Mathematical, Theoretical and Phenomenological Challenges beyond the Standard Model

A

= dpbT4 = -<'3p$p$

&'2p$

(26)

and A , . . .

= a,&$ *

A

= -iw&$ - [SP,&]$

+w / @ .

= -iw(3p$)

(27)

This is the transformation law of a vector field:

i p v p = -pGpvp- jwvp +w p q p .

(28)

For a tensor or spinor field we define the transformation law as follows: L

A

S ~ T A= - t p a p F A - i,,,TA

where M,'," isfies

+W / M , " A " P ~ ,

(29)

is a representation of the undeformed Lorentz agebra. It sat-

[MP', M"'

1=ll~Xj-p"

+l

-l

l g " ~ ~X l l ~ " ~ g X

l g X ~ ~ " ,

(30)

where qp' is the metric depending on the algebra, Kronecker symbol for SU(n) or Minkowski metric for SU(1,n - 1). It is easy to see that the transformations (29) represent the algebra (23). For the bialgebra we have to specify the comultiplication. For the translations comultiplication is straightforward: &(FA

8F B ) = -

~ P A ( ~ ~8) F(BF) ~

= (&FA) 8 F B )

+FA 8 (&FBI.

(31)

For the Lorentz transformations we have t o use the comultiplication (24). We obtain:

8, ( F A@?B)

2i ( 6 p u W / - ~ p u ~ u p )a p T A 8 8 p F B . A , .

= (8LFA)@?B + F A @ ( 8 L F B ) +

(32) The compatibility of the algebraic relations with the comultiplication can again be verified. We have established a tensor calculus on tensor and spinor fields. After these considerations it is clear that the Klein-Gordon equation and the Dirac equation are covariant

128

1. Klein-Gordon equation:

The sign of m2 depends on the metric qp” 2. Dirac equation:

4

transforms like a spinor and the y’s are t h e usual y matrices. where Invariant Lagrangian with interaction terms can be constructed with the above tensor calculus for tensor and spinor fields.

References 1. M. Dimitrijevit, L. Jonke, L. Moller, E. Tsouchnika, J. Wess and M. Wohlgenannt, Eur. Phys. J. C31, 129 (2003), hep-th/0307149; M. Dimitrijevit, F. Meyer, L. Moller and J. Wess, Eur. Phys. J. C36, 117 (2004), hepth/0310116; M. Dimitrijevit, L. Moller and E. Tsouchnika, Derivatives, forms and vector fields o n the K-deformed Euclidean space, submitted to J . Phys. A, hep-th/0404224. 2. J. Wess and B. Zumino, Nucl. Phys. Proc. Suppl. B18, 3002 (1991); S. L. Woronowic, Commun. Math. Phys. 122, 125 (1989). 3. A. Nowicki, E. Sorace and M. Tarlini, Phys. Lett. B302, 419 (1993), h e p th/9212065; J. Lukierski, H. Ruegg and W. Ruhl, Phys. Lett. B313, 357 (1993). 4. Y. I. Manin, Commun. Math. Phys. 123,163 (1989). 5. C.S. Chu and P.M. Ho, Nucl. Phys. B550, 151 (1999), hep-th/9812219; V. Schomerus, JHEP 9906,030(1999), hep-th/9903205; J. Madore, S. Schraml, P. Schupp and J. Wess, Eur. Phys. J . C16, 161 (2000), hep-th/0001203. 6. B. JurEo, S. Schraml, P. Schupp and J. Wess, Eur. Phys. J . C17,521 (2000), hepth/0006246. 7. J . Lukierski, A. Nowicki, H. Ruegg and V.N. Tolstoy, Phys. Lett. B264, 331 (1991); J. Lukierski, A. Nowicki and H. Ruegg, Phys. Lett. B293,344 (1992). 8. A. Klimyk and K. Schmiidgen, Quantum Groups and Their Representations, Springer (1997). 9. J. Wess, Gauge theory beyond gauge theory, prepared for Workshop on the Quantum Structure of Spacetime and the Geometric Nature of Fundamental Interactions (1st Workshop of RTN Network and 34th International Symposium Ahrenshoop on the Theory of Elementary Particle), Berlin, Germany, 4-10 Oct 2000, Fortsch.Phys. 49,377 (2001). 10. E. Abe, Hopf Algebras, Cambridge University Press (1980). 11. F. Koch and E. Tsouchnika, Construction of 0-Poincare‘ algebras and their invariants on Mo, to be published. 12. M. Chaichian, P.P. Kulish, K. Nishijima and A. Tureanu On a LorentzInvariant Interpretation of Noncommutative Space- Time and Its Implications on Noncommutative QFT, hep-th/0408069.

11. SHORT LECTURES

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DEFORMED COHERENT STATE SOLUTION TO MULTIPARTICLE STOCHASTIC PROCESSES

B. ANEVA INRNE, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria E-mail: boyka. anevaocern. ch Deformed algebraic generalized coherent states are considered as the q-analogues of the conventional undeformed harmonic-oscillator algebra squeezed states. It is shown that the boundary vectors in the matrix-product states approach t o multiparticle diffusion processes are deformed coherent or squeezed states of a deformed harmonic-oscillator algebra. A deformed squeezed and coherent states solution to the n-species stochastic diffusion boundary problem is proposed and studied.

1. Introduction By origin coherent states are quantum states, but at the same time they are parametrized by points in the phase space of a classical system.lY2 This makes them very suitable for the study of systems where one encounters a relationship between classical and quantum descriptions. From this point of view, interacting many-particle systems with stochastic dynamics provide an appropriate playground to enhance the utility of generalized coherent states. A stochastic process is described in terms of a master equation for the probability distribution P(si,t ) of a stochastic variable si = 0,1,2, ...,n - 1 at a site i = 1,2, ...,L of a linear chain. A configuration on the lattice at a time t is determined by the set of occupation numbers s1,sp, ..., SL and a transition to another configuration s' during an infinitesimal time step dt is given by the probability r(s,s')dt. In the bulk dynamics is restricted to changes of configuration at two adjacent sites only, and the two-site rates F = rff,i, j , k,1 = 0,1,2, ..., n - 1 are independent from the position in the bulk. At the boundaries, i.e. sites 1 and L , additional processes can take place with single-site rates LL and RL, i, k = 0,1, ..., n - 1. In the matrix-product states approach to stochastic dynamics3i4 the stationary probability distribution of a system with nearest-neighbour in131

132

teraction in the bulk and single-site boundary terms can be expressed as a product of (or a trace over) matrices that form a representation of a quadratic algebra determined by the dynamics of the process. For diffusion processes that will be considered in this paper, r;: = g i k and the n-species diffusion quadratic algebra has the form g i k D i D k -9kiDkDi = X k D i

-XiDk,

(1)

where g i k and s k i are positive (or zero) probability rates, xi are c-numbers and i, k = 0,1, ...,n - 1. (No summation over repeated indices in Eq. (I).) The algebra has a Fock representation in an auxiliary Hilbert space where the n generators D act as operators. For open sytems (with boundary processes) the stationary probability distribution is related to a matrix element in the auxiliary vector space P(S1,..*> SL)

with respect to the vectors) . 1 ditions

(Wl(LFDk 4-

Xi)

= (wlDslDs2...DsL ).I

,

(2)

and (wl, determined by the boundary con= 0,

(@Dk

- xi)/.)

=0,

(3)

where the x-numbers sum up to zero, because of the form of the boundary rate matrices n-1

n- 1

L; = -

E L"

3'

j=O

Ri

n- 1

Rj,

=j=O

E X i

=o.

(4)

i=O

These relations simply mean that one associates with an occupation number si = k at position i a matrix Dsi = Dk (i = 1 , 2 , ..., L; k = 0,1, ...,n - 1) if a site i is occupied by a k-type particle. The number of all possible configurations of an n-species stochastic system on a chain of L sites is nL and this is the dimension in the configuration space of the stationary probability distribution as a state vector. Each component of this vector, i.e. the (unnormalized) steady-state weight of a given configuration, is (a trace or) an expectation value in the auxiliary space given by Eq. (2). The quadratic algebra reduces the number of independent components to only monomials symmetrized upon using the relations (1). In the known examples of exactly soluble 2- and 3-species models, through the matrix product ansatz, the solution of the quadratic algebra is provided by a deformed bosonic oscillator algebra, if both g i k and g k i differ from zero, or by infinite-dimensional matrices, if g i k = 0. It can be shown that in the general n c a ~ e ,if~all ? ~parameters x i are equal to zero on the

133

right hand side of Eq. (l),the homogeneous quadratic algebra defines a multiparameter quantized non-commutative space realized equivalently as a q-deformed Heisenberg algebra'>' of n oscillators. Alternatively a solution of the non-homogeneous algebra with 2-terms on the right hand side of Eq. ( 1 ) can be found trough a relation to a lower-dimensional quantum space realized equivalently as a lower-dimensional deformed Heisenberg algebra. Proposition 1.1. The boundary vectors with respect to which one determines the stationary probability distribution of the n-species diffusion process are generalized, coherent or squeezed states of the deformed Heisenberg algebra underlying the algebraic solution of the corresponding quadratic algebra. We first review the known basic properties of the deformed oscillator coherent states and then define a deformed squeezed state of a pair of deformed oscillators by analogy with the conventional squeezed states as the eigenstate of the deformed boson operators linear combination and study their squeezing properties. Such a q-generalization of the conventional undeformed squeezed states is not known. As a physical application of the deformed coherent and the considered squeezed states we obtain the boundary problem solution of the general n-species stochastic diffusion process.

2. Coherent States of a q-Deformed Heisenberg Algebra We consider an associative algebra with defining relations

aa+ - qa+a = 1,

qNa+ = qa+qN,

N

- -1

4 a-9

(5)

aqN,

where 0 < q < 1 is a real parameter and a+a = = [ N ] . A Fock representation is obtained in a Hilbert space spanned by the orthonormal basis %lo) = In), n = 0, 1,2, ... and (nln') = dnn,: N

alO) = 0 ,

+

aln) = [n]'/21n - 1) ,

+

a+ln) = [n 1]1/21n 1). ( 6 )

The Hilbert space consists of all elements = Cr=o fnln) with complex fn and finite norm with respect to the scalar product = C,"==, The q-deformed oscillator algebra has a Bargmann-Fock representation on the Hilbert space of entire analytic functions. Generalized or q-deformed coherent s t a t e ~ are ~ ~ defined '~ as the eigenstates of the deformed annihilation operator a and are labelled by a con-

If)

(flf)

lfnI2.

134

tinuous (in general complex) variable z:

&.

These vectors belong to the Hilbert space for 1 . ~ 1<~ [m] = The scalar product of two coherent states for different values of the parameter z is non-vanishing

)'.I(

=

c !I.[ O0

(52')"

-

,

-- egZ ,

0

and they can be properly normalized with the help of the q-exponent on the RHS of Eq. (8):

The q-deformed coherent states reduce to the conventional coherent states of a one-dimensional Heisenberg algebra in the limit q -+ 1-. These generalized coherent states carry the basic characteristics of the conventional ones, namely continuity and completeness (resolution of unity I = J Iz)(zI exp,(-lz12)diz). Hence one can expand any state in the coherent states If) = Jdilz) exp,(-1z12)(21 f) and thus

(zla+lf)

= .f (> .

7

(.I.lf)

= Qf

d

(.I

7

( W l f )= z-&f (2)

7

(10)

which is the Bargmann-Fock representation of the deformed oscillators and number operator. 3. Squeezed States of a Deformed Oscillator Algebra

Following the analogy with the (conventional) squeezed oscillator states,11v12generated by the action of a squeezed operator through a linear transformation, we are lead by this idea t o keep the linear structure of the deformed squeezing operator and assume an analogous definition. Proposition 3.1. Let a7a+ and qN generate a deformed Heisenberg algebra with the equivalent f o r m of defining relations [a7

.+I

= qN7

qN a = q - l a q N 7

qNa+ = q a + q N .

(11)

Then there is a two-parameter-dependent linear map to a pair of "quasioscillators" with a "quasiparticle" number operator N

A = pa + ua+ ,

A+=jia++Da.

(12)

135

These operators generate a deformed Heisenberg algebra with relations

[A,A+]= $/ ,

$/A'

qNA = q - ' A $ / ,

= qA'$/,

(13)

provided 4."/ = (lp12 - Iv12)qN.In the limit q + 1- the relation between the parameters of the conventional squeezed state is rec0vered.l' In the deformed "quasi"-oscillator algebra Fock representation space with a vacuum lo), one can define a normalizable coherent state I<), as the eigenvector of the annihilation operator A:

Proposition 3.2. A squeezed state of the deformed creation and annihilation operators is a normalized solution of the eigenvalue equation: (pa

+ .+)Cl ,

p, 4

s

= CIC, P, 4

s

=4

0 s .

(15)

This proposition is motivated by the analogy with the non-deformed case and by the fact that such normalized eigenstate vectors of the written above linear combination of q-deformed oscillators appear in the solution of the boundary problem of a many-particle non-equilibrium system. To show the effects of squeezing we consider the Hermitian quadrature operators x = h ( a a+) and p = &(a - a+), where the boson operators obey the relations of the form (5) and consequently the operators x and p satisfy the deformed canonical commutation relation [ z , p ] = i q N . The variances (62)' = ((x - (x))') and similary (Sp)' = ( ( p - (p))') in any state obey a generalized Heisenberg-Robertson inequality of the form (Sx)2(6p)22 i l ( [ x , p ] ) l ' . One can calculate now the variances in the deformed coherent states 12) and finds that the deformed uncertainties are equal:

+

1

( 6 x 1= ~ ( ~ p ) ' = 5 exp, ( ( q - 1)1z1')

.

(16)

The deformed coherent states are thus states of equal uncertainties only. Minimum-uncertainty states are labeled by the value z for which the qexponent in Eq. (16) has a minimum. In the limit q -+ 1- for 0 < )z)' < 00 the equality of the undeformed uncertainties in the Glauber coherent states is recovered. We proceed further with the discussion of the algebraic states I<, p,v,q ) which reveal stronger squeezing properties, generalizing thus the undeformed case. To calculate the uncertainties with respect to these states

136

we first write the inverse of the linear map in Eq. (12)

a=

P 1P12 -

-

U

ITA- IPI2- 1Ul2

A',

a+=

-U

Id2- l V l 2

A+

1PI2 - 1Ul2

A+, (17) \

I

where lp12 - 1uI2 # 0, being the Jacobian of the linear transformation (12). Exploring the eigenvalue properties of the normalized coherent eigenstates of A,

(CIAIC)S = 4-7

(CIA+lC)S =

c

7

(CI4NlOs

= (CIq[%

IC)S

= ep-l)'['z

, (18)

we calculate the corresponding mean values. This yields a non-equality of the q-deformed uncertainties which read explicitly

At this step we recall the known definition of a squeezed state13 requiring for one of the variances to be smaller than the equal uncertainties common minimal value determined by the equality (16) as the minimum in the variable z of the q-exponential function. F'rom the analyses of the function 'z1 exp, ( ( q - 1)1<12)it follows that the minimum of this function is the finite

+

limit (l+(l-q))T for I() is a squeezed state if

+

&.Hence, according to the expression (19),

which is satisfied provided the parameters (in general complex) p , v of the linear transformation (12) are chosen in such a way that O<

IP - Y l 2 (1PI2- 14")"

and thus the criterion

(W2 holds. The ratio at the very right hand side of Eq. (22) is the basic hypergeometric series 1a.0((1- q)ICI2;q , (q - 1)). Alternatively, from Eq. (20)

137

is satisfied if

which gives

For 0 < q < 1 the values p = f u are not admissible. The inequality (23) (or (26)) together with the condition (22) (or (25)) for the parameters p,Y, q define the eigenstates I<) of the linear combination pa va+ of the deformed boson operators as generalized squeezed states. In the limit q -+ 1- the corresponding expressions for the x , p uncertainties with respect to the conventional harmonic oscillator squeezed states12 are recovered. This analogy with the squeezing properties of the quadratures of the boson creation and annihilation operators justifies, in our opinion, the proposed definition of a q-deformed squeezed state in Eq. (15) as a q-generalization of the conventional squeezed states.

c,

+

4. Deformed Squeezed and Coherent State Solution to the n-Species Diffusion Process We consider a diffusion process with n species on a chain of L sites with nearest-neighbour interaction with exclusion, i.e. a site can be either empty or occupied by a particle of a given type. On successive sites the species i and k exchange places with probability g i k d t , where i, k = 0,1,2, ...,n - 1. With i < k, g i k are the probability rates of hopping to the left, and S k i to the right. The process is totally asymmetric if all jumps occur in one direction only, and partially asymmetric if there is a different non-zero probability of both left and right hopping. In most studied examples of open systems one considers phase transitions inducing boundary processes when a particle of type k, k = 1 , 2 , ..., n - 1 is added with a rate LO, and/or removed with a rate LE at the left end of the chain, and it is removed with a rate RE and/or added with a rate Ri at the right end of the chain. The advantage of the matrix-product state approach is that important physical quantities such as multiparticle correlation functions, currents, density profiles, phase diagrams can be obtained from the representations of the quadratic algebra (1). The problem to be solved is to find matrix representations of the quadratic algebra consistent with the boundary conditions (3), namely that the combinations (LF& xi) and ( R f D k - X i )

+

138

have common vectors with eigenvalue zero, where the only nonvanishing boundary rates are L,k,Ljh, RE, Rjh, k = 1 , 2 , ..., n - 1. The algebra for the n-species open asymmetric exclusion process of a diffusion system coupled at both boundaries to external reservoirs of particles of k e d density has the form

Dn-lDO

- qDoDn-l =

20 -

Xn- 1 Do

Qn-l,O

where k , 1 = 1 , 2 , ..., n - 2, xo 4'-

QO,n-1

Sn-l,o

,

+ xn-l

q k l = -gkl ,

7

Qn-l,O

= 0 and

q k = - gk0 =%.

glk

gOk

gk,n-l

(28)

The equalities in the last formula, together with the relations g k = QOk = g k , n - l i g O k - gkO = g k , n - l - gn-1,k = g o p - 1 - gn-l,O, yield a mapping to the commutation relations of a q-deformed Heisenberg algebra of n - 1 oscillators a h , a:, k = 0 , 1 , 2 , ...,n - 2. A solution is obtained by a shift of the oscillators ao, a$

and by the identification of the rest of the generators with the remaining n - 2 creation operators a t

Dk,

k = 1 , 2 ,...)n - 2

For the phase transition inducing boundary processes, the system (3) for the boundary vectors reduces to the equations:

for k = 1 , 2 , ...n - 2, and the pair of equations

139

Making use of the explicit solution for Dn-l and DO as shifted deformed oscillators (with 20 = -21 = l),we rewrite Eq. (32) as

The latter equations, in accordance with Eq. (15), determine the boundary vectors as squeezed coherent states of the deformed boson operators ao, a$ corresponding to the eigenvalues

The explicit form of these vectors is readily written, namely (wl = a

-3.w

2.w

w

and ).I = eP- Cn=o~~nllIn)* We therefore conclude: the left and right boundary vectors are squeezed coherent states of the shifted deformed annihilation and creation operators D,-1 and Do,associated with the non-zero boundary parameters xn-l and 2 0 , and with eigenvalues depending on the right and left boundary rates: W n

I.( Cn=om

e q

n .

where the eigenvalues u and w are given by Eqs. (39-40). From the inverse linear maps, with R;-lL;-l - L;j.-'REPl # 0 , we obtain

with the help of which the mean values of the generators

DO, Dn-l and the

140

rest Dk for k = 1,2, ..., n - 2 are readily found

With these expressions at hand, it is easy to calculate the expectation value of any monomial of the form (wID,,D s2...Ds~1v) (where DSi = Dj for i = 1,2, ...,L, j = 0,1,2, ..., n - l),which enters the stationary probability distribution, the current, the correlation functions. One first makes use of the algebra to bring all generators Dk for k = 1 , 2 , ...,n - 2 to the very right or to the very left, which results in an expression of the expectation value as a power in DOand Dn-l. Then one writes the arbitrary power of Do, Dn-l as a normally ordered product of A and A+ to obtain, upon using the eigenvalue properties of the latter, an expression for the relevant physical quantity in terms of the probability-rate-dependent boundary eigenvalues v and w. Our results generalize the known solved examples. We note that if the boundary processes are such that there are only incoming particles of ( n- 1)th-type at the left boundary and only outgoing ( n - 1)th-type particles at the right boundary, i.e. Lt-l = R:-l = 0 in (75), then the eigenstate equations define the boundary vectors Iv) and (wl as q-deformed coherent states. Using the eigenvalue properties of the latter one can likewise obtain the physical quantities of interest for the system. The value q # 0 corresponds to a partially asymmetric while q = 0 to a totally asymmetric diffusion in the bulk of the n - 1-type particle. The deformed oscillator coherent states defined for 0 < q < 1 and for q = 0 provide a unified description of both the partially and the totally asymmetric hopping of a given type of particle.

Acknowledgements Most of this work was completed during a stay at the LMU, Munich and the author is very grateful to Julius Wess for the opportunity t o join his

141

theory group at the Physics Department there. The author is happy t,o take part in the organization and to participate the BW2003. References 1. A. M. Perelomov, Generalized Coherent states and their applications, Springer, NY, (1986). 2. W. M. Zhang, D. H. Feng and R. Gilmore, Rev. Mod. Phys. 6 2 , 867 (1990). 3. B. Derrida, M. R. Evans, V. Hakim and V. Pasquier, J. Phys. A 2 6 , 1493 (1991). 4. K. Krebs and S. Sandow, J. Phys. A 3 0 , 3163 (1997). 5. A. Isaev, P. Pyatov and V. Rittenberg, cond-matt/0103603. 6. B.Aneva, J. Phys.A35, 859 (2002). 7. J. Wess and B. Zumino, Nucl. Phys.(Proc. Suppl.) B 1 8 , 302 (1991). 8. D. Fairlie and C. Zachos, Phys. Lett. B256, 43 (1991). 9. M. Aric and D. D. Coon, J. Math. Phys. 17, 524 (1976). 10. B. Jurco, Lett. Math. Phys.21, 51 (1991). 11. D. Stoler, Phys. Rev D 1 , 3217 (1970); D 4 , 2308 (1971). 12. H. P. Yuen, Phys. RevA13, 2226 (1976). 13. K. Wodkiewicz and J. Eberly, J . Opt. SOC.Am.B2, 458 (1985).

NON-COMMUTATIVE GUTS, STANDARD MODEL AND C , P, T PROPERTIES FROM SEIBERG-WITTEN MAP

P. ASCHIERI Dipartimento d i Scienze e Tecnologie Avanzate, Universitci del Piemonte Orientale and INFN, Piazza Ambrosoli 5, I-15100, Alessandm‘a, Italy and Max-Planck-Institut fur Physik, Fohringer Ring 6, 0-80805 Munchen and Sektion Physik, Universitat Miinchen, Theresienstmpe 37, 0-80333 Munchen, Germany E-mail: [email protected] Noncommutative (NC) generalizations of Yang-Mills (YM) theories using SeibergWitten (SW) map are in general not unique. We study these ambiguities and see that SO(10) GUT, at first order in the noncommutativity parameter 8, is unique and therefore is a truly unified theory, while SU(5) is not. We then present the noncommutative Standard Model compatible with SO(10) GUT. We next study the reality, hermiticity and C, P, T properties of the Seiberg-Witten map and of these noncommutative actions at all orders in 8. This allows t o compare the Standard Model (SM) discussed in Ref. 1 with the present GUT inspired one.

1. Introduction There are different examples of noncommutative theories. We here concentrate on the case where noncommutativity is described by a constant parameter 6””. The commutation relations among the coordinates read [x” x”]= x” * x” - x” * x” = i6””, where the star product between functions f , g is given by f * g = f e:op” g . We do not claim that spacetime has exactly this noncommutativity, rather we are interested in investigating a mathematically sound gauge theory based on this easiest noncommutative structure. General aspects of this noncommutative theory will then probably be in common with more refined choices of 9. In particular, the c

+

142

143

choice P” = constant breakes the Lorentz group in a spontaneous way; in a bigger theory we would like to consider Bp”” (or the related B field) dynamical and not frozen to a constant value, thus recovering Lorentz covariance. One can also consider gauge theories with 0 nondynamical but frozen to a particular nonconstant value, linear in the coordinates, such that one has a (kappa) deformed Poincark symmetry, see Ref. 2. Using Seiberg-Witten map13 that relates commutative gauge fields to noncommutative ones in such a way that commutative gauge transformations are mapped in NC gauge transformations, one can construct NC gauge theories with arbirary gauge g r o u p ~ . ~These t ~ theories are invariant under both commutative and noncommutative gauge transformations. Along these lines noncommutative generalizations of the standard model and GUT theories have been studied.ly6 The SW map and the product allow us to expand these noncommutative actions order by order in 0 and to express them in terms of ordinary commutative fields so that one can then study the physics properties of these &expanded commutative actions, see for example Ref. 7. It turns out that given a commutative YM theory, SW map and commutative/noncommutative gauge invariance are in general not enough in order to single out a unique noncommutative generalization of the original YM theory. One can follow different criteria in order to select a specific noncommutative generalization. We here focus on a classical analysis, in particular imposing the constraint that the noncommutative generalization of the Standard Model should be compatible with noncommutative GUT theories. Another issue would be to single out a noncommutative SM or GUT that is well behaved at the quantum level. We refer to the problems relative to renormalization, see for example Ref. 8. On the other hand chiral gauge anomalies are absent in these model^.^ In this talk, following Ref. 6 , we present a general study of the ambiguities that appear when constructing NCYM theories. We then see that at first order in 0 there is no ambiguity in SO(10) NCYM theory. In particular no triple gauge bosons coupling of the kind BFFF is present. We further study the noncommutative SM compatible with SO(10). We next study the reality, hermiticity, charge conjugation, parity and time reversal properties of the SW map and of &expanded NCYM theories. This constraints the possible freedom in the choice of a “good” SW map. In Ref. 10 the C, P,T properties of NCQED were studied assuming the usual C, P and T transformations also for noncommutative fields. We here show that the usual C, P,T transformation on commutative spinors and

*

144

nonabelian gauge potentials imply, via SW map, the same C, P,T transformations for the noncommutative spinors and gauge potentials. We also see that CPT is always a symmetry of noncommutative actions. In Ref. 11 CPT is studied more axiomatically. The reality property of the SW map is then used to analyze the difference between the SM in Ref. 1 and the GUT inspired SM proposed here. It is a basic one, and can be studied also in a QED model. While in Ref. 1, and in general in the literature, left and right handed components of a noncommutative spinor field are built with the same SW map, we here use and advocate a different choice: if noncommutative left handed fermions are built with the SW map then their right handed companions should be built with the -0 SW map; this implies that both noncommutative ~ ! J L and q!JcL = -202 ~!JR* are built with the +6’ SW map. In other words, with this choice, noncommutativity does not distinguish between a left handed fermion and a left handed antifermion, but does distinguish between fermions with different chirality. This appears to be the only choice compatible with GUT theories.

2. Seiberg-Witten Map and NC Particle Models Consider an ordinary “commutative” YM action with gauge group G, and one fermion multiplet, J d42 $ Tr(F,,F,”) +Ti@!$. This action is gauge invariant under 6% = ipa(A)!$ where pa is the representation of G determined by !$. Following Ref. 5 the noncommutative generalization of this action is given by

where the noncommutative field strength @ is defined by d v z , - iff[&, A^,].The covariant derivative is given by

F,,

h

=

a,A,

-

The action (1) is invariant under the noncommutative gauge transformations

6G = i p a ( X )* 5 , 62, = a,A + iq[A, A,]. h

A

h

-The fields A, and X are functions of the commutative fields A, !$

and the noncommutativity parameter

(3) !$,A

via the SW map.3 At first order in

145

0 we have

&[A, e] = At

+ ~1 e P v { AdVP, A ~+} ~ B P " { F P ~ , +A ,o(e2), }

(4)

1

QA,A, el = A + , e y a , n ,

A , ) + o(e2), (5) @[*,A,O]= @ +1T ~ ~ " ~ ~ ( A , ) ~ ~ @ + ~ ~ ~ Y [ P Y IO(e2) ( A ~. () 6, )~ ~ ( A ,

+

A

8 In terms of the commutative fields the action (1) is also invariant under the ordinary gauge transformation SAP = d,A i [ A , A , ] ,6Q = i p @ ( A ) @ .In Eq. (1) the information on the gauge group G is through the dependence of the noncommutative fields on the commutative ones. The commutative gauge potential A and gauge parameter A are valued in the G Lie algebra, A = AaTa,A = AaTa;and from Eqs, (4)and ( 5 ) it follows that and are valued in the universal enveloping algebra of the G Lie algebra. However, due to the SW map, the degrees of freedom of are the same as that of A. Similarly to A^, also P is valued in the universal enveloping algebra of G. Now expression (1) is ambiguous because in Tr(FPV* p P v ) we have not specified the representation p(Ta). We can render explicit the ambiguity in (1) by writing

+

A^

A^

1

-Tr(&, g2

* W )=

c

cpTr(p(Fpv)* p(E'"V)) ,

(7)

P

where the sum is extended over all unitary irreducible and inequivalent representations p of G. The real coefficients cp parametrize the ambiguity in Eq. (7). They are constrained by requiring that in the commutative limit, 0 4 0, (7) becomes the correctly normalized commutative gauge kinetic term. The ambiguity (7) in the action (1) can also be studied by expanding (7) in terms of the commutative fields *,A , F . At first order in 0 we have A

Sgauge

.1

dimG

4g2

a=l

=--]d4x

FZ,FaPv

where 1 1 -DpabC Tr(p(Ta){p(Tb), p ( T C ) }= ) A(p)"r(ta{tb,tc}) TA(p)dabc.(9) 2 Here ta denotes the fundamental representation, and we are using that the completely symmetric DpabCtensor in the representation p is proportional

146

to the dabc one defined by the fundamental representation. In particular for all simple Lie groups, except S U ( N ) with N 2 3 , we have DZbC= 0 for any representation p. Thus from Eq. (8) we see that at first order in 8 the ambiguity (7) is present just for S U ( N ) Lie groups. Among the possible representations that one can choose in Eq. (7) there are two natural ones. The fermion representation and the adjoint representation. The adjoint representation is particularly appealing if we just have a pure gauge action, then, since only the structure constants appear in the commutative gauge kinetic term CaFivFaP”, a possible choice is indeed to consider only the adjoint representation. This is a minimal choice in the sense that in this case only structure constants enter (7). (It can be shown6 that in this case the gauge action is even in 0). If we also have matter fields then from Eq. (2) we see that we must consider the particle representation pq given by the multiplet Q (and inherited by G). In Eq. (7) one could then make the minimal choice of selecting just the pa representation. Along the lines of the above NCYM theories framework we now examine the SO(lO), the S U ( 5 ) and the Standard Model noncommutative gauge theories.

2.1. Noncommutative SO( 10) We consider only one fermion generation: the 16-dimensional spinor representation of SO(10) usually denoted 16+ (no relevant new effects appear considering all three families). We write the left handed multiplet as

VT

where i is the S U ( 3 ) color index and = -ia2 v ~ is * the charge conjugate of the neutrino particle V R (not present in the Standard Model). The gauge and fermion sector of noncommutative SO(10) is then simply obtained by replacing with 5; in Eq. (1). Notice that no linear term in 8, i.e. no cubic term in F can appear. This is so because SO(10) is anomaly free: D a b c = 0 for all p. In other words, at first order in 8, noncommutative SO(10) gauge theory is unique.

G

2.2. Noncommutative SU (5) The fermionic sector of S U ( 5 ) has the $Pr,multiplet that transforms in the 3 of S U ( 5 ) and the X L multiplet that transforms according to the 10 of S U ( 5 ) . In this case we expect that the adjoint, the 3 and the 10 representations enter in (7). In principle one can consider the coefficients

147

q # c10, i.e. while the (qCL, X L ) fermion rep. is 5 @ 10, in (7) the weights cp of the 5 and the 10 can possibly be not the same. It turns out that cPDzbc# 0 in (8). We see that, already at first only if q # c10 then order in the noncommutativity parameter 0, noncommutative S U ( 5 ) gauge

cp

theory is not uniquely determined by the gauge coupling constant g, but also by the value of C , cpDzbc. Thus S U ( 5 ) is not a truly unified theory in a noncommutative setting. It is tempting t o set q = c1O so that exactly the fermion representation 10 enters (8). We then have C , C,,D,”~“ = 0, (however this relation is not protected by symmetries).

s@

2.3. (GUT Inspired) Noncommutative Standard Model One proceeds similarly for the SM gauge group. The full ambiguity of the gauge kinetic term is given in Ref. 6. About the fermion kinetic term, the fermion vector Q L is constructed from Q L = (ui, di , -up , d p , v,e- , e f ) L . The covariant derivative is as in Eq. (2), with Q + Q L and with A, = A $ T A , where { T A } = {Y,Tf,TA}are the generators of U(1) 8 S U ( 2 ) @ S U ( 3 ) . The fermion kinetic term is then as in Eq. (1). This Standard Model is built using only left handed fermions and antifermions. We call it GUT inspired because its noncommutative structure can be embedded in SO(10) GUT. Indeed Q L and differ just by the extra neutrino v% = -ia2 v ~ ;* moreover under an infinitesimal gauge transformation all fermions in Q L transform with on the left. This GUT inspired Standard Model differs from the one considered in Ref. 1; indeed here we started from the chiral vector Q L , while there the vector 9’= ( u i , d i ,u k d k , v ~ , e LeR) , is considered. In the commutative caSe JV@W= JE@QL, but in the h

Qi

x,

h

noncommutative case (see later) this is no more true:

J QI

- A

#J -

-

*

@ QL ; if we change 0 into -0 in the right handed sector of J @ * @ @ I , -

- A

A

then the two expressions coincide. Finally it is a natural choice t o consider in the SM gauge kinetic term only the adjoint rep. and the fermion rep., we then have that a t first order in 0 there are no modifications t o the SM gauge kinetic term. This is so because the fermion rep. is anomaly free: DAA’A” = 0, and because for Pferrnlon U(1) the adjoint rep. is trivial. 2.4. Higgs Sectors

While the noncommutative Higgs kinetic and potential terms are given by

(E,@ * E,? + p2J-t * 4-A & + * &*$t * 3,

(11)

148

a noncommutative version of the SM and GUT Yukawa terms is not straigh-

forward and requires the introduction of the hybrid Seiberg-Witten maps on fermions. A typical noncommutative Yukawa term then reads

--H

Jt * L L * e;l + -31-

where L, =

(z)

herrn. wnj. ,

. Under an infinitesimal U(1) 8 SU(2) @I SU(3) gauge

4

-31

-

4

-

3

1

transformation A, L, transforms as 6 L, = ++(A) * L, - ZL, *pe;,(A). We see that in the hybrid SW map appears both on the left and on the right of the fermions, moreover the representation of is inherited from -31 the Higgs and fermions that sandwich L , . The Yukawa term (12) is thus invariant under noncommutative gauge transformations. Of course in the 9 -+ 0 limit we recover the usual gauge transformation for the leptons. An explicit formula for the hybrid SW map at first order in 9 is in Refs. 1, 6. The Yukawa terms (12) differ from those studied in Ref. 1. There the

A

A i

hybrid SW map is considered on 4, in particular there q!~ is not invariant under SU(3) gauge transformations (and this implies that in Ref. 1 gluons couple directly to the Higgs field). The Higgs sector in the SO(10) and S U(5 ) models can be constructed with similar techniques.6

3. Hermiticity and Reality Properties of SW Map From Eq. (4) we see that if A is hermitian then A^ is also hermitian. Actually, to all orders in 9, and can be chosen hermitian if A and A are hermitian. Otherwise stated, SW map can be chosen to be compatible with hermitian conjugation. Compatibility of SW map with complex conjugation reads, -* a*=*, (13)

A^

where = SW[@,p,p(A),O]denotes the SW map of @ constructed with the representation p,p of the potential A , and the SW map of the complex conjugate spinor @* is defined by h

@*

(14)

SW[@*,p,p* ( A ) ,-6'1,

where p,p. is the representation conjugate to ~ , p Notice . ~ that in Eq. (14) the noncommutativity parameter 0 appears with opposite sign with respect aGiven the group element g =e i A = eiAaT" we h a v E ( g ) E are real, we have pq* (A) = -pp(A) , p q - (A)= -pq(A) .

a

and, since Aa, A"

149

-

-

to the 8 in the SW map of 9.Similarly to (13) we have pa* ( A ) = -p*(A)

:nd p * * ( R ) = - p ~ r ( R ) where p w ( A ) = A^[pq8(A),-8]and p*.(h) = A [ ~ Q * ( A ) , ~ ~ * ( RThe ) , - proof ~ ] . of Eq. (13) relies on showing that the SW differential equations3 (obtained by requiring that gauge equivalence classes of the gauge theory with noncommutativity 8 68, correspond to gauge equivalence classes of the gauge theory with noncommutativity 8 ) are themselves compatible with complex conjugation.6 One proceeds similarly for the case of hermitian conjugation.

+

3.1. Noncommutativity and Chirality We can now discuss a further ambiguity of noncommutative gauge theories, and resolve it by requiring compatibility with grand unified theories. For simplicity we consider noncommutative QED. Let ?I, be a 4-component Dirac spinor, and decompose it into its Weil spinors $L and $R. Their charge conjugate spinors are $Lc = $% = -iu2$; and $Rc = $% = iaz$i. Consider the noncommutative left-handed spinor $L = SW[$L,p+L ( A ) ,81, we then have the k8 choice A

'& = sw[$R,P+'R ( A ) *'9] 1

(15)

1

for the right handed one. In the literature, the choice +8 is usually considered so that for the 4-component Dirac spinor $ we can write = SW[$, A , 81, d$ = ix * We here advocate the opposite choice

6

6.

h

(-8) in Eq. (15). Indeed from Eq. (13) we have that therefore

GLC = -202

-*

$R

h

so that with the -8 choice in Eq. (15), both left handed fermions $ L , h

-

and

-

qLC

are associated with 8 , while the right handed ones $ R , GRC are associated with -8. In GUT theories we have multiplets of definite chirality and therefore this is the natural choice to consider in this setting. These observations allow us to compare QED+ with QED-, the two different QED theories obtained with the two different f 0 choices (15). This difference immediately extends to the fermion kinetic terms of nonabelian gauge theories and allows us to compare the NCSM discussed in Ref. 1 with the present GUT compatible one. We have (up to gauge kinetic terms)

150

where the GUT inspired QED- is obtained using the left handed spinor h

(:c)

so that

$f = sW[$f,p+f(A),8].Now -t

A

from

h

n *

$Lc

=

-Pt

- '

2 ~ 2 1 4 and ~

- P e p

from u matrix algebra we have S $Lc *i@qLC = J $R i@ $R , where we have emphasized that we are using the -8 convention in the SW map by writing w p instead of - . We conclude that in order to obtain QEDfrom QED+ we just need to change 8 into -8 in the right handed fermion sector of QED+. *OP

4. C , P, T Properties of SW Map and of NCYM Actions

Using compatibility of SW map with complex conjugation and the tensorial properties of SW map (i.e. that SW map preserves the space-time index) one can study the properties of SW map with respect to the C, P and T operations. In particular these same expressions as in the commutative case holds:

h*

-CP

G c p = i a 2 @ L , QR

-*

-CP

=-iazQ~, A,

-A

-

=(-Ao,Ai),

(18)

where the action of the P and C operators on spinors is given by

-T

while the time inversion is given by QL = SW[QT,p q L ~ ( A TB T) ,, d T , -i] . In these expressions we have written explicitly the dependence on the partial derivatives, and the imaginary unit i in the last slot marks that the coefficients in the SW map are in general complex coefficients. The -i in the last expression means that we are considering the complex conjugates of the coefficients in the SW map, this is so because T is antilinear and multiplicative. Relations (17) and (18) hold provided that Owv transforms under C, P, T as a U(1) field strenght FPy.If we choose +8 in Eq. (15) then parity and charge conjugation sepatately assume the same expression as in the commutative case. Now we discuss the transformations properties of NCYM actions under C, P and T . With the +8 choice (15) we have that NCYM actions are invariant under C, P and T iff in the commutative limit they are invariant. On the other hand, with the -8 choice NCYM actions are invariant under CP and T iff in the commutative limit they are invariant. For the fermion kinetic term these statements are a straighforward consequence of

s QL-t

*

151

@ g= JQL-t

-

@QL. Since ? transforms like F under CP and T , and in the +8 case also under C and P separately, the C, P,T properties of the gauge kinetic term Ti-(@*@) = Tr(@F)easily follow. Inspection of the fermion gauge bosons interaction term leads also to the same conclusion. We have studied the C, P and T symmetry properties of NCYM actions where 6' transforms under C, P and T as a field strength. Viceversa, if we keep 8 fixed under C, P and T transformations, we in general have that NCYM theories break C, P and T symmetries. Finally, U(1) field strength is invariant under the combined CPT transformation, and therefore 8 does not change. This implies that CPT is always a symmetry of NCYM actions.

s

Acknowledgments

It is a pleasure to thank the organizers for the nice and stimulating atmosphere at the conference and its efficient organization. References 1. X. Calmet, B. Jurco, P. Schupp, J. Wess and M. Wohlgenannt, Eur. Phys. J. C23,363 (2002), hep-ph/Olllll5. 2. M. Dimitrijevic, F. Meyer, L. Moller and J. Wess, Gauge theon'es on the kappa-Minkowski space time, hep- th/03 10116. 3. N. Seiberg and E. Witten, JHEP 9909,032 (1999), hep-th/9908142. 4. J. Madore, S. Schraml, P. Schupp and J. Wess, Eur. Phys. J . C16, 161 (2000), h e p th/0001203; B. Jurco, S. Schraml, P. Schupp and J. Wess, Eur. Phys. J. C17,521 (2000), hepth/0006246. 5. B. Jurco, L. Moller, S. Schraml, P. Schupp and J. Wess, Eur. Phys. J. C21, 383 (2001), hepth/0104153. 6. P. Aschieri, B. Jurco, P. Schupp and J. Wess, Nucl. Phys. B651,45 (2003), hepth/0205214. 7. P. Schupp, J. Trampetic, J. Wess and G. Raffelt, The photon neutrino interaction in non-commutative gauge field theory and astrophysical bounds, hepph/02 12292. 8. M. Buric and V. Radovanovic, JHEP 0402, 040 (2004), hep-th/0401103. 9. F. Brandt, C. P. Martin and F. R. Ruiz, JHEP 0307, 068 (2003), hepth/0307292. 10. M. M. Sheikh-Jabbari, Phys. Rev. Lett. 84, 5265 (2000), hep-th/0001167. 11. M. Chaichian, K. Nishijima and A. Tureanu, Phys. Lett. B568, 146 (2003), hep-th/0209008 ; L. Alvarez-Gaume and M. A. Vazquez-Mozo, Nucl. Phys. B668,293 (2003), hepth/0305093; N. Mahajan, Phys. Lett. B569,85 (2003), hep-th/0305105.

SEESAW, SUSY AND SO(10)

BORUT BAJC JoEef Stefan Institute Jamova 39 1001 Ljubljana, Slovenia E-mail: borut. [email protected]

The seesaw mechanism can explain why the neutrino masses are so tiny with respect to the charged fermion masses. In the canonical version it is the presence of the right-handed neutrino that is responsible for it. In renormalizable grand unified theories with left-right gauge symmetry it is possible to show quite generically that there is another type of seesaw contribution, mediated by a heavy weak triplet. In this talk I will show that such non-canonical seesaw mechanism can very nicely connect b - T Yukawa unification with the large atmospheric neutrino mixing angle in the context of a SO(10) grand unified theory. Also, a fit to the available low energy masses and mixings points towards the domination of this non-canonical contribution with respect t o the canonical one. Finally I will explicitly present the minimal supersymmetric SO(10) model which has all the above nice features.

1. Introduction

The charged and neutral fermion sectors in the Standard Model (SM) have quite different structures: 1) though very different among themselves, the charged fermions have much larger masses than neutrinos, 2) the mixing angles in the quark sector tend to be much smaller than the corresponding ones in the leptonic (neutrino) sector. While the first point can be easily accounted with the famous seesaw mechanism,l the second issue is more controversial and its solution still debated. The first nontrivial framework that could be predictive in both charged fermion and neutrino sectors is SO(10) grandunification. This is because it automatically incorporates the righthanded neutrino, thus providing for a theory of the seesaw mechanism. Most important, the model connects the neutrino and charged fermion mass matrices, providing some nontrivial relations between them at the grandunification scale. Then, with the assumption of the desert, motivated by gauge coupling unification, one can extrapolate the values of these masses and mixings down to low energies 152

153

and here compare them with experimental data. I will describe in some details the origin and form of the neutrino mass matrix, and connect it to charged fermion masses in the context of a SO(10) grand unified theory. 2. Seesaw

The left-handed neutrino u (EUL) is a component of a weak SU(2)L doublet with B - L = -1. If left-right symmetry is assumed, than a right-handed neutrino uc (s Cv,') must exist as part of a doublet of S u ( 2 ) ~gauge symmetry, with B - L = +l. Since uc is a standard model singlet, its mass is given by the scale of S u ( 2 ) ~breaking, which can be achieved in a renorrnalizable version by the vev of a S u ( 2 ) ~ triplet (with B - L = -2), MvR O( (AR). This, Majorana mass, must then be added to the Dirac mass MUDoc (a) with a a su(2)Lxsu(2)R bidoublet:

+

Lm = - y c T M v R ~ ' U ~ ~ M , + , Uh.c. . After integrating out the heavy uc one gets the famous seesawl formula for the light neutrino mass:

Mu = -MTDM;: Mv, ,

(2) which clearly connects the smallness of the neutrino mass with the largeness of the S u ( 2 ) ~breaking scale. There is however another contribution to the seesaw,2 several times neglected, but as we will see, of great potential importance. It comes essentially from the following arguments: as we said, the large right-handed Majorana mass comes from the vev of a S u ( 2 ) ~ triplet via the term u C T A ~ v C , so due to left-right symmetry an analog term uTALu must exist, with AL a triplet under S u ( 2 ) ~ .With these two interaction terms plus the usual Dirac term uCT@uone can easily show that the one-loop box diagram contribution to A R @ ~ AisLUV divergent, so that such a term must be present already at tree level. Thus, the potential for the triplets looks like

+

V = - M 2 (A; +A:) ARGAL, (3) with M a large mass (not much less than MGUT).The last term represents a tadpole for the lefthanded triplet:

154

since (AR) M M and (@) M M w . So the term

"T(wu gives another contribution to the seesaw. All together we thus have

+

MN = -MTD M&l MvD My,, where the first term represents the type I or canonical seesaw formula mediated by the S u ( 2 ) singlet ~ u C ,while the second term is the S u ( 2 ) triplet ~ contribution t o the type I1 or non-canonical see-saw formula. Of course, in general, the matrices in generation space M,,, M,, and MVD are arbitrary, so the above formula is not very useful. It is thus important t o connect the above matrices to the charged fermion sector, which is experimentally better known. For this one needs a framework. We will choose the most economical one, i.e. a SO(10) grand unified theory.

3. SO(10) Although GUTS are not theories of flavours, they still put some constraints on the possible Yukawa interactions, since different SM fields live in the same representations. In SO(10) all the light fermions of each generation plus the right-handed neutrino are grouped together in the 16-dimensional spinorial representation, while in the most economical version the two light Higgs doublets live in a fundamental 10-dimensional complex Higgs representation. There is thus only one Yukawa matrix in generation space (3 x 3). An O(3) rotation of the 16's in generation space can diagonalize it, giving one good prediction (yb = yT = yt) and several bad ones (equality of charged lepton, up and down quark Yukawas of the second and first generations, and no mixing at all). A nontrivial quark mixing could be achieved adding a new 10-dimensional Higgs, but that would not help in improving the relations among Yukawas of the first two generations. To see it, one can remember that the 10-dimensional Higgs gets a vev in the ( 2 , 2 , 1 ) direction of the Pati-Salam S U ( ~ ) L X S U ( ~ ) R X Ssubgroup. U ( ~ ) ~ This means that such a vev cannot break SU(4)c and thus the equality between leptons and quarks. Thus, a nontrivial multiplet in SU(4)c (and bidoublet in the left-right sector) is needed to get such a splitting. An ideal possibility is given by the Higgs in the representation 126 (5 index antisymmetric and anti-self-dual) of SO(10): its vev in the SU(3)c colour singlet state of

155

the (2,2,15) direction is a possible candidate. The Yukawa terms in the Lagrangian can thus be written as3

These terms can be decomposed in the usual Pati-Salam basis as

+ (1,2,z), 126~ = (1,3,lO) + ( 3 , 1 , m )+ (2,2,15)+ (1,1,6), 16 = (2,1,4)

(9)

(10)

where the fields in boldface have SU(3)cxU(l),, singlets and can thus generate a nonzero vev. The two colour singlet bidoublets are actually needed to develop a vev in order t o fit the light fermion masses, as we have just seen. On top of that, the S u ( 2 ) triplet ~ (1,3,10) above is ideally suited to give a large Majorana mass to the heavy right-handed neutrino v c from (2,1,z): from here we can see the double role that the 126-dimensional Higgs can play. Finally, the S u ( 2 ) ~triplet in (3,1,m), if nonzero as conjectured in the previous section, can contribute to the type I1 see-saw formula. Denoting uydd and u2";dG the vevs from the bidoublets in 1 0 and ~ 1268 and WL,R the vevs from the L or R triplets, one gets from (7) the following expressions for the fermion masses:

+ uY26y126 M D = W t ~ ) y l O + v&vjy126 MU = u;bylO

Mv,

= ~yoY10- 3vy2,y126

ME = V f o y l O

7

(11)

7

(12) (13)

7

- 3v&?6y126

(14)

1

MVR= ((1,31 10)m)Y126 = WRY126 MVL= ((37 1,m)m)y126 = VLy126 . 7

(15)

( 16)

Notice that the factor -3 from the 126~ contribution in the lepton sector with respect to the quark one comes because the SU(3)c colour singlet direction in 15 of SU(4)c is proportional to B - L c( diug(l,l, 1,-3). Also, the above relations are valid at the GUT scale only, so that the known experimental values of the masses and mixings must first be run up to that energy using the renormalization group equations. Expressions (11) and (12) are needed to evaluate the matrices Y10 and Y126 in terms of the better known Mu and MO. Their expressions are then

156

used in (14) as well as in the various neutrino matrices (13), (15) and (16) to be used in (6). Defining

one gets two matrix equations:

4. Type I Versus Type I1 Seesaw What we want to find out is which type of seesaw is compatible with data. To make the analysis simpler, let us study the situation of the 2nd and 3rd generations only, as well as no CP violation (real parameter^).^ Equation (19) has 6 known masses ( n ~ mt,c, ~ , ~m b,, + ) , one known mixing angle (0, = O c b ) , but 3 unknown parameters, x,y and the angle between the orthogonal matrices that diagonalize ME and M D (call it 6,). Since the matrices are all symmetric (due t o its SO(10) origin), we have 3 equations, just enough. After determining these parameters one can attack equation (19). We are interested mainly in the leptonic mixing angle and this is determined as a function of one single parameter, a. In the limit a -+ 00 one remains with a pure type I seesaw, while in the opposite case a -+ 0 and the seesaw is of type 11. To get a feeling, let us first consider the idealized situation of small second generation masses (0 M m2 << ms), but still finite quark mixing 0, (although << 1). It is easy to show that the leptonic atmospheric mixing angle is given by

tan201 = with

sin 20, 2sin2 8, - A '

157

1 mb - mr [-5a (1 - 4a) €1, E = 1-9a m b In the limit a -+ 00 one gets A = (5 4 ~ ) / 9 ,while the opposite case a + 0 gives A = E . Since the parameter E is experimentally small (approximate b - T unification), the small a regime is the one that predicts a large atmospheric angle: type I1 seesaw is thus favoured in these type of models. Now we restore a finite mass of the second generation, but keep in mind that they are small, i.e. that

+

A=---

+

There are two solutions: lStsolution (small OD): writing in a schematic way (coefficients of order 1 are not explicitly written) the neutrino mass matrix is 6 6/E hdN = -a! (6,€ l,€)

+

(: :)

where the first (second) matrix is the type I (11) seesaw contribution. The atmospheric mixing angle is tan281 x

If we further assume that

E

+

6 (1 a / € ) 6(1 + a ) E a/€. M O(6), we get

81 x O(1)

+ +

* a 5 O(62) .

(25)

The dominant type I seesaw ( a ---t m) is clearly excluded, since it would predict a small mixing angle 81 M O(6). Notice that without b--7 unification (small E ) 81 would be small. Znd solution (large OD): here the relevant matrices are 62

M N o - f f ( 6 6, 1

+

(; i)

and the atmospheric mixing angle tan281 M

1+Cd l+a

158

so that now a large angle is easily obtained:

We see that both type I1 or the mixed seesaw are compatible with data, but in no case can the type I ( a -+ m) seesaw dominate and predict a large atmospheric mixing angle. In this case the b - r unification parameter E needs not be small. It is possible to understand why a large atmospheric mixing angle emerges in type I1 ~ e e s a w :the ~ point is that the triplet contribution to the Majorana neutrino mass is proportional t o Y l 2 6 (16), but from (12) and (14) this Yukawa is proportional to the difference M D - M E , so that in pure type I1 seesaw

Assuming small mixings, one immediately obtains approximately

so that an unexpected connection between large atmospheric neutrino angle and b - r unification emerges from type I1 seesaw and the assumption of small large

Batm

u b-r

unification.

(31)

Notice however, that in the large OD case b - r unification is not really needed to get a large atmospheric mixing angle. 5 . SUSY

Up to now we did not specify which is the grandunification theory we were considering: the only requirements were that the Yukawas came from the interactions of matter 16-dimensional representations with two complex Higgses - 1 0 and ~ 126~, and that they got some nonzero vevs in the relevant directions. In order to show that this is possible and to find these vevs as functions of parameters in the Lagrangian, we need to write down an explicit model. I will here shortly describe the minimal supersymmetric

159

SO(10) model, which has all the nice above features and is still consistent with data. First of all, why supersymmetry? There are different reasons to believe that supersymmetry is present in some way or another. The first is of course the (in)famous hierarchy problem, i.e. why are scales in nature so different (for example MW and MGUT),since it is known that quantum corrections tend to destabilize them. One of the few possibilities is to consider supersymmetric extensions, in which such quantum corrections are miracolously stable. The second reason is the origin of the spontaneous breaking of electroweak symmetry: although all scalars could have a positive mass square at a very high scale (Mplanck ?), the coupling of the SM Higgs to the large top Yukawa makes the Higgs mass tachyonic, thus triggering its vev.6 Finally, the presence of the supersymmetric partners in the TeV region modifies the beta functions of the SM gauge couplings exactly in the right way to get (under the assumption of desert) unification at the scale of approximately 10l6 GeV.7 As we saw, in order to fit the fermion masses we need two Higgs representations, 1 0 and ~ 126~. In supersymmetry, the presence of 126 is particularly welcome, since its vev does not break R-parity. In fact (1,3, lo), which vev gives a large mass to the right-handed neutrino, has B - L = -2. R-parity is given by

R = (-1)3(B-L)+2S

(32)

and since the spin S of any vev is 0, the vc mass is R-parity even and so does not break it at the large scale.8 One can show that this is true all the way to the electroweak scale, i.e. R-parity is exact.g This means, among other things that the lightest supersymmetric partner is stable and is thus a good candidate for dark matter. Now, in supersymmetry we need more than just 1 0 and ~ 126~, since the right-handed neutrino mass must be very large, on the order of the GUT scale or so. This would strongly break supersymmetry by the D-terms, so another Higgs - 1 2 6 (5 ~ index antisymmetric and self-dual) - must be used to cancel it. SO(10) symmetry allow the matter 16 to be coupled in the superpotential only to 10,126 and 120 (3 index antisymmetric) dimensional representations, so this new 1 2 6 cannot ~ change (7). Finally, we need to get two (practically) massless Higgs doublets, while heavy all colour triplets. This can be achieved by first introducing the Higgs representation 2 1 0 ~(4 index antisymmetric) and then by one fine-tuning of the parameters in the superpotential.

160

Such a renormalizable theory, with three matter 16, and with the Higgs and 2 1 0 ~(see Ref. 10) has the correct sector made of 1 0 ~ 1, 2 6 ~1268 , symmetry breaking pattern'' and has been shown to be the minimal grand unified theory,12 i.e. the most predictive one. 6. Further Developments The easy example with two generations and real parameters is instructive, and it can be generalized for a more realistic three generations set-up. It has to be said however, that one should not take the results in this case too seriously, since various uncontrolled corrections (like for example soft susy breaking terms) can in principle take place for the tiny first generation masses. The three generation generali~ation'~ roughly confirms the consistency of the type I1 seesaw with data, and predicts a quite large Ue3 x 0.15. Type I see-saw domination seems however still possible, especially after CP violating phases are considered.l4 An interesting and necessary check of the model is to calculate the proton decat rates of d = 5 operators. A preliminary though incomplete study15 shows that these decays could easily be too fast and thus rule out the minimal model. The question is of course of great importance, but a detailed analyses requires the knowledge of the SO(10) Clebsch-Gordan coefficients, which are now available.16 Another issue is the gauge coupling unification: recently it has been shown, that at least in some of the relevant regions of parameter space the threshold corrections are under control in spite of the large representations involved.17

Acknowledgments

It is a pleasure to thank the organizers of the conference for the stimulating environment and especially Goran Djordjevib for his excellent work. I thank my friends Charan Aulakh, Alejandra Melfo, Goran Senjanovib, and Francesco Vissani for collaboration. This work is supported by the Ministry of Science, Sport and Education of the Republic of Slovenia.

References 1. T. Yanagida, Proceedings of the Workshop o n Unified Theories and Baryon Number in the Universe, Tsukuba, Japan (1979) (edited by A. Sawada and

161

2.

3. 4. 5. 6.

7.

8.

9.

10. 11.

12. 13. 14.

15. 16.

17.

A. Sugamoto, KEK Report No. 79-18, Tsukuba); S. L. Glashow, Quarks and Leptons, Cargkse (1979), eds. M. LBvy, et al., (Plenum 1980 New York); M. Gell-Mann, P. Ramond and R. Slansky, Proceedings of the Supergravity Stony Brook Workshop, New York, (1979), eds. P. Van Niewenhuizen and D. Freeman (North-Holland, Amsterdam); R. N. Mohapatra and G. Senjanovit, Phys. Rev. Lett. 44, 912 (1980). G. Lazarides, Q. Shafi and C. Wetterich, Nucl. Phys. B181, 287 (1981); R. N. Mohapatra and G. Senjanovit, Phys. Rev. D23, 165 (1981). K. S. Babu and R. N. Mohapatra, Phys. Rev. Lett. 70, 2845 (1993). B. Bajc, G. Senjanovit and F. Vissani, hep-ph/0402140. B. Bajc, G. Senjanovik and F. Vissani, Phys. Rev. Lett. 90, 051802 (2003); B. Bajc, G. Senjanovit and F. Vissani, hep-ph/0110310. K. Inoue, A. Kakuto, H. Komatsu and S. Takeshita, Prog. Theor. Phys. 68, 927 (1982), [Erratum-ibid. 70 (1983) 3301; L. Alvarez-Gaume, J. Polchinski and M. B. Wise, Nucl. Phys. B221, 495 (1983). S. Dimopoulos, S. Raby and F. Wilczek, Phys. Rev. D24, 1618 (1981); L. E. Ibanez and G. G. RQSS, Phys. Lett. B105, 439 (1981); M. B. Einhorn and D. R. T. Jones, Nucl. Phys. B196,475 (1982); W. J. Marciano and G. Senjanovit, Phys. Rev. D25, 3092 (1982). D. G. Lee and R. N. Mohapatra, Phys. Rev. D51, 1353 (1995); C. S. Aulakh, B. Bajc, A. Melfo, A. RGin and G. Senjanovit, Nucl. Phys. B597, 89 (2001). C. S. Aulakh, K. Benakli and G. Senjanovic, Phys. Rev. Lett. 79,2188 (1997); C. S. Aulakh, A. Melfo and G. Senjanovik, Phys. Rev. D57, 4174 (1998); C. S. Aulakh, A. Melfo, A. R S i n and G. Senjanovik, Phys. Lett. B459, 557 (1999). C. S. Aulakh and R. N. Mohapatra, Phys. Rev. D28,217 (1983); T. E. Clark, T. K. Kuo and N. Nakagawa, Phys. Lett. B115, 26 (1982). X. G. He and S. Meljanac, Phys. Rev. D40, 2098 (1989); D. G. Lee, Phys. Rev. D49, 1417 (1994). C. S. Aulakh, B. Bajc, A. Melfo, G. Senjanovit and F. Vissani, Phys. Lett. B588, 196 (2004). H. S. Goh, R. N. Mohapatra and S. P. Ng, Phys. Lett. 33570, 215 (2003), and Phys. Rev. D68, 115008 (2003). K. Matsuda, Y. Koide, T. Fukuyama and H. Nishiura, Phys. Rev. D65, 033008 (2002) [Erratum-ibid. D 65 (2002) 0799041; T. Fukuyama and N. Okada, JHEP 0211, 011 (2002); B. Dutta, Y. Mimura and R. N. Mohapatra, hepph/0402113. H. S. Goh, R. N. Mohapatra, S. Nasri and S. P. Ng, Phys. Lett. B587, 105 (2004). C. S. Aulakh and A. Girdhar, hep-ph/0204097; T. Fukuyama, A. Ilakovac, T. Kikuchi, S. Meljanac and N. Okada, hep-ph/0401213; B. Bajc, A. Melfo, G. Senjanovik and F. Vissani, hepph/0402122. T. F'ukuyama, A. Ilakovac, T. Kikuchi, S. Meljanac and N. Okada, hep-ph/0405300. C. S. Aulakh and A. Girdhar, hep-ph/0405074.

ON THE DYNAMICS OF BMN OPERATORS OF FINITE SIZE AND THE MODEL OF STRING BITS

S. BELLUCCI AND C. SOCHICHIU* INFN - Laboratori Nazionali d i Frascati, Via E. Fermi 40, 00044 Frascati, Italy E-mail: [email protected]

We consider the discretization effects of a string bit model simulating the nearBMN operators in the super-Yang-Mills model. The fermionic sector of this model is altered by the so called species doubling. We analyze the possibilities to cure this disease and propose an alternative formulation of the fermionic sector free from the above drawbacks. Also we propose a formulation of string bits with exact supersymmetry, which produces however an even number of continuous strings in the limit J -+ m.

1. Introduction AdS/CFT correspondence is originally formulated1y2 as a duality relation between string theory on Ads5 x S5 space and conformal theory of N = 4 Super Yang-Mills (SYM) on four dimensional Minkowski space, which is the boundary of Ads (see Ref. 3 for a review). Being a true duality, AdS/CFT correspondence relates the weakly coupled SYM regime to the strongly coupled string theory and vice versa. This makes it a strong predictive tool, however, a difficult one to test, since there are no means to probe either SYM or string theory at strong coupling. Beyond this, the solution of string theory on Ads background is not known (although some progress towards this was recently achieved, see Ref. 4 and references therein). A few years ago, Berenstein-Malda~ena-Nastase~-~ proposed to consider the particular limit of Ads5 x S5 geometry known as pp-wave back*On leave from: Bogoliubov Lab. Theor. Phys., JINR, 141980 Dubna, Moscow Reg., RUSSIA and Institutul de Fizics Aplicatg AS, str. Academiei, nr. 5, Chiginh, MD2028 MOLDOVA. 162

163

ground, on which the string theory is s o l ~ a b l e This . ~ ~ ~situation of the overlap of the applicability of both string theory and SYM was further extended t o spinning string backgrounds10-12 (see also the contribution of G. Arutyunov to the present proceedings). In the absence of a more adequate tool to describe string theory near the BMN limit it was proposed t o use the string bit mode1.13-15 This model consists of the set of point-like interacting particles - string bits which in the limit of the number of particles going t o infinity is expected t o produce the continuous pp-wave string. Later on, however, the exact description in terms of spin chains was found in the planar limit of SYM.16>17 Nevertheless, string bits remained an efficient tool t o take care of nonplanarity which correspond, via AdS/CFT, to string production. Although the string bit model was quite useful in describing the dynamics of the bosonic sector in the near BMN limit, it suffers from internal inconsistencies due t o the fermionic spectrum doubling. In the theory of discrete fermions there is a well-known no-go theorem due to Nielsen-Ninomiya" which limits the possibilities of having discrete fermions. Therefore, one should make sure that these limits do not interfere with the AdS/CFT correspondence. For this purpose, one needs to prove the existence of a discrete supersymmetric model which in the continuum model reduces to the pp-wave string. This is the aim of the present note. It is based mainly on the Refs. 19, 20 where this problem was approached (for a staggered fermion approach see also Ref. 21).

2. Fermionic Doubling After fixing the permutation symmetry, the one-string sector of the string bit model is given by the following Hamiltonian:

with commutation relations

where i = 1,. . . , 8 are vector and, respectively, a = 1,. . . ,8 spinor indices of SO(8) appearing in the light-cone quantization of the pp-wave string. II

164

is the matrix in SO(8) spinor space given in terms of 16 x 16 dimensional y-matrices in chiral representation by

n2=1.

r I = y1-2 y y3-4 y,

(3)

The Hamiltonian (l), together with the shift operator

where

afn

= (l/a)(fn+i - f n ) are generated by the “supercharges” {&a, Qb}

= % b ( H -k P ) ,

where

Q =a

{ Q a , Qb} = 2 d a b ( H - P ) ,

C [piyien - ~ L ~ T I B+, arcnyien1 ,

(5)

(64

n

In spite of the fact that the supercharges generate according to Eq. (5) both the Hamiltonian and the shift operator, the supersymmetry fails because the supercharges do not commute

{&,&I# 0.

(7)

The fermionic doubling is better seen if we pass to the Fourier mode description

where fn stands for zn, p,, 8, and fin while f k represents their Fourier modes. In terms of Fourier modes the fermionic part of the Hamiltonian has the form

where the doubling is visible due to additional zeroes of the sin function in Eq. (9). Due to the fact that modes around these additional zeroes have low energy, they survive and contribute in the continuum limit, though they correspond to fast oscillating fields. In contrast, in the bosonic part one has a factor with the cosine function which becomes large in the continuum limit and one has no doubling there. Therefore, contrary to what can be expected, supersymmetry is not restored for a + 0.

165

In Ref. 21 it was proposed to use a staggered fermion approach in order to cure the fermion doubling. It consists in placing only a half number of fermions, e.g. in odd points ( n = 2k 1) one has only 8, while in even ( n = 2 k ) there are only 8,. Due to doubling the number of fermions in the continuum limit is the correct one. The supersymmetry is also restored in the continuum limit. However, at any finite lattice size it is still broken.

+

3. Almost supersymmetry

In fact, it appears possible to make the theory supersymmetric with the exception of a single mode by properly choosing the lattice derivative function. If this mode is placed in the high energy region of the model one may expect that it does not contribute in the continuum limit. On the lattice one can have various definitions of the derivative of a field. The only criterion is that the lattice derivative should give the usual derivative in the continuum limit. In general the lattice derivative could be written as the operator m

an,

where = condition that

8,

are the elements of the derivative matrix describes a lattice derivative reads

a.

The

Earn= 0, ESmm= 1, m

m

and

In terms of the Fourier mode expansion this means that the Fourier transshould have the limit form

&

&la+)

-+ ipk = 27rik.

Substituting the forward lattice derivatives d in Eq. (6) by a generic derivative 8 and requiring that we get the same bosonic part of the Hamiltonian, we find the condition of restoration of supersymmetry to be

S = i(a*a)f.

(13)

The condition (13) is trivially satisfied in the continuum limit, since (in Fourier mode) dk = -ipk is an anti-Hermitian operator. In contrast,

166

on the lattice 6' has both Hermitian and anti-Hermitian p a r h a Therefore, extracting the square root in Eq. (13) is a nontrivial procedure also because of its ambiguity. In terms of Fourier modes equation (13) is reduced to the extraction of the square root for each mode

where f k = f l is the sign, which can be chosen separately for each mode. The natural choice f k = sgn k yields a jump in the derivative function at k = f J / 2 . Also, at this point, the solution (14) fails to satisfy equation (13). The jump of the derivative function is related t o the nonlocality of the fermionic derivative

an,

2

= -(-l)n-m a

n(n-m) cos( 5 )sin( 7)

cos(

5 ) - cos(q).

Although Eq. (15) looks nonlocal and ugly, one can checkb that its square is a local operator given by

which is the next-to-neighbor second derivative. Let us see that the ambiguity of the momentum function does not affect physical observables like the energy. 4. Fermion Doubling/Nondoubling in the BMN

Correspondence Earlier we argued that having a well-defined supersymmetric discrete model is indispensable for confirming the self-consistence of the BMN limit in the fermionic sector. Now we want to define precisely the quantities to be compared. In the mode expansion of the pp-wave string described by Met~aev'?~ one has the energy of each mode given by the square root

aIn general, the Hermitian part of the derivative is the one responsible for fermionic doubling. bThe simplest way to do this is to use the Fourier transform.

167

where X is the 't Hooft coupling (for simplicity, other parameters here are put to unity). On the other hand, the Yang-Mills contribution at k loops yields a quantity of order X 2 k which corresponds to the k-th term in the expansion of the square root ( 1 7 ) . The planar Yang-Mills contribution up to three loops nowadays is well known and is described by integrable spin chains. 16,1722 N

In the string bit model the argument of the square root is replaced by 19,20

zn

N

41 + X 2 d k 2 .

(18)

Let us consider, for definiteness, the contribution of order X2 which corresponds to the Yang-Mills one-loop approximation. Then, the energy of the string bit mode in this approximation is given by 1-loop wk

N

A282 = P ( a * a ) k ,

(19)

where 8 is the usual bosonic forward lattice derivative afn = (l/a)(f,+l f n ) , while a * is the backward one a*f, = ( l / u ) ( f n- fn-l). The one loop energy (19) corresponds to the following term in the fermionic Hamiltonian: n

n

n

a2

where we integrated by parts after using Eq. (16), in order to express in terms of next neighbor derivatives a and a * . Let us note that the fields entering in Eq. (20) are inverse Fourier transforms of Hamiltonian eigenmodes rather than the original fermionic fields On and &. The transformation which relates them is a singular one, what makes it possible for the Hamiltonian to be ill-defined in terms of On and &, although in terms of there is no such problem. In particular, this means that the eigenvalues of the fermionic Hamiltonian are well-defined and compatible with supersymmetry in spite of the discontinuity in Eq. (14).

$A*)

$i*)

5. Doubling and Supersymmetry

In the last section we have shown that one can consistently define the supersymmetric model with the right spectrum, provided one does not insist

168

on the worldsheet ferrnionic structure. Let us show now that one can have more. Namely, let us construct an exactly supersymmetric string bit model. As we have established, supersymmetry is present when the fermionic discrete derivative and the bosonic one d satisfy the relation (13). The problem arises when we try to solve equation (13) for the fermionic 6 with a given bosonic d, corresponding to the next neighbor lattice derivative. In this section we want to show that the problem disappears if one relaxes the latter condition and allows also for an arbitrary bosonic derivative with unambiguous square root (13). Since such a derivative should vanish at the end of the Brillouin zone (k = f J / 2 , in the case of an even number of bits), the bosonic sector should be also doubled! In other words, since we are unable to avoid the fermionic doubling without the alteration of the model let us allow the doubling also for the bosons, then one can expect to have supersymmetry. The simplest choice for a “fermionizable” bosonic derivative function dk can be obtained by choosing it to be the “doubled” fermionic one

a

1 arcneu, = - sin (27rka) . 2a

(21)

This corresponds to the symmetric derivative of the bosonic fields, anewfn =

(1/2a)(fn+1 - fn-1).

Since the derivative (21) is purely imaginary, it solves also Eq. (13) for the fermionic derivative: = Pew. Then, the supersymmetric Hamiltonian is a combination of the bosonic part with derivative (21) and the naive fermionic part

a

which generates the supersymmetry algebra (5) together with the supercharges (6) and the shift operator (4), where we put 0 = = Pew. Let us note that no even bit n in Eq. (22) interacts with an odd one in both fermionic and bosonic sectors. Therefore, the model is doubled in the bosonic sector as well as in the fermionic one. This is confirmed by the inspection of the bosonic Hamiltonian which now reads

a

169

As it can be seen from Eq. (23), sin2(2rka) has two zeroes: one at the origin k = 0 and one at the edge of the Brillouin zone k = fJ. Each of these zeroes contributes in the continuum limit a superstring. Therefore one ends up with two superstrings. One can consider a derivative with any even number 2s of zeroes of the derivative function &. In this case one will have 2s superstrings in the continuum limit. Let us note that, since the supersymmetry is present at any stage of the discrete model, the continuum theory is also supersymmetric.

6. Discussion In this note we reviewed the fermionic spectrum of the string bit model. In the naive formulation the model suffers from inconsistencies due to the fermion spectrum doubling and supersymmetry breaking. Supersymmetry is not restored automatically in the continuum limit. One can “optimize” the fermionic sector for the given form of the bosonic Hamiltonian, in order for supersymmetry breaking to be minimal. In fact, one can have supersymmetry well-defined on all modes but one placed on the edge of the Brillouin zone. We have shown that this discrepancy does not affect, however, the physical spectrum of the model, since it depends on the square of the fermionic derivative which is a well-defined function. In the case one allows doubling also for bosonic part, one can construct a fully supersymmetric discrete theory. The continuum limit of this model is expected to contain a pair or, in general, any even number of superstrings. It is interesting to find out if one can use a correlated doubling removing procedure for both the bosonic and the fermionic sector, like the Wilson generalization or some other procedure, in order to get a single copy of string in the continuum limit and preserve supersymmetry. We hope to return to this elsewhere. Finally, our analysis leads us to the conclusion that the obstructions on the existence of discrete supersymmetric models are not enough to make the BMN limit in SYM theory problematic. Beyond this, there is no topological nor any other obstruction to this, since the theory is not chiral. In this sense, this case is similar to SU(2) chiral fermions on the lattice, where one can have a discrete theory preserving all perturbative ~ y m r n e t r i e s . ~ ~

Acknowledgements This work is supported by Russian grant for support of leading scientific schools 2052.2003.1, NATO Collaborative Linkage Grant PST.CLG.

170

97938, INTAS-00-00254 grant, RF Presidential grants MD-252.2003.02, NS-1252.2003.2, INTAS grant 03-51-6346, RFBR-DFG grant 436 RYS 113/669/0-2, RFBR grant 03-02-16193 and the European Community’s Human Potential Programme under contract HPRN-CT-2000-00131 Quantum Spacetime. We thank G. Arutyunov, N. Beisert and J. Plefka for useful discussions and the organizers of the Conference for warm hospitality and creative atmosphere. References 1. J. M. Maldacena, Adv. Theor. Math. Phys. 2,231 (1998), hep-th/9711200. 2. S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Phys. Lett. B428, 105 (1998), hepth/9802109. 3. 0. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri, and Y . 0 2 , Phys. Rept. 323,183 (ZOOO), hep-th/9905111. 4. I. Bena, J. Polchinski, and R. Roiban, Phys. Rev. D69, 046002 (2004), hepth/0305116. 5. D. Berenstein, J. M. Maldacena, and H. Nastase, JHEP 04, 013 (2002), hepth/0202021. 6. D. Berenstein and H. Nastase, On lightcone string field theory from super Yang-Mills and holography, hep-th/0205048. 7. D. Berenstein, E. Gava, J. M. Maldacena, K. S. Narain, and H. Nastase, Open strings on plane waves and their Yang-Mills duals, hepth/0203249. 8. R. R. Metsaev, Nucl. Phys. B625, 70 (2002), hep-th/0112044. 9. R. R. Metsaev and A. A. Tseytlin, Phys. Rev. D65, 126004 (2002), hepth/0202109. 10. S. Frolov and A. A. Tseytlin, Nucl. Phys. B668, 77 (2003), hep-th/0304255. 11. G. Arutyunov, S. Frolov, J. RUSSO,and A. A. Tseytlin, Nucl. Phys. B671,3 (2003), hep-th/0307191. 12. A. A. Tseytlin, Spinning strings and AdS/CFT duality, hep-th/0311139. 13. H. Verlinde, Bits, matrices and 1/N, hep-th/0206059. 14. D. Vaman and H. Verlinde, Bit strings from N = 4 gauge theory, h e p th/0209215. 15. J.-G. Zhou, pp-wave string interactions from string bit model, h e p th10208232. 16. J. A. Minahan and K. Zarembo, JHEP 03,013 (2003), hep-th/0212208. 17. N. Beisert and M. Staudacher, The n/ = 4 SYM integrable super spin chain, hepth/0307042. 18. H. B. Nielsen and M. Ninomiya, Phys. Lett. B105, 219 (1981). 19. S. Bellucci and C. Sochichiu, Phys. Lett. B571, 92 (2003), hep-th10307253. 20. S. Bellucci and C. Sochichiu, Phys. Lett. B564, 115 (2003), hep-th/0302104. 21. U. Danielsson, F. Kristiansson, M. Lubcke, and K. Zarembo, String bits without doubling, hepth/0306147. 22. N. Beisert, The su(2--3) dynamic spin chain, hep-th/0310252. 23. C . Sochichiu, Phys. Lett. B422, 227 (1998), hep-lat/9710009.

DIVERGENCIES IN &EXPANDED NONCOMMUTATIVE S U ( 2 ) YANG-MILLS THEORY

M. BURIC AND

v. RADOVANOVIC

Faculty of Physics, University of Belgrade P.O. Box 368, 11001 Belgrade, Serbia and Montenegro E-mail: [email protected], [email protected] We analyze the divergent part of the one-loop effective action for the noncommutative SU(2) gauge theory coupled to the fermions in the fundamental representation. We show that the divergencies in the 2-point and the 3-point functions in the 8linear order can be renormalized, while the divergence in the 4-point fermionic function cannot.

1. Introduction Although discussed for quite some time, the question of renormalizability of field theories on noncommutative (NC) R4 has not been settled in a satisfactory way yet. Noncommutativity of the coordinates, i.e., a relation of the type

[P, 27 = i

(1) puts the lower bound on the coordinate measurements, so one would expect that it also implies a natural ultraviolet cut-off and acts as a regulator. However, this idea has not been successfully implemented in models. Usually one represents the noncommutative field &(?) by a function &(z) on R4 and encodes the noncommutativity in the multiplication rule (*-product). For example, for constant P”, the field multiplication is given by the Moyal-Weyl product: i OPY

8

e y

=+?iqz)x(u)/v+”

. (2) This product is nonlocal, and so is the field theory defined by it. In this paper we study the renormalizability of NC S U ( 2 ) in the so called &expanded approach. As shown by Seiberg and Witten,4 noncommutative and commutative gauge theories are in some aspects equivalent. This equivalence is realized by a mapping relating the representation

4 ( x ) * x ( z ) = e5

171

172

of noncommutative gauge symmetry to the fields carrying the representation of its commutative counterpart. The Seiberg-Witten (SW) map is given as a series in powers of 0,". Classical action also can be expanded in 6: in the zero-th order it reduces to the action of ordinary gauge theory; additional terms can be treated as couplings. Renormalizability of the &expanded NC gauge theories has been addressed in Refs. 5-8 for the gauge group U (1) with and without fermionic matter and for its supersymmetric extension. Here we discuss the SU(2) theory coupled to fermions in the fundamental representation. Although the lagrangian of the SU(2) gauge theory technically differs from the abelian U(l), the conclusion concerning renormalizability is the same one. If fermions are massive, theory is not renormalizable. For massless fermions the theory is 'almost renormalizable', meaning that there is only one divergent term in the effective action which cannot be absorbed by the SW field redefinition scheme. The method which we use to calculate the divergent contributions is the background field method. As it was thoroughly explained in Refs. 8, 9, we skip the technical points here, referring also to the standard literature.'' The calculations for SU(2) are quite involved so we discuss only the 0-linear order; we find the divergent parts of the 2-point, 3-point and fermionic 4point functions. Previously, the results for U(1) in the 6-linear order were given in Refs. 5, 6; for the 2-point functions they were extended in Ref. 8 to the 02-order.

2. The Model The general construction of gauge theories on noncommutative space and their relation to the SW map were introduced in Refs. 1-3. The classical action for NC SU(2) Yang-Mills theory is given by

S=

J

's

d4x I$ * (iypD, - rn)G - 4

where the noncommutative field strength

F,,

= a,A,

-

a,A,

-

F,,

d4x Tr (F,,

*Fpy),

is defined with

i(A, *A, - A, *A,),

and the covariant derivative is

If we denote the commutative fields by

(3)

(4)

173

Dpd = q L $

- iApd

(7)

1

(matter, $(z) is in the fundamental representation of SU(2): T” = %), the noncommutative fields are related to the commutative ones by the SW map A p ( x ) = A p ( x )-

41 8’”” { A p ( x )&Ap(z) , + J ’ v p ( x ) } +...

(8)

1

1 i $(x) = $(z) - - ~ p u A p ( x ) d u $ ( z )4e””Ap(X)Av(x)$(x) . . . 2

+

+

to the first order in 0 . Inserting the expansion (8) into Eq. (3), we get the classical action in &linear order:3

To simplify the notation we introduced the symbol A:;; defined as A:,”,’ = 6:6;6;

- 6;6!6;

+ (cyclic c@y)

= - E ~ ~ ~ ~ E , ~(12) ~ x .

The bosonic &linear term S 1 , A vanishes (unlike in the U ( 1 ) case) because it is proportional to the symmetric coefficients dab‘ = Tr ({T”,Tb} Tc),and for S U ( 2 ) these coefficients are zero for all irreducible representations. In the functional integration we will treat ( A p l$) as a multiplet, so we want both gauge and matter fields to be real (or to be complex). Therefore, we write the Dirac spinor $J in terms of the Majorana spinors +1,2. For the charge-conjugated spinor QC = CqT Majorana spinors are given by $ 1 , ~ = ~(.ICI*$J~); vice versa: .1c, = $1 +id2 . To express the action in terms of Majorana spinors we have to use explicitly the form of Pauli matrices (i.e., the representation of the group generators). CTZ is antisymmetric and D ~ , C Tare Q symmetric matrices and therefore A: couples to Majorana spinors

174

A;. The action reads:

differently from A:,

This is the initial point for the quantization.

3. One-loop Effectiwe Action Background field method is one of the standard methods t o obtain divergent and finite quantum contributions to the classical action.1° In the first step, one expands fields around their classical configuration, i.e. splits the fields into the background (classical) part and the quantum correction: A,

--+

A,

+ A,

,

$1,2

+

A,’ + *1,2.

(15)

Quantum fields are denoted here by A,, Q1,2. The functional integration over the quantum fields in the generating functional is then performed; the effective action, r, is the Legendre transformation of the generating functional. In the saddle-point approximation, the integration gives: I’[Ap,$l,$zl = S[A,,$i,$2]

+ i S d e t logS(2)[A,,$1,$zl.

(16)

Sdet denotes the functional superdeterminant and S(’) is the second functional derivative of the classical action. For polynomial interactions, the second derivative can be obtained from the quadratic part of the action; it is an expression of the type:

175

where we wrote B instead of S(2)as, in fact, we have to include the gauge fixing term, too. We use the background field analogue of the Feynman't Hooft gauge,"

S,,

=

-12 /d4z(D,A,a)2,

with the gauge k i n g parameter equal to 1. D, is the background covariant derivative, D,A"" = d,A"" PbcA:Auc . To calculate the one-loop correction

+

i I"') = 1 log Sdet B = - STr logB 2 2 perturbatively, one expands logB. In our notation B can be written as a 3 x 3 block matrix

B11 Bl2 B13

B31 B32 B33 The submatrices B12, B13, B21 and B31 are Grassmann-odd, while the rest are Grassmann-even; the supertrace is defined by STrB = TrBll TrB22 - TrB33. B depends on the classical fields. One should keep in mind that A; (for a = 1,2,3) is a triplet, while $1 and $2 are dublets of the SU(2) group. In the absence of the interaction, B is the inverse propagator; in general case, one can separate the kinetic part: $gapS"bO 0 0 i0) O ip) + M .

B=(

We are to expand logB around identity I = diag(g,,bab, 1,l).To achieve this, we multiply B by C:ll 2 0

c = (o-i? 0 0

0 0

-ip

),

and then for the one-loop correction we obtain i I?(') = - STr log(BC) + STr log C-l 2 2 i i = - STr log(1 O-lMC) STr log C-' STr log 0 . (19) 2 2 The second and the third terms, being independent on the fields, can be included in the infinite renormalization. Note that now the propagator for

+

+

+1

176

all fields is 0 - l . The operator O-lMC defines the rules in the perturbation expansion. To get the structure of the expansion more clearly, we decompose MC into the sum

MC = NO

+ + + + T2 +T3 N1

N2

Tl

(20)

with respect to the number of fields - indices denote their number in a given term. NO,N1 and N2 originate from the commutative theory, while Ti, T2 and T3 are the noncommutative interactions linear in 6'. One can read No.. .T3 from the action (13-14), after the separation of the part quadratic in the quantum fields. The details of the structure of (20) are given in Ref. 9; we shall omit them here. Let us just mention that all operators have the same form Qii Q12

Qi3

Q31 -Q23

Q22

which is a consequence of the fact that the action was originally in Dirac spinors. In the case of the massless fermions all matrices simplify; No = 0.

4. Divergencies Introducing the decomposition (20) for the one-loop correction (19) we get i 00 (-)n+l r(l)= -2i s n log (I + O - ~ M C )= -2 n n=l

C

XSTr

~

(o-%J+ o-W1+ 0-1N2 + 0-1T1 + 0-9, + O-1T3y

Our notation allows to easily extract the contributions from different terms in this expansion. Parts of the effective action which give the 2-point functions have two classical fields. This means that the sum of indices in the monomials which we are interested in is equal to 2. Since there is an operator with the index 0, in principle, infinitely many terms contribute to the n-point functions. However, as we are calculating the divergencies only, we will need just finite number of terms: O-'No behaves as p-l in the momentum space, and the integrals become convergent for ( O - l N ~ ) kof a high enough degree. For the 2-point function in the 6'-linear order, potentially divergent terms are: STr(0-1 N1 O-lT1), STr( (O-lNoO-lNi 0-N1 - NO)0-2'1 ) , STr (0-NO0-T2) and STr ((0-No) 0-N1

+

'

177

'

+O- N0O-l NICI-~NO + I T 1 N1( 0-1 N o ) ~ O - ' T ~Adding ). their contributions with the appropriate coefficients after the dimensional regularization we obtain the divergent part

We shall need the covariant form of this formula; in m = 0 case it reads

The result ( 2 2 ) is nice: comparing it with the &linear correction for the 2-point functions in noncommutative electrodynamics18 we see that in the SU(2) case also, only fermionic propagator gets a correction; the correction is up to a factor the same as the one for U(1). This has further consequences. In NC QED we argued that the massive terms obstruct renormalization, as only for the case m = 0 one can redefine fields in such a way that the divergent terms disappear. The analysis can be repeated for SU(2) without change. Hence, we come to the conclusion: the NC gauge theories with massive fermions are not renormalizable. For this reason in the calculations of 3-point and 4-point functions we focus to the massless case. One might add that the calculations become so cumbersome that otherwise they would hardly be doable. The divergent parts of the 3-point functions are in the terms STr ((O-1N1)20-1T1) and STr (0-1N10-1T2). After a tedious but straightforward calculation one obtains the result which we present in its covariant form:

+ 2i47,FvaDV + i$7,paFva)$ 1 + -58 Gmpp4757P(D,Fa99 - fp"ap475YsYa(DaFa9$ 16 1 + 8 E,"ap(24757PFPaDp7b + 4rs7"DpFP")i)) *

(24)

4-fermionic vertex occured to be very important for the discussion of renormalizability. The corresponding 4-point function is relatively easy to find: to this end one can put A; = 0 in N 1 . . .T3. The divergent part

178

comes from STr ((O-1N1)20-1T2) and STr ((O-1N1)30-1Z’1); one gets

The equations (23), (24) and (25) are the principal results of this paper. 5 . Discussion

As it is well known,12 SW map is not unique. The redefinition of fields which it allows changes the action by terms of the form: d4x ( D , F p ” ) A P ) ,

AS$) =

/

+ iidn)ip$),

d4x (sip!D@)

(26)

(27)

written for the case of massless fermions. Here A?) and !D(n) are gauge covariant expressions of the n-th order in 0. The important thing in Eqs. (26 and 27) is that, except for the fields Fp” and lc,, a t least one derivative must be present. The obtained divergencies in Eqs. (23), (24) and (25) are such that they cannot be subtracted by the usual counterterms. However, if they were of the types (26-27), we could include them in the field redefinitions; thus the theory would be renormalizable in a generalized sense. Analyzing the divergencies, we see that the situation with the NC SU(2) is pretty much the same as with the electrodynamics. We already noted that the propagator correction (23) breaks the renormalizability, unless m = 0. The 3-point functions in the massless case present no problem, too. Gluon 3and 4-vertices get no quantum correction in the &linear order - there is no classical gluon vertex in Eq. (14) in that order, either. In this respect the behavior is again similar t o U(1), where 0-linear 3-photon vertices did exist: quantum one-loop corrections were precisely of the same form as the corresponding classical vertices.6 Further, the fermion-gluon 3-vertex (24) is already written in the form (27) with

(

+ ir;2uppFvp+ i ~ 3 ~ ~ , , , , , ~+564upVD2)lc,. F~~ (28)

i P E r (= l ) Op” 61Fpy

We observe that the fermions in the renormalized theory would be redefined via the gluon fields, i.e., noncommutativity would be mixed with (or partly immersed into) the gauge interactions.

179

However, there is a divergent term which spoils the renormalizability: the 4-point function (25). It predicts the current-current interaction, and there is no simple way to circumvent this coupling induced by noncommutativity. In fact, from the dimensional analysis we can see that in the massless case the propagators in NC gauge theories are renormalizable to all orders. Namely, for gauge bosons the n-th order corrections are of the form e . . . e A a ... a ~ , * k n

while for fermions they are e...eqya w wa $ . n

k

The power counting gives the number of derivatives k: for the gluon propagator k = 2 + 2n; for the fermions, k = 1 2n. This shows that one can, in all orders, transform (29-30) into the desired forms (26-27). The use of the background field method guarantees the gauge covariance. On the other hand, the vertices are potentially problematic. From the power counting we see that in the linear order the ‘wrong’ vertex could be

+

e(13;r$)2 ,

(31)

while in the quadratic order we could have, e.g.,

e 2 ~, 4 e2(&q2F.

(32)

These terms contain no derivatives and therefore break the SW generalized renormalization scheme. The term (31) is present for both U(1) and SU(2) theories coupled to fermions. An interesting fact, however, is that in both theories it has the same form (a different coefficient), namely e ~ ” q L ” p u 4 Y 5 Y ‘ ~4 Y P $

1

(33)

whereas the other combinations allowed by covariance, as, e.g. e ~ ’ ~ p v p u ~ y g y$oqyp+$ + or e ~ j y ~ y ,$$ y,,$, never show up. This opens the possibility that this divergence might cancel in a gauge theory based on the product of gauge groups. However, we are not in favor of a theory which needs too much fine tuning. Thus we are inclined to interpret our results (and the previous one^^^^^^) as an indication that NC gauge theories coupled to fermions are not renormalizable. But before a definite conclusion, one should certainly check whether the specific €J2-corrections,as 4A vertex (32), vanish. The

180

presence of the €J2F4divergence would prove non-renormalizability, possibly even for the pure gauge theories. It would also be interesting t o understand if there is some systematics in the behavior of various divergent terms.

Acknowledgment This work is supported by the project 1468 of the Serbian Ministry of Science, Technology and Development.

References 1. J. Madore, A n Introduction to Noncommutative Differential Geometry and its Physical Applications, Camb. Univ. Press, Cambridge (1999). 2. J. Madore, S. Schraml, P. Schupp and J. Wess, EUT.Phys. Jour. C16, 161 (2000). 3. B. Jureo, L. Moller, S. Schraml, P. Schupp and J. Wess, Eur. Phys. J. C21, 383 (2001). 4. N. Seiberg and E. Witten, JHEP 9909, 032 (1999). 5. A. A. Bichl, J. M. Grimstrup, H. Grosse, L. Popp, M. Schweeda and R. Wulkenhar, JHEP 0106, 013 (2001). 6. R. Wulkenhaar, JHEP 0203, 024 (2002). 7. A. A. Bichl, M. Ertl, A. Gerhold, J. M. Grimstrup, H. Grosse, L. Popp, V. Putz, M. Schweda and R. Wulkenhaar, Non-commutative U ( 1) Super- YangMills Theory: Perturbative Self-Energy Corrections, hep-th/0203141. 8. M. Burit and V. Radovanovit, JHEP 0210, 074 (2002). 9. M. Burit and V. RadovanoviC, JHEP 0402, 040 (2004). 10. M. E. Peskin and D. V. Schroeder, A n Introduction to Quantum Field Theory, Addison-Wesley, Reading, MA (1995). 11. A. G. Barvinsky and G. A. Vilkovisky, Phys. Rev. 119, 1 (1985). 12. T. Asakawa and I. Kishimoto, Comments on Gauge Equivalence in Noncommutative Theory, hep-th/9909139.

HETEROTIC STRING COMPACTIFICATIONS WITH FLUXES

G. L. CARDOSO, G. CURIO, G. DALL'AGATA AND D. LUST Humboldt Uniwersitiit z u Berlin, Institut fur Physik, Newtonstrasse 15, 12489 Berlin, Germany E-mail: gcardoso, curio, dallagat, [email protected] We review some of our work on compactifications of heterotic string theory to four dimensions in the presence of H-flux.

1. Introduction String compactifications in the presence of fluxes have been revived recently as an appealing way to address the moduli problem. Turning on fluxes in the ten-dimensional string theories produces, at the level of the effective four-dimensional action, a potential. Upon minimization of this potential one finds new vacua with (generically) less moduli. Moreover, one expects to find Minkowski vacua only when the deformation of the internal manifold balances the presence of the fluxes. Supersymmetric compactifications of the heterotic string in the presence of three-form H-flux requires the six-dimensional compactification manifold to be n~n-Kahler.'-~ More precisely, one finds that a non-vanishing H-flux is associated to a non-vanishing exterior derivative of the complex structure J,5

1 H = --e-'+ 2

* d(e84J).

(1)

This condition, which follows from the analysis of the supersymmetry rules, can also be understood from the perspective of rewriting the tendimensional effective action of heterotic string theory as a sum of squares.6 This rewriting results in the following term (among others),

181

182

This makes it clear that in order to obtain a vanishing four-dimensional effective potential V = 0, the condition (1) has to be imposed. Therefore, one obtains supersymmetric Minkowski vacua by fixing the (2,l) + (1,2) Hodge components of the three-form flux, H(’y1) and H(l>’),in terms of the internal geometry. The other Hodge components, i.e. H(3t0)and H(073), must vanish in order to have unbroken supersymmetry. In the following, let us review the rewriting leading to Eq. (2).

2. Fluxes and Torsion

The bosonic part of the Lagrangean up to second order in a’ is given by7

This action is written in the string frame and its fermionic completion makes it supersymmetric using the three-form Bianchi identity given by

dH = a’ (tr R+ A R+ - t r F A F ) ,

(4)

where the curvature RS is the generalized Riemann curvature built from the generalized connection V+ (i.e. from wf = w H ) . In the search for a BPS rewriting of Eq. ( 3 ) , 6 we will assume that the ten-dimensional space is given by the warped product of four-dimensional Minkowski spacetime with a six-dimensional internal space admitting an SU(3) structure. In order to consistently obtain that setting to zero the BPS-like squares implies a solution to the equations of motion, we also impose that the only degrees of freedom for the various fields are given by expectation values on the internal space and are functions only of the internal coordinates. To simplify the discussion we limit ourselves to the case with dilaton and warp factor identified, i.e. q5 = A, but the generalization of the following results is straightforward. After various manipulations, the action (3) can

183

be written as6

-

14 J d6y &eg+

Nmnpgmqgnrgps N~~~

In this expression the traces are taken with respect to the fiber indices a, b, . . ., whereas the Hodge type refers to the base indices m, n,. . . of the curvatures. The other geometrical objects appearing in the above expression are the Lee-form 3 0 JAdJ = - J"" qrnJnpldxP , (6) 2 the Nijenhuis tensor

NmnP= JmqdiqJnlP- Jnq6jq J,,',

(7)

and the generalized curvature R, which is constructed using the Bismut connection built from the standard Levi-Civita connection and a totally antisymmetric torsion T B proportional to the complex structure,

The action ( 5 ) will now be used to find the conditions determining the background geometry by demanding the vanishing of ( 5 ) . Setting the squares to zero yields 0

0

the vanishing of the Nijenhuis tensor the vanishing of some components of the generalized Riemann curvature constructed from the V+ connection, the vanishing of

1 d4+-0=O, 8

(9)

184

the vanishing of

1 H+-*e-'+d(e'+J) 2 0

=0,

(10)

the vanishing of

The vanishing of the Nijenhuis tensor states that the internal manifold is complex. The conditions on the R+ curvature can be translated into the requirement of SU(3) holonomy for the V- connection. The proof requires the identity b R :

cd

= R z ab - (dH)abcd

(11)

7

which relates the R+ and R- curvatures with the base and fiber indices swapped. Using this identity and the fact that d H gives higher order terms in a', the conditions on the base indices of R+ become conditions on the R- fiber indices, to lowest order in a', R-

(2,o) = R- (0,2)

= JabR-

ab

=0

'

(12)

These conditions precisely state that the generalized curvature R- is in the adjoint representation of SU(3) c SO(6) and therefore its holonomy group is contained in SU(3). The conditions in the gauge sector are that the gauge field strength is of type (1,l)and J traceless. On a complex manifold, the condition (10) yields

=0 , H(3,0)+(0>3)

(13)

where we also used (9). On the solution, R+ A R+ and F A F are of type (2,2). Therefore, the Bianchi identity (4) implies that (dH)(311)f(193)= 0, which is indeed satisfied. We conclude our discussion with remarks about the possible superpotential describing such vacua in the effective four-dimensional theory. It has been argued that a candidate superpotential describing the N = 1 vacua of the heterotic theory in the presence of fluxes is given by6,'

where R denotes a holomorphic (3,O)-form on Ms.This conjecture is motivated by the hope of getting the conditions (13) out of W = DW = 0.

185

For instance, assuming that H is constant under variation of the almost complex structure moduli and that &s2 = kiR xi,where xi are a set of ( 2 , l ) forms and ki are constants over the internal manifold, one obtains t h a t t h e (1,2) part of H $ d J is vanishing. On a complex manifold, this yields the locking of the t h r e e f o r m flux onto the geometry, namely

+

+

H(ly2)+ 8 J = 0 , and since H is a real form, it follows that

References 1. A. Strominger, Nucl. Phys. B274, 253 (1986). 2. G. L. Cardoso, G. Curio, G. Dall’Agata, D. Lust, P. Manousselis and

G. Zoupanos, Nucl. Phys. B652, 5 (2003). 3. K. Becker, M. Becker, K. Dasgupta and P. S. Green, JHEP 04, 007 (2003). 4. J. P. Gauntlett, D. Martelli and D. Waldram, Phys. Rev. D69,086002 (2004). 5. J. P. Gauntlett, N. w. Kim, D. Martelli and D. Waldram, JHEP 0111, 018 (2001). 6. G. L. Cardoso, G. Curio, G. Dall’Agata and D. Lust, JHEP 0310,004 (2003). 7. E.A. Bergshoeff and M. de Roo, Nucl. Phys. B328, 439 (1989). 8. K. Becker, M. Becker, K. Dasgupta and S. Prokushkin, Nucl. Phys. B666, 144 (2003).

SYMMETRIES AND SUPERSYMMETRIES OF THE DIRAC-TYPE OPERATORS ON EUCLIDEAN TAUB-NUT SPACE

I. I. COTAESCU West University of Tamigoara, V. P6rvan Ave. 4, RO-1900 Timigoara, Romania E-mail: [email protected]

M. VISINESCU Department of Theoretical Physics, National Institute for Physics and Nuclear Engineering, P.O.Box M. G.-6, Magurele, Bucharest, Romania E-mail: [email protected]

The role of the Killing-Yano tensors in the construction of the Dirac-type operators is pointed out. The general results are applied to the case of the four-dimensional Euclidean Taub-Newman-Unti-Tamburino space. Three new Dirac-type operators, equivalent to the standard Dirac operator, are constructed from the covariantly constant Killing-Yano tensors of this space. The Dirac operators are related among themselves through continuous or discrete transformations, In this space there is also a non covariantly constant Killing-Yano tensor connected with hidden symmetries. The Runge-Lenz operator for the Dirac equation in this background is written down pointing out its algebraic properties.

1. Introduction The (skew-symmetric) Killing-Yano (K-Y) tensors, that were first intrcduced by Yanol from purely mathematical reasons, are profoundly connected to the supersymmetric classical and quantum mechanics on curved manifolds where such tensors do exist.2 The K-Y tensors play an important role in theories with spin and especially in the Dirac theory on curved spacetimes where they produce first order differential operators, called Dirac-type operators, which anticommute with the standard Dirac one, D,.3 Another virtue of the K-Y tensors is that they enter as square roots in the struc186

187

ture of several second rank Stackel-Killing tensors that generate conserved quantities in classical mechanics or conserved operators which commute with D,. The construction of Carter and McLenaghan depended upon the remarkable fact that the (symmetric) Stackel-Killing tensor K,, involved in the constant of motion quadratic in the four-momentum p p

has a certain square root in terms of K-Y tensors f,,: Kpv =

The K-Y tensor here is a 2-form

fpu

fpxf.,A . .

(2)

= -f,,, which satisfies the equation

These attributes of the K-Y tensors lead to an efficient mechanism of supersymmetry, especially when the Stackel-Killing tensor K,, in Eq. (1) is proportional with the metric tensor gPv and the corresponding K-Y tensors in Eq. (2) are covariantly constant. Then each tensor of this type, fi, gives rise to a Dirac-type operator, Di, representing a supercharge of the superalgebra {Di, Dj} 0: D2&. The general results are applied to the case of the four-dimensional Euclidean Taub-Newman-Unti-Tamburino (Taub-NUT) space. The Euclidean Taub-NUT metric is involved in many modern studies in physics. This metric might give rise to the gravitational analog of the Yang-Mills i n ~ t a n t o nThe . ~ Kaluza-Klein monopole of Gross and Perry5 and of Sorkin‘ was obtained by embedding the Taub-NUT gravitational instanton into fivedimensional Kaluza-Klein theory. On the other hand, in the long-distant limit, the relative motion of two monopoles is approximately described by the geodesics of this space.? The Euclidean Taub-NUT space which is a hyper-Kahler manifold possessing three covariantly constant K-Y tensors. Using these covariantly constant Killing-Yano tensors it is possible to construct new Dirac-type operators3 which anticommute with the standard Dirac ~ p e r a t o r . * ~ ~ The Taub-NUT space also possesses a Killing-Yano tensor which is not covariantly constant. The corresponding non-standard operator, constructed with the general rule3 anticommutes with the standard Dirac operator but is not equivalent to it.lo This non-standard Dirac operator is connected with the hidden symmetries of the space allowing the construction of a conserved vector operator analogous to the Runge-Lenz vector of the Kepler problem. We shall discuss the behavior of this operator under

188

discrete transformations pointing out that the hidden symmetries are in some sense decoupled from the discrete symmetries studied here. 2. Dirac Equation on a Curved Background In what follows we shall consider the Dirac operator on a curved background which has the form (4)

D, = y p V p .

In this expression the Dirac matrices y,,are defined in local coordinates by the anticommutation relations {y,,y’} = 2gp’I and V, denotes the canonical covariant derivative for spinors. Carter and McLenaghan showed that in the theory of Dirac fermions for any isometry with Killing vector R, there is an appropriate ~ p e r a t o r : ~

which commutes with the standard Dirac operator (4). Moreover, each Killing-Yano tensor f p u produces a non-standard Dirac operator of the form A

1

D, = - i Y Y f ( f , ” V u - p ” r V p u ; p )

(6)

1

which anticommutes with the standard Dirac operator D,.

3. Dirac Operators of the Taub-NUT Space Let us consider the Taub-NUT space and the chart with Cartesian coordinates z@(p, v = 1,2,3,4) having the line element

+

+

where 2 denotes the three-vector 2 = ( T , 0, cp), (dZ)2 = ( d ~ l ) ( ~ d ~ ~ ) ~ ( d ~ ’ and ) ~ A’ is the gauge field of a monopole divA’=O,

-

2 B =rotA=p-.

r3

(8)

The real number p is the parameter of the theory which enters in the form of the functions P+r f ( r ) = g-l(T) = V-’(r) = -. (9) r

189

In the Taub-NUT geometry there are four Killing vector^.'^^^^ On the other hand, in the Taub-NUT geometry there are known to exist four KillingYano tensors of valence 2. The first three are covariantly constant

D,

fr' = 0 ,

i, j , k = 1 , 2 , 3 .

( 10)

These first three Killing-Yano tensors of the Taub-NUT space16 are rather special since they are covariantly constant. The f i define three anticommuting complex structures of the Taub-NUT manifold, their components realizing the quaternion algebra fifj+ fjfi

=-2&,

fifj

- f j f i = - 2 & . .k f k .

(11)

The existence of these Killing-Yano tensors is linked to the hyper-Kahler geometry of the manifold and shows directly the relation between the geometry and the N = 4 supersymmetric extension of the theory.2*18 For the theory of the Dirac operators in Cartesian charts of the TaubNUT space, it is convenient to consider the local frames given by tetrad fields .a(.) and 2"(x) while the four Dirac matrices y" satisfy {y", yP} = 2S@, where all of them are self-adjoint. In addition, we consider the matrix y5 = yiyiyiyd which is denoted by yo in Kaluza-Klein theory explicitly involving the time." The standard Dirac operator of the theory without explicit mass term is defined D, = y"V&, where the spin covariant derivatives with local indices, V", depend on the momentum operators, Pi = -i(ai - A&) and P4 = -4,and spin connection,12such that the Hamiltonian 0 p e r a t o r l ~ 7 ~ ~

can be expressed in terms of Pauli operators,

involving the Pauli matrices, ~ i .These operators give the (scalar) KleinGordon operator of the Taub-NUT space: A = -V,gp""V, = We specify that here the star superscript is a mere notation that does not represent the Hermitian conjugation because we are using a non-unitary representation of the algebra of Dirac operators. Of course, this is equivalent to the unitary representation where all of these operators are self-adjoint.12

190

Moreover we can give a physical interpretation of these Killing-Yano tensors defining the spin-like operators,

that have similar properties to those of the Pauli matrices. In the pseudoclassical description of a Dirac particle27l8the covariantly constant KillingYano tensors correspond to components of the spin which are separately conserved. Here, since the Pauli matrices commute with the Klein-Gordon operator, the spin-like operators (14) commute with H 2 . Remarkable the existence of the Killing-Yano tensors allows one to construct Dirac-type operators3

which anticommute with D , and y5 and commute with H.l0 Another Dirac operator can be defined using the fourth Killing-Yano tensor, but this will be discussed separately in Sec. 6. 4. Equivalent Representations

In Ref. 12 we have shown that in the massless case the operators Qi (i = 1,2,3) and the new supercharge QO = iD, = iy5H form the basis of an N = 4 superalgebra obeying the anticommutation relations {QA, Q B ) = ~

~ A B H A,B ~ ,... , =0,L2,3,

(16)

linked t o the hyper-Kahler geometric structure of the Taub-NUT space. In addition, we associate to each Dirac operator Q A its own Hamiltonian operator QA = - i y ' Q ~ , obtaining thus another set of supercharges

Qo =H ,

Qi = i [ H , C i ] ,

(17)

which obey the same anticommutation relations as Eq. (16). Thus we find that there are two similar superalgebras of operators with precise physical meaning. Obviously, since all of these operators must be self-adjoint we have to work only with unitary representations of these superalgebras, up to an equivalence. The concrete form of these supercharges depends on the representation of the Dirac matrices which can be changed at any time with the help of a non singular operator T , such that all of the 4 x 4 matrix operators of the Dirac theory transform as X -+ X' = T X T - l . In this way one obtains

191

an equivalent representation which preserves the commutation and the anticommutation relations. We used such transformations for pointing out that the convenient representations are equivalent to a unitary one.12 We note that some properties of the transformations changing representations in theories with two Dirac operators and their possible new applications are discussed in a paper by K1i~hevich.l~ For example, simple and convenient transformations can be chosen of the form:

U(d

E SU(2). where O(p,f) = e - i P 6 ( f i E U(2) = U(1) 8 SU(2) with This is because among these transformations one could find those linking equivalent Dirac operators. It is interesting to observe that the SU(2) transformations are generated just by the above defined spin-like operators as

<

+

If we take now = 2 9 5 with 151 = 1 and cp E [O,.rr], we find that +

-

..

~ (= e-iE.d/2 0 = 1 2 cos cp - is.:sin

cp ,

(20)

and after a little calculation we can write the concrete action of Eq. (19) as

Qb = U ( f ) Q o U + ( d = QO cos cp Q: = U ( d Q i U ' ( d

+

niQi sin cp

+

= Qi coscp - (niQo

,

&ijknjQk)

(21) sincp.

(22)

Hereby we see that the supercharges are mixed among themselves in linear combinations involving only real coefficients. In addition, we observe that these transformations correspond to an irreducible representation since the supercharges transform like the real components of a Pauli spinor. In other words, the usual SU(2) transformations $Q 4 = U+(()$Q of the spinor-operator

$b

'Q=

QO - iQ3

(Q2-iQl)

give just the transformations (21) and (22).

'

192

5. Discrete Transformations Let us focus now only on the transformations which transform the supercharges QA among themselves without to affect their form. From (21) and (22) we see that there exists particular transformations, Qk

, k = 1,2,3 ,

= UkQou:

(24)

where the matrix u k = diag(-ick112) is given by -ick E s u ( 2 ) . In addition, we consider the parity operator P = P-l = -y5 which changes the sign of supercharges, PQAP=-QA,

A=0,1,2,3.

(25)

Then it is not hard to verify that: the identity I = 1 4 , P and the sets of matrices u k and P u k (k = 1,2,3) form a discrete group of order eight the multiplication table of which is determined by the following rules

P2= I ,

Puk = UkP,

u12 = u22

U1U2 = U3 ,

= u32 = P ,

(26)

U2U1 = PU3 , ... etc.

We denote this group by GQ since it is a realization of the quaternion group Q which is isomorphic with the dicyclic group (2, 2, 2).20721In the usual representation of the y-matricesl8 its operators are defined by proper unitary matrices (which satisfy G-' = G+ and detG = 1,V G E GQ) constructed using the elements f l 2 , fiel, f i e 2 , fie3 of the natural realization of Q as a discrete subgroup of SU (2). 6. Hidden Symmetries and the Fifth Dirac Operator

In the Taub-NUT space, in addition to the above discussed covariantly constant Killing-Yano tensors, there exists a fourth Killing-Yano tensor , fY = 2p(dx+cosOdq) A d r + 2 r ( 2 r $ p ) ( l + L)sinedOAdp, P

having a non-vanishing covariant derivative r fy,.o;p = 2 ( 1 + -)r sin 0 . P

(27)

(28)

In Taub-NUT space there is a conserved vector analogous to the RungeLenz vector of the Kepler-type p r ~ b l e m l ~ ? ~ ~ (29)

193

where the conserved energy is E = ~ g , , x ~ x " . The components Kip" involved with the Runge-Lenz vector (29) are Stackel-Killing tensors satisfying the equations Ki(,v;~)= 0, Kip" = Ki",, and they can be expressed as symmetrized products of the Killing-Yano tensors f i , f y and Killing vector~.~~>~~ As in the case of the Dirac operators (15), one can use f Y for defining the fifth Dirac operator

called here the non-standard or hidden Dirac operator t o emphasize the connection with the hidden symmetry of the Taub-NUT problem. It is denoted by Q,' to point out its relation to the standard Dirac operator since it can be put in the form

Q,'

= iT [Qo

P

(:

g,v-l)] 0

9

where B, = 3 .Z / r . We showed that QF commutes with QO = H and anticommutes with QO and y5.lo This operator is important because it allowed us to derive the explicit form of the Runge-Lenz operator, 2, of the Dirac field in Taub-NUT background establishing its properties." We recall that the components of the conserved total angular momentum, f, and the operators Ri = F-lKi with F2= P42 - H 2 are just the generators of the dynamical algebra of the Dirac theory in Taub-NUT b a ~ k g r o u n d . ' ~ ? ~ ~ Starting with Q,' we can construct a new orbit, Ry, of GQ defining

(for k = 1 , 2 , 3 ) and observing that P Q Z P = -QZ , A = 0 , 1 , 2 , 3 . From the explicit form (32) we deduce that, in contrast with the operators of the orbits RQ and those of the orbit RY have more involved algebraic properties. We can convince that calculating

a~, +

H2(Q,')2 = H4

(33)

and it is worth comparing it with Eq. (16). The Dirac-type operators Q A are characterized by the fact that their quanta1 anticommutator close on the square of the Hamiltonian of the theory. No such expectation applies to the non-standard, hidden Dirac operators QZ which close on a combination of different conserved operators. Also from Eq. (33) it results that ( Q f ; ) 2#

194

(QZ)2 if A # B (because ? does not commute with u k ) . Moreover, one can show that the commutators [QX, Q';] have complicated forms which can not be expressed in terms of operators Q,'. Therefore, neither the commutator nor the anticommutator of the pairs of operators of this orbit do not lead t o significant algebraic results as the anticommutation relations (16) of the operators Q A , ( A = 0 , 1 , 2 , 3 ) . The hidden symmetries of the Taub-NUT geometry are encapsulated in the non-trivial Stackel-Killing tensors Kip",(i = 1,2,3). For the Dirac theory the construction of the Runge-Lenz operator can be done using products among the Dirac-type operators Q y and Qi. Let us define the operator:8

Ni = m { Q Y , The components of the operator lowing commutation relations

Q i } - JiP4.

fl commutes with H

[Ni, p4] = 0 , [Ni, Jj] = i & i j k N k , [Ni, Qo] = 0 [Ni, Q j ] = i E i j k Q k p 4 9

[Ni, N j ] = i S i j k J k F 2 +

(34) and satisfy the fol-

>

(35)

2 -&ijkQiH,

2

where F 2 = P42 - H 2 . In order to put the last commutator in a form close to that from the scalar case,16917 we can redefine the components of the Runge-Lenz operator, g , as follows:

Ki = Ni

+ -21H - l ( F

- P4)Qi,

having the desired commutation relation:8

7. Concluding Remarks In the study of the Dirac equation in curved spaces, it has been proved that the Killing-Yano tensors play an essential role in the construction of new Dirac-type operators. The Dirac-type operators constructed with the aid of covariantly constant Killing-Yano tensors are equivalent with the standard Dirac operator. The non-covariantly constant Killing-Yano tensors generates non-standard Dirac operators which are not equivalent to the standard Dirac operator and they are associated with the hidden symmetries of the space.

195

The Taub-NUT space has a special geometry where the covariantly constant Killing-Yano tensors exist by virtue of the metric being self-dual and the Dirac-type operators generated by them are equivalent with the standard one. The fourth Killing-Yano tensor fY which is not covariantly constant exists by virtue of the metric being of type D.The corresponding non-standard or hidden Dirac operator does not close on H as it can be seen from Eq. (33) and is not equivalent to the Dirac-type operators. As it was mentioned, it is associated with the hidden symmetries of the space allowing the construction of the conserved vector-operator analogous to the Runge-Lenz vector of the Kepler problem. Acknowledgments

M.V. thanks Professor Goran Djordjevic for his kind invitation and pleasant hospitality. This work is partially supported by a grant of the Romanian Academy. References 1. K. Yano, Ann. Math. 55,328 (1952). 2. G. W. Gibbons, R. H. Rietdijk and J. W. van Holten, Nucl. Phys. B404,42 (1993). 3. B. Carter and R. G. McLenaghan, Phys. Rev. D19,1093 (1979). 4. S. W . Hawking, Phys. Lett. A60,81 (1977). 5. D. J. Gross and M. J. Perry, Nucl. Phys. B226,29 (1983). 6. R. Sorkin, Phys. Rev. Lett. 51, 87 (1983). 7. M. F. Atiyah and N. J. Hitchin, The geometry and dynamics of magnetic monopoles, Princeton, Princeton University Press (1987). 8. I. I. Cotbscu and M. Visinescu, Gen. Rel. Grav. 35,389 (2003). 9. I. I. Cotbscu and M. Visinescu, Class. Quant. Grav. 21, 11 (2004). 10. I. I. Cotbscu and M. Visinescu, Phys. Lett. B502,229 (2001). 11. I. I. Cot&scu and M. Visinescu, Mod. Phys. Lett., A15, 145 (2000). 12. I. I. Cotbscu and M. Visinescu, Int. J. Mod. Phys. A16,1743 (2001). 13. I. I. Cotbscu and M. Visinescu, Class. Quant. Grav. 18,3383 (2001). 14. I. I. Cotbscu and M. Visinescu, J. Math. Phys. 43, 2978 (2002). 15. M. Visinescu, Int. J. Mod. Phys. A17,1049 (2002). 16. G. R. Gibbons and P. J. Ruback, Comrnun. Math. Phys. 115,267 (1988). 17. G. W. Gibbons and N. S. Manton, Nucl. Phys. B274,183 (1986). 18. J. W. van Holten, Phys. Lett. B342,47 (1995). 19. V. V. Klishevich, Class, Quant. Grav. 17,305 (2000). 20. A. 0. Barut and R. b c z k a , Theory of Group Representations and Applications, Warszawa, PWN (1977). 21. H. S. M. Coxeter and W. 0. J. Moser, Generators and Relations for Discrete Groups, Berlin, Springer-Verlag (1965).

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22. D. Vaman and M. Visinescu, Phys. Rev. D54,1398 (1996). 23. D. Vaman and M. Visinescu, Fortschr. Phys. 47, 493 (1999).

REAL AND P-ADIC ASPECTS OF QUANTIZATION OF TACHYONS

G.S. DJORDJEVIk AND LJ. NESIC Department of Physics, Faculty of Sciences P.O.box 22.4, 18001 NiS, Serbia and Montenegro E-mail: gorandj, [email protected]. yu A simplified model of tachyon matter in classical and quantum mechanics is constructed. p-Adic path integral quantization of the model is considered. Recent results in using padic analysis, as well as perspectives of an adelic generalization, in the investigation of tachyons are briefly discussed. In particular, the perturbative approach in path integral quantization is proposed.

1. Introduction An increasing number of researchers are testing again an (almost twenty year) old idea that padics and padic string theory' can be useful in attempts to understand ordinary string theory, D-brane solutions and various aspects of tachyon^.^?^ Possible cosmological implications4 are also an interesting matter. pAdic strings have many properties similar to ordinary strings, but p adic ones are much simpler. For example, one can find an exact action for this theory, as well as for practically all variations of padic string theory. In addition, it turns out that padic string could be a very useful model for testing Sen% conjecture on tachyon condensation in open string field theory (see e.g. Ref. 5). Corresponding padic classical solution in string field theory can be explicitly found. An important fact should be also considered, that is, padic string field theory in the p 4 1 limit reduces to the tachyon effective action.6 It might be the case that results for padic strings are applicable t o boundary string field theory. The next reason for further investigation is similarity between padic D-branes (without problems with strong coupling problems) and the results found in vacuum cubic string filed theory. It is interesting that the effective energy-momentum tensor is equivalent to that of nonrotating dust.' 197

198

Generally speaking, after almost twenty years of “discoveringL‘ padic strings, our understanding of physics on padic spaces is still very poor. From many point of view, including pedagogical one, it is very useful to understand (quantum) mechanical analogies of new, unfamiliar objects, including tachyons. Considering “padic tachyons L L we stress results in foundation of padic quantum mechanics, padic quantum cosmology and their connection with standard theory on real numbers (and corresponding spaces: Minnkowsky, Riemman ...) and adelic quantum theory. It has been noted that path integrals are extremely useful in this a p proach. Following S. Kar’s’ idea on the possibility of the examination of zero dimensional theory of the field theory of (real) tachyon matter, we extend this idea to the padic case. In addition, we note possibilities for adelic quantum treating of tachyon matter. After some mathematical background in Section 2, we present padic path integrals in Section 3. Section 4 is devoted to padic strings and tachyons. Simple quantum mechanical analog of padic tachyons is considered in Section 5. The paper is ended by a short conclusion and suggestion for further research.

2. pAdic Numbers and Related Analysis Let us recall that all numerical experimental results belong to the field of rational numbers Q. The completion of this field with respect to the standard norm I (absolute value) leads to the field of real numbers R = Qoo. Completion of Q with respect to the padic norms yields the fields of padic numbers Q p ( p is a prime number). Each non-trivial norm (valuation) on Q, due to the Ostrowski theorem, is equivalent either to a padic norm I I p or to the absolute value function. Any padic number x E Q p can be presented as an expansiong

lo

x = x”(x0 + x1p

+ x2p2 + . * . ),

v E 2,

(1) where xi = 0,1, . . . , p - 1. pAdic norm of any term in (1) is P-(”+~). The padic norm is the nonarchimedean (ultrametric) one. There are a lot of exotic features of padic spaces. For example, any point of a disc &( a ) = {x E Qp : Ix - alp 5 p”} can be treated as its center. It also leads to the total disconnectedness of padic spaces. For the foundation of the path integral approach on padic spaces it is important to stress that no natural ordering on Q p exists. However, one can define a linear order as follows: x < y if lxlp < Iylp, or when 1xIp = Iylp, when there is an index m 2 0 such that the following is satisfied:

199

z o = yo,z1 = yl, ..., x ,-1

= ym-l, z , < ym.l0 Generally speaking, there are two analysis over Q p . One of them is connected with map 4 : Q p -+ Q p , and the second one is related to the map ?c, : Q p + C. In the case of padic valued function, derivatives of 4(z) are defined as in the real case, using padic norm instead of the absolute value. pAdic valued definite integrals are defined for analytic functions M

n=O as follows:

Jd

M

b

4(t)dt =

4n x(bn+l

-ant1). (3) n=O In the case of mapping Q p -+ C , standard derivatives are not possible, and several different types of pseudodifferential operators have been introduced.’~~~ Contrary, there is a well defined integral with the Haar measure. Of special importance is Gauss integral

~ where xp(u) = exp(27ri{u},) is a padic additive character, and { u } denotes the fractional part of u E Q p . X,(CY) is an arithmetic, complex-valued function.’ To explore the existence of a vacuum state in padic quantum theory we need

where 2, is the ring of padic integers (a set of all padics z, lzlp 5 1) and R is the characteristic function of 2,. It should be noted that R is the simplest vacuum state in padic Quantum Theory. There is quite enough similarity between real numbers and padics and the corresponding analysis for the so called adelic approach in mathematics” and physics (see e.g. Ref. 4), that in a sense unifies real and all padic number fields. 3. Path Integral in Ordinary and p-Adic Quantum Mechanics According to Feynman’s idea,13quantum transition from a space-time point (z’,t’) to another (d’”’’) is a superposition of motions along all possible

200

paths connecting these two points. Let us remind the corresponding probaz*i s bility amplitude is (x”,t”(x’,t’) = C , e T [,I. Dynamical evolution of any quantum-mechanical system, described by a wave function $ ( x , t ) ,is given by $(XI’,

t”) = J

K(x”,t”;X I , t’)$(x‘, t’)dx‘,

Q-

(6)

where K(x”,t”;X I , t’) is a kernel of the unitary evolution operator U(t”,t‘). In Feynman’s formulation of quantum mechanics, K(x”,t”;x’, t’) was postulated to be the path integral

where x” = q ( t ” ) and x’ = q(t’). As we know, for a classical action S(X”,t”;x’, t’),which is a polynomial quadratic in XI‘ and X I , the corresponding kernel K (for one-dimensional quantum system) reads

( i 823 > + exp (2ai -S(X”, t“;x‘, t’) K(X”,t”;2 1 , t’) = -hax~~ax/ h It can be rewritten in the form more suitable for generalization (at least from the number theory point of view)

,/-

where & = = lalz2A,(-a). In (9), x,(a) = exp(-27ria) is an additive character of the field of real numbers R. D-dimensional generalization of the transition amplitude is:

where ,A

is defined as

(,) and x = (x,), a = 1 , 2 , . . . , D. By defining AO this A,-function satisfies the same properties as A.,

= 1, one can see that

201

In padic quantum mechanics dynamical differential equation of the Schrodinger type does not exist and padic quantum dynamics is defined by the kernel Kp(x”, t”;x’,t’) of the evolution operator:

$,(x”, t”) = Up(t”,t’)$,(x’, t’) =

s,,

Kp(x”,t”;x’,t’)$,(z’, t’)dx’. (12)

All general properties which hold for the kernel K(x”,t”; t’) in standard quantum mechanics also hold in padic case, where integration is now over Q,. pAdic generalization of (7) for a harmonic oscillator was done in” starting from

( h E Q and q , t E Q,). In (13) d q ( t ) is the Haar measure and padic path integral is the limit of a multiple Haar integral. This approach was extended in Ref. 14. A rather general path integral approach, valid for analytical classical solutions, was developed for quadratic padic quantum systems in Ref. 15 (in one-dimension)

Kp(x”,t”;

t’) = A,

1

s2s

(-2hm)

1 4 xp(--S(x”, 1t”;x‘, t’)) l%mlp h a2S

(14) and Ref. 16 (two-dimensional case). The obtained padic result (14) has the same form as (11) in the real case. The higher-dimensional padic kernel was also c0nsidered.l’ Considering real-ordinary and all padic quantum mechanics on the same foot, adelic formulation is also possib1e.l’ It could be a good starting point to consider quantum mechanical analog of “real“, “padic“ and, possibly, ‘adelic“ tachyons . 4. p-Adic Strings and Tachyons

The padic open string theory can be deduced from ordinary bosonic open string theory on a D-brane by replacing the integral over the real worldsheet by padic integra1.l’ A tachyon was defined as a particle that travels faster than light, and consequently has negative mass2. Surely, it is not a convincing case for the tachyon. Quantum field theory offers a much better framework for considering such a pretty exotic physical model. If we would carry out

202

perturbative quantization of the scalar field by expanding the potential around $J = 0, and ignore higher (cubic, ...) terms in the action, we would find a particle-like state with mass2 = V”(0).In the case of V”(0)< 0 we have again a particle with negative mass2, i.e., a tachyon. The physical interpretation is that the potential V ( 4 ) has a maximum at the origin and hence a small displacement of 4 will make it grows exponentially. It is associated with the instability of the system and a breakdown of the theory. Conventional formulation of string theory uses a first quantized formalism. In this formulation one can get a state-particle with negative muss2, i.e. tachyons. The simplest case appears in 26 dimensional bosonic string theory. This approach is, unfortunately, not suitable for testing tachyon‘s solutions,2 but there are supersting theories defined in (9+1) dimensions that have tachyon free closed string spectrum. In addition, some string theories contain open string excitations with appropriate boundary conditions at the two ends of the string. So, one can ask: is there a stable minimum of the tachyon potential around which it is possible to quantize the theory. In the last few years there many papers devoted to this problem have made some progress, but we will not consider them in this paper. In padic string theory all tree level amplitudes involving tachyons in the external states can be computed. The padic (open) string theory is obtained from ordinary bosonic (open) string theory on a D-brane by replacing the integrallg over the real world-sheet coordinates by padic integral associated with a prime number p. There have been somewhat different approaches, but we will not consider the constructions of all these theories. Let us see the exact effective action for the ptachyon field. It is described by the lagrangian21

This form, obtained by computing Koba-Nielsen amplitudes for a prime p, makes sense for all (integer) values of p . The classical equation of motion derived from (15) is

Besides the trivial constant solutions $J = 0,1, a soliton solution is admitted. The equations separate in the arguments and for any spatial

203

direction x we get

a gaussian lump whose amplitude and spread are ~ o r r e l a t e d . ~

5. Quantum Mechanical Analogue of Tachyon Matter Now, we will concentrate on a relatively new field theory - the field theory of tachyon matter was proposed by Sen a few years ago.2o The derivation of its action is based on a rather involved argument. The obtained form is pretty strange and different from the actions we used to be familiar ones

Let us consider padic analogue of the above action, originally considered as real one, i.e. 700 = -1, ,,v = ,S where p , v = 1,2, ..., n. T ( x ) is a padic scalar tachyon field and V ( T )is the tachyon potential: V ( T )= exp(-crT/2). In the bosonic case, n = 25, cr = 1, and for superstring n = 9, Q = fi.The square root appearing in action (18) (and its multiplication to tachyon potential) makes this theory so unusual. Here we examine a lower (zero-dimensional) mechanical analogue of the field theory of padic tachyon matter (whatever it would physically mean). As usually, the correspondence can be obtained by the correspondence xi + t , T + x , V ( T )+ V ( x ) .The corresponding zero-dimensional action reads

so = -

I

dtV(X)drn,

(19)

where integration has to be performed over padic time. From the above action it is not difficult to get the classical equation of motion

where function f ( x ) denotes

) . ( f

=

---.1 d V

v dx

Partial differentiation of padic valued function is well defined, although in this case it can be replaced by the ordinary one, because V = V ( x ) .

204

Keeping in mind that exponential padic function (the tachyon potential V ( z )= exp(-ax) should be understood as an analytic function with corresponding radius of convergenceg r l/p, we obtain as in the real case N

?+a* 2 = a .

By the replacements motion

j. = ~ y a, y =

mji

6 and

(21) = g we get the equation of

+ pg2 = mg,

(22) which describes motion of a particle of mass m moving in a constant (say gravitational, Newtonian) field with quadratic friction. It is interesting that this equation can be derived from the (padic) action

Surprisingly or not, the zero dimensional analog of the (Sen's) field theory of tachyon matter offers an action integral formulation for the system under gravity in the presence of (quadratic) damping. The solution of the equation of motion (22) reads Y = y o + - lm n(

2P

:),

g-Pv2 g-,v

with initial t = 0 conditions for position y(0) = yo and velocity w(0) = wo. This solution has the same form in the real and padic case, but the radius of convergence is rather different. Faced with the increasing interest in various aspects of tachyon field theory, including its padic aspect, this connection with the field theory of tachyons thorough action integral formulation seems worth mentioning and examining in general. Also, quantization of the theory in path integral language might be very useful and, as we know, very general (for real, p adic and adelic path integrals see, e.g. Ref. 17). However, a kernel of the operator of evolution that corresponds to the action (23) is still unknown, even in the real case. Because of that the square root and exponential for small P should be expanded. If we treat p padicaly small, in respect to padic norm, we obtain

205

We have already calculated the path integral for the particle in constant external field.14 Here we have a slightly changed form (y” = Y(T), = Y(O), h = 1)

It makes it possible to check the existence of the simplest tachyonic vacuum state (invariant in respect t o the evolution operator), of the corresponding quantum mechanical model, i. e.

1

Kp(Yll,7; d ,O)fl(lY’lp)dY’. (27) IY’ll Using (5) and (26) we find that for the existence of the “ground“ (52) state of (quantum) padic tachyons (here some technical details and case p=2 are omitted) the following is necessary 5 1 for 5

52(lY”lp) =

1, or 12y” -

$1

5 1 for P

I &l p >

15

(s) I I5I P

1. Possible physical implication on

constraints for quantities related to the starting tachyon action (18) will be discussed elsewhere. The existence of R state opens the “door“ for further adelic generalization and investigation of higher-excited states. As in the real case8 a quadratic damping effect could enter explicitly into the play treating it as a perturbation over classical solution of the equation for a particle in constant external field without friction. Damping effect would be ascertained and understood through its dependence of ,B term.22 6. Conclusion

In this paper we show that quantum mechanical simplification of the tachyon field theory, besides the real case, is possible in a padic context. Also, an adelic generalization looks possible, i. e. without some obvious principal obstacles. Path integral formulation of zero-dimensional padic tachyons has been done and some ”minimal” conditions for their existence have been found. Of course, how much this approach could be useful for deeper understanding of the whole string theory and of its tachyon sector requires time and further, in-depth research. The fact, that the exact effective tachyon action in the usual string theory is not known, while in

206

padic string theory it is, is quite enough motivation for this and similar and investigation. We would propose a few promising lines for further investigation. The exact formula for quadratic quantum padic systems in two and more dimensions17 could be useful for multidimensional generalization of padic tachyons. It is tempting to extend our approach t o 1 1 dimensional field theories, even nonlinear field theories would be here quite nontrivial problem. Finally, padic string theory could be a very useful guide to difficult question in the usual string theory. It requires deeper understanding of padic string theory itself, especially of closed padic strings (strings on p adic valued worldsheet and target space as well). It is a worthwhile task to explore padic strings in nontrivial backgrounds. It will naturally lead to noncommutative formulation on padic quantum theory and examination of the corresponding Moyal-product, introduced in a context of the Noncommutative Adelic Quantum Mechanics.23 Recently, the Moyal-product has been applied in the calculation of the Noncommutative Solitons in padic string theory.24

+

Acknowledgments The research of both authors was partially supported by the Serbian Ministry of Science and Technology Project No 1643. A part of this work was completed during a stay of G. Djordjevic at LMU-Munich supported by DFG Project "Noncommutative space-time structure-Cooperation with Balkan Countries". Warm hospitality of Prof. J. Wess is kindly acknowledged.

References 1. I.V. Volovich, Theor. Math. Phys. 71 (1987) 337. 2. A. Sen, Tachyon Dynamics in Open String Theory, hepth/0410103. 3. D. Ghoshal and T. Kawano, Towards p-Adic String in Constant B-Field, hepth/0409311. 4. G. S. Djordjevic, B. Dragovich, Lj. D. Nesic and I. V. Volovich, Int. J.Mod.Phys. A17 (2002) 1413. 5. D. Ghoshal and A. Sen, N d P h y s . B584 (2000) 300. 6. A.A. Gerasimov and S.L. Shatashvili, JHEP 0010 (2000) 034. 7. A. Sen, Znt. J . Mod.Phys. A18 (2003) 4869. 8. S. Kar, A simple mechanical analog of the field theory of tachyon matter, [hep-th/0210108],

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9. V. S. Vladimirov, I. V, Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics, World Scientific, Singapore 1994. 10. E. I. Zelenov, J. Math. Phys. 3 2 (1991) 147. 11. D. Dimitrijevic, G. S. Djordjevic and LJ. Nesic, Fourier Transformation and Pseudodifferential Operator with Rational Part, Proceedings of the Fifth General Conference of the Balkan Physical Union BPU-5, Vrnjacka Banja, Serbia and Montenegro, August 25-29, (2003) 1231. 12. I. M. Gel’fand, M. I. Graev and I. I. Piatetskii-Shapiro, Representation Theory and Automorphic Functions (Saunders, London, 1966). 13. R. P. Feynman, Rev. Mod. Phys. 20 (1948) 367; R. P. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals, McGraw Hill, New York, 1965. 14. G. S. Djordjevik and B. Dragovich, O n p-Adic Functional Integration, in Proc. of the I1 Math. Conf. in PriStina, PriStina (Yugoslavia) (1996) 221. 15. G. Djordjevik, B. Dragovich, Mod. Phys. Lett A12 (1997) 1455. 16. G.S. Djordjevic, B. Dragovich and Lj. Nesic, p-Adic Feynmads Path Itegrals, Int. Conference: FILOMAT 2001,Nis, 26-29 August 2001. FILOMAT 15 (2001) 323. 17. G. S. DJordjevic and LJ. Nesic, Path Integrals f o r Quadratic Lagrangians in Two and More Dimensions, Proceedings of the Fifth General Conference of the Balkan Physical Union BPU-5, Vrnjacka Banja, Serbia and Montenegro, August 25-29, (2003) 1207. 18. G. Djordjevic, B. Dragovich and Lj. Nesic, Infinite Dimensional Analysis, Quantum Probability and Related Topics 6 (2003) 179. 19. P. G. 0. Freund and E. Witten, Phys. Lett. B199 (198’7) 191. 20. A. Sen, JHEP 0204 (2002) 048. 21. L. Brekke, P.G.O. Freund, M. Olson and E. Witten, Nucl. Phys. B302 (1988) 365. 22. G.S. Djordjevic and Lj. NeSic, Quantum p-tachyons, work in progress. 23. G.S. Djordjevic, B. Dragovich and Lj. Nesic, Adelic Quantum Mechanics: Nonarchimedean and Noncommutative Aspects, Proceedings of the NATO ARW ”NONCOMMUTATIVE STRUCTURES IN MATHEMATICS AND PHYSICS”, Kiev, Ukraine, September 2000, Eds. S. Duplij and J. Wess, Kluwer. Publ. (2001) 401-415. 24. D. Ghoshal, JHEP 0409 (2004) 041.

SKEW-SYMMETRIC LAX POLYNOMIAL MATRICES AND INTEGRABLE RIGID BODY SYSTEMS

V. DRAGOVIC AND B. GAJIC Mathematical Institute SANU Kneza Mihaila 35, 11 000 Belgrade, Serbia and Montenegro E-mail: [email protected], [email protected] Skew-symmetric matrix Lax polynomials are considered. Few rigid body systems with such representations are presented. Lagrange bitop is completely integrable four-dimensional rigid body system. Algebro-geometric integration procedure, using Lax representation, for that system is performed. This integration is based on deep facts from the theory of Prym varieties such as the Mumford relation and Mumford-Dalalyan theory. Class of isoholomorphic integrable systems is established and importation class of its perturbations is observed, generalizing classical Hess-Appel’rot system.

1. Introduction

Lax representation is one of the most powerful tools in the modern theory of integrable systems. The Lax equation

,with L(X),A(X) being matrix polynomials in so called spectral parameter X were studied by different authors in the middle of seventies. Approach, based on the Baker-Akhiezer functions, is developed by Dubrovin (see Ref. 1). Different version of that theory, based on Ref. 2 is presented by Adler and van Moerbeke in Ref. 3. Both of the theories were applied to the rigid body motion in Refs. 4,5, 3, 6 and 7. Recently in Refs. 8 and 9 we have found Lax representation for the classical Hess-Appel’rot system of rigid body, and for a new completely integrable four-dimensional rigid body system that we called Lagrange bitop. It appears that in both cases matrices L and A are skew-symmetric. This was not the case in Manakov top.4 In the case of Lagrange bitop matrix 208

209

L(X) satisfies (in proper basis) Ll2 = L2l = L34 = L43 = 0,

(1)

which is excluded in both theories (Dubrovin’s and Adler-van Moerbeke’s). Analysis of the spectral curve and the Baker-Akhiezer function shows that dynamics of the system is related to certain Prym variety ll (which splits according to the Mumford-Dalalyan theory and evolution of divisors of some meromorphic differentials 0;. Then the condition (1) requires that some of these differentials have to be holomorphic during the whole evolution. Compatibility of this requirement with dynamics put a strong constraint on the spectral curve: its theta divisor should contain some torus. The conditions (1)and holomorphicity conditions create a new situation from the point of view of the existing integration techniques. Such systems we call isoholomorphic systems (see Ref. 9). Recently, in Ref. 13 we have constructed families of partially integrable rigid body systems with skew-symmetric Lax matrices. These families represent higher-dimensional generalizations of the Hess-Appel’rot system in any dimension. They are certain perturbations of n-dimensional Lagrange top and Lagrange bitop. 2. Lagrange bitop

We will consider motion of a heavy n-dimensional rigid body fixed at a point. Equations on semidirect product so(n) x sa(n) in moving frame are introduced in Ref. 6

Ail = [ M R I + [F’X],

= [F’RI 7

(2)

+

where the matrix I is diagonal, diag(l1,. . . ,In).Here Mij = (Ii Ij)Rij E so(n) is the kinetic momentum, R E so(n) is the angular velocity, x E so(n) is a given constant matrix (describing a generalized center of the mass), r E so(n). Then Ii Ij are the principal inertia momenta. The Lagrange bitop is system defined by (see Ref. 9):

+

0 13 = 1 4 = b and ’‘=I2=’

x12

.=(-:2

0

with the conditions a

# b,

0

0

0

x3 O4)

-x34

0

# 0 , l x l ~# l 1x341.

x12,x34

(3)

210

Proposition 2.1. T h e equations of m o t i o n ( 2 ) under the conditions (3) have a n L - A pair representation L(X) = [L(X),A(X)] ,(see Refs. 8,9) where

L(X) = X2C+ AM and C = ( u

+ r,

A(X)

= XX

+ R,

(4)

+b ) ~ .

One can observe that both leading terms in the operators L and A (matrices C and x) are skewsymmetric, while in Refs. 1 and 14 C is symmetric and M is skew-symmetric. Before analyzing spectral properties of the matrices L(X),we will change the coordinates in order to diagonalize the matrix C. In this new basis the matrices L(X) have the form L(X) = U-lL(X)V,

The spectral polynomial p(X, p ) = det

(E(X) - p . 1) has the form

where

P(X) = AX4 + BX3 + DX2+EX

+ F,

&(A)

= GX4

+ HA3 + I X 2 + J X + K .

21 1

Their coefficients

A = C t 2 + C i 4 = (C+,C+)+(C-.,C-), B=2C34M34+2C12M12 = 2 ( ( C + , M + ) + ( C - , M - ) ) ,

D

=~ =

3 + 22c12r12 ~ + 2c34r34

, 2+, M ? ~+ M ; ~+ M ? ~+ ~

(M+,M+)+(M-,M-)+2((c+,r+)+(c-,r-)),

+ + 2r14w4 + 2 r 2 3 ~+223r 2 4 ~+224r 3 4 ~ 3 4 = 2 (F+, M+) + F-, M-)), F= + ri3+ r:4 + r;3 + r;4+ ri4= (r+,r+)+ (r-,r-), E

= 2r12~1 2r13w3 2

G = c 1 2 c 3 4 = (c+, c-),

+

H = c34M12 + c12M34 = (c+, M - ) (c-,M + ) , I = c34r12r34c12M12M34 M23M14 - M13M24 = (C+,F-) + (C-,F+) + ( M + , M - ) , J = ~~~r~~ + ~~~r~~ + ~~~r~~~~~r~~ - r 1 3 ~-2r42 4 ~ 1 3

+

+

+

+

W+W + w-,r+), r34r12 + r23r14 - r13r24 = (r+,r-). integrals of motion of the system (a), (3). =

K are

=

M+, M- E R3 which correspond to

Mij E

We used two vectors so(4) according to

(7)

Here M$ are the j-th coordinates of the vector M+. The system (2), (3) is Hamiltonian with the Hamiltonian function

1 2

= -(M13013+M14fl14+M23023+M12012 +M34034) Sx12r12 +x34r34*

The algebra so(4) x so(4) is 12-dimensional. The general orbits of the coadjoint action are 8-dimensional. According to Ref. 6, the Casimir functions are coefficients of A', A, A4 in the polynomials [ d e t z ( A ) ] 1 / 2and - I2T T ( L ( A ) )and ~ they are J , K , E , F . Nontrivial integrals of motion are B , D, H, I , and they are in involution. When 1x121 = 1x341, then 2H = B or 2H = -B and there are only 3 independent integrals in involution. Thus, for 1x121 # 1x341, the system (2), (3) is completely integrable in the Liouville

212

Ratiu introduced in Ref. 6 two families of integrable Euler-Poisson equations: the generalized symmetric case, defined by the conditions

x arbitrary;

I1 = . . . = I,,,

and the generalized Lagrange case is defined by I1 =

IZ = a, I3 = ... = I,,

= b, xij = 0 if ( i 7 j )

6 ((17 2)7 (271)).

The system (2), (3) doesn’t fall in any of those families and together with them it makes the complete list of systems with the L operator of the form

+

L ( A ) = A2C AM

+ r.

More precisely, if x 1 2 # 0 then the Euler-Poisson equations (2) could be written in the form ( 4 ) (with arbitrary C ) if and only if the equations (2) describe the generalized symmetric case, the generalized Lagrange case or the Lagrange bitop, including the case X ~ = Z f ~ 3 4 . ~ ~ ’ Starting from the well-known decomposition so(4) = so(3) CB so(3), introducing

1 1 M2 = -(M+ - M - ) M I = -(M+ + M - ) 2 2 (and similar for R, r,x),where M+, M- are defined with (7), equations (2) become

a1+ rl x xl), = 2(hf2 x a2+r2x x 2 ) ,

A& = 2 p 1 x

r1= 2(r1 x nl), r2= 2(r2 x 02),

(8)

and

As it is shown in Ref. 9 after changing of variables equations (8) can be written in the form: 2i; = Pi(?&),

i = 1,2,

P ~ ( u=) -4u3 - 4uZBi+ 4uCi + Di,

i = 1,2;

where the constants Bi, Ci, Di, i = 1,2 depends of the first integrals and the parameters of the system. For more details see Ref. 9. From the previous relations, we have

213

So, the integration of the system (8) leads to the functions associated with the elliptic curves El, E2, given with:

Ei : y2 = P ~ ( u )i, = 1 , 2 .

(9)

3. Properties of spectral curve The L(X) matrix (4) for the Lagrange bitop (2), (3) is a quadratic polynomial in the spectral parameter X with matrix coefficients. There are two general theories describing the isospectral deformations with matrix coefficients. The first one, based on the Baker-Akhiezer function is developed by Dubr0vin.l The other one, given by Adler and van Moerbeke in Ref. 3 was based on Ref. 2. Both of these theories were applied to the rigid body motion in Refs. 4, 5 and 3, 6, 7, respectively. As it is shown in Ref. 9, non of these two theories can be directly applied in our case. So, we are going to make certain modifications, and then we will integrate the system (2), (3). As usual in the algebro-geometric integration, we consider the spectral curve

I? : det i ( A ) - p . 1) = 0.

(

By using ( 5 ) , ( 6 ) , we have

r : p4+p2(A;, +

+ 4p3p3*+ 4p4p4*)+[A12A34+ 2i(p3*p4- p3~4*>i2 = 0.

There is an involution (T : (A, p ) -+ (A, - p ) on the curve r, which corresponds to the skew symmetry of the matrix L(X). Denote the factor-curve by rl = r/o.

Lemma 3.1. (a) The curve rl is a smooth hyperelliptic curve of the genus g(r1) = 3. The spectral curve is a double covering of r l . The arithmetic genus of is ga(r)= 9. (b) The spectral curve r has f o u r ordinary double points Si, i = 1,. . . , 4 . The genus of its normalization is five. (c) The singular points Si of the curve are fixed points of the involution (T. The involution o exchanges the two branches of I' at Si . In general, whenever matrix L(X)is antisymmetric, the spectral curve is reducible in odd-dimensional case and singular in even-dimensional case.

214

Before starting the study of the analytic properties of the BakerAkhiezer function, let us give the formulae for (nonnormalized) eigenvectors of the matrix L(A). An eigen-vector of L , i.e. such that L(A)f = pf is given by

f1 =

+ p2)(iA34

- p ) - 2p(P3P3*

f 2 = W P 3 - iP4)(iP$

+ P4P4*) + 2 A 1 2 ( @ 3 f l i - P 4 P ; ) ,

+Pi),

+ '$4) [(iAi2- P ) ( i A 3 4 - P ) + 2i(P3Pi P;P4)] , f4 = (iP: + P i ) [(iAi2+ P ) ( i A 3 4 - P ) -t2i(P3Pi - P;P4)] . f 3 = (-P3

-

Corollary 3.1. The eigenvectors f' normalized by the condition

f: + f; + f; + fi = 1, f',

have different values in the points Sl,S,!l E points SiE I?.

which cover the singular

Following ideas of Dubrovin and Krichever (see Refs. 15, 16, 1, and bibliography therein) , we consider the next eigen-problem

where $k are the eigenvectors with the eigenvalue p k . Then $ k ( t , A) form 4 x 4 matrix with components $ i ( t ,A). Denote by cp: the corresponding inverse matrix. Let us introduce gj(t1 (A, P k ) )

=

$i(t,A)

'

$(t, A)

or, in other words g ( t ) = $ k ( t ) 18 p(t)k. Matrix g is of rank 1,and we have d$/at = --A$, dcp/dt = cpA, dgldt = [ g ,A].w e can consider vector-functions $ k ( t , A) = ($i(t,A), ...,$:( t , A ) ) ~ T

as one vector- function $(t,(A, p ) ) = ($'(t, (A, p ) ) , ...,7,b4(t,(A, p ) ) ) on the curve r defined with lCli(t,(A, p k ) ) = $ i ( t , A). The same we have for the matrix 9:. The relations for the divisors of zeroes and poles of the functions $2 i cpi in the afine part of the curve I? are:

(g;), = d j ( t )

+ dZ(t) - D,

-

D$,

(10)

where D, is the ramification divisor over X plane (see Ref. 1) and D , is the divisor of singular points, D: 5 D,. One can easily calculate deg D , = 16, degD, = 8.

215

Lemma 3.2.

(a) The matrix g has a representation

+

+

where a1 = L , a2 = P E L 2 ,a3 = P L L3. (b) For the Lax matrix L and for X i such that &(Xi) = 0 , it holds a3 = 0. (c) The matrix g has no poles at the singular points of the curve F. From now on we will consider all the functions in this section as functions on the normalization ? of the curve F. The matrix elements gj(t,(A, P k ) ) are meromorphic functions on the curve I?. We need their asymptotics in the neighborhoods of the points Pk, which cover the point X = 00. Let be the eigenvector of the matrix z ( X ) normalized in Pk by the condition = 1, and let be the inverse matrix for We will also use another decomposition of matrix elements of g : 9;.= dJ:p$ = $i+$.It is an immediate consequence of the proportionality of the vectors dJk and (and p k and g k ) . Using asymptotics of the functions i g$ in neighborhoods of the points Pk we getg that the divisors of matrix elements of g are

'& 4;

4:.

+:

I .

&

(9;) = 2

4;

+ J j - D, + 2 (PI + P2 + P3 + P4)

-

where the divisors c&, & are of the same degree deg & thet are given with:

+ P2, 2' = d1 + P2,

21 = d1

+ Pi, 2' = d2 + P i ,

2 2 = d2

Pa - Pj,

=

deg

&

=

5 and

+ P4, 2 4 = d4 + P3, J3 = d3 + P4, J4 = d4 + P3. 23

= d3

Let us denote with @ ( tA) , the fundamental solution of

(g+

A ( h ) ) q t , A)

= 0,

normalized with Q ( T ) = 1. Then, if we introduce functions

'$(t,T,(A,

Pk))

@ t ( hA)h"(T,(A, P k ) ) ,

= S

where h" are the eigenvectors of L(A) normalized by the condition h s ( t ,(A, p k ) ) = 1, it follows that

c,

216

Proposition 3.1. Ref. 9)

The functions

Gi satisfy the following properties (see Gi

(a) I n the a f i n e part o f f ' the function has 4 time-dependent zeroes which belong to the divisor d i ( t ) defined by formula (lo), and 8 time-independent poles, e.q.

(Gi(t,T , (A, p k ) ) ) a = d i ( t ) - 9, (b) At the points follows:

4 , the functions

di@,

7, (A,

4 2

degV = 8.

have essential sangularities as

P ) ) = exp [-(t -

G i ( t ,7, (A, P ) )

where Rk are given with

and di are holomorphic in a neighborhood of Pk,

Gi(~,7,(X,p)) = h i ( 7 , ( A , p ) ) , di(t,7,Pk) = S t + v : ( t ) z + 0 ( z 2 ) , with

Observeg that the following relation takes place on the Jacobian Jx(f'):

+

A(dj(t) odi(t))= A(dj(7)

+odj(7))

where A is the Abel map from the curve f' to Jac(f'). Thus, the vectors A ( d i ( t ) ) belong to some translation of the Prym variety Il = Pryrn(f'lr1). More details concerning the Prym varieties one can find in Refs. 10, 17 and 12. The natural question arises to compare two two-dimensional tori Il and El x Ez, where the elliptic curves Ei are defined in (9). 4. Geometry of the Prym variety II and Mumford's relation

Lemma 4.1.

The curves Ei are Jacobians of the curves

C1 :

v2 = P(A) + Q(A), 2

C2 : v2 =

Ci

-- &(A). 2

given by:'

217

Theorem 4.1. (For details see Ref. 9) (a) The Prymian II is isomorphic to the product of the curves Ei:

n = Jac(C1) x Jac(C2). The curve f' is the desingularization of rl x p CZ and C1 x p r l .

(b) (c) The canonical polarization divisor E of II satisfies

E = El x

0 2

+ 0 1 x Ez,

where 0i is the theta - divisor of Ei. Theorem 4.1 is based on Mumford-Dalalyan theory and it gives the connection between the curves E l , E2 and the Prym variety II. The Baker - Akhiezer function Q satisfies usual conditions of normalized 4-point function on the curve of genus g = 5 with the divisor 2, of degree d e g B = 8, see Refs. 16, 14. It was proven by Dubrovin in the case of general position, that 0; = g i j d X , i = 1, ..., 4 are a meromorphic differentials having poles at Pi and Pj, with residues vj and -v! respectively. But here we have that the four differentials O i l Oq, fli, 0: are holomorphic during the whole e v o l ~ t i o n . ~ We can say that the condition (1) implies isoholomorphicity. Let us recall the general formulae for v from14

where U = C X ( ~ ) U is( ~ certain ) linear combination of b periods U ( i )of the differentials of the second kind Og) , which have pole of order two at Pi; X i are nonzero scalars, and

(Here v is an arbitrary odd non-degenerate characteristics.) Thus, from

vz1 = 2112 = v43 = 2134 = 0 ,

(12)

using (11)we get that holomorphicity of some of the differentials 0; implies that the theta divisor of the spectral curve contains some tori (see ref. 9). In a case of spectral curve which is a double unramified covering

T : Fj r l ;

218

with g(r1) = g, g(p) = 2g - 1, as we have here, it is really satisfied that the theta divisor contains a torus, see Ref. 10. More precisely, following," let us denote by II- the set

IT-

= L E Pic2g-2?INmL = Krl, ho(L)is odd},

{

where Krl is the canonical class of the curve l?l and N m : Picf' -+ P i c r l is the norm map, see Refs. 10, 12 for details. For us, it is crucial that ITis a translate of the Prym variety II and that Mumford's relationlo holds

II- c QF. Complete formulae for Baker-Akhiezer function are given in Ref. 9. 5 . Conclusion

New family of rigid body systems with skew-symmetric Lax matrices have been constructed recently in Ref. 13. These systems are higher-dimensional generalizations of Hess-Appel'rot system constructed in any dimension. They are certain perturbations of higher-dimensional Lagrange tops. The four-dimensional Hess-Appel'rot system is perturbation of Lagrange bitop and belongs to the class of isoholomorphic systems. Thus, for the integration of the system we need to apply approach given it this paper.

Acknowledgement The research of both authors was partially supported by the Serbian Ministry of Science and Technology Project No 1643.

References B. A. Dubrovin, Funk. Analiz i ego prilozheniya 11,28 (1977 [in Russian]). P. van Moerbeke and D. Mumford, Acta Math. 143,93 (1979). M. Adler and P. van Moerbeke, Advances in Math. 38,318 (1980). S. V. Manakov, Funkc. Anal. Appl. 10, 93 (1976) [in Russian]. 0. I. Bogoyavlensky, Soviet Acad Izvestya 48,883 (1984) [in Russian]. T. Ratiu, American Journal of Math 104,409 (1982). T. Ratiu and P. van Moerbeke, 32,211 (1982). V. DragoviC, B. GajiC: An L-A pair for the Hess-Apel'rot system and a new integrable case for the Euler-Poisson equations on so(4) x so(4). Roy. SOC.of Edinburgh: Proc A 131,845 (2001). 9. V. DragoviC, B. GajiC, American Journal of Math., (2004), (to appear) 10. D. Mumford, A collection ofpapers dedicated to Lipman Bers (Acad. Press.) New York, 325 (1974). 1. 2. 3. 4. 5. 6. 7. 8.

219

11. S. G. Dalalyan, Uspekhi Math. Naukh 29, 165 (1974) [in Russian]. 12. V. V. Shokurov, Algebraic Geometry 111, 219 (Berlin: Springer-Verlag, 1998). 13. V. DragoviC, B. GajiC, ”Systems of Hess-Appel’rot type”, Preprint SISSA, (2004). 14. B. A. Dubrovin, Uspekhi Math. Nauk. 36,11 (1981) [in Russian]. 15. I. M. Krichever, Uspekhi Math. Naukh 32, 183 (1977). 16. B. A. Dubrovin, I. M. Krichever and S. P. Novikov, Dynamical systems l V , Berlin: Springer-Verlag, 173. 17. J. D. Fay, Lecture Notes in Mathematics vol. 352, Springer-Verlag, (1973).

SUPERSYMMETRIC QUANTUM FIELD THEORIES

D. R. GRIGORE Department of Theoretical Physics, Institute of Atomic Physics, Bucharest-Miigurele, Romcinia E-mail: [email protected] We consider some supersymmetric multiplets in a purely quantum framework. A crucial point is to ensure the positivity of the scalar product in the Hilbert space of the quantum system. For the vector multiplet we obtain some discrepancies with respect to the literature in the expression of the super-propagator and we prove that the model is consistent only for positive mass. The gauge structure is constructed purely deductive and leads to the necessity of introducing scalar ghost superfields, in analogy t o the usual gauge theories. Then we consider a supersymmetric extension of quantum gauge theory based on a vector multiplet containing supersymmetric partners of spin 3/2 for the vector fields. As an application we consider the supersymmetric electroweak theory. The resulting self-couplings of the gauge bosons agree with the standard model up to a divergence.

1. Introduction

The supersymmetric gauge theories are constructed using the so-called vector supersymmetric multiplet.’ The justification for this choice comes from the analysis of the unitary irreducible representations of the N = 1 supersymmetric extension of the Poincark group; there are two irreducible massive representations R l p N [m,01 @ [m,1/21 @ [m,1/21 @ [m,I], and 01 [m,1/2]@[m,11@[m, 11 @ [m,3/21, containing a spin 1 system; here [m,s] is the irreducible representation of mass m and spin s of the Poincark group.) The standard vector multiplet is constructed such that the oneparticle subspace of the Fock space carries the “simplest” representation

-

%/2.

We use entirely the quantum f r a m e ~ o r k ~avoiding -~ the usual approach based on quantizing a classical supersymmetric theory. In this way we do not have the complications associated to the proper mathematical definition of a super-manifold and we do not need a quantization procedure. A rigorous treatment of the perturbative aspects of these models can be pro220

221

vided using the Epstein-Glaser framework. We start this program analyzing the layout of the model, that is the construction of the quantum multiplet, its gauge structure and the expression of the interaction Lagrangian (or,in the language of perturbation theory, the first order chronological product). The main results are the following. The vector model is consistent only for positive mass. The origin of this result comes from the condition of positivity of the scalar product in the Hilbert space. In the standard literature one constructs the Hilbert space by applying polynomials in the free fields of the model on the vacuum. (In the language of the reconstruction theorem from axiomatic field theory this amounts to the construction of the Borchers algebra). However, this is not enough: one needs to provide the expression of the scalar product and prove that it is positively defined. The comparison with the literature dedicated to this subject shows that this point has not been analyzed. The structure of the Hilbert space of the model is never described explicitly. One can infer it only from the Feynman rules, more precisely the expressions of the (super)propagators. The point is that the assumption that the BRST quantization procedures gives automatically a positively defined Hilbert space structure is wrong. An explicit check should be made and this leads to some conditions on the mass of the supersymmetric multiplet. (For another point of view concerning supersymmetric positivity see Ref. 5). We can determine the gauge structure of this model in a deductive way: it coincides essentially with the expression from the literature but, because the mass of the multiplet is positive, we need to introduce some scalar ghost superfields. We are also able to determine the general expressions for the Feynman super-propagators in a purely deductive way; some discrepancy with the standard literature appears. Similar results are available for extended supersymmetries.6 Next we construct a new vector multiplet based on the representation R1. This multiplet exists for all values of the mass, its gauge structure is very similar to the gauge structure of the usual gauge theories and the interaction Lagrangian can be obtained by simply replacing fields by superfields in the Lagrangian of the standard model. In this way we obtain a supersymmetric extension of the standard model. One can conclude that the new vector multiplet proposed for the first time in Ref. 4 and based only on chiral superfields is a more natural object and it remains as a serious candidate for a possible supersymmetric extension of the standard model.

222

2. Quantum Supersymmetric Theory We remind here the definition of a N = 1 supersymmetric theory in a pure quantum context. Suppose that we have a quantum theory of free fields; this means that we have the following construction: (a) H is a Hilbert space of Fock type (associated to some one-particle Hilbert space describing some choice of elementary particles) with the scalar product (., .); (b) R E H is a special vector called the vacuum; (c) U a ,is~a unitary irreducible representation of inSL(2,C) the universal covering group of the proper orthochronous Poincark group such that a E R4 is translation in the Minkowski space and A E SL(2,C); (d) b j , j = 1,.. . ,NB (resp. f ~ ,A = 1,.. . , N F ) are the quantum free fields of integer (resp. halfinteger) spin. We assume that the fields are linearly independent up to equations of motion and that the equations of motion do not connect distinct fields. In practice, one considers only particles of spin s 5 2. For the standard vector model we consider only the case s 5 1. For the new vector model we consider the more unusual case 1 5 s 5 3/2. If one considers higher-spin fields (more precisely s 2 l), as we have done in Refs. 4, 7, it is necessary to extend somewhat this framework: gauge fields must be considered and we will describe them in the indefinite metric approach (Gupta-Bleuler). That is, we assume the existence of the following objects: (a) A gauge charge operator Q verifying Q 2 = 0 ; (b) A non-degenerate sesqui-linear form < . > which becomes positively defined when restricted to a factor Hilbert space K e r ( Q ) / l m ( Q ) Q ; is called gauge charge and we denote by At the adjoint of the operator A with respect to < ., . >; (c) Hphys = K e r ( Q ) / l m ( Q )will be the physical space of our problem; (d) The interaction Lagrangian t ( x ) is some Wick polynomial acting in the total Hilbert space H and verifying the conditions a,

[Q,t(x)l=ia#’(x)

(1)

for some Wick polynomials t”(x). This condition guarantees that the interaction Lagrangian t ( x ) factorises to the physical Hilbert space K e r ( Q ) / l m ( Q )in the adiabatic limit, i.e. after integration over x; the condition (1) is equivalent to the usual condition of (free) current conservation. The condition (1) has far reaching physical consequences: under some reasonable additional assumptions one can prove that the usual expression of the interaction Lagrangian for a Yang-Mills model is unique, up to trivial terms. It is desirable to generalize this scheme to supersymmetric theories. We note here that our framework is different from the usual treatment of quantum gauge invariance based on the construction of a classical

223

field theory with BRST invariance, some quantization procedure and the quantum action principle imposed on the Green functions. It seems that for ordinary gauge theories the two formalisms give identical results. But this is not the case for the supersymmetric gauge theories. This makes our alternative approach far from being superfluous. Now we define the notion of supersymmetry invariance of the system of Bosonic and Fermionic fields considered above. Suppose that in the Hilbert space H we also have the operators Q a , a = 1 , 2 such that: (i) the following relations are verified:

[&a, P p ]

UiIQaUA = AabQb ,

= 0,

(4)

here Pp are the infinitesimal generators of the translation group and Q b (&b)t; (ii) The following commutation relations are true: i [ Q a , b ] =p(a)f >

{&a,

f}= q(a)b,

=

(5)

where b (resp. f ) is the collection of all integer (resp. half-integer) spin fields and p , q are matrix-valued polynomials in the partial derivatives 8, (with constant coefficients). These relations express the tensor properties of the fields with respect to (infinitesimal) supersymmetry transformations. If these conditions are true we say that Qa are super-charges and b, f are forming a supersymmetric multiplet. As emphasized in Ref. 4, the matrix-valued operators p and q are subject to various constraints: ( 1 ) From the compatibility of (5) with Lorentz transformations it follows that these polynomials are Lorentz covariant. (2) Next, we start from the fact that the Hilbert space of the model is generated by vectors of the type 9 = b(zp) f(x,) R E H . It follows that the relations (3) are true if the left hand sides commute with all the fields of the model. Using the (graded) Jacobi identities we obtain:

n

n

[&a, [ Q b , W ] ]

[&a,

[QG,

w ] l + [&a,

[&a,

= -(a

~ 1 =1 -2i

++

b) ,

apw;

7

(6)

here w = b , f and [.,.I is the graded commutator. (3) The equation of motion are supersymmetric invariant. (4) The (anti)commutation relations have the implication that one and the same vector from the Hilbert space H can be expressed in distinct ways. This means that the supercharges are

224

well defined via (3) iflsome new consistency relations are valid following again from graded Jacobi identities; the non-trivial ones are of the form:

[b(z),{ f ( ~ 7 &all ) = -{f(y)

[&a, b(z)l}

.

(7)

(5) If a gauge supercharge Q is present in the model, then it is usually determined by relations of the type (5) involving ghost fields also, so it means that we must impose consistency relations of the same type as above. Moreover, it is desirable to have

{Q,Qa) = 0 , {Q,Q E } =

(8)

0;

this implies that the supersymmetric charges Qa and Qa factorizes to the physical Hilbert space Hphys = K e r ( Q ) / l m ( Q ) . These new consistency relations are of the type ( 6 ) with one of the supercharges replaced by the gauge charge: {&a,

[Q,bl}

= - {Q,[&a, bl>

>

[&a,

{Q,f}] = - [Q,{ & a , f}]

*

(9)

( 6 ) A relation of the type (7) must be also valid for the gauge charge:

[b(z),{ f ( ~Q)1 ) , = -{f(~),[Q,b ( z ) l ) .

(10)

(7) To have Q2 = 0 we must also impose

{ Q , [Q,bl)

=0

[Q,{ Q ,f)] = 0 .

(11)

(8) In the presence of a gauge structure one can relax the condition (3). Indeed, every operator A which (anti)commutes with the gauge charge Q factorizes t o an operator 7r(A) : K e r ( Q ) / l m ( Q ) -+ K e r ( Q ) / l m ( Q ) . In particular, the supercharges anticommute with Q by construction and Pp also commute with Q. So, we can impose Eq. (3) only on the physical Hilbert space7 i.e. we replace Qa -+ 7r(Q,), QE -+ 7 r ( & ) , Pp -+ r(P,>. All these conditions are of pure quantum nature i.e. they can be understood only for a pure quantum model. Some of them do not have a classical analogue so we can interpret the obstacles in constructing supersymmetric quantum models (associated to some classical supersymmetric theories) as some quantum anomalies. It seems to be an essential point to describe supersymmetric theories in superspace. We do this in the following way. We consider the space HG = G 8 H where G is a Grassmann algebra generated by Weyl anticommuting spinors 8, and their complex conjugates & = (8,). and perform a Klein transform such that the Grassmann parameters 0, are anticommuting with all Fermionic fields, the supercharges and the gauge charge.

225

The field operators acting in HG are called superfields. Quite generally one associates to every Wick polynomial w (x) its supersymmetric extension W = S(W) according t o the formula:

W ( x ,8,e) E exp ( i P Q , - i6'Q8)

W(Z)

exp (-ida&,

+ ie"Qa) ,

(12)

and we interpret the exponential as a (finite) Taylor series. Of special interest are the superfields constructed as in Refs. 2, 8 according to the preceding formulz taking w = b, f . It is a remarkable fact that only such type of superfields are really necessary, so in the following, when referring to superfields we mean expressions given by Eq. (12). We will call them super-Bose and respectively super-Fermi fields. For convenience we will denote frequently the ensemble of Minkowski and Grassmann variables by = @,8,s). Moreover, one postulates that the interaction Lagrangian t should be of the form

x

(13) for some supersymmetric Wick polynomial T . This hypothesis makes possible the generalization of the Epstein-Glaser approach to the supersymmetric case as it is showed in Ref. 4. Concerning the gauge invariance of the model there are two possible attitudes. One is to impose only Eq. (1); this "minimal" possibility is certainly consistent from the physical point of view but in general one loses the unicity results concerning the interaction Lagrangian. One can hope to keep this unicity result if one finds out a supersymmetric generalization of Eq. (1). A natural candidate would be the relation:

[Q,T ( X ) ]= ia,T.(X)

+ .. . ,

(14)

where by . . . we mean total divergence expressions in the Grassmann variables. It is clear that Eq. (14) implies Eq. (1) but not conversely. We call Eq. (14) the condition of supersymmetric gauge invariance because it involves only superfields in contrast to Eq. (1). In Ref. 4 we have showed that the stronger condition (14) can be imposed for the 01-vector model and indeed the unicity argument concerning the interaction Lagrangian holds. However for the Ol,z vector model the situation is not so good. If one uses only the LLminimal" gauge invariance condition (1) then one loses the unicity of the interaction Lagrangian. (However some other limitations might come from the condition of gauge invariance in higher orders of perturbation theory.) If one tries to impose the supersymmetric version

226

(14) one finds out that the usual expressions for the interaction Lagrangian suggested by the literature do not fulfill it. Of course, it is in principle possible to find alternative expression for the interaction Lagrangian such that Eq. (14) is true, but this possibility seems to be rather unprobable. So our results concerning the construction of the interaction Lagrangian for the R 1 p vector model must be considered as a criticism of the traditional approaches. The construction of the ghost and anti-ghost multiplets (which are needed in the construction of supersymmetric gauge theories) is done by inverting the spin-statistics assignment: the integer (resp. half-integer) spin fields have Fermi-Dirac (resp. Bose-Einstein) statistics.

3. The Vector Multiplet By definition, the vector multiplet has the following content: the Bosonic fields are some (real) scalars b ( j ) , j = 1,.. . ,s and a real vector field b,; the Fermionic fields are some Majorana fields of spin one-half f(A), A = 1 , . . . ,f . Let us consider that C is one of the scalar fields b(j) (or a linear combination of them). Now we define the following superfield: V E s(C) using formula (12); it is clear that one has the reality condition V t = V. Moreover the generic expression of V must be V(X,

e,B) = C ( X ) + ex(.) + ~z(x)+ e2 4 ( ~+)e-2 4t (x) +(e&7) .,(x) + e2 E ~ ( x+) e2 ex(^) + e2g2 d ( ~ ) ,

(15)

where the fields appearing in the expansion must linear combinations of the basic fields taking into account Lorentz properties and statistics. The action of the supercharges on the components of the multiplet is given in a compact way if we introduce the covariant derivative operators

acting on superfields. Then one imposes

i [ ~ V ( X~, 8,,

e)]= ~ , v 8,( e)~, ,

i[Q,,v

( e,8)] ~ , = D , V ( ~ ,e , e )(17)

and determines the action of the supercharges on the component fields by expanding both sided in the Grassmann variables. The explicit formulae are given in Ref. 7 and are compatible with the Jacoby identities (6); they also coincide with the formulae from the l i t e r a t ~ r e ~ after ~ ' ~some field redefinitions.

227

If C verifies the Klein-Gordon equation (for mass m ) , then the superfield V verifies the Klein-Gordon equation so all the components of the multiplet are verifying Klein-Gordon equation of mass rn. These equations are compatible with the supersymmetry action, i.e., they are left invariant by the supercharges Qa and Q f i . The multiplet (C,4l ZI,, d, A, x a )is irreducible; in particular it follows that the indices j and A take four values, C and d are real scalar fields of mass m, b, is a real vector field of mass m, 4 is a complex scalar field of mass m and xa and A, are Dirac fields of mass m. We now determine the supercommutator of the vector field. Let us consider the causal commutator

[ V ( X l ) V(X2)I , = -iD(X1; X2) 1 .

(18)

The expression D ( X 1 ; X 2 ) is a distribution in the variables xj and a polynomial in the Grassmann variables B j , j = 1 , 2 and verifies the following properties: (a) it is Poincarh covariant; in particular it depends only on the difference 2 1 - 2 2 ; (b) it has causal support; (c) verifies Klein-Gordon equation; (d) verifies the Hermiticity condition: D(X1;X2) = -D(X2; X I ) ; (e) verifies the antisymmetry condition: D(X2;X I ) = -D(X1; X2); and (f) verifies the consistency condition:

+

(0: D2)D(X1;X2) = 0 .

(19)

All properties except (f) are immediate. The property (f) follows from the Jacobi identity [&a,

[V(X1)1V(X2)11+ XI), [V(X2)7&all

+ [V(X2),

[&a,

V(X1)ll = 0

(20) and formula (17). It is easy to prove that the generic solution of the conditions (a)-(f) of the preceding proposition is a linear combination of four expressions. The condition of positivity gives an important result, namely the vector multiplet exists only for m > 0 and also some constraints on the coefficients of the development. If we compare our results with the usual quantization procedure by means of path integrals we can find out the corresponding values of these coefficients and it turns out that the constraints are not verified. One can argue that some of the fields are gauged away in the Wess-Zumino gauge; however, Wess-Zumino gauge can be proved to be in conflict with the supersymmetric transformation rules. Now we mention the (quantum) gauge structure of the vector model. In ordinary quantum gauge theory, one gauges away the unphysical degrees of freedom of a vector field v, using ghost fields. Suppose that the vector field is of positive mass

228

m; then one enlarges the Hilbert space with three ghost fields u,ii, 4 such that: (a) all three are scalar fields; (b) the fields u,ii are Fermionic and 4 is Bosonic. (c) all them have the same mass m as the vector field. (d) the ut = u, ut - = --u;and (e) the Hermiticity properties are; q5t = 4, commutation relations are:

[dW,d(Y)l = -2

{u(x),qY)} = -i Om(a:- Y) 7 (21)

- Y) 7

and the rest of the (anti)commutators are zero. Then one introduces the gauge charge Q according to:

QR=O,

[Q,up] = i a p u ,

Qt=Q,

[Q,41 = i m u ,

{Q,u}= 0 , {Q,ii} = -2 (dpup + m 4). (22) It can be proved that this gauge charge is well defined by these relations i.e. it is compatible with the (anti)commutation relations. Moreover, one has Q2 = 0 so the factor space K e r ( Q ) / l m ( Q )makes sense; it can be proved that this is the physical space of an ensemble of identical particles of spin 1. For details see Refs. 11, 12. One can generalize this structure in the supersymmetric context if one introduces ghost superfields. One obtains essentially the expressions from the literature;" the main difference is due to the fact that we have m > 0. One can compare this gauge structure to the classical gauge structure;' there is no straightforward correspondence, but at least for ordinary Yang-Mills theories, one finds out that the analogy is rather miraculous. It is natural to try the same idea in a supersymmetric context. However, we have found a negative result (the details are in Ref. 7). There are other multiplets associated to the vector multiplet as the linear multiplet and the rotor multiplet. The algebraic structure of these multiplets is also clarified in Ref. 7, but there are obstacles in the construction of a consistent supersymmetric extension of the standard model also in these cases.

4. The New Vector Multiplet We will consider here a new vector multiplet which has the nice property that the corresponding gauge structure is similar to the usual gauge theories. If this model is consistent with the phenomenology it brings new physics. The new vector multiplet is obtained from the Wess-Zumino multiplet if we transform the scalar (resp. the spinor) field in a vector (resp. a Rarita-Schwinger) field wp,Gpa by simply adding the index p ; this multiplet is subject to the following consistency conditions: (a) Hermiticity

229

= Gpii; (b) all fields verify Klein-Gordon (resp. Dirac) equation of mass M . (c) the action of the supercharges can be obtained from the corresponding formulz for the Wess-Zumino multiplet:

[Qalupl = 0 ,

i [Qa,ut1 = 2

+pa

= CJl6avup ;

(23) and (d) (anti)commutation relations can be obtained starting from the commutation relation for the vector field u, and using the consistency relation (10). We call this new multiplet the RS vector multiplet. The associated superfield can be easily constructed using the supersymmetric extension formula. The gauge structure of this multiplet can be obtained if we replace fields by superfields v, + v,, q5 + @, u U, G + 0 where Ul 0 are ghost and anti-ghost multiplets; all these multiplets are of the same positive mass M . One can obtain the action of the supercharge on the field components if one substitutes in the preceding relations the expressions of the superfields in terms of component the fields and the Grassmann variables. It also can be showed that, as for the usual gauge case, the factor space K e r ( Q ) / l m ( Q )describes a system of identical I;21 super-symmetric systems. So, the analogy with the usual gauge case is really remarkable. Moreover, it is quite easy to obtain consistent gauge invariant couplings if we substitute fields by superfields in the known formulz.12 One can prove4 that in this way a consistent supersymmetric extension of the standard model can be obtained. {&a, $pb}

=

EabMup

7

{Qal

$p6}

-

References 1. 2. 3. 4. 5. 6.

7. 8. 9. 10.

11.

S. Ferrara, 0. Piguet, Nucl. Phys. B93,261 (1975). F. Constantinescu, M. Gut, G. Scharf, Ann. Phys. 11,335 (2002). D. R. Grigore, European Phys. Journ. C21,732 (2001). D. R. Grigore, G. Scharf, Annalen der Physik 12,5 (2003). F. Constantinescu, Lett. Math. Phys. 62,111 (2002). D. R. Grigore, G. Scharf, Quantum Extended Supersymmetries, h e p th/0303176, to appear in Annalen der Physik (2003). D. R. Grigore, G. Scharf, Annalen der Physik 12,643 (2003). F. Constantinescu, G. Scharf, Causal Approach to Supersymmetry: Chiral Superfields, hep-th/0106090. P. P. Srivastava, Supersymmetry, Superfields and Supergmuity: an Introduction, IOP Publ. (1986). S. J. Gates Jr., M. T. Grisaru, M. RoEek, W. Siegel, Superspace or One Thousand and One Lessons in Supersymmetry, Cummings (1983), hepth/0108200. D. R. Grigore, Journ. Phys. A33, 8443 (2000).

230

12. G. Scharf, Quantum Gauge Theories: A Tme Ghost Story, Wiley (2001).

PARASTATISTICS ALGEBRAS AND COMBINATORICS

T. POPOV Institute f o r Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, bld. Tsarigradsko chauske 72, 1784 Sofia, Bulgaria E-mail: [email protected]. bg We consider the algebras spanned by the creation parafermionic and parabosonic operators which give rise t o generalized parastatistics Fock spaces. The basis of such a generalized Fock space can be labelled by Young tableaux which are combinatorial objects. By means of quantum deformations a nice combinatorial structure of the algebra of the plactic monoid that lies behind the parastatistics is revealed.

1. Introduction Wigner' was the first to remark, back in 1950, that the quantum mechanical equations of motion allow for a commutation relations more general than the canonical commutation relations

A few years latter (in 1953) Green wrote the relations' now known under the name parastatistics commutation relations

Let us define the parafermi algebra p$ and the parabose algebra p B to be the associative algebra (over C ) generated by the creation at and annihilation ai operators with relations (l),(2) and (3) with the upper and lower sign, respectively. These are not all parastatistic relations, one obtains new ones by hermitean conjugation. We shall only consider quantum systems with finite number D degrees of freedom, i, j, k = 1 , . . . ,D. 23 I

232

The parastatistics commutation relations are generalization of the cannonical commutation relations

Indeed, the quadratic relations (4) imply the trilinear parastatistics relations ( l ) , (2) and (3). The representations of the parafermionic and p% parabosonic algebras relevant in physics are labelled by the non-negative integer number p , called the order of parastatistics. A creation operator u! acting on the vacuum state creates one particle state, and the annihilation operator ai annihilates it. One has

ps

uiu; 10) = p6ij 10)

.

(5)

The representation p = 0 is trivial (ui = ul = 0) and corresponds to the vacuum representation. The case p = 1 yields particular representation of (p%) in which the relations (4) hold and one retrieves the ordinary fermionic (bosonic) Fock space. For p 2 2 fermionic generalized Fock spaces are characterized by the property that one can accomodate up to p identical particles in a state. This property is a manifestation of a generalized exclusion principle and the Greens parastatistics2 is the first consistent example of generalized statistics. We denote by (p%+) the associative algebra, subalgebra of ps(p%) generated by the creation operators ui subject of the relations (3). Every generalized parafermionic (parabosonic) Fock space labelled by integer p arises as a quotient of the the creation algebra ( p S + ) , i e . , by imposing further pdependent conditions. Remark The parafermionic parabosonic p% algebra with D degrees of freedom has been identified with the universal enveloping algebra U(so(2D 1)),3 U ( o ~ p ( 1 1 2 D ) )respectively. ~ However, we shall not make use of these identifications in what follows.

ps

ps+

ps+

ps,

+

2. Homogeneous Algebras

A homogeneous algebra of degree N or N-homogeneous algebra is an algebra of the form5

A = A ( E ,R ) = T ( E ) / ( R ) ,

(6)

where E is a finite-dimensional vector space over C, T ( E ) is the tensor algebra of E and ( R )is the two-sided ideal of T ( E )generated by a vector

233

subspace R of E @ N .The homogeneity of ( R ) implies that A is a graded algebra A = e n E ~ A with n A, = Em'" for n < N and

A,=E@'"/

E @ r @ R @ E @for s n>N,

(7)

r+s=n-N

where we have set E@'O= C. Thus A is a graded algebra A = anE~.Rn which is connected ( A= C), generated in degree 1 and such that the A, are finite-dimensional vector spaces. When N = 2 or N = 3 we shall speak of quadratic or cubic algebras respectively. We shall refer to the subspace R as relation space. We set E = @:,Cul N C D . When the parastatistical order p = 1 then the symmetrical S ( E ) = en>oSnE and the exterior A ( E )= @,>o - An E algebras provide the basises of the bosonic and fermionic Fock space respectively. The grading is the number of particles in a state. The creation algebras pS+ and p%+ are cubic algebras. The creation parafermionic algebra pS+ is the cubic algebra6

PS+ = A ( E ,R ) = T ( E ) / ( " x Y, l @ l .I@) Yl t E E , (8) where [x,y]@= x @ y - y @ x and the subspace R is associated to the 7

2 1

relations (3). The creation parabosonic algebra p%+ is the cubic algebra6

P%+

.I@)

= A(E1 R) = T ( E ) / ( [ { x , Y ) @ ,

7

x1 Y l z E E

1

(9)

where the R means that instead of commutators one has t o take supercommutators, ie., the relation space R is the super-counterpart of R. The parabosons are super-parafermions. Note that in this approach the parafermions are even and the parabosons are odd. Choosing a basis in E l which means fixing the isomorphism E N C D ,we identify the linear group G L ( E )with GL(D,C). One has a natural action of the linear group GL(D,C) (denoted GL(D) from now on) on the linear space E N C D given by left matrix multiplication of the column vector ( G z ) ~= Gixj,

2 = X ~ U ;E

E,

G E GL(D) .

The natural action of GL(D) can be continued t o every tensor power

E@'"as G(x~@ ~2 @ . . . x,) = G z @ ~ G x 8.. ~ . Gx,

.

The natural action of the linear group G L ( D ) on E@' preserves the subspaces R and R, ie., they are invariant submodules. It follows that GL(D)-action passes to the quotients pS+ and p%+ respecting the grading.

234

ps+

This implies that G L ( D )acts on and p%+ by automorphisms, i.e., the algebras p$+ and p%+ are modules of the linear group GL(D). 3. The Symmetric and the Linear Group We briefly recall some general facts about the intimately related representation theories of the symmetric and the linear group. Let X = (X1,Xz ,... Xk) with A1 2 Xz 2 .. . 2 X k > 0 be partition of k

n,

C X i = n.

A partition X of n will be identified with a Young diagram

i=l

X with n boxes and k rows, where X i is the length of the i-th row. The number of boxes in X is denoted by 1x1. The filling of a Young diagram with numbers non-decreasing on rows (from left t o right) and strictly increasing on columns (from top t o bottom) is called a Young tableau. A standard Young tableau is a Young tableau strictly increasing in rows, i.e., without repetition in the entries. The group ring C[Sn]is endowed with structure of a right (left) S -, module in a natural way. The regular representation is the representation of S, into C[S,]. The dimension fx of an irreducible representation Sx of the symmetric group S, (with X partition of n) is equal t o the number of the standard tableaux with shape A. It is well known fact from the theory of finite groups that the regular representation contains all irreducible representations and each representation enters with multiplicity equal t o its dimension. Therefore the left S,-module C[Sn]has the decomposition into a direct sum of modules

C[Sn]N

@ (SX)@fX,

fx

= dim Sx ,

(10)

IXI=n

where Sx are the irreducible S,-modules. The Young projectors Yx(T) are idempotent elements of the group algebra C[Sn]of the symmetric group S,. They are labelled by a partition X and a standard tableau T with shape A. The projectors Yx(T)form an orthogonal system

Yx(T)Yx,(T’) = SAA/STT’YX(T).

(11)

When acting from the right on C[Sn]the Young projectors give rise t o the left irreducible representations of the symmetric group S,

S’ = c[s~IY~(T), 1x1 = n .

(12)

235

The Sn-modules SX arising through different Young projectors Yx(T)with one and the same X axe isomorphic. One can endow the tensor degree EBn with a right action of Sn which permutes the slots in E B n . The right &-action commute with the natural left GL(D)-action on E@l"(Schur-Weylduality). Hence the projection E B n Y ~ ( Tis) stable under the natural left GL(D)-action. The physical meaning of the right &-action is a place permutation of the creation operators a+ while the left So-action (which is a subgroup of GL(D)) operates as particle permutations. Schur module is the irreducible representation E X given by

E X21 E@1"Yx(T),

1x1 = 72.

(13)

The Schur modules arising through Young projectors with one and the same X are isomorphic. So the finite-dimensional irreducible representations E X of the linear group GL(D) are parametrized by Young diagrams too. The character & ( E x ) of the representation E X is the Schur polynomial associated to a partition X

and therefore the decomposition of the GL(D)-module EBn

EBn = @ (EX)@fA IXI=n

follows from the well known formula

4. The Algebras p B + and p 5 + as GL(D)-modules

The standard tableaux for X = ( 2 , l ) are two, thus for n = 3 one has @ E(3) = r\3E @ E(2,1)@ E(2,1)@ S 3 E . E@3 E(l,l,l)@ E(2y1)@ E(2~1)

The Jacobi and super-Jacobi identities

b, [Y,.IF1 + [Y,[z, 471 + b, b,YlFl = 0 imply that the GL(D)-modules R and R (corresponding to p$+ and p%+ ) are neither symmetric nor antisymmetric. It follows that R and R are two orthogonal GL(D)-modules associated to the Young diagram ( 2 , l )

RN

N

I?.

(16)

236

Let T be the maximal element in S3, 7 = slszsl = S Z S ~ S Z . One can choose the Young projectors qz,l)(T1)= Y + and Y&)(T2) = Y - to be the eigenvalues of the place permutation action of T , the so called flip (flip(u@bbc) = ( a @ b @ c ) T = c @ b @ u )

f l i p ( y * ) = y * T = &Y* ,

(17)

which determines Y + and Y - uniquely as S('tl)-modules. The corresponding Schur modules isomorphic to E('?l) are

R = EB3Y+,

R = EB3Y-.

(18)

This simple observation will be very useful later.

Theorem 4.1. Each irreducible representations E x of GL(D) appears exactly once in the decomposition of creation parafermionic pg+ and parabosonic p?B+ algebras

pg+ N @EX N p?B+.

x In the classical textbook on paras tat is tic^^ this result has a tedious proof. Chaturvedi' seems to be the first who used combinatorial identities in parastatistics (in this context). Here we give a short proof to the theorem which boils down to an exercise in the Fulton's Proof: Let us endow the space E @ A'E with the bracket defined by [z, y] = z A y if z and y are both in E and [z,y] = 0 otherwise. The so defined bracket is a Lie bracket and E @ A'E is a graded Lie algebra for this bracket if one ascribes the degree 1to the elements of E and the degree 2 to the elements of A'E. By definition pg+ is the universal enveloping algebra of the graded Lie algebra E @ A'E

pg+ = U ( E CB 2 ~ ) . In view of the PoincarbBirkoff-Witt theorem U ( E@ A'E) is isomorphic as graded vector space and as a graded coalgebra to the symmetric algebra

S ( E @ A'E) = S ( E ) @ S(A'E).

(19)

The character of the symmetric algebra S ( E )over the space E (of degree 1

n (l--zi)-l. It follows from (19) that the character D

elements) is ch S ( E ) =

i=l

of pg+ is given by the left hand side of the Schur combinatorial identity

237

while the right hand side is a sum (over all Young diagrams A) of the characters of Schur modules which implies the decomposition of In view of isomorphisms (16) and p%+ must be isomorphic as GL(D)modules which ends the proof. Remark. The direct proof" of the decomposition of p%+ uses the PoincarB-Birkhoff-Witt theorem for super-Lie algebras. One of the definitions of Schur polynomial is

ps+.

ps+

sx(z) =

czT,

where

zT = z1

... zg

,

(21)

T

where ai(T)is the number of times the entry i appears in the Young tableau T and the sum is over all Young tableaux T which are fillings of the diagram X with numbers from the set (1,. . . , D}. Therefore the Young tableaux T with shape X are in 1-1 correspondence with the monomials in the Schur polynomial s~(z),hence with the basis of the representation EX.Due to the homogeneity of the Schur polynomials sx(t,.

. . ,t ) = tlX1sX(l,. . . , 1) = tl'l dim E X ,

one obtains the PoincarQ series of the algebra p$+ (and also of p%+)

5 . The Plactic Algebra

We now introduce another cubic algebra coming from the combinatorics. The set of Young tableaux can be endowed with a structure of associative monoid, the so called plactic monoid" (see also Ref. 9). Every Young tableau T can be encoded with a word written with the entries of T and one can define algebraic operations on these words, i.e., on the set of Young tableaux. The plactic monoid is generated by ordered set of elements { e l , . . . ,e g } (for tableaux with entries in (1,. . . ,D})subjects of the Knuth relations

eizei3ei, = ei,eil ei3 if

il

< i 2 5 i3

= ei3eilei2 if

il

5 i2 < i 3

eilei3ei,

I

(23)

Let us choose the basis ( e i ) of the linear space C D 21 E to be the canonical basis of C D . The set { e l , . . . , e g } is ordered with the natural ordering

238 el < e:! < ..- < e D . To the Knuth relations we associate in an obvious way the subspace Rv c EB3 which generates the ideal (Rv).The plactic algebra p is the cubic algebra6

'$ = A ( C D ,Rv).

(24)

In contrast to R and R, the relation space Rv depends on the basis ( e k ) and on the ordering of ( e k ) . There is no natural action of GL(D) on $ I because GL(D)-action spoils the ordering. We have seen that the Schur module E X has a linear basis labelled by Young tableaux which are fillings of the diagram X with entries out of the set { 1,2,. . . ,dim E } . On the other hand the homogeneous independent elements of the plactic monoid (and so the plactic algebra) are identified with Young tableaux. We conclude that the Poincarh series of the plactic algebra and the creation parafermionic (parabosonic) p5+ ( ~ 2 3 ' ) algebra coincide

P&)

= Ppg+( t )= PPB+( t ).

(25)

This is not just a coincidence, it turns out that the plactic algebra can be obtained by means of the deformation of the algebras pS+ and pB+. 6. Hecke Algebra and the Quantum Linear Group The Hecke algebra '&(q) is the algebra generated from bi, i = 1,.. . ,n - 1 and the unit element with relations

bibi+lbi = bi+lbibi+l bibj = bjbi b: = 1 ( 4 - q-l)bi

+

Z=l, ..., n - 2 , Ji - j l 2 2, i = l , ...,n - 1 .

These are the same relations as for the symmetric group Sn except for the last one which is relaxation of .s: = 1. When q is not root of unity the Hecke algebra NHn(q) is isomorphic to the group algebra C[Sn]

which allows to index the idempotents in 7-lFln(q) called generalized Young projectors Y:(T)I2 in the same way as Yx(T),ie., again by standard tableaux. One has the q-analog of Eq. (11)

Yx"(T)Y,4(T') = bXA'bTT/YXQ(T).

(27)

239

An irreducible ',Jf,(q)-module stems from the counterpart of (12)

'FIX((?) = ',Jfn(q)YXQ(T) 7

1x1 = n .

The right action of 'H,(q) commutes with the left action of the quantum linear group GL,(D) (quantum Schur-Weyl duality). An irreducible GL,(D)module or q-Schur module is defined in the manner of (14)

1x1 = n .

E A ( q )= E@"YZ(T), 7. Deformation of pS+ and p B + and $!3

Let T, be the maximal element in ' , J f 3 ( q ) . The quantum flip, which is the place permutation (ie., the right) action of T,, has two idempotent eigenvalues Yq* q-flip(Y") = Y P f T , = fY,*.

Yqf are generalized Young projectors Y&)(T).

(28)

We now define two

GL,(D)-modules isomorphic to the q-Schur module E(271)(q) R, = E@'3Yq+,

R -E@3yQ-.

(29)

4-

Deformed creation parafermionic p$,f and deformed creation parabosonic p23: algebra are the following cubic algebras"

Ps:

= A(& Rq)

,

p%;

ii,) .

= A(E,

An outcome of lengthy calculations (using a representation of the Hecke algebra given by a R-matrix of Hecke type) are the explicit expressions for the relations of ps: and p23: (see Ref. 10 and also Ref. 6 for D = 2)

t t t t bf2,[ais,ai1lrlq2 + daf,, [ai2,aillrl =0 r.13, .lll,,al2I42-1- q"al,, "Z',lr,az',] = 0

< i2 I i3 il I i2 < i3 il

}

*

(30)

At the classical point q = 1 with the help of the (super)Jacobi identity we recover the relations (3). In the application of the quantum groups in two-dimensional statistical mechanics, the parameter q has merely the sense of temperature, q = e - h . The point q = 0 corresponding to the absolute zero temperature T = 0 is a singular point for the GL,(D)-symmetry. Figuratively speaking at q = 0 the symmetry is frozen. Nevertheless, the relations (30) are regular at q = 0 and the evaluation in this point called the crystal limit yields

240

Comparing the crystal limit of the pTi-relations (the upper sign in (30)) with the Knuth relations (23) we come to the conclusion that

is a crystal limit of the deformed creation that is, the plactic algebra parafermionic algebra p T i . By analogy the crystal limit of pBl is an algebra that might be called the super-plactic algebra

One can speculate that the ordinary parastatistics is the high temperature limit T 4 00, i e . , q = 1 of the more general deformed (or quantum) parastatistics. Then the algebra of the (super)plactic monoid is the zero temperature limit q = 0.

Acknowledgments

I wish to thank to Michel Dubois-Violette for many inspiring discussions. References 1. E. Wigner, Phys. Rev. 77,711 (1950). 2. H.S. Green, Phys. Rev. 90, 270 (1953). 3. C. Ryan, E.C.G. Sudarshan, NucZ. Phys. 47, 207 (1963). 4. A. Ganchev, T. Palev, J . Math. Phys. 21,797 (1980). 5. R. Berger, M. Dubois-Violette, M. Wambst, J . Algebra 261,172 (2003). 6. M. Dubois-Violette, T. Popov, Lett. Math. Phys. 61,159 (2002). 7. Y. Ohnuki, S. Kamefuchi, Quantum field theory and parastatistics, SpringerVerlag (1982). 8. S. Chaturvedi, hepth/9509150. 9. W. filton, Young tableaux, Cambridge University Press (1997). 10. T. Popov, Ph.D. thesis, http://qcd.th.u-psud.fr/preprints_labo/physiquemath/art2OO3/. 11. A. Lascoux, M.P. Schutzenberger, Quaderni de “La ricerca scientzfica” 109, Roma, CNR 129 (1981). 12. D. Gurevich, Algebra i Analiz 2, 119 (1990).

NONCOMMUTATIVE D-BRANES ON GROUP MANIFOLDS

J. PAWELCZYK Institute of Theoretical Physics, Warsaw University, H o i a 69, PL-00-681 Warsaw, Poland E-mail: [email protected]

H. STEINACKER Institut f u r theoretische Physik, LMU Munchen, Theresienstr. 37, 0-80333Munchen, Germany E-mail: Harold.Steinacker@physik. uni-muenchen. de

We propose an algebraic description of (untwisted) D-branes on compact group manifolds G using quantum algebras related to U,(g). It reproduces the known characteristics of D-branes in the WZW models, in particular, their configurations in G, energies, and the set of harmonics.

1. Introduction This report is a brief review of the quantum algebraic description of Dbranes on group manifolds as proposed in Ref. 1. The structure of D-branes in a B field background has attracted much attention recently. The case of flat branes in a constant B background has been studied extensively and leads to quantum spaces with a MoyalWeyl star product. A rather different situation is given by D-branes on compact Lie groups G, which carry a B field which is not closed. It is known from their conformal field theory (CFT) descriptions2 that stable branes are given by certain conjugacy classes in the group manifold. On the other hand, it is expected that these branes are formed as bound states of DO-branes. Attempting to unify these various approaches, we proposed in Ref. 3 a matrix description of D-branes on S U ( 2 ) . This was generalized in Ref. 1,giving a simple and compact description of all (untwisted) D-branes on group manifolds G in terms of quantum algebras related to the quantized universal enveloping algebra U,(g).The main result is that a model based 241

242

on the reflection equation (RE) algebra leads to precisely the same branes as the WZW model. It not only reproduces their configurations in G, i.e. the positions of the corresponding conjugacy classes, but also gives the same (noncommutative) algebra of functions on the branes.

2. CFT and the Classical Description of Untwisted D-branes The CFT description is given in terms of a WZW model, which is specified by a compact group group G and an integer level k.4 We concentrate on the case G = S U ( N ) ,but all constructions work for S O ( N ) and U S p ( N ) as well. The WZW branes can be described by certain boundary states of the Hilbert space of closed strings. We consider here only "symmetrypreserving branes" (untwisted branes), given by the Cardy (boundary) states. They are labeled2v5by a finite set of integral weights

X E I'z = {A

E

P+; X.8 5 k } ,

(1)

(here 8 is the highest root of g), corresponding to integrable irreps of G. Hence untwisted branes are in one-to-one correspondence with X E P l . The energy of the brane X is given by

The CFT also contains the description of branes as quantum manifolds, in terms of boundary primary fields. Their number is finite for any compact WZW model. In the k -+ oa limit, these boundary primaries generate the (noncommutative) algebra of functions on the b r a n e ~ ,see ~ )also ~ Section 4. For finite k , the corresponding algebra, as given in Ref. 6, is not associative. It becomes associative after "twisting", so that it can be considered as algebra of functions of a quantum manifold. Then the primaries become modules of the quantum group U,(g). On a semi-classical level, the D-branes are simply conjugacy classes of the group manifold, of the form

C ( t ) = {gtg-';

g E G} .

(3)

One can assume that t belongs to a maximal torus T of G. These conjugacy classes are invariant under the adjoint action of the vector subgroup Gv

243

GLx GR of the group of motions on G. This reflects the breaking 5~X

~ -+ R

iV.

A lot of information about the spaces C ( t ) can be obtained from the harmonic analysis, i.e. by decomposing scalar fields on C ( t ) into harmonics under the action of the (vector) symmetry G v . One finds

F(C(t))2 @ muzt,,( K t ) v,.

(4)

XEP+

Here X runs over all dominant integral weights P+, V, is the corresponding highest-weight G-module, and r n u l t ~is~the ) dimension of the subspace of V,+ = V,, which is invariant under the stabilizer Kt o f t . As discussed above, there is only a finite set of stable D-branes on G (up to global motions) in the CFT description, one for each integral weight X E P:. They correspond to C ( t x ) for

t,

tn

= q2(HA+HP)

(5)

q=ek+sv.

Here gv is the dual Coxeter number. The location of these branes in G is encoded in s,

= t r(gn) = tr(t") ,

g E C(t), n E

N,

(6)

which are invariant under the adjoint action. For the classes C ( t x ) , they can be easily calculated: s, = trVN ( 42 7 G P + H A ) ) =

c

q274P+A)+

(7)

UCvN

where VN is the defining representation. The s, completely characterize C ( t x ) , and their quantum analogs (11) can be calculated exactly. An equivalent characterization of these conjugacy classes is provided by a characteristic equation of the form P x ( M ) = 0.l 3. Quantum Algebras and Symmetries for Branes We now define the quantum space describing G and its branes in terms of a non-commutative algebra M , which transforms under a quantum symmetry. The quantized algebra M of functions on G is generated by elements Mj with indices i,j in the defining representation of G, subject to some commutation relations and constraints. The relations are given by the socalled reflection equation (RE),' which in a short notation reads R2iMiRi2M2 = M2R21MiRi2.

(8)

244

Here R is the R matrix of U,(g)in the defining representation. For q = 1, this reduces to [M:, Mf]= 0. Because M should be a quantization of G, there must be further constraints. In the case G = S U ( N ) , these are det,(M) = 1, where det, is the so-called quantum determinant (12), and suitable reality conditions imposed on the generators M j . The RE appeared more than 10 years ago in the context of the boundary integrable model^.^ M is covariant under the the transformation

M; -+ ( s - w t ) ; ,

(9)

where si and ti generate algebras BL and GR respectively, which both coincide with the well-known quantum groups Fun,(G), as defined in Ref. 8; i.e. s2slR = RslsZ, t2tlR = Rtlt2 etc. This (co)action is consistent with RE if we impose that (the matrix elements of) s and t commute with M , and in addition satisfy s2tlR = Rtls2. Formally, M is then a right 8' GR - comodule algebra; seeRef. 1 for further details. Notice that Eq. (9) is a quantum analog of the action of the classical isometry group GL x GR on classical group element g. Furthermore, GL B R GR can be mapped to a vector Hopf algebra GV with generators T , by si @I 1 -+ rj and 18 tj -+ T ; . The (co)action of Gv on M is then

M;

-+ (r-lMr);.

.

(10)

The (generic) central elements of the algebra (8) are given by c, = t r q ( M n )E trv,,,

(M"

W)

E

M ,

(11)

where v = T ( q - 2 N p ) is a numerical matrix which satisfies S'(T) = W - ~ T W for the generator T of Gv. These elements c, are independent for n = 1 , 2 , ...,rank(G). One can also show that the c, are invariant under Gv. As we shall see, the c, for n = 1,...rank(G) fix the position of the brane configuration on the group manifold, i.e. they are quantum analogs of the s n , Eq. (7). There is another central term det,(M), the quantum determinant, which is invariant under the full chiral quantum algebra GL BR GR. Hence we can impose the constraint det,(M) = 1.

(12)

Furthermore, there is a realization (algebra homomorphism) of the RE algebra (8) in terms of the algebra U,(g), given by M = (T 8 i d ) ( R z i R i z ) where T is the defining representation. Equation (8) is then a consequence of the Yang-Baxter equation for R.8

245

4. Representations of M and Quantum D-branes We claim that the quantized orbits corresponding to the D-branes of interest here are described by irreps 7rx of M . On any irrep, the Casimirs c, (11) take distinct values, i.e. they become constraints. In view of their explicit form, an irrep of M should be considered as quantization of a conjugacy class C ( t ) , whose position depends on c,. To confirm this interpretation, one can calculate the position of the branes on the group manifold, and study their geometry by performing the harmonic analysis on the branes. The “good” irreps of the algebra M coincide with the highest weight representations Vx of U,(g)for X E P z : they are ~ n i t a r y have , ~ positive quantum-dimension, and are in one-to-one correspondence with the integrable modules of the affine Lie algebra g. As shown in Ref. 1,these irreps 7 r ~ of M for X E P z describe precisely the stable D-branes C ( t x ) , denoted by Dx. It is an algebra of maps from Vx to Vx which transforms under the quantum adjoint action of U,(g).For “small” weights, this algebra coincides with Mat(Vx). There is clearly a one-to-one correspondence between the (untwisted) branes in string theory and these Dx,since both are labeled by X E P z . 4.1. Position of DX

The values of the Casimirs c, on Dx are as follows:1

Here XN is the highest weight of the defining representation VN, and the sum in Eq. (14)goes over all v E VN such that X v lies in P z . The value of q ( X ) agrees precisely with the corresponding value (7) of s1 on C ( t x ) . For n 2 2, the agreement of cn(X) with sn on C ( t x ) is only approximate, becoming exact for large A. The discrepancy can be blamed to operator-ordering ambiguities. M also satisfies a characteristic equation1 similar to the classical one. Therefore the position and “size” of the branes essentially agrees with the results from CFT. Furthermore, the energy of the D-brane is given by the quantum dimension of the representation space

+

vx

246

4.2. The Space of Harmonics on DA

For simplicity, assume that X is not too large. Then

Dx

2

Mut(Vx) = Vx @ V i % CB,N;~+V, ,

(16)

since the tensor product is completely reducible. Here N f x + are the usual fusion rules of 8. This has a simple geometrical meaning if p is small enough: comparing with Eq. (4),one can show' that

D x 2 F(C(ti))1

(17)

. differs slightly from up t o some cutoff in p, where t i = e x p ( 2 ~ i h ) This Eq. (5), by a shift X + X p. The structure of harmonics on Dx is however in complete agreement with the CFT results. Moreover, it is known5 that the structure constants of the corresponding boundary operators are essentially given by the 6 j symbols of V,(g), which in turn are precisely the structure constants of the algebra of functions on Dx. Therefore, our quantum algebraic description not only reproduces the correct set of boundary fields, but also essentially captures their algebra in (B)CFT. Particularly interesting examples of degenerate conjugacy classes are the complex projective spaces @PN-l, which in the simplest case of G = SU(2) become (q-deformed) fuzzy spheres" Si,N.

+

References 1. J. Pawelczyk, H. Steinacker, H. Nucl. Phys. B638, 433 (2002). 2. A. Yu. Alekseev, V. Schomerus, Phys. Rev. D60, 061901 (1999). 3. J. Pawelczyk and H.Steinacker, JHEP 0112,018 (2001). 4. J. Fuchs, AfJine Lie Algebras and Quantum Groups, Cambridge University Press (1992). 5. G. Felder, J. Frohlich, J. Fuchs, C. Schweigert, J. Geom. Phys. 34, 162 (2000).

6. A. Yu. Alekseev, A. Recknagel and V. Schomerus, JHEP 9909, 023 (1999). 7. E. Sklyanin, J. Phys. A21,2375 (1988). 8. L. D. Faddeev, N. Yu. Reshetikhin and L. A. Takhtajan, Algebra Anal. 1, 178 (1989). 9. H. Steinacker, Rev. Math. Phys. 13, No. 8, 1035 (2001). 10. H. Grosse, J. Madore, H. Steinacker, J. Geom. Phys. 38, 308 (2001).

HIGH-ENERGY BOUNDS ON THE SCATTERING AMPLITUDE IN NONCOMMUTATIVE QUANTUM FIELD THEORY

A. TUREANU

P.

Department of Physical Sciences, and Helsinki Institute of Physics, 0. Box 64, 00014 University of Helsinki, Helsinki, Finland E-mail: [email protected]

In the framework of quantum field theory (QFT) on noncommutative (NC) space time with SO(1,l) x SO(2) symmetry, which is the feature arising when one has only space-space noncommutativity (&i = 0), we prove that the Jost-LehmannDyson representation, based on the causality condition usually taken in connection with this symmetry, leads to the mere impossibility of drawing any conclusion on the analyticity of the 2 + 2-scattering amplitude in cos 8,0 being the scattering angle. A physical choice of the causality condition rescues the situation and, as a result, an analog of Lehmann’s ellipse as domain of analyticity in cos 8 is obtained. However, the enlargement of this analyticity domain to Martin’s ellipse and the derivation of the Froissart bound for the total cross-section in NC QFT is possible only in the special case when the incoming momentum is orthogonal to the NC plane. This is the first example of a nonlocal theory in which the cross-sections are subject t o a high-energy bound. For the general configuration of the direction of the incoming particle, although the scattering amplitude is still analytic in the Lehmann ellipse, no bound on the total cross-section has been derived. This is due to the lack of a simple unitarity constraint on the partial-wave amplitudes.

1. Introduction The development of QFT on NC space-time, especially after the seminal work of Seiberg and Witten,l which showed that the NC QFT arises from string theory, has triggered lately the interest also towards the formulation of an axiomatic approach to the subject. Consequently, the analytical properties of scattering amplitude in energy E and forward dispersion relations have been ~ o n s i d e r e d ,Wightman ~>~ functions have been introduced and the CPT theorem has been p r o ~ e n , and ~ ’ ~as well attempts towards a proof of 247

248

the spin-statistics theorem have been made.5a In the axiomatic approach to commutative QFT, one of the fundamental results consisted of the rigorous proof of the Froissart bound on the high-energy behaviour of the scattering amplitude, based on its analyticity properties.lOyll Here we aim at obtaining the analog of this bound when the space-time is noncommutative. Such an achievement, besides being topical in itself, will also prove fruitful in the conceptual understanding of subtle issues, such as causality, in nonlocal theories to which the NC QFTs belong.12 In the following we shall consider NC QFT on a space-time with the commutation relation [ Z p , ZCYI

= iepv

7

(1)

where 8,, is an antisymmetric constant matrix (for a review, see, e.g., Refs. 13, 14). Such NC theories violate Lorentz invariance, while translational invariance still holds. We can always choose the system of coordinates, such that 813 = 823 = 0 and OI2 = -OZ1 = 8. Then, for the particular case of space-space noncommutativity, i.e. 8oi = 0 , the theory is invariant under the subgroup SO(1,l) x SO(2) of the Lorentz group. The requirement that time be commutative (8oi = 0 ) discards the well-known problems with the unitarity15 of the NC theories and with ~ a u s a l i t y . ~ ~ ? ~ ~ As well, the Boi = 0 case allows a proper definition of the S - m a t r i ~ . ~ In the conventional (commutative) QFT, the Froissart bound was first obtained1' using the conjectured Mandelstam representation (double dispersion relation)," which assumes analyticity in the entire E and cos@ complex planes. The Froissart bound,

expresses the upper limit of the total cross-section ctot as a function of the CMS energy E , when E -+ cy). However, such an analyticity or equivalently the double dispersion relation has not been proven, while smaller domains of analyticity in cos 0 were already known.lg One of the main ingredients in rigorously obtaining the Froissart bound is the Jost-Lehmann-Dyson representation20'21 of the Fourier transform of the matrix element of the commutator of currents, which is based on aIn the context of the Lagrangian approach to NC QFT,the CPT and spin-statistics theorems have been proven in general in Ref. 6; for CPT invariance in NC QED,see Refs. 7, 8, and in NC Standard Model Ref. 9.

249

the causality as well as the spectral conditions (for an overall review, see Ref. 22). Based on this integral representation, one obtains the domain of analyticity of the scattering amplitude in cos0. This domain proves However, the domain to be an ellipse - the so-called Lehmann’s e1lip~e.l~ of analyticity in c o s 0 can be enlarged to the so-called Martin’s ellipse by using the dispersion relations satisfied by the scattering amplitude and the unitarity constraint on the partial-wave amplitudes. Using this larger domain of analyticity, the Froissart bound (2) was rigorously provenll (for a review, see Ref. 23). In NC QFT with 8oi = 0 we shall follow the same path for the derivation of the high-energy bound on the scattering amplitude, starting from the Jost-Lehmann-Dyson representation and adapting the derivation to the new symmetry SO(1,l) x SO(2) and to the nonlocality of the NC theory. In Section 2 we derive the Jost-Lehmann-Dyson representation satisfying the light-wedge (instead of light-cone) causality condition, which has been used so far, being inspired by the above symmetry. In Section 3 we show that no analyticity of the scattering amplitude in cos 0 can be obtained in such a case. However, with a newly introduced causality condition, based on physical arguments, we obtain from the Jost-Lehmann-Dyson representation a domain of analyticity in cos 0 , which coincides with the Lehmann ellipse. In Section 4 we show that the extension of this analyticity domain to Martin’s ellipse is possible in the case of the incoming particle momentum orthogonal t o the NC plane (x1,x2), which eventually enables us to derive rigorously the analog of the Froissart-Martin bound (2) for the total cross-section. The general configuration of incoming particle momentum is also discussed, together with the problems which arise. Section 5 is devoted to conclusion and discussions. 2. Jost-Lehmann-Dyson Representation The Jost-Lehmann-Dyson representation2’t21 is the integral representation for the Fourier transform of the matrix element of the commutator of currents:

where X

X

f ( x ) = ( p ’ l [ j l ( ~ ) , j 2 ( - ~ ) I l p7 )

(2)

satisfying the causality and spectral conditions. The process considered is the 2 -+ 2 scalar particles scattering, k + p -+ k’ + p ’ , and j , and j 2 are the

250

scalar currents corresponding to the incoming and outgoing particles with momenta k and k' (see also Refs. 22, 24). For NC Q F T with SO(1 , l ) x SO(2) symmetry, in Ref. 25 a new causality condition was proposed, involving (instead of the light-cone) the light-wedge corresponding to the coordinates xo and x3, which form a two-dimensional space with the S O ( 1 , l ) symmetry. Accordingly we shall require the vanishing of the commutator of two currents (in general, observables) a t space-like separations in the sense of S O ( 1 , l ) as: X

X

for ~ 2 = x ~ -< ox . ~

[ j l ( 2- ) , j 2 ( - - 2) ] = 0 ,

(3)

The spectral condition compatible with (3) would require now that the physical momenta be in the forward light-wedge:

p2 =pE

-pi

> 0 and

po

>0.

(4)

The spectral condition (4) will impose restrictions on f ( q ) (see Ref. 12 for details), in the sense that f ( q ) = 0 in the region outside the hyperbola Po -

Jm<

qo

< -Po

+b G .

(5)

To derive the Jost-Lehmann-Dyson representation, we further consider the 6-dimensional space-time with the Minkowskian metric (+, -, -, -, -, -) (for details of the derivation, see Ref. 12). Using the standard mathematical procedure,28 one obtains the Jost-Lehmann-Dyson representation in NC QFT, satisfying the light-wedge causality condition

(3): f(q)=

/

d4udK26(qo

- uo)b[(qo- uo)2 - (q3 - u3)2 - K21

x d(q1 - u l N q 2 - u2)4('zL,K 2 ) 7

(6)

where + ( u , K ~ ) = -*. Equivalently, denoting ii = (uo,us),Eq. (6) can be written as:

The function 4(G,41,q 2 , K ~ is ) an arbitrary function, except that the requirement of spectral condition determines a domain in which +(ii, q1, q 2 , K ~ =) 0. This domain is outside the region where the 6 function in (7) vanishes, i.e.

(@- i i ) 2 - K 2 = 0 ,

(8)

25 1

but with 4 in the region given by ( 5 ) , where f(q) = 0. Putting together (8) and (5), we obtain the domain out of which $(ii, 41, 4 2 , K ~ =) 0: a)

1 -(F 2

+@I)

fii are in the forward light-wedge (cf. (4));

(9)

For the purpose of expressing the scattering amplitude, we actually need the Fourier transform fR(4) of the retarded commutator, fR(x) = e(xo)f(x) = ( ~ ' ~ ~ ( ~ 0 ) [ ~ 1 ( ~ ) , ~, 2 ( - ~ ) (10) 1 ~ ~ )

which is obtained in the form:12

Compared to the usual Jost-Lehmann-Dyson representation,

the expression (11) is essentially different in the sense that the arbitrary function $ now depends on 41 and 4 2 . This feature will have further crucial implications in the discussion of analyticity of the scattering amplitude in cos 8. 3. Analyticity of the Scattering Amplitude in cos 0. Lehmann's Ellipse

In the center-of-mass system (CMS) and in a set in which the incoming particles are along the vector = ( O , O , e),b the scattering amplitude in NC QFT depends still on only two variables, the CM energy E and the cosine of the scattering angle, cos 8 (for a discussion about the number of variables in the scattering amplitude for a general type of noncommutativity see Ref. 26).

6

bThe "magnetic" vector p' is defined as pi = iEijk6'jk The terminology stems from the antisymmetric background field B,, (analogous to F,, in $ED), which gives rise to , essentially proportional to B p y (see, e.g., noncommutativity in string theory, with 8 Ref. 1).

252

In terms of the Jost-Lehmann-Dyson representation, the scattering amplitude is written as (cf. Ref. 22 for commutative case):

where +(G, K’, ...) is a function of its SO(1,l)- and SO(2)-invariant variables: - u:, (ko PO)^ - (k3 - ~ 3 ) ~(ki , PI)^ (k2 + ~ 2 ) ~(k:, (kh The function q5 is zero in a certain domain, determined by the causal and spectral conditions, but otherwise arbitrary. For the discussion of analyticity of M ( E ,cos 0 ) in cos 8,it is of crucial importance that all dependence on cos 8 be contained in the denominator of (13). But, since the arbitrary function q5 depends now on (k’- p’)1,2, it also depends on cos 8. This makes impossible the mere consideration of any analyticity property of the scattering amplitude in cos 8. Since the Jost-Lehmann-Dyson representation reflects the effect of the causal and spectral axioms, we notice that the hypotheses (3) and (4) used for the present derivation of JLD representation are too weak, in the sense of their physical implications, since they allow for a much larger physical region, by not at all taking into account the effect of the NC coordinates x1 and 22.

+

+

ui

3.1. Causality in NC QFT In the following, we shall challenge the causality condition f(x) = 0 , for

z2 = x i - x i < 0 ,

(14)

which takes into account only the variables connected with the SO(1,l) symmetry. This causality condition would be suitable in the case when nonlocality in NC variables x1 and 2 2 is infinite, which is not the case on a space with 8. The the commutation relation [xI,x2] = 28, which implies AxlAx2 fact that in the causality condition (14) the coordinates x1 and 2 2 do not enter means that the propagation of a signal in this plane is instantaneous: no matter how far apart two events, are in the noncommutative coordinates, the allowed region for correlation is given by only the condition xg -xi > 0, which involves the propagation of a signal only in the x3-direction, while the time for the propagation along 21- and xz-directions is totally ignored. Admitting that the scale of nonlocality in x1 and 2 2 is 1 i.e. the propagation of interaction in the noncommutative coordinates is instanta-

-

- 4,

253

neous only within this distance 1, we can argue, based on these physical arguments, that the locality condition should indeed be given by:

f(x) = o

, for z2 - (xf+ xi - 1 2 )

3 xi

+ x; - 1’)

- xi - (xf


or, equivalently,

f(x) = 0 , for xi - xi - (x: + xi) < -12, where l 2 is a constant proportional to NC parameter 8. When l2 becomes the usual Iocality condition. Correspondingly, the spectral condition will read as Pi -Pi - (Pf + P i ) 2 0 ,

PO

>0,

(15) -+ 0,

(15)

(16)

since there is no noncommutativity in momentum space. In fact, the consideration of nonlocal theories of the type (15) was initiated by W i g h t m a x ~who , ~ ~ asked the concrete question: whether the vanishing of the commutator of fields (or observables), i.e. f(x) = 0, for xg - x: -xi -xi < -12, would imply its vanishing for xi - x: -xi - xi < 0. It was proven later28-30 (see also Ref. 31) that, indeed, in a quantum field theory which satisfies the translational invariance and the spectral axiom (16), the nonlocal commutativity 2 f(z)= 0 , for xo - xf - xi - xi < -12

implies the local commutativity

f(x) = 0 ,

for xi - xf - xi - xi < 0 .

(17)

This powerful theorem, which does not require Lorentz invariance, can be applied in the noncommutative case, since the hypotheses are fulfilled, with the conclusion that the causality properties of a QFT with space-space noncommutativity are physically identical to those of the corresponding commutative QFT. It is then obvious that the Jost-Lehmann-Dyson representation (12) obtained in the commutative case holds also on the NC space. Consequently, the NC two-particle4two-particle scattering amplitude will have the same form as in the commutative case:

This leads to the analyticity of the NC scattering amplitude in cos 0 in the analog of the Lehmann ellipse, which behaves at high energies E the same

254

way as in the commutative case, i.e. with the semi-major axis as yr, = (COSO),,,

=1

const +E4 .

4. Enlargement of the Domain of Analyticity in cos 0 and Use of Unitarity. Martin's Ellipse Two more ingredients are needed in order to enlarge the domain of analyticity in cos O to the Martin's ellipse and to obtain the F'roissart-Martin bound: the dispersion relations and the unitarity constraint on the partialwave a m p l i t ~ d e s . ~ ~ Imposing the physical nonlocal commutativity condition (15) and reducing it to the local commutativity (17), by using the theorem due to Wightman, Vladimirov and Petrina, leads straightforwardly to the usual forward dispersion relation also in the NC case with a general direction. As for the unitarity constraint on the partial wave amplitudes, the problem has been dealt with in Ref. 26, for a general case of noncommutativity O p v , Qoi # 0. For space-space noncommutativity (Ooi = 0), the scattering amplitude depends, besides the center-of-mass energy, E , on three angular variables. In a system were we take the incoming momentum @in the z-direction, these variables are the polar angles of the outgoing particle momentum, 0 and 4,and the angle a between the vector and the incoming momentum. The partial-wave expansion in this case reads:

p

A ( E ,074,a ) =

C (21' + 1)a~~~rn(E)J'irn(@, 4)8t(cosa)

7

(20)

l,l',m

where J'i, are the spherical harmonics and Pp are the Legendre polynomials. Imposing the unitarity condition directly on (20) or using the general formulas given in Ref. 26, it can be shown that a simple unitarity constraint, which involves single partial-wave amplitudes one at a time, cannot be obtained in general, but only in a setting where the incoming momentum is orthogonal to the NC plane (equivalently it is parallel to the vector In this case, the amplitude depends only on one angle, 0 , and the unitarity constraint is reduced to the well-known one of the commutative cases, i.e.

p).

p,

For this particular setting, @ 11 it is then straightforward, following the prescription developed for commutative QFT, to enlarge the analyticity

255

domain of scattering amplitude to Martin’s ellipse with the semi-major axis at high energies as

and subsequently obtain the NC analog of the Froissart-Martin bound on the total cross-section, in the CMS and for ji 11

8:

E . atot(E)5 c In2 EO

(23)

Thus, the unitarity constraint on the partial-wave amplitudes distinguishes a particular setting (6 11 /?) in which the Lehmann’s ellipse can be enlarged to the Martin’s ellipse and Froissart-Martin bound can be obtained. Nevertheless, this does not exclude the possibility of obtaining a rigorous high-energy bound on the cross-section for p’N and the issue deserves further investigation.

8,

5. Conclusion and Discussions In this paper we have tackled the problem of high energy bounds on the twoparticle+two-particle scattering ampIitude in NC QFT and obtained that, using the causal and spectral conditions (3) and (4) proposed in Ref. 25 for NC theories, it is impossible to draw any conclusion about the analyticity of the scattering amplitude (13) in cos0. However, the physical observation that nonlocality in the noncommuting coordinates is not infinite brought us to imposing a new causality condition (15). We proved that the new causality condition is formally identical t o the one corresponding to the commutative case (17), using the Wightman-Vladimirov-Petrina t h e ~ r e m . ~ ~Thus, - ~ ’ the scattering amplitude in NC QFT is proved to be analytical in cos 0 in the Lehmann ellipse, just as in the commutