Compacti"cation on ¹ and a$ne EK  symmetry 5. Mass formulae on skew tori with gauge backgrounds 5.1. Skew tori and Sl(d, 9) invariance 5.2. T-duality decomposition and invariant mass formula 5.3. T-duality spectral #ow 5.4. U-duality spectral #ows 5.5. A digression on Iwasawa decomposition 5.6. Particle multiplet and U-duality invariant mass formula 5.7. String multiplet and U-duality invariant tension formula

* Corresponding author.  Present address: Nordita, Blegdamsvej 17, DK-2100 Copenhagen.  UniteH mixte CNRS UMR 7644. E-mail address: [email protected] (N.A. Obers), [email protected] (B. Pioline) (N.A. Obers) 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 0 4 - 6

149 152 154 156 157 159 161 164 166 167 169 170 170 171 172 173 175 176 177

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225 5.8. Application to R couplings 5.9. Half-BPS conditions and quarterBPS states 6. Matrix gauge theory 6.1. Discrete light-cone quantization 6.2. Why is Matrix theory correct? 6.3. Compacti"cation and Matrix gauge theory 6.4. Matrix gauge theory on ¹ 6.5. Matrix gauge theory on ¹ 6.6. Matrix gauge theory on ¹ 6.7. Dictionary between M-theory and Matrix gauge theory 6.8. Comparison of M-theory and Matrix gauge theory SUSY 6.9. SYM masses from M-theory masses 7. U-duality symmetry of Matrix gauge theory 7.1. Weyl transformations in Matrix gauge theory 7.2. U-duality multiplets of Matrix gauge theory 7.3. Gauge backgrounds in Matrix gauge theory 7.4. Sl(3, 9);Sl(2, 9)-invariant mass formula for N"4 SYM in 3#1 dimensions

177 178 181 182 183 184 185 185 186 187 187 189 190 191 193

7.5. Sl(5, 9)-invariant mass formula for (2,0) theory on M5-brane 7.6. SO(5, 5, 9)-invariant mass formula for non-critical string theory on NS5-brane 7.7. Extended U-duality symmetry and Lorentz invariance 7.8. Nahm-type duality and interpretation of rank Acknowledgements Appendix A. BPS mass formulae A.1. Gamma matrix theory A.2. A general con"guration of KK-M2-M5 on ¹ A.3. A general con"guration of D0, D2, D4branes on ¹ A.4. A general con"guration of KK}w}NS5 on ¹ Appendix B. The d"8 string/momentum multiplet Appendix C. Matrix gauge theory on ¹ References Note added in proof

115

200 201 202 203 205 206 206 206 207 208 209 211 214 223

197 199

Abstract This work is intended as a pedagogical introduction to M-theory and to its maximally supersymmetric toroidal compacti"cations, in the frameworks of 11D supergravity, type II string theory and M(atrix) theory. U-duality is used as the main tool and guideline in uncovering the spectrum of BPS states. We review the 11D supergravity algebra and elementary 1/2-BPS solutions, discuss T-duality in the perturbative and non-perturbative sectors from an algebraic point of view, and apply the same tools to the analysis of U-duality at the level of the e!ective action and of the BPS spectrum, with a particular emphasis on Weyl and Borel generators. We derive the U-duality multiplets of BPS particles and strings, U-duality invariant mass formulae for 1/2- and 1/4-BPS states for general toroidal compacti"cations on skew tori with gauge backgrounds, and U-duality multiplets of constraints for states to preserve a given fraction of supersymmetry. A number of mysterious states are encountered in D43, whose existence is implied by T-duality and 11D Lorentz invariance. We then move to the M(atrix) theory point of view, give an introduction to Discrete Light-Cone Quantization (DLCQ) in general and DLCQ of M-theory in particular. We discuss the realization of U-duality as electric}magnetic dualities of the Matrix gauge theory, display the Matrix gauge theory BPS spectrum in detail, and discuss the conjectured extended U-duality group in this scheme.  1999 Elsevier Science B.V. All rights reserved. PACS: 11.11.Kc; 11.25.!w; 11.27.#d; 11.30.!j; 2.20.Rt Keywords: M-theory; U-duality; T-duality; BPS spectrum; Matrix theory; Gauge theory

116

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

1. Introduction 1.1. Setting the scene Since its invention in the late 1960s, string theory has grown up in a tumultuous history of unexpected paradigm shifts and deceptive lulls. Not the least of these storms was the discovery that the "ve anomaly-free perturbative superstring theories were as many glances on a single elevendimensional theoria incognita, soon baptized M-theory, awaiting a better name [304,314]. The genus expansion of each string theory corresponds to a di!erent perturbative series in a particular limit g P0 in the M-theory parameter space, much in the same way as the genus expansion arises Q in 't Hooft large-N, "xed-g N regime of Yang}Mills theory [298]. M-theory can be de"ned by the 7+ superstring expansions on each patch, and the superstring (perturbative or non-perturbative) dualities allow a translation from one patch to another, in a way analogous to the de"nition of a di!erential manifold by charts and transition functions. This analogy overlooks the fact that string theories are only de"ned as asymptotic series in g P0, and some analyticity is therefore Q required to move into the bulk of parameter space. This de"nition has been e!ective in uncovering a number of features of M-theory, or rather its BPS sector, which behaves in a controlled way under analytic continuation at "nite-g . In Q particular, M-theory is required to contain Cremmer, Julia and Scherk's eleven-dimensional supergravity [74] in order to account for the Kaluza}Klein-like tower of type IIA D0-branes as excitations carrying momentum along the eleventh dimension of radius R &g, as shown by Q Q Townsend and Witten [304,314]; it should also contain membrane and "vebrane states, in order to reproduce the D2- and D4-brane, as well as the NS5-brane and the type IIA `fundamentala string. Which of these states is elementary is not decided yet, although M2-branes and D0-branes are favourite candidates [84,22]. It may even turn out that none of them may be required, and that 11D SUGRA may emerge as the low-energy limit of a non-gravitational theory [160]. While the dualities between string theories relate di!erent languages for the same physics, the symmetries of string theory provide a powerful guide into M-theory, which is believed to hold beyond the BPS sector. The best established of them is certainly T-duality, which identi"es seemingly distinct string backgrounds with isometries (see for instance Refs. [3,132] and references therein). Throughout this review, we shall restrict ourselves to maximally supersymmetric type II or M theories, and accordingly T-duality will reduce to the inversion of a radius on a d-dimensional torus. To be more precise, a T-duality maps to each other type IIA and type IIB string theories compacti"ed on circles with inverse radii, while a T-symmetry consists of an even number of such inversions (together with Kalb}Ramond spectral #ows to which we shall return), and therefore corresponds to a symmetry of type II string theories and of their M-theory extension. As we shall recall, such T-symmetries on a torus ¹B generate a SO(d, d, 9) discrete symmetry group, the continuous version of which SO(d, d, 1) appears as a symmetry of the low-energy e!ective action. On the other hand, the action of 11D or type IIA supergravity compacti"ed on a torus ¹B as well as the equations of motion of uncompacti"ed type IIB supergravity have for long been known to exhibit continuous non-compact global symmetries, namely the exceptional symmetry E (1) of BB  Having emphasized this point, we shall henceforth omit the distinction between dualities and symmetries.

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

117

Cremmer and Julia and the Sl(2, 1) symmetry of Schwarz and West, respectively [70,178,273,318]. These symmetries transform the scalar "elds and in general do not preserve the weak coupling regime, which puts them out of reach of perturbation theory, in contrast to the well established target-space T-duality. In analogy with the electric}magnetic Sl(2, 9) Montonen}Olive}Sen duality of four-dimensional N"4 super Yang}Mills theory [226,277], Hull and Townsend have proposed [172] that a discrete subgroup E (9) (resp. Sl(2, 9) ) remains as an exact quantum symmetry of M-theory compacti"ed BB on a torus ¹B (resp. of ten-dimensional type IIB string theory and compacti"cations thereof). The two statements are actually equivalent, since after compacti"cation on a circle the type IIB string theory becomes equivalent under T-duality on the (say) tenth direction to type IIA, and the symmetry E (9) can be obtained by intertwining the Sl(2, 9) non-perturbative symmetry with the BB T-duality SO(d!1, d!1, 9). Conversely, the Sl(2, 9) symmetry of type IIB theory can be obtained from the M-theory description as the modular group of the 2-torus in the tenth and eleventh directions [16,269], and is a particular subgroup of the modular group Sl(d, 9) of the d-torus. This, being a remnant of eleven-dimensional di!eomorphism invariance after compacti"cation on the torus ¹B, has to be an exact symmetry as soon as M-theory contains the graviton. The T-duality symmetry SO(d!1, d!1, 9) is however not manifest in the M-theory picture. All in all, the U-duality group reads E (9)"Sl(d, 9) ( ) SO(d!1, d!1, 9) , (1.1) BB where the symbol ( ) denotes the group generated by the two non-commuting subgroups. The structure of the group (1.1) will be discussed at length in this review, and a set of Weyl and Borel generators will be identi"ed. The former preserve the rectangularity of the torus and the vanishing of the gauge background, while the latter allow a move to arbitrary tori. States are classi"ed into representations of the U-duality group E (9), whether BPS or not, and we will BB derive U-duality invariant mass and tension formulae for 1/2- and 1/4-BPS states, as well as conditions for a state to preserve a given fraction of the supersymmetries. Besides the entertaining encounter with discrete exceptional groups, this will actually teach us about the spectrum of M-theory, since the more M-theory is compacti"ed, the more degrees of freedom come into play. In particular, we will show the need to include states with masses that behave as 1/gL, n53, which are Q unconventional in perturbative string theory. An important application of these results is the exact determination of certain physical amplitudes in M-theory, such as the four-graviton R coupling, which can be interpreted as traces over M-theory BPS states [20,139,248]. The weak coupling analysis of these exact couplings provides a very useful insight into the rules of semi-classical calculus in string theory [19,32,137,143,194,241,250]. A proposal has recently been put forward by Banks, Fischler, Shenker and Susskind to de"ne M-theory ab initio on the (discrete) light front, as the large-N limit of a supersymmetric matrix model given by the dimensional reduction of 10D ;(N) super Yang}Mills (SYM) theory to 0#1 dimension [22,296]. This model also describes low-energy interactions of D0-branes induced by open string #uctuations [80,98,181], and, as shown by Seiberg, arises from considering M-theory on the light front as a particular limit of M-theory compacti"ed on a circle, i.e. type IIA  The "rst example of string duality actually appeared in the context of heterotic string theory [116].

118

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

theory [274]. D0-branes are therefore identi"ed as the partons of M-theory in this framework. This proposal has passed numerous tests, and has been shown to incorporate membrane and (transverse) "vebrane solutions, and to reproduce 11D SUGRA computations. The invariance under eleven-dimensional Lorentz invariance remains, however, to be demonstrated (see [212] for a step in that direction). Upon compacti"cation of d dimensions, the D0-branes interact by open strings wrapped many times around the compacti"cation manifold, and the in"nite-dimensional quantum mechanics can be rephrased as a gauge theory in d#1 dimensions [124,301]. This dramatic increase of degrees of freedom certainly removes part of the appeal of the proposal, but becomes even more serious for d54, where the gauge theory loses its asymptotic freedom and becomes ill-de"ned at small distances. We will brie#y discuss the proposals for extending this de"nition to d"4, 5. We will also discuss the interpretation of M-theory BPS states in the gauge theory, and show the occurrence of unconventional states with energy 1/gL , n52. Despite these di$culties, the Matrix gauge theory 7+ gives a nice understanding of U-duality as the electric}magnetic duality of the gauge theory, together with the modular group of the torus on which it lives [108,124,295]. The interpretation of "nite-N matrix theory as the compacti"cation of M-theory on a light-like circle implies that the U-duality group E (9) be enlarged to E (9) [52,169,170,239]; we will show that this extra BB B>B> symmetry mixes the rank N of the gauge group with charges in a way reminiscent of Nahm duality. All these features are guidelines for a hypothetical fundamental de"nition of Matrix gauge theory. 1.2. Sources and omissions This review is intended as a pedagogical introduction to M-theory, from the point of view of its 11D SUGRA low-energy limit, its strongly coupled type II string description, and its purported M(atrix) theory de"nition. It is restricted to maximally supersymmetric toroidally compacti"ed M-theory, and uses U-duality as the main tool to uncover the part of the spectrum that is annihilated by half or a quarter of the 32 supersymmetries. The exposition mainly follows [108,239,250], but relies heavily on [89,274,307,314]. It is usefully supplemented by other presentations on supergravity solutions [292,306,307], M(atrix) theory [21,49,92], D-branes [18,252,263,302], string dualities [85,96,233,271,280,313] and perturbative string theory [193,242] and general introductions [14,249,272]. The following topics are beyond the scope of this work: E Black hole entropy: the modelisation of extremal black-holes by D-brane bound states has allowed a description of their microscopic degrees of freedom and a derivation of their Bekenstein}Hawking entropy [293] (see [219,247] for reviews). The latter can be related to a U-duality invariant of the black hole charges [5,9,77,164,184], and U-duality can even allow the control of non-extremal states [290]. E Gauge dynamics: the study of D-brane con"gurations has also led to a qualitative understanding of gauge theories dynamics as world-volume dynamics of these objects; see [130] for a thorough review. We will mainly consider con"gurations of parallel branes, as describing the Matrix gauge theory description of M-theory on the light cone. E BPS-saturated amplitudes: A special class of terms in the e!ective actions of M-theory and string theory receives contributions from BPS states only. We will brie#y discuss an application of the M-theory mass formulae that we derived to the computation of exact R couplings in Section 5.8,

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

E

E

E

E

E

E

E

119

and refer to the existing literature for more details on the exact non-perturbative computation of these couplings, and their interpretation at weak coupling as a sum of instanton e!ects. Relevant references include [32,241,276] for two-derivative terms in N"2 type II strings, [12,144,153,154,319,324,325] for four-derivative terms in type II/heterotic theories, [19,20,194] for F (and related) terms in type I/heterotic theories, and [12,46,137}139,142,143,187,195, 248,250] for R (and related) terms in type IIB/M-theory. In"nite series of higher-derivative BPS-saturated RFE\ or RHE\, and RL terms have also been computed or conjectured in Refs. [11,221,47,186,262] and [264]. Scattering amplitudes: in order to validate the M(atrix) theory conjecture of BFSS, a number of scattering computations have been carried out both in the Matrix model and in 11D supergravity; they have shown agreement up to two loops, see for instance [30,31,64,94,110,188,224, 240,253,303]. This agreement is better than naively expected, and indicates the existence of nonrenormalization theorems [243] for these interactions. D-instanton matrix model: an alternative formulation of M-theory as a statistical matrix model has been proposed by Ishibashi, Kawai, Kitazawa and Tsuchiya [176]. It has so far not been developed to the same extent as the BFSS proposal, and in particular the origin of U-duality has not been explicited. See Refs. [4,13,51,61}63,111,112,120,121,157,196,198,220,266,297] for further discussion. Twelve dimensions and beyond: the structure of U-duality symmetry has led to speculate on the existence of a 12D [25,28,237,310] or higher [24,26,27,235,236,261,289] dimensional parent of M-theory, with extra time directions. The N"2 heterotic strings suggest an appealing construction of this theory (see [222] for a review). However, the full higher-dimensional Lorentz symmetry is partially reduced to its U-duality subgroup, and its usefulness remains unclear at present. We shall, however, encounter in Section 4.6 a tantalizing hint for an extra time-like direction with `lengtha l. N String networks: a construction of 1/4-BPS states based on three-string junctions [58,81,271] has been suggested [284], that reproduces the U-duality invariant mass formula in 8 dimensions [48,48,203]. These solutions have been constructed from the M2-brane [201,223] and their dynamics discussed in [59,259], but their supergravity description is still unclear. Non-commutative geometry: it has been argued that non-commutative geometry [67] is the appropriate framework to discuss D-brane dynamics, and is even required in the presence of Kalb}Ramond two-form background [65,68,97]. This description incorporates T-duality [256,267] and even U-duality in its Born}Infeld generalization [159]. It should in particular (see Section 7.8) extend Nahm's duality of ordinary two-dimensional Maxwell theory to higherdimensional cases [232]. Related discussions can be found in Refs. [42,69,158,185,206,208,228]. Gauged supergravity: 11D SUGRA possesses maximally supersymmetric backgrounds other than tori, namely compacti"cations on products of spheres and anti-deSitter spaces [117]. These correspond to the near-horizon geometry of M2- and M5-branes, and have been argued to provide a dual description to the gauge theory on these extended objects [218]. They will be ignored in this review. String theories with non maximal supersymmetry: the E ;E heterotic string and type I string   can be obtained from M-theory by orbifold compacti"cation [162,163], while the SO(32) heterotic string is related to E ;E by a T-duality, or to type I string theory by a non  perturbative duality [254]. M(atrix) theory descriptions have been proposed both in the

120

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

heterotic [23,88,134,161,182,189,199,200,210,211,230,257] and the type II [44,87,88, 115,135,190,192] cases, as well as on non-orientable surfaces [191,317], and will not be treated here. E Non-BPS states: The study of stable non-BPS states has been iniated in [283] and further examined [34,281,282,285]. It would be interesting to investigate the implications of U-duality symmetry on the spectrum of non-BPS states. 1.3. Outline Section 2 introduces the superalgebra and fundamental BPS states of M-theory in the context of 11D SUGRA and type IIA/B superstring theories. T-duality is recalled and revisited in Section 3 from an algebraic point of view, at the level of the e!ective action and of the perturbative and non-perturbative BPS spectrum. The same techniques are used in Section 4 to introduce U-duality and its action on the spectrum of particles and strings, restricting to Weyl generators. Borel generators are incorporated in Section 5, where U-duality invariant mass and tension formulae for general toroidal compacti"cation with arbitrary gauge backgrounds are derived, as well as U-duality multiplets of BPS constraints. Section 6 introduces matrix gauge theory as the discrete light-cone quantization of M-theory following an argument by Seiberg, and discusses the dictionary between M-theory and Matrix gauge theory. The U-duality symmetry is "nally discussed in Section 7 from the perspective of the Matrix gauge theory, as well as the extended U-duality symmetry arising from the extra light-like direction.

2. M-theory and BPS states 2.1. M-theory and type IIA string theory M-theory was originally introduced as the strong coupling limit of type IIA superstring theory. The latter has been argued [304,314] to dynamically generate an extra compact dimension at "nite coupling of radius R &g in units of an eleven-dimensional Planck length l : Q Q N R /l "g, Q N Q

l"g l , N QQ

(2.1)

where 1/l"a denotes the string tension and g its coupling constant. The strong coupling limit Q Q g PR should therefore exhibit eleven-dimensional N"1 super-PoincareH invariance. Q While a consistent eleven-dimensional theory of quantum gravity is still missing, it has been known for a long time that type IIA supergravity can be obtained from the eleven-dimensional N"1 supergravity of Cremmer, Julia and Scherk, by dimensional reduction on a circle. M-theory is therefore required to reduce at energies much smaller than 1/l to 11D SUGRA, in the same way N as type IIA (or type IIB) superstring theory reduces to type IIA [60,128,175] (or type IIB [165,268,273]) supergravity at energies much smaller than 1/l (which is also smaller than both the Q ten-dimensional Planck mass g\/l and the eleven-dimensional Planck mass g\/l at weak Q Q Q Q

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

121

coupling). This is summarized in the following diagram: M-theory

type IIA string theory

P 1

11D supergravity P 10D type IIA supergravity 1 where compacti"cation on a circle occurs from left to right and the energy decreases from top to bottom. The matching relations (2.1) can be easily obtained by studying the Kaluza}Klein reduction of 11D SUGRA, described by the action










l (2 1 CdCdC S " dx(!g R! N (dC) #  l 2 ) 3 48 N

(2.2)

up to fermionic terms that we will ignore in the following. In addition to the usual Einstein}Hilbert term involving the scalar curvature R of the metric g , the action contains a kinetic term for the +, 3-form gauge potential C (which we shall often denote by C ) as well as a topological +,0  Wess}Zumino term required by supersymmetry. The action (2.2) does not contain any dimensionless parameter, and the normalization of the Wess}Zumino term with respect to the Einstein term is "xed by supersymmetry. The dependence on the Planck length l has been reinstated by N dimensional analysis, with the following conventions: [dx]"0, [g

]"2, +,

[(!g]"11, [R]"!2, [C

]"0, +,0

[d]"0 ,

(2.3)

relegating the dimension to the metric only. In particular, the relation between the elevendimensional Planck length and Newton's constant is given by i "l/(2(2n)). We will generally  N ignore all numerical factors. Dimensional reduction is carried out by substituting an ansatz ds "R(dxQ#A dxI)#ds  Q I 

(2.4)

for the metric, where R stands for the #uctuating radius of compacti"cation (as measured in the Q eleven-dimensional metric) and A describes the Kaluza}Klein ;(1) gauge "eld arising from the isometry along xQ, and splitting the three-form C in a two-form B "C and a 3-form C . +,0 IJ IJQ IJM Only the zero Fourier component (i.e. the zero Kaluza}Klein momentum part) of these "elds along xQ is kept. On dimensional grounds the scalar curvature becomes




RR  Q #R(dA) , R(g )"R(g )# Q +, IJ R Q  The subscript s is used to indicate that string theory is obtained in this way.

(2.5)

122

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

so that the reduced action reads








RR  l 1 Q #R(dA)#l(dC)# N (dB) S " dxR (!g R# Q N  l Q R R Q Q N



# BdCdC .

(2.6)

On the other hand, the low-energy limit of type IIA string theory is given (in the string frame) by the action








1 l l l S " dx(!g e\( R#4(R )! Q (dB) ! Q (dA)! Q (dC) '' l 12 4 48 Q



# BdCdC ,

 (2.7)

which describes the dynamics of the (bosonic) massless sector NS-NS: g , B , , (2.8a) IJ IJ R-R: A , C (2.8b) I IJM denoting the metric, antisymmetric tensor and dilaton from the Neveu}Schwarz sector, and the one- and three-form gauge potentials from the Ramond sector (indices k run over 1 to 10). Ramond p-form gauge "elds will be generically denoted by R . The dependence on the string length l is N Q again instated on dimensional grounds, while the dependence on the coupling g"e( (2.9) Q stems from the fact that the two-derivative action originates from string tree level (hence the e\( factor), with each Ramond "eld coming with an additional power of e(, ensuring the correct Maxwell and Bianchi identities (see [255] for a recent discussion). In particular, the ten-dimensional Newton's constant is given by i "gl. Identifying the dilaton with the scalar modulus  Q Q ln R up to a numerical factor, and matching the two actions (2.6) and (2.7) leads to the relations Q 1 1 1 R 1 R 1 R Q" , Q" , Q" , " , (2.10) l l l l gl R l gl l Q Q QN Q Q N Q N Q N obtained by comparing the terms R, dB, dA and dC, respectively in Eqs. (2.6) and (2.7). Two of these four relations turn out to be redundant as a consequence of supersymmetry, and they can be reduced to the matching relations already stated in Eq. (2.1), or equivalently R 1 Q" , (2.11) l l N Q which summarize the relation between the M-theory parameters +l , R , and the string theory N Q parameters +l , g ,. Q Q R "l g , Q Q Q

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

123

Using Eqs. (2.11) and (2.9) in the metric (2.4) we "nd the alternative form ds ds "e((dxQ#A dxI)#e\(  , (2.12) I l l N Q which will be used to relate low-energy solutions of M-theory and type IIA string theory. 2.2. M-theory superalgebra and BPS states While M-theory has to reduce to 11D SUGRA in the low-energy limit, little is known about its microscopic degrees of freedom. It is however postulated that the N"1 supersymmetry of 11D SUGRA should be valid at any energy, and the spectrum is therefore organized into representations of the super-PoincareH algebra [305]: 1 1 ) Z+,./0 , +Q , Q ,"(CC+) Z # (CC ) Z+,# (CC +, ?@ +,./0 ?@ ? @ ?@ + 2 5!

(2.13a)

(2.13b) [Q , Z+2]"0 . ? Here Q denotes the 32-component Majorana spinor generating the supersymmetry (see [202] ? for a general account on spinorial representations), and C 2 the antisymmetric product of +, C matrices, i.e. C C 2 for distinct indices and zero otherwise. See Appendix A.1 for our gamma + , matrix conventions. In addition to the usual translation operator P , which we denoted by Z for uniformity, the + + right-hand side of Eq. (2.13a) contains `central chargesa Z+, and Z+,012 in non-trivial representations of the Lorentz group. These charges appear as irreducible representations (irreps) in the decomposition 528"11#55#462 of the symmetric tensor product +Q , Q ,, and the simplest ? @ assumption is that they should commute with the SUSY charges Q  (their commutation properties ? with the Lorentz generators are encoded in their index structure). They can be identi"ed as the electric and magnetic charges of extended objects [82,166] with respect to the gauge potential C and the metric g and their Kaluza}Klein descendants. +,. +, The various components of the central charges, their corresponding potentials, as well as the nature of the solution, are summarized in Table 1. Here, E denotes the six-form dual to C and   K the 7-form dual to the Kaluza}Klein gauge potential g after compactifying the '_'+,./012 '+ direction I. This hints toward the existence of extended states charged under these gauge "elds, namely 2-branes, 5-branes, 6-branes and 9-branes. The 9-branes, which are not charged under a gauge potential, are not dynamical and correspond to the `end-of-the-worlda branes in compacti"cations of M-theory with lower supersymmetry [162]. They will not be further considered in this review, but we will shortly return to the M2, M5 and KK6-brane.

 It is possible to introduce non-Abelian relations while still preserving the Jacobi identity [288], but the status of this possibility is still unclear.  No antisymmetry is assumed for indices separated by a semi-colon. The peculiar index structure K "K ensures that the seven-form indices M,2,¹ are distinct from the compact direction I, and the _ '_'+,./012 double occurrence of I has the same origin as the square radius R in Eq. (2.28).

124

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

Table 1 M-theory central charges, gauge "elds and extended objects Z 

Z '

Z'(

Z'()*+

Z'

Z'()*

g  mass

g ' momentum

C '( M2-brane

E '()*+ M5-brane

none 9-brane

K +_+,./012 KK6-brane

The generic representation of the superalgebra (2.13) is generated by the action of 16 fermionic creation operators on a vacuum "02 in a given representation of the Lorentz group; it is therefore 2-dimensional, i.e. contains 32768 bosonic states and 32768 fermionic states. The positivity of the matrix 10"+Q , Q ,"02 implies a bound on the rest mass Z known as the Bogomolny bound. When ? @  this bound is saturated, part of the supersymmetries annihilate the vacuum "02: (2.14) Z+,2(CC 2)?@Q "02"0 , +, @ 8 resulting in a reduced degeneracy. Equation (2.14) requires that the 32;32 matrix Z+,2(C 2)?@ +, has at least one zero eigenvalue, and implies in particular the BPS condition






(2.15) det (Z ) C)?@ "0 , ?@ 8 which determines the rest mass Z in terms of the other charges.  The dimension can be further reduced if the zero eigenvalue is degenerate, and this requires more relations between the various charges. Since only Z contributes to the trace on the right-hand side  of Eq. (2.13a), the maximum number of zero eigenvalues is 16, corresponding to a state annihilated by half the supersymmetries, or in short a 1/2-BPS state. Because of its reduced dimension, a BPS state with smallest charge cannot decay, except if it can pair up with another state of opposite charge to make a representation twice as long [316]. These states can therefore be followed at strong coupling (in the M-theory language, this means for arbitrary geometries of the compacti"cation manifold) and serve as the basis for many duality checks. As an illustration, we wish to investigate the case where, besides the mass M"Z , only the  two-form central charges Z'( do not vanish. This will be later interpreted as an arbitrary superposition of M2-branes. We therefore have to solve the eigenvalue equation: Ce"Me, C,Z'(C . '( Squaring this equation yields

(2.16)

C"Z'(Z'(#Z'(Z)*C >M , (2.17) '()* where the symbol > denotes the equality when acting on e. The space of solutions now depends on the value of k'()*,Z '(Z)* "ZZ. If k"0, Eq. (2.17) implies (Z'()"M and C"M. Since Tr C"0, the 32;32 matrix C has 16 eigenvalues M and 16 eigenvalues !M, and therefore

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

125

Eq. (2.16) is satis"ed for a dimension-16 space of vectors e. The state with charges Z'( is therefore annihilated by half the supersymmetry generators Q , and has a mass ? M"Z'(Z'( . 

(2.18)

The condition ZZ"0 means that the antisymmetric charge matrix Z'( has rank 2, i.e. that only parallel M2-branes are superposed. If on the other hand ZZO0, we may rewrite Eq. (2.17) as Ce"(M!M)e, C"k'()*C ,  '()*

(2.19)

and we are lead back to an equation similar to Eq. (2.16). Squaring again yields C"(k'()*)#(k ) k)'()*C #(kk)'()*+,./C '()* '()*+,./ >(M!M) , 

(2.20)

where (k ) k)'()*"k'(+,k)*+,. As before, if k ) k"kk"0, this equation implies (k'()*)"(M!M)"C. Since Tr C"0, Eq. (2.19) is satis"ed by half the supersymmetries,  but Eq. (2.16) by a quarter only. We therefore get a 1/4-BPS state with mass squared: M"Z'(Z'(#(k'()*k'()* ,

(2.21a)

k'()*"Z '(Z)* .

(2.21b)

This expression reduces to Eq. (2.18) for a 1/2-BPS state, i.e. when k'()*"0. On the other hand, if k ) k or kkO0 do not vanish, the state is at most 1/8-BPS and we have to carry the same analysis one step further. Note that the conditions k ) kO0 (resp. kkO0) can only be satis"ed when d56 (resp. d58), in agreement with the absence of 1/8-BPS states in more than "ve space-time dimensions. 2.3. BPS solutions of 11D SUGRA In want of a microscopic formulation of M-theory (or of non-perturbative type IIA string theory), it is certainly di$cult to determine what representations of the eleven-dimensional PoincareH superalgebra actually occur in the spectrum. However, this is achievable for BPS states, since supersymmetry protects these from quantum e!ects and in particular determines their exact mass formula. They can be studied at arbitrarily low energy, and in particular in the 11D SUGRA limit of M-theory. Instead of describing the equations implied by the BPS condition on the supergravity con"guration, we refer the reader to existing reviews in the literature [102,104,292,306,307], and content ourselves with recalling the four 1/2-BPS standard solutions: the pp-wave and three extended solutions, the membrane (or M2-brane), "vebrane (M5-brane) and the Kaluza}Klein monopole, also known as the KK6-brane. The eleven-dimensional metric describing the extended solutions splits into two parts: the world-volume, denoted by EN, including the time and p world-volume directions, and the

126

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

transverse Euclidean part E\N. These solutions are given in terms of a harmonic function H on the transverse space, which we choose as a single pole k , H(r)"1# r\N

(2.22)

although any superposition of such poles would do (this is stating the no-force condition between static BPS states; the constant shift in Eq. (2.22) ensures the asymptotic #atness of space-time, required for a soliton interpretation). The constant k depends on Newton's constant i and on the  p-brane tension, and is quantized by the requirement that the space-time be smooth (we will henceforth choose the smaller quantum). The pp-wave and KK6-brane solutions only involve the metric, and read [167,304] pp-wave: ds "!dt#do#(H!1)(dt#do)#ds(E) ,  k H"1# . r

(2.23a) (2.23b)

KK6-brane: ds "ds(E)#ds (y) ,  2,

(2.24a)

ds "HdyGdyG#H\(dt #< (y)dyG), i"1, 2, 3 , 2, 2, G

(2.24b)

k

;<" ) H, H"1# . "y"

(2.24c)

The KK6-brane solution is analogous to the "ve-dimensional Kaluza}Klein monopole [291], and is built out from the four-dimensional Taub}NUT gravitational instanton (see Ref. [107] for a review of this topic), which is asymptotically of the form 1;S, where t is the compact 2, coordinate of S with period 2nR. Consequently, this solution only arises when at least one direction is compact. It is localized in the four Taub}NUT directions, as should be the case for a 6-brane, and magnetically charged under the graviphoton g . It can be considered as the I2, electromagnetic dual of a pp-wave, electrically charged under the graviphoton arising after compacti"cation on a circle of radius R. pp-waves in compact directions will be called indi!erently Kaluza}Klein excitations or momentum states. The corresponding solutions for the M2- and M5-brane read [106,146]: M2-brane: ds "H\ds(E)#Hds(E) , 

(2.25a)

dC "Vol(E)dH\ , 

(2.25b)

i T k"   . 3X 

(2.25c)

k H"1# , r

 The name pp-wave stands for plane fronted wave with parallel rays [53]. The solution (2.23a), (2.23b) was generalized in [39] to include excitations of the three-form potential.

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

127

M5-brane: ds "H\ds(E)#Hds(E) , (2.26a)  dC "夹 dH , (2.26b)   i T k (2.26c) H"1# , k"   , 3X r  which also show that the M2-brane (resp. M5-) is electrically (resp. magnetically) charged under the 3-form gauge potential. The symbol 夹 denotes Hodge duality in q dimensions, and X the volume O L of the sphere SL with unit radius: 2nL>  X" . (2.27) L C(L>)  The tensions (or mass per unit world-volume) of these four basic BPS con"gurations can be easily evaluated from ADM boundary integrals and Dirac quantization, or more easily yet by dimensional analysis: R 1 KK6-brane: T " , KK-state: T " ,   R l N (2.28) 1 1 M5-brane: T " . M2-brane: T " ,  l  l N N The tension (i.e. mass) of the pp-wave with momentum along a compact direction of radius R (occasionally denoted as R ) is the one expected for a massless particle in eleven dimensions; the 2, tension of the KK6-brane is easily obtained from the latter by electric}magnetic duality, after reading o! from Eq. (2.6) the Kaluza}Klein gauge coupling 1/g "R/l: )) N R T . (2.29) T" "  g l N )) All these BPS states have been inferred from a classical analysis of 11D supergravity. They should in principle arise from a microscopic de"nition of M-theory, which would allow a full account of their interactions. Nevertheless, it is still possible to formulate their dynamics in terms of their collective coordinates which result from the breaking of global symmetries in the presence of the soliton [127]. Supersymmetry gives an important guideline, since (the unbroken) half of the 32 supercharges has to be realized linearly on the world-volume, while the other half is realized non-linearly. This "xes the dynamics of the M5-brane to be described in terms of the chiral (2, 0) six-dimensional tensor theory [57], while the membrane is described by the 2#1 supermembrane action [40,84]. Unfortunately, the quantization of these two theories remains a challenge. As for the KK6-brane, the description of its dynamics is still an unsettled problem [152]. 2.4. Reduction to type IIA BPS solutions Upon compacti"cation on a circle (with periodic boundary conditions on the fermion "elds), the supersymmetry algebra is una!ected and the generators merely decompose under the reduced Lorentz group. The 32-component Majorana spinor Q decomposes into two 16-component ? Majorana}Weyl spinors of SO(1,9) with opposite chiralities, and the N"1 supersymmetry in 11D

128

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

Table 2 Type IIA central charges, gauge "elds and extended objects Z

ZGH

ZG

ZGHIJK

ZGHIJ

ZG

Z

ZGHIJ

ZGHI

A  D0

C GH D2

B G F1

E GHIJK NS5

R GHIJ D4

none D8

none 9-brane

K K_KNOPQ KK5

R JKLNOP D6

gives rise to non-chiral N"2 supersymmetry in 10D. However, it is convenient not to separate the two chiralities explicitly, and rewrite the supersymmetry algebra as 1 +Q , Q ,"(CCI) P #(CC ) Z# (CC ) ZIJ#(CC C ) ZI IJ ?@ I Q ?@ ? @ ?@ I Q ?@ 2 1 1 # (CC ) ZIJMNO# (CC C ) ZIJMN , IJMNO ?@ IJMN Q ?@ 5! 4!

(2.30)

where the eleventh Gamma matrix C is identi"ed with the 10D chirality operator C C 2C . The Q    eleven-dimensional central charges give rise to the charges Z, ZI, ZIJ, ZIJMN, ZIJMNO whose interpretation is summarized in Table 2, where we omitted the momentum charge P . In this table, I K denotes the 6-form dual to g after compacti"cation of the direction m. K_KLNOPQR IK Under Kaluza}Klein reduction, the BPS solutions of 11D SUGRA yield BPS solutions of type IIA supergravity. This reduction can, however, be carried out only if the eleventh dimension is a Killing vector of the con"guration. This is automatically obeyed if the eleventh direction is chosen along the world-volume EN, and reduces the eleven-dimensional p-brane to a tendimensional (p!1)-brane with tension T "RT ; this procedure is called diagonal or double N\ N reduction [103], and we shall call the resulting solutions wrapped or longitudinal branes. One may also want to choose the eleventh direction transverse to the brane, but this is not an isometry, since the dependence of the harmonic function H on the transverse coordinates is non-trivial. However, this can be easily evaded by using the superposition property of BPS states, and constructing a continuous stack of parallel p-branes along the eleventh direction. The harmonic function on E\N turns into an harmonic function on E\N:





1 dxQ & . \N o\N \[(xQ)#o] 

(2.31)

We therefore obtain an unwrapped or transverse p-brane in ten dimensions with the same tension T as the one we started with. This procedure is usually called vertical or direct reduction. It has N also been proposed to reduce along the isometry that arises when the sphere S\N in the transverse space E\N is odd-dimensional, hence a ;(1) Hopf "bration [105], but the status of the solutions obtained by this angular reduction is still unclear. Applying this procedure to the four M-theory BPS con"gurations, with tensions given in Eq. (2.28), we "nd, after using the relations (2.11), the set of BPS states of type IIA string theory listed in Table 3. Table 3 shows that we recover the set of all 1/2 BPS solutions of type IIA string theory, which include the KK excitations, the fundamental string and the set of solitonic states

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

129

Table 3 Relation between M-theory and type IIA BPS states M-theory

Mass/tension

Type IIA

Longitudinal M2-brane

R 1 T " Q"  l l N Q 1 1 T " "  l g l N QQ R 1 T " Q"  l g l N QQ 1 1 T " "  l gl N Q Q 1 1 T " "  R gl Q QQ 1 1 T " "  R R G G R R R T " Q 2," 2,  l gl N Q Q R 1 T " Q"  l g l N QQ R R T " 2," 2,  gl l Q Q N

F-string

Transverse M2-brane Longitudinal M5-brane Transverse M5-brane Longitudinal KK mode Transverse KK mode Longitudinal KK6-brane KK6-brane with R "R 2, Q Transverse KK6-brane

D2-brane D4-brane NS5-brane D0-brane KK mode KK5-brane D6-brane 6-brane 

comprised by the NS5-brane, KK5-brane and the Dp-branes with p"0, 2, 4, 6. The NS5-brane is a solitonic solution that is magnetically charged under the Neveu}Schwarz B-"eld [57]. The Dp-branes are solitonic solutions, electrically charged under the RR gauge potentials R (or N> magnetically under R ) [251]. The tension of these BPS states does not receive any quantum \N corrections perturbative or non-perturbative, which is why these objects are useful when considering non-perturbative dualities. States electrically (resp. magnetically) charged under the Neveu} Schwarz gauge "elds have tensions that scale with the string coupling constant as g (resp. 1/g), Q Q whereas states charged under the Ramond "elds have tensions that scale as 1/g . Q The last line in Table 3 is an unconventional solution, which we call a 6-brane, obtained by  vertical reduction of the KK6-brane in a direction in the 1 part of the Taub}NUT space [52]. The integration involved in building up the stack is logarithmically divergent, and, if regularized, yields a non-asymptotically #at space. However, as we will see in more detail in Section 4.9, at the algebraic level this solution is required by U-duality symmetry. At that point we will also explain  The letter D stands for the Dirichlet boundary conditions in the 9!p directions orthogonal to the world-volume of the Dp-brane, which force the open strings to move on this (p#1)-dimensional hyperplane.  There is also an 8-brane coupling to a nine-form, whose expectation value is related to the cosmological constant [36,38,251].

130

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

our nomenclature for this (and other) non-conventional solutions. It is also interesting to note that all the tensions obtained above are not independent, since they follow from the basic relations (2.11). This already hints at the presence of a larger structure that relates all these states, a fact that we will establish using the conjectured U-duality symmetry of compacti"ed M-theory. The dimensional reduction can also be carried out at the level of the supergravity con"guration itself. For example, using the relation (2.12) between the 11D metric and 10D string metric, one "nds that a solution with 11D metric of the form ds "HGds(EN)#HHds(E\N)  yields two 10D solutions with metric and dilaton ds "H?ds(ENY)#H@ds(E\NY), e\("HA ,  where diagonal: p"p!1, vertical: p"p,

i 3i 3i a" , b"j# , c"! , 2 2 2

3j 3j j a"i# , b" , c"! , 2 2 2

(2.32)

(2.33a)

(2.33b) (2.33c)

for diagonal and vertical reduction respectively. As explained in the beginning of this subsection, in the "rst case the harmonic function is the same as the original one, and in the second case it is a harmonic function on a transverse space with one dimension less. The reduction of the gauge potentials can be worked out similarly. The resulting 10D type IIA con"gurations are then described by the following solutions: F-string: ds "H\ds(E)#ds(E) ,  k B "H\, e\("H, H"1#  r NS5-brane: ds "ds(E)#Hds(E) ,  k dB"夹 dH, e\("H\, H"1# ,  r Dp-brane: ds "H\ds(EN)#Hds(E\N) ,  k e\("HN\, H"1# , r\N

(2.34a) (2.34b) (2.35a) (2.35b) (2.36a) (2.36b)

FN>"Vol(EN)dH\, p"0, 1, 2 , (2.36c) C F\N"夹 dH, p"4, 5, 6 , (2.36d) K \N F"F#F, p"3 , (2.36e) C K where, for completeness, we have included the Dp-brane solutions for all p"0,2, 6, although we note that only even p occurs in type IIA. The subscripts e and m indicate whether the p-branes are electrically or magnetically charged under the indicated "elds. One also "nds the ten-dimensional

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

131

gravitational solutions, consisting of the pp-waves and KK5-brane, which have a metric analogous to the eleven-dimensional case (see Eqs. (2.23a), (2.23b) and (2.24a)}(2.24c)), with harmonic functions on a transverse space with one dimension less. Of course, one may explicitly verify that all of these solutions are indeed solutions of the tree-level action (2.7). In contrast to the M2-brane and M5-brane, the dynamics of Dp-branes has a nice and tractable description as (p#1)-dimensional hyperplanes on which open strings can end and exchange momentum with [251]. The integration of open string #uctuations around a single D-brane at tree level yields the Born}Infeld action [17,56,205],



1 dN>me\((g( #BK #l F . (2.37) S " Q ' lN> Q Here, the hatted "elds g( , BK stand for the pullbacks of the bulk metric and antisymmetric tensor to the world-volume of the brane, and F is the "eld strength of the ;(1) gauge "eld living on the brane. The coupling to the RR gauge potentials is given by the topological term [95,140]



S "i e K >JQ $R , 00

(2.38)

where R" R denotes the total RR potential. N N In the zero-slope limit, the Born}Infeld action becomes the action of a supersymmetric Maxwell theory with 16 supercharges. In the presence of N coinciding D-branes the world-volume gauge symmetry gets enhanced from ;(1), to ;(N), as a consequence of zero mass strings stretching between di!erent D-branes [315]. The non-Abelian analogue of the Born}Infeld action is not known, although some partial Abelianization is available [309], but its zero-slope limit is still given by ;(N) super-Yang}Mills theory. 2.5. T-duality and type IIA/B string theory So far, we have discussed M-theory and its relation to type IIA string theory. In this subsection, we turn to type IIB string theory and its relation, via T-duality, to type IIA [78,93]. We "rst recall that the massless sector of type IIB consists of the same Neveu}Schwarz "elds (2.8a) as the type IIA string, but the Ramond gauge potentials of type IIB now include a 0-form (scalar), a 2-form and a 4-form with self-dual "eld strength, a, B , D , (2.39) IJ IJMN with 夹D "D . The low-energy e!ective action has a form similar to that in Eq. (2.7), with the   appropriate "eld strengths of the even-form RR potentials in Eq. (2.39), as long as the 4-form is not included. The standard 1/2-BPS solutions of type IIB are the fundamental string, NS5-brane, Dp-branes with odd p, pp-waves and KK5-brane.

 There is also a gravitational term required for the cancellation of anomalies [136], but it does not contribute on #at backgrounds.  A local covariant action for the self-dual four-form can be written with the help of auxiliary "elds [79], but for most purposes the equations of motion are su$cient.

132

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

In order to describe the precise T-duality mapping, we again write the ten-dimensional metric as a ;(1) "bration ds "R(dx#A dxI)#g dxIdxJ, k, l"0,2, 8 .  I IJ

(2.40)

T-duality on the direction 9 relates the "elds in the type IIA and type IIB theories in the Neveu}Schwarz sector as l l ¹ : R Q , g  Q g , A B , Q R Q I I  R

B B !A B #A B IJ IJ I J J I

(2.41)

leaving g and the string length l invariant. The Ramond gauge potentials are furthermore IJ Q identi"ed on both sides according to ¹ : Rdx ) R#dxR, R" R ,  N N

(2.42)

where ) and  denote the interior and exterior products, respectively. In other words, the 9 index is added to the antisymmetric indices of R when absent, or deleted if it was already present. These identi"cations actually receive corrections when BO0, and the precise mapping is [39,109,140] e RPdx ) (e R)#dx(e R)

(2.43)

in accord with the T-duality covariance of the RR coupling in Eq. (2.38). Whereas one T-duality maps the type IIA string theory to IIB and should be thought of as a change of variables, an even number of dualities corresponds to a global symmetry of either type IIA or type IIB theories. This symmetry will be discussed in Section 3, and its non-perturbative extension in Section 4. The action on the BPS spectrum can again be easily worked out, at the level of tension formulae or of the supergravity solutions themselves. As implied by the exchange of the Kaluza}Klein and Kalb}Ramond gauge "elds A and B , states with momentum along the 9th direction are I I interchanged with fundamental string winding around the same direction. On the other hand, T-duality exchanges Neumann and Dirichlet boundary conditions on the open string world-sheet along the 9th direction, mapping Dp-branes to D(p#1)- or D(p!1)-branes, depending on the orientation of the world-volume with respect to x [35,78]. This of course agrees with the mapping of Ramond gauge potentials in Eq. (2.42). Similarly, NS5-branes are invariant or exchanged with KK5-branes, according to whether they are wrapped or unwrapped, respectively [109,244]. This can also be easily seen by applying the transformation (2.41) to the tension formulae, as summarized in Table 4 for a T-duality ¹ on an arbitrary compact dimension with radius R . G G T-duality can then be used to translate the relation between strongly coupled type IIA theory and M-theory in type IIB terms. In this way, it is found that the type IIB string theory is obtained  Whereas the worldvolume dynamics of type IIB NS5- and D5-branes is described by a non-chiral (1,1) vector multiplet, the type IIB KK5-brane is chiral and supports a (2,0) tensor multiplet. Indeed, it is T-dual to the chiral type IIA NS5-brane [10]. On the other hand, the type IIA KK5-brane, dual to the type IIB NS5-brane, is nonchiral.

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

133

Table 4 T-duality of type II BPS states Type IIA (B)

Tension

¹ -dual tension G

Type IIB (A)

KK mode

1 M" R G

R M" G l Q

Winding mode

Wrapped Dp-brane

R " G N\ g lN> QQ R T " G  gl Q Q 1 T "  gl Q Q

1 " N\ g lN QQ R T " G  gl Q Q R T " G  gl Q Q

Unwrapped D(p!1)-brane

Wrapped NS5-brane Unwrapped NS5-brane

T

T

Wrapped NS5-brane Unwrapped KK5-brane

by compactifying M-theory on a two-torus ¹, with vanishing area, and a complex structure q equated to the type IIB complex coupling parameter [269]: i q"a# . g Q

(2.44)

Here, a is the expectation value of the Ramond scalar and g the type IIB string coupling. Q We focus for simplicity on the case where the torus is rectangular, so that q is purely imaginary and hence the RR scalar a vanishes. In this case, the relation between the M-theory parameters and type IIB parameters reads R l l g " Q , l" N , R " N , Q R Q R RR  Q Q 

(2.45)

where R , R are the radii of the M-theory torus and R the radius of the type IIB 9th direction. Q  The uncompacti"ed type IIB theory is obtained in the limit (R , R )PR, keeping R /R "xed. Q  Q  From Eq. (2.45), we can then identify the type IIB BPS states to those of M-theory compacti"ed on ¹. The results are displayed in Table 5 for states still existing in uncompacti"ed type IIB theory, and in Table 6 for states existing only for "nite values of R . As in Table 3, we see in the last entry of Table 6 a non-standard BPS state with tension scaling as g\, which we have called a 7 -brane. As this brane will turn out to be related to the D7-brane by Q  S-duality (see Section 4.5) it may also be referred to as a (1,0) 7-brane. This and other non-standard solutions will be discussed in more detail in Section 4.9.

3. T-duality and toroidal compacti5cation Having discussed how dualities of string theory lead to the idea of a more fundamental eleven-dimensional M-theory, we now turn to the symmetries that this theory should exhibit, with

134

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

Table 5 Relations between M-theory and type IIB BPS states M-theory

Mass/tension

Type IIB

M2-brane wrapped around xQ

R 1 Q" l l N Q R 1 " l g l N QQ RR 1 Q " l g l N QQ RR 1 Q " l g l N QQ RR 1  Q" l gl N Q Q

Fundamental string

M2-brane wrapped around x M5-brane wrapped on xQ, x KK6-brane wrapped on x, charged under g

IQ

KK6-brane wrapped on xQ, charged under g I

D1-brane (D-string) D3-brane D5-brane NS5-brane

Table 6 More relations between M-theory and type IIB BPS states M-theory

Mass/tension

Type IIB

M2-brane wrapped on xQ, x

R R 1  Q" l R N 1 R " l gl N Q Q 1 R " l g l N QQ R R Q" l g l N QQ R R " gl l Q Q N R R Q" l g l N QQ R R Q" gl l N Q Q

KK mode

Unwrapped M5-brane Unwrapped M2-brane M5-brane wrapped on xQ M5-brane wrapped on x Unwrapped KK6-brane, charged under g IQ Unwrapped KK6-brane, charged under g I

KK5-brane with R "R 2, Q Wrapped D3-brane Wrapped D5-brane Wrapped NS5-brane Wrapped D7-brane Wrapped 7 -brane 

the hope of getting more insight into its underlying structure. For this purpose, it is convenient to consider compacti"cations on tori, which have the advantage of preserving a maximal amount of the original super-PoincareH symmetries, while bringing in degrees of freedom from extended states in eleven dimensions in a still manageable way.

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

135

The approach here is similar to the one that was taken for the perturbative string itself, where the study of T-duality in toroidal compacti"cations revealed the existence of spontaneously broken `stringya gauge symmetries (see [132] for a review). Given the analogy between the two problems, we shall "rst review in this section how T-duality in string theory appears at the level of the low-energy e!ective action and of the spectrum, with a particular emphasis on the brane spectrum. We shall then apply the same techniques in Sections 4 and 5 in order to discuss U-duality in M-theory. 3.1. Continuous symmetry of the ewective action Compacti"cation of string theory on a torus ¹B can be easily worked out at the level of the low-energy e!ective action, by substituting an ansatz similar to Eq. (2.4) ds "g (dxG#AG dxI)(dxH#AH dxJ)#g dxIdxJ ,  GH I J IJ i, j"1,2, d, k, l"0,2, (9!d)

(3.1a) (3.1b)

in the ten-dimensional action








l 1 (3.2) S " dx(!g e\( R#4(R )! Q (dB) ,  l 12 Q where we omitted Ramond and fermion terms. We have also split the ten-dimensional two-form B into d(d!1)/2 scalars B , d vectors B and a two-form B . GH GI IJ Concentrating on the scalar sector, and rede"ning the dilaton as <e\("lBe\(B where Q <"(det g  is the volume of the internal metric, we obtain













1 1 1 S " d\Bx(!g e\(B 4(R )# Tr RgRg\# Tr g\RBg\RB .  l\B B 4 4 Q This can be rewritten as



1 1 S " d\Bx(!g e\(B 4(R )# Tr RMRM\ ,  l\B B 8 Q where M is the 2d;2d symmetric matrix




M"



g\

g\B

!Bg\

g!Bg\B




, MgM"g, g"

(3.3)

(3.4)

(



B ,

(3.5) B orthogonal for the signature (d, d) metric g. The scalars g and B therefore parametrize a symmetGH GH ric manifold SO(d, d, 1) H" UM , SO(d);SO(d)

(

(3.6)

where SO(d);SO(d) is the maximal compact subgroup of SO(d, d, 1). The matrix M is more properly thought of as the SO(d);SO(d) invariant M"VV built out from the vielbein  g denotes the internal metric g , except in the space-time volume element (!g multiplying the action density. GH

136

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

in SO(d, d, 1)

corresponding to the Iwasawa decomposition of SO(d, d, 1), as will be discussed in more detail in Section 4.2. The two-derivative action for the scalars g , B , is therefore invariant [217] under GH GH B the action MPXMX of X3O(d, d, 1), and so is the entire two-derivative action in the Neveu}Schwarz sector, if the 2d gauge "elds AG and B transform altogether as a vector under I GI O(d, d, 1), the dilaton , metric g and two-form B being invariant. B IJ IJ The action on the Ramond sector is more complicated, since the Ramond scalars and one-forms transform as a spinor (resp. conjugate spinor) of SO(d, d, 1), with the chirality depending on whether we consider type IIA or IIB. Elements of O(d, d, 1) with (!1) determinant #ip the chirality of spinors; they therefore are not symmetries of the action in the Ramond sector, but dualities, exchanging type IIA and type IIB theories. Indeed it is easy to see that the RP1/R dualities that we discussed in Section 2.5 belong to this class of transformations. The tree-level e!ective action is therefore invariant under the continuous symmetry SO(d, d, 1), which extends the symmetry Sl(d, 1) that would be present in the dimensional reduction of any Lorentz-invariant "eld theory. 3.2. Charge quantization and T-duality symmetry Owing to the occurrence of particles charged under the gauge "elds AG and B , the continuous I GI symmetry SO(d, d, 1) can, however, not exist at the quantum level. For instance, perturbative string states have integer momenta m and winding numbers mG under these gauge "elds, lying in an even G self-dual Lorentzian lattice C . The 1/2-BPS states are obtained when the world-sheet oscillators N aR and aR are not excited, and satisfy the mass formula and matching condition IL IL M"mMm"(m #B mH)gGI(m #B mJ)#mGg mH , G GH I IJ GH

(3.8a)

""m"""0 ,

(3.8b)

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

137

where m"(m , mG) is the vector of charges, ""m"""2m mG its Lorentzian square-norm and M is the G G moduli matrix given in Eq. (3.5). On the other hand, 1/4-BPS states are obtained when the world-sheet oscillators are excited on the holomorphic (or antiholomorphic) side only, and have mass M"mMm#" #m#",

(3.9)

where the norm """m""" is equated to the left or right oscillator number by the matching conditions. Only the discrete subgroup preserving C can be a quantum symmetry, and this group is O(d, d, 9), N the set of integer-valued O(d, d, 1) matrices. In particular, the subgroup Sl(d, 1) of SO(d, d, 1) is reduced to the modular group of the torus Sl(d, 9), an obvious consequence of momentum quantization in compact spaces. In addition to this perturbative spectrum, type II string theory also admits a variety of D-branes, which are charged under the Ramond gauge potentials. Their charges take value in another lattice, C , and transform as a spinor under SO(d, d, 1). Again, the determinant (!1) elements of O(d, d, 9) " #ip the chirality of spinors, and therefore do not preserve C . As we shall see shortly, SO(d, d, 9) " however does preserve the lattice of D-brane charges. This is in agreement with the fact that this group can be seen as the Weyl group of the extended gauge symmetries that appear at particular points in the torus moduli space, and are spontaneously broken elsewhere [131]. 3.3. Weyl and Borel generators In order to better understand the structure of the T-duality symmetry, it is useful to isolate a set of generating elements of SO(d, d, 9). We de"ne Weyl elements as the ones that preserve the conditions g "Rd , B "0 , (3.10) GH G GH GH that is square tori with vanishing two-form background, and Borel elements as the ones that do not. Weyl generators include the exchanges of radii S : R R , which belong to the Sl(d, 9) GH G H modular group, as well as the simultaneous inversions of two radii ¹ : (R , R )P(1/R , 1/R ). GH G H H G We choose the following minimal set of Weyl generators: S : R R , i"1,2, d!1 , (3.11a) G G G> g 1 1 Q , , . (3.11b) ¹ : (g , R , R )  Q   R R R R     For convenience, we followed the double T-duality on directions 1 and 2 by an exchange of the two radii, included the action on the coupling constant and set the string length l to 1. Altogether, the Q Weyl group of SO(d, d, 9) is the "nite group






W(SO(d, d))"9 ( )S (3.12)  B generated by the T-duality transformation ¹ and the permutation group S of the d directions of B the torus.  The Weyl group of SO(d, d) can actually be written as the semi-direct product S ( (9 )B\, where the commuting B  9 's are the double inversions of R and, say, R .  G 

138

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

On the other hand, Borel generators include the Borel elements of the modular subgroup, acting as c Pc #c on the homology lattice of the lattice, as well as the integer shifts of the expectation G G H value of the two-form in the internal directions B PB #1. Any element in SO(d, d, 9) can be GH GH reached by a sequence of these transformations. Weyl and Borel generators can be given a more precise de"nition as operators on the weight space of the Lie group or algebra under consideration (see for instance Ref. [173] for an introduction to the relevant group theory). Weyl generators correspond to orthogonal re#ections with respect to planes normal to any root and generate a "nite discrete group, while Borel generators act on the weight lattice by translation by a positive root. Any "nite-dimensional irreducible representation (of the complex Lie algebra) can then be obtained by action of the Borel group on a, so-called, highest-weight vector, and splits into orbits of the Weyl group with de"nite lengths. 3.4. Weyl generators and Weyl reyections Weyl generators encode the simplest and most interesting part of T-duality. It is very easy to study the structure of the "nite group they generate, by viewing them as orthogonal re#ections in a vector space (the weight space) generated by the logarithms of the radii. More precisely, let us represent the scalar moduli (ln g , ln R ,2, ln R ) as a form u on a vector space < with basis Q  B B> e , e ,2, e , and associate to any weight vector j"xe #xe #2#xBe , its tension   B   B    B (3.13) T"e6PH7"gV RV RV 2RV . Q   B The vector j should be seen as labelling a state in the BPS spectrum, with tension T. The generators (3.11a) and (3.11b) are then implemented as linear operators on < with matrices B> 1 1




S" G

1 1




!1

!1

, ¹"

!1

!1



.

(3.14)

( ( B\ B\ These operators S and ¹ in Eq. (3.14) are easily seen to be orthogonal with respect to the G signature (!#2#) metric ds"!(dx)#(dxG)#dx(dx#2#dxB) ,

(3.15)

and correspond to Weyl re#ections a)j jPo (j)"j!2 a ? a)a

(3.16)

 From this point of view, Weyl generators are not properly speaking elements of the group, but can be lifted to generators thereof, at the cost of introducing 9 phases in their action on the step operators E . See for instance Appendix  ? B in Ref. [207], for a discussion of this issue in the physics literature.  One could omit the x coordinate since g can be absorbed by a power of the invariant Planck length “R /g, but we Q G Q include it for later convenience.

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

139

with respect to planes normal to the vectors a "e !e , i"1,2, d!1 (3.17a) G G> G a "e #e . (3.17b)    The group generated by S and ¹ is therefore a Coxeter group, familiar from the theory of Lie G algebras (see [173] for an introduction, and [119,174] for a full account). Its structure can be characterized by the matrix of scalar products of these roots: (a )"(a )"2 , (3.18a) G  a ) a "a ) a "!1 . (3.18b) G G>   This precisely reproduces the Cartan matrix D of the T-duality group SO(d, d, 1), summarized in B the Dynkin diagram: (3.19) The only delicate point is that the signature of the metric (3.15) on < is not positive-de"nite. B> This can be easily evaded by noting that the invariance of Newton's constant “R /g implies that G Q all roots are orthogonal to the vector d"e #2#e !2e (3.20)  B  with negative proper length d"!(d#4), so that the re#ections actually restrict to the hyperplane < normal to d: B d ) x"x"0 . (3.21) The Lorentz metric on < then restricts to a positive-de"nite metric g "d on < . The dualities B> GH GH B S and ¹ therefore generate the Coxeter group D , which is the same as the Weyl group of the Lie G B algebra of SO(d, d, 1). In order to distinguish the various real and discrete forms of D , one needs to B take into account the Borel generators, which we defer to Section 3.7. The Dynkin diagram (3.19) allows a number of simple observations. We may recognize the Dynkin diagram A of the Lorentz group Sl(d, 1) (denoted with #), extended with the root B\ * into the Dynkin diagram of the T-duality symmetry SO(d, d, 1). T-duality between type IIA and type IIB corresponds to the outer automorphism acting as a re#ection along the horizontal axis of the Dynkin diagram. The chain denoted with *'s represents a dual Sl(d, 1) subgroup, which is nothing but the Lorentz group on the type IIB T-dual torus. The full T-duality group is generated by these two non-commuting Lorentz groups of the torus and the dual torus. Decompacti"cation of the torus ¹B into ¹B\ is achieved by dropping the rightmost root, which reduces D to D . When the root a is reached, the diagram disconnects into two pieces, B B\  corresponding to the identity SO(2, 2, 1)"Sl(2, 1);Sl(2, 1), or to the decomposition of the torus moduli space into the ¹ and ; upper half-planes. Finally, for d"1 the T-duality group SO(1, 1, 9) becomes trivial, while the generator of O(1, 1, 9) corresponds to the inversion R1/R, not a symmetry of either type IIA or type IIB theories.  The extra 9 exchanging the two Sl(2, 1) factors belongs to O(2, 2, 1) but not to SO(2, 2, 1). 

140

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

3.5. BPS spectrum and highest weights Having proved that the transformations S and ¹ indeed generate the Weyl group of SO(d, d, 9), G we can use the same formalism to investigate the orbit of the various BPS states of string theory. According to Eq. (3.13) the mass or tension can be represented as a weight vector in < , and one B> should let Weyl and Borel generators act on it to obtain the full orbit. Each orbit admits a highest weight from which all other elements can be reached by a sequence of Weyl and Borel generators (Weyl generators alone are not su$cient, because they preserve the length of the weight). All highest weights can be written as linear combinations with positive integer coe$cients of the fundamental weights R j"e !e PM "  ,   " g Q

(3.22a)

R R j"e #e !2e PM "   ,    ,1 g Q

(3.22b)

R 2R B\ , jB\"e #2#e !2e PM 2 "  (3.22c)  B\   ,1 g Q 1 jB\"e #2#e !2e &!e PM " , (3.22d)  B\  B $ R B 1 j"!e PM " (3.22e)  " g Q dual to the simple roots, that is jG ) a "!d . We used the symbol & for equality modulo the H GH invariant vector d in Eq. (3.20), and the notation F, D and NS for fundamental, Dirichlet and Neveu}Schwarz states, respectively, depending on the power of the coupling constant involved, and w for each wrapped direction (the notation wF is justi"ed by the fact that the Kaluza}Klein states are in the same multiplet as the string winding states). This is summarized in the Dynkin diagram

(3.23)

which shows the highest weights associated to each node of the Dynkin diagram. In particular, we see from Eq. (3.23) that the type IIA D-particle mass (M"1/g l ) lies in the QQ spinor representation dual to a , just as do the type IIB D-string tension (T "1/g l) and   QQ  The minus sign shows that we are really considering lowest-weight vectors, but we shall keep this abuse of language.

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

141

D-instanton action (T "1/g ), whereas the type IIB D-particle mass (M"R /g l) and type IIA \ Q G QQ D-string tension (T "R /g l) and D-instanton action (T"R /g l ) transform in the spinor  G QQ G QQ representation dual to a , of opposite chirality. On the other hand, the Kaluza}Klein states lie in  a vector representation. All highest-weight representations can be obtained from the tensor product of these `extremea (from the point of view of the Dynkin diagram) representations. T-duality on a single radius exchanges the two spinor representations, as it should. 3.6. Weyl-invariant ewective action In the previous subsections, we have discussed how the Weyl group of SO(d, d) arises as the "nite group generated by the permutations and double T-duality (3.11a) and (3.11b), whereas the low-energy action itself is invariant under the continuous group SO(d, d, 1). This has been checked in the scalar sector in Eq. (3.4), by direct reduction of the 10D e!ective action on ¹B. It is however possible to rewrite the full action in a manifestly Weyl-invariant way, by a step-by-step reduction from 10D, as was originally developed in Ref. [213] in the context of 11D supergravity. This procedure leads to a clear identi"cation of `dilatonica scalars, which appear through exponential factors in the action and include the dilaton g and the radii R of the torus, versus `Peccei}Quinna Q G scalars which have constant shift symmetries and are better thought of as 0-forms with a 1-form "eld strength. , with internal indices Each "eld strength FN gives rise to "eld strengths of lower degree FO G2GO i 2i (given by the exterior derivative of a (q!1)-form up to Chern}Simons corrections), while  O the metric gives rise to Kaluza}Klein two-form "eld strengths FG and one-form "eld strengths FG, i(j, of the vielbein components in the upper triangular gauge H g "E. E/ g , (3.24a) +, + , ./

(3.24b) where EJ denotes the vielbein in the uncompacti"ed directions. The action (2.7) in the Neveu} I Schwarz sector then takes the simple form:












 

< RR  R  G # GFG R#(R )# S " d\Bx(!g ,1\B gl R R H Q Q G H GH G




l  l  Q F # (R FG)#(lF)# Q F # , G Q R G R R GH G G H G G GH

(3.25)

142

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

where the "rst "ve terms come from the reduction of the Einstein}Hilbert term and the last three terms from the kinetic term of the two-form. Putting the forms of the same degree together, we see that their coe$cients form the Weyl orbit U , of the string tension (F), the Weyl orbit U of the Kaluza}Klein and winding states (F), Q H )) H and the set of positive roots U "+e $e , i(j, (F). We can therefore rewrite the action in the > G H ? Weyl-invariant form:





< R#Ru ) Ru# e\6P?7(F) S " d\Bx(!g ? ,1\B gl Q Q ?ZU>



# e\6PH7(F)# e\6PH7(F) , H H HZUQ HZU))

(3.26)

where u"(ln g , ln R ,2, ln R ) is the vector of dilatonic scalars, 1u, j2 the duality bracket in Q  B Eq. (3.13) and Ru ) Ru the Weyl-invariant kinetic term obtained from the non-diagonal metric (3.15). A diagonal metric on the dilatonic scalars is recovered upon going to the Einstein frame. The Weyl group acts by permuting the various weights appearing in Eq. (3.26), and the invariance in the gauge sector is therefore manifest. As for the scalars, the set of positive roots U is > not invariant under Weyl re#ections, but the Peccei}Quinn scalars undergo non-linear transformations APe\6P?7A that compensate the sign change [215]. The Peccei}Quinn scalars therefore appear as displacements along the positive (non-compact) roots. Together with the dilatonic (non-compact) scalars u, they generate the solvable Lie subalgebra that forms the tangent space of the moduli space H [6}8,308]. We have so far concentrated on the Neveu}Schwarz sector, but the same reasoning can be applied to the full type II action. The T-duality Weyl symmetry can, however, be exhibited only by dualizing the p-form gauge "elds GN"dRN\ into lower rank (10!d!p)-form gauge "elds when possible, and keeping them together when their dual when the self-duality condition 10!d!p"p is satis"ed. We then obtain, for the action of the Ramond "elds





< e\6PH7(G)# e\6PH7(G) S " d\Bx(!g H H 00 gl U Q Q HZ "' HZU"



# e\6PH7(G)# e\6P H7(G) , H H HZU" HZU"

(3.27)

where U , U , U , U denote the Weyl orbits with highest weight 1/g R , 1/g l , R /g l, 1/g l, "' " " " Q G QQ G QQ QQ respectively, corresponding in turn to the two spinor representations. 3.7. Spectral yow and Borel generators Having discussed the structure of the Weyl group we now want to investigate the full SO(d, d, 9) symmetry. For this purpose, it is instructive to go back to the perturbative multiplet of Kaluza}Klein and winding states. The action of the Weyl group on the highest weight 1/R of the B vector representation generates an orbit of 2d elements, 1/R and R . However, a particle can have G G

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

143

any number of momentum excitation along each axis, and wind along any cycle of the torus ¹B. It is therefore described by integer momenta m and winding numbers mG, so that its mass on an G arbitrary torus reads M"m gGHm #mGg mH, i, j"1,2, d , (3.28) G H GH when B "0. This mass formula is then invariant under modular transformations cGPcG#*AG cH GH H of the torus, i.e. integer shifts AG PAG #*AG of the o!-diagonal term of the metric (no sum on i) H H H ds"R(dxG#AG dxH)#g dxHdxI , (3.29) B G H HI upon transforming the momenta and winding as m Pm !*AG m , mIPmI#dI*AG mH. (3.30) I I I G G H This transformation generates a spectral yow on the lattice of charges m and mG. G In addition, being charged under the gauge potential B , the momentum of the particle shifts IG according to m Pm "m #B mH, yielding the mass (3.8a) and (3.8b). From this, we see that the G G G GH Borel generator B PB #*B induces a spectral #ow GH GH GH m Pm #*B mH, mIPmI. (3.31) I I HI The two spectral #ows (3.30) and (3.31) can be understood in a uni"ed way as translations on the weight lattice by positive roots. Indeed, the set of all positive roots of SO(d, d) includes the Sl(d) roots e !e , i(j, images of the simple roots a "e !e , 14i4d!1 under the Weyl group H G G G> G S of Sl(d), as well as the roots e #e , which are images of the T-duality simple root a "e #e . B G H    The translation by a root e !e generates in"nitesimal rotations in the (i, j) plane: H G *"!e 2"!*AG "!e 2, *"e 2"dI*AG "e 2 (3.32) I I G I G H H equivalent to the spectral #ow in Eq. (3.30), whereas translations by a root e #e generate an G H in"nitesimal B shift: GH *"!e 2"*B "e 2, *"e 2"0 (3.33) I HI H I as in Eq. (3.31). The moduli AG and B can therefore be identi"ed as displacements on the moduli H GH space H along the positive roots e !e and e #e . We note that the two displacements do not G H G H necessarily commute and that only integer shifts are symmetries of the charge lattice. 3.8. D-branes and T-duality invariant mass In order to study the analogous properties of the D-brane states, we may try to write down the moduli matrix M 3SO(d, d, 1)/SO(d);SO(d) in the spinorial representation and look for the 1 transformations of charges that leave the mass mM m invariant, when now m is a spinor of 1 D-brane charges. It is in fact much easier to study the D-brane con"guration itself and compute its Born}Infeld mass [149,250].  The Borel generators E actually either translate the weight vectors j or annihilate them. ?

144

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

BPS D-brane states are obtained by wrapping Dp-branes on a supersymmetric p-cycle of the compacti"cation manifold. In the case of a torus ¹B, this is simply a straight cycle, and in the static gauge the embedding is speci"ed by a set of integer (winding) numbers NG : ? XG"NG p?, i"1,2, d, a"1,2, p , (3.34) ? where p? and XG are the space-like world-volume and embedding coordinates respectively. The numbers NG can, however, be changed by a world-volume di!eomorphism, and one should instead ? look at the invariant mGHIJ"e?@ABNG NH NINJ , (3.35) ? @ A B where we restricted to p"4 for illustrative purposes. mGHIJ is a four-form integer charge that speci"es the four-cycle in ¹B. In addition, the D-brane supports a ;(1) gauge "eld that can be characterized by the invariants 1 mGH" e?@ABNG NH F , ? @ AB 2

1 m" e?@ABF F , ?@ AB 8

(3.36)

which are again integer-valued, because of the #ux and instanton-number integrality. The charges N"+m, mGH, mGHIJ,2, constitute precisely the right number to make a spinor representation of SO(d, d, 9) when p"d or p"d#1 (depending on the type of theory and dimensionality of the torus); indeed, the spinor representation of SO(d, d) decomposes under Sl(d) as a sum of even or odd forms, depending on the chirality of the spinor. The Chern}Simons coupling (2.38) can be rewritten in terms of these charges (up to corrections when BO0) as



1 1 e K >?Y$R"mR # mGHR # mGHIJR #2  2 GH 4! GHIJ

(3.37)

so that (for p"4) the instanton number m can be identi"ed as the D0-brane charge, the #ux mGH as the D2-brane charge and mGHIJ as the D4-brane charge. Con"gurations with mO0 exist in SYM theory on a torus, even for a ;(1) gauge group, and correspond to torons [147,148,299]. The mass of the wrapped D-brane can be evaluated by using the Born}Infeld action (2.37), and depends only on the parametrization-independent integer charges m, mGH, mGHIJ,2 . Explicitly, we obtain, for p"d, the T-duality invariant mass formula: 1 1 1 m # (m GH)# (m GHIJ)#2 , M" 2gl 4!gl gl Q Q Q Q Q Q m "m#mGHB #mGHIJB B #2 , GH  GH IJ  m GH"mGH#mIJGHB #2 , IJ  m GHIJ"mGHIJ#2 ,  This expression was originally derived in Ref. [250] by a sequence of T-dualities and covariantizations.

(3.38a) (3.38b) (3.38c) (3.38d)

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

145

where the dots stand for the obvious extra terms when d54. A similar expression holds for p"d#1 and yields the tension of D-strings: 1 1 1 (m G)# (m GHI)# (m GHIJJ)#2, T"  gl 3!gl 5!gl Q Q Q Q Q Q m G"mG#mHIGB #mHIJKGB B #2, HI  HI JK  m GHI"mGHI#mJKGHIB #2, JK  m GHIJK"mGHIJK#2,

(3.39a) (3.39b) (3.39c) (3.39d)

where the integer charges read, e.g. for p"5, mGHIJK"e?@ABCNG NH NINJ NK , (3.40a) ? @ A B C mGHI"e?@ABCNG NH NIF , (3.40b)  ? @ A BC NG F F . (3.40c) mG"e  ?@ABC ? @A BC The mass formulae (3.38a)}(3.38d) and (3.39a)}(3.39d) hold for 1/2-BPS states only; they are the analogues of Eqs. (3.8a) and (3.8b) for the two spinor representations of SO(d, d). They can be derived by analysing the BPS eigenvalue equation in a similar way as in Section 2.2. This analysis is carried out in Appendix A.3, and yields, in addition, the conditions for the state to be 1/2-BPS, as well as the extra contribution to the mass in the 1/4-BPS case. In the d46 case, we "nd a set of conditions: kGHIJ,m GHmIJ #mmGHIJ"0 ,

(3.41a)

kG_HIJKL,mG HmIJKL #mmGHIJKL"0 ,

(3.41b)

kGH_IJKLNO,nGHnIJKLNO#nGH IJnKLNO "0

(3.41c)

analogous to the level-matching condition ""m"""0 on the perturbative states. In contrast to the latter, they have a very clear geometric origin, since they can be derived by expressing the charges m in terms of the integer numbers NG (Eqs. (3.38a)}(3.38d)). For d"6, they transform in ? a 15#36#15"66 irrep of the T-duality group SO(6, 6, 9). The last line in Eqs. (3.41a)}(3.41c) drops when d"5, giving a 5#5"10 irrep of SO(5, 5, 9). When d"4, only the k"mm#mm,0 component remains, which is a singlet under SO(4, 4, 9). When the conditions n"0 in Eqs. (3.41a)}(3.41c) are not met, the state is at most 1/4-BPS, and its mass receives an extra contribution, e.g. for d"5:







1 1 1 1 1 m # (m GH)# (m GHIJ)# (kI GHIJ)# (kI G_HIJKL) , M" 4!l 2l 4!l 5!l gl Q Q Q Q Q Q where the shifted charges are given by kI GHIJ"kGHIJ#B kK_LGHIJ, KL

kI G_HIJKLN"kG_HIJKLN .

(3.42)

(3.43)

146

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

For d"6, there are still conditions to be imposed in order for the state to be 1/4-BPS instead of simply 1/8-BPS, which are now cubic in the charges m and transform as a 32 of SO(6, 6, 9) (see Appendix A and Section 5.9).

4. U-duality in toroidal compacti5cations of M-theory T-duality is only a small part of the symmetries of toroidally compacti"ed string theory, namely the part visible in perturbation theory. We shall now extend the techniques of Section 3 in order to study the algebraic structure of the non-perturbative symmetries, which go under the name of U-duality. In this section, we focus on the subgroup of the U-duality symmetry that preserves compacti"cations on rectangular tori with vanishing expectation values of the gauge potentials. The most general case of non-rectangular tori with gauge potentials, for which the full U-duality symmetry can be exhibited, is discussed in the next section. 4.1. Continuous R-symmetries of the superalgebra As in our presentation of uncompacti"ed M-theory in Section 2, the superalgebra o!ers a convenient starting point to discuss the symmetries of M-theory compacti"ed on a torus ¹B. The N"1, 11D supersymmetry algebra is preserved under toroidal compacti"cation: the generators Q merely decompose as bispinor representations of the unbroken group SO(1, 10!d);SO(d), and ? form an N-extended super-PoincareH algebra in dimensions D"11!d. The "rst factor SO(1, 10!d) corresponds to the Lorentz group in the uncompacti"ed dimensions and is actually part of the superalgebra, while the second only acts as an automorphism thereof, and is also known as an R-symmetry. There can be automorphisms beyond the obvious SO(d) symmetry, however, and these are expected to be symmetries of the "eld theory. This symmetry enhancement can be observed at the level of the Cli!ord algebra itself [177,215]. The Gamma matrices C , M"0, d#1,2,10 of eleven-dimensional supersymmetry can be kept + to form a (reducible) Cli!ord algebra of SO(1, 10!d), while the matrices C , I"1,2, d form an ' internal Cli!ord algebra. Note that we have chosen here, in contrast to the notation of the rest of the review, the internal indices running from 1 to d. The generators C generate the SO(d) '( R-symmetry, but they can be supplemented by generators C to form the Lie algebra of a larger ' R-symmetry group SO(d#1). It was the attempt to exhibit the SO(8) symmetry of 11D SUGRA compacti"ed on ¹ that led to the discovery of hidden symmetries [71]. The R-symmetry group is actually larger still. Consider the algebra generated by C , C , C , C , where the subscripts denote the number of antisymmetric internal indices, and     the corresponding generators are dropped when the number of internal directions is insu$cient: E For d"2, the only generator C "C generates a ;(1) R-symmetry. '(   The R-symmetry is actually part of the local supersymmetry, but we are only interested in its global #at limit.  This is the basis for the Twelve-dimensional S-theory proposal [25]. It is important that these generators commute with the momentum charge CC . I

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

147

Table 7 Classi"cation of the supercharges and central charges w.r.t the Lorentz/R-symmetry group SO(1, 10!d);H. Irreps of H are in bold face. Charges in parenthesis are PoincareH -dualized (moved) into charges in square brackets. Adapted from Ref. [25] d

Q? ?

p"0 Z', Z'( Z'()*+

p"1 ZI, ZI' ZI'()*

p"2 ZI' ZIJ'()

p"3 ZIJM'(

p"4 ZIJMN'

p"5 ZIJMNO

H

1

($, 16)

1#0 #0

1#1 #0

1 #0

0

1

1> #1\

1

2

(2, 16)

2#1 #0 "2#1

1#2 #0 "2#1

1#0 "1

1

[1] #2 "2#1

(1) move

SO(2)

3

(2, 8>) #(2, 8\)

3#3 #0 "3>#3\

1#3 #0 "3#1

1#1 "1#1

3#[1] "3#1

3> #3\ "3>#3\

(1) move

SO(2) ;;(1)

4

(4, 8)

4#6 #0 "10

1#4 #1 "5#1

1#4 #[1] "5#1

6#[4] "10

(4) move

(1) move

SO(5)

5

(4, 4 ) #(4 , 4)

5#10 #1 "(4,4)

1#5 #5#[1] "(5, 1) #(1, 5) #2(1, 1)

1#10 #[5] "(4, 4)

10>#10\ "(10, 1) #(1, 10)

(5) move

(1) move

SO(5) ;SO(5)

6

(8, 4)

6#15 #6 #[1] "27#1

1#6 #15#[6] "27#1

1#20 #[15] "36

(15) move

(6) move

(1) move

;Sp(8)

7

(8>, 2) #(8\, 2 )

7#21 #21 #[7] "28 A

1#7 #35#[21] "63#1

1!#35! "36 A

(21) move

(7) move

0

S;(8)

8

(16, 2)

8#28 #56 #[28] "120

1#8#70 #[1#56] "135#1

(1#56) move

(28) move

0

0

SO(16)

E For d"3, C and C commute, and generate an SO(3);;(1) symmetry.   E For d"4, C "C C , where C is the space-time or internal chirality (see Eq. (A.1)) and,  >  > together with C , generates an SO(5) symmetry.  E For d"5, C $C C generate two commuting SO(5) subgroups.  > 

148

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

E For d"6, C appears in the commutator [C , C ] and a ;Sp(8) is generated.    E For d"7 (resp. d"8) the generator C comes into play and one obtains an S;(8);;(1) (resp.  SO(16)) R-symmetry group. The various R-symmetry groups are summarized in the right column of Table 7, which furthermore gives the decomposition of the 528 central charges on the right-hand side of Eq. (2.13a) under the Lorentz group SO(1, 10!d) in the uncompact directions and the R-symmetry group. The various columns correspond to distinct SO(1, 10!d) representations, after dualizing (moving) central charges into charges with less indices when possible. In all these cases, the superalgebra can be recast in a form manifestly invariant under the R-symmetry. Here we collect the cases D"4, 5, 6, including the central charges, which transform linearly under the R-symmetry: E For D"4 (d"7), the 32 supercharges split into 8 complex Weyl spinors transforming as an 8  8 of S;(8): +Q , Q Q M ,"pI Q P d M , ?@ I  ? @

(4.1a)

+Q , Q ,"e Z , ? @ ?@  +Q  M , Q Q M ,"e  Q ZHM M , ? @ ?@ 

(4.1b) (4.1c)

where k"0, 1, 2, 3 are SO(3, 1) vector indices, a, a"1, 2 are Weyl spinor indices, and A, AM "1, 2, 8 are 8, 8 indices of S;(8). The central charges are incorporated into a complex antisymmetric matrix Z .  E For D"5 (d"6), the 32 supercharges split into 8 Dirac spinors of SO(4, 1), transforming in the fundamental representation of ;Sp(8). The N"8 superalgebra in a ;Sp(8) basis is +Q , Q ,"P (CcI) X #C Z , ? @ I ?@  ?@ 

(4.2)

where k"0, 1, 2, 3, 4 are SO(4, 1) vector indices, a"1, 2, 3, 4 are Dirac spinor indices, A"1,2,8 are indices in the 8 of ;Sp(8), and X is the invariant symplectic form and Z is the central   charge matrix. E For D"6 (d"5), the 32 supercharges form 4 complex spinors transforming in the (4, 1)#(1, 4) of SO(5);SO(5) and the superalgebra takes the form +Q?, Q@ ,"u?@cI p , ? @ ?@ I

(4.3)

+Q?, QM @ ,"d Z?@ , ? @ ?@

(4.4)

where a, b"1,2, 4 are SO(5) spinor indices and u?@ is an invariant antisymmetric matrix, from the local isomorphism SO(5)";Sp(4). The 16 central charges are incorporated in a matrix Z?@ transforming as a bispinor under the R-symmetry SO(5);SO(5) and satisfying the reality condition ZH"uZuR. The R-symmetries that we have discussed here will be of use in the next section to determine the scalar manifold of the compacti"ed 11D SUGRA and hence the global symmetries.

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

149

4.2. Continuous symmetries of the ewective action In our discussion, in Section 3.1, of the continuous symmetry of the e!ective action of the toroidally compacti"ed type IIA theory, we have intentionally focused our attention on the Neveu}Schwarz sector, and have brie#y described how the Ramond "elds would transform under the symmetries of the Neveu}Schwarz scalar manifold. The distinction between Neveu}Schwarz and Ramond sectors is however an artefact of perturbation theory and, as we discussed in Section 2, the two sets of "elds are uni"ed in the 11D SUGRA description. They mix under the eleven-dimensional Lorentz symmetries unbroken by the compacti"cation on ¹B, namely Sl(d, 1). The low-energy e!ective action therefore admits a continuous symmetry group G containing B SO(d!1, d!1, 1) ( ) Sl(d, 1) , (4.5) where the symbol ( ) denotes the group generated by the two non-commuting subgroups. As found by Cremmer and Julia [70,178], the groups G turn out to correspond to the E series, listed in B BB Table 8. The notation E denotes a particular non-compact form of the exceptional group E , namely its BB B normal real form, and from now on this distinction will be omitted. As evident from their Dynkin diagrams shown in Table 9, the groups E form an increasing family, whose members are related by B a process of group disintegration re#ecting the decompacti"cation of one compact direction in ¹B. This is displayed in Table 9, and will be discussed more fully in the next subsection. The occurrence of these groups can be understood by "tting the number of scalar "elds (including the duals of forms of higher degree) to the dimension of a coset space G /H , where H is B B B the R-symmetry of the superalgebra described in the previous section. In order to have a positive metric for the scalars, it is necessary that H be the maximal compact subgroup of G . Together B B with the dimension of the scalar manifold, this su$ces to determine G . B Scalar "elds arise from the internal components of the metric g of the torus ¹B, and from the '( expectation value of the three-form gauge "eld C on ¹B; they also arise from the expectation '() value E on ¹B of the six-form dual to C in eleven dimensions, or equivalently the '()*+, +,. expectation value of the scalar dual to the three-form C in D"5, the axion scalar dual to the IJM two-form C in D"4, or to the one-form C in D"3; similarly, the Kaluza}Klein gauge IJ' I'( potentials g can be dualized in D"3 into scalars K , which can be interpreted as the expectation I' ' value K on ¹B of the magnetic gauge potential dual to g in eleven dimensions. The '_()*+,./0 +, counting is summarized in Table 10. The factor 1> appearing in D"10 and D"9 corresponds to the type IIA dilaton, and generates a scaling symmetry of the e!ective action, called trombonne symmetry in Ref. [76]. Note that a quite di!erent U-duality group would be inferred if one did not dualize the Ramond "elds into "elds with less indices [72,214], or if one would consider Euclidean supergravities [75,171]. An analogous counting has been performed in Tables 11 and 12 for one-form and two-form potentials, inducing particle and string electric charges, respectively. The latter can be put in  Note that d has been upgraded by one unit with respect to the previous section.  The normal real form has all its Cartan generators and positive roots non-compact, and is the maximal noncompact real form of the complex algebra E (") [129,155]. B

150

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

Table 8 Cremmer}Julia symmetry groups and their maximal compact subgroups D

d

G "E B BB

H B

10 9 8 7 6 5 4 3

1 2 3 4 5 6 7 8

1> Sl(2, 1);1> Sl(3, 1);Sl(2, 1) Sl(5, 1) SO(5, 5, 1) E  E  E 

1 ;(1) SO(3);;(1) SO(5) SO(5);SO(5) ;Sp(8) S;(8) SO(16)

one-to-one correspondence to the central charges of the supersymmetry algebra discussed in the previous section, with two exceptions. Firstly, the Lorentz-invariant central charge Z in "ve dimensions, where 024 denote the "ve space-time dimensions, does not correspond to any one-form potential [25,29]. This truncation of the superalgebra is consistent with U-duality and is of no concern, except for the twelve-dimensional origin of M-theory. Secondly, there are only 120 Lorentz singlet central charges in D"3 for 128 gauge potentials (equivalently, there are only 64 Lorentz vector charges in D"4 for 70 two-form gauge "elds). As we shall see shortly, U-duality implies that there should in fact be 248 electric charges in D"3 (133 string charges in D"4), yielding a linear representation of the duality group E (resp. E ). Of course, the notion of electric   charge is ill-de"ned in D"3, where a one-form (or a two-form in D"4) is PoincareH -dual to a zero-form and a particle (or a string) to an instanton. Another manifestation of the pathology of the D"3 case is the non-asymptotic #atness of the point-like solitons (or string-like in D"4), and the logarithmic divergence of the kernel of the Laplacian in the transverse directions. In spite of these di$culties, we shall pursue the algebraic analysis of these cases in the hope that they can be resolved. If the charges m under the gauge "elds can be put in one-to-one correspondence with the central charges Z, they are nevertheless not equal: the gauge charges are integer-quantized, as we will discuss in the next subsection, whereas the central charges are moduli-dependent linear combinations of the latter: Z"V ) m ,

(4.6)

where V is an element in the group G containing the moduli dependence; it is de"ned up to the left B action of the compact subgroup K"H , inducing an R-symmetry transformation on Z. B The local H gauge invariance can be conveniently gauge-"xed thanks to the Iwasawa decompoB sition (see for instance [197,227]) V"k ) a ) n3K ) A ) N

(4.7)

 Equivalently, the central charges Z, Z2 transform as a vector in six space-time dimensions. These charges could be attributed to a KK6-brane, if only the KK6-brane did not need six compact directions to yield a string, and seven to yield a particle state.

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

151

Table 9 Dynkin diagrams of the E series. The group disintegration proceeds by omitting the rightmost node. The integers shown B are the Coxeter labels, that is the coordinates of the highest root on all simple roots

of G into the maximal compact K, Abelian A and nilpotent N. A natural gauge is obtained by B taking K"1, in which case the `vielbeina V becomes a (generalized) upper triangular matrix V"a ) n. The Abelian factor A is parametrized by the `dilatonic scalarsa, namely the radii of the internal torus, whereas the nilpotent factor N incorporates the `gauge scalarsa, namely the expectation values of the gauge "elds (including the o!-diagonal metric, three-form and their duals) on the torus. G acts on the charges m from the left and on V from the right. The transformed B

152

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

Table 10 Scalar counting and scalar manifolds in compacti"ed M-theory D

d

g

10 9 8 7 6 5 4 3

1 2 3 4 5 6 7 8

1 3 6 10 15 21 28 36

C 

1 4 10 20 35 56

E 

1 7 28

K _

8

Total

Scalar manifold

1 3 7 14 25 42 70 128

1> Sl(2, 1)/;(1);1> Sl(3, 1)/SO(3);Sl(2,1)/;(1) Sl(5,1)/SO(5) SO(5,5,1)/SO(5);SO(5) E /;Sp(8)  E /S;(8)  E /SO(16) 

V can then be brought back into an upper triangular form by a moduli-dependent R-symmetry compensating transformation on the left. This implies that the central charges Z transform nonlinearly under the continuous U-duality group G . For the case of T-duality in type II string theory B this decomposition is given in Eq. (3.7). In Section 5, we shall obtain an explicit parametrization of V in terms of the shape of the torus and the various gauge backgrounds. 4.3. Charge quantization and U-duality As in the case of T-duality, the continuous symmetry E (1) of the two-derivative e!ective BB action cannot be a symmetry of the quantum theory: the gauge potentials transform non-trivially under E , and the continuous symmetry is therefore broken by the existence of states charged under B these potentials. At best there can remain a discrete subgroup E (9), which leaves the lattice of BB charges invariant. For one thing, a subset of the charges corresponds to the Kaluza}Klein momentum along the internal torus, and are therefore constrained to lie in the reciprocal lattice of the torus. Another subset of charges corresponds to the wrapping numbers of extended objects around cycles of ¹B, and are then constrained to lie in the homology lattice of ¹B. A way to determine the remaining discrete subgroup is to consider M-theory compacti"ed to D"4 dimensions, in which case PoincareH duality exchanges gauge one-forms with their magnetic duals [172]. In this dimension, Dirac}Zwanziger charge quantization takes the usual form mGn!mGn 39 (4.8) G G for two particles of electric and magnetic charges mG and n respectively, and i runs from 1 to 28, as G read o! from Table 11. This condition is invariant under the electric}magnetic duality Sp(56, 9), under which (mG, n ) transforms as a vector. The exact symmetry group is therefore at most G E (9)LE (1)5Sp(56, 9) . (4.9)   This translates into a condition on E (9) for d47 by the embedding E (9)LE (9). A similar BB BB  condition can be obtained in D"3, where all one-forms are dual to scalars. The condition (4.9) requires a precise knowledge of the embedding of E (1) in Sp(46, 1).  Instead, we shall take another approach, and postulate that the U-duality group of M-theory

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

153

Table 11 Vectors and particle charge representations in compacti"ed M-theory D

d

g

10 9 8 7 6 5 4 3

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8

C  1 3 6 10 15 21 28

E 

1 6 21 56

K _

7 36

Total

Charge representation

1 3 6 10 16 27 56 128

1 3 of Sl(2) (3, 2) of Sl(3);Sl(2) 10 of Sl(5) 16 of SO(5, 5) 27 of E  56 of E  248 of E 

Total

Charge representation

1 2 3 5 10 27 70

1 2 of Sl(2) (3, 1) of Sl(3);Sl(2) 5 of Sl(5) 10 of SO(5,5) 27 of E  133 of E 

Table 12 Two-forms and string charge representations in compacti"ed M-theory D

d

10 9 8 7 6 5 4

1 2 3 4 5 6 7

g

C  1 2 3 4 5 6 7

E 

1 5 15 35

K _

6 28

compacti"ed on a torus ¹B is generated by the T-duality SO(d!1, d!1, 9) of type IIA string theory compacti"ed on ¹B\, and by the modular group Sl(d, 9) of the torus ¹B: E (9)"SO(d!1, d!1, 9) ( ) Sl(d, 9) . (4.10) BB The former was argued to be a non-perturbative symmetry of type IIA string theory, as discussed in the previous section, while the latter is the remnant of eleven-dimensional general reparametrization invariance, after compacti"cation on a torus ¹B: it is therefore guaranteed to hold, as long as M-theory, whatever its formulation, contains the graviton in its spectrum. The above construct is therefore the minimal U-duality group, and since it preserves the symplectic condition (4.8) also the maximal one. In the d"2 case, the U-duality group (4.10) is the modular group Sl(2, 9) of the M-theory torus, which in particular contains the exchange of R and R ; translated in type IIB variables, this is Q  simply the Sl(2, 9) S-duality of type IIB theory (in 9 or 10 dimensions), which contains the strong-weak coupling duality g P1/g , as can be seen from Eq. (2.45). Note that we do not expect Q Q  A veri"cation of this statement requires a precise knowledge of the branching functions of Sp(56) into E . 

154

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

Table 13 Discrete subgroups of E B D

d

E (1) BB

E (9) BB

10 9 8 7 6 5 4 3

1 2 3 4 5 6 7 8

1 Sl(2, 1) Sl(3, 1);Sl(2, 1) Sl(5, 1) SO(5, 5, 1) E (1)  E (1)  E (1) 

1 Sl(2, 9) Sl(3, 9);Sl(2, 9) Sl(5, 9) SO(5, 5, 9) E (9)  E (9)  E (9) 

any quantum symmetry from the trombonne symmetry factor 1>. For d"3, the T-duality group splits into two factors Sl(2, 9);Sl(2, 9), one of which is a subgroup of the modular group Sl(3, 9) of the M-theory torus ¹. The de"nition (4.10) therefore yields E (9)"Sl(3, 9);Sl(2, 9) and is the  natural discrete group of E . For d"4, SO(3, 3, 9) is isomorphic to a Sl(4, 9) (in the same way as  SO(6)&S;(4)), which does not commute with the modular group Sl(4, 9) of M-theory on a torus ¹. Altogether, they make the Sl(5, 9) subgroup of E (1)"Sl(5, 1). For d"5, we obtain the  SO(5, 5, 9) subgroup of E (1)"SO(5, 5, 1). For d56, this provides a dexnition of the discrete  subgroups of the exceptional groups E (1). These groups are summarized in the rather BB tautological Table 13. We note that it is crucial that the groups E be non-compact in order BB for an in"nite discrete group to exist. The maximal non-compact form is also required in order that all representations be real (i.e. that the mass of a particle and its anti-particle be equal, see Section 4.8). 4.4. Weyl and Borel generators A set of generators of the U-duality group can easily be obtained by conjugating the T-duality generators under Sl(d, 9). The Weyl generators now include the exchange of the eleven-dimensional radius R with any radius of the string-theory torus ¹B\, in addition to the exchange of Q the string-theory torus directions among themselves and T-duality on two directions thereof. It is interesting to rephrase the latter in M-theory variables, using relations (2.1), (3.11a) and (3.11b): l l l l (4.11) ¹ : R P N , R P N , R P N , lP N . H RR Q RR N RRR GH G RR Q G G H G H Q H Q These relations are symmetric under permutation of i, j, s indices, and using an R R transformaI Q tion, we are free to choose i, j, s along any direction of the M-theory torus ¹B. The M-theory  This is particularly interesting in the d59 case, where we obtain discrete versions of a$ne and hyperbolic groups, see Section 4.6.

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

155

T-duality therefore reads l l l l N ¹ : R P N , R P N , R P N , lP (4.12) '() ' R R ( R R ) RR N RRR ( ) ) ' ' ( ' ( ) and in particular involves three directions, contrary to the naive expectation. We emphasize that the above equation summarizes the non-trivial part of U-duality, and arises as a mixture of T-duality and S-duality transformations. It can in particular be used to derive [322] the duality between the heterotic string compacti"ed on ¹ and type IIA compacti"ed on K in the  Horava}Witten picture, and thus unify all vacua with 16 supersymmetries. We however restrict ourselves to the maximally supersymmetric case in this review. The Weyl group can be written in a way, similar to Eq. (3.12): W(E )"9 ( )S (4.13) B  B but it should be borne in mind that the algebraic relations between the 9 symmetry ¹ and the   permutations S are di!erent from those of the T-duality generators ¹ and S ; in addition '(  GH d di!ers by one unit from the one we used there. We also note that the transformations ¹ and '() S preserve Newton's constant '( “R < 1 '" 0 , " (4.14) l l i N N B where we have de"ned < to be the volume of the M-theory compacti"cation torus. 0 On the other hand, the Borel generators now include a generator c Pc #c that mixes the G G Q eleven-dimensional direction with the other ones, as well as the T-duality spectral #ow B PB #1, from which, by an R R conjugation, we can reach the more general M-theory GH GH Q G spectral yow C : C PC #1 . (4.15) '() '() '() We should also include a set of generators shifting the other scalars from the dual gauge potentials, as explained in Section 4.2: E : E PE #1 , (4.16a) '()*+, '()*+, '()*+, K : K PK #1 . (4.16b) '_()*+,./0 '_()*+,./0 '_()*+,./0 These scalars and corresponding shifts are needed for d56 and d58 respectively. For d59, as will become clear in Section 4.6, the enlargement of the symmetry group to an a$ne or Kac}Moody symmetry requires an in"nite number of such Borel generators. As we shall see in Section 5.4, the Borel generators (4.16) can be obtained from commutators of C transformations. '()  This equation holds for d53 only; when d(3 the 9 symmetry (4.12) collapses and only the permutation group  S remains. B  As discussed in Section 5.4, the C shift actually has to be accompanied by E and K shifts to be a symmetry of the equations of motion.

156

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

Table 14 S-dual type IIB BPS states State

Tension

S-dual

Dual state

D1-brane

1 g l QQ 1 g l QQ 1 g l QQ R gl Q Q 1 g l QQ 1 g l QQ

1 l Q 1 g l QQ 1 gl Q Q R gl Q Q 1 gl Q Q 1 gl Q Q

F-string

D3-brane D5-brane KK5-brane D7-brane D9-brane

D3-brane NS5-brane KK5-brane 7 -brane  9 -brane 

4.5. Type IIB BPS states and S-duality Before studying the structure of the U-duality group, we shall pause and brie#y discuss the action of the extra Weyl generator R  R on the type IIB side. Using the identi"cation (2.45) to Q  convert to type IIB variables, this action inverts the coupling constant and rescales the string length as 1 l  lg , (4.17) g  , Q Q Q Q g Q in such a way that Newton's constant 1/(gl) is invariant. Its action on the BPS spectrum can be Q Q straightforwardly obtained by working out the action on the masses or tensions, and is summarized in Table 14. In this table, we have displayed the action of the 9 Weyl element only. Under more general  duality transformations, the fundamental string and the NS5-brane generate orbits of so called (p, q) strings and (p, q) "ve-branes. The former can be seen as a bound state of p fundamental strings and q D1-branes, or (in the Euclidean case) as a coherent superposition of q D1-branes with p instantons [195]. The (p, q) "ve-branes similarly correspond to bound states of p NS5-branes and q D5-branes. On the other hand, the action of S-duality on the D7 and D9-brane yields states with tension 1/g and 1/g, respectively. These exotic states will be discussed in Section 4.9, where our Q Q nomenclature will be explained as well. Again, such states have less than three transverse dimensions, and do not preserve the asymptotic #atness of space-time and the asymptotic constant value of the scalar "elds. In particular, the D7-brane generates a monodromy qPq#1 in the complex scalar q at in"nity. Its images under S-duality then generate a more general Sl(2, 9)

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

157

monodromy




M"



1!pq

p

!q

1#pq

(4.18)

ascribable to a (p, q) 7-brane. We "nally remark that the relations in Table 14 can also be veri"ed directly using the R  R #ip and the M-theory/IIB identi"cations as (un)wrapped M-theory Q  branes, given in Tables 5 and 6. 4.6. Weyl generators and Weyl reyections In order to understand the occurrence of the E U-duality group, we shall now apply the same BB technique as in the T-duality case and investigate the group generated by the Weyl generators. We choose as a minimal set of Weyl generators the exchange of the M-theory torus directions S : R  R , where I"1,2, d!1, as well as the T-duality ¹"¹ on directions 1, 2, 3 of ' ' '>  the M-theory torus. Adapting the construction of Ref. [108] and Section 3.4, we represent with basis the monomials u"(ln l, ln R , ln R ,2, ln R ) as a form on a vector space < N   B B> e , e , e ,2, e , and associate to any weight vector j"xe #xe #2#xBe its `tensiona    B   B T"e6PH7"lVRVRV2RVB . (4.19) N   B The generators S and ¹ can then be implemented as linear operators on < , with matrix ' B> 2 1 1 1 1 !1 !1 !1 1 , ¹" !1 !1 . (4.20) !1 S" ' 1 !1 !1 !1 ( B\ ( B\ The operators S and ¹ in Eq. (4.20) are easily seen to be orthogonal with respect to the Lorentz ' metric











ds"!(dx)#(dx') ,

(4.21)

and correspond to Weyl re#ections a)j a jPo (j)"j!2 ? a)a

(4.22)

along planes orthogonal to the vectors a "e !e , I"1,2, d!1, ' '> '

a "e #e #e !e .     

(4.23)

 It has also been proposed that the IIB 7-branes transform as a triplet of Sl(2, 9) [225].  In Ref. [108], the discussion was carried out from the gauge theory side, and the U-duality invariant (4.14) was used to eliminate the vector e , except when d"9. This vector can, however, be kept for any d, and, as we shall momentarily  see, appears as an extra time-like direction.  T actually has the dimension of a p-brane tension T , with p"!3x!x!2!xB!1. N

158

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

It is very striking that l appears on the same footing as the other radii R , but with a minus sign in N ' the metric: it can be interpreted as the radius of an extra time-like direction, much in the spirit of certain proposals about F-theory [25,310]. The only non-vanishing (Lorentzian) scalar products of these roots turn out to be (a )"(a )"2 , a ) a "a ) a "!1 '  ' '>  

(4.24)

summarized in the Dynkin diagram: * "



(4.25)

# ! ! ! !2! .     B\ This is precisely the Dynkin diagram of E as shown in Table 9, in agreement with the analysis B based on moduli counting. In Eq. (4.25) it is easy to recognize the diagrams of the SO(d!1, d!1, 9) (denoted by *'s) and Sl(d, 9) (denoted by #'s) subgroups. The branching of the Sl(d) diagram on the third root re#ects the action of T-duality on three directions. The full diagram can be built from the M-theory Lorentz group Sl(d, 9) denoted by #'s, and from the type IIB Lorentz group Sl(d!1, 9) generated by the roots a , a ,2, a . Under decompacti"cation, the rightmost root has to be dropped, so   B\ that E disintegrates into E . When the root at the intersection is reached, the diagram falls B B\ into two pieces, corresponding to the two Sl(2) and Sl(3) subgroups in D"8. The root a itself  disappears for d"2, leaving only the root a of Sl(2, 1).  Again, the action of the Weyl group on < is reducible, at least for d48. Indeed, the B> invariance of Newton's constant “R /l implies that the roots are all orthogonal to the vector ' N d"e #2#e !3e ,  B 

(4.26)

with proper length d"d!9, so that the re#ections actually restrict to the hyperplane < normal B to d: x#2#xB#3x"0 .

(4.27)

The Lorentz metric on < restricts to a metric g "d !1/9 on < , which is positive-de"nite for B> '( '( B d48, so that S and ¹ indeed generate the Weyl group of the Lie algebra E (1). The order and ' B number of roots of these groups are recalled in Table 15 [174]. When d"9, however, the invariant vector d becomes null, so that < no longer splits into B> d and its orthogonal space; the generators act on the entire Lorentzian vector space < , and the B>  From this point of view, the U-duality is a consequence of general coordinate invariance in M and type IIB theories [204].  There is a notable exception for d"8, where E disintegrates into E ;Sl(2). This is because the extended Dynkin   diagram of E has an extra root connected to a . Only Sl(2) singlets remain in the spectrum, however. The same happens   in d"4, where E "Sl(3);Sl(2) in E "Sl(5) is not a maximal embedding.  

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

159

Table 15 Order and number of roots of E Weyl groups B d

2

3

4

5

6

7

8

9

E B Order Roots

A  2 2

A ;A   6;2 6#2

A  5! 20

D  25! 40

E  23 5 72

E  23 5 7 126

E  2357 240

EK  R R

generators S and ¹ no longer span a "nite group. Instead, they correspond to the Weyl group of ' the azne Lie algebra E "EK . This is in agreement with the occurrence of in"nitely many   conserved currents in D"2 space-time dimensions. This case requires a speci"c treatment and will be discussed in Section 4.12. For d'9, that is compacti"cation to a line or a point, the situation is even more dramatic, with the occurrence of the hyperbolic Kac}Moody algebras E and E ,   about which very little is known. The reader should go to [126,179,180,234] for further discussion and references. 4.7. BPS spectrum and highest weights Pursuing the parallel with our presentation on T-duality, we now discuss the representations of the U-duality Weyl group. The fundamental weights dual to the roots a ,2, a , a are easily  B\  computed: R j"e !e PT "  ,    l N R R j"e #e !2e PT "   ,     l N R R R j"e #e #e !3e PT "    ,      l N R R R R j"e #2#e !3e PT "     ,     l N 2 R R jB\"e #2#e !3e PT " 2 B\ ,  B\  \B l N 1 jB\"e #2#e !3e &!e PM" ,  B\  B R B 1 j"!e PT " ,   l N

(4.28a) (4.28b) (4.28c) (4.28d)

(4.28e) (4.28f) (4.28g)

160

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

where the symbol & in Eq. (4.28f) denotes equality modulo d, that is up to a power of the invariant Planck length. In the above equations, we have translated the weight vectors into monomials, and interpreted it as the tension T of a p-brane: N> E The weight jB\ corresponds to the Kaluza}Klein states, with mass 1/R , as well as its ' U-duality descendants. We shall name its orbit the particle multiplet, or yux multiplet, for reasons that will become apparent in Section 6.9. E The weight j on the other hand has dimension 1/¸, and corresponds to the tension of a membrane wrapped on the direction 1: it will go under the name of string multiplet, or momentum multiplet. The latter name will also become clear in Section 6.9. E The weight j is the highest weight of the membrane multiplet containing the fundamental membrane with tension 1/l, together with its descendants. N E The weights j and j both correspond to threebrane tensions T and T . Even though they   are inequivalent under the Weyl group, it turns out that j is a descendant of j under the full U-duality group. The U-duality orbit of the state with tension T is therefore a subset of the  orbit of the state with tension T , and j is the true highest-weight vector of the threebrane  multiplet. E The same holds for j associated to a membrane tension T and descendant of the highest  weight j of the membrane multiplet under U-duality, as well as for j and j. E The weight j corresponds to a "vebrane tension T , but is again not the highest weight of the  xvebrane multiplet, which is instead a non-fundamental weight: 1 T " Pj"!2e "2j .   l N

(4.29)

Similarly, the weight j corresponds to a fourbrane tension T , and is not the highest weight  of the fourbrane multiplet, which is instead a non-fundamental weight: R T " Pj"e !2e "j#j .    l N

(4.30)

E Finally, the instanton multiplet does not appear in Eqs. (4.28a)}(4.28g). An instanton con"guration can be obtained by wrapping a membrane on a three-cycle, and corresponds to a weight vector R R R T "   Pj"a . \  l N

(4.31)

Since this vector is a simple root, it corresponds to a multiplet in the adjoint representation. It is, however, not the highest weight of the U-duality multiplet, which is instead the highest root t whose expansion coe$cients on the base of the simple roots are given by the Coxeter labels  We should, however, warn the reader that it is not the representation arising in non-perturbative couplings, as we shall discuss in Section 5.8.

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

161

in Table 9. An explicit computation gives d"4: t"d!j!j ,

(4.32a)

d"5: t"d!j ,

(4.32b)

d"6: t"d!j ,

(4.32c)

d"7: t"d!j ,

(4.32d)

d"8: t"d!j .

(4.32e)

Since the fundamental weights jG are dual to the simple roots a , it is clear that t ) a "d , ' ' G' where I is the index appearing on j in Eqs. (4.32a)}(4.32e) at a given d, and moreover it can be easily checked that t"2. The highest root can therefore be added as an extra root in the Dynkin diagrams in Table 9, and turns them into extended Dynkin diagrams. The previous considerations are summarized in the diagram 1 l N "

(4.33)

R R R R R R R R R R 1 !  !   !    !2! l l l l R N N N N B where we have indicated the highest weight associated to each node of the Dynkin diagram. For simplicity, we shall henceforth focus our attention on the particle and string multiplets, corresponding to the rightmost node with weight jB\ and leftmost node with weight j, respectively. 4.8. The particle alias yux multiplet The full particle multiplet can be obtained by acting with Weyl and Borel transformations on the Kaluza}Klein state with mass 1/R . Instead of working out the precise transformation of the ' supergravity con"gurations, we can restrict ourselves to considering the masses of the various states in the multiplet. We note that the action of S and ¹ on the dilatonic scalars R is '( '() ' independent of the dimension d of the torus, so that we can work out the maximally compacti"ed case D"3, and obtain the higher-dimensional cases by simply deleting states that require too many di!erent directions on ¹B to exist. The results are displayed in Table 16, where distinct letters stand for distinct indices. The states are organized in representations of the Sl(8, 9) modular group of the torus ¹. These representations arrange themselves in shells with increasing power of l; since l is invariant under Sl(8, 9), this N N corresponds to the grading with respect to the simple root a . Generalized T-duality ¹ may  '()  See Ref. [216] for the construction of U-duality multiplets of p-brane solutions, and Ref. [113] for a discussion of the continuous U-duality orbits of p-brane solutions.

162

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

Table 16 Particle/#ux multiplet 248 of E  Mass M

Sl(8) irrep

Charge

1 R ' RR ' ( l N RRR R R ' ( ) * + l N RR R R R R R < ' ( ) * + , ., 8 0 l l N N RRR R R R R R ' ( ) * + , . / l N RRR RR R R R ' ( ) * + , . / l N RRR RR R RR ' ( ) * + , . / l N

8

m 

28

m

56

m

1#63

m_

56

m_

28

m_

8

m__

move from one shell to the next or previous one, whereas S acts within each shell. Eight states '( with mass < /l have been added in the middle line, corresponding to zero-length weights that 0 N cannot be reached from the length-2 highest state. These states are, however, necessary in order to get a complete representation of the modular group Sl(8, 9), and can be reached by a Borel transformation in Sl(8, 9). They can be thought of as the eight ways to resolve the radius that appears squared in the mass of the other states on the same line into a product of two distinct radii. This is not required for the other lines, since all squares can be absorbed with a power of Newton's constant. In the last column of Table 16, we have indicated the representation of Sl(8, 9) that yields the same dimension. The superscripts denote the number of antisymmetric indices, and no symmetry property is assumed across a semicolon. In other words, m_ correspond to the << where < is the de"ning representation of Sl(d). These representations are precisely dual to those under which the various gauge vectors transform (see Table 11); they actually correspond to the charges of the BPS state under these ;(1) gauge symmetries (see also Section 4.2). They generalize the D-brane charges we discussed in Section 3. Altogether, these states sum up to 248, the adjoint representation of E , which indeed decomposes in the indicated way under the branching  Sl(8)LE . The occurrence of the adjoint representation simply follows from the last equality of  Eqs. (4.32a)}(4.32e) identifying the fundamental weight j with the highest root of E .  The "rst three lines in Table 16 have an obvious interpretation. The state with mass 1/R is ' simply the Kaluza}Klein excitation on the dimension I, and m denotes the vector of integer ' momentum charges. The state with mass R R /l is the membrane wrapped on a two-cycle ¹ of ' ( N the compacti"cation torus ¹B, and the two-form m'( labels the precise two-cycle, just as in the

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

163

Table 17 Particle/#ux multiplets of E B D

d

U-duality group

irrep

Sl(d) content

10 9 8 7 6 5 4 3

1 2 3 4 5 6 7 8

1 Sl(2, 9) Sl(3, 9);Sl(2, 9) Sl(5, 9) SO(5, 5, 9) E (9)  E (9)  E (9) 

1 3 (3, 2) 10 16 27 56 248

1 2#1 3#3 4#6 5#10#1 6#15#6 7#21#21#7 2(8#28#56)#63#1

D-brane case of the previous section. The third line corresponds to the "vebrane wrapped on the "ve-cycle labelled by m'()*+. The states on the fourth line are more interesting. The "rst of them involves one square radius, and therefore does not exist in uncompacti"ed eleven dimensions. It is simply the KK6-brane with Taub}NUT direction along R and wrapped along the directions J ' to P. The second state with mass < /l, however, does exist in eleven uncompacti"ed directions, 0 N and has the tension of a would-be 8-brane. Its asymptotic space-time is however not #at, but logarithmically divergent. The status of this solution is unclear at present, together with that of the following lines of the table. These states only appear as particles in D"3, with the peculiarities that we have already mentioned. Upon decompacti"cation, the last two lines in Table 16 disappear since they require eight distinct radii, and the particle multiplet reduces to a representation of the corresponding U-duality group, as indicated in Table 17. When d54, the representation remains the one dual to the rightmost root. For d"3, the U-duality group disconnects into Sl(3) and Sl(2), and !e becomes B equal to j#j instead of being equal to j, as in other cases. Consequently, the particle multiplet transforms as a (3, 2) representation of U-duality. The full particle multiplet on ¹B can be easily decomposed in representations of the U-duality group E (9) in one dimension higher by separating the states in Table 16 according to their B\B\ dependence on the decompacti"ed radius R (which gives a gradation with respect to the simple B root a ). We obtain the general decomposition B\ MB"1" M" (T T )" T " (T )" , (4.34) \        where we have denoted the multiplets as in Eqs. (4.28a)}(4.28g) and speci"ed the power of R in B subscript. The notation (T ) means twice the fundamental weight associated to T . The   multiplets on the right-hand side of Eq. (4.34) become empty as d decreases. In particular, we note that the particle multiplet on ¹B decomposes into a singlet, corresponding to the Kaluza}Klein excitation around the decompacti"ed direction xB, as well as a particle and a string multiplet on ¹B\, depending on whether the state was wrapped around xB. There are also a number of additional states that appear for d56, to which we shall come back in Section 7.7.  For d"8, this is 248"156(1331)561.

164

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

As a side remark, we note that Table 16 is symmetric under re#ection with respect to the middle line: for each state with mass M there is a state with mass M satisfying




<  0 , (4.35) l N where < is the volume of the eight-torus. In particular, the lowest weight is equal to minus the 0 highest weight, modulo the invariant vector d. This is a general property of real representations of compact groups, and indeed 248 is the adjoint representation of E , therefore real. The same also  holds for the 56 representation of E in the d"7 case. However, whether or not real with respect to  the compact real form of the group E ("), all the representations appearing in Table 17 are real as B representations of the non-compact group E , as is required by the existence of an anti-particle for BB each particle. This is obvious for d44; for d"5, it is equivalent to the statement that the spinor of SO(8) is real, since the reality properties of spinors of SO(p, q) depend only on p!q mod 8. This property is a characteristic feature of the representations of the normally non compact real form. MM"

4.9. T-duality decomposition and exotic states In order to make contact with string theory solutions, it is useful to decompose the particle multiplet into irreducible representations of the T-duality group SO(d!1, d!1, 9). This can be simply carried out by distinguishing whether the indices lie along the eleventh dimension or not, and substituting the matching relations (2.1). Since T-duality commutes with the grading in powers of the string coupling g , the various irreps are then sorted out according to the dependence of the Q mass of the states on g . Table 18 summarizes the decomposition of the particle/#ux multiplet for Q M-theory on ¹ into irreducible representations of the SO(7,7) T-duality symmetry group of type IIA string theory on ¹, as well as the Sl(8) (resp. Sl(7)) modular group of the M-theory (resp. string theory) torus. The masses of the states in the ath column depend on the string coupling constant as 1/g?\, and Q are given by 1 R , G, (4.36a) R l G Q 1 1 RR RR R R RRR RR R , G H, G H I J, G H I J K L , (4.36b) S :  g l l l l Q Q Q Q Q 1 R R R R R < RR R R R R < R R G H I J K, 0, G H I J K L, 0 G H , (4.36c) S#AS: l l l l g Q Q Q Q Q < R R R R R R R R R < (4.36d) S : 0 G, G H I, G H I J K, 0 , l l l gl l Q Q Q Q Q Q <  l 0 Q ,R , <: (4.36e) gl R G Q Q G where < denotes the volume of the string-theory seven-torus. At level 1/g we observe the usual 0 Q KK and winding states of the string and the level 1/g reproduces the D0-, D2-, D4- and D6-branes. Q <:


















N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

165

Table 18 Branching of the d"8 particle multiplet into irreps of Sl(8) and SO(7, 7). The entries in the table denote the irreps under the common Sl(7) subgroup of Sl(8) and SO(7, 7) 248(E )MSO(7,7)  6 Sl(8) 8 (m )  28 (m) 56 (m) 1#63 (m_) 56 (m_) 28 (m_) 8 (m__)

14 (<)

64 (S ) 

7 (m )  7 (mQ)

1 (m ) Q 21 (m) 35 (mQ) 7 (mQ_Q)

1#91 (SAS)

21 (m) 1#1#48 (mQ_, m_Q) 21 (mQ_Q)

64 (S )

14 (<)

7 (m_) 35 (m_Q) 21 (mQ_Q) 1 (mQ_Q_Q)

7 (m_Q) 7 (m_Q_Q)

At level 1/g, the NS5 and KK5-brane appear together with two new types of state, a 7 -brane and Q  a 5-brane. Our nomenclature displays on-line the number of spatial world-volume directions, i.e.  the number of radii appearing linearly in the mass; the superscript speci"es the number of directions (if non-zero) that appear quadratic, cubic, etc., listed from the right to the left. The subscript denotes the inverse power of the string coupling appearing in the mass formula; for example, in this convention the KK5-brane is a 5-brane. According to this notation, we "nd at  level 1/g a 6-, 4-, 2- and 0-brane. Their masses are related to those of the even Dp-branes, by Q     the type IIA (on ¹) mirror symmetry




MM"

<  0 , gl Q Q

(4.37)

which follows from the M-theory mirror symmetry relation (4.35). Finally, at level 1/g, a 1- and Q  a 0-brane are obtained, whose masses are related to those of the KK and winding states by  Eq. (4.37). At this point a few remarks are in order on the new type IIA states that appear in Eqs. (4.36a)}(4.36e). The 7 - and 5-brane, with mass proportional to 1/g have a conventional   Q dependence on the string coupling, but no supergravity solutions are known for these states. In addition, a variety of states with exotic dependence on the string coupling, 1/g and 1/g, are Q Q observed. They arise from M-theory states with mass diverging as 1/l or faster. It is not clear what N the meaning of these new states in M-theory and type IIA string theory is. These states cannot be accommodated in weakly coupled string theory where the most singular behaviour is expected to be 1/g, corresponding to Neveu}Schwarz solitons. A higher power would imply a contribution of Q a Riemann surface with Euler characteristic s'2. Another way to see this is by considering the gravitational "eld created by these objects, which scales as Mi : since i &g, states whose mass   Q goes like gL, n42 create a vanishing or at most "nite gravitational "eld in the weak coupling limit, Q allowing for a #at space description in the spirit of D-branes. On the other hand, when n'2, the surrounding space becomes in"nitely curved at weak coupling, and these states do not correspond

166

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

Table 19 String/momentum multiplet 133 of E  Tension T mass

Sl(7) irrep

Charge

R ' l N RRR R ' ( ) * l N RR R R R R < ' ( ) * + ,, 7 0 l l N N RRR R R R R ' ( ) * + , . l N RRR RR R R ' ( ) * + , . l N

7

n

35

n

1#48

n_

35

n_

7

n_

to solitons anymore. In fact, the simplest of these states, namely 6, can be obtained by constructing  an array of Kaluza}Klein along a non-compact direction of the Taub-NUT space, and wrapping the worldvolume directions on the string theory torus ¹ [52]. The summation of the poles in the harmonic function is logarithmically divergent, implying that the asymptotic space-time is logarithmically divergent as well. This is the rule and not the exception for a pointlike state in 3 space-time dimensions (since the Laplacian in the two transverse coordinates has a logarithmic kernel), and the conventional states with an asymptotically #at space-time are simply con"gurations with a vanishing charge. The same issue arises for p-branes in p#3 dimensions (or less). We emphasize, though, that our present purpose is to examine the consequences at the algebraic level of the presence of the conjectured U-duality, which does require these exotic states. The supergravity solutions describing these states can in principle by computed using the known duality relations, which indeed do not preserve the asymptotic #atness of the metric. 4.10. The string alias momentum multiplet The same analysis can be carried out for the string multiplet, by applying a sequence of Weyl re#ections on the highest weight R /l describing the wrapped membrane. After adding a multiplet ' N of length 2 and 35 zero-weights for Sl(8, 9) invariance, we obtain a 3875 representation of E . The  precise content of this representation is displayed in Appendix B; instead, we display in Table 19 the more manageable result for the d"7 case, where the string multiplet transforms as a 133 adjoint representation of E . The occurrence of the adjoint representation is again understood  from Eq. (4.32d) relating the fundamental weight j to the highest root t.

 This construction "rst appeared in the context of the conifold singularity in the hypermultiplet moduli space [241].

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

167

Table 20 String/momentum multiplets of E B D

d

U-duality group

irrep

Sl(d) content

10 9 8 7 6 5 4 3

1 2 3 4 5 6 7 8

1 Sl(2, 9) Sl(3, 9);Sl(2, 9) Sl(5, 9) SO(5, 5, 9) E (9)  E (9)  E (9) 

1 2 (3, 1) 5 10 27 133 3875

1 2 3 4#1 5#5 6#15#6 7#35#49#35#7 8#70#2

These states have the same interpretation as the states in the particle multiplet, but for wrapping one dimension less of the world-volume. In other words, the states in the particle multiplet can be obtained by wrapping strings on one dimension more } except for the Kaluza}Klein state, which is a genuine point-like (or wave-like, rather) object. We note again that Table 19 is symmetric under re#ection with respect to its middle line, in agreement with the reality of the 133 adjoint representation of E .  The string multiplet in higher dimensions is simply obtained by dropping the states that require too many di!erent radii, as displayed in Table 20; in all cases, it corresponds to the representation dual to  string multiplet is distinct from the 27 particle the leftmost root a . We note that in d"6 the 27  multiplet, but is related to it by an outer automorphism of E corresponding to the 9 symmetry of its   Dynkin diagram. We also note, for later use, that in all cases the string multiplet representation arises in the symmetric tensor product of two particle multiplets, i.e. (M M)T always contains a singlet. Q Like the particle multiplet, the full string multiplet on ¹B can be easily decomposed in representations of the U-duality group E (9) in one dimension higher by using the gradaB\B\ tion in powers of the decompacti"ed radius R  B TB"T " (T T )" T " , (4.38)       where we have denoted the multiplets as in Eqs. (4.28a)}(4.28g) and again speci"ed the power of R in subscripts. In particular, we note that the string multiplet on ¹B decomposes into a string and B a membrane multiplet on ¹B\, depending on whether the state was wrapped around xB. There are also a number of additional states that disappear for d46. As in the previous subsection, we give the branching of the d"7 string multiplet in terms of irreps of the T-duality SO(6, 6, 9) as well as the modular groups Sl(7, 9) and Sl(6, 9) of the M-theory and string theory tori in Table 21. 4.11. Weyl-invariant ewective action As in our discussion of T-duality, we would now like to write the supergravity action (2.2) in a manifestly Weyl-invariant form. This has been carried out in Refs. [213,215], a simpli"ed version  For d"7, this is 133"27(781)27 .

168

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

Table 21 Branching of the d"7 string multiplet into irreps of Sl(7) and SO(6,6). The entries in the table denote the irreps under the common Sl(6) subgroup of Sl(7) and SO(6,6) 133 (E )MSO(6,6)  6 Sl(7) 7 (n) 35 (n) 1#48 (n_) 35 (n_) 7 (n_)

1 (S)

32 (S )

1 (nQ)

6 (n) 20 (nQ) 6 (nQ_Q)

1#66 (SAS)

15 (n) 1#1#35 (nQ_,n_Q) 15 (nQ_Q)

32 (S )

1 (S)

6 (n_) 20 (n_Q) 6 (nQ_Q)

1 (n_Q)

of which will be presented here. As in Eq. (3.25), we decompose the eleven-dimensional "eld strength F and metric in lower-degree forms. The action then takes the simple form:













< RR  R  G # GFG # (R FG) R# S " d\Bx(!g G H \B l R R N G H GH G G l  l  l  N F # N F #(lF)# N F # , (4.39) N R G R R GH RRR G G G H G H I G GH GHI where the "rst line comes from the reduction of the Einstein}Hilbert term and the second from the kinetic term of the three-form. In Eq. (4.39) we again recognize in front of the one-form "eld strength F and F the positive roots e !e and e #e #e !e , in front of the two-form "eld strength F the weights !e of G H G H I  G the particle multiplet, in front of the three-form "eld strength F the weights e !e of the string G  multiplet, and in front of the four-form "eld strength F the weight !e of the membrane  multiplet. However, these weights do not form complete orbits: it is necessary to dualize the "eld strengths FN into lower-degree "eld strengths F\B\N so as to display the Weyl symmetry. In the 2p"11!d case, both the "eld strength and its dual should be kept. Alternatively, all "eld strengths may be doubled with their duals, and display an even larger symmetry [72,73]. We then obtain a manifestly Weyl-invariant action:


















< R#Ru ) Ru# e\6P?7(F)# e\6PH7(F) S " d\Bx(!g ? H \B l N ?ZU> HZU 



(4.40) # e\6PH7(F)# e\6PH7(F)#2 , H H HZU 
  HZU  where u"(ln l, ln R ,2, ln R ) is the vector of dilatonic scalars (whose "rst component is N  B non-dynamical), 1u, j2"x ln l#x ln R #2 is the duality bracket (4.19) and Ru ) Ru the N  Weyl-invariant kinetic term (Rl"0) obtained from the metric (4.21). In addition to the equations N of motion from (4.40), the duality equations FN"*F\B\N should also be imposed. As in the

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

169

case of T-duality, the set of positive roots U is not invariant under Weyl re#ections, but the > Peccei}Quinn scalars undergo non-linear transformations APe\6P?7A that compensate for the sign change [215]. 4.12. Compactixcation on ¹ and azne EK symmetry  As we pointed out in Section 4.6, the compacti"cation on a nine-torus ¹ to two space-time dimensions gives rise to a qualitative change in the U-duality group: the invariant vector d in Eq. (4.26) corresponding to the dimensionless Newton constant becomes light-like w.r.t. the Lorentzian metric !(dx)#(dx)#2#(dx), so that the action of the U-duality group generated by S and ¹ in Eq. (4.20) cannot be restricted to its orthogonal subspace. Instead, it generates the Weyl ' group of the EK a$ne algebra, as was shown in Ref. [108]; we shall recast their construction in the  notation of this review, at the same time settling several issues. In order to see the a$ne symmetry EK arise, we simply note that the Dynkin diagram of E (see   Table 9) is nothing but the extended Dynkin diagram of E , where the additional root with Coxeter  label 1 corresponds to a "e !e . The roots a , a ,2, a generate the E horizontal Lie algebra,        whereas a and d"  e !3e are the extra dimensions needed to represent the central charge '   ' K and degree D generators of the standard construction of a$ne Lie algebras (see e.g. Ref. [118]). To make the identi"cation precise, we recall that the simple roots of an a$ne Lie algebra GK can be chosen as a( "(a , 0, 0), I"1,2, r, a( "(!t, 0, 1) (4.41) ' '  in the basis (k, k, d) of the Minkovskian weight space < "1P#1 with norm k#2kd. Here, P> t is the highest root of G, r is the rank of G, k is the a$ne level, and d the ¸ eigenvalue. In the case  at hand, we have G"E so r"8 and want to "nd the change of basis between the roots  a , I"0,2, 8 and null vector d of our formalism and the standard roots a( , I"0,2, 8 and ' ' vectors c"(0, 0, 1), i"(0, 1, 0). From Eqs. (4.32a)}(4.32e) we have, t"e #2#e #2e !3e "d!a ,      so that, comparing with Eq. (4.41), we can identify d with c"(0, 0, 1) and

(4.42)

a( "a , I"1,2,7 , (4.43a) ' ' a( "a , (4.43b)   a( "a . (4.43c)   The vector i"(0, 1, 0) can be easily calculated from the requirements that i"i ) a( "0, ' I"1,2, 8 and i ) d"1: 1 d i" (!e !2!e #e #3e )"e ! .      2 2

(4.44)

 In order to keep with the standard notation, the simple roots of the Lie algebra are now labelled by subscripts ranging from 1 to r, as opposed to our notation for the simple roots of the U-duality groups E , which carry labels 0 P to r!1.

170

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

The level k and degree d of any weight vector j3< can now be obtained from the products d ) j  and d ) i, respectively, and they both have a simple interpretation: k"d ) j"x#2#x#3x is simply the length dimension of the associated monomial “RV'lV, and ' N d"i ) j"x!k/2

(4.45)

(4.46)

counts the power of R appearing in the same monomial, up to a shift k/2. This was expected,  since the horizontal subalgebra E LEK does not a!ect R and by de"nition commutes with ¸ .     ¸ (n'0) generators, on the other hand, bring additional powers of R and increase the \L  degree d. In particular, the ¸ eigenvalues are integer-spaced, as they should.  We proceed by considering the particle/#ux and string/momentum multiplets introduced in Sections 4.8 and 4.10, with highest weights jB\"!e and j"e !e , respectively (see Eqs.    (4.28a)}(4.28g)). The particle multiplet is therefore a level !1 representation with trivial ground state k"0 (that is, in the chiral block of the identity). A bit of experimentation reveals the "rst Sl(9) representations occurring in the particle multiplet: m__, m__, m__, m__, m__, m___,2

(4.47)

with tensions scaling from 1/l to 1/l, in addition to the representations already present in d"8, N N given in Table 16. However, the full orbit is in"nite. On the other hand, the string multiplet is a level !2 representation with ground state in the 3875 of E . In both cases, the representations are  in"nite-dimensional, and need to be supplemented with weights of smaller length as in the E and  E cases. The instanton multiplet, on the other hand, is a level-0 representation of EK , with   a non-singlet ground state in the adjoint of E . This makes it obvious that the usual unitarity  restrictions for compact a$ne Lie algebras do not apply in our case. This concludes our analysis of the d"9 case, and we now restrict ourselves to the better understood d48 case.

5. Mass formulae on skew tori with gauge backgrounds We would now like to generalize the mass formulae of the U-duality multiplets obtained so far for rectangular tori and vanishing gauge potentials to the more general case of skew tori and arbitrary gauge potentials, which will exhibit the full U-duality group. This will also allow a better understanding of the action of Borel generators on the BPS spectrum. We will concentrate on the d"7 #ux multiplet, but the same method applies to the other multiplets. 5.1. Skew tori and Sl(d, 9) invariance We have already argued that BPS states could be labelled by a set of tensors of integer charges describing their various momenta and wrappings. In particular, for the case of the d"7 #ux multiplet, the charges m , m, m, m_ 

(5.1)

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

171

describe the Kaluza}Klein momentum, membrane, "vebrane and KK6-brane wrappings. The position of the index has been chosen in such a way that we obtain the correct mass by contracting each of them with the vector of radii R' or inverse radii 1/R . Note that for d"7 the tensor m_ is ' really a tensor m, but the extra seven indices account for an extra factor of the volume in the tension. Of course, a BPS state with generic charges m will not be 1/2-BPS state in general (for d55): some quadratic conditions on m have to be imposed, as already discussed in Sections 2.2 and 3.8. We shall henceforth assume these conditions ful"lled, deferring the study of the latter to Section 5.9. The 1/2-BPS state mass formula for a non-diagonal metric g can be straightforwardly obtained '( by replacing contractions with the vector of radii by contractions with the metric, and inserting the proper symmetry factor and power of the Planck length on dimensional grounds: M"(m )#(m)#(m)#(m_)  1 1 m'()*+g g g g g m,./01#2 . "m g'(m # m'(g g m)*# ') (* ', (. )/ *0 +1 ' ( 2!l 5!l N N

(5.2)

This formula is invariant under Sl(d, 9), but not yet under the T-duality subgroup SO(d!1, d!1, 9) of the U-duality group. It only holds when the expectation value of the various gauge "elds on the torus vanish. To reinstate the dependence on the three-form C , we apply the '() following strategy. E Decompose the #ux multiplet as a sum of T-duality irreps. E Include the correct coupling to the NS two-form "eld B using the T-duality invariant mass GH formulae. E Study the T-duality spectral #ow BPB#*B. E Covariantize this #ow under Sl(d, 9) into a CPC#*C #ow. E Integrate the CPC#*C #ow to obtain the U-duality invariant mass formula. 5.2. T-duality decomposition and invariant mass formula We have already discussed the "rst step in Section 4.9, and we only need to restrict ourselves to the case d"7. Table 18 then truncates to its upper left-hand corner displayed in Table 22, as can be read from the d"7 particle multiplet mass formula (5.2) written with s and i indices:



M"

 

 

 



(m) m (mQ) (m) (mQ_Q) (m_Q) Q #(m ) # (mQ)# # # # #  g g g g g g Q Q Q Q Q Q

(5.3)

corresponding to three SO(6, 6) irreps, <"(m , mQ) momentum and winding , 

(5.4a)

S"(m , m, mQ, mQ_Q) D0-, D2-, D4-, D6-brane , Q

(5.4b)

<"(m, m_Q) NS5-brane and KK5-brane .

(5.4c)

172

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

Table 22 Branching of d"7 particle multiplet into irreps of Sl(7) and SO(6, 6). The entries in the table denote the irreps under the common Sl(6) subgroup of Sl(7) and SO(6, 6) 56(E )MSO(6,6)  6 Sl(7)

12 (<)

32 (S ) 

7 (m )  21 (m) 21 (m) 7 (m_)

6 (m )  6 (mQ)

1 (m ) Q 15 (m) 15 (mQ) 1 (mQ_Q)

12 (<)

6 (m) 6 (m_Q)

We can now use the T-duality invariant mass formulae for the T-duality irreps that we obtained in Section 3. In terms of the present charges, they read schematically (in units of l ) Q M "(m #B mQH)gGI(m #B mQJ)#mQGg mQH , (5.5a) 4 G GH I IJ GH 1 M " [(m #mB #mQB#mQ_QB)#(m#mQB #mQ_QB)      1 g Q Q #(mQ#mQ_QB )#(mQ_Q)] , (5.5b)  1 M " [(m#m_QB )#(m_Q)] , (5.5c) 4Y g  Q where we used the vector and spinor representation mass formulae (3.8a), (3.8b) and (3.38a)}(3.38d). Adding the three contributions M+ together, we now obtain the #ux multiplet mass formula 41 4Y, for vanishing values of the Ramond "elds and arbitrary B-"eld:

 

   



m  (m ) Q #(m ) # (m Q)# (5.6a)  g g Q Q (m Q) (m ) (m Q_Q) (m _Q) # # # , (5.6b) # g g g g Q Q Q Q where the tilded charges are shifted to incorporate the e!ect of the two-form as in (5.5), so that for instance M"

1 1 1 m "m # B m# BmQ# BmQ_Q Q Q 2  8  48 



(5.7)

is the shift in the D0-brane charge. 5.3. T-duality spectral yow In Sections 3.7 and 3.8 we have already discussed the spectral #ow B PB #*B in the GH GH GH vectorial and spinorial representations. We only need to rephrase this #ow in terms of the present

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

173

charges: m Pm #*B mQH, mQGPmQG , G G HG S : m Pm #*B mGH, mGHPmGH#*B mQIJGH Q Q  GH  IJ mQGHIJPmQGHIJ#*B mQ_QKLGHIJ, mQ_QPmQ_Q ,  KL <: mGHIJKPmGHIJK!*B mL_QNGHIJK LN m_QPm_Q . <:

(5.8)

The #ow indeed acts as an automorphism on the charge lattice, and in particular the charges cannot be restricted to positive integers (except for m_). This fact will be of use in Section 7.8. Alternatively, the above spectral #ow can be recast into a system of di!erential equations for the shifted charges m , e.g. for the spinor representation we have Rm GH Rm Q "m GH, "m QGHIJ   RB RB IJ GH (5.9) S : Rm QGHIJ Rm Q_QGHIJKL "m Q_QGHIJKL, "0 .  RB RB KL NO This system can be integrated to yield the spinor representation mass formula; the constants of integration correspond to the integer charges m. The integrability of this system of di!erential equations follows from the commutativity of the spectral #ow. 5.4. U-duality spectral yows The mass formula (5.6) obtained so far is invariant under T-duality and holds for vanishing values of Ramond gauge backgrounds. In order to obtain a U-duality invariant mass formula, we have to allow expectation values of the M-theory gauge three-form C , which extends the '() Neveu}Schwarz two-form B "C ; the expectation value of the Ramond one-form is already GH QGH incorporated as the o!-diagonal metric component A "g /RO0. For d56, one should also G QG Q allow expectation values of the six-form E (PoincareH -dual to C in eleven dimensions). In '()*+, '() string-theory language, this corresponds to the Ramond "ve-form E and the Neveu}Schwarz Q six-form dual to B in ten dimensions. IJ In order to reinstate the C dependence in mass formula we covariantize the B "C spectral '() GH QGH #ow (5.8) under Sl(d, 9), with the result that m Pm #*C m() , ' '  ()' m'(Pm'(#*C m)*+'( ,  )*+ m'()*+Pm'()*+#*C m,_./'()*+ ,  ,./ m_Pm_ .

(5.10)

 For d"8, we also need to include the form K , which in string-theory language includes the Ramond seven-form _ K , along with a K form. Q_Q _Q

174

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

Here, however, the C spectral #ow turns out to be non-integrable. De"ning '() as the #ow induced by the shift C PC #*C , we have the commutator '() '() '() [ '(), *+,]"20 '()*+, , (5.11) where '()*+, is the #ow induced by the shift E PE #*E : '()*+, '()*+, '()*+, m Pm # *E m()*+, , ' ' r ()*+,' m)_*+,./'( , m'(Pm'(# *E r )*+,./ mPm ,

(5.12)

m_Pm_ . The non-integrability (5.11) of the C-#ow can be understood as a consequence of the Chern}Simons interaction in the 11D supergravity action Eq. (2.2) [72,73]: the equation of motion for C reads 1 d*F # F F "0 ,  2   so that the dual "eld strength of F has a Chern}Simons term  1 F ,*F "dE ! C F .    2  

(5.13)

(5.14)

The equation of motion (5.14) is invariant under the gauge transformations dC "K ,  

1 dE "K ! K C ,   2  

(5.15)

for closed K and K . Restricting to constant shifts, this reproduces the commutation relations   (5.11). An equivalent statement holds in D"3, where the C and E shifts close on a K shift.   _ The non-integrability of the system (5.10) can therefore be evaded by combining the *C shift  with a *E shift  1 1 *E " C *C , (5.16) 5! '()*+, 12 '() *+,

upon which the resulting #ow

'()" '()!10C

)*+'() (5.17) )*+ becomes integrable. The extra shift is invisible in the type IIA picture for zero RR potentials since it does not contribute to the T-duality spectral #ow. We emphasize again that these extra terms are  For d"8 there is also a non-trivial commutator [72] between the C and E #ow, closing onto the K #ow, which   _ induces further shifts.

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

175

generated as a consequence of the integrability of the #ow, which we take as a guiding principle for reconstructing the invariant mass formula. The explicit form of the resulting #ow equations that follow from (5.17) is then given by [239] 1

()*m " m ()d* , ' ' 2

(5.18a)

1

)*+m '(" m )*+'( , 6

(5.18b)

1

,./m '()*+" m ,_./'()*+ , 2

(5.18c)

012m '_()*+,./"0 ,

(5.18d)

1

()*+,.m " m ()*+,d. , ' 5! '

(5.19a)

1

)*+,./m '(" m )_*+,./'( , 5!

(5.19b)

,./012m '()*+"0 ,

(5.19c)

012345m '_()*+,./"0 ,

(5.19d)

which now can be integrated, as will be shown in Section 5.6. 5.5. A digression on Iwasawa decomposition In order to understand the non-commutativity of the spectral #ow from another perspective, it is worthwhile coming back to a simpler example of a non-compact group, namely the prototypical G(1)"Sl(n, 1) group. The Iwasawa decomposition (4.7) then takes the form g"k ) a ) n3K ) A ) N ,

(5.20)

where K"SO(n, 1) is the maximal compact subgroup of G(1), A is the Abelian group of diagonal matrices with determinant 1 and N is the nilpotent group of upper triangular matrices. The factor k is absorbed in the coset G(1)/K, and the coset space is really parametrized by A ) N. Now the subgroup of G(9) leaving A invariant is nothing but the Weyl group S of permutations L of entries of A, whereas that leaving N invariant is the Borel group of integer-valued upper triangular matrices with 1's on the diagonal. The latter is graded by the distance away from the diagonal, in the sense that [B , B ]LB , (5.21) N NY N>NY where B is the subset of upper triangular matrices with 1's on the diagonal and other non-zero N entries on the pth diagonal only. In particular, B is a non-compact Abelian subgroup when N p'n/2.

176

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

Returning to the case at hand, we see that ,  (and _ in the d"8 case) are analogous to the B , B (and B ) Borel generators of Sl(3) (or Sl(4)). More precisely, they correspond to the grading    of the root lattice of E with respect to the simple root a extending the Sl(d, 9) Lorentz subgroup to B  the full E (9) subgroup, or in other words the grading of the adjoint representation in powers of BB l. This can be seen from Table 16 for d"8 since, in this case, the particle multiplet happens to be N in the adjoint representation 248 of E . For d(8, this can also be seen from the Coxeter label a of   a in Table 9, i.e the a component of the highest root of E : the degree p of all the positive roots   B then runs from 0 (corresponding to the g Borel generators) to a , with intermediate values 1 for '(  the C #ow, 2 for E and 3 for K .   _ We "nally note that, in the notation of Eq. (5.20), the mass formula we are seeking takes the form, M"m R(a ) n)R(a ) n)m ,

(5.22)

where m is the vector of integer charges transforming in the appropriate linear representation R of E (1). BB 5.6. Particle multiplet and U-duality invariant mass formula The #ow (5.18) can be integrated to obtain the E (9)-invariant mass formula for the particle  multiplet of M-theory compacti"ed on a torus ¹ with arbitrary shape and gauge background. The result is: 1 1 1 (m )# (m )# (m _) , M"(m )#  5! l 7! l 2! l N N N where the shifted charges depend on the gauge potentials as




 

(5.23)

1 1 1 m()*+, m "m # C m()# C C # E ()* +,' ' ' 2 ()' 4! 5! ()*+,'




#

1 1 C C C # C E m(_)*+,./0 , 3!4! ()* +,. /0' 2 ) 5! ()* +,./0'






(5.24a)

1 1 1 m)*+'(# C C # E m)_*+,./'( , m '("m'(# C 4! )*+ ,./ 5! )*+,./ 3! )*+

(5.24b)

1 m '()*+"m'()*+# C m,_./'()*+ , 2 ,./

(5.24c)

m '_()*+,./"m'_()*+,./ .

(5.24d)

The shifts induced by the expectation values of C and E give an explicit parametrization of the   upper triangular vielbein V in terms of the physical compacti"cation parameters (see Eq. (4.6)). The mass formula (5.23) is now invariant under T-duality, besides the manifest Sl(d, 9) symmetry.  V is actually upper triangular in blocks, because we did not decompose the metric g in a product of upper '( triangular vielbeins.

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

177

As an illustration, we can look at the shift in T-duality vector charge mQ implied by the above equation: m Q#A m "mQ#A m#(C #A B )mQ#(E #C B #A B B )mQ_Q      Q      #(A C )m#(E #C#A E #A B C )m_Q . (5.25)      Q    The second line precisely involves the tensor product of the charge spinor representation S with the spinor representation made up by the Ramond moduli. In fact, to see that the set (A ,C #A B , E #C B #A B B ) transforms as a spinor, one may simply note that it is     Q      precisely the combination that appears in the expansion in powers of F of the T-duality invariant D-brane coupling e >JQ R. Formula (5.23) reduces to the d"5 result of Ref. [89] for vanishing expectation values of the gauge backgrounds (see also [294]). 5.7. String multiplet and U-duality invariant tension formula Exactly the same analysis can be done for the momentum multiplet. We give here the result for d"6. The contributing charges n, n, n_ decompose into SO(6, 6) T-duality multiplets I"(nQ),

S"(n, nQ, nQ_Q),

<"(n, n_Q) ,

(5.26)

and we obtain the E

(9)-invariant tension formula for the d"6 string multiplet:  1 1 1 T" (n )# (n )# (n _) , l l l N N N where the shifted charges are





(5.27)

n 

" n#C n#(C C #E )n_ ,     n  " n#C n_ ,  n _ " n_ .

(5.28)

The combinatorial factors and explicit index contractions are easily reinstated in this equation by comparison with Eq. (5.24a). This yields the parametrization of the vielbein V of Eq. (4.6) in the representation appropriate to the string multiplet. 5.8. Application to R couplings As an illustration of the result (5.27), we display the d45 string multiplet invariant tension formula. Because of antisymmetry, only the charges n and n contribute, so that the tension of the string multiplet is given by









1 1 1 g n+# n+,./C T" n'# n'()*C ()* '+ ,./ 3! 3! l N 1 # n'()*g g g g n+,./ . '+ (, ). */ 4! l N

(5.29)

178

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

This is precisely the U-duality invariant quantity that was obtained in the study of instanton corrections to R corrections in type II theories in Ref. [195], where it was conjectured that the coupling for d"5 is given by the SO(5, 5, 9) Eisenstein series



2n<  dt K < K [T]\" e\LTR , (5.30) A" t ' '()* l l ' '()* N  N L L L L where T is given by the tension formula (5.29), and < denotes the volume of the M-theory torus ¹B. As will become clear in the next Subsection (see Eq. (5.38a)), the sum has to be restricted to integers such that n 'n()*+ "0, in order to pick up the contribution of half-BPS states only. The generalisation of this construction to compacti"cation of M-theory in lower dimensions was addressed in Ref. [238]. Under Poisson resummation on the charge nQ, the U-duality invariant function (5.30) exhibits a sum of instanton e!ects of order e\EQ, corresponding to the D0-branes (with charge n) and D2-branes (with charge nQ), but there is also a contribution of the extra charge n of order e\EQ . The NS5-brane does not yield any instanton on ¹, so these e!ects seem rather mysterious. On the other hand, we may interpret Eq. (5.30) as a sum of loops from all perturbative and nonperturbative strings. The occurrence of the NS5-brane of the string multiplet is then no longer surprising. 5.9. Half-BPS conditions and quarter-BPS states The U-duality mass formulae (5.23) and (5.27) that we have obtained only hold for 1/2-BPS states, and require particular conditions on the various integer charges. These conditions can be obtained from a precise analysis of the BPS eigenvalue equation, as in Section 2.2, or from a sequence of U-dualities from the perturbative level-matching condition ""m"""0 in Eqs. (3.8a) and (3.8b). In analogy to the latter condition, they should be quadratic in the integer charges, be moduli-independent, and constitute a representation of the U-duality group E (9), appearing in BB the symmetric tensor product of two charge multiplets. We have already noticed in Section 4.10 that the string multiplet always appears in the symmetric product of two particle multiplets, and indeed all the computations in Appendix A point to the fact that the 1/2-BPS condition on the particle multiplet is the string multiplet constructed out of the particle charges. This has also been observed in Ref. [113], where it was shown that for d"7 the 1/2-BPS conditions on the 56 particle multiplet were transforming in a 133 adjoint representation of E , which is the corresponding string multiplet.  In order to extract the precise conditions, it is convenient to consider the branching under the ST-duality group: E MSO(6, 6);Sl(2) ,  56"(12, 2)#(32, 1) , 133"(1, 3)#(32, 2)#(66, 1),

(5.31)

 The naive inclusion of the KK6-brane as an extra ! Z'()*+, term does not seem, however, to yield '()*+, a U-duality invariant mass formula by this method.

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

179

where the 32 correspond to the D-brane charges m , m, mQ, mQ_Q and the two 12 to the Q Kaluza}Klein and winding charges m , mQ and the NS5-brane/KK5-brane charges m, m_Q  respectively (see also Tables 22 and 21). The 133 in the symmetric tensor product 56  56 of two Q particle multiplets is therefore (1312  12,3)#(32312  32, 2)#(66332  32#1212,1) . Q Q

(5.32)

So as to work out the tensor products in Eq. (5.32), it is advisable to consider the further branching SO(6, 6)MSl(6);O(1, 1) , 12"6 #6 ,  \ 32"1 #15 #15 #1 ,  \ \  66"15 #1 #35 #15 . \   

(5.33)

The decomposition of the 133 conditions in terms of the various Sl(6)LSO(6, 6) charges is therefore 1 : kQ,m mQ   32 : 

66 : 

 

k,m m#m mQ  Q kQ,m mQ#mQm  kQ_Q,m mQ_Q#mQmQ 

(5.34b)

k,m mQ#mm#m m Q  k_Q,mmQ#m mQ_Q#mQm#m m_Q Q  kQ_Q,mmQ_Q#mQmQ#mQm_Q

(5.34c)

1 : kQ_,mQm#m mQ_Q  Q 32

1

: \

(5.34a)



k_,mm#m_Qm Q k_Q,mmQ#m_Qm

(5.34d)

(5.34e)

kQ_Q,mmQ_Q#m_QmQ

: k_Q,mm_Q \

(5.34f )

where the subindex denotes the SO(1, 1)LSl(2) charge, and the contractions are the obvious ones. In particular, for D-brane charges only, the condition 66 reduces to the one introduced in Section  3.8. The condition 1 is the familiar perturbative level-matching condition, whereas 1 is the  \ analogous condition on NS5}KK5 bound states. The other conditions mix di!erent T-duality multiplets. For example, the spinor constraints (5.34b) and (5.34e) are composed of products of D-brane charges with either KK- and winding charges or NS5- and KK5-brane charges.

180

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

As suggested by the index structure of the conditions k in Eqs. (5.34a)}(5.34f), the constraints combine in a string or momentum multiplet as k"m m ,  k"m m#mm ,  k_"m m_#mm ,  k_"mm_#mm , k_"mm_ .

(5.35a) (5.35b) (5.35c) (5.35d) (5.35e)

If these composite charges do not vanish, the state is at most 1/4-BPS, in which case its mass formula is given by (5.36) M"M(m)#([T(k)] ,  where M (m) and T(k) are given by the half-BPS mass and tension formulae (5.23) and (5.27).  Noting from Eq. (4.34) that the string multiplet T appears in the decompacti"cation of the  particle multiplet M, we can obtain the half-BPS condition on the string multiplet by allowing non-zero mQ, mQ, m_Q charges only, where s denotes a "xed direction on the torus: kQ_Q"mQmQ ,

(5.37a)

kQ_Q"mQm_Q#mQmQ ,

(5.37b)

kQ_Q"mQm_Q

(5.37c)

and identifying these charges with the n, n, n_ charges of the string multiplet in one dimension lower. We therefore obtain a multiplet of half-BPS conditions k"nn ,

(5.38a)

k_"nn_#nn ,

(5.38b)

k_"nn_ .

(5.38c)

This is easily seen to transform as a T multiplet, as can also be inferred from the decomposition  (4.38) at level 2 of the string multiplet under decompacti"cation. For d"6, this is a 27 quadratic condition on the 27 string multiplet of E , whereas for d"5 only the "rst condition remains, giving  a singlet condition on the 10 multiplet of SO(5, 5). For d(5, a BPS string state is automatically 1/2-BPS, while for d"7 the T condition transforms as a 1539 of E . The tension of a 1/4-BPS   string can also be obtained by decompactifying one direction in Eq. (5.36), and has an analogous structure T"T(n)#([T (k)] ,  

(5.39)

 One could have alternatively derived these conditions from the branching E MSl(7), but the one we used is more  constrained and convenient.

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

181

where T (n) and T (k) are given by the half-BPS tension (5.27) and the half-BPS 3-brane tension,   which can be worked out easily. For d56 (resp. d55), there still remain conditions to be imposed on the particle multiplet (resp. string multiplet) in order for the state to be 1/4-BPS and not 1/8. In the d"7 case, it should be required that the 56 in the third symmetric tensor power of the 56 particle charges vanishes [113]. For d"6, this reduces to the statement that the singlet in 27 should vanish. This condition is empty for d45. We shall however not investigate the 1/8-BPS case any further, and refer to Appendix A.4 for the 1/8-BPS mass formula of a NS5}KK-winding bound state in d"6 (D"5). In contrast to 1/2-BPS states, 1/4-BPS and 1/8-BPS states in general have a non-trivial degeneracy and therefore entropy, which still has to be a U-duality invariant quantity depending on the charges m [5,77,90,164,184]. This allows non-trivial checks on U-duality and predictions on BPS bound states, which we shall only mention here [278,311,312].

6. Matrix gauge theory The de"nition of M-theory as the strong-coupling limit of type IIA string theory and the "nite energy extension of the eleven-dimensional SUGRA does not allow the systematic computation of S-matrix elements, since type IIA theory is only de"ned through its perturbative expansion and 11D SUGRA is severely non-renormalizable. In Ref. [22], Banks, Fischler, Susskind and Shenker (BFSS) formulated a proposal for a non-perturbative de"nition of M-theory, in which M-theory in the in"nite momentum frame (IMF) with IMF momentum P"N/R, is related to the supersymmetric quantum mechanics of N;N Hermitian matrices in the large-N limit, the same as the one describing the interactions of N D0-branes induced by #uctuations of open strings. Despite the powerful constraints of supersymmetry, it is still a formidable problem to solve this quantum mechanics in the large-N limit. As was argued by Susskind [296], sense can however be made of the "nite-N Matrix gauge theory, as describing the Discrete Light-Cone Quantization (DLCQ) of M-theory, that is quantization on a light-like circle. This stronger conjecture has been further motivated in Ref. [274], relating through an in"nite Lorentz boost the compacti"cation of M-theory on a light-like circle to compacti"cation on a vanishing space-like circle, i.e. to type IIA string theory in the presence of D0-branes. This argument gives a general prescription for compacti"cation of M-theory (see also Sen's argument Ref. [279]), and we shall brie#y go through it in this section. Upon toroidal compacti"cation on ¹B, the extra degrees of freedom brought in by the wrapping modes of the open strings extend this quantum mechanics to a quantum "eld theory, namely a ;(N) Yang}Mills theory with 16 supersymmetries on the T-dual torus ¹I B in the large-N limit [124,301]. This prescription is consistent up to d43, but breaks down for compacti"cation on higher-dimensional tori, owing to the ill-de"nition of SYM theory at short distances. Several proposals have been made as to how to supplement the SYM theory with additional degrees of freedom while still avoiding the coupling to gravity, which will be brie#y discussed in this section. Besides their relevance for M-theory compacti"cation, these theories are also interesting theories in their own right, as non-trivial interacting theories in higher dimensions.  This model was "rst introduced in Ref. [66,320,321].

182

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

Our aim is to provide the background to discuss in Section 7 the implications of U-duality for the Matrix gauge theory describing toroidal compacti"cation of M-theory. The relation between the M-theory compacti"cation moduli, including gauge backgrounds, and Matrix gauge-theory parameters will be obtained, as well as the spectrum of excitations that Matrix gauge theory should exhibit in order to describe compacti"ed M-theory. This will leave open the issue of what is the correct Matrix gauge theory reproducing these features. 6.1. Discrete light-cone quantization The "nite-N conjecture of Ref. [296] is formulated in the framework of the DLCQ, the essentials of which we review "rst. In "eld theory, one generally uses equal-time (t"x) quantization, which breaks PoincareH invariance, but preserves invariance under the kinematical generators consisting of spatial rotations and translations. However, an alternative quantization procedure exists, in which the theory is quantized with respect to the proper time x>"(x#x)/(2, which is referred to as light-cone quantization. In this case, the transverse translations PG and rotations ¸GH, as well as the longitudinal momentum P> and the boosts ¸\G, ¸>\ do not depend on the dynamics, while the generator P\ generates the translations in the x> direction and plays the role of the Hamiltonian. The usual dispersion relation H"(PGP #M in equal-time quantization, is G replaced in the light-cone quantization by PGP #M G , P\" 2P>

(6.1)

exhibiting Galilean invariance on the transverse space. Particles, with positive energy P\'0, necessarily have positive longitudinal momentum P>, while antiparticles will have negative P>. The vacuum of P\ is hence reduced to the Fock-space state "02, and the negative-norm ghost states are decoupled as well. This simpli"cation of the theory is at the expense of instantaneous non-local interactions due to the P>"0 pole in Eq. (6.1). Discrete light-cone quantization proceeds by compactifying the longitudinal direction x\ on a circle of radius R : J x Kx #2nR . (6.2) \ \ J This results into a quantization of the longitudinal momentum of any particle i according to n (6.3) P>" G . G R J Because the total momentum is conserved, the Hilbert space decomposes into "nite-dimensional superselection sectors labelled by N" n . Note that the "nite dimension does not require G imposing any ultraviolet cut-o! on the eigenvalues n , but follows from the condition n '0. G G It is important to note that, because the x\ direction is a light-like direction, the length R of the J radius is not invariant, but can be modi"ed by a Lorentz boost ¸>\,



x

P

x




cosh b

!sinh b

x

!sinh b

cosh b

x

,

(6.4)

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

183

which amounts to R Pe@R , P\Pe@P\, P>Pe\@P> . (6.5) J J This implies that the Hamiltonian P> depends on the radius R through an over-all factor J P\"R H , (6.6) J , so that the mass M"2P>P\ is independent of R . J 6.2. Why is Matrix theory correct? Following Ref. [274], we will now derive the Hamiltonian H describing the DLCQ of , M-theory, and obtain the BFSS Matrix-theory conjecture. The basic idea is to consider the compacti"cation on the light-like circle as Lorentz-equivalent to a limit of a compacti"cation on a space-like circle. Acting with a boost (6.4) on an ordinary space-like circle, we "nd




cosh b




!sinh b

0









R !1#e\@ 1 !RJ " J P , (6.7) !sinh b cosh b R 1#e@ R (2 (2 Q J where R "R e\@. Sending bPR while keeping R "nite, we see that the light-like circle is Q J J Lorentz-equivalent to a space-like circle of radius R P0. Q In order to keep the energy "nite, which from Eq. (6.6) and on dimensional ground scales as R /l, J N we should also rescale the Planck length (and any other length) as l "e\@l . Altogether, NQ N M-theory with Planck length l on the light-like circle of radius R in the momentum P>"N/R N J J sector is equivalent to M-theory with Planck length l on the space-like circle of radius R in the NQ Q momentum P"N/R sector, with Q R "R e\@, l "e\@l (6.8) Q J NQ N in the limit bPR. Eliminating b, we obtain the following scaling limit: R R R P0, M" Q " J""xed . (6.9) Q l l NQ N Following Ref. [274], we shall denote the latter theory as M I theory. Since the space-like circle R shrinks to zero in l units, this relates the DLCQ of M-theory to Q NQ weakly coupled type IIA string theory in the presence of N D0-branes carrying the momentum along the vanishing compact dimension. Using Eq. (2.1), the scaling limit becomes R g "(R M), a"l" Q , R P0, Q Q Q Q M

M""xed .

(6.10)

In particular, g and a go to zero, so that the bulk degrees of freedom decouple, and only the Q leading-order Yang}Mills interactions between D0-branes remain. This validates the BFSS conjecture, up to the possible ambiguities in the light-like limit bPR [50,156]. Several di$culties have also been shown to arise for compacti"cation on curved manifolds [100,101], but since we are only concerned with toroidal compacti"cations, we will ignore these issues.

184

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

6.3. Compactixcation and Matrix gauge theory For toroidal compacti"cations of M-theory, we consider the same scaling limit as in Eq. (6.9), and keep the torus size constant in Planck length units, that is




R  R Q , r " ' ""xed . (6.11) R "r ' l ' ' M NQ However, comparing Eqs. (6.9) and (6.11), we "nd that the size of the torus goes to zero in the scaling limit. To avoid this it is convenient to consider the theory on the T-dual torus ¹I B, obtained by a maximal T-duality in all d directions. From Eq. (2.5), this has the e!ect that,



IIA with N D0-branesP

IIA with N Dd-branes

d"even ,

IIB with N Dd-branes

d"odd .

(6.12)

Using the maximal T-duality transformation “B ¹ , with ¹ given in Eq. (2.5), the type II ' ' ' parameters then become R 1 (R M)\B , a"l" Q , RI " , (6.13a) g" Q Q M ' rM Q “r ' ' R R R P0, M" Q ""xed, r " ' ""xed , (6.13b) Q ' l l NQ NQ so that, in particular, the size of the dual torus is "xed in the scaling limit. We will sometimes refer to the type II theory in this T-dual picture as the II I -theory. The behaviour of the string coupling in the scaling limit is now di!erent according to the dimension of the torus:



0,

d(3 ,

g P Finite, Q R,

d"3 ,

(6.14)

d'3 .

In particular for d(3 we still have weakly coupled type IIA or IIB string theory in the presence of N Dd-branes, so that M-theory is described by the SYM theory with 16 supercharges living on the world-volume of the N Dd-branes. The gauge coupling constant of this Matrix gauge theory and the radii s of the torus on which the D-branes are wrapped read ' 1 M\B , < ,“ r , s "RI " (6.15) g "g lB\" P ' ' ' 7+ QQ rM < ' P ' showing, in particular, that g is "nite in the scaling limit. 7+ The special case of Matrix theory on a circle (d"1) yields (after an S-duality transforming the background D1-strings into fundamental strings) Matrix string theory [91,92,229], in which an identi"cation between the large-N limit of two-dimensional N"8 supersymmetric YM theory and type IIA string theory is established. We will not further discuss this topic here, and refer to

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

185

Refs. [114,287] for the next case d"2 and its relation to type IIB string theory. Moving on to the case d"3, the same conclusion as in the d(3 case continues to hold, since although the string coupling is "nite, the string length goes to zero so that loop corrections are suppressed in the aP0 limit. Consequently, the d"3 Matrix gauge theory is N"4 supersymmetric Yang}Mills theory. For d'3, however, the coupling g blows up, and the weakly coupled string description of the Q D-branes is no longer valid. This coincides with the fact that the Yang}Mills theory becomes non-renormalizable and strongly coupled in the UV. Hence, in order to de"ne a consistent quantum theory, one needs to supplement the theory with additional degrees of freedom. In the following we brie#y review the proposals for d"4 and d"5, and show the complication that arises for d"6. These proposals follow from the above prescription, using further duality symmetries, which will be examined in more detail in Section 7. Other decoupling limits have been considered in [168]. 6.4. Matrix gauge theory on ¹ In the case d"4, it follows from (6.12) that the e!ective theory is 4#1 SYM coming from the type IIA D4-brane world-volume theory. In the scaling limit the type IIA theory becomes strongly coupled and using the correspondence between strongly coupled IIA theory and M-theory a new eleventh dimension is generated, which plays the role of a "fth space dimension in the gauge theory [43,45,260]. Using Eqs. (2.11) and (6.13a), the radius and 11D Planck length are 1 , lI "gl "RM\<\ . RI "g l " N Q Q Q P Q Q M< P

(6.16)

Moreover, comparing with Eq. (6.15) we "nd that the radius RI is in fact equal to the YM coupling constant RI "g . 7+

(6.17)

Hence, in the scaling limit (6.9), the Planck length lI goes to zero so that the bulk degrees of N freedom decouple, while the radius RI remains "nite. The N type IIA D4-branes become N M5branes wrapped around the extra radius RI , and M-theory on ¹;S is then described by the (2,0) world-volume theory of N M5-branes, wrapped on ¹ and the extra radius RI , related to the Yang}Mills coupling constant by Eq. (6.17). The proper formulation of this theory is still unclear, but Matrix light-cone descriptions have been proposed in Refs. [2,15,54,125,183,209,275] and the low-energy formulation studied in Refs. [122,145]. In particular, at energies of order 1/g the 7+ Kaluza}Klein states along the extra circle come into play. They can be identi"ed as instantons of 4D SYM lifted as particles in the (4#1)-dimensional gauge theory. Additional evidence for this conjecture that follows from the U-duality symmetry will be discussed in Section 7. 6.5. Matrix gauge theory on ¹ In the case d"5, we have N type IIB D5-branes at strong string coupling, so that it is useful to perform an S-duality that maps the D5-branes to NS5-branes. Using Eqs. (4.17) and (6.13a), we "nd

186

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

that the string coupling and length become 1 1 lK "g l" , RK "RI . (6.18) g( " "(R M)< , Q P Q Q Q M< ' ' Q g P Q Moreover, comparing with Eq. (6.15), we "nd that the string tension is related to the gauge coupling constant by lK "g . (6.19) Q 7+ The string coupling g( goes to zero in the scaling limit, so that the bulk modes are decoupled Q from those localized on the NS5-branes. However, the string theory on the NS5-branes is still non-trivial, and has a "nite string tension in the scaling limit [45]. As a consequence, we "nd that M-theory on ¹;S is described by a theory of non-critical strings propagating on the NS5-brane world-volume with a tension related to the gauge coupling by Eq. (6.19). The proper formulation of this theory is still unclear, but light-cone Matrix formulations have been proposed [275,286]. The string can be identi"ed with a 4D Yang}Mills instanton lifted to 1#5 dimensions. This description is close but not identical to the proposal in Refs. [89,90] according to which the (1/4-) BPS sector of M-theory should be described by the (1/2-) BPS excitations of the M5-brane, whose dynamics would be described by a (ground-state) non-critical micro-string theory on its six-dimensional world-volume. In particular, the theory on the type IIB NS5-brane is non-chiral, whereas that on the M5-brane is chiral. We refer the reader to the work of [92] for a discussion of these two approaches. 6.6. Matrix gauge theory on ¹ Finally, we discuss the problems that arise for d"6, in which case we have N type IIA D6-branes at strong coupling. As in the d"4 case, an eleventh dimension opens up, and we "nd M-theory compacti"ed on a circle of radius RI with 1 1 , lI "gl " . (6.20) RI "g l " N Q Q M< Q Q RM< P Q P The N D6-branes actually correspond to N coinciding Kaluza}Klein monopoles with Taub}NUT direction along the eleventh direction, and as RI PR, the monopoles shrink to zero size and reduce to an A singularity in the eleven-dimensional metric. It was suggested in Ref. [152] that the , bulk dynamics still decouples from the (6#1)-dimensional world-volume, and that the latter can be described in the IMF by the m(atrix) quantum mechanics of N D0-branes inside N ten dimensional Kaluza}Klein monopole, in the large-N limit. This is very reminiscent to the BFSS  description of M-theory, but the quantum mechanics is now a matrix model with eight supersymmetries and corresponds to the Coulomb phase of the quiver gauge theory in 0#1 dimensions associated to the Dynkin diagram A [99]. In other words, this is a sigma model with vector , multiplets in the adjoint representation of [;(N )]B, and hypermultiplets in bifundamental  representations (N , NM ) of ;(N ) ;;(N ) for k"12N, with ;(N ) denoting the k-th copy   I  I> I of ;(N ) and ;(N ) identi"ed with ;(N ) ; this model is restricted to its Coulomb phase, where   ,

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

187

the hypers have no expectation value. In the low-energy limit, it is expected to reduce to SYM in 1#6 dimensions, with gauge coupling lI "g . (6.21) N 7+ Other approaches have been proposed in Refs. [55,123]. We shall come back to the d"6 case in Section 7 when we display the BPS states in terms of the low-energy SYM theory. 6.7. Dictionary between M-theory and Matrix gauge theory We "nally give the dictionary that allows us to go from M-theory on ¹B;S (with S a light-like circle) in the sector P>"N/R and Matrix gauge theory on ¹I B. This can be obtained by solving J Eqs. (6.13a) and (6.15), for the parameters (s , N, g ) of the ;(N) Matrix gauge theory in terms of ' 7+ the parameters (R , R , l ) of M-theory compacti"cation on ¹B;S: ' J N l (6.22a) s" N , ' RR J ' lB\ g " N . (6.22b) 7+ RB\“R J ' For completeness we also give the inverse relations







1 R <  R<  N J Q R" , l" J Q , P>" , (6.23) ' s g N g R ' 7+ 7+ J where we have de"ned < "“ s as the volume of the dual torus on which the Matrix gauge Q '' theory lives. 6.8. Comparison of M-theory and Matrix gauge theory SUSY In order to describe the M-theory BPS states from the point of view of the gauge theory, we need to understand how the space-time supersymmetry translates to the brane world-volume. This is in complete analogy with the perturbative string in the Ramond}Neveu}Schwarz formalism, in which space-time supersymmetry emerges from world-sheet supersymmetry (see [141], Section 5.2), and the case of the M5-brane has been thoroughly discussed in Ref. [89]. We will abstract their argument and discuss the case of a general 1/2-BPS brane, whether D, M, KK or otherwise, referring to that work for computational details. In the presence of a p-brane, the breaking of the 11D N"1 space-time supersymmetry is only spontaneous. The unbroken SUSY charges generate a superalgebra on the world-volume of the brane, whereas the broken ones generate fermionic zero modes. The "xing of the reparametrization invariance on the world-volume is most easily done in the light-cone gauge. The 32-component supercharge Q then decomposes as a (spinor, spinor, spinor) of the unbroken Lorentz group ?  or a sum of, depending on the parity of p.

188

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

SO(1,1);SO(p!1);SO(10!p). The algebra is graded by the eigenvalue $1/2 of the generator of SO(1,1), so that the unbroken generators Q> have charge #1/2 and the broken ones Q\ charge ?? ?? !1/2, where a is the spinorial index of the SO(10!p) R-symmetry and a the spinorial index of the SO(p!1) Lorentz world-volume symmetry. The anticommutation relations then take the form +Q>, Q>,"H#P#Z>> ,

(6.24a)

+Q>, Q\,"p#Z ,

(6.24b)

+Q\, Q\,"Z\ \ .

(6.24c)

In this expression, H and P are the world-volume Hamiltonian and momentum, Z>>, Z, Z\ \ some possible central charges and p is the transverse momentum. A contraction of the central charges with the appropriate Gamma matrices is also assumed. In the following, we absorb the momentum in the charges Z>>, and set p"0 by considering a particle at rest in the transverse directions. The central charges Z and Z!! are simply a renaming of the Z+,, Z+,./0 central charges of the 11D superalgebra (2.13a). As their indices show, Z is a singlet of SO(1, 1), whereas Z>> and Z\ \ combine in a vector of SO(1, 1); Z is therefore identi"ed with the Z'(, Z'()*+ charges, whereas Z!! correspond to the Z', Z'()*+ charges, where as usual I, J,2, are directions on the torus and 1 is the space direction combined with the time direction on the light cone. In other words, Z is identi"ed with the particle charges, whereas Z!! correspond to the string charges. In order for the superalgebra (6.24a) to reproduce the space-time superalgebra (2.13a) with particle charges only, we therefore need to impose Z!!"0 on the physical states. This is the analogue of the ¸ "¸M level-matching condition.   The broken generators Q\ and the central charges Z are given by the fermionic and bosonic zero modes only. On the other hand, the unbroken generators as well as the central charges Z!! have a non-zero-mode contribution: Q>"Q>#QK >, Z!!"Z!!#ZK !!, H"H #HK .   

(6.25)

The zero-mode part of the generators Q> is built out of the bosonic and fermionic zero modes  Z and Q\, and anticommutes with the oscillator part QK >. It generates the same algebra as in Eq. (6.24a), while the oscillator parts generate the same algebra on their own and anticommute with the zero-mode broken generators Q\"Q\. The level-matching conditions Z!!"0 are achieved  through a cancellation of the zero-mode part, quadratic in the particle charges Z, and the oscillator parts. Let us now consider the Hamiltonian H. Because of supersymmetry, both H and HK are positive  operators and for given zero modes Z, the supersymmetric ground state is given by the condition HK "0, or QK >"02"0. This state is therefore annihilated by all the QK > supersymmetries, i.e. half the space-time supersymmetries, and must have vanishing ZK !! charge, i.e. from the level-matching conditions Z!!"(Z)"0. This condition is, in less detail, the 1/2-BPS condition k"0 with  k de"ned as in Eq. (2.21b). The energy of this state is given by the zero-mode part H "+Q>, Q>,    quadratic in the particle charges Z. This is equivalent to the mass formula Eq. (2.18) for 1/2-BPS states in space-time.

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

189

On the other hand, BPS states preserving 1/4 of the space-time supersymmetry are only annihilated by half the world-volume supercharges QK >, and their energy is shifted by the non-zeromode contribution HK . The latter is quadratic in the non-zero-mode part of the string charges ZK !!"!Z!!"!(Z), therefore quartic in the particle charge. This is precisely what was  found in Eq. (2.21a): E"M(Z)#([T(K)] .

(6.26)

This equation has a simple interpretation: the quadratic term corresponds to the 1/2-BPS bound state between the heavy mass M p-brane and the mass M particle, with binding energy N M , (6.27) E"(M#M!M K N M N N whereas the second corresponds to a 1/4-BPS bound state between the p-brane and the mass M"RT of the string with tension T wrapped on the circle R: E"(M #M)!M "RT . (6.28) N N There is therefore a complete identity between (i) the space-time supersymmetry algebra and particle spectrum in the absence of the p-brane, (ii) the p-brane world-volume gauge theory and (iii) the bound states of the p-brane with other particles. This also holds at the level of space-time "eld con"gurations, which can be seen as con"gurations on the world-volume [37,58]. 6.9. SYM masses from M-theory masses We shall now explicit the correspondence of the previous subsection in the D-brane case, relevant for Matrix gauge theory, and relate the energies in the Yang}Mills theory to the masses in space-time. This has been discussed in particular in Refs. [114,124,133,295]. Based on the last interpretation as bound states of the N background D-branes with other particles, we identify R"R and M "P "N/R , where R is the radius of the light-like direction, and "x the J N > J J normalization of the Yang}Mills energies as R E " JM(Z)#R ([T(K)] . J 7+ N

(6.29)

We then proceed by using the dictionary (6.22a) and (6.22b) to obtain the Yang}Mills energy of the BPS states we discussed previously. We now apply these considerations to the highest-weight states of the two U-duality multiplets of Sections 4.8 and 4.10. The highest-weight state of the particle multiplet is a Kaluza}Klein excitation on the Ith direction, which becomes, after the maximal T-duality, an NS-winding state bound to the background Dd-brane. Hence, it is a bound state with non-zero binding energy, and using Eq. (6.27) we "nd (1/R ) g s ' " 7+ ' , E " 7+ N< N/R Q J

(6.30)

190

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

where in the second step we used the dictionary (6.22a) and (6.22b) to translate to Matrix gauge theory variables. This is the energy of a state in the gauge theory carrying electric #ux in the Ith direction. For this reason, the particle multiplet is also called the yux multiplet. Next we turn to the highest-weight state of the string multiplet, wrapped on the light-like direction R . The highest weight is a membrane wrapped on R and R , which becomes, after the J ' J maximal T-duality, a Kaluza}Klein state bound to the background Dd-brane. These two states form a bound state at threshold and according to (6.28) we have 1 RR (6.31) E " J '" , 7+ s l ' N where Eqs. (6.22a) and (6.22b) was used again in the second step. This is the energy of a massless particle with momentum along the Ith direction in the gauge theory, so that we may alternatively call the string multiplet the momentum multiplet from the point of view of Matrix gauge theory. This translation can be carried out for all other members of the U-duality multiplets, and since U-duality preserves the supersymmetry properties of the bound state, one "nds the following general relation between SYM masses and M-theory masses: particle/#ux multiplet: E

R " JM , 7+ N

(6.32a)

string/momentum multiplet: E

"R T . (6.32b) 7+ J  In Section 7.2, we will explicitly see for the cases d"3, 4, 5 that indeed all non-zero binding energy and threshold bound states appear in the particle/#ux and string/momentum multiplets, respectively. Finally, we remark that the equalities in the two equations (6.30) and (6.31) can be solved to yield the dictionary (6.22a) and (6.22b), so that the comparison of these two types of energy quanta gives a convenient shortcut.

7. U-duality symmetry of Matrix gauge theory If any of the previously discussed Matrix gauge theories purports to describe compacti"ed Matrix gauge theory, it should certainly exhibit U-duality invariance. In this section, we wish to investigate the implications of U-duality on the Matrix gauge theory at the algebraic level, irrespective of its precise realization. To this end we use the dictionary (6.22a) and (6.22b) between compacti"ed M-theory and Matrix gauge theory. We "rst recast the Weyl transformations of the U-duality group (see Section 4.4) in the gauge-theory language and interpret them as generalized electric}magnetic dualities of the gauge theory. Then, we translate the U-duality multiplets of Sections 4.8 and 4.10 in matrix gauge theory and discuss the interpretation of the states. Finally, we use the results of Section 5.4 to discuss the realization of the full U-duality group in Matrix gauge theory and in particular the couplings induced by non-vanishing gauge potentials. At the end of this section a more speculative aspect of "nite-N matrix gauge theory is discussed. By promoting the rank N to an ordinary charge, we show the existence of an E (9) action on B>B>

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

191

the spectrum of BPS states. In this way, we "nd that the conjectured extended U-duality symmetry of matrix theory on ¹B in DLCQ has an implementation as action of E (9) on the BPS B>B> spectrum, as demanded by eleven-dimensional Lorentz invariance. 7.1. Weyl transformations in Matrix gauge theory The discussion of Matrix gauge theory from M-theory in Section 6 has been restricted to rectangular tori with vanishing gauge potentials, so that we "rst focus on the transformations in the Weyl subgroup of the U-duality group W(E (9))"9 ( )S . (7.1) BB  B The permutation group S that interchanges the radii R of the M-theory torus obviously still B ' permutes the radii s of the Matrix gauge theory T-dual torus. On the other hand, the generalized ' in Eq. (4.12), using the dictionary (6.22a) and (6.22b), translates into the following T-duality ¹ '() transformation of the Matrix gauge theory parameters:



gB\ g P 7+ , 7+ =B\

S : '(E )

s Ps , ? ? g s P 7+s , ? = ?

=,“ s , ?$' ( ) ? a"I, J, K ,

(7.2)

aOI, J, K .

For d"3 the transformation (7.2) is precisely the (Weyl subgroup of ) S-duality symmetry of N"4 SYM in 3#1 dimensions [124,295]: g P1/g , (7.3) 7+ 7+ obtained for zero theta angle. The transformation (7.2) generalizes this symmetry to the case d'3, by acting as S-duality in the (3#1)-dimensional theory obtained by reducing the Matrix gauge theory in d#1 dimensions to the directions I, J, K and the time only [108]. Indeed, the coupling constant for the e!ective (3#1)-dimensional gauge theory reads = 1 " , g g 7+  and the transformation (7.2) becomes

(7.4)

(g , s , s )P(1/g , s , g s ) . (7.5)  ? ?  ?  ? To summarize, we see that from the point of view of the Matrix gauge theory the U-dualities are accounted for by the modular group of the torus on which the gauge theory lives (yielding the  We restrict to the case d53; the case d"1 has trivial Weyl group, while for the case d"2 there is only the permutation symmetry S . 

192

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

Sl(d, 9) subgroup) as well as by generalized electric}magnetic dualities (implementing the Tdualities of type IIA string). We now discuss in more detail the d"4, 5, 6 cases, in order to give more support to the proposals discussed in Section 6. Explicitly, one obtains d"4: S

d"5: S

'()

'()





g  s , 7+ ? s Ps , ? ?

aOI, J, K , a"I, J, K ,

g Pg , 7+ 7+ g a, bOI, J, K , s P 7+, ? s @ s Ps , a"I, J, K , ? ?

(7.6a)

(7.6b)



g g P 7+ , a, b, cOI, J, K , 7+ s s s ?@A d"6: S '() g (7.6c) s P 7+ , ? ss @A s Ps , a"I, J, K . ? ? For d"4 we see that Eq. (7.6a) induces a permutation of the YM coupling constant with the radii, in accordance with the interpretation (6.17) of the YM coupling constant as an extra radius. For d"5, Eq. (7.6b) takes the form of a T-duality symmetry (2.41) of the non-critical string theory living on the type IIB NS5-brane world-volume with the YM coupling related to the string length as in Eq. (6.19). Finally, for d"6, we see by comparing Eq. (7.6c) with the U-duality transformation in Eq. (4.12) that we recover the symmetry transformation ¹ in M-theory with the YM coupling '() constant related to the Planck length by Eq. (6.21). At this point, it is also instructive to recall the full U-duality groups for toroidal compacti"cations of M-theory, as summarized in Table 8, and discuss their interpretation in view of the Matrix gauge theories for d"3, 4, 5 (see Table 23). For d"3, the Sl(3, 9);Sl(2, 9) U-duality symmetry is the product of the (full) S-duality and the reparametrization group of the three-torus. For d"4, the Sl(5, 9) symmetry is the modular group of the "ve-torus, corroborating the interpretation of this case as the (2, 0) theory on the M5-brane [260]. Finally, for d"5 the SO(5, 5, 9) symmetry should be interpreted as the T-duality symmetry of the string theory living on the NS5-brane [45,89]. The E (9) symmetry is by no means obvious in the IMF description discussed in Section 6.6, but this  is expected since part of it are Lorentz transformations broken by the IMF quantization. The interpretation of the exceptional groups E (9), d"7, 8 is not obvious either, since a consistent BB quantum description for these cases is lacking as well. In Sections 7.4}7.6, the precise identi"cation of the full U-duality groups for d"3, 4, 5 will be discussed in further detail. Note also that as we are considering M-theory compacti"ed on a torus  From the point of view of type IIB theory, it can be shown that the latter also account for the restoration of the transverse Lorentz invariance [287].

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

193

Table 23 Interpretation of U-duality in matrix gauge theory D

d

U-duality

Origin

8 7 6 D45

3 4 5 d56

Sl(3, 9);Sl(2, 9) Sl(5, 9) SO(5, 5, 9) E (9) BB

S-duality;symmetry of ¹ Symmetry of ¹ of M5-brane T-duality symmetry on NS5-brane Unclear

times a light-like circle, it has been conjectured that the E (9) U-duality symmetry should be BB extended to E (9), as a consequence of Lorentz invariance. This extended U-duality B>B> symmetry will be discussed in Section 7.7. Finally, we can translate the U-duality invariant Newton constant (4.14) in the Matrix gauge theory language. The most convenient form is obtained by writing




< 
(7.7)

which depends on the invariant D-dimensional Planck length and the radius of the light-like circle, invariant under the E (9) transformations acting on the transverse space. Again, in agreement BB with the Matrix gauge theory descriptions, we see that for d"3 the invariant I "1/< is related  Q to the volume < of the three-torus; for d"4 the invariant I "1/< g is related to the total Q  Q 7+ volume of the "ve-torus, constructed from the four-torus with and the extra radius RI "g ; for 7+ d"5 the invariant I "1/g is related to the "nite string tension ¹"1/g of the string theory.  7+ 7+ Finally, note also that for d"6 the U-duality invariant I "< /g is related to the 5-D Planck  Q 7+ length, when using l"g . N 7+ 7.2. U-duality multiplets of Matrix gauge theory We now turn to translating the U-duality multiplets of Sections 4.8 and 4.10 in the Matrix gauge theory picture. To this end we use the dictionary (6.22a), (6.22b) and the mass relations in Eqs. (6.32a) and (6.32b). Equivalently, one may start with the highest-weight states corresponding to electric #ux (6.30) and momentum states (6.31) in the Matrix gauge theory and subsequently act with the transformations (7.2) of the Weyl subgroup. Of course these two methods lead to the same result, which are summarized in Tables 24 and 25, for the particle/#ux and string/momentum multiplet respectively. As a compromise between explicitness and complexity, we have chosen to write down the content for d"7 in the "rst case, and for d"6 in the latter case. The tables list the mass M in M-theory variables, the corresponding gauge theory energy E and their associated 7+ charges, obtained from the M-theory charges by raising lower indices or lowering upper indices. In Table 24, the "rst entry corresponds to a state with electric #ux in the Ith direction, while the second one carries magnetic #ux in the I, J direction. The "rst entry in Table 25, is a KK state of the gauge theory, while the second one is a YM instanton in 3#1 dimensions, lifted to d#1 dimensions. For d55, new states appear. As a further illustration, we take a closer look at the

194

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

Table 24 Flux multiplet (56 of E ) for Matrix gauge theory on ¹  M

E 7+

Charge

1 R ' RR ' ( l N RRR R R ' ( ) * + l N R ;R R R R R R R ' ( ) * + , . / l N

g s 7+ ' N< Q < Q Ng (s s ) 7+ ' ( < Q Ng (s s s s s ) 7+ ' ( ) * + < Q Ng (s ; s s s s s s s ) 7+ ' ( ) * + , . /

m m  m  m _

Table 25 Momentum multiplet (133 of E ) for Matrix gauge theory on ¹  M

E 7+

Charge

RR J ' l N RR R R R J ' ( ) * l N R R ;R R R R R R J ' ( ) * + , . l N R R R R ;R R R R R R R J ' ( ) * + , . / 0 1 l N R R R R R R R ;R R R R R R R J ' ( ) * + , . / 0 1 2 3 4 l N

1 s '

n 

< Q g s s s s 7+ ' ( ) * < Q g s ;s s s s s s 7+ ' ( ) * + , . < Q g s s s ;s s s s s s s 7+ ' ( ) * + , . / 0 1 < Q g s s s s s s ; s s s s s s s 7+ ' ( ) * + , . / 0 1 2 3 4

n  n _ n _ n _

special cases d"3, 4, 5, 6, which can be obtained from the tables by omitting those states that have too many compacti"ed dimensions. The Tables 26}29 list the content of each of the two multiplets for these cases [108,239,265], including the M-theory mass, the YM energy, the multiplicity of each type of state and its interpretation both in the Matrix gauge theory and as a bound state with the I Dd-branes. For d"4, 5 we have also added a column giving the bound-state N background type II interpretation in the M5- and NS5-brane theories, respectively. A few comments on these tables are in order. E A number of states in the Matrix gauge theory have a uniformly valid interpretation as bound states with the background Dd-branes, namely, for the #ux multiplet, electric #ux"Dd}NS-winding bound state ,

(7.8a)

magnetic #ux"Dd}D(d!2) bound state

(7.8b)

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

195

Table 26 Flux and momentum multiplet for d"3: (3,2) and (3,1) of Sl(3);Sl(2) M

E 7+

C

YM state

b.s. of N D3

1 R '

g s 7+ ' N< Q

3

electric #ux

NS-w

RR ' ( l N

< Q Ng (s s ) 7+ ' (

3

magnetic #ux

D1

RR J ' l N

1 s '

3

momentum

KK

Table 27 Flux and momentum multiplet for d"4: 10 and 5 of Sl(5) M

E 7+

C

YM state

b.s. of N D4

b.s. of N M5

1 R '

g s 7+ ' N< Q

4

electric #ux

NS-w

M2

RR ' ( l N

< Q Ng (s s ) 7+ ' (

6

magnetic #ux

D2

RR J ' l N

1 s '

4

momentum

KK

R< J 0 l N

1 g 7+

1

YM particle

D0

KK

and for the momentum multiplet, KK momentum"Dd}KK bound state ,

(7.9a)

YM state"Dd}D(d!4) bound state ,

(7.9b)

where the YM state denote the 4D Yang}Mills instanton lifted to d#1 dimensions. The correspondences in Eq. (7.8a), (7.8b), (7.9a) and (7.9b) were noted in Refs. [95,245,315]. E In the d"3 case, only perturbative states are observed in Table 26. E For d"4 one non-perturbative state occurs in Table 27, which corresponds precisely to momentum along the dynamically generated "fth direction, i.e. to a Yang}Mills instanton lifted to 4#1 dimensions [260]. From the M5-brane point of view, the #ux multiplet describes the M2-brane excitations, while the momentum multiplet comprises the KK states, as indicated in the last column.

196

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

Table 28 Flux and momentum multiplet for d"5: 16 and 10 of SO(5,5) M

E 7+

C

YM state

b.s. of N D5

b.s. of N NS5

1 R '

g Ms 7 ' N< Q

5

electric #ux

NS-w

D1

RR ' ( l N

< Q Ng (s s ) 7+ ' (

10

magnetic #ux

D3

D3

< 0 l N

< Q Ng 7+

1

new sector

NS5

D5

RR J ' l N

1 s '

5

momentum

KK

KK

R< J 0 R l 'N

s ' g 7+

5

YM string

D1

NS-w

Table 29 Flux and momentum multiplet for d"6: 27 and 27 of E  M

D 7+

C

YM state

b.s. of N D6

1 R '

g s 7+ ' N< Q

6

electric #ux

NS-w

RR ' ( l N

< Q Ng (s s ) 7+ ' (

15

magnetic #ux

D4

< 0 R l 'N

< s Q' Ng 7+

6

new sector

KK5

RR J ' l N

1 s '

6

momentum

KK

R< J 0 R R l ' (N

ss '( g 7+

15

YM membrane

D2

RR < J ' 0 l N

< Q g s 7+ '

6

new sector

NS5

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

197

E For the case d"5 in Table 28, we focus on the last column obtained by S-duality from the D5-brane picture of the II I theory. The YM string in the momentum multiplet arises in this case from the wound strings on the NS5-brane. The wrapped transverse "vebrane on ¹ appears as a bound state of D5-branes with the background NS5-branes, with non-zero binding energy (since it is related by electric}magnetic duality to the D1-NS1 bound state). It corresponds to a new sector in the Matrix gauge theory Hilbert space, with energy scaling as 1/g . This state 7+ does not correspond to any known con"guration of the 1#5 gauge theory, but may be understood as a magnetic #ux along one ordinary dimension together with the dynamically generated dimension in a 1#4 gauge theory obtained by reducing the original one on a circle [151]. For the d"6 case, we see from Table 29 that all BPS states of type IIA theory on ¹ are involved in the bound states of the #ux and momentum multiplet, except for the D6-D0 `bound statea. It has been argued that the latter forms a non-supersymmetric resonance with the unconventional mass relation [86,300]: M"(M#M) . " "

(7.10)

As a consequence we expect to "nd a state in the gauge theory with energy E "(M #M)!M KMM . 7+ ," " ," " ,"

(7.11)

Using the corresponding D-brane masses and the relation g "g l we then obtain 7+ QQ < E "N Q "NI .  7+ g 7+

(7.12)

In the last step we have expressed the mass in terms of the U-duality invariant (7.7), explicitly showing that this extra state transforms as a singlet under the U-duality group E (9). Since the  D0-brane is mapped onto a D6-brane under the maximal T-duality, the space-time interpretation of this extra U-duality multiplet follows from the M-theory origin of the D6-brane, i.e. the state is KK6-brane with the TN direction along the light-like direction. The corresponding data of this extra singlet are summarized in Table 30. The d"7 case is discussed in Appendix C, and exhibits a number of similar states (with M-theory masses depending on multiple factors of R ) as the extra J singlet in d"6. 7.3. Gauge backgrounds in Matrix gauge theory Our discussion of the Matrix gauge theory U-duality symmetries and mass formulae has so far been restricted to the rectangular-torus case, with zero expectation values for the M-theory gauge potentials. However, gauge backgrounds in M-theory yield moduli, and should have a counterpart as couplings in the Matrix gauge theory. As a simple example, consider "rst M-theory on ¹, in which case we can switch on an expectation value for the component C of the three-form. 

198

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

Table 30 Additional multiplet for d"6: 1 of E  M

E 7+

C

YM state

b.s. of N D6

R< J 0 l N

N< Q g 7+

1

new sector

D0

Together with the volume < of ¹, it forms a complex scalar q"C

< #i ,  l N

(7.13)

which transforms as a modular parameter under the subgroup Sl(2, 9) of the U-duality group Sl(3, 9);Sl(2, 9) [143]. On the other hand, according to Eq. (6.22a) the volume is identi"ed in the Matrix gauge theory with 1/g , which together with the theta angle forms a complex scalar 7+ 4n h , S" #i g 2n 7+

(7.14)

transforming as a modular parameter under the electric}magnetic duality group Sl(2, 9). One should therefore identify C with h, or in other words the three-form background induces  a topological coupling FF on the D3-brane world-volume. This can be derived more generally for any d by making use of Seiberg's argument and the well-known coupling of Ramond gauge "elds to the D-brane world-volume. Details of the derivation can be found in Ref. [239] and we only quote the result which is that the expectation value of the three-form induces the following topological coupling in Matrix gauge theory:



SC"C'() dt

F F . ' ()

(7.15)

I B

2

This coupling reduces to the h term (7.14) for d"3 and was conjectured in Ref. [33]. As we now show, the coupling (7.15) can also be inferred from the U-duality invariant mass formulae. To see this, we "rst translate the general U-duality invariant mass formulae (5.23) into the gauge theory language using Eqs. (6.22a), (6.22b) and (6.32a) (6.32b), restricting to d46 for simplicity:

 











g <  <  Q (m )# Q (m ) E " 7+ (m )# 7+ N<   g g Q 7+ 7+ <  <  Q (n )# Q (n ) # (n )#   _ g g 7+ 7+

(7.16)

in which we have added the #ux multiplet and momentum multiplet together, as was argued in Section 6.8. Index contractions are performed with the dual metric g "g'(l/R, and upper '( N J

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

199

Table 31 Matrix gauge theory interpretation of three-form potential D

d

C '()

Interpretation

8 7 6 D45

3 4 5 d56

1 4 10 (B ) 

h-parameter O!-diagonal component A of ¹-metric ' B -background "eld of string theory on 5-brane '( Unclear

(lower) indices in the M-theory picture have become lower (upper) indices in the Matrix gauge theory picture. We also recall that < is the volume of the dual torus ¹I B on which the Matrix gauge Q theory lives. The expression of shifted charges is then given by m "m#Cm #(CC#E)m , (7.17a)   m "m #Cm , (7.17b)    m "m , (7.17c)   n "n #Cn #(CC#E)n , (7.17d)    _ n "n #Cn , (7.17e)   _ n "n . (7.17f) _ _ As we will see below, the linear shift in C is in agreement with the coupling obtained in Eq. (7.15). As a preview, the interpretation of the C coupling in the various Matrix gauge theories is summarized in Table 31. We will discuss these formulae in further detail for d"3, 4, 5 below. There is as yet no derivation of the coupling of the E gauge potential to the Matrix gauge theory.  7.4. Sl(3, 9);Sl(2, 9)-invariant mass formula for N"4 SYM in 3#1 dimensions As a "rst case, we consider the mass formula (7.16) for d"3,









g 1 1 E " 7+ m'# C'()m g m*# C*+,m 7+ N< () '* +, 2 2 Q < # Q (m g ')g (*m )#(n g '(n . (7.18) '( )* ' ( Ng 7+ This includes the energy of the electric #ux m' (i.e. the momentum conjugate to F ) and the ' magnetic #ux m "F in the diagonal Abelian subgroup of ;(N), together with the energy of '( '( a massless excitation with quantized momentum n . We observe that the e!ect of the M-theory ' background value of the three-form C is to shift the electric #ux m', which is a manifestation of the  Witten e!ect and indicates that the coupling of C to gauge theory occurs through the topological  term (7.15). Indeed, the only e!ect of such a coupling is to shift the momentum conjugate to R A by  ' a quantity C'()F . ()

200

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

Moreover, introducing the dual magnetic charge m' "e'()m and setting C'()"he'(), the H  () mass formula (7.18) can be written in the alternative form






1 1 g (m'#hm' )g (m(#hm( )# m' g m( #(n g '(n , (7.19) E " H '( H ' ( 7+ N< 7+ g H '( H 7+ Q which manifestly exhibits the Sl(2, 9) S-duality symmetry as well as the Sl(3, 9) modular group of the three-torus. 7.5. Sl(5, 9 )-invariant mass formula for (2,0) theory on M5-brane Moving on to the case d"4, an extra momentum charge n appears in (7.16), which corresponds  to the momentum along the dynamically generated 5th dimension. After some algebra, the total mass (7.16) can be rewritten in a manifestly U-duality (Sl(5, 9))-invariant form: 1 m g g m!"#(n g  n , (7.20) E " ! "  7+ N<  where A, B,2"1,2, 5 and < "< g is the volume of the "ve-dimensional torus. Here, the  Q 7+ two-form and vector charges m , n on the "ve-torus are related to the original set on the  four-torus by m'"m', n "n , ' '

1 m'(" e'()*m , )* 2 1 n " e'()*n ,  4! '()*

I"1, 2, 4, I"1, 2, 4,

(7.21a) (7.21b)

where the charge m is the quantized #ux (in the diagonal Abelian group) conjugate to the two-form gauge "eld that lives on the (5#1)-dimensional world-volume, and n is simply the  momentum along the direction A. The gauge potential C combines with the gauge coupling and '() the ¹ metric to make the metric on ¹: ds"RI (dx#A'dx )#ds ,  '  1 C()* . RI "g , A " e 7+ ' 3! '()*

(7.22a) (7.22b)

In particular, it is seen that the three-form potential plays the role of the o!-diagonal component of the "ve-dimensional metric relevant to the M5-brane. As a check, we recall that the bosonic part of the M5-brane action can be written in a noncovariant form by solving the self-duality condition after singling out a special ("fth) space-like direction and integrating the resulting equations of motion [1,246,270]. In particular, it contains the coupling GH 1 HI IJHI MN , L"! e IJHMN G 4

(7.23a)

1 HI IJ" eIJMHNH , k, l"0,2, 4 , MHN 6

(7.23b)

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

201

which precisely reproduces, upon the identi"cations in (7.22), the topological coupling (7.15) in the e!ective (4#1)-dimensional SYM theory, where the "eld strength F is identi"ed with the dual IJ "eld strength HI . Finally, we note that E in (7.20) depends on the volume of ¹ through an IJ 7+ overall factor <\, in agreement with the scale invariance of the conjectured (5#1)-dimensional  (2,0) theory. 7.6. SO(5, 5,9 )-invariant mass formula for non-critical string theory on NS5-brane Finally, we consider the case d"5, for which according to the reasoning in Section 6.5 the Matrix gauge theory should correspond to a non-critical string theory on the type IIB NS5-brane with vanishing string coupling g(  and "nite string tension lK "g . After some algebra, the mass Q Q 7+ formula (7.16) can be rewritten in the manifestly U-duality (SO(5, 5)) invariant form 1 M(D1, D3, D5)#(M(KK, F1) , (7.24) E " 7+ NM ,1 where M "< /g( lK  is the mass of the background NS5-brane. ,1 Q Q Q The second part of (7.24) involves the momentum (n ) and winding (n, dual to n ) excitations of   the strings living on the world-volume, which form the vector representation 10 of the SO(5, 5) T-duality group. The corresponding invariant mass 1 M(KK, F1)"(n #B n)# (n)   lK  Q directly follows from the second part of (7.16), using the identi"cation 1 C)*+ B " e '( 3! '()*+

(7.25)

(7.26)

for the background antisymmetric tensor "eld in terms of the components of the three-form gauge potential on the "ve-torus. The "rst term in (7.24) involves the D-brane excitations arising from the charges (m, m , m ) that   can be dualized into (m, m, m). It exhibits the correct invariant mass Eqs. (3.39a)}(3.39d) for a spinor representation of SO(5, 5):






m   m   m  # # , g( lK  g( lK  g( lK  QQ QQ QQ m "m#B m#Bm, m "m#B m ,    where we again used the identi"cation (7.26). As a further check, let us note that the Green}Schwarz term M(D1, D3, D5)"



dxBFF ,

 g( cancels out in the following formulae. Q

(7.27a) (7.27b)

(7.28)

202

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

in the e!ective action of the six-dimensional string theory, correctly gives the topological term (7.15) after using the relation (7.26) between the background string theory B-"eld and the vacuum expectation values of the M-theory three-form. 7.7. Extended U-duality symmetry and Lorentz invariance M(atrix) theory still lacks a proof of eleven-dimensional Lorentz covariance to shorten its name to M-theory. In the original conjecture, this feature was credited to the large-N in"nite-momentum limit. The much stronger Discrete Light Cone (DLC) conjecture, if correct, allows Lorentz invariance to be checked at "nite N } or rather at "nite N's, since the non-manifest Lorentz generators mix distinct N superselection sectors. In particular, M(atrix) theory on ¹B in the DLC should exhibit a U-duality E (9), if it is assumed that U-duality is una!ected by light-like B> compacti"cations [52,169,239]. In this section, we show that an action of E (9) on the B>B> M-theory BPS spectrum can be de"ned when we include the light-like circle S on an equal footing with the space-like torus ¹B. In the presence of an extra (light-like) compact direction of radius R , the states from the string J multiplet in Table 19 can be wrapped to yield extra particles in the spectrum that join the already existing states from the particle multiplet in Table 16. We have summarized in Table 32 the various particles obtained in the case d"7. It clearly appears that altogether, the d"7 particle and string multiplets build a particle multiplet of the d"8 U-duality group, whose charges M are obtained from the particle m and string n charges through the relations m "M ,   m"M, n"MJ,

m_"M_, n_"M_J ,

m"M, n"MJ,

m"M, n"M_J ,

m"M_, n"M_J ,

(7.29)

where we have denoted the light-cone direction by an index l. This is not quite correct, however, since in particular there is no candidate for the M state, which would correspond to a Kaluza} J Klein excitation along the light-like direction. Obviously, this missing charge is nothing but the rank of the gauge group N"M , (7.30) J which indeed denotes the momentum along the light-like direction, and should therefore be considered as a charge on the same footing as the others, labelling the vacuum of some M(eta) theory on which the eleven-dimensional Lorentz group is represented. This charge has to be invariant under the U-duality group E (9), but it gets mixed with other charges under BB E (9). B>B> While N is the only missing charge for d45, there is still, for d56, an extra missing U-duality singlet N"MJ_J  As usual, the same relations hold for d(7 by dropping the tensors with too many antisymmetric indices.

(7.31)

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

203

Table 32 Particle multiplet and string multiplet wrapped on R for d"7. Together with the rank multiplets, they form the d"8 J particle multiplet Particle multiplet

Charge

1 R ' RR ' ( l N RRR R R ' ( ) * + l N RR R R R R R ' ( ) * + , . l N

m (7)  m(21) m(21) m_(7)

String multiplet

RR J ( l N RR R R R J ' ( ) * l N R RR R R R R J ' ) * + , . l N R RRR R R R R J ' ( ) * + , . l N R RRR RR R R J ' ( ) * + , . l N

Charge

Missing charges

Ext. part.

N(1)

M (8) 

n(7)

M(28)

n(35)

M(56)

n_(49)

N(7), N(1)

M_(64)

n_(35)

N_(21)

M_(56)

n_(7)

N_(21)

M_(28)

N__(7), N_(1)

M__(8)

which can be interpreted as the D6}D0 bound state of Eq. (7.12). For d"7, one needs even more extra charges, namely N_,MJ_J, N,MJ_J, N_,MJ_J, N__,M_J_J ,

(7.32)

which form the 56 of E , isomorphic to the particle multiplet of Table 22, as well as the two singlets  N"MJ_, N_"MJ_J_J , (7.33) for which Table 37 gives the bound-state interpretation as well. These extra charges along with N were referred to in [239] as the rank multiplet. The results are summarized in the Table 33, which lists, for all d's, the dimensions of the particle and string multiplet, as well as the rank multiplets that are needed to complete the "rst two into the particle multiplet of the d#1 case. We note that the above discussion follows immediately from the decomposition (4.34) of the particle multiplet of E into the particle and string multiplet of E plus extra irreps for BB B\B\ d56. In particular, the extra representations that appear are nothing but the extra charges forming the rank multiplet. If we omit the light-like direction, we indeed see an extra T " for  d"6; for d"7 we have the extra representations T " , T " and (T )" , whose subscripts are in     precise correspondence with the number of times the light-like direction appears in the charges of Eqs. (7.31)}(7.33). 7.8. Nahm-type duality and interpretation of rank To see the physical signi"cance of the U-duality enhancement, we discuss the extra generators in E (9). First there is the Weyl generator, exchanging the light-cone direction with a chosen B>B>

204

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

Table 33 Flux, momentum and rank multiplets

D

d

U-duality E (9) B

Flux +m,

Mom. +n,

Rank +N,

Total +M,

10 9 8 7 6 5 4 3

1 2 3 4 5 6 7 8

1 Sl(2) Sl(3);Sl(2) Sl(5) SO(5, 5) E  E  E 

1 3 (3, 2) 10 16 27 56 248

1 2 (3, 1) 5 10 27 133 3875

1 1 1 1 1 1#1 56#1#1#1 R

3 6 10 16 27 56 248 R

direction I on ¹B: R R . (7.34) J ' The action of this Weyl transformation leaves the other R 's and l invariant. In particular, ( N Newton's constant in 11!(d#1) dimensions < R <B\ 1 " 0 J"RB\ Q J l gB\ i N is invariant under U-duality. In terms of matrix gauge theory, this means




(7.35)




R B\ R g , s Ps , s P J s , JOI . (7.36) g P J 7+ ' ' ( 7+ R R ( ' ' Note that the transformed parameters depend on the original ones and on R . On the other hand, J the only dependence of the gauge theory on R should be through a multiplicative factor in the J Hamiltonian, since R can be rescaled by a Lorentz boost (see Eq. (6.4)). This leaves open the J question of how the M(eta) theory itself depends on R . J The action on the charges follows from the exchange of the I and l indices, so that restricting to d"6 for simplicity, we have Nm , nm', nm', n_m_' . (7.37) ' In particular, the rank N of the gauge group is exchanged with the electric #ux m , whereas the ' momenta are exchanged with magnetic #uxes. This is reminiscent of Nahm duality, relating (at the classical level) a ;(N) gauge theory on ¹ with background #ux m to a ;(m) gauge theory on the dual torus with background #ux N [231]. In the context of higher-dimensional Yang}Mills theories, this symmetry was "rst observed at the level of the multiplicities of the BPS spectrum of SYM in 1#3 dimensions [150], and extended in the context of Matrix theory on ¹B in Refs. [52,83,170,239]. Non-commutative geometry may provide the correct framework for this duality [232].

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

205

The other generator is the Borel generator, C PC ,#*C (7.38) J() J() J() which is obtained from the usual E (9) shifts by conjugation under Nahm-type duality. It is BB therefore not an independent generator, but still gives a spectral #ow on the BPS spectrum NPN#*C m, m Pm #*C n, mPm#*C n, mPm#*C n_ . (7.39) J   J J J In particular, this implies that states with negative N need to be incorporated in the M(eta) theory if it is to be E (9)-invariant. This is somewhat surprising since the DLC quantization selects B>B> N'0, and it seems to require a revision both of the interpretation of N as the rank of a gauge theory and of the relation between N and the light-cone momentum P>. Finally, let us comment in some more generality on the occurrence of this extended U-duality group. At least at low energies, the Matrix gauge theory describing the DLCQ of M-theory compacti"ed on ¹B is nothing but the gauge theory on the N Dd-brane wrapped on ¹B. The latter is certainly invariant under the T-duality SO(d, d, 9), and not only SO(d!1, d!1, 9) ( ) Sl(d). Its spectrum of excitations, or equivalently bound states, is therefore invariant under SO(d, d, 9), and very plausibly under the extended duality group E (9). On the other hand, we have B>B> expanded the bound-state mass in the limit where the N Dd-branes are much heavier than their bound partners, whereas T-duality can exchange the Dd-branes with some of their excitations. SO(d, d, 9) is therefore explicitly broken, and E (9) is broken to E (9). The invariance of B>B> BB the mass spectrum can be restored by using the full non-commutative Born}Infeld dynamics instead of its small a Yang}Mills limit [68]. While not relevant for M(atrix) theory anymore, interesting insights can certainly be obtained by studying these stringy gauge theories.

Acknowledgements The material presented in this review grew from lectures given by the authors on various occasions during the 1997}1998 academic year, including CERN-TH Workshop and journal club, Nordita and NBI, Tours meeting of the Working Group on integrable systems and string theory, Ecole Normale SupeH rieure and CEA SPhT seminars, Amsterdam Summer Workshop on String Theory and Black Holes, Hamburg Workshop on Conformal Field Theory of D-branes, Corfu Summer Institute on Elementary Particle Physics. We are very grateful to the organizers for invitation and support. It is our pleasure to thank E. Rabinovici for an enjoyable collaboration and early participation to this work. We are grateful to C. Bachas, I. Bars, D. Bernard, J. de Boer, E. Cremmer, E. Eyras, K. FoK rger, A. Giveon, F. Hacquebord, M. Halpern, C. Hofman, K. Hori, R. Iengo, B. Julia, E. Kiritsis, M. Krogh, D. Olive, S. Ramgoolam, H. Samtleben, C. Schweigert, K. Stelle, W. Taylor, P. di Vecchia, E. Verlinde and G. Zwart, for useful remarks or discussions. This work is supported in part by the EEC under the TMR contract ERBFMRX-CT96-0090 and ERBFMRX-CT96-0045.  See Ref. [258] for a discussion of DLCQ with negative light-cone momentum.

206

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

Appendix A. BPS mass formulae Here, we analyse the BPS eigenvalue equation (2.14) for various choices of non-vanishing central charges. This gives a check on the mass formulae obtained on the basis of duality, and yields the conditions on the charges for a state to preserve a given fraction of supersymmetry. A.1. Gamma matrix theory In order to maintain manifest eleven-dimensional Lorentz invariance, we use the 11D Cli!ord algebra [C , C ]"2g , with signature (!,#,2), even after compacti"cation. The matrices + , +, C are then 32;32 real symmetric except for the charge conjugation matrix C"C , which is real +  antisymmetric. All products of Gamma matrices are traceless except for C C 2C C "1 , (A.1)    Q where we denote by s the eleventh direction. We de"ne C 2"C C 2 if the p indices M, N,2 + , +, are distinct, zero otherwise, and abbreviate it as C . We have N

(A.2) (C C )"(!1) N\ (C )"(!1) N ,  ,  N N where the p indices are non-zero and the square brackets denote the integer part. Furthermore, C C #(!)NOC C " N O O N



I N>O\IYN\O

C C !(!)NOC C " N O O N



C , N>O\I

I N>O\\IYN\O

C

N>O\\I

(A.3a) ,

(A.3b)

with no restrictions on the p#q indices. On the right-hand side of Eq. (A.3a) (resp. Eq. (A.3b)), a contraction between the "rst 2k (resp. 2k#1) indices of C and the "rst 2k (resp. 2k#1) indices N of C is implied. In particular, N [C , C ]"C , (A.4)  N N since C generates Lorentz rotations.  A.2. A general conxguration of KK-M2-M5 on ¹ Here we consider M-theory compacti"ed on ¹, and allow for non-vanishing central charges Z , ' Z ,Z , where the indices I, J,2 are internal indices on ¹. We therefore look for solutions to '( '()*+ the eigenvalue equation Ce"Me , C,Z C'#Z'(C #Z'()*+C . ' '( '()*+ Squaring this equation, we obtain Z Z +C , C ,!Z'(Z +C , C ,#Z'()*+Z,./01+C ,C , ' ( ' ( )* '( )* '()*+ ,./01 #2Z Z()[C', C ]#2Z Z()*+,+C', C ,!2Z'(Z)*+,.[C , C ]>M , ' () ' ()*+, '( )*+,.

(A.5)

(A.6)

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

207

where the symbol > denotes the equality when acting on e. Using the identities (A.3a), (A.3b), this reduces to (Z )#(Z'()#(Z'()*+)##Z Z'(C #(Z Z+'()*#Z'(Z)*)C >1 . ' ( ' + '()*

(A.7)

A 1/2-BPS state is obtained under the conditions k',Z Z'("0 , (

(A.8a)

k'()*,Z Z+'()*#Z'(Z)*"0 , +

(A.8b)

which indeed form a string multiplet 10 of E "SO(5, 5), and has a mass given by  M"(Z )#(Z'()#(Z'()*+).  '

(A.9)

k'()*/4!, C "C and rewrite If the conditions are not satis"ed, we can de"ne k "e ' '()*+   Eq. (A.7) as k'C #k C C' > M!M . ' '  

(A.10)

Note that the SO(5, 5) vector (k , k') is null: k k'"0. Squaring again yields the 1/4-BPS state mass ' ' formula M"(Z )#(Z'()#(Z'()*+)#((k')#(k ) . ' '

(A.11)

This result can be straightforwardly made invariant under the full U-duality group by including the couplings to the gauge potentials through the lower charges as found for the particle and string multiplet in Eqs. (5.23) and (5.27). A.3. A general conxguration of D0, D2, D4-branes on ¹ We now consider the D-brane sector of M-theory on ¹, that is a general con"guration of D0, D2, D4-branes. The eigenvalue equation becomes Ce"Me ,

(A.12a)

C,ZC #ZGHC #ZGHIJC , Q GH GHIJQ

(A.12b)

where Z, ZGH, ZGHIJ denote the D0, D2, D4-brane charges, respectively, and i, j,2 run from 1 to 5. Squaring this equation, we obtain 2Z#ZGHZIJ+C , C ,#ZGHIJZKLNO+C C ,#4Z ZGHIJC GH IJ GHIJ KLNO GHIJ #2ZGHZIJKL[C ,C ]C > M . GH IJKL Q

(A.13)

Using identities (A.3a), (A.3b), this becomes Z#(ZGH)#(ZGHIJ)#kGHIJC #(k)GHIJC > M , GHIJ GHIJQ

(A.14)

208

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

where we de"ned kGHIJ,Z GHZIJ #ZZGHIJ ,

(A.15a)

kYGHIJ,ZK GZHIJ K .

(A.15b)

The second combination can be rewritten on ¹ as a form kG_HIJKL"ZG HZIJKL . Then, k and k_ can be dualized into a 10 null vector (k , kG) of the T-duality group SO(5, 5). A state with k"k"0 is G 1/2-BPS with mass M"(Z)#(ZGH)#(ZGHIJ) . 

(A.16)

If these conditions are not met, we can rewrite Eq. (A.14) as k CGC #kCGC C > M!M , G  G  Q 

(A.17)

implying a mass formula M"(Z)#(ZGH)#(ZGHIJ)#2((kG)#(k ) G

(A.18)

or, in terms of the natural undualized charges, M"(Z)#(ZGH)#(ZGHIJ)#2((kGHIJ)#(kG_HIJKL) .

(A.19)

A.4. A general conxguration of KK}w}NS5 on ¹ Finally, we consider the Neveu}Schwarz sector of the theory considered in the Appendix A.3, namely the bound states of NS5-branes, winding and Kaluza}Klein states. The eigenvalue equation then reads (z CG#zGC #zGHIJKC )e"Me . G QG GHIJK

(A.20)

Taking the square gives z#(zG)#(z )#2zzGC #2zzGC !2C zGz > M , G G QG Q G

(A.21)

so the 1/2-BPS conditions appear to be zzG"zz "zGz "0 . G G

(A.22)

This agrees with the vanishing of the entropy zzGz and its "rst derivatives, as obtained in Ref. [113]. G We can go further and "nd the complete 1/8-BPS mass formula: multiply Eq. (A.20) by zC :  !zz C !zzGC !ze > zMC G G QG 

(A.23)

and combine with Eq. (A.21) to obtain: (!z#(zG)#(z )#2zMC !2zGz C ) > M . G  G Q

(A.24)

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

209

Now C and C commute, are traceless and square to 1, so this is a second-order equation: Q  !z#(zG)#(z )$2zM$2zGz > M , (A.25) G G with solutions M"$z$((z $zG) G or, equivalently:

(A.26)

(A.27) M"z#(z )#(zG)#2"z zG"#2"z"((z )#(zG)#2"z zG" . G G G G This reduces to the usual mass formula for perturbative string states (z"0) and for KK}NS5 or w}NS5 bound states. For momentum and winding charges along a single direction, this reduces to M"$z$z $z, in agreement with the identi"cation of central charges in Ref. [113]. The  U-duality invariant generalization of this mass formula is however unclear.

Appendix B. The d"8 string / momentum multiplet For completeness, we give in Table 34, the content of the string/momentum multiplet for d"8 in the 3875 of E . It comprises the 2160 states in the Weyl orbit of the highest weight R /l of length  G N 4, together with 7 copies of the 240 weights of length 2 with tension < T" 0;(d"8 particle multiplet) , l N as well as 35 zero weights with tension

(B.1)




<  0 . (B.2) l N As in d"7, the resulting multiplet exhibits a mirror symmetry, which relates each state with tension R?\/l?, a"1,2, 6 to another state with tension R\?/l\? through the relation N N T"




<  0 , (B.3) l N where < is the volume of the eight-torus. For this reason, Table 34 only gives the explicit form of 0 the tensions for the lower half a"1,2, 5 and the self-mirror part a"6. The second column gives the Sl(8) irreps at each level graded by 1/l?, while the last column lists the corresponding charges. N Here the notation is as follows: a semicolon denotes an ordinary tensor product as before (so in general contains more than one Sl(8) irrep); two superscripts (p; q) grouped within parentheses and separated by a semicolon denote the irrep, whose Young tableau is formed by juxtaposition of a column with p rows and one with q rows. As an aid to the reader, we give the charges of the dual states at level l\?: N a"1: n___, a"2: n___, a"3: n___ , (B.4) MM"

a"4: n___, n___, a"5: n___, n___ .

210

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

Table 34 String/momentum multiplet 3875 of E  Mass

Sl(8) irrep

Charge

R ' l N RRR R ' ( ) * l N RR R R R R < ' ( ) * + ,, 7 0 l R l N 'N RRR R R R R ' ( ) * + , . l N < R < R R 0 ', 7 0 ' ( l l N N RRR RR R R ' ( ) * + , . l N < RR R R < R R R R R 0 ' ( ) *, 7 0 ' ( ) * + l l N N < RRR R R R < RR R R R R R < 0 ' ( ) * + ,, 7 0 ' ( ) * + , ., 35 0 l l l N N N

8

n

70

n

8#216

n_

28#36#420

n_, n__

56#168#404

n_, n__

1#63#720

n__, n__

Finally, we display the decomposition of the d"8 string multiplet under the T-duality subgroup group SO(7, 7, 9). Here, we may again restrict to those states with (type IIA) tensions M&1/g?, Q a"024, for each of which there is a dual state with tension M related to it by <  MM" 0 , (B.5) gl Q Q where < stands for the seven-dimensional type IIA torus. The type IIA mirror symmetry (B.5) 0 easily follows from Eq. (B.3) using the M-theory/type IIA connection in Eq. (2.11). The results are summarized in Table 35. Here the type IIA states in the a-th column, have a tension proportional to 1/g?\. The "rst Q column is the singlet irrep formed by the fundamental string. The second column is the spinor irrep consisting of Dp"0-, 2-, 4-, 6-branes, with one unwrapped world-volume direction. The third column can be decomposed into the SO(7, 7) irreps 378"13;91104, and contains, together with NS5 and KK5 with one unwrapped direction, many non-standard states with tension &1/g. Q The fourth column contains the representation 896"14  64 formed by tensor product of vector and spinor representation, and has states with tension &1/g. The "fth column consists of 1197 Q states with tension &1/g. The set of duals of these states includes states with tension up to 1/g, all Q Q of which are at present far from understood. We note that the string state with tension < R/l is presumably related to the conjectured 0 ' N M9-brane [36,41], which would be more properly called M8-brane. In fact, for d"9 there will be




N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

211

Table 35 Branching of the d"8 string multiplet into representations of Sl(8) and SO(7, 7). The entries in the table denote the common Sl(7) reps. The full table can be reconstructed using mirror symmetry in the point with Sl(7) representation 539 3875(E )MSO(7,7)  6 Sl(8) 8 70 224 484 728 847 728 484 $

1

64

1

7 35 21 1

378

896

35 154 154 35

1197

49 294 292 154 7

35 294 539 294 35

896

2

7 154 292 294 $

2 2 \

a corresponding particle with mass < R/l, where < is now the volume of the nine-torus. 0 ' N 0 Taking R "R "l g , this reduces to the mass of the type IIA D8-brane, while taking R in one of ' Q Q Q Q the other world-volume directions gives an 8-brane with exotic mass < R/(lg). Vertical 0 G Q Q reduction, on the other hand, would give a type IIA 9-brane.

Appendix C. Matrix gauge theory on ¹ In this appendix we discuss in some detail the Matrix gauge theory on ¹, performing the analysis of Section 7.2 for the case d"7. For our discussion, it will be useful to "rst consider the type IIB states obtained from the set of type IIA states in Eqs. (4.36a)}(4.36e), by performing a maximal T-duality on the seven-torus. Using Eq. (2.41), we "nd the following T-duality multiplets for type IIB string theory compacti"ed on ¹ R 1 G, , l R Q G 1 < RR R RR RRR R 0, G H I J K, G H I, G , S : g l l l l Q Q Q Q Q 1 < R R < RR R R R R R R R R R 0 G H, 8 0, G H I J K L, G H I J K , S#AS: g l l l l Q Q Q Q Q < RRR RR R RRR R RR 0 G H I J K L, G H I J, G H, 1 , S :  gl l l l Q Q Q Q Q <  l 0 <: R, Q . GR gl Q Q G

<:


















(C.1a) (C.1b) (C.1c) (C.1d) (C.1e)

212

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

Table 36 Flux and momentum multiplet for d"7: 56 and 133 of E  M

E 7+

1 R ' RR ' ( l N < 0 R R l ' (N < R 0 ' l N

g s 7+ ' N< Q < Q Ng (s s ) 7+ ' ( < (s s ) Q '( Ng 7+ < Q Ng s 7+ '

RR J ' l N R< J 0 R R R l ' ( )N RR < J ' 0 R l (N RR R R < J ' ( ) 0 l N R < J 0 R l 'N

1 s ' sss '() g 7+ <s Q( g s 7+ ' < Q g s s s 7+ ' ( ) <s Q ' g 7+

C

YM state

b.s. of N D7

electric #ux

NS-w

21

magnetic #ux

D5

21

new sector

5 

7

new sector

1 

7

KK momentum

KK

35

YM threebrane

D3

49

new sector

KK5, 7

35

new sector

3 

7

new sector

0 

7



Here we have given the states in each multiplet in the order in which they are obtained from the corresponding type IIA states. At the levels 1/g?, with a even, we obtain the same set of states as in Q type IIA. At odd level, however, the spinor representations are interchanged, so that at level 1/g we Q obtain the odd Dp branes, while at level 1/g we "nd the set of p\N branes with p"1, 3, 5, 7. Q  We also give the S-duality structure of the type IIB states (C.1). Using Eq. (4.17), the following list of S-duality singlets (appearing at each level) is found: KK, D3, 7 , KK5, 3, 0 .    The remaining states pair up into S-duality doublets

(C.2)

F1}D1, D5}NS5, D7}7 , 5}5, 1}1 . (C.3)      The M-theory mass, gauge-theory energy and bound-state interpretation of the #ux and momentum multiplets is given in Table 36. Comparing the states in the last column of this table with the total set of 1/2 BPS states (C.1) for type IIB on ¹, we note that there is a large number of states that do not appear. In analogy with the extra D6}D0 multiplet (a singlet) that appeared for d"6, we can construct in this case an extra multiplet that contains the D7}D1 bound state, for which we conjecture (by T-duality) the same

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

213

Table 37 Additional multiplets for d"7: 56, 1 and 1 of E  C

YM state

b.s. of N D7

7

new sector

D1

21

new sector

NS5

21

new sector

5 

7

new sector

1 

< Q g 7+

1

new sector

7 

N< Q g 7+

1

new sector

7 

M

E 7+

R< J 0 R l 'N RR R < J ' ( 0 l N R< J 0 R R l ' (N RR < J ' 0 l N

N<s Q ' g 7+ N< Q g(s s ) 7+ ' ( N< (s s ) Q '( g 7+ N< Q g s 7+ '

R< J 0 l N R< J 0 l N

bound-state mass formula as in Eq. (7.10), so that <s E "MM "N Q ' , (C.4) 7+ " ," g 7+ where we used g "g l. The relevant data of the corresponding U-duality multiplet, which forms 7+ QQ the 56 of E , is given in Table 37. The easiest way to obtain this table, starting with the  gauge-theory mass (C.4) obtained for the D7}D1 bound state, is by noticing that this state is, up to a multiplicative U-duality invariant factor I (see Eq. (7.7)) and up to a power of 1/3, exactly  analogous to the #ux multiplet of Table 36. Note that the 56 states are precisely the S-dual states of those involved in the #ux multiplet bound states. The bound states relevant to the momentum multiplet, on the other hand, involve S-duality singlets. Besides the D7 itself, this leaves two more possible states left in the type IIB, which can form a bound state with the D7, namely the two 7-branes with mass < < Q . Q, (C.5) gl gl Q Q Q Q For the "rst one, we know already from the momentum multiplet that the mass relation is < E "M" Q "I (C.6) 7+  g 7+ and hence a U-duality singlet. For the second state in (C.5), we deduce the mass relation by the requirement that the bound-state energy be such that E "[M? #M?]?!M KM?M\? 7+ ," ," ,"

(C.7)

214

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

(i) can be written in gauge-theory variables, and (ii) is a U-duality singlet. Either of these requirements yields a"1/2, and we are left with a gauge-theory state with energy < E "M M "N Q "NI . 7+  ,"  g 7+ The singlets in Eqs. (C.6) and (C.8) are also given in Table 37.

(C.8)

References [1] M. Aganagic, J. Park, C. Popescu, J.H. Schwarz, World volume action of the M theory "ve-brane, Nucl. Phys. B 496 (1997) 191}214. hep-th/9701166. [2] O. Aharony, M. Berkooz, S. Kachru, N. Seiberg, E. Silverstein, Matrix description of interacting theories in six dimensions, Adv. Theor. Math. Phys. 1 (1998) 148}157. hep-th/9707079. [3] E. Alvarez, L. Alvarez-Gaume, Y. Lozano, An introduction to T duality in string theory, Nucl. Phys. Proc. (Suppl.) 41 (1995) 1}20. hep-th/9410237. [4] J. Ambjorn, L. Chekhov, The NBI matrix model of IIB superstrings, J. High Energy Phys. 12 (1998) 007. hep-th/9805212. [5] L. Andrianopoli, R. D'Auria, S. Ferrara, Five-dimensional U duality, black hole entropy and topological invariants, Phys. Lett. B 411 (1997) 39}45. hep-th/9705024. [6] L. Andrianopoli, R. D'Auria, S. Ferrara, U duality and central charges in various dimensions revisited, Int. J. Mod. Phys. A 13 (1998) 431}490. hep-th/9612105. [7] L. Andrianopoli, R. D'Auria, S. Ferrara, P. FreH , R. Minasian, M. Trigiante, Solvable Lie algebras in type IIA, type IIB and M theories, Nucl. Phys. B 493 (1997) 249}280. hep-th/9612202. [8] L. Andrianopoli, R. D'Auria, S. Ferrara, P. FreH , M. Trigiante, R-R scalars, U duality and solvable Lie algebras, Nucl. Phys. B 496 (1997) 617}629. hep-th/9611014. [9] L. Andrianopoli, R. D'Auria, S. Ferrara, P. FreH , M. Trigiante, E duality, BPS black hole evolution and "xed  scalars, Nucl. Phys. B 509 (1998) 463}518. hep-th/9707087. [10] I. Antoniadis, S. Ferrara, R. Minasian, K.S. Narain, R couplings in M and type II theories on Calabi}Yau spaces, Nucl. Phys. B 507 (1997) 571}588. hep-th/9707013. [11] I. Antoniadis, E. Gava, K.S. Narain, T.R. Taylor, N"2 type II heterotic duality and higher derivative F terms, Nucl. Phys. B 455 (1995) 109}130. hep-th/9507115. [12] I. Antoniadis, B. Pioline, T.R. Taylor, Calculable e\H e!ects, Nucl. Phys. B 512 (1998) 61}78. hep-th/9707222. [13] H. Aoki, S. Iso, H. Kawai, Y. Kitazawa, T. Tada, Space-time structures from IIB matrix model, Prog. Theor. Phys. 99 (1998) 713}746. hep-th/9802085. [14] R. Argurio, Brane physics in M theory, PhD thesis, UniversiteH Libre de Bruxelles, 1998. hep-th/9807171. hep-th/9807171. [15] R. Argurio, L. Houart, Little theories in six dimensions and seven dimensions, Nucl. Phys. B 517 (1998) 205}226. hep-th/9710027. [16] P.S. Aspinwall, Some relationships between dualities in string theory, Nucl. Phys. Proc. (Suppl.) 46 (1996) 30}38. hep-th/9508154. [17] C. Bachas, D-brane dynamics, Phys. Lett. B 374 (1996) 37}42. hep-th/9511043. [18] C. Bachas, Lectures on D-branes, 1998. hep-th/9806199. [19] C. Bachas, C. Fabre, E. Kiritsis, N.A. Obers, P. Vanhove, Heterotic/type I duality and D-brane instantons, Nucl. Phys. B 509 (1998) 33}52. hep-th/9707126. [20] C. Bachas, E. Kiritsis, F terms in N"4 string vacua, Nucl. Phys. Proc. Suppl. B 55 (1997) 194}199. hepth/9611205. [21] T. Banks, Matrix theory, Nucl. Phys. Proc. Suppl. 67 (1998) 180}224. hep-th/9710231. [22] T. Banks, W. Fischler, S.H. Shenker, L. Susskind, M theory as a matrix model: a conjecture, Phys. Rev. D 55 (1997) 5112}5128. hep-th/9610043. [23] T. Banks, L. Motl, Heterotic strings from matrices, J. High Energy Phys. 12 (1997) 004. hep-th/9703218.

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

215

[24] I. Bars, Algebraic structure of S theory, 1996. hep-th/9608061. [25] I. Bars, Supersymmetry, p-brane duality and hidden space-time dimensions, Phys. Rev. D 54 (1996) 5203}5210. hep-th/9604139. [26] I. Bars, A case for fourteen dimensions, Phys. Lett. B 403 (1997) 257}264. hep-th/9704054. [27] I. Bars, S theory, Phys. Rev. D 55 (1997) 2373}2381. hep-th/9607112. [28] I. Bars, C. Kounnas, Theories with two times, Phys. Lett. B 402 (1997) 25}32. hep-th/9703060. [29] I. Bars, S. Yankielowicz, U duality multiplets and nonperturbative superstring states, Phys. Rev. D 53 (1996) 4489}4498. hep-th/9511098. [30] K. Becker, M. Becker, A two-loop test of M(atrix) theory, Nucl. Phys. B 506 (1997) 48}60. hep-th/9705091. [31] K. Becker, M. Becker, J. Polchinski, A.Tseytlin, Higher order graviton scattering in M(atrix) theory, Phys. Rev. D 56 (1997) 3174}3178. hep-th/9706072. [32] K. Becker, M. Becker, A. Strominger, Five-branes, membranes and nonperturbative string theory, Nucl. Phys. B 456 (1995) 130}152. hep-th/9507158. [33] D. Berenstein, R. Corrado, J. Distler, On the moduli spaces of M(atrix) theory compacti"cations, Nucl. Phys. B 503 (1997) 239}255. hep-th/9704087. [34] O. Bergman, M.R. Gaberdiel, Stable non BPS D particles, Phys. Lett. B 441 (1998) 133. hep-th/9806155. [35] E. Bergshoe!, M. de Roo, D-branes and T duality, Phys. Lett. B 380 (1996) 265}272. hep-th/9603123. [36] E. Bergshoe!, M. de Roo, M. Green, G. Papadopoulos, P. Townsend, Duality of type-II 7-branes and 8-branes, Nucl. Phys. B 470 (1996) 113}135. [37] E. Bergshoe!, J. Gomis, P.K. Townsend, M-brane intersections from world volume superalgebras, Phys. Lett. B 421 (1998) 109}118. hep-th/9711043. [38] E. Bergshoe!, M.B. Green, G. Papadopoulos, P.K. Townsend, The IIA supereight-brane, 1995. hep-th/9511079. [39] E. Bergshoe!, C. Hull, T. Ortin, Duality in the type II superstring e!ective action, Nucl. Phys. B 451 (1995) 547}578. hep-th/9504081. [40] E. Bergshoe!, E. Sezgin, P.K. Townsend, Supermembranes and eleven-dimensional supergravity, Phys. Lett. B 189 (1987) 75}78. [41] E. Bergshoe!, J. van der Schaar, On M9-branes, Class. Quant. Grav. 16 (1999) 23. hep-th/9806069. [42] M. Berkooz, Nonlocal "eld theories and the non-commutative torus, Phys. Lett. B 430 (1998) 237}241. hep-th/9802069. [43] M. Berkooz, String dualities from matrix theory, A summary, Nucl. Phys. Proc. (Suppl.) 68 (1998) 374}380. hepth/9712012. [44] M. Berkooz, M.Rozali, String dualities from matrix theory, Nucl. Phys. B 516 (1998) 229. hep-th/9705175. [45] M. Berkooz, M. Rozali, N. Seiberg, Matrix description of M theory on ¹ and ¹, Phys. Lett. B 408 (1997) 105}110. hep-th/9704089. [46] N. Berkovits, Construction of R terms in N"2, D"8 superspace, Nucl. Phys. B 514 (1998) 191}203. hepth/9709116. [47] N. Berkovits, C. Vafa, Type IIB R HE\ conjectures, Nucl. Phys. B 533 (1998) 181}198. hep-th/9803145. [48] S. Bhattacharyya, A. Kumar, S. Mukhopadhyay, String network and U duality, Phys. Rev. Lett. 81 (1998) 754}757. hep-th/9801141. [49] D. Bigatti, L. Susskind, Review of matrix theory, 1997. hep-th/9712072. [50] A. Bilal, DLCQ of M theory as the light-like limit, Phys. Lett. B 435 (1998) 312}318. hep-th/9805070. [51] A.K. Biswas, A. Kumar, G. Sengupta, Oscillating D strings from IIB matrix theory, Phys. Rev. D 57 (1998) 5141}5149. hep-th/9711040. [52] M. Blau, M. O'Loughlin, Aspects of U duality in matrix theory, Nucl. Phys. B 525 (1998) 182}214. hep-th/9712047. [53] H. Brinkmann, Proc. Natl. Acad. Sci. US 9 (1923) 1. [54] J. Brodie, S. Ramgoolam, On matrix models of M "ve-branes, Nucl. Phys. B 521 (1998) 139}160. hep-th/9711001. [55] I. Brunner, A. Karch, Matrix description of M theory on ¹, Phys. Lett. B 416 (1998) 67}74. hep-th/9707259. [56] C.G. Callan, C. Lovelace, C.R. Nappi, S.A. Yost, Loop corrections to superstring equations of motion, Nucl. Phys. B 308 (1988) 221. [57] C.G. Callan Jr., J.A. Harvey, A. Strominger, Worldbrane actions for string solitons, Nucl. Phys. B 367 (1991) 60}82. [58] C.G. Callan Jr., J.M. Maldacena, Brane death and dynamics from the Born}Infeld action, Nucl. Phys. B 513 (1998) 198. hep-th/9708147. [59] C.G. Callan Jr., L. Thorlacius, World sheet dynamics of string junctions, Nucl. Phys. B 534 (1998) 121}136. hep-th/9803097.

216

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

[60] I.C.G. Campbell, P.C. West, N"2, D"10 nonchiral supergravity and its spontaneous compacti"cation, Nucl. Phys. B 243 (1984) 112. [61] L. Chekhov, K. Zarembo, E!ective action and measure in matrix model of IIB superstrings, Mod. Phys. Lett. A 12 (1997) 2331}2340. hep-th/9705014. [62] I. Chepelev, Y. Makeenko, K. Zarembo, Properties of D-branes in matrix model of IIB superstring, Phys. Lett. B 400 (1997) 43}51. hep-th/9701151. [63] I. Chepelev, A.A. Tseytlin, Interactions of type IIB D-branes from D instanton matrix model, Nucl. Phys. B 511 (1998) 629}646. hep-th/9705120. [64] I. Chepelev, A.A. Tseytlin, Long-distance interactions of branes, Correspondence between supergravity and super Yang}Mills descriptions, Nucl. Phys. B 515 (1998) 73}113. hep-th/9709087. [65] Y.-K.E. Cheung, M. Krogh, Non-commutative geometry from 0-branes in a background B "eld, Nucl. Phys. B 528 (1998) 185}196. hep-th/9803031. [66] M. Claudson, M.B. Halpern, Supersymmetric ground state wave functions, Nucl. Phys. B 250 (1985) 689. [67] A. Connes, Non-commutative Geometry, Academic Press, New York, 1994. [68] A. Connes, M.R. Douglas, A. Schwarz, Non-commutative geometry and matrix theory, Compacti"cation on tori, J. High Energy Phys. 2 (1998) 3. hep-th/9711162. [69] A. Connes, D. Kreimer, Hopf algebras, renormalization and non-commutative geometry, Commun. Math. Phys. 199 (1998) 203}242. hep-th/9808042. [70] E. Cremmer, &Supergravities in 5 dimensions', in: S.W. Hawking, M. Rocek (Eds.), Superspace and Supergravity, Cambridge University Press, Cambridge, 1981. [71] E. Cremmer, B. Julia, The SO(8) supergravity, Nucl. Phys. B 159 (1979) 141. [72] E. Cremmer, B. Julia, H. Lu, C.N. Pope, Dualization of dualities. 1, Nucl. Phys. B 523 (1998) 73}144. hepth/9710119. [73] E. Cremmer, B. Julia, H. Lu, C.N. Pope, Dualization of dualities. 2. Twisted selfduality of doubled "elds, and superdualities, Nucl. Phys. B 535 (1998) 242}292. hep-th/9806106. [74] E. Cremmer, B. Julia, J. Scherk, Supergravity theory in eleven dimensions, Phys. Lett. B 76 (1978) 409}412. [75] E. Cremmer, I.V. Lavrinenko, H. Lu, C.N. Pope, K.S. Stelle, T.A. Tran, Euclidean signature supergravities, dualities and instantons, Nucl. Phys. B 534 (1998) 40}82. hep-th/9803259. [76] E. Cremmer, H. Lu, C.N. Pope, K.S. Stelle, Spectrum generating symmetries for BPS solitons, Nucl. Phys. B 520 (1998) 132}156. hep-th/9707207. [77] M. Cvetic, C.M. Hull, Black holes and U duality, Nucl. Phys. B 480 (1996) 296}316. hep-th/9606193. [78] J. Dai, R.G. Leigh, J. Polchinski, New connections between string theories, Mod. Phys. Lett. A 4 (1989) 2073}2083. [79] G. Dall'Agata, K. Lechner, D. Sorokin, Covariant actions for the bosonic sector of d"10 IIB supergravity, Class. Quant. Grav. 14 (1997) L195}L198. hep-th/9707044. [80] U.H. Danielsson, G. Ferretti, B. Sundborg, D particle dynamics and bound states, Int. J. Mod. Phys. A 11 (1996) 5463}5478. hep-th/9603081. [81] K. Dasgupta, S. Mukhi, BPS nature of three string junctions, Phys. Lett. B 423 (1998) 261. hep-th/9711094. [82] J.A. de Azcarraga, J.P. Gauntlett, J.M. Izquierdo, P.K. Townsend, Topological extensions of the supersymmetry algebra for extended objects, Phys. Rev. Lett. 63 (1989) 2443. [83] R. de Mello Koch, J.P. Rodrigues, Duality and symmetries of the equations of motion, Phys. Lett. B 432 (1998) 83}89. hep-th/9709089. [84] B. de Wit, J. Hoppe, H. Nicolai, On the quantum mechanics of supermembranes, Nucl. Phys. B 305 (1988) 545. [85] B. de Wit, J. Louis, Supersymmetry and dualities in various dimensions, 1997. hep-th/9801132. [86] A. Dhar, G. Mandal, Probing four-dimensional nonsupersymmetric black holes carrying D0-brane and D6-brane charges, Nucl. Phys. B 531 (1998) 256}274. hep-th/9803004. [87] D.-E. Diaconescu, J. Gomis, Matrix description of heterotic theory on K , Phys. Lett. B 433 (1998) 35}42.  hep-th/9711105. [88] D.-E. Diaconescu, J. Gomis, Neveu}Schwarz "ve-branes and matrix string theory on K , Phys. Lett. B 426 (1998)  287}293. hep-th/9710124. [89] R. Dijkgraaf, E. Verlinde, H. Verlinde, BPS quantization of the "ve-brane, Nucl. Phys. B 486 (1997) 89}113. hep-th/9604055. [90] R. Dijkgraaf, E. Verlinde, H. Verlinde, BPS spectrum of the "ve-brane and black hole entropy, Nucl. Phys. B 486 (1997) 77}88. hep-th/9603126.

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

217

[91] R. Dijkgraaf, E. Verlinde, H. Verlinde, Matrix string theory, Nucl. Phys. B 500 (1997) 43}61. hep-th/9703030. [92] R. Dijkgraaf, E. Verlinde, H. Verlinde, Notes on matrix and micro strings, Nucl. Phys. Proc. (Suppl.) 62 (1998) 348}362. hep-th/9709107. [93] M. Dine, P. Huet, N. Seiberg, Large and small radius in string theory, Nucl. Phys. B 322 (1989) 301. [94] M. Dine, A. Rajaraman, Multigraviton scattering in the matrix model, Phys. Lett. B 425 (1998) 77}85. hepth/9710174. [95] M.R. Douglas, Branes within branes, 1995. hep-th/9512077. [96] M.R. Douglas, Superstring dualities, Dirichlet branes and the small scale structure of space, 1996. hep-th/9610041. [97] M.R. Douglas, C. Hull, D-branes and the non-commutative torus, J. High Energy Phys. 2 (1998) 8. hepth/9711165. [98] M.R. Douglas, D. Kabat, P. Pouliot, S.H. Shenker, D-branes and short distances in string theory, Nucl. Phys. B 485 (1997) 85}127. hep-th/9608024. [99] M.R. Douglas, G. Moore, D-branes, quivers, and ALE instantons, 1996. hep-th/9603167. [100] M.R. Douglas, H. Ooguri, Why matrix theory is hard, Phys. Lett. B 425 (1998) 71}76. hep-th/9710178. [101] M.R. Douglas, H. Ooguri, S.H. Shenker, Issues in (M)atrix model compacti"cation, Phys. Lett. B 402 (1997) 36}42. hep-th/9702203. [102] M.J. Du!, Supermembranes, 1996. hep-th/9611203. [103] M.J. Du!, P.S. Howe, T. Inami, K.S. Stelle, Superstrings in D"10 from supermembranes in D"11, Phys. Lett. B 191 (1987) 70. [104] M.J. Du!, R.R. Khuri, J.X. Lu, String solitons, Phys. Rep. 259 (1995) 213}326. hep-th/9412184. [105] M.J. Du!, H. Lu, C.N. Pope, AdS ;S untwisted, Nucl. Phys. B 532 (1998) 181}209. hep-th/9803061.  [106] M.J. Du!, K.S. Stelle, Multimembrane solutions of d"11 supergravity, Phys. Lett. B 253 (1991) 113}118. [107] T. Eguchi, P.B. Gilkey, A.J. Hanson, Gravitation, gauge theories and di!erential geometry, Phys. Rep. 66 (1980) 213. [108] S. Elitzur, A. Giveon, D. Kutasov, E. Rabinovici, Algebraic aspects of matrix theory on ¹B, Nucl. Phys. B 509 (1998) 122}144. hep-th/9707217. [109] E. Eyras, B. Janssen, Y. Lozano, Five-branes, KK monopoles and T duality, Nucl. Phys. B 531 (1998) 275}301. hep-th/9806169. [110] M. Fabbrichesi, G. Ferretti, R. Iengo, Supergravity and matrix theory do not disagree on multigraviton scattering, J. High Energy Phys. 6 (1998) 2. hep-th/9806018. [111] A. Fayyazuddin, Y. Makeenko, P. Olesen, D.J. Smith, K. Zarembo, Towards a nonperturbative formulation of IIB superstrings by matrix models, Nucl. Phys. B 499 (1997) 159}182. hep-th/9703038. [112] A. Fayyazuddin, D.J. Smith, P-brane solutions in IKKT IIB matrix theory, Mod. Phys. Lett. A 12 (1997) 1447}1454. hep-th/9701168. [113] S. Ferrara, J. Maldacena, Branes, central charges and U duality invariant BPS conditions, Class. Quant. Grav. 15 (1998) 749}758. hep-th/9706097. [114] W. Fischler, E. Halyo, A. Rajaraman, L. Susskind, The incredible shrinking torus, Nucl. Phys. B 501 (1997) 409}426. hep-th/9703102. [115] W. Fischler, A. Rajaraman, M(atrix) string theory on K , Phys. Lett. B 411 (1997) 53}58. hep-th/9704123.  [116] A. Font, L.E. Ibanez, D. Lust, F. Quevedo, Strong}weak coupling duality and nonperturbative e!ects in string theory, Phys. Lett. B 249 (1990) 35}43. [117] P.G.O. Freund, M.A. Rubin, Dynamics of dimensional reduction, Phys. Lett. B 97 (1980) 233. [118] J. Fuchs, A$ne Lie Algebras and Quantum Groups, Cambridge University Press, Cambridge, 1992. [119] J. Fuchs, C. Schweigert, Symmetries, Lie Algebras and Representations, Cambridge University Press, Cambridge, 1997. [120] M. Fukuma, H. Kawai, Y. Kitazawa, A. Tsuchiya, String "eld theory from IIB matrix model, Nucl. Phys. B 510 (1998) 158}174. hep-th/9705128. [121] M. Fukuma, H. Kawai, Y. Kitazawa, A. Tsuchiya, String "eld theory from IIB matrix model, Nucl. Phys. Proc. (Suppl.) 68 (1998) 153. [122] O. Ganor, L. Motl, Equations of the (2,0) theory and knitted "ve-branes, J. High Energy Phys. 5 (1998) 9. hep-th/9803108. [123] O.J. Ganor, On the M(atrix) model for M theory on ¹, Nucl. Phys. B 528 (1998) 133}155. hep-th/9709139. [124] O.J. Ganor, S. Ramgoolam, W. Taylor IV, Branes, #uxes and duality in M(atrix) theory, Nucl. Phys. B 492 (1997) 191}204. hep-th/9611202.

218

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

[125] O.J. Ganor, S. Sethi, New perspectives on Yang}Mills theories with sixteen supersymmetries, J. High Energy Phys. 1 (1998) 7. hep-th/9712071. [126] R.W. Gebert, H. Nicolai, E(10) for beginners, 1994. hep-th/9411188. [127] J.L. Gervais, A. Jevicki, B. Sakita, Collective coordinate method for quantization of extended systems, Phys. Rep. 23 (1976) 281. [128] F. Giani, M. Pernici, N"2 supergravity in ten-dimensions, Phys. Rev. D 30 (1984) 325. [129] R. Gilmore, Lie groups, Lie Algebras and Some of their Applications, Wiley, New York, 1974. [130] A. Giveon, D. Kutasov, Brane dynamics and gauge theory, Rev. Mod. Phys., in press. hep-th/9802067. [131] A. Giveon, N. Malkin, E. Rabinovici, On discrete symmetries and fundamental domains of target space, Phys. Lett. B 238 (1990) 57. [132] A. Giveon, M. Porrati, E. Rabinovici, Target space duality in string theory, Phys. Rep. 244 (1994) 77}202. hep-th/9401139. [133] R. Gopakumar, BPS states in matrix strings, Nucl. Phys. B 507 (1997) 609}620. hep-th/9704030. [134] S. Govindarajan, Heterotic M(atrix) theory at generic points in narain moduli space, Nucl. Phys. B 507 (1997) 589. hep-th/9707164. [135] S. Govindarajan, A note on M(atrix) theory in seven-dimensions with eight supercharges, Phys. Rev. D 56 (1997) 5276}5278. hep-th/9705113. [136] M. Green, J.A. Harvey, G. Moore, I-brane in#ow and anomalous couplings on D-branes, Class. Quant. Grav. 14 (1997) 47}52. hep-th/9605033. [137] M.B. Green, M. Gutperle, E!ects of D instantons, Nucl. Phys. B 498 (1997) 195}227. hep-th/9701093. [138] M.B. Green, M. Gutperle, H. Kwon, Sixteen fermion and related terms in M theory on ¹, Phys. Lett. B 421 (1998) 149}161. hep-th/9710151. [139] M.B. Green, M. Gutperle, P. Vanhove, One loop in eleven-dimensions, Phys. Lett. B 409 (1997) 177}184. hep-th/9706175. [140] M.B. Green, C.M. Hull, P.K. Townsend, D-brane Wess}Zumino actions, T duality and the cosmological constant, Phys. Lett. B 382 (1996) 65}72. hep-th/9604119. [141] M.B. Green, J.H. Schwarz, E. Witten, Superstring Theory, vol. 1, Introduction, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1987. [142] M.B. Green, S. Sethi, Supersymmetry constraints on type IIB supergravity, 1998. hep-th/9808061. [143] M.B. Green, P. Vanhove, D-instantons, strings and M theory, Phys. Lett. B 408 (1997) 122}134. hep-th/9704145. [144] A. Gregori, E. Kiritsis, C. Kounnas, N.A. Obers, P.M. Petropoulos, B. Pioline, R corrections and nonperturbative dualities of N"4 string ground states, Nucl. Phys. B 510 (1998) 423}476. hep-th/9708062. [145] C. Grojean, J. Mourad, Superconformal 6-d (2,0) theories in superspace, Class. Quant. Grav. 15 (1998) 3397}3409. hep-th/9807055. [146] R. Gueven, Black p-brane solutions of D"11 supergravity theory, Phys. Lett. B 276 (1992) 49}55. [147] Z. Guralnik, S. Ramgoolam, Torons and D-brane bound states, Nucl. Phys. B 499 (1997) 241}252. hep-th/9702099. [148] Z. Guralnik, S. Ramgoolam, From 0-branes to torons, Nucl. Phys. B 521 (1998) 129}138. hep-th/9708089. [149] Y.S.H. Ishikawa, Y. Matsuo, K. Sugiyama, BPS mass spectrum from D-brane action, Phys. Lett. B 388 (1996) 296}302. hep-th/9605023. [150] F. Hacquebord, H. Verlinde, Duality symmetry of N"4 Yang}Mills theory on ¹, Nucl. Phys. B 508 (1997) 609}622. hep-th/9707179. [151] E. Halyo, A proposal for the wrapped transverse "ve-brane in M(atrix) theory, 1997. hep-th/9704086. [152] A. Hanany, G. Lifschytz, M(atrix) theory on ¹ and a m(atrix) theory description of KK monopoles, Nucl. Phys. B 519 (1998) 195}213. hep-th/9708037. [153] J.A. Harvey, G. Moore, Five-brane instantons and R couplings in N"4 string theory, Phys. Rev. D 57 (1998) 2323. hep-th/9610237. [154] J.A. Harvey, G. Moore, On the algebras of BPS states, Comm. Math. Phys. 197 (1998) 489. hep-th/9609017. [155] S. Helgason, Di!erential Geometry and Symmetric Spaces, Academic Press, New York, 1962. [156] S. Hellerman, J. Polchinski, Compacti"cation in the lightlike limit, 1997. hep-th/9711037. [157] P.-M. Ho, Y.-S. Wu, IIB/M duality and longitudinal membranes in M(atrix) theory, Phys. Rev. D 57 (1998) 2571}2579. hep-th/9703016. [158] P.-M. Ho, Y.-S. Wu, Non-commutative gauge theories in matrix theory, Phys. Rev. D 58 (1998) 066003. hep-th/9801147.

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

219

[159] C. Hofman, E. Verlinde, G. Zwart, U-duality invariance of the four-dimensional Born}Infeld theory, J. High Energy Phys. 10 (1998) 20. hep-th/9808128. [160] P. Horava, M theory as a holographic "eld theory, 1997. hep-th/9712130. [161] P. Horava, Matrix theory and heterotic strings on tori, Nucl. Phys. B 505 (1997) 84}108. hep-th/9705055. [162] P. Horava, E. Witten, Eleven-dimensional supergravity on a manifold with boundary, Nucl. Phys. B 475 (1996) 94}114. hep-th/9603142. [163] P. Horava, E. Witten, Heterotic and type I string dynamics from eleven-dimensions, Nucl. Phys. B 460 (1996) 506}524. hep-th/9510209. [164] G.T. Horowitz, D.A. Lowe, J.M. Maldacena, Statistical entropy of nonextremal four-dimensional black holes and U duality, Phys. Rev. Lett. 77 (1996) 430}433. hep-th/9603195. [165] P.S. Howe, P.C. West, The complete N"2, D"10 supergravity, Nucl. Phys. B 238 (1984) 181. [166] J. Hughes, J. Polchinski, Partially broken global supersymmetry and the superstring, Nucl. Phys. B 278 (1986) 147. [167] C.M. Hull, Exact pp wave solutions of eleven-dimensional supergravity, Phys. Lett. 139 B (1984) 39. [168] C.M. Hull, Decoupling limits in M theory, Nucl. Phys. B 529 (1998) 207}224. hep-th/9803124. [169] C.M. Hull, Matrix theory, U duality and toroidal compacti"cations of M theory, J. High Energy Phys. 10 (1998) 11. hep-th/9711179. [170] C.M. Hull, U duality BPS spectrum of super Yang}Mills theory and M theory, J. High Energy Phys. 7 (1998) 18. hep-th/9712075. [171] C.M. Hull, B. Julia, Duality and moduli spaces for timelike reductions, Nucl. Phys. B 534 (1998) 250}260. hep-th/9803239. [172] C.M. Hull, P.K. Townsend, Enhanced gauge symmetries in superstring theories, Nucl. Phys. B 451 (1995) 525}546. hep-th/9505073. [173] J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, Heidelberg, 1972. [174] J.E. Humphreys, Re#ection groups and Coxeter Groups, Number 29 in Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1990. [175] M. Huq, M.A. Namazie, Kaluza}Klein supergravity in ten-dimensions, Class. Quant. Grav. 2 (1985) 293. [176] N. Ishibashi, H. Kawai, Y. Kitazawa, A. Tsuchiya, A large N reduced model as superstring, Nucl. Phys. B 498 (1997) 467}491. hep-th/9612115. [177] B. Julia, Extra dimensions, Recent progress using old ideas, 1979. [178] B. Julia, Group disintegrations, in: S.W. Hawking, M. Rocek (Eds.), Superspace and Supergravity, Cambridge University Press, Cambridge, 1981. [179] B. Julia, In"nite Lie algebras in physics, 1981. [180] B. Julia, Supergeometry and Kac}Moody algebras, in: Publications of the Mathematical Sciences Research Institute, vol. 3, Springer, Heidelberg, 1983. [181] D. Kabat, P. Pouliot, A comment on zero-brane quantum mechanics, Phys. Rev. Lett. 77 (1996) 1004}1007. hep-th/9603127. [182] D. Kabat, S.-J. Rey, Wilson lines and T duality in heterotic M(atrix) theory, Nucl. Phys. B 508 (1997) 535}568. hep-th/9707099. [183] S. Kachru, Y. Oz, Z. Yin, Matrix description of intersecting M-5 branes, J. High Energy Phys. 11 (1998) 4. hep-th/9803050. [184] R. Kallosh, B. Kol, e symmetric area of the black hole horizon, Phys. Rev. D 53 (1996) 5344}5348. hep th/9602014. [185] T. Kawano, K. Okuyama, Matrix theory on non-commutative torus, Phys. Lett. B 433 (1998) 29}34. hepth/9803044. [186] A. Kehagias, H. Partouche, D instanton corrections as (p, q) string e!ects and nonrenormalization theorems, Int. J. Mod. Phys. A 13 (1998) 5075}5092. hep-th/9712164. [187] A. Kehagias, H. Partouche, The exact quartic e!ective action for the type IIB superstring, Phys. Lett. B 422 (1998) 109}116. hep-th/9710023. [188] E. Keski-Vakkuri, P. Kraus, Short distance contributions to graviton-graviton scattering; matrix theory versus supergravity, Nucl. Phys. B 529 (1998) 246}258. hep-th/9712013. [189] N. Kim, S.-J. Rey, M(atrix) theory on an orbifold and twisted membrane, Nucl. Phys. B 504 (1997) 189}213. hep-th/9701139. [190] N. Kim, S.-J. Rey, M(atrix) theory on ¹/9 orbifold and twisted zero-brane, 1997. hep-th/9710245. 

220

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

[191] N. Kim, S.-J. Rey, Nonorientable m(atrix) theory, 1997. hep-th/9710192. [192] N. Kim, S.-J. Rey, M(atrix) theory on ¹/9 orbifold and "ve-branes, Nucl. Phys. B 534 (1998) 155}182. hep th/9705132. [193] E. Kiritsis, Introduction to superstring theory, Leuven Univ. Press, Leuven, 1998. hep-th/9709062. [194] E. Kiritsis, N.A. Obers, Heterotic type I duality in D(10-dimensions, threshold corrections and D instantons, J. High Energy Phys. 10 (1997) 4. hep-th/9709058. [195] E. Kiritsis, B. Pioline, On R threshold corrections in IIB string theory and (p, q) string instantons, Nucl. Phys. B 508 (1997) 509}534. hep-th/9707018. [196] N. Kitsunezaki, J. Nishimura, Unitary IIB matrix model and the dynamical generation of the space-time, Nucl. Phys. B 526 (1998) 351}377. hep-th/9707162. [197] A.W. Knapp, Representation theory of semisimple groups: an overview based on examples, Princeton Mathematical Series, vol. 36, Princeton University Press, Princeton, NJ, 1986. [198] C.F. Kristjansen, P. Olesen, A possible IIB superstring matrix model with Euler characteristic and a double scaling limit, Phys. Lett. B 405 (1997) 45. hep-th/9704017. [199] M. Krogh, Heterotic matrix theory with wilson lines on the lightlike circle, Nucl. Phys. B 541 (1999) 98. hepth/9803088. [200] M. Krogh, A matrix model for heterotic Spin(32)/9 and type I string theory, Nucl. Phys. B 541 (1999) 87. hep th/9801034. [201] M. Krogh, S. Lee, String network from M theory, Nucl. Phys. B 516 (1998) 241}254. hep-th/9712050. [202] T. Kugo, P. Townsend, Supersymmetry and the division algebras, Nucl. Phys. B 221 (1983) 357. [203] A. Kumar, S. Mukhopadhyay, Supersymmetry and U-brane networks, 1998. hep-th/9806126. [204] I.V. Lavrinenko, H. Lu, C.N. Pope, T.A. Tran, U duality as general coordinate transformations, and space-time geometry, 1998. hep-th/9807006. [205] R.G. Leigh, Dirac}Born}Infeld action from Dirichlet sigma model, Mod. Phys. Lett. A 4 (1989) 2767. [206] R.G. Leigh, M. Rozali, A note on six-dimensional gauge theories, Phys. Lett. B 433 (1998) 43}48. hep-th/9712168. [207] W. Lerche, A.N. Schellekens, N.P. Warner, Lattices and strings, Phys. Rep. 177 (1989) 1. [208] M. Li, Comments on supersymmetric Yang}Mills theory on a non-commutative torus, 1998. hep-th/9802052. [209] A. Losev, G. Moore, S.L. Shatashvili, M & m's, Nucl. Phys. B 522 (1998) 105}124. hep-th/9707250. [210] D.A. Lowe, Heterotic matrix string theory, Phys. Lett. B 403 (1997) 243}249. hep-th/9704041. [211] D.A. Lowe, E ; E small instantons in matrix theory, Nucl. Phys. B 519 (1998) 180. hep-th/9709015.   [212] D.A. Lowe, Eleven-dimensional Lorentz symmetry from SUSY quantum mechanics, J. High Energy Phys. 10 (1998) 3. hep-th/9807229. [213] H. Lu, C.N. Pope, p-brane solitons in maximal supergravities, Nucl. Phys. B 465 (1996) 127}156. hep-th/9512012. [214] H. Lu, C.N. Pope, T duality and U duality in toroidally compacti"ed strings, Nucl. Phys. B 510 (1998) 139}157. hep-th/9701177. [215] H. Lu, C.N. Pope, K.S. Stelle, Weyl group invariance and p-brane multiplets, Nucl. Phys. B 476 (1996) 89}117. hep-th/9602140. [216] J.X. Lu, S. Roy, U duality p-branes in diverse dimensions, Nucl. Phys. B 538 (1999) 149. hep-th/9805180. [217] J. Maharana, J.H. Schwarz, Noncompact symmetries in string theory, Nucl. Phys. B 390 (1993) 3}32. hep-th/9207016. [218] J. Maldacena, The large N limit of superconformal "eld theories and supergravity, Adv. Theor. Phys. 2 (1998) 231}252. hep-th/9711200. [219] J.M. Maldacena, Black holes in string theory, PhD thesis, Princeton University, Princeton, NJ, 1996. hepth/9607235. hep-th/9607235. [220] B.P. Mandal, S. Mukhopadhyay, D-brane interaction in the type IIB matrix model, Phys. Lett. B 419 (1998) 62}72. hep-th/9709098. [221] M. Marino, G. Moore, Counting higher genus curves in a Calabi}Yau manifold, 1998. hep-th/9808131. [222] E. Martinec, M theory and N"2 strings, 1997. hep-th/9710122. [223] Y. Matsuo, K. Okuyama, BPS condition of string junction from M theory, Phys. Lett. B 426 (1998) 294}296. hep-th/9712070. [224] J. McCarthy, L. Susskind, A. Wilkins, Large N and the Dine}Rajaraman problem, Phys. Lett. B 437 (1998) 62}68. hep-th/9806136. [225] P. Meessen, T. Ortin, An Sl(2, Z) multiplet of nine-dimensional type II supergravity theories, Nucl. Phys. B 541 (1999) 195. hep-th/9806120.

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

221

[226] C. Montonen, D. Olive, Magnetic monopoles as gauge particles? Phys. Lett. B 72 (1977) 117. [227] G. Moore, Finite in all directions, 1993. hep-th/9305139. [228] B. Morariu, B. Zumino, Super Yang}Mills on the non-commutative torus, Relativity, Particle Physics and Cosmology, World Scienti"c, Singapore, in press. hep-th/9807198. [229] L. Motl, Proposals on nonperturbative superstring interactions, 1997. hep-th/9701025. [230] L. Motl, L. Susskind, Finite N heterotic matrix models and discrete light cone quantization, 1997. hep-th/9708083. [231] W. Nahm, Monopoles in quantum "eld theory, in: N. Craigie (Ed.), Monopole meeting, Trieste, 1982. [232] N. Nekrasov, A. Schwarz, Instantons on non-commutative R and (2,0) superconformal six-dimensional theory, Commun. Math. Phys. 198 (1998) 689}703. hep-th/9802068. [233] H. Nicolai, On M theory, 1997. hep-th/9801090. [234] H. Nicolai, H. Samtleben, Integrability and canonical structure of d"2, N"16 supergravity, Nucl. Phys. B 533 (1998) 210}242. hep-th/9804152. [235] H. Nishino, Supergravity theories in D greater than or equal to 12 coupled to super p-branes, 1998. hep-th/9807199. [236] H. Nishino, Supersymmetric Yang}Mills theories in D512, Nucl. Phys. B 523 (1998) 450}464. hep-th/9708064. [237] H. Nishino, E. Sezgin, Supersymmetric Yang}Mills equations in (10#2)-dimensions, Phys. Lett. B 388 (1996) 569}576. hep-th/9607185. [238] N.A. Obers, B. Pioline, Eisenstein series and string thresholds, hep-th/9903113. [239] N.A. Obers, B. Pioline, E. Rabinovici, M theory and U duality on ¹B with gauge backgrounds, Nucl. Phys. B 525 (1998) 163}181. hep-th/9712084. [240] Y. Okawa, T. Yoneya, Multibody interactions of D particles in supergravity and matrix theory, 1998. hepth/9806108. [241] H. Ooguri, C. Vafa, Summing up D instantons, Phys. Rev. Lett. 77 (1996) 3296}3298. hep-th/9608079. [242] H. Ooguri, Z. Yin, TASI lectures on perturbative string theories, 1996. hep-th/9612254. [243] S. Paban, S. Sethi, M. Stern, Supersymmetry and higher derivative terms in the e!ective action of Yang}Mills theories, J. High Energy Phys. 6 (1998) 12. hep-th/9806028. [244] G. Papadopoulos, T duality and the world volume solitons of "ve-branes and KK monopoles, Phys. Lett. B 434 (1998) 277}284. hep-th/9712162. [245] G. Papadopoulos, P.K. Townsend, Kaluza}Klein on the brane, Phys. Lett. B 393 (1997) 59}64. hep-th/9609095. [246] P. Pasti, D. Sorokin, M. Tonin, Covariant action for a D"11 "ve-brane with the chiral "eld, Phys. Lett. B 398 (1997) 41}46. hep-th/9701037. [247] A.W. Peet, The Bekenstein formula and string theory (N-brane theory), Class. Quant. Grav. 15 (1998) 3291}3338. hep-th/9712253. [248] B. Pioline, D-e!ects in toroidally compacti"ed type II string theory, 1997. hep-th/9712155. [249] B. Pioline, Aspects non perturbatifs de la theH orie des supercordes, PhD thesis, UniversiteH Paris 6, 1998. hep-th/9806123. hep-th/9806123. [250] B. Pioline, E. Kiritsis, U duality and D-brane combinatorics, Phys. Lett. B 418 (1998) 61}69. hep-th/9710078. [251] J. Polchinski, Dirichlet branes and Ramond}Ramond charges, Phys. Rev. Lett. 75 (1995) 4724}4727. hepth/9510017. [252] J. Polchinski, TASI lectures on D-branes, 1996. hep-th/9611050. [253] J. Polchinski, P. Pouliot, Membrane scattering with M momentum transfer, Phys. Rev. D 56 (1997) 6601}6606. hep-th/9704029. [254] J. Polchinski, E. Witten, Evidence for heterotic-type I string duality, Nucl. Phys. B 460 (1996) 525}540. hep-th/9510169. [255] D. Polyakov, RR-Dilaton interaction in a type IIB superstring, Nucl. Phys. B 468 (1996) 155}162. hep-th/9512028. [256] M. Rei!el, A. Schwarz, Morita equivalence of multidimensional non-commutative tori, 1998. math.QA/9803057. [257] S.-J. Rey, Heterotic M(atrix) strings and their interactions, Nucl. Phys. B 502 (1997) 170. hep-th/9704158. [258] S.-J. Rey, M(atrix) theory on the negative light front, 1997. hep-th/9712055. [259] S.-J. Rey, J.-T. Yee, BPS dynamics of triple (p, q) string junction, Nucl. Phys. B 526 (1998) 229}240. hep-th/9711202. [260] M. Rozali, Matrix theory and U duality in seven-dimensions, Phys. Lett. B 400 (1997) 260}264. hep-th/9702136. [261] I. Rudychev, E. Sezgin, P. Sundell, Supersymmetry in dimensions beyond eleven, Nucl. Phys. Proc. (Suppl.) 68 (1998) 285}294. hep-th/9711127. [262] J.G. Russo, An ansatz for a nonperturbative four graviton amplitude in type IIB superstring theory, Phys. Lett. B 417 (1998) 253. hep-th/9707241.

222

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

[263] J.G. Russo, BPS bound states, supermembranes, and T duality in M theory, 1997. hep-th/9703118. [264] J.G. Russo, A.A. Tseytlin, One loop four graviton amplitude in eleven-dimensional supergravity, Nucl. Phys. B 508 (1997) 245. hep-th/9707134. [265] S. Ryang, BPS spectrum and Nahm duality of matrix theory compacti"cations, Phys. Lett. B 433 (1998) 279}286. hep-th/9803198. [266] B. Sathiapalan, Fundamental strings and D strings in the IIB matrix model, Mod. Phys. Lett. A 12 (1997) 1301}1315. hep-th/9703133. [267] A. Schwarz, Morita equivalence and duality, Nucl. Phys. B 534 (1998) 720}738. hep-th/9805034. [268] J.H. Schwarz, Covariant "eld equations of chiral N"2 D"10 supergravity, Nucl. Phys. B 226 (1983) 269. [269] J.H. Schwarz, An Sl(2, 9) multiplet of type IIB superstrings, Phys. Lett. B 360 (1995) 13}18. hep-th/9508143. [270] J.H. Schwarz, Coupling a self-dual tensor to gravity in six dimensions, Phys. Lett. B 395 (1997) 191}195. hep-th/9701008. [271] J.H. Schwarz, Lectures on superstring and M theory dualities, Given at ICTP spring school and at TASI summer school, Nucl. Phys. Proc. (Suppl.) B 55 (1997) 1}32. hep-th/9607201. [272] J.H. Schwarz, From superstrings to M theory, 1998, Phys. Rep., in press. hep-th/9807135. [273] J.H. Schwarz, P.C. West, Symmetries and transformations of chiral N"2 D"10 supergravity, Phys. Lett. B 126 (1983) 301. [274] N. Seiberg, Why is the matrix model correct? Phys. Rev. Lett. 79 (1997) 3577}3580. hep-th/9710009. [275] N. Seiberg, S. Sethi, Comments on Neveu}Schwarz "ve-branes, Adv. Theor. Math. Phys. 1 (1998) 259. hep-th/9708085. [276] N. Seiberg, S. Shenker, Hypermultiplet moduli space and string compacti"cation to three-dimensions, Phys. Lett. B 388 (1996) 521}523. hep-th/9608086. [277] A. Sen, Dyon-monopole bound states, selfdual harmonic forms on the multi-monopole moduli space, and S¸(2, 9) invariance in string theory. Phys. Lett. B 329 (1994) 217}221. hep-th/9402032. [278] A. Sen, U duality and intersecting Dirichlet branes, Phys. Rev. D 53 (1996) 2874}2894. hep-th/9511026. [279] A. Sen, D0-branes on ¹L and matrix theory, Adv. Theor. Math. Phys. 2 (1998) 51}59. hep-th/9709220. [280] A. Sen, An introduction to nonperturbative string theory, 1998. hep-th/9802051. [281] A. Sen, SO(32) spinors of type I and other solitons on brane}anti-brane pair, J. High Energy Phys. 9 (1998) 23. hep-th/9808141. [282] A. Sen, Stable non BPS bound states of BPS D-branes, J. High Energy Phys. 8 (1998) 10. hep-th/9805019. [283] A. Sen, Stable non BPS states in string theory, J. High Energy Phys. 6 (1998) 7. hep-th/9803194. [284] A. Sen, String network, J. High Energy Phys. 3 (1998) 5. hep-th/9711130. [285] A. Sen, Tachyon condensation on the brane anti-brane system, J. High Energy Phys. 8 (1998) 12. hep-th/9805170. [286] S. Sethi, The matrix formulation of type IIB "ve-branes, Nucl. Phys. B 523 (1998) 158}170. hep-th/9710005. [287] S. Sethi, L. Susskind, Rotational invariance in the M(atrix) formulation of type IIB theory, Phys. Lett. B 400 (1997) 265}268. hep-th/9702101. [288] E. Sezgin, The M algebra, Phys. Lett. B 392 (1997) 323}331. hep-th/9609086. [289] E. Sezgin, Super Yang}Mills in (11,3)-dimensions, Phys. Lett. B 403 (1997) 265}272. hep-th/9703123. [290] K. Sfetsos, K. Skenderis, Microscopic derivation of the Bekenstein}Hawking entropy formula for nonextremal black holes, Nucl. Phys. B 517 (1998) 179}204. hep-th/9711138. [291] R.D. Sorkin, Kaluza}Klein monopole, Phys. Rev. Lett. 51 (1983) 87. [292] K.S. Stelle, Lectures on supergravity p-branes, 1996. hep-th/9701088. hep-th/9803116. [293] A. Strominger, C. Vafa, Microscopic origin of the Bekenstein}Hawking entropy, Phys. Lett. B 379 (1996) 99}104. hep-th/9601029. [294] Y. Sugawara, K. Sugiyama, D-brane analyses for BPS mass spectra and U duality, Nucl. Phys. B 522 (1998) 158}192. hep-th/9707205. [295] L. Susskind, T duality in M(atrix) theory and S duality in "eld theory, 1996. hep-th/9611164. [296] L. Susskind, Another conjecture about M(atrix) theory, 1997. hep-th/9704080. [297] T. Suyama, A. Tsuchiya, Exact results in n "2 IIB matrix model, Prog. Theor. Phys. 99 (1998) 321}325. A hep-th/9711073. [298] G. 't Hooft, A planar diagram theory for strong interactions, Nucl. Phys. B 72 (1974) 461. [299] G. 't Hooft, A property of electric and magnetic #ux in non-Abelian gauge theories, Nucl. Phys. B 153 (1979) 141. [300] W. Taylor IV, Adhering zero-branes to six-branes and eight-branes, Nucl. Phys. B 508 (1997) 122}132. hepth/9705116.

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

223

[301] W. Taylor IV, D-brane "eld theory on compact spaces, Phys. Lett. B 394 (1997) 283}287. hep-th/9611042. [302] W. Taylor IV, Lectures on D-branes, gauge theory and M(atrices), 1997. hep-th/9801182. [303] W. Taylor IV, M.V. Raamsdonk, Three graviton scattering in matrix theory revisited, Phys. Lett. B 438 (1998) 248}254. hep-th/9806066. [304] P.K. Townsend, The eleven-dimensional supermembrane revisited, Phys. Lett. B 350 (1995) 184}187. hepth/9501068. [305] P.K. Townsend, p-brane democracy, 1995. hep-th/9507048. [306] P.K. Townsend, Four lectures on M theory, 1996. hep-th/9612121. [307] P.K. Townsend, M theory from its superalgebra, 1997. hep-th/9712004. [308] M. Trigiante, Dualities in supergravity and solvable Lie algebras, PhD thesis, University of Wales, 1998. hep-th/9801144. hep-th/9801144. [309] A.A. Tseytlin, On non-Abelian generalization of Born}Infeld action in string theory, Nucl. Phys. B 501 (1997) 41}52. hep-th/9701125. [310] C. Vafa, Evidence for F theory, Nucl. Phys. B 469 (1996) 403}418. hep-th/9602022. [311] C. Vafa, Gas of D-branes and Hagedorn density of BPS states, Nucl. Phys. B 463 (1996) 415}419. hep-th/9511088. [312] C. Vafa, Instantons on D-branes, Nucl. Phys. B 463 (1996) 435}442. hep-th/9512078. [313] C. Vafa, Lectures on strings and dualities, 1997. hep-th/9702201. [314] E. Witten, String theories in various dimensions, Nucl. Phys. B 443 (1995) 85. hep-th/9503124; hep-th/9507121. [315] E. Witten, Bound states of strings and p-branes, Nucl. Phys. B 460 (1996) 335}350. hep-th/9510135. [316] E. Witten, D. Olive, Supersymmetry algebras that include topological charges, Phys. Lett. B 78 (1978) 97. [317] G. Zwart, Matrix theory on nonorientable surfaces, Phys. Lett. B 429 (1998) 27}34. hep-th/9710057. [318] B.L. Julia, hep-th/9805083. [319] A. Gregori, C. Kounnas, P.M. Petropoulos, Nucl. Phys. B 537 (1999) 317. hep-th/9808024. [320] R. Flume, Ann. Phys. 164 (1985) 189. [321] M. Baake, M. Reinicke, V. Rittenberg, J. Math. Phys. 26 (1985) 1070. [322] I. Antoniades, B. Pioline, hep-th/9902055. [323] T. Banks, W. Fischler, L. Motl, J. High Energy Phys. 1 (1999) 18. [324] A. Gregori, C. Kounnas, P.M. Petropoulos, hep-th/9901117. [325] A. Gregori, C. Kounnas, J. Rizos, hep-th/9901123. [326] S. Chaudhuri, G. Hockney, J.D. Lykken, Phys. Rev. Lett. 75 (1995) 2264. hep-th/9505054. [327] A. Sen, Int. J. Mod. Phys. A 9 (1994) 3707. hep-th/9402002. [328] A. Sen, Phys. Lett. B 329 (1994) 217. hep-th/9402032. [329] N. Marcus, J.H. Schwarz, Nucl. Phys. B 228 (1983) 145. [330] A. Sen, Nucl. Phys. B 434 (1995) 179. hep-th/9408083. [331] A. Sen, Nucl. Phys. B 447 (1995) 62. hep-th/9503057. [332] O. Ganor, hep-th/9903110.

Note added in proof Boundaries of M-theory moduli space. As we discussed in Section 2, M-theory arises in the strong coupling regime of type II string theory, and reduces at low energy to 11D supergravity. It is important to determine what portion of the M-theory moduli space are covered by these weakly coupled descriptions, and thus what room is left for truly M-theoretic dynamics. The techniques we developed in Section 4 allow us to easily answer this question, "rst addressed in Ref. [314] for compacti"cations down to D54, and recently for D"2 in Ref. [323]. We "rst consider the case D'2, and consider an asymptotic direction in the moduli space, represented by an arbitrarily large weight vector j in the weight space < , see Section 4.6. Modulo U-duality, j can be chosen B> in the fundamental Weyl chamber j ) a'0 for all positive roots a. This corresponds to choosing R (R (2(R ,   B

R R R 'l ,    N

224

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

where the inequalities are understood to be large inequalities, in order to have a maximal degeneration in the moduli space [314]. The 11D supergravity description is valid provided all radii are larger than the Planck length, i.e. l (R . N  On the other hand, when the radius R is much smaller than l , we can have a type IIA description  N with weak coupling g"(R /l ), provided all radii are larger than the string length l"l/R : Q  N Q N  IIA: R (l , R R 'l .  N   N If this is not the case, then we may instead try a type IIB description with weak coupling g "R /R , same string length l"l/R and 10th radius R "l/R R . The IIB radii R and Q   Q N  N   R ,2, R are larger than the string length provided R R (l and R R 'l , and it is not  B   N   N di$cult to see that, using the above relation, the "rst implies R (l , and the second is automati N cally satis"ed. The type IIB description thus hold in the region 11D SUGRA:

R (l , R R (l .  N   N The weakly coupled 11D supergravity, type IIA and type IIB descriptions therefore cover, up to U-duality, the entire asymptotic moduli space of M-theory on ¹B, d'2. Of course, these descriptions fail when any of the large inequalities above become approximate equalities, hence the need for a more fundamental de"nition of M-theory. On the contrary, when D42, there are asymptotic sectors of the moduli space where no perturbative description is possible. Indeed, the weight space < is now intrinsically Minkovskian, and the light-cone j ) j"!(x)# (xG)"0 separates B> the moduli space into three sectors that can never be related to each other by U-duality. For instance, the above 11D supergravity region has xG'x/3 for all i, so that j ) j'0. It therefore sits in the interior of the future light-cone if x'0, or past light-cone if x(0 (one may choose x"0 by working in l units). In fact, the time-like region can be shown to have a weakly coupled 11D N supergravity, type IIA or type IIB description, whereas the spacelike region can be argued to be cosmologically forbidden by the holographic principle [323]. U-duality in theories with 16 supercharges. In this review, we have concentrated on dualities of maximally supersymmetric theories obtained by toroidal compacti"cation of M-theory. Our techniques extend in a straightforward way to theories with 16 supersymmetries, since the two-derivative e!ective action, which lies at the basis of the symmetry and duality maps, is still protected from quantum corrections. These vacua can be obtained in a variety of ways, including toroidal compacti"cation of the E ;E heterotic and the Spin(32)/9 heterotic or type I string,    compacti"cation of type IIA or IIB strings on a K surface, or orientifolds of type II strings, and all  these constructions are conjectured to be perturbatively or non-perturbatively dual descriptions of the same vacuum (there also exists `reduced ranka models with a smaller number of vector multiplets which are not equivalent to the ones above [326], but we shall ignore this possibility here). These vacua exhibit duality symmetries analogous to the U-duality of M-theory, which are most easily understood in the framework of heterotic string compacti"ed on a torus ¹B. The latter has a perturbative T-duality symmetry group O(d, d#16, 9) acting as an automorphism of the even self-dual compacti"cation lattice C , and identifying points in the moduli space B B> 1>;SO(d, d#16, 1)/[SO(d);SO(d#16)]. Here, the 1> factor denotes the T-duality invariant dilaton g "g (lB /< , where g and l are the heterotic coupling and string scale and < the & & B &B & & B IIB:

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

225

volume of ¹B. The factor 1> is enhanced to a Sl(2, 1)/;(1) factor in four dimensions, where the Kalb-Ramond two-form B can be dualized into an axionic "eld b and combined with the IJ four-dimensional coupling into a modular parameter S"b#i< /g l . Accordingly, there is  && a non-perturbative symmetry SP!1/S [327,328] which transforms the heterotic string length and coupling (for b"0) as < g l g P  . l P & &, & & g l < &&  Compactifying further on a circle of radius R , all 30 vectors can be dualized into scalars, while it is  easy to see that the symmetry (1) exchanges the 3D dilaton 1/g "< R /g l with the radius in &   && heterotic units R /l [330]. The latter also mixes with the six internal radii under the T-duality  & O(7, 23, 9), so that S-duality combines with the perturbative T-duality into a larger U-duality O(8, 24, 9), acting on the moduli space SO(8, 24, 1)/[SO(8);SO(24)] [329,330]. As in the case of M-theory on ¹B, d'8, the symmetry becomes an a$ne OK (8, 24, 9) [331] in D"2 and culminates for D"1 in the hyperbolic group DE [332] with Dynkin diagram 

where the numbers denote the dimension D at which the root starts appearing. From the point of view of type II strings, this U-duality is already non-trivial in D"6, where it implies the mirror symmetry SO(4, 20, 9) of K , and D"5, where it is a genuine non-perturbative symmetry. The  S-duality corresponding to the root 4 is completely perturbative on the type II side, where it corresponds to a T-duality. On the heterotic side, it implies gauge symmetry enhancement at particular values of the 3D coupling, analogous to the perturbative enhancement at the self-dual radius R "¸ .  & U-duality representations and Unitarity. As we have shown in Section 4, BPS states carry charges that transform under "nite-dimensional representations of the non-compact U-duality group E (1), or rather its discrete subgroup E (9), and one may worry whether this con#icts with BB BB unitarity, which requires in"nite-dimensional representations. In fact, the states in the Hilbert space are not labelled by weights "j 2 but by arbitrary integer combinations " mGj 2 taking value in G G the weight lattice. The Weyl generators simply permute the charges mG (up to phases), but the Borel generators induce spectral #ow that can increase mG arbitrarily, thus yielding an in"nite-dimensional representation. Likewise, the fact that the usual bound on the level for a$ne U-duality symmetries does not hold as noted at the end of Section 4.12 does not con#ict with unitarity.

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

227

ELECTRODYNAMICS OF NEUTRON STARS

F. Curtis MICHEL , Hui LI
Department of Space Physics and Astronomy, Rice University, Houston, TX 77251, USA
Theoretical Astrophysics, T-6, MS B288, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO

Physics Reports 318 (1999) 227}297

Electrodynamics of neutron stars F. Curtis Michel *, Hui Li
Department of Space Physics and Astronomy, Rice University, Houston, TX 77251, USA,
Theoretical Astrophysics, T-6, MS B288, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Received December 1998; editor: M.P. Kamionkowski

Contents 1. A brief history of radio pulsars 2. Our basic approach 3. Gravitating and rotating conductors 3.1. Rotating spherical conductor in B "eld 4. Fields in the aligned case 4.1. The `induceda electric quadrupole 4.2. Rigid corotation 4.3. The required central charge 4.4. The total charge of the star 4.5. The required external quadrupole 4.6. Surface charge redistribution: the electrosphere 4.7. Earnshaw's theorem 4.8. The totally "lled magnetosphere (TFM) solution 4.9. Numerical solutions 4.10. Pair production 4.11. Image charge 4.12. Boundary conditions 4.13. Relevance to pulsars 5. The orthogonal rotator 5.1. The induced electric quadrupole 5.2. Conducting star; a second `induceda electric quadrupole 5.3. Rigid corotation 5.4. Plasma distribution about the orthogonal rotator

230 231 232 233 235 236 237 237 239 239 240 241 241 242 243 245 245 247 248 248 249 251 252

6. The inclined rotator 6.1. Summary of external near "elds 6.2. Locus of trapping regions 6.3. Trapping regions 7. The Deutsch "elds 8. The particle motions 9. Pickup and acceleration in planar waves (exact) 9.1. Nonrelativistic case 9.2. The textbook solution 9.3. Particle pickup from rest (nonrelativistic case again) 9.4. Nonlinear terms 9.5. Relativistic Lorentz force 9.6. Covariant version 10. Relativistic exact solutions 10.1. Circular vs. linear polarization 10.2. Resonance with a static uniform B 10.3. Motion in space 10.4. Particle not starting at rest 10.5. Plasma dispersion e!ects 11. Motion in realistic "elds 11.1. Decreasing wave amplitude (still planar) 11.2. hO0 case 11.3. Motion in spherical waves 11.4. Motion in Deutsch "elds 11.5. Role of a non-zero B P

* Corresponding author. E-mail addresses: [email protected] (F.C. Michel), [email protected] (H. Li) 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 0 2 - 2

252 253 254 255 255 263 264 265 266 267 268 269 270 272 273 274 276 277 278 278 278 281 282 288 290

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297 11.6. 11.7. 11.8. 11.9.

Ions, positrons vs. electrons Plasma dispersion e!ects Comparison with MHD wind solutions Radiation reaction and quantum corrections

292 293 294

12. Conclusions Acknowledgements References

229 296 297 297

296

Abstract Although the standard model for radio pulsars is a rotating magnetized neutron star and the vast majority (if not all) of pulsars are thought to have appreciable inclination angles between the spin and magnetization axes, most theoretical papers use simpli"ed "eld models (e.g., aligned spin axis and magnetic dipole axis). Deutsch long ago gave exact (in vacuo) closed expressions for these inclined "elds (modulo some typos and oversights), but these expressions were rather clumsy and required extensive hand processing to convert into ordinary functions of radius and angle for the electromagnetic "elds. Moreover, these expressions were e!ectively written down by inspection (no details of the derivations given), which leaves the reader with little physical understanding of where the various electric and magnetic "eld components come from, particularly near the neutron star surface where many models assume the radio emission is generated. Finally, rather little analysis of what these "elds implied was given beyond speculation that they could accelerate cosmic rays. As pulsar models become more sophisticated, it seems important that all researchers use a consistent set of underlying "elds, which we hope to present here, as well as understand why these "elds are present. It is also interesting to know what happens to charged particles from the star that move in these "elds. Close to the star, ambient particles tend to simply E;B drift around the star with the same rotational velocity as the star itself. But far from the star, charged particles are accelerated away in the wave zone, as was "rst pointed out by Ostriker and Gunn. We expand their calculations using more general "elds and elucidate the particle's dynamics accordingly. Very e$cient acceleration is observed even for particles starting at '10 light-cylinder distances. We also stress the e!ects of a non-zero radial magnetic "eld. Electrons are accelerated to much higher energies than, say, protons (not to the same energy as when the two cross a "xed potential drop). We pay particular attention to particles accelerated along the spin axis (particles that might be involved in jet formation). An important limitation to the present work is the neglect of collective radiation reaction. Single particle radiation reaction (e.g., Compton scattering of the wave #ux) is not an accurate estimate of the forces on a plasma. We are working on remedying this limitation.  1999 Elsevier Science B.V. All rights reserved. PACS: 97.60.Jd; 97.60.Gb

230

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

1. A brief history of radio pulsars The "rst radio pulsar (then simply `pulsarsa before similar objects radiating in X-rays were found) was discovered in 1967 and published after some delay [3]. Although the list of known pulsars is now of the order of 700, the original discoveries are still being observed in detail because they are among the brightest. New discoveries usually require intensive computer processing to pull their signals from the noise, and it is correspondingly laborious to study their properties. In general, pulsars (we will not discuss the X-ray versions, which are thought to radiate by an entirely di!erent mechanism) are bright at low frequencies (e.g., 400 MHz) and dim at high frequencies (e.g., 4000 MHz). The pulses typically consist of one or more `componentsa closely packed together and separated by a very much longer interval of weak to non-detectable emission. At present, the shortest interval between successive pulses is about 1.5;10\ s and the longest is about 5 s. Electrons in the interstellar medium delay the lower frequency parts of the pulse relative to the higher frequencies, and removing this relative delay (i.e., e!ectively lining up the pulses) yields the `dispersion measurea, which is simply the line-of-sight path-integrated electron density, which in turn yields a rough distance estimate. There seems to be little or no observable di!erence between the fastest and the slowest, which pretty much excludes two of the three usual astronomical suspects for causing periodicity: orbital motion or physical oscillation in size. The shortest periods are within a factor of about 2 (or less) of the fastest that a nucleon-degenerate collapsed star (`neutrona star) can rotate. If the power source is the rotational energy, then one expects and indeed "nds that the periods of all pulsars are increasing. The only (so far) plausible reason for a tiny (a solar mass within 10 km) dense #ywheel to lose energy is if it is magnetized and is emitting magnetic dipole radiation. An elementary estimate then gives a magnetic "eld of about 10 gauss (G) for the magnetization. The fastest (`milliseconda) pulsars necessarily have much weaker "elds of about 10 G (if they had the stronger "eld, they would spin down too quickly to hang around to be observed still spinning that fast). A histogram of numbers versus magnetic "eld suggests a bimodal distribution clustered about these two values. Why there should be two distinct preferred magnetic "eld strengths is a lively topic of speculation. `Weak "eld pulsara would arguably be a better term than `millisecond pulsara, but it is rare for astronomical terminology to overcome casual usage. The term `neutron stara is similarly a bit of a misnomer since there are a number of compositional transitions expected with depth, but the usual theoretical estimates have neutrons outnumbering protons by about 8 : 1. The main reason is that if the nucleons are resisting gravity by degeneracy pressure, electrons at the same densities have huge Fermi temperatures, which drive electron capture by the protons (inverse beta decay) to remove electrons and favor neutrons. One expected source of neutron stars is in the collapse of the cores of massive stars (one supernova model) and indeed some young radio pulsars are found inside of some young supernova remnants. Pulsar ages are estimated from the observed slowing down, PQ , and the usual rule of thumb, q +P/PQ , which for the speci"c case of magnetic dipole radiation is q "P/2PQ .   The discovery of a pulsar within the Crab Nebula, generally thought to be the remnant of a historical supernova in 1054 AD, qualitatively solved the mystery of why continuum optical

 Unless contaminated by external gravitational forces on the neutron star.

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

231

radiation from that nebula is highly polarized and what powers it; evidently a magnetized plasma #ows from the pulsar (not simply large amplitude electromagnetic waves in vacuo) and accumulates within a conducing shell of ejecta (visible in line emission). This general deductive analysis by elimination of what pulsars are (rotating magnetized neutron stars) seems broadly consistent with such observations.

2. Our basic approach The obvious question (`Why are they emitting radio waves?a) has not yet been answered. A series of theoretical models have been forwarded, which have either been found to wrong or to be charitably described as `incompletea. A generic (and non-committal) model would pose that there are some sort of discharges within the pulsar magnetosphere. We are then seeing these discharges rotate in and out of view. A serious theoretical problem is that a bright low frequency source from a tiny object vastly exceeds the black body emissivity at any plausible `temperaturea and the spectrum is exactly the inverse of a hot black body (falling with frequency instead of rising). The radiation must then be coherent to begin with and then it must pass through a highly transparent medium to avoid being thermalized. A simple model having these properties would be discharges that bunched particles of one sign. At long wavelengths, the bunch radiates intensely as a single (highly charged) particle and at short wavelengths one eventually gets incoherent radiation from the individual particles (weaker emission by a factor of N, the number of particles in the bunch). This behavior is qualitatively just the spectral signature seen. The next alternative is some sort of maser action excited by counter-streaming in the (putative) discharges. Although we are not without preferences [4], such details are far beyond the basics we hope to lay out here. Most phenomenological models start with the assumption that discharges exist, usually posited to be con"ned to the vicinity of the magnetic poles, where the B "eld is usually taken to be dipolar with the magnetic axis inclined to the rotation axis. Our approach here is to start at the opposite limit, namely assume by contradiction that the inclined rotator has no discharges, try to "nd such solutions, and hope to "nd a paradoxical inconsistency. We would then know where discharges are required and why. For example, in the case of perfect alignment (an attractive theoretical simpli"cation) we already "nd quiescent solutions with no obvious requirement for discharges other than one-shot transient ones as the system relaxes to a more stable con"guration. This "nding suggests that inclination is not only required to produce the `light-housea rotation of the discharges (i.e., the pulses), but is essential for maintaining the activity to begin with. Not everyone is happy with this situation, and a few theorists still try to get an aligned rotator to do something. But in 15 years, not a single researcher in the "eld has come forward and shown how an inactive solution might be transformed into an active one. Although our subject of interest is astrophysical in nature, the application closely parallels some interesting paradoxes in laboratory physics. In classical mechanics, the motions of bodies is

 More complicated "elds are an obvious possibility, but pulsars are so ubiquitous (birthrates + supernova rates) that we cannot require much beyond the simplest topology.

232

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

essentially trivial until one comes to the rotational motions, which require much more care in understanding. In the same way, electromagnetism is fairly straightforward even when uniform motions are added, but when rotation is added, textbook treatments are either non-existent or sometimes muddled at best [5]. For example, is rotating the observer equivalent to rotating the system, as it is for uniform translation? (No!) Does anything happen if you rotate a bar magnet about its axis? If you rotate a conductor and suddenly stop it, will a current #ow? If you magnetize an object, is it set into rotation? If you stand a conductor in a gravitational "eld, do the conduction electrons `settlea toward the bottom? Some of these questions may be familiar, some may seem exotic. When radio pulsars were discovered in 1967, an intense observational and theoretical e!ort soon zeroed down on these astrophysical objects as being rotating neutron stars having huge magnetic "elds [4,6]. We are therefore back to one of the above questions, what happens if you rotate a magnet? [7]. An essential "rst step to understanding neutron star electrodynamics is an understanding of how the (assumed) conducting interior adjusts to rotation through its own magnetic "eld. Although the rotating magnetized neutron star model for pulsars seems a stable paradigm, the theoretical situation has been one of ups and downs, with very plausible models having proven to be unphysical. Given that three decades have passed without a theoretical breakthrough, it seems likely that enough di!erent things are happening in connection with the generation of (highly coherent) radio emission from pulsars that it is correspondingly unlikely that one can simply guess at what is happening. Accordingly, it seems implausible that we can understand how pulsars function without "rst gaining some elementary levels of understanding of what should be happening in their vicinity.

3. Gravitating and rotating conductors For rotating magnetized neutron starts, the Lorentz forces (on an electron, say) are huge compared to the gravitational forces, which in turn are usually orders of magnitude larger than the centrifugal forces. Nevertheless, these inertial forces are often dominant. A star such as our Sun can be regarded as a fully ionized plasma and hence an excellent conductor. The simplest model for a conductor is the free electron model. If we apply this model to the Sun, it would predict that the free electrons would attempt to `falla to the center of the Sun. Such motion would produce a radial charge-separation electric "eld that in steady state would have the value E "!m g/"e" (wrong) . P C

(1)

In other words, the Sun would become negatively charged to keep the electrons from falling inwards against gravity. In fact, this estimate has the wrong sign and is several orders of magnitude

 Even near the so-called light cylinder distance, where it is often erroneously assumed that particles would `havea to rotate at the speed of light were they not slung away by the centrifugal force.

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

233

too small! Both the ions (mainly protons) and the electrons are better approximated as being gases, and under thermal equilibrium the electrons move much faster than the ions. A better approximation would be to assume that the electrons are essentially unbound gravitationally, in which case the Sun needs a positive charge to retain them (or will become positively charged by their loss). The situation becomes more extreme when we examine electron-degenerate stars (white dwarfs) and nucleon-degenerate ones (neutron stars) where the large electron Fermi energies require even larger stellar charges to retain the electrons. If we now examine rotating stellar objects, we run into some semantical ambiguity of how we would exert a torque on either gas. It is easier to return to the laboratory scale where we can directly rotate (say) a metal sphere. Now the torques are distributed among the ions by the lattice forces. But in the free electron model the electrons do not see the lattice forces, and therefore the lattice will try to rotate through the electrons. The resultant ion current will produce a timedependent (growing) magnetic "eld, and the accompanying induction electric "eld will then oppose the lattice rotation and transfer some of the torque to the electrons. Once a new steady state rotation has been achieved, a weak magnetic "eld of order X+u "eB/m (2)  C is roughly su$cient to tie the electrons to the lattice. For the fastest known pulsar (X+4000 rad/s), this "eld would be only about 2;10\ G. Ohmic dissipation would eventually remove this "eld (and any need for it, since the particles are now all corotating). The point here is to emphasize that self-gravitation and rotation make for some deviations from our classic views of `classicala bulk objects. Adding magnetic "elds makes for even more profound changes. 3.1. Rotating spherical conductor in B xeld We now consider the case of a rotating spherical conductor in B "eld. The simplest symmetry is for the rotation axis to parallel a uniform external magnetic "eld. Although we assume the sphere to be conducting, the usual boundary condition at the surface, U"const. ,

(3)

is incorrect. If the potential on the surface were constant (as appropriate for stationary conductors), the electric "eld inside would be identically zero (the desired result for a stationary conductor). But the magnetic "eld inside a rotating conductor provides an unbalanced Lorentz force F"$eV;B ,

(4)

where V"X;r

 To our knowledge this toy problem has not been worked out exactly.

(5)

234

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

is the rotational velocity. The consequence of this force is to separate the charges axially. Such a charge separation will clearly lead to an electric "eld of some sort. This e!ect is well-known in plasma physics, where it is usually termed the MHD (MagnetoHydroDynamic) condition and written variously as E#V;B"0

(6)

E;B . V" B

(7)

or

The latter is more general since it is always possible provided that E(cB, while the "rst is only possible if additionally E ) B"0. The appearance of this electric "eld is perplexing to some physicists given that there is no induction, which is exactly correct. The resultant electric "elds are indeed curl free. The short answer is that the conduction charges need to see both a B and an E "eld such that the E;B particle drift exactly matches the rotation rate of the star (we are largely repeating arguments in [4, Section 4.2]). If this electric "eld were absent, the Lorentz force on the corotating charges would move all the electrons outward (for the choice B ) X'0) and all the ions inward, which would immediately create an electric "eld. But one does not need to follow the time-dependent particle dynamics, and their consequences, one simply requires of the interior that it ends up with E ) B"0, one MHD condition. If we hook a stationary wire between the pole and equator of this rotating sphere, current will #ow; this is simply a (fat) Faraday disk. Some textbooks even "nesse this point (e.g., [8]) by imagining that one can transform into a rotating frame where the sphere is stationary but now the wire is moving in the magnetic "eld and `cuttinga lines of force. If the magnet were an insulator (which is entirely possible in the laboratory, since magnetization arises from electron spin and not from electron conduction) the Lorentz force would be zero to "rst order on the neutral constituents. In their frame of reference the atoms would see an electric "eld which would cause some "nite polarization and the latter would produce some (much weaker) electric "eld to be seen in the stationary frame. In the case of pulsars, even this e!ect would probably produce interestingly strong electric "elds, but we will follow convention and simply assume the neutron star to be conducting, as it is expected to be from other arguments. Altogether then, the Lorentz forces do the `inducinga, and the terminology comes from the `unipolar inductora: the battery-like action of a metal disk rotating in an external magnetic "eld when the rim is shorted to the center, as shown in Fig. 1. Actually, there is a deep irony here because this rotating disk was apparently "rst suggested [10] by Faraday (and sometimes called a `Faraday diska), presumably as an example of Faraday's law of induction!

 We generally reserve upper case for bulk quantities and lower case for particle attributes, unless esthetically distracting (e.g., coordinates).  A somewhat dangerous assumption given that the rotating frame is non-inertial [9].

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

235

Fig. 1. Faraday disk, a rotating disk in a magnetic "eld develops a potential between the axis and the periphery, an elementary dynamo.

In the case of radio pulsars, the magnetic "elds we will be interested in are estimated to be of the order of 10 G. This estimate originally came from the assumption that the magnetic "eld is orthogonal and that the neutron star (an object nominally of about 1.4 solar masses and about 10 km radius) is slowing down owing to magnetic dipole radiation as if radiating into a vacuum [2]. The reader can simply regard such numbers as scaling estimates, quite apart from detailed justi"cation. In any event the now-standard estimate } for all practical purposes the dexnition } is [11] B"3.2;10(PPQ )  G

(8)

for the pulsar period P in seconds, with PQ being the dimensionless slowing down rate. Pulsars have periods in a wide range but mostly near one second, while the characteristic slowing down time is a few millions of years and hence a representative PQ +10\ giving a magnetic "eld of about 10 G. Consequently, we will operate in a quite di!erent parameter space from that of rotating magnets in the laboratory! Note also that the age of the galaxy itself is more like 10 years, and therefore there must be a very large number of no-longer-seen pulsars.

4. Fields in the aligned case First we discuss the vacuum electrodynamic "elds expected about rotating magnetized stars and touch on some of the non-vacuum cases, where the neutron star is cloaked with (non neutral) plasma. The model for the interior magnetic "eld is irrelevant to the (assumed dipolar) external "eld. A simple choice is to simply imagine that there is a point dipole at the center of the star. Alternatively, the interior "eld could be modeled as being uniform, although no one expects that either model would hold for an actual pulsar. Neither model should make any di!erence to what is happening in the exterior around the star, but certain points are easiest made in one or the other

236

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

picture. If a central dipole is aligned with the rotation axis, the "elds are simply a B "2B cos h (all r) , P  r a sin h (all r) , B "B F  r

(9) (10)

where a is the stellar radius, and for positive B the magnetic "eld points up out of the North  (h"0) pole, exactly opposite to the magnetic "eld of the Earth. Rotation of the conducting star through its own magnetic "eld (or perhaps less confusing, the rotation of each electron through the magnetic "eld created by all the other electrons) forms in essence a `fata Faraday disk. Here, B is  the "eld strength at the equator, which is also that estimated in Eq. (8). 4.1. The `induceda electric quadrupole There are a number of ways of solving for the internal electric "elds required to transform the corotational motion of the particles into E;B drift. One is the observation that in steady state the magnetic "eld lines must be equipotentials so that the electric "eld has no parallel component with which to drive currents (again, the so-called MHD condition in plasma physics). This condition can certainly be imposed on simple models if not necessarily on an active pulsar, where currents would be expected to circulate. Indeed, the simplest possibility works exactly, namely where the electrostatic potential inside is given by a U"U sin h (inside star) , r

(11)

where r"¸ sin h

(12)

is an equation for dipole magnetic "eld lines, ¸ being the equatorial (maximum) distance of the magnetic "eld line. Here the scaling requires (as we will establish) U "B aX . (13)   Again, B is the equatorial (dipole) magnetic "eld strength, exactly the quantity estimated in Eq. (8).  Here positive X in Eq. (13) corresponds to counter-clockwise rotation as viewed looking down along the North pole, as is the case for the Earth's rotation. However, positive B is opposite to that  of the Earth but similar to that of Jupiter. These choices of sign (both positive or both negative) are a popular theoretical choice if it is assumed that radio emission comes from electrons accelerated over the magnetic polar caps. But this assumption gets us ahead of our discussion.

 Ironically, many theorists have labored for years under the impression that this estimate corresponded to the polar magnetic "eld strength!

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

237

It is direct to calculate that the resultant internal electric "elds from the potential in Eq. (11) a E "U sin h (inside) , P  r

(14)

a E "!2U sin h cos h (inside) , F  r

(15)

which from Eq. (9) and (10) give E ) B"0. The interior "elds therefore satisfy our requirement that they do not accelerate corotating particles. 4.2. Rigid corotation As a "nal check, let us calculate the drift velocity V ,E;B/B inside the star. Only the " azimuthal component is nonzero, a (E;B) "U B sin h(1#3 cos h) , (   r

(16)

and since a B"B (4 cos h#sin h)  r

(17)

we "nd U r (18) < "  sin h"rX sin h " B a  as required for rigid rotation. This exercise is useful for checking that a consistent de"nition of U has been used (especially with at least one factor of 2 #oating around, namely whether B is the   equatorial or polar magnetic "eld strength). The di$culty with this solution, which corresponds to a global solution, is the "nite extent of the star. The required charge density is a o"e  ) E"2U e (1!3 cos h) ,    r

(19)

and the angular distribution is recognized to be the Legendre polynomial P (cos h), characteristic  of a quadrupole distribution. But while it is necessary to have this charge distribution, which would arise from charge separation inside the conducting star, the solution is not su$cient when the charge distribution is truncated at the stellar radius, a. The missing charge outside the star is still required to give E ) B"0 inside the star and must be replaced with something else that does just that. But before resolving that point, we should make one additional point about the internal charge distribution. 4.3. The required central charge A quadrupole charge distribution (Eq. (19)) has no net charge, but the 1/r dependence of U in Eq. (11) requires a monopole moment, in addition to the distributed quadrupole. The value of the

238

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

charge is simply given by integrating E in Eq. (14) over a sphere using Gauss's law which gives P Q"8ne U a/3 , (20)   hence 2 a U " U .

  3  r

(21)

The existence of this central charge is without question, although it is often neglected even in some theoretical papers. But the above evaluation does not explain why this central charge is required. This required central charge is more explicit if we turn to the alternative internal magnetization model: a uniform internal magnetic "eld (B "2B ). Now E;B corotational drift requires an axial X  electric "eld which points toward the axis and increases linearly with axial distance, E "!2XB o (inside) , M  corresponding to a uniform internal charge density

(22)

o"e  ) E"!4e XB ,    which gives (again) the net charge within the volume of the star

(23)

16n e XB a"!2Q (inside) . Q "!  3  

(24)

Now that the magnetic "eld changes from the uniform "eld inside to pure dipole outside, the associated electric "eld changes from Eq. (22) to Eq. (14). Such a transition demands a `surfacea charge which is p"e (E!E )"3e XB a sin h"e XB a[2#(1!3 cos h)] . (25)  P P     Integrating this over the whole sphere, we get a required total charge for making the transition to be Q "8ne U a"3Q . (26)    Thus, this `surfacea charge proportional to sin h can be understood in terms of the sum of a uniform positive charge (Eq. (26)) and the accompanying quadrupole distribution. These then cause the star to have an e!ective `centrala charge which is Q"Q #Q and the `internala   quadrupole moment as if it simply had a point dipole at the center as indicated by Eq. (20) in the model where the magnetic dipole is shrunk to a point at the stellar center. As a note of clari"cation, the `surfacea charge given in Eq. (25) is di!erent from the usual de"nition of surface charge since it is required by the transition of uniform to dipolar "eld

 Even though this expression is given as an inside "eld, it is the correct expression for the immediate outside electric "eld also since the electrostatic potential is directly proportional to the magnetic "eld line function for a rigidly rotating star.

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

239

con"gurations and cannot be altered. As discussed in the following sections, there will also be the induced surface charge whose motion is subject to the surface electric "elds, which can (and will) #y o! the star. 4.4. The total charge of the star The important point here is that the net internal charge Q of the star is not a free parameter but is determined by the magnetic dipole "eld and the condition E ) B"0 inside the star. If the star is not charged initially, a neutralizing charge !Q will consequently appears on the surface. This neutralizing surface charge is again subject to the surface electric "elds, which, as we will show later, are so strong that no surface charges could be maintained. Then these surface charges will #y o! the star. If one considers a whole system consisting of the star and its surrounding plasma, then the surrounding plasma must contain a net amount of charge !Q to ensure charge neutrality. But nothing said that the star had to be initially neutral, so the net system charge is actually arbitrary. The casual assumption that the overall system charge be zero probably contributes to theorists sometimes neglecting the intrinsic internal stellar charge Q. We will discuss the e!ects of varying total charge on the distribution of plasmas surrounding a star in the following sections. 4.5. The required external quadrupole So far we have the elements of a solution, but no complete solution if the quadrupolar charge distribution is truncated beyond r'a. For a magnet (sometimes called a `terrellaa if in the form of a sphere) rotated in the laboratory, the rest of the solution is no mystery, it simply corresponds the above pieces: the central charge and the quadrupolar space charge inside the sphere plus a surface charge. The surface charge is itself required to be quadrupolar since it replaces the missing external quadrupole charge distribution plus an arbitrary uniform surface charge (e.g., add !Q if zero total charge desired). The requirement for this external quadrupolar charge distribution follows again from the condition E ) B"0 inside the star. Let's just go through the derivation in small steps. At the surface, we can reinterpret the distributed quadrupole as a vacuum internal quadrupole generated by the external distributed charge and a vacuum external quadrupole generated by the internal distributed charge. In other words (here we normalize U a,1 and r"1 is the surface)  sin h A B " # (1!3 cos h)#Cr(1!3 cos h) , (27) r r r where the A term is the central monopole, B is the quadrupole potential due to charge separation inside the star, and C is the vacuum quadrupole potential due to charges external to r (e.g., a surface charge). What we are trying to determine here is what happens if the third component is absent, but for the moment we will assume it to be there. The coe$cients A, B, and C are determined by the condition that the above equation holds and that E ) B"0 on the surface (owing to the di!ering r-dependences, the "elds of each are di!erent). Then we have for the potential at the surface (r"1) sin h"A#B(1!3 cos h)#C(1!3 cos h)

(28)

240

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

or A"2/3 and B#C"1/3. Each electric "eld component has a di!erent dependence on r, but must give the same electric "eld at r"1, so we can determine the coe$cients by di!erentiating the potential, Eq. (27) with respect to r and then setting r"1: !sin h"!A!3B(1!3 cos h)#2C(1!3 cos h)

(29)

which again requires A"2/3, but 3B!2C"1/3, hence B"1/5 and C"2/15. We are now prepared to give the resultant electric potential for the case that the external distributed charge is missing. For example, if the rotation were of a spherical magnet in the laboratory, the distributed charge outside of the surface would simply be replaced by a quadrupolar surface charge that gave the same Cr(1!3 cos h) potential inside. Then at every point, E ) B"0 inside the magnet. If instead we go to the case of a neutron star with nominal 10 G "eld, radius of 10 km, and a rotation rate of 6 rad/s, the induced electrostatic "elds are of the order of 6;10 V/m. Not only is this a huge "eld, but we can see that an electron would be accelerated to relativistic energies over a few microns. Consequently, it has become common to assume that the work function of the neutron star is e!ectively zero, whereas for laboratory parameters the work function of a rotating terrella is e!ectively in"nite. A neutron star would then di!er fundamentally from the Earth in that there is no `grounda available. The neutron star has just the net charge Q and no other charges can be stored on it to vary that number. And no surface charge can be maintained! How then can we have E ) B"0 inside without a surface charge? Clearly we then need an external quadrupolar space charge around the star.

4.6. Surface charge redistribution: the electrosphere One way to provide the external quadrupole is simply to imagine that the internal charge separation simply extends beyond the star and to in"nity. Now instead of the surface charge (terrella solution) we simply have the quadrupole charge separation density everywhere, even where there is no conductor to supply it. Mathematically this solution is "ne but not physically. If we imagine the neutron star to be spun up, with the surface charge appearing brie#y on the surface before being torn o!, what would we expect? Again with our sign convention, positive XB  corresponds to negative particles over the poles and positive particles in the equatorial plane. The positive particles would then appear at the equator and would have no place to go beyond following, to "rst order, the magnetic "eld lines. Since these lines are tightly closed, the positive particles could depart no further than about half a radius away from the surface. The quadrupole charge changes sign at cos h"1/3 or sin h"2/3 and from the dipolar "eld line equation Eq. (12) we have ¸"3/2. The negative particles also move on closed magnetic "eld lines (except from the exact poles) and we will now show that there is a trapping region right above the poles. It is easiest to locate the trapping zone in the case where we start with the surface charge on the star, even if this will not be the "nal state. Then the power law dependence on C in Eq. (27) above to r\ gives the

 We will show that the electric "elds act to pull surface charge o! regardless of sign.

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

241

external "eld from the surface charge, and the potential is then (1!3 cos h) 2 . U" # 3r 3r

(30)

We then satisfy E ) B"0 inside the star, but now there is a trapping region starting at r"a(3 above the pole (where RU/Rr"0) and arching to the surface. This trapping region is created by the net charge that the neutron star has, which is positive and attracts the electrons back to the star. However, the electrons are accelerated o! the star by the quadrupole "eld, so they simply move out until the quadrupole "eld dies o! su$ciently with distance (much faster than the monopole, of course). Thus any negative charges that might be introduced into the polar regions, or any negative charges that might escape the surface charge, will simply be trapped. The natural expectation of single charged plasma trapping is that the plasma forms a non-neutral plasma, a topic of extensive research in the laboratory [12}14]. Determining the exact distribution of such plasma is, however, a somewhat di$cult task. But "lling only portions of the volume about the neutron star with charge suggests the generic name `electrospherea. Early pulsar theory obsessed with the quadrupolar electric "eld `driving the electrons awaya, and neglect of the central positive charge guaranteed that the trapping feature would be missed. In the same way, the equatorial plane is also a locus of E ) B"0 and in this case positive particles would accumulate here (repulsion by the central charge driving them to the most distance points on their magnetic "eld lines). In general, non neutral plasmas collect initially at loci of E ) B"0. 4.7. Earnshaw's theorem A standard proof in electrostatics texts is that one cannot trap particles in a static electric "eld because in the space outside of point charge, we have

U"0

(31)

and such a U cannot have a local maxima or minima. But this theorem is not very robust, given that it fails if the slightest assumption is changed, such as adding other forces or introducing time dependence. Thus an electron is trapped in a classical atom (centrifugal force), in a Penning trap (magnetic force), in a Paul trap (oscillating electric "elds), and in an aligned rotating magnetized neutron star (huge magnetic "elds). The electrostatic trapping is only along the magnetic "eld lines, but the magnetic "eld itself inhibits escape in the tangential directions. 4.8. The totally xlled magnetosphere (TFM ) solution Historically, this complexity was entirely missed and it was originally proposed [15] that the quadrupolar charge density inside the star simply continued as such to `in"nitya. The putative source of the external space charge was again particles from the surface. Unfortunately the apparent advantages of this assumption outweighed the de"ciencies, because then this external quadrupolar charge distribution would have to rigidly corotate with the star (that was the boundary condition to begin with) and therefore at a distance c/X a physically impossible region

242

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

would be encountered beyond which the plasma particles would have to rotate faster than the speed of light. This distance was interpreted as the transition to a centrifugally driven wind-zone, which is evident in terminology that labels this distance as being the `light-cylinder distancea, `cylindera referring to locus where centrifugal force would become arbitrarily large in relation to the spin axis. Since a wind would drain particles from the surroundings, mainly on magnetic "eld lines leading to the polar regions, it was additionally assumed that charges would have to be accelerated from the poles to replace those lost. Much of the subsequent work concentrated on how such a polar acceleration region might produce coherent radio emission (the radio luminosity of radio pulsars being far too large to be incoherent and yet be emitted from such small areas of a neutron star). This diversion of e!ort away from fundamentals helped the model persist, and even today a few workers still struggle to try to make it work [16}18]. The fact that the particles drained from the polar regions would all be electrons (so assumed because accelerated electrons radiate most easily), and therefore that charge conservation should be violated, was regarded as a technical detail to somehow be "xed in this otherwise promising model. To a newcomer, it might seem astonishing that } in models for an actual radio pulsar, which surely must be an arcing, discharging, electromagnetic monster } something as comparatively benign in e!ect as centrifugal forces were imagined to be the dominant ones! 4.9. Numerical solutions The electrosphere would generically be a dome of (negative) charge over each magnetic pole and a torus of (positive) charge around the equatorial zone. If these zones were to be expanded as much as possible, one should asymptotically approximate the entirely "lled magnetosphere solution. The dome and torus con"guration has been shown in numerical simulations [7] and reproduced in Fig. 2, which shows the case in which the net charge in the electrosphere is zero (thus the total system charge is again Q). The simulation was intentionally unsophisticated, with the surface charge replaced by a series of charged rings on the surface, which were released one at a time and allowed to "nd an equilibrium position. As more rings are released, the surface charge has to be readjusted to keep E ) B"0 inside the star. But as more charge accumulates outside, the electrosphere itself starts to remove the need for a surface charge and ultimately the simulation terminates naturally when there is no more surface charge to be removed. As we have been discussing here, it is fairly obvious that `startinga the star with a surface charge and then releasing it will result in the positive particles from the equatorial zone simply popping out a short distance and being trapped by the intense closed magnetic "eld lines there. If the electrons are approximated as following magnetic "eld lines, there is only one "eld line on which they could escape, but that point is moot because there is a trapping region for electrons directly above the polar caps, which they occupy to form the domes. The build-up of the dome and torus are shown in Fig. 3. The electrons are attracted by the positive intrinsic charge of the star but repelled by the rotationally induced quadrupole electric "eld, which would drive them o! the surface to begin with. But the monopolar electric "eld component will always dominate at some "nite distance, thus they cannot escape. And the positive particles, which are repelled from the star, are trapped on the tightly closed equatorial "eld lines. The plasma density is far too low (+10 cm\ [4]) for interchange instabilities to grow. Escape seems only possible if the total

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

243

Fig. 2. Non neutral space charge distribution about an aligned rotator, as originally calculated [7].

Fig. 3. Formation of the disk/torus electrosphere. The "rst "gure is after one step, the second after 25 steps, and the "nal "gure has terminated after 164 steps.

system charge is somehow driven to zero, which corresponds approximately to a con"guration with all dome (extending to in"nity) and virtually no torus. 4.10. Pair production Even before the discovery of pulsars, it was realized [19] that gamma rays could convert into electron}positron pairs in strong magnetic "elds. In active pulsars (as opposed to these idealized models) it is entirely possible that such pair production is essential in providing current carriers to power the activity. The important constraint has been that particles were assumed to be available only from the surface (positive particles would then be ions). If pair production e!ectively creates particles in vacuum regions (positive particles would then be positrons) the surface-origin

244

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

constraint is lifted. Such a source of ionization would surely modify the dome/torus con"gurations [20]. The process usually envisioned is energetic particles radiating curvature radiation to produce the gamma rays, which then convert to provide more energetic particles. In some models the successive particles are proportionately less energetic and in others an electric "eld maintains their energy. However, such a process is tightly localized to the vicinity of the neutron star because (1) it requires an intense magnetic "eld and the dipole "elds weaken rapidly (1/r) with distance, and (2) because the surfaces of the dome and torus are E ) B"0 surfaces, so any accelerating electric "eld will vanish if the system tries to "ll up and asymptotically approach the TFM solution. Consequently one expects modi"cation of the dome/torus con"guration, but it does not seem a plausible mechanism to drive activity in the aligned model [20]. Given that these non-neutral plasma distributions may be unfamiliar, it is probably worthwhile to go over Fig. 2 in some detail. First, notice that the dome and torus truncate abruptly as is evident even in this coarse discrete-particle simulation. These vacuum-to-plasma discontinuities are typical of non-neutral plasmas and contrast sharply with the tendency of quasi-neutral plasmas to "ll all space by ambipolar di!usion along magnetic "eld lines. The charge density does not feather o! smoothly to zero. Notice that the magnetic "eld lines (dashed) cut across these surfaces; it is not necessary that the "eld lines parallel the surfaces as is a possibility for stable density discontinuities in the quasi-neutral case. Next the decline in density of the dome is evident with increasing height. The charge distribution here is exactly that same as for the TFM (totally "lled magnetopause) solution Eq. (19) but truncated at the discontinuities. The magnetic "eld lines have to be equipotentials where the plasma `shortsa them electrically to the surface, and the dome plasma must also rigidly corotate with the star. The equipotential lines are the solid ones and smoothly joint on to the magnetic "eld lines (dashed) inside the dome, as required, but outside they become more like spherical shells about the center of the star indicating the dominance of the central charge. The density should increase somewhat toward the axis, but our representation in terms of charged rings gives the opposite appearance, since the rings e!ectively represent wedges of charge density. The surrounding vacuum, which "lls essentially all space is labeled `Ha after a work by Holloway [21], wherein he pointed out that if one removed equatorial positive particles from the TFM solution on magnetic "eld lines leading to the (assumed negative) polar regions, there is no plausible way that they could be replaced and therefore a gap would have to form. It turns out that some solutions are all gap! The torus is a bit more complicated and one can see what appears to be a dark band in the middle of it. What is happening here is that the magnetic "eld lines leading to the star only cut through part of the torus, and again here the charge density has to be the same as the TFM solution and the motion has to be corotation. But the outer parts are on magnetic "eld lines that pass through vacuum before reaching the surface and these are `open-circuiteda. Consequently, the plasma on the open-circuited magnetic "eld lines does not match that of the TFM solution, and the plasma here rotates somewhat faster than the star itself. Far from being a nuisance, this behavior is essential! If both the dome and torus had the same density as the TFM solution, then the internal quadrupole moment that they would generate would be less than that

 Holloway's argument is misunderstood in numerous recent publications: it would be invalid if any ionization process were present in the gap, but one can "nd Holloway cited as justixcation for gaps having just such action taking place.

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

245

given by the TFM solution and the latter provides exactly that required for E ) B"0 inside of the star. We have thrown e!ectively away both positive and negative particles from the TFM solution and the quadrupole moment must fall, unless there is an additional charge density added somewhere (here, the super-rotating part of the torus). Thus the system automatically "nds a somewhat sophisticated solution. On the other hand, the trapping has counter-intuitive properties. For example, a stray negative charge introduced near to the negative dome is `attracteda to the dome (actually it is attracted to the central positive charge more than it is repelled by the dome electrons). Ditto for positive particles above the torus. Moreover, these charged volumes cannot generally be neutralized. If a positive charge were introduced into the negative dome, it would see no macroscopic parallel electric "eld in "rst order, but in second order, the electrons have to be held up against gravity, so there will be a weak parallel electric "eld present to do this. But the very same "eld will accelerate the positive charge to the surface. Alternatively, the positive particle could have enough velocity to drift up out of the dome in which case it would be accelerated to the torus. The dome cannot be `neutralizeda. 4.11. Image charge The way conductivity is handled here di!ers slightly from the standard textbook application in inertial systems. There the conductor is simply taken to be an equipotential (E"0 inside). For our rotating systems, we have instead E ) B"0 inside. Given those conditions, in either system the introduction of a new test charge would produce image charges to appear on the surfaces of the conductors. But there is no discussion of image charges insofar as the charges in the dome and torus are concerned. The di!erence is in the assumption that charges can always leave the surface, so any invocation of image charges is therefore a temporary one, with the "nal state free of such charges. Thus the dome/torus con"guration could, if one wished, be thought of having an image charge on the stellar surface (this would be a perfect quadrupole) in addition to which the star itself has the quadrupole surface charge needed to keep E ) B"0 inside. The two image charges are equal and opposite, corresponding to the resulting solution having no image charge. In electrostatics, there are no parallel space charge con"gurations owing to Earnshaw's theorem: they cannot be stable. 4.12. Boundary conditions In numerical simulations, one actually has at least two limiting choices. One strategy is to keep the surface charge and release it (numerically) to form the electrosphere. But since the quadrupolar moment from the charges diminishes as they recede from the star, it is necessary to keep adding to the surface charge distribution as well, to keep E ) B"0 inside the magnet (star). Thus a number of release cycles are necessary to asymptotically approach an equilibrium solution. The other choice, which is simpler in the aligned case, is just to discard the surface charge, and use the radial electric "eld as an indicator of how much surface charge there should have been and create the appropriate charges there (simulating currents #owing to the surface, not the creating of charge per se). Now one just continues, in either case, until no surface charge or no radial electric "eld is present, and one has driven E ) B"0 in the interior and by force balance E ) B"0 where ever there are charges.

246

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

The only di!erence is the starting electric potential, which is






2 a 1 a U "U # (1!3 cos h) 1!  3 r 3 r

(32)

if the surface charge is present to begin with (E ) B"0 inside the star), the external (vacuum) electric "elds would be 2 a a (1!3 cos h) (outside) , E " U #U  r P 3  r

(33)

a E "!2U sin h cos h (outside) . F  r

(34)

These "elds give E ) BO0 at just above the surface (since we pass from E ) B"0 inside through this surface charge), and we "nd again from Eq. (9) and Eq. (10), setting r"a, that 4U B E ) B"   cos h(1!3 cos h) (surface) , 3a

(35)

showing (as advertised earlier) that the forces act to remove whichever sign of surface charge sitting on the surface. Alternatively, we could instead assume the surface charge is missing in which case






2 a 1 a # (1!3 cos h) , U "U ,  3 r 5 r

(36)

but the star is `nakeda and now does not have E ) B"0 in the interior. The disadvantage here is that it is less clear where and in what order charges should be introduced, and since we can have full families of solutions depending on the overall system charge (the central charge plus whatever excess might reside in the electrosphere) one risks getting a slightly di!erent equilibrium solution. In the case where we start with the surface charge, we can simply add a second uniform surface charge (which is invisible to the interior) and then start the process, which results in a unique family of solutions [22]. In Fig. 4 we show how the dome and torus vary as one changes the overall system charge. These con"gurations are each unique because the charges all start at the surface and follow magnetic "eld lines to their equilibrium positions. Unless one adds additional sources of ionization (most popularly, the pair-production cascades) or somehow allows particles to cross magnetic "eld lines, one will always get the same one-parameter family of con"gurations. In any case, once the electrosphere has been set up, the surface charge is gone and E ) B"0 in the interior. For reasons special to the oblique rotator (to be discussed next), we shall prefer the choice in which the star initially has the required surface charge, which is subsequently released. As a "nal point, we show what happens when a totally "lled magnetosphere is truncated at some "nite distance. Since truncation removes space charge that is essential for E ) B"0 in the interior, the resultant con"guration is immediately rendered unstable and collapses into the dome and torus con"guration, as shown in Fig. 5.

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

247

Fig. 4. Dome and torus for varying system charges. The central charge is taken to be #15 units.

Fig. 5. Collapse of the totally "lled magnetosphere when truncated. The line in the "rst "gure is the zero charge density line separating minus from plus charges.

Even if we were to truncate the TFM at the `light cylindera, it would simply collapse to a stationary dome/torus con"guration. Consequently, the TFM is not a solution of the aligned model! 4.13. Relevance to pulsars From time to time the aligned model(s) have come under attack for the wrong reasons, namely that they would not pulse, owing to the axial symmetry. But pulsation has generally been assumed to be due to a "nite angle between spin and dipole axis. The key assumption was that the physics would be `more or less the samea, despite such misalignment. Now however, the assumption seems quite the opposite (at least in these quarters), namely that the physics will be signi"cantly di!erent! If an aligned rotator simply functions as an ion/electron trap on a huge scale, it is not too promising as a pulsar model. Curiously, a number of ide& es xxes are left over from the aligned model: (1) radio emission comes up out of the poles, (2) centrifugal forces are dominant, and (3) that it would even be active. The "rst consequence of dropping axial symmetry is that another important force is to be

248

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

found at the `light cylindera, namely the ponderomotive force of the large amplitude electromagnetic waves on the particles surrounding the pulsar. Indeed, energy radiated by such waves is where the magnetic "eld estimates (Eq. (8)) came from in the "rst place. However, if one assumes a wind, then the dipolar "eld lines would be forced open and the magnetic energy carried away by the wind gives exactly the same scaling. Thus, the energy density at the light cylinder scales as B , where  B "B a/R , with R ,c/X, and the energy #ux at c through the surrounding area 4nR scales      as BaX, exactly the same as for dipole radiation. Given the zeroth order defects in the aligned  model insofar as pulsar function goes, we will examine the other limit of the inclined rotator, an orthogonal rotator.

5. The orthogonal rotator Here we calculate the electric and magnetic "elds near the star for the case that the dipole and spin axes are orthogonal. Tilting the magnetic "eld (modeled as a central point dipole) through 903 gives the "eld components [4, Section 5.2], a sin h cos( !Xt) (all r) , B "2B P  r

(37)

a B "!B cos h cos( !Xt) (all r) , F  r

(38)

a B "B sin( !Xt) (all r) . (  r

(39)

Now however the time dependence of the magnetic "eld directly induces electric "elds as the next step of approximation (`truea induction if one wishes to make that distinction). 5.1. The induced electric quadrupole These induction electric "elds (which are now only part of the total electric xeld) are E "0, P

(induction "elds only) (all r) ,

(40)

a E "!U cos( !Xt) (all r) , F  r

(41)

a E "U cos h sin( !Xt) (all r) . (  r

(42)

These equations are not full solutions of Maxwell's equations but only the leading terms in the power series in rX/c, which must terminate in the 1/r wave "elds. Thus we implicitly assume aX/c;1 here, although it is not necessary to make this approximation, and further on we will give the full solutions, including the radiation "elds, that would "ll all of space if the space around

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

249

a rotating magnetized neutron star were a perfect vacuum (only possible if surface charge is retained, as we have seen). Rather than derive the above components, we can simply check them, as for example starting with the -component RB ( ;E) "! ( (all r) ( Rt

(43)

we obtain from Eqs. (39) and (41) a a cos( !Xt)"XB cos( !Xt) (all r) U  r  r

(44)

so we require once again the boundary condition U "XaB . The remaining induction equations   can easily be shown in directly the same way to be satis"ed. The time-dependence of these electric "elds themselves give the displacement currents that provide the magnetic part of the outgoing wave "elds, but we neglect that near the star. Because we will no longer need to take into account the time-dependence of the "elds, we will write !Xt, where is the longitude on the star, with "0 the longitude of the magnetic    moment of the star. We have not yet included the fact that the star is a conductor. 5.2. Conducting star; a second `induceda electric quadrupole One can see that the star being a conductor is important by calculating for the interior (as well as exterior) "elds that (Eqs. (38), (39), (41) and (42)) a cos h (all r) , E ) B"B U   r

(45)

and therefore the induction electric "eld would drive currents in the star. Consequently, an internal quadrupole must appear to kill o! this non-zero E ) B below the surface. Again, this electric "eld appears spontaneously because it is `induceda by the non-zero Lorentz force that would be present otherwise. The required quadrupolar potential can be shown from direct calculation to be a U"!U sin h cos h cos (inside) r 

(46)

which gives the additional electric "elds a E "!U sin h cos h cos (inside) , P  r 

(47)

a (sinh!cosh) cos (inside), E "!U  F  r

(48)

a E "!U cos h sin (inside) . (  r 

(49)

250

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

If we now calculate just the part of E ) B contributed by these "elds, we "nd a cos h (inside) . E ) B"!B U   r

(50)

Thus we see that the desired term canceling Eq. (45) has been obtained, and again the internal MHD condition will be satis"ed. Just as in the case of the aligned rotator, the internal quadrupole "eld requires a charge separation everywhere inside the star, while in fact the conducting star ends at the surface. In exactly the same way (except here there is no monopole component because this potential averages to zero), we conclude that we need an internal quadrupole (b) and an external quadrupole (c) such that sin hcos h cos

sin h cos h cos

#cr sin h cos h cos (inside) "b !  r r

(51)

so immediately (at r"1) b#c"!1 and di!erentiating with respect to r sin h cos h cos

sin h cos h cos

#2cr sin h cos h cos

"!3b  r r

(52)

giving !3b#2c"1, and hence b"!3/5 and c"!2/5. At this point we will examine just these leading near-"eld terms. Again from the point of view of numerical simulation, one has the choice of providing a surface charge to replace the missing space-charge beyond the star or simply leave the surface naked and tolerate a "nite E ) B inside the star until the electrosphere is set up. It proves to be somewhat less confusing to provide the surface charge because the external quadrupole "eld then cancels the induction "eld at the magnetic polar caps and thereby removes a nonzero component of E ) B at the surface of the star that cannot actually pull particles from the star. Including such a term can make assessing where particles are removed on the basis of the surface value of E ) B very confusing, but this is a technical point mainly of interest to someone doing numerical simulations. If we keep the surface charge, then outside of the star we have a potential a sin h cos h cos (outside) U "!U  1!  r

(53)

which di!ers only in the power of r necessary for a vacuum solution external to the star. This di!erence alters the external electric "elds to a sin h cos h cos (outside) , E "!3U  P  r

(54)

a (sin h!cos h)cos (outside) , E "!U  F  r

(55)

a E "!U cos h sin (outside) , (  r 

(56)

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

251

and when we add these to the induction electric "elds (starting with Eq. (47)), we "nally get a sin h cos h cos (outside) , E "!3U  P  r







a a cos 2h! cos (outside) , E "U  F  r r



a a ! cos h sin (outside) . E "U  (  r r

(57) (58) (59)

If we now calculate the value of E ) B just above the surface we get 4U B E ) B"!   sin h cos h cos (outside) .  a

(60)

Thus the action is to pull the surface charge o! the surface and exactly matches where the surface charge is and what sign it has ( just as we obtained in the aligned case). Surface charge, regardless of sign, is always pulled ow. The charges appear at the surface in the "rst place because (by assumption) they cannot go any further. And there has to be a surface to terminate any rotating system lest material move faster than c. The total electric "eld inside is then the sum of the induction "eld and the quadrupole "eld, hence a sin h cos h cos (total int. "eld) (inside) , E "!U  P  r

(61)

a E "!2U sin h cos (inside) , F  r 

(62)

E "0 (inside) . (63) ( Notice that E vanishes at h"0 both inside and outside, as needed to eliminate a `surfacea F contribution to E ) B from the induction electric "eld. 5.3. Rigid corotation Let us now calculate V ,E;B/B inside the star, for the special case of "elds in the "0 "  plane (for reasons to become apparent) a sin(1#3 sin h) (inside) (E;B) "U B (   r

(64)

a B"B (1#3 sin h) (inside)  r

(65)

while

252

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

giving U r (66) < "  sin h"rX sin h (inside) " B a  and again rigid corotation of charges with the star, as required. In the orthogonal plane (cos "0), the electric "eld vanishes and the magnetic "eld is parallel  to the corotational velocity vector. There is no corotational E;B drift. Here the charges corotate for the same reason that they would in an unmagnetized conducting sphere, not because the Lorentz force gives them a drift velocity but because they get in the way of their corotating neighbors. Corotation is not simply E;B drift in the orthogonal (or inclined) case. 5.4. Plasma distribution about the orthogonal rotator The problem of numerically simulating the non-neutral plasma distribution about the orthogonal rotator have not yet been addressed. This problem lacks, of course, the symmetry that reduces a three-dimensional simulation to a two-dimensional one. Qualitatively it is clear that the quadrupolar surface charge will again leave the surface but not escape the system. Thus what was two domes and a torus for the aligned case now deform and the torus splits in two to give a total of 4 domes of alternating charge girding the star. For purposes of illustration, we can repeat the `GJa treatment [15] by again assuming that the internal charge-separation density continues right through the surface to become a space-charge density and that the entire space becomes the `insidea solution. In the divergence of the (inside) electric "eld, only Eq. (62) contributes and we immediately get the (inside) charge-separation density [23,24] a sin h cos h cos . o"!6e U    r

(67)

If we simply extend this solution to everywhere outside the star aH la` GJ, we would de"nitely get an active system since the plasma would be driven out of the system by wave acceleration near the wave zone. Moreover, equal numbers of positive and negative charges could be driven away (`solvinga the current closure problem). But at the same time, the polar cap charge densities and accelerating "elds vanish at the polar caps. Note again that o is directly proportional to B : X o"!2e XB . (68)  X Work is in progress to do these simulations for the orthogonal and also inclined rotator discussed below. Since pulsars must have (according to the conventional wisdom) inclined dipolar magnetic "elds to even act as rotating `light-housesa, the inclined case showing the transition between the two limits is of particular interest.

6. The inclined rotator An inclined magnetic dipole is simply a linear superposition of the aligned and orthogonal limits. The only possible misstep here concerns E ) B"0 inside the star because now we have two

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

253

components to each, E"EA#EO (Aligned and Orthogonal) and B"BA#BO, so we need to con"rm that the cross-terms EA ) BO#EO ) BA"0 as well. Although the individual terms EA ) BA and EO ) BO vanish, the later condition is not guaranteed in general. However, it should hold because each pair is orthogonal and in the same ratio < . Thus for the orthogonal electric "eld " components (Eq. (61)) and aligned magnetic "eld (Eq. (9)) we have a EO ) BA"!2U B sin h cos

  r 

(69)

while for the aligned electric "eld (Eq. (14)) and oblique magnetic "eld (Eq. (37)) we have a EA ) BO"2U B sin h cos

  r 

(70)

and we see that indeed one can superimpose the two "elds while retaining rigid corotation and zero "eld-aligned electric "elds inside the star. 6.1. Summary of external near xelds Here we list the electric and magnetic "elds outside the surface of a rotator inclined an angle m (see Fig. 6) with respect to the rotation axis and having a canceling surface charge that guarantees E ) B"0 inside the star. Note that even though the interior satis"es the MHD condition, the exterior does not because one passes through the surface-charge layer. Thus if one creates and releases particles from the surface, the resultant internal quadrupole will violate the MHD condition and the surface-charge distribution will have to be reduced or modi"ed. A simple numerical strategy would be to calculate, after having released a number of particles, their quadrupole moment inside the star and appropriately reduce that due to surface charges until

Fig. 6. Inclined rotator with magnetic moment directed along M, rotating about the inertial axis z relative to axes x and y. The inclination angel m is "xed as changes. The usual polar angle h is measured from z (not illustrated).

254

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

(to some "nite accuracy) one cannot release another charge (each `chargea being a huge one compared to the electron charge, of course). The magnetic "elds are given from Eqs. (9) and (37), a (cos m cos h#sin m sin h cos ) , B "2B  P  r

(71)

a B "B (cos m sin h!sin m cos h cos ) , F  r 

(72)

a sin m sin . B "B  (  r

(73)

The associated electric "elds are then given from Eqs. (33) and (57)






a a a 2 E "XaB cos m #cos m (1!3 cosh)!3 sin m sin h cos h cos , P  3  r r r




 

a a a cos 2h! cos , E "XaB !2 cos m sin h cos h#sin m  F  r r r






a a E "XaB sin m ! cos h sin

(   r r

(74)

(75)

(76)

6.2. Locus of trapping regions In the case of E;B (or at least E ;B), it is well known that particles mainly experience , acceleration along the B "eld (JE ) B) while the acceleration perpendicular to B averages to zero. Hence, E ) B"0 de"nes the force-free surfaces. If we concentrate on the forces just above the surface of the star, the values of E ) B simplify signi"cantly and we get 3a E)B "cosm cos h(1!3 cosh)#sin m cos m sin h(6 sin h!5)cos

 4U B   !3 sin m sin h cos h cos . 

(77)

If we wish to know the distribution in space above the surface, then we have (for "0)  M#N

a "0 , r

(78)

where M" cos m(cos m cos h#sin m sin h)!sin m(cos m sin h!sin m cos h) ,  N"!4 cos m cosh!sinm cos h(5!4 cosh)#sin m cos m sin h(1!8 cos h) .

(79) (80)

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

255

Fig. 7. Cross sections of force-free surfaces (thick solid lines) around a star (thin solid circle) using vacuum "elds for various angles between spin and magnetic dipole axes (dashed line). An electric dipole from the central charge  cos m is  also included. As more and more charges are `releaseda from the star surface, they will modify the "elds around the star and `"ll upa these force-free surfaces. Consequently, `domesa and `torusa will form instead.

The locus of E ) B"0 is shown in Fig. 7. Notice the di!erence between our results and those in Thielheim and Wolfsteller [25] where they omitted the central charge. Note that even in this most general case, the locus of zero charge density is the locus of zero B , since X o"!2e XB (81)  X once again. 6.3. Trapping regions Notice in the aligned case that the trapping locus over the magnetic polar caps is a circle quite close to the surface. This trapping locus is what anchors the domes. In the equatorial plane, the plane is the trapping surface, and the positive particles would all accumulate here were it not for their self-repulsion so they form a torus instead [7].

7. The Deutsch 5elds The full solution of the outside "elds generated by a rotating magnetized sphere in vacuum with angle m between the spin and magnetic dipole axes (Fig. 6) has been given by Deutsch [1], with the (implicit) assumption that surface charges can somehow be maintained. Here, we give his original

256

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

solutions (in complex variables) for the sake of completeness (adopting changes in notations and correcting several typos)

  
  




a h /o cos m cos h#  sin m sin h e ( , B "2B P  r (h /o)  ? a o o h B "B cos m sin h# h #  sin m cos h e ( , h # F  r   oh #h h o   ?  ? o o h B "B h #  i sin m e ( , (82) h cos 2h# (  oh #h   h o   ?  ? h 1 a o  sin m sin 2h e ( , cos m(3 cos 2h#1)#3 E "E ! P  o 2 r oh #h   ? o oh #h a h   cos 2h!  sin m e ( , E "E ! cos m sin 2h# F  oh #h o r (h )   ? ? o oh #h h   !  i sin m cos h e ( , E "E (83) (  oh #h o (h )   ? ? where o"rX/c, a"aX/c, " !Xt, and E ,XaB . h , h and h , h are spherical Bessel        functions of the third kind with argument o and their derivatives, respectively. The subscript a means that the functions enclosed are evaluated at o"a. Speci"cally, we have






 


 

 
 

  
 
 





 



1 i e M , h " ! !  o o 2 2 1 h " #i !  o o o

e M ,

1 3 3 h " ! #i !  o o o h " 

e M ,

9 9 1 4 ! #i ! o o o o

e M .

(84)

It is apparent that the "elds are a sum of the aligned component (terms with cos m) and the orthogonal component (terms with sin m e (). Furthermore, orthogonal component radiates, thus all of them have a phase term e (>M\? where the extra e M\? term comes from the spherical Bessel functions (e.g., terms such as h /(h ) ). Hence, the "elds (F) can be written as  ? F"F(aligned)#F(dipole)#F(quadrupole) , (85)

 The analysis here is largely repeating Section 5.2 of Ref. [4], except for a few typos in Ref. [4]: Eq. (16d) should be multiplied by a factor of 2 and the multiplication factors (before and after Eq. (25)) di!er for B and E by a factor of a (cf. our Eq. (88)).

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

257

where F(aligned) is given as a B "2B cos m cos h , P  r a cos m sin h , B "B F  r B "0 , ( a E "E cos m(1!3 cosh) , P  r a E "!E cos m sin 2h , F  r E "0 , (

(86)

F(dipole) is h /o  sin m sin h e ( , B "2B P  (h /o)  ?







h o h #  sin m cos h e ( , B "B  F  h o  ? o h h #  i sin m e ( , B " (  h o  ? E "0 , P h  sin m e ( , E "!E F  (h ) ? h  i sin m cos h e ( E "!E (  (h ) ? and F(quadrupole) is B "0 , P





 

o B "B h sin m cos h e ( , F  oh #h    ? o h cos 2h i sin m e ( , B "B  (  oh #h   ?

(87)

258

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297






 



h o  sin m sin 2h e ( , E "E 3 P  oh #h o   ?

 

o oh #h   cos 2h sin m e ( , E "E F  oh #h o   ? o oh #h   i sin m cos h e ( . E "E (  oh #h o   ?

(88)

Finally, adding the electrostatic monopole discussed above, we can recast Deutsch's solutions Eqs. (82) and (83) as, after taking real parts, a +cos m cos h#sin m sin h[d cos t#d sin t], , B "2B   P  r a +cos m sin h!sin m cos h[(q #d )cos t#(q #d )sin t], , B "B     F  r a sin m+![q cos 2h#d ]cos t#[q cos 2h#d ]sin t, , B "B     (  r

 

(89)



a 3 a 2 cos m# cos m(1!3 cos h)! sin m sin 2h[q cos t#q sin t] , E "E   P  r 3 r o



a a ! cos m sin 2h#sin m[(q cos 2h!d )cos t#(q cos 2h!d )sin t] , E "E     F  r r a sin m cos h+(q !d )cos t!(q !d )sin t, , E "E     (  r where t" #o!a (again, the extra term o!a comes from the Bessel functions) and  ao#1 , d "  a#1 o!a d " ,  a#1 1#ao!o , d "  a#1

 A subroutine on solving these "elds around a neutron star is available by sending an email to [email protected].

(90)

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

259

(o!1)a#o d " ,  a#1 3o(6a!a)#(3!o)(6a!3a) , q "  a!3a#36 (3!o)(a!6a)#3o(6a!3a) q " ,  a!3a#36 (o!6o)(a!6a)#(6!3o)(6a!3a) q " ,  o(a!3a#36) (6!3o)(a!6a)#(6o!o)(6a!3a) , q "  o(a!3a#36)

(91)

where all the d and q terms come from F(dipole) and F(quadrupole), respectively. For those who G G prefer Gaussian units over SI units, change E to E /c in all the above equations.   We will now discuss these "elds in near (o&a) and far (oa<1) regions. First, in the near zone, the leading (considering the comparative importance) terms are d +1, 

d +1, 

q +a/2, 

q +(a/o), 

which give (noting that t+ )  a (cos m cos h#sin m sin h cos ) , B "2B  P  r a B "B (cos m sin h!sin m cos h cos ) , F  r  a B "B sin m sin , (  r 






2 a a a cos m #cos m (1!3 cos h)!3sin m sin h cos h cos , E "E  P  3 r r r




 

a a a E "E !2 cos m sin h cos h#sin m cos 2 h! cos , F   r r r






a a E "E sin m ! cos h sin .  (  r r

(92)

These are the same as Eq. (71)}(76), which show that the nearby B "elds are dominated by the aligned component and part of the dipole component (J1/r), whereas all components are present in E "elds.

260

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

The asymptotic far "elds (o<1) can be obtained by noting that the only leading components are d +o, 

d +!o 

(93)

which give us a X B "2B sin m sin h sin t , P  r c a X B "B sin m cos h cos t , F  r c a X sin m sin t , B "!B (  r c a 2 E "E cos m , P  r 3 a X E "!E sin m sin t , F  r c a X E "!E sin m cos h cos t . (  r c

(94)

Except for the electric monopole due to the central charge, these are the standard textbook wave "elds. Fig. 8 depicts some open "eld lines when m is nonzero. In Fig. 9, we show the deviation from static dipole magnetic "eld lines when the star is rotating by starting at the same footpoints on the star surface. Fig. 10 shows the role of electric dipole from the central charge. Fig. 11 compares the shape of the polar cap of a tilted rotating dipole with that of an aligned dipole. The polar cap is de"ned by the footpoints of those magnetic "eld lines whose maximum distance to the rotation axis is R . Fig. 12 shows the selected closed and open "eld lines of an orthogonal rotator and how they  `crossa the light-cylinder distance. Fig. 13 shows the structure at large distances. In the plane perpendicular to the rotation axis and containing the rotating dipole, curiously, there are only two open "eld lines which are the Archimedes spirals. As we will show immediately below, these two spirals are nearly the minimum of "B" at each r. The rest of "eld lines (with o<1) all appear to converge to these two `nulla lines. This behavior is rather counter-intuitive as one generally expects all open "eld lines in this plane spiral around to in"nity. This derives from the consideration that the dominant B term falls as 1/r while B drops as 1/r, so the "eld line equation gives P ( dr/d "const., which is a spiral. To show this explicitly, using Eq. (94) for an orthogonal m"n/2 rotator and in the plane of h"n/2, one actually "nds 2 sin t "!2 oB /B "! P ( sin t

(95)

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

261

Fig. 8. Single polar "eld lines (h"m and "0) for m"53, 453, and 853 from top to bottom, respectively. These "eld lines (as well as their near neighbors) are presumably the `opena "eld lines that go out to in"nity. The rotation axis is up and the magnetic axis is tilted to the right. All dimensions are in units of R and parameters similar to Crab pulsar are  used.

262

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

Fig. 9. 3-D and a side-view (2D) of two closed "eld lines (solid lines) starting with "0 and h+m$(a/R )  from a titled rotator (m"453). The dashed lines represent the case if the star is not rotating. The rotation axis is up and the magnetic axis is titled to the right. All dimensions are in units of R and parameters similar to Crab pulsar  are used. Fig. 10. The electric "eld lines near the star with (upper panel) and without the central charge (lower panel). Here, m"453, "0 and two panels have the same set of initial h angles. It is clear that the radial component from the electric dipole `pushesa the "eld lines out. The rotation axis is up and the magnetic axis is titled to the right (dashed line). All dimensions are in units of R and parameters similar to Crab pulsar are used. 

which implies that "*o"""*r/R ""2* . 

(96)

In other words, the distance between the same "eld line is 4n instead of 2n as expected for one wavelength. Hence the casual expectation that all open "eld lines spiral around is not right. To do this correctly, one has to realize that the `correctiona terms in B become important when sin tP0.

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

263

Fig. 11. The shape of the polar cap for m"453. The magnetic axis is at (0, 0) and pointing out of the paper, and the rotation axis is tilted to the left. Filled dots indicate the footpoints of the "eld lines with the maximum distance to the rotation axis being R . Solid line represents an aligned dipole case. Parameters similar to Crab pulsar are used.  Fig. 12. Selected closed and open "eld lines for an orthogonal rotator. All "eld lines are in the plane perpendicular to the rotation axis and containing the rotating dipole. (The dipole axis is pointing to the right.) As the distance from the star increases, the radial component of the open "eld lines decreases, so that they all appear to converge to two `nulla "eld lines, which are the locus of (nearly) minimum B. The null "eld lines form two Archimedes spirals, separated by n, and going out to in"nity (not shown here). Note that the radial component of the open "eld lines should never be exactly zero. All dimensions are in units of R and a period of 1 ms is assumed in order to resolve the size of the star. 

Using the full expressions from Eq. (89), we need the following ratio oB 2(a cos t#sin t) P"! (97) B (a#1/o)cos t#sin t ( to be !1 for an outgoing spiral. Hence, we "nd that the Archimedes spirals are only possible for speci"c t, t"arctan(!a#1/o) or

n#arctan(!a#1/o) .

(98)

These are the two spirals seen in Fig. 13. Note that with these t, B and B are nearly zero, thus the P ( spirals are indeed almost the minimum of "B" at that distance r.

8. The particle motions Having discussed the sort of "elds that we would expect about an oblique rotator in vacuo, we will examine some of the particle dynamics in such "elds. Particle acceleration (and radiation) in

264

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

Fig. 13. Shown is the only two Archimedes spirals that are allowed by the magnetic "eld structure in the equatorial wave zone. Same parameters as in Fig. 12 are used.

the intense pulsar far "elds has been an important topic over the years due to its possible implications for cosmic rays and the excitation of plerions [26,27]. These considerations have become even more important with the Hubble Space Telescope discovery that the pulsar in the Crab Nebula seems to have a jet coming out along what seems to be the spin axis, which is at "rst invisible but then excites a knot of emission very near the pulsar. At about 10 times further out the jet seems to create a shock wave, evidenced by a second knot that moves chaotically around. Beyond this apparent shock one "nd a very long jet visible most easily in X-rays [28]. The nature of the "rst knot is quite open since one would not expect a #ow to shock twice. If one uses standard estimates of magnetic "eld strength in the Nebula, one would expect that the magnetic "eld at this "rst knot to be about 10\ G. This "eld would not be a static "eld but the "eld of a circularly polarized wave being emitted by the rotating pulsar. There are important questions about the stability of electromagnetic waves from a pulsar [29,30] and one possibility is that there is wave to particle energy transfer at the distance of the "rst knot. An alternative is that there is a resonance with the gyrofrequency of the particles in a longitudinal (static) component of the magnetic "eld with the rotation rate of the waves. The latter consideration inspired a careful examination of such a process, which we review in increasing steps of detail. We start with the planar wave in order to understand particle's general behavior, then we discuss the particle motion in the full Deutsch "elds. However, we must emphasize that the following is not intended as a review of wave-particle interactions [31], a subject that ranges over a vast landscape, but simply provides a text-book level analysis to provide a standardized starting point for electromagnetic wave emission from pulsars. 9. Pickup and acceleration in planar waves (exact) A particularly simple case is when particles start at the axis of rotation of an orthogonal magnetic dipole. Along this axis, the radiated electromagnetic waves will be circularly polarized

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

265

and the huge intensity of such "elds (in the case of pulsar modeling) should sweep up any plasma near the wave zone and drive it away from the neutron star. If we approximate the wave as a plane wave, a number of interesting phenomena appears: (1) the solutions are exact and analytic, (2) the acceleration is only temporary, and the wave cyclically reabsorbs the particle energy, (3) the critical parameter in determining a particle's energy is the ratio of wave frequency and particle cyclotron frequency, (4) resonance takes place when the frequency of the wave (u) equals the particle cyclotron frequency in the steady axial "eld component, i.e., u"u because for particles starting , from rest, c(1!b )"1. , 9.1. Nonrelativistic case For a circularly polarized wave moving in the #z direction we will have oscillating E and B "elds in both the x and y directions. Thus the Lorentz force reads m

dv V"e(E #0!v B ) , V X W dt

(99)

m

dv W"e(E #v B !0) , W X V dt

(100)

m

dv X"e(0#v B !v B ) . V W W V dt

(101)

(Later we will include a static B component.) Let us select the case where the "elds rotate X counterclockwise in the x!y plane, as shown in Fig. 14 where the instantaneous E;B drift velocity would be in the #z direction, and the test particle is an electron (e"!e ). The rotating  "elds can be written (propagation in vacuo), E "cB cos , V ,

Fig. 14. The circularly polarized wave "elds as seen looking in the !z direction.

(102)

266

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

E "cB sin , W ,

(103)

B "!B sin , V ,

(104)

B "B cos , W ,

(105)

"ut!kz# , 

(106)

with

where is an arbitrary phase factor, which will henceforth be set to zero.  9.2. The textbook solution The standard analytic approach is to look for harmonic solutions. Here we have set B "0 X (we will put it back in almost immediately). The obvious solution can be gotten from inspection since the equilibrium solution will have * orthogonal to E, which means that the velocity is parallel to the wave magnetic "eld and those components of the Lorentz force vanishes. Thus the electron will only see the electric "eld and will in e!ect act like a particle on a string being swung around at the rotational rate u with the centrifugal force balancing the electric force (Fig. 15): mv u"e E"e cB , ,   ,

(107)

Fig. 15. Inhomogeneous solution for electron in rotating wave "elds. Here the velocity is orthogonal to E such that centrifugal and electric forces balance. Consequently, * parallels B and the Lorentz forces all balance with an arbitrary velocity in the z direction. But this is not the general solution.

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

267

where e is the magnitude of the electron charge (in order that we can de"ne the cyclotron  frequency as a positive quantity without negative signs) and v is the circular speed, , u (108) v "c , , , u where u is the non-relativistic cyclotron frequency at the wave's magnetic "eld, , e B u ,  ,. , m

(109)

Notice immediately that one could easily have v'c, at low enough wave frequency in which case we would need to return and do the problem relativistically. This observation is a central consideration here. Continuing with the nonrelativistic limit for the moment, we can see that the acceleration along the z axis is zero and hence v can be any constant. Explicitly writing out this X solution, we have v "!v sin , (110) V , v "v cos , (111) W , v "const . (112) X We can explicitly substitute these into the Lorentz equations and see that they are satis"ed. However, if we put some numbers in here we can see that this solution is preposterous! Even at the distance of the apparent shock along the spin axis of the pulsar (the second, much more distant knot), the expected wave magnetic "eld would be of the order of B +10\ G or an electron , cyclotron frequency of about u +2;10. Since the rotational frequency of the Crab pulsar is , roughly 200 rad/s, we see that the perpendicular velocity would have to be 200c! Obviously a non-relativistic treatment is insu$cient and the simplest reinterpretation is that the Lorentz factor should enter as a change in e!ective mass, mPcm in Eq. (107), and therefore c+200. But even with this quick "x we still have a problem: where could the electron get an energy of 50 MeV simply in order to run around in a circle? But textbooks give derivations just like the one above. What is the problem? The problem is that the answer is incomplete, it is only the inhomogeneous part of the solution. The inhomogeneous part is the part having no free parameters. Thus the particle initial condition cannot be varied. That is, of course, exactly what the homogeneous part of the solution is for. The homogeneous part is implicitly discarded at various places in many textbooks because the author is not interested in it for one reason or another (e.g., a harmonically bound electron in an electromagnetic wave to illustrate scattering, etc.). But it breeds a bad habit as illustrated here. The most likely initial condition is for a wave to overrun an electron at rest, not an electron that happens to have energy 50 MeV and moving in just the right direction. 9.3. Particle pickup from rest (nonrelativistic case again) Now consider the case where a charged particle suddenly "nds itself at rest at some phase of such a wave instead of dutifully circling. Nowhere is the particle at rest in the above solutions. But this

268

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

solution is the standard one found in standard plasma textbooks about plasma motion in a circularly polarized wave (with v "0 in addition). It is plausible that the mean velocities might X be zero before the wave is introduced. But the typical argument is that these velocities are somehow zero after the wave is introduced, which we immediately show to be incorrect. The simple answer for the motion starting from rest is that, if we see the circular motion in a moving system with the same velocity v , the particle will repeatedly come to rest (at "0). It is , immediately clear then that the x and y velocities are instead v "!v sin , V ,

(113)

v "v (cos !1) . W ,

(114)

However, now the particle has a uniform component of motion, so as it moves, it sees the alternating B which causes it to oscillate up and down along the z axis, since now the Lorentz force V reads m

dv X"e(v B )"ev B sin

, V , , dt

(115)

and therefore v "v (1!cos ) , X 

(116)

where again we need a constant term if the particle started from rest (at "0), and




u  u v "v ,"c , .  ,u u

(117)

Thus an electron is accelerated forward with the wave. This acceleration is rarely seen in the standard plasma physics textbook treatments because the mean velocities are all assumed to be zero after the wave is introduced. It is tempting to view this acceleration as being due to `radiation pressurea, but as we will see, it is really a coherent wave-particle interaction since the particle will return to rest. 9.4. Nonlinear terms Having uniform and harmonic components to v is not entirely innocuous because the term kz X depends on z which hitherto was an independent variable. A constant z velocity simply changes the value of k (nonrelativistic equivalent of the Doppler shift), but a harmonic term in k makes it time dependent, and our whole treatment becomes approximate. However, the time derivative of v now V becomes d dv dv dv V"(u!kv ) V V" X d

dt d

dt

(118)

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

269

and as long as c"u/k, the potentially nonlinear term v B cancels out because we now have X W dv m(u!kv ) V"eB (c!v )cos

X d

, X

(119)

and our harmonic solutions go through exactly as before, even when v itself is non-uniform as is X the case for particle pickup. The z component of the Lorentz force itself becomes nonlinear, but in an integrable form; dv m(u!kv ) X"e v B sin

 , , X d

(120)

which can immediately be integrated to give the quadratic equation






1 m u! kv v "e v B (1!cos ) ,  , , 2 X X

(121)

where the unity is the constant of integration needed if v starts from rest (at "0). If we neglect X the non-linear term, we already have seen that v has amplitude v , and if we label b "v /c the X  LP X above value, the relativistic value becomes 1 b"1!(1!2b +b # b #2.  2  

(122)

It may look as if this expression becomes complex for b '1/2, but in the relativistic treatment, LP m becomes cm (sort of, we will do the relativistic treatment exactly in the following section), and it is easy to show that 2b/c cannot exceed unity. As an aside, when the equations are non-linear, we no longer have the usual classi"cation of inhomogeneous and homogeneous solutions. The analog of the inhomogeneous solution is called the singular solution, and the rest are the general solutions. The singular solution has the same properties as the inhomogeneous solution: no degrees of freedom. 9.5. Relativistic Lorentz force The Lorentz force gives the time derivatives of the particle momenta, which in the relativistic treatment become cmv or equivalently mccb, so we now have mc

d(cb ) V "e(E #0!v B ) , V X W dt

(123)

mc

d(cb ) W "e(E #v B !0) , W X V dt

(124)

mc

d(cb ) X "e(0#v B !v B ) , V W W V dt

(125)

270

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

and the fourth (energy) equation is mc

dc "e(v E #v E #0) . V V W W dt

(126)

9.6. Covariant version Super"cially, the "rst three come from the Lorentz force simply by multiplying every m by c and the fourth equation simply says that work is done on the particle at rate * ) E. In covariant notation, the four equations are one four-vector equation, m

du? e " F?@u , @ c ds

(127)

where a and b run over the four time plus coordinate components. Repeated indices (here b) are summed over all four components. The four-velocity is dx? u?, & c(c, v ) & c(c, v , v , v ) , G V W X ds

(128)

where we contrast the indexed vectors with their explicit components. Here i is the corresponding ordinary mathematical index running over just the three coordinates. These indices are often numbered (e.g., 0, 1, 2, 3) for no good reason, given that ordinarily vector components are just subscripted with the obvious axis symbols x, y, z. The down-index (covariant) four-velocity just di!ers by a sign, u "c(c,!v ) , @ G which gives the invariant (scalar) equation






u?u ,c"c c! v "c(c!v) , G ? G so as usual the Lorentz factor is 1 . c" (1!v/c

(129)

(130)

(131)

Since the ordinary velocity is dx v, G, G dt

(132)

it also follows from the de"nition of the four-velocity, Eq. (128), that dt c" . ds

(133)

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

271

Note that relativity does not supplant the ordinary velocity, which keeps the meaning it has always had: how fast our coordinates are crossed according to our clocks. If *"0, dt"ds so ds is the time interval for a clock moving with (say) a particle, which by de"nition has to be the same for all observers, hence the term `propera time. `Propera velocity (or four-velocity) is how fast our coordinates are crossed according to the moving clock and is not limited to below c, a common misconception. The electromagnetic "eld tensor corresponds to the electromagnetic "elds seen in the local rest system, and then (again writing "rst time and the spatial components) FR?"(0,!E ) G

(134)

and (e being the totally antisymmetric tensor that de"nes the usual cross-products and curls) GHI






F?H" E ,!c e B . G GHI I I

(135)

(the electric "eld sign changes because F?@"!F@? and the sum is somewhat gratuitous given that k must simply be whichever i and j are not). For a"t we obtain mc

dc e " (!E )(!cv ) , H H ds c H

(136)

and since dt d d d " "c , dt ds ds dt

(137)

one set of Lorentz factors cancel leaving (after multiplying by c) mc

dc "e* ) E . dt

(138)

In the same way, for a"i we obtain






e d(cv ) e G " (cc)(E )# (!cv ) !c e B G H GHI I c c ds H I and again one set of Lorentz factors cancel to yield m

m

d(c*) "e(E#*;B) . dt

(139)

(140)

Note that Eq. (138) must follow from Eq. (140) simply by taking the scalar product of the latter with *, which leads to the unobvious but true identity that * ) d(c*)"d(c) .

(141)

272

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

10. Relativistic exact solutions Returning to the issue of analytic solutions, we see that just as in the non-relativistic case, we have factors (c!v ) that cancel out and we have that X d(cb ) V "ecB cos , (142) mcu , d

which has the exact solution cb "!g sin , V where we de"ne u g, , u

(143)

(144)

since we have seen this ratio appears repeatedly. The companion component is again cb "g(cos !1) , (145) W for particles picked up in the wave. The two equations for z-motion and energy conservation, Eqs. (125) and (126), are remarkable in that they are identical to one another within a factor of c. Thus we can divide the one into the other and obtain d(cb ) X "1 , dc

(146)

which immediately gives, for the initial conditions (particle starting from rest) b "0 and c"1, X cb "c!1 . (147) X This result provides an important constraint on the system. Since we have explicit expressions for all three components of momentum, we can use the identity c(b#b#b),c!1 V W X and substitute into the left hand side of Eq. (147) gives g(sin #cos !2 cos #1)#(c!1)"c!1

(148)

(149)

which simpli"es to c"1#g(1!cos ) .

(150)

We now can solve for the z momentum obtaining cb "g(1!cos ) , X

(151)

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

273

which are exactly the non-relativistic solutions. The only remaining step to "nding a complete solution is to solve for (t), which is given from u d

"u(1!b )" . X c dt

(152)

Hence ut"(1#g) !g sin .

(153)

What these equations say is that the solutions are perfectly periodic but the period is increased by a factor 1#g

(154)

owing to the electron (or positron) `sur"nga the wave, for once the particle is relativistic, the circularly polarized wave looks almost linearly polarized as the particle moves with it and is steadily accelerated. Notice however that there is nothing relativistic about the solutions themselves, they are equally valid for g;1. We can check this result for c (since we never used the energy term, Eq. (126), per se) by direct di!erentiation of Eq. (150), dc d

u "g sin "g sin (u!k b )"g sin , A X dt dt c

(155)

which is indeed the energy equation multiplied by c (one factor of g is brought in by the constant part of cb and ug is just eB /m). W , The reason we make this point is that our relativistic results are perhaps counter-intuitive. For g;1 (non-relativistic) the larger g the higher the x and y velocities. But this behavior reverses for g<1, where the Lorentz factor increases as g but cb only increases as g. Thus the velocities in the V x}y plane decrease as the particle gets accelerated to ever higher energies. Mathematically, this decrease means that the momentum in the z direction hogs all of the energy. Physically, what happens is that the magnetic "elds become so strong that the particle is almost immediately curved from the initial x-direction into the z-direction. The fact that the orbits become smaller may be signi"cant to design of laboratory acceleration of particles to large energies. 10.1. Circular vs. linear polarization It is not necessary that the waves be perfectly circularly polarized. If we attenuate the E and V B components by a factor of a, the only change is reduce the x-component of moment by the same W factor, cb "!ag sin , V

(156)

 A relevant problem is the particle motion in constant, static E and B cross "elds (as compared to wave "eld) with E"cB (e.g., [32]). If starting from rest, a particle gets a tremendous boost along the direction of E;B whereas it obtains no momentum along B and little along E. Interestingly, 1!b "1/c, same as in the wave case. X

274

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

while the z and energy equations stay in the same "xed ratio. Thus 1 c"1#g(1!cos )# g(1!a) sin . 2

(157)

Notice that, as can be seen by turning o! completely one or the other components, that there is a phase dependence to the maximum energy the particle ever attains, being a maximum if the particle is picked up when the (now linear) wave has its maximum "eld strength, and a minimum if picked up at a null. 10.2. Resonance with a static uniform B A classical plasma physics exercise is the e!ect of a resonance in the dispersion relation for a circularly polarized wave propagating along the magnetic "eld direction in a uniformly magnetized plasma. Rather than repeat the non-relativistic treatment, we will simply modify appropriately the relativistic treatment above by adding in a static B component to the Lorentz force. The x and X y Lorentz forces become, after rearrangement of terms d(cb ) V (1!b )"!g(1!b ) cos !hb , X X W d

(158)

d(cb ) W (1!b )"!g(1!b ) sin #hb , X X V d

(159)

where u e B h,  X,  , u mu

(160)

and h is just the dimensionless frequency ratio analog to g. Now we see that the (1!b ) terms no X longer cancel out. However, the remaining two Lorentz force terms are completely unchanged and we still have the extraordinary (and general) relationship that for pickup from rest 1 1!b " , X c

(161)

so multiplying through by c gives d(cb ) V "!g cos !hcb , W d

(162)

d(cb ) W "!g sin #hcb . V d

(163)

We see that we have a very simple driven linear coupled di!erential equation in just the two variables cb and cb . Although h and g would seem to be on similar standings, this expectation is V W

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

275

not correct because we can divide out g from the equation and all that does is to rescale the two variables. Thus, we can symbolically rewrite the equations in such normalized form, dm "!cos !hg , d

(164)

dg "!sin #hm , d

(165)

where m"cb /g and g"cb /g. These equations are identical to the non-relativistic ones! It is a bit V W simpler to di!erentiate one of the equations with respect to and eliminate one of the variables, giving for example dg dm " sin !h "(1#h) sin !hm . d

d 

(166)

The resonance has nothing to do with the (1#h) sin term but follows from the harmonic solution being at frequency when h"1 in the hm term. The textbook approach is as usual to look for harmonic expressions for m and g, giving the solutions m"!a sin ,

(167)

g"a cos ,

(168)

where 1 a" 1!h

(169)

and one then sees the resonance at h"1 corresponding to u"u . Although this resonance  condition is the obvious solution in the non-relativistic limit, it is a bit surprising to have exactly the same solution in the relativistic case where the particles `surf a along with the wave and one would not have been surprised to "nd the resonance Doppler shifted by one or more Lorentz factors. The homogeneous solutions are just those at frequency h and we have then for pickup at rest at

"0 g cb "! (sin !sin h ) , V 1!h

(170)

g cb " (cos !cos h ) , W 1!h

(171)

g [1!cos (1!h) ] . cb " X (1!h)

(172)

276

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

Fig. 16. Precession of the trajectory in velocity space for a non-zero longitudinal magnetic "eld.

The seemingly singular point at h"1 does not really `blow upa the solutions because when hP1, the solutions become: cb "!g cos , V cb "!g sin , W 1 cb " g  . X 2

(173) (174) (175)

These Cartesian solutions somewhat obscure what the particle is doing in velocity space, which is to execute a circle for h"0 but the starting point (m"g"0) is not at the center of the circle but is a point on the circle (obvious in retrospect), and for small h the circle precesses about this point, as shown in Fig. 16. 10.3. Motion in space The progress of the particle along z can easily be gotten because dz dz d dz dz u " " u(1!b )" X dt d dt d

d c

(176)

but we also have that 1 b "1! X c

(177)

so all together u dz "c!1 . c d

(178)

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

277

However, it is easy to see that substituting arbitrary values for g and m into the Lorentz identity, Eq. (148) gives g 1 c"1# [(cb )#(cb )]"1# (m#g) V W 2 2

(179)

so we "nd that



c g (m#g) d , z" u 2

(180)

which, for h"0, has the solution, c z" g( !sin ) . u

(181)

Even though we opened our analysis with a discussion directed at electrons, the key conclusions (cf. Eqs. (179) and (180)) only depend on g, which is the same for electrons and positrons. Likewise, solutions can be generalized to protons by simply replacing m with the proton rest mass, and we C see immediately that protons or other ions would be accelerated to much lower energies. 10.4. Particle not starting at rest If a particle starts with a non-zero b but b "b "0, we de"ne c(1!b )"c (1!b )"c , W X  X  X V or b "(1!c)/(1#c) and c "(1#c)/2c, one gets      X g (sin !sin h ) , (182) cb "! V 1!h g cb " (cos !cos h ) , W 1!h

(183)

1 g cb " [1!cos (1!h) ]#c b  , (184) X (1!h) c  X  c(1!b )"c , (185) X  where h"h/c .  Alternatively, if b "0 but b "b O0, we de"ne c "c . Then cb "((c!1)/2"d . The V W   V   X solutions are (again, h"h/c ):  g cb "d [ cos (h )!sin (h )]# [ sin (h )!sin ] , (186) V  1!h g cb "d [ cos (h )#sin (h )]! [ cos (h )!cos ] , W  1!h

(187)

278

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297






g g gd  sin (h!1) , cb " d # [1!cos (h!1) ]! X c (h!1)  h!1 c (h!1)  

(188)

c(1!b )"c . X 

(189)

In both cases, the resonant condition changes from h"1 to h/c "1, which is obvious from the  requirement u!kv "u /c. X  10.5. Plasma dispersion ewects An additionally curious feature of these equations that the reader may have noticed is that we have assumed that u"ck at a number of places in the discussion. If we had a number of particles picked up in place of a single one, the natural expectation would be that the coherent conduction currents from these particles would modify the phase velocity of the wave, just as in the case of an ordinary non-relativistic plasma. In every case, the same ratio u/k replaces c in every step of the derivation, so the same results obtain even for a plasma dispersed wave. For our discussion of particle motion in intense "elds, the linear assumption implicit in even discussing a dispersion relation becomes inapplicable. However, for other applications this observation may be of some use.

11. Motion in realistic 5elds Having discussed the particle motion in constant wave "elds, we move on to more complicated situations. 11.1. Decreasing wave amplitude (still planar) Given that we can solve for z as the phase advances (e.g., Eq. (180)), we can ask what happens when the wave "elds E and B (parameterized as g) and radial B "eld (h) vary with distance, as would be the case for spherical waves instead of plane waves. To our knowledge, this second step of complication cannot be done analytically. However, since the underlying equations of motion become linear when we take the phase to be the independent variable, we have the advantage that numerical solutions of linear equations mimic the exact solutions. Thus a harmonic oscillator solved numerically (with appropriately small steps) gives exactly harmonic solutions, the only di!erence being that the oscillation frequency is slightly di!erent from the continuum case (the di!erence vanishes as the step size vanishes). The important point then is that there is no true `errora in the numerical solution insofar as generic behavior goes. Consequently, introducing additional (non-linear) terms does not introduce additional errors but rather is the only step that introduces potential errors so long as we are basically interested in the generic behavior of systems and do not require a precision simulation of a speci"c system. Neglecting "rst h, it is easy to show that the e!ect of reducing g, the wave amplitude, with distance causes the persistent y component of velocity to approach a constant "xed value which is

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

279

slightly more than half its maximum value (if one instead models the variation in terms of a monotonic dependence on rather than z, the approach turns out to be more symmetrical toward exactly 1/2). Fig. 17 shows this process. What is happening physically is that the wave having become weaker is no longer able to bring the particle back to rest and ultimately the wave becomes an irrelevant perturbation with all three components of momentum approaching "xed values. The physics is somewhat di!erent if h, the magnetic "eld component parallel to the propagation direction, is non-zero, because the persistent component of velocity now circles in the B "eld as shown previously in Fig. 16. The e!ect of precession in a declining wave "eld is X illustrated in Fig. 18. Thus one has a trade-o! depending on which is the faster, the decline in g or the precession induced by h. If the former dominates, the system will approach nonzero values as just discussed, but if the latter dominates the system will approach zero energy! It is straightforward to solve the set of equations (e.g., Eqs. (162), (163), (179)) numerically by introducing





z g"g  ,  z

(190)

z  h"h  ,  z

(191)

as would be the simplest behavior expected for the decline with distance from a pulsar for a wave "eld and a frozen-in radial component of magnetic "eld, respectively, with z being the distance along the spin axis. Using parameters believed to be appropriate for the Crab pulsar, we have




e B a  g "   "9.5;10  mX R 

(192)

Fig. 17. Lorentz factor of a particle moving in a plane wave whose strength declines along the particle trajectory. Fig. 18. Precession in velocity space of a particle in a declining wave "eld moving parallel to a static magnetic "eld.

280

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

Fig. 19. Electron (or positron) energy development as a function of distance z!z (starting from rest at various initial  distances z ) accelerated in a plane wave whose amplitude decreases as 1/z. Parameters appropriate for Crab pulsar are  used. The asymptotic energy and the pivoting distance beyond which electron shows large oscillations in energy are all in good agreement with the analytic estimates. The initial enormous energy gain comes from the fact that particles are picked up by and are almost in phase with the wave. The "rst decrease in energy marks the "rst time when a particle is out of phase with the wave.

at the wave zone distance (sometimes called the `light-cylindera distance) R "c/X (hereafter,  all distances are in units of R ).  Fig. 19 shows the acceleration of electrons (or positrons) starting with di!erent initial g or e!ectively, di!erent distances from the pulsar along the rotation axis. The parameter h is chosen to be zero in all these runs. The overall trend is clear that the further away a particle starts, the lower its "nal energy is. There also seems to be a `pivotinga behavior in electron's energy development in that the further away a particle starts, the larger the `oscillationsa in its energy as a function of distance. (Such behavior is more clearly seen in the spherical wave cases considered in the following sections.) To qualitatively understand Fig. 19, recall that when previously g is a constant (and h"0), the particle's energy is a periodic function of z with period 2ng. In other words, a particle starting from rest reaches its maximum energy when *z"ng. Now that g decreases as 1/z, we can de"ne a critical distance z by equating  z "n(g /z ) or z "(ng )+3;10. (193)       The term comes from earlier models discussed above that assumed centrifugal forces, not electromagnetic forces, were dominant in pulsar activity. In that view, we could technically still be inside the `light-cylindera if on the spin axis.

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

281

Thus, we are faced with two possibilities of either z (z or z 'z , which are equivalent     to z (*z and z '*z, respectively. The pivoting behavior can be understood as follows:   Particles starting inside the critical distance z will be continuously accelerated until they  reach beyond z where particle starts to become out of phase with the wave and the strength  of the wave has decreased enough. That the actual `turning pointsa in Fig. 19 are larger than z  can be understood as overshooting since particles already have enormous energies as they advance each decade in z. Particles starting beyond z , on the other hand, will reach their  highest energy when *z"z!z +n(g /z ). For example, at roughly the distance to knot 2    (the possible shock) for the Crab, we have z "10, and *z&3;10;z , so g(z) is e!ectively the   same as g(z ).  The maximum energy a particle can obtain in such "elds can be estimated again from the constant g case (which is 2g). Similarly,





g  c +2  +2;10, z (z ,  

 z 

(194)

g  c +2  , z 'z .  

 z 

(195)

The actual "nal energies shown in Fig. 19 are higher than the above estimate when z (z due to   the continuous acceleration between z and z . For z 'z , the above equation gives a good     agreement. Lower particle energies also enable waves to `take backa most of the particle energy as the particle moves in and out of phase with the wave, as evidenced by the large amplitude oscillations in c(z).

11.2. hO0 case By letting h "g at the wave zone distance, we can explore the e!ects due to a nonzero B .   X Since h&1/z, the condition for h"1 will occur at z&3;10. On the other hand, the particle dynamics is still controlled by staying in phase with the wave due to the large wave amplitude (i.e., g&1/z), the Lorentz force due to p B is comparatively insigni"cant until the particle VW X starts to slip out of phase with the wave. By that time, the distance is so large (z5z ) that  wave becomes so weak that it could not strongly a!ect particle motion anymore. When starting at z"10, Fig. 20 shows that there is essentially no di!erence in electron energy gain with hO0. Fig. 21 compares the electron and positron energy gain with hO0. Again, only little di!erence is seen. Although the resonance does not play a role in the particle's dynamics, if a particle is starting very close to the wave zone, the B component will become important. A distance can be roughly X estimated by balancing the two Lorentz force components, g /z&p g /z, which gives z&p .  VW  VW So, during the acceleration phase, if p exceeds z, the particle will experience gyration around B , VW X thus limiting the magnitude of the transverse momentum. Consequently, the particle "nal energy is reduced also.

282

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

Fig. 20. Upper panel shows the energy development of an electron accelerated in a planar wave as a function of distance z!z starting from rest at z "10. Lower panel shows the corresponding evolution of the transverse momentum.   Parameters appropriate for Crab pulsar are used. The solid and dashed curves are for h"0 and h"g /z, respectively.  It is clear that there is hardly any di!erence between these two cases.

11.3. Motion in spherical waves The true "elds from a rotating pulsar will be spherical instead of planar at large distances. As we will show, there is a subtle but important "rst order error if the spherical wave is treated as simply a plane wave whose strength drops appropriately with distance (i.e., the discussion in the proceeding section). For an orthogonal rotator (sin m"1), the pure spherical vacuum wave along the rotation axis (cos h"1) can be expressed as B "E "0 , P P

(196)

1 B "!E /c"B cos t , F (  R

(197)

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

283

Fig. 21. Upper panel shows the electron and positron energy development accelerated in a planar wave as a function of distance z!z starting from rest at z "10. Lower panel indicates the changes in the transverse momentum.   Parameters appropriate for Crab pulsar are used. The solid and dashed curves are for electron and positron, respectively. Again, h"g /z. There is only very little di!erence between electrons and positrons due to a non-zero h. 

1 B "E /c"!B sin t , ( F  R

(198)

where R"r/R is the dimensionless radial distance, B "B (a/R ) is the B "eld at     R , t" #R!Xt" ! , and "Xt!R. For the sake of completeness, we write out the  Q Q Lorentz equation of a charged particle q in spherical coordinates by taking the dot product of e( , e( , P F and e( with dp/dt"q(E#*;B), since it proves fruitful to think of the particle motion in +r, h, , ( rather than +x, y, z, as in the planar case. Substituting the above expressions for "eld components, we can write the three components plus energy equation as (see also [33]). p#p q 1 dp (!g P" F (p sin t#p cos h cos t) ,  "q" R F ( R ds

(199)

dp p cot h!p p q 1 F" ( P F!g (c!p ) sin t ,  "q" R P ds R

(200)

dp p p cot h#p p q 1 ("! F ( P (!g (c!p ) cos h cos t ,  "q" R P ds R

(201)

284

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

dc q 1 "!g (p sin t#p cos h cos t) ,  ( ds "q" R F

(202)

p#p dw d(c!p ) P "! F (, , ds R ds

(203)

where we de"ne here the dimensionless proper time interval ds"X dt/c, the dimensionless momentum components p "cb "dR/ds, p "cb "R dh/ds, and p "cb "R sin h d /ds. A P P F F ( ( new variable w"c!p is used to indicate whether a particle hogs most of its energy in radial comP ponent or whether it is gaining energy in the transverse components. The "rst terms on the right-hand side are geometric terms due to the fact that e( ) de( /dt is usually not zero, or they can be thought of G H as inertial terms (e.g., centrifugal and Coriolis forces) in the non-Cartesian coordinate system. It is tempting to solve the equations directly in spherical rather than Cartesian coordinates, which is indeed convenient for particles starting with large h, say, from the equator. But for particles starting along the rotation axis (h"0), there is an awkward singularity (cot h) in the h and

momentum equations, which complicates numerical calculations. One way to get around is to solve the Cartesian equations when h is very small but switch to spherical coordinates otherwise. Ostriker and Gunn [2] considered the special case with particles starting at the equator (h"n/2), e!ectively removing this singularity. This set of Lorentz equation proves to be harder to solve than its counterpart in the planar wave case, mainly due to the fact that one has to track the phase term t" ! very carefully since it is  a function of both time and position. The time derivative of a quantity X can be expressed as dX/ds"(dX/d )(d /ds) and d /ds"c(1!b ). In the planar wave case, is not de"ned (thus is    P a constant e!ectively), and c(1!b ),1, which allows a great simpli"cation by transferring time X derivatives into phase derivatives. As shown in Eq. (203), c(1!b ) is not a constant in the spherical P wave. In fact, as we will show below, c(1!b ) varies from 1 to &1/c. P On the other hand, Ostriker and Gunn [2] argued that the phase does not change appreciably  for particles close to the light-cylinder and with this important simpli"cation, they were able to obtain some analytic solutions for a test particle picked up by a plane-polarized spherical wave from the equatorial plane (h"n/2). Here, by rigorously tracking the phase and solving the full set of equations, we are able to con"rm the key assumption used in [2] and explore parameter regimes when this assumption is not applicable. For example, when particles start at large distances, they experience large variations in the angle . Another example is when the full Deutsch "elds are used, which will be discussed in the next section. An additional motivation for accurate motion determination is the consequent implications for calculating its radiation in the pulsar "elds. Fig. 22 shows the particle energy in a spherical wave as a function of R starting with di!erent heights on the spin axis z . Again, we see a `pivotinga behavior in the particle energy development  depending on whether z (z or not, where z is the same as Eq. (193). Particles starting within    z are continuously accelerated without any decrease in energy whereas particles starting with  z 'z show large energy `oscillationsa, and the run with z "z +4;10 indicates the transition     between these two types of motion.  The codes used to solve the equation of motion in this and following sections can be obtained by contacting Hui Li at [email protected].

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

285

Fig. 22. Electron (or positron) energy development as a function of distance r!r accelerated in a spherical wave whose  amplitude decreases as 1/r. Parameters appropriate for Crab pulsar are used with m"n/2 and particles are placed at rest on the rotation axis with various initial heights z . All length scales are in units of R +1.5;10 cm. For   z (z "3;10, particles reach maximum energy in &2r and no energy decrease is seen because particles have not yet  A  been overtaken by another cycle of the wave. Note the di!erences when compared with the plane wave case (Fig. 19).

In order to better understand the results in Fig. 22, we can describe the initial motion of an electron in a pure spherical wave by "rst considering a tiny increase in "Xt!R from 0Pe.  This makes tP!e which implies E +!1 and E +0. Thus the angle changes from 0Pn/2 ( F as soon as electron moves away from the axis, which then makes t+n/2!e. Consequently, sin t+1 and cos t+0. The equation of motion of an electron can then be further simpli"ed (making use the fact that p ;p , p ) as ( P F p#g p dp  F , P+ F (204) R ds !p p #g w dp P F  , F+ R ds

(205)

dc g p +  F , R ds

(206)

p dw +! F . R ds

(207)

286

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

Armed with these simple equations, we note the following features of particle motion in a pure spherical wave: First, the physics for this strong particle acceleration follows because particles stay more in phase with the wave. Note that even though the wave is circularly polarized along the spin axis, elliptically polarized at some intermediate h angle and plane polarized at the spin equator, a particle picked up from rest essentially sees a plane polarized wave (e.g., E +!1 and E +0 in F ( the above example). So the analysis by [2] applies here. Eq. (207) shows that c(1!b ) is driven from P 1 at t"0 to ;1 at later time, giving 1 , b P1! P 2c or equivalently, b , b P0. Notice that the equivalent expression for a plane wave is F ( 1 b "1! . X c

(208)

(209)

When using parameters typical of the Crab pulsar nebula, the extra power of c at large values of this parameter makes a huge di!erence in the asymptotic behavior, with the e!ect that a particle (even though it is moving at less than the speed of light), will for all practical purposes never be overtaken by another cycle of the wave, i.e., it will be at such huge distances that interaction with the local interstellar medium will become dominant. For example, in order for to increase by n,  Xt has to be &2nc, which would require a distance of &100 kpc (!) if c&10. This behavior is not what one would get for a plane wave whose strength is made to drop, because there the particle ends up with a non-zero transverse momentum, and the proper velocity parallel to the (plane) wavefront does not represent the proper velocity parallel to the curved wavefront, which is asymptotic to the total proper velocity of the particle, as illustrated in Fig. 23. Treating the wave as plane gives us instead Eq. (209) so that particle is constrained in z propagation and slips out of phase with the wave at Xt&nc. Second, the total amount of energy a particle can get depends on the transverse momentum p and p . This is not surprising since an electron has to move parallel to the electric "eld to gain F ( energy. Above equations indicate that for particles starting from rest, p will increase "rst, so will F

Fig. 23. Edge-on view of plane and spherical wave showing that the velocity vector is signi"cantly closer to the spherical wave when there is a velocity component parallel to the wave front.

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

287

Fig. 24. The details of an electron's energy development and position changes for z "100 (left panel) and z "10 (right   panel) respectively. Notice the di!erent scales for these two cases. All other parameters are the same as in Fig. 22. Note that electron reaches its maximum energy when p is the largest. Also, c!p "c(1!b )+1/(2c )&10\ and F P P  &3;10\, respectively.

c and p (w+1). The continuous growth of c depends on p being positive. But Eq. (205) implies P F that p will reach a maximum when g w"p p , from which point p starts to decrease. The F  P F F decrease of p will eventually diminish the increase of c and the decrease of w implies that essentially F all the particle's momentum is in e( direction. Consequently, particle starts `coastinga in energy P with p , p , and w all asymptotically being zero. F ( Fig. 24 shows in greater detail the evolution of several key quantities with z "100 (left panel)  and z "10 (knot 2 distance in Crab nebula, right panel). One can see that for z "100,   p stays positive so particle energy c never decreases. This is not true for z "10, in which case F  the wave is much weaker, a particle, even though still gains quite an amount of energy, repeatedly goes out of phase with the wave (note also the oscillations in p and p ) and conseF ( quently will be brought to rest and be reaccelerated, until the particle has gone far enough that the wave is too weak to bring particle completely to rest. In both cases, the main particle motion in real space is straight up along the spin axis. For z (z , there is a tiny   translation away from the rotation axis with no `circulationa since stays essentially constant. Whereas for z 'z , there is a rotation about the axis that is translated away from the spin axis   (due to positive p ). F Third, particles obtain the same asymptotic energy as long as z (z and they reach that energy   at R&2z . This is quite di!erent from the planar wave case (i.e., comparing Figs. 19 and 22). To 

288

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

estimate the asymptotic particle energy in spherical wave, we can take the ratio of Eqs. (206) and (205), dc g p  F + dp g w!p p F  P F

(210)

which at early time (w+1 and p p ;g ) gives c+p. On the other hand, p will reach its P F  F F maximum when g w"p p , which implies p +g /c. Equating c+p and p +g /c, we get  P F F  F F  c+g+2;10 

(211)

which is in good agreement with Fig. 22. Note that this dependence is quite di!erent from the planar wave case in Eq. (194). In estimating the "nal energy, there is no distance scale involved (as long as z (z ). We can roughly estimate the distance where c is reached by noting that  

 ds"Xdt/c+dR/c. Substituting this into Eq. (206), we get dR dc + N*R+R . g R 

(212)

This is again in agreement with Fig. 22. Note that c in spherical wave is smaller than the planar  waves because particle motion is slightly more limited in transverse directions, whereas transverse motion/displacement is necessary for particle to gain energy from the E "eld. This di!erence is evident by comparing the amplitude of transverse momentum in Figs. 20 and 24. The results given in Eq. (211) also agree with those in [2], where cJg(1!R /R). When   *R"R!R ;R , this indicates that c increases as *R, which is in perfect agreement with   Fig. 22. For z 'z , the "nal particle energy is determined by &2(g /z ), which is the same as in the     planar wave case (cf. Eq. (195)) for the same physical reason. This again agrees with Fig. 22. 11.4. Motion in Deutsch xelds The Deutsch "elds at R
(213)

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

289

It is straightforward to write down the equation of motion of a charged particle q in Deutsch "elds (without radiation reaction). De"ne B/R"B/B and E/R"E/(cB ), it reads   dp p#p q 1 P" F (#g (cE #p B !p B ) ,  P F ( ( F ds R "q" R

(214)

p cot h!p p dp q 1 P F#g F" ( (cE #p B !p B ) ,  "q" R F ( P P ( R ds

(215)

p p cot h#p p dp q 1 P (#g ("! F ( (cE #p B !p B ) ,  ( P F F P R ds "q" R

(216)

q 1 dc "g (p E #p E #p E ) ,  "q" R P P F F ( ( ds

(217)

dw p#p q 1 (#g "! F (!wE #p (E !B )#p (E #B )) .  P F F ( ( ( F ds R "q" R

(218)

The Deutsch "elds are asymptotically pure spherical vacuum waves for su$ciently large R, so only when particles start close to R are there signi"cant di!erences. So one would expect the same  particle behavior as described in spherical wave case when R is su$ciently large (we will quantify this statement soon). We again consider an orthogonal rotator (sin m"1) with an electron starting on the spin axis. Fig. 25 shows particle's energy development in Deutsch "elds. Again, the physics for particle acceleration is unchanged compared to the pure spherical wave case. And indeed, particle's behavior is the same as in a spherical wave when z '10.  Comparing with the pure spherical wave (cf. Fig. 22), Fig. 25, however, depicts marked di!erence when z 410 (the change is obviously continuous). To understand this di!erence, we again follow  the arguments in deriving Eqs. (204)}(207) where t"n/2$e. We obtain p#g p dp  F , P+ F R ds

(219)

dp !p p #g w P F  , F+ R ds

(220)

dc g p +  F , ds R

(221)

dw p g p +! F #  F (B !E ) , ( F ds R R

(222)

which are the same as Eqs. (204)}(207) except the last equation on dw/ds where we keep an extra term. Using the expressions given in Section 7, it can be rigorously shown that B !E + ( F sin t/R+1/R, diwerent from spherical wave where B !E ,0. Even though this di!erence is ( F

290

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

Fig. 25. Similar to Fig. 22 but in Deutsch "eld of an orthogonal rotator instead of a pure spherical wave. Particles are more strongly accelerated in a Deutsch "eld when starting relatively close by (z 410). 

much smaller than the strength of the "eld itself (1/R vs. 1/R), it constitutes an important contribution in Eq. (222) because g /R can still be much larger than p . This seemingly `smalla  F di!erence is solely responsible for the large increase in c when z 410 in Fig. 25. We now discuss  why this is the case. Eq. (222) guarantees that w"c!p should increase initially since g /R'p . This is contrary P  F to the spherical wave case where w is always decreasing. Physically it means that not all the energy is going into the radial momentum, and the increase in w makes the maximum p even larger. As F discussed before, the larger the p , the higher the c. As both p and R increase, it comes to a point F F when g /R"p (or dw/ds"0) which implies R+(g /p )+10 if p &10. This is consistent  F  F F with Fig. 25 in which the run of z "10 already shows a slightly higher "nal c. Eventually, w starts  to decrease and becomes much less than one (i.e., radial component hogs all the energy), particle starts to coasting with the wave with no transverse momentum. Fig. 26 compares the di!erent behavior for an electron in Deutsch "elds and spherical wave with the same initial z "100. It  con"rms the above analytic estimates. 11.5. Role of a non-zero B

P

The e!ect of a non-zero B is unimportant in the cases we considered here because particles are P picked up at the rotation axis (i.e., sin h+h;1). This, however, is not the case if a particle starts at the equatorial plane of the rotation axis.

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

291

Fig. 26. A detailed comparison of an electron's energy development (starting at z "100) in a Deutsch "eld (solid lines)  and a pure spherical wave (dashed lines). The fact that c!p can be much larger than 1 results in a much larger p , thus P F rendering higher asymptotic particle energy in the Deutsch "eld.

In the planar wave case, we have discussed the e!ect of a nonzero B by pointing out the X possibility of resonance via c(1!b )"(eB /m)/X, or, h"eB /mX"1 because c(1!b ) is X X X X always 1. We also showed that this resonance does not really in#uence the dynamics of particle motion. In the spherical wave (or Deutsch "eld) case, there is an equivalent resonant condition due to a non-zero B , P eB c(1!b )" P . (223) P mX We can see that the resonant condition changes from h"1 in the planar wave case to h"c(1!b )+1/2c;1 for spherical waves. Using c&2;10, this requires an exceedingly small P B (&3;10\ G when X+200 rad/s), which is impractical. Physically, it means that a particle is P moving along B so close to the speed of light, the wave frequency seen by the particle is very small. P The actual B will always be much larger than the required B given above for resonance to P P occur. Though B does not in#uence the particle dynamics via resonance, it could potentially have P a strong e!ect on a particle motion by forcing the particle into gyration if p and p are large F ( enough. Imagine instead of B decreases continuously as 1/r, it approaches a constant after some P "nite distance, then depending on the strength of B and the magnitude of transverse momentum, it P will reduce the particle "nal energy by limiting the maximum transverse momentum. For this e!ect

292

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

to occur, the Lorentz force components due to p B and p B have to be important during the F P ( P acceleration phase. Otherwise, during the `coastinga stage, even if B is comparable to B and B , it P F ( will not make any di!erence because p and p are essential zero. F ( 11.6. Ions, positrons vs. electrons For an orthogonal rotator, the energy development of electrons and positrons is essentially indistinguishable when z '100. For an inclined rotator (mOn/2), the electric dipole "eld E due  P to the central charge (2/3 cos m) cannot be completely ignored even when launching particles in the wave zone. One tends to think that the presence of a positive E helps (hinders) positrons (electrons) P gaining energy. But the gained energy is in the radial component, which hinders (helps) positrons (electrons) gaining transverse momentum. So, these two e!ects tend to cancel each other, and the net di!erence in "nal energy between an electron and a positron is less than 10% when z "100 (not shown here). This dipole "eld, however, could be important for particles injected at  the wave zone. We have also studied the proton motion in the same Deutsch "eld and Fig. 27 indicates that they follow basically the same behavior as electrons except to reduce g by m /m . Thus, protons  N C asymptotically reach a Lorentz factor of &10 (i.e., &10 eV) versus c&2;10 (i.e., &10 eV) for electrons. This is very di!erent from parallel electric "eld acceleration where electrons and ions are expected to gain the same energies on average. Another e!ect which is not discussed here is that the electron motion in real space is di!erent from that of positron, even though they gain the same energy (probably at the same rate). This

Fig. 27. Similar to Fig. 25 but for protons. The asymptotic energy is approximately reduced by a factor of (m /m ).  

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

293

might produce charge separation in space between electrons and positrons as they are accelerated away. 11.7. Plasma dispersion ewects So far, we have been studying the dynamics of a test particle injected into the wave zone of a vacuum wave "eld (E"cB) from an inclined magnetic rotator. Thus, the generic picture is that strong waves `draga particles with themselves and accelerate them to extremely high energies. Obviously when enough particles are injected into this "eld, wave energy will likely be absorbed by particles and instead one has a relativistic MHD #ow (see the next section). Ostriker and Gunn [2] pointed out that vacuum-like propagation of the wave requires the wave frequency to exceed the e!ective plasma frequency, which implies an upper limit of the particle injection rate into the wave zone, above which MHD theory should apply. Arons [34] further discussed this point and argued that most pulsar theories predicted higher particle injection rates. Another argument that has been discussed in literature is that even if the particle injection rate is low enough so that the wave frequency is still above the plasma frequency, the wave will have a phase speed E/B'c. Since the basic mechanism discussed in previous sections relies so critically on the `phase-lockinga between the particle and the wave, even a little plasma will `destroya this matching. Here, we want to show that particles actually are accelerated even more strongly when the wave has E/B'c, contrary to the common conception. We will follow the approach in Ref. [2] since it is analytically much simpler and tractable. Near the equatorial plane of the rotation, we modify the plane-polarized spherical wave as the following: B "E "0 , P P B "E "0 , F (

(224) (225)

1 B "!B sin t , (  R

(226)

1 E F"!f B sin t ,  R c

(227)

where we have introduced a factor f (51) to mimic the e!ects of plasma dispersion causing E/B'c. The simpli"ed Lorentz force on an electron is then p g dp P+ F # p , R R F ds

(228)

!p p g dp P F#  ( fc!p ) , F" P R R ds

(229)

dc g "  fp , R F ds

(230)

p g dw d(c!p ) P "! F #( f!1) p . , ds R R F ds

(231)

294

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

Note the additional positive ( f!1) term in dw/ds. This is very similar to the Deutsch "eld case (cf. Eq. (222)) where w"c!p is shown to increase "rst (cf. Fig. 26). Physically, the larger E (or f'1) P ensures that a larger transverse momentum p will be reached, hence the particle's "nal energy will F increase. Carrying out the detailed derivations, one "nds that p +(2wc#( f !1)c) , (232) F where the "rst term on the right-hand-side is the original Ostriker and Gunn solution. If we approximate f"1#e with e;1, then we have two possibilities. One is that w"c!p 'ec or P e(1!p /c for all c, then the perturbation f on the wave is so small that it gives essentially the P same results as if E/B"c. On the other hand, if e'1!p /c, then the rate of increase of p can be P F much higher. Consequently, we get c+(2e f g ln(R/R ) ,   which can be compared to the result from [2]

(233)

c+g(1!R /R) . (234)   Note the di!erent power dependence on g .  Let's use the Crab pulsar as an example. We will assume e"10\. As a particle is accelerated from rest, the initial behavior should be the same as in a pure vacuum wave until ec&1, after which p is dominated by the ( f !1)c term in Eq. (232). So, there will be an enhanced acceleration after F c'1/e&10. Furthermore, during this enhanced acceleration, particle's c should increase as *R"R!R as predicted by Eq. (233). This is di!erent from Eq. (234) where c increases as *R.  Eq. (234) also implies that acceleration essentially stops when, say, R"2R , where Eq. (233) will  predict a continuing acceleration for R
(236)

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

295

Fig. 28. A comparison between the electron acceleration in a pure vacuum wave (E/B"c) and a wave mediated by some plasma (E/B'c). Parameters similar to Crab pulsar are used.

Kennel and Coroniti [36] have applied such models to the Crab pulsar, but have chosen to use a di!erent de"nition (which we designate p*), p p*, . c

(237)

Here the dimensionless ratio p* measures the relative energy #ux in the "elds versus the particles. The motivation seems to come from the idea that observation might more easily constrain p* (which in turn determines the power input from the pulsar to the nebula) than p (which requires an estimate of both B and the injection rate). However, from a theoretical point of view p signi"es a boundary condition for the #ow while p* speci"es a solution that results from this boundary condition. As a result, considerable confusion has been propagated in follow-on papers: if a paper speci"es p, the goal is likely to be to calculate in e!ect the asymptotic Lorentz factor, while if a paper speci"es p*, it is assuming that the relative power outputs are known, and the goal may or may not to be to see if that number can be theoretically justi"ed. For the Crab pulsar parameters, both the MHD and the particle pickup estimates are similar, c+10, but we have to regard this similarity to be something of a coincidence. For a single particle introduced into a vacuum electromagnetic wave from a vacuum rotator, both p and p* would be in"nite. Indeed, we could turn the problem around and calculate what particle #ux could be

296

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

tolerated before the wave energy was largely transfered to the particles. Then in a sense we would be looking for solutions with p*+1. Such wave-particle calculations are of high interest, but given that the phase-locking is such a delicate but important issue even in the vacuum case, and given that particle loading will perturb the wave phase as part of the loading mechanism, all the present calculations can serve for is as a template for future many particle or MHD-like calculations. 11.9. Radiation reaction and quantum corrections Since particles are accelerated to extremely high energies (c&10}10) and the magnetic "eld is reasonably strong, it is natural to ask whether particle energy gain process is balanced by radiation damping long before the particle gets these extreme energies. Shen [37] presented a thorough discussion of these issues and the validity of classical electrodynamics with possible quantum corrections. The radiation reaction of relativistic particles depends on the magnitude and direction of acceleration, regardless of the speci"c cause of acceleration. If particles always start from rest, Shen [37] has argued convincingly that the radiation reaction (damping) e!ect is almost negligible. The main reason for this is that the radiative damping force depends on both the particle's energy and the angle between the particle's motion and the "eld. The particle tends to align itself to have minimum perpendicular acceleration. This is not the case when the particle enters the "elds with a large Lorentz factor. However, the issue of single-particle radiation reaction is something of a red herring given that su$cient particle #uxes should be leaving an active pulsar to act like a plasma and the coherent radiation reaction on a plasma is generally huge compared to Thomson scattering o! the individual particles. Since the quasi-linear plasma theory is not even close to being valid for the large amplitude waves from pulsars, entirely new approximations will have to be developed.

12. Conclusions Our intention for Part I (Sections 1}7) is to provide a standard set of background "elds as starting points for pulsar theoretical modeling. It is not an intention to be dictatorial in the sense that these "elds must be used, but rather again to establish a standard for comparison. If an author wishes for whatever reason to omit yet again the central charge on an aligned rotator, so be it but please at least explicitly admit it and not leave it to the reader to puzzle through the results before suddenly realizing that omission. Insofar as Part II (Sections 8}11) goes, here again we hope to at least establish the minimal consequences of plasma from the pulsar being injected into the "elds of Part I. Even there we have been able to do little more than scratch the surface. Issues remain unresolved such as how the non neutral plasma is arrayed about an inclined rotator, how to best model pair production, where transitions from corotation in the inner magnetosphere to wave pickup in the outer magnetosphere takes place, and ultimately, what do any of these models have to do with the radio pulsars that they are expected to simulate.

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

297

Acknowledgements Useful discussions with Jon Weisheit and Stirling Colgate are gratefully acknowledged. FCM thanks the support of a grant from NASA, NAG-5-3070, and HL's research is supported by an Oppenheimer Fellowship at LANL.

References [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] [31] [32] [33] [34] [35] [36] [37]

A.J. Deutsch, Ann. d'Astrophysique 18 (1955) 1. J.P. Ostriker, J.E. Gunn, Ap. J. 157 (1969) 1395. A. Hewish, S.J. Bell, J.D.M. Pilkington, P.F. Scott, R.A. Collins, Nature 217 (1968) 709. F.C. Michel, Theory of Neutron Star Magnetospheres, University of Chicago Press, Chicago, 1991. W.K.H. Panofsky, M. Phillips, Classical Electricity and Magnetism, Addison-Wesley, Reading, MA, 1955, 344 pp. F.C. Michel, Rev. Mod. Phys. 54 (1982) 1. J. Krause-Polstor!, F.C. Michel, MNRAS 213 (1985) 43. L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media, Reed Edu. and Prof. Ltd, 1984, 220 pp. L.I. Schi!, Proc. Natl. Acad. Sci. 25 (1939) 391. R.A.R. Tricker, Contributions of Faraday and Maxwell to Electrical Science, Pergamon Press, Oxford, 1966, 31 pp. R.N. Manchester, J.H. Taylor, Pulsars, Freeman, San Francisco, 1977. J.H. Malmberg, J.S. deGrassie, Phys. Rev. Lett. 35 (1975) 577. L.R. Brewer, J.D. Prestage, J.J. Bollinger, W.M. Itano, D.J. Larson, D.J. Wineland, Phys. Rev. A 38 (1988) 859. J.J. Heinzen, J.J. Bollinger, F.L. Moore, W.M. Itano, D.J. Wineland, Phys. Rev. Lett. 66 (1991) 2080. P. Goldreich, W.H. Julian, Astrophys. J. 157 (1969) 869. L. Mestel, M.H.L. Pryce, Mon. Not. R. Astron. Soc. 254 (1992) 355. A. Musmilov, A.K. Harding, Astrophys. J. 485 (1997) 735. V.S. Beskin, A.V. Gurevisch, I.N. Istomen, Astrophys. Space Sci. 146 (1988) 205. T. Erber, Rev. Mod. Phys. 38 (1966) 626. P.D. Thacker, MS Thesis, Rice University, 1998. N.J. Holloway, Nature (Phys. Sci.) 246 (1973) 6. P.D. Thacker, F.C. Michel, I.A. Smith, Revista Mexicana de Astronomia y Astro"sica Serie de Conferenias, 1997. J.L. Parish, Astrophys. J. 193 (1974) 225. J.H. Cohen, A. Rosenblum, Astrophys. Space Sci. 16 (1972) 130. K.O. Thielheim, H. Wolfsteller, Astrophys. J. (Suppl.) 71 (1989) 583. J.E. Gunn, J.P. Ostriker, Astrophys. J. 165 (1971) 523. C.F. Kennel, R. Pellat, J. Plasma Phys. 15 (1976) 335. J.J. Hester et al., Astrophys. J. 448 (1995) 240. F.V. Coroniti, Astrophys. J. 349 (1990) 538. F.C. Michel, Astrophys. J. 431 (1994) 397. H. Karimabadi, K. Akimoto, N. Omidi, C.R. Menyuk, Phys. Fluids B 2 (1990) 606. L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields, Butterworth-Heinemann, London, 1975, 58 pp. K.O. Thielheim, Astrophys. J. 409 (1993) 333. J. Arons, in: G. Setti, G. Spada, A. W. Wolfendale (Eds.), Origins of Cosmic Rays, Reidel, Dordrecht, 1981, 175 pp. F.C. Michel, Astrophys. J. 158 (1969) 727. C.F. Kennel, F.V. Coroniti, Astrophys. J. 283 (1984) 694. C.S. Shen, Phys. Rev. D 17 (1978) 434.

299

CONTENTS VOLUME 318 F.T. Arecchi, S. Boccaletti, P. Ramazza. Pattern formation and competition in nonlinear optics

Nos. 1}2, p. 1

F.A. Escobedo, J.J. de Pablo. Molecular simulation of polymeric networks and gels: phase behavior and swelling

No. 3, p. 85

N.A. Obers, B. Pioline. U-duality and M-theory F.C. Michel, H. Li. Electrodynamics of neutron stars

Nos. 4}5, p. 113 No. 6, p. 227

CONTENTS VOLUMES 311}317 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 D. Youm. Black holes and solitons in strong theory J. Main. Use of harmonic inversion techniques in semiclassical quantization and analysis of quantum spectra N. KonjevicH . Plasma broadening and shifting of non-hydrogenic spectral lines: present status and applications M. Beneke. Renormalons A. Leike. The phenomenology of extra neutral gauge bosons R. Balian, H. Flocard, M. VeH neH roni. Variational extensions of BCS theory

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. 3, p. 109 313, No. 4, p. 171 313, No. 5, p. 237 313, No. 6, p. 293 314, Nos. 1}2, p. 1 314, No. 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. 409 316, Nos. 1}3, p. 1 316, Nos. 4}5, p. 233 316, No. 6, p. 339 317, Nos. 1}2, p. 1 317, Nos. 3}4, p. 143 317, Nos. 5}6, p. 251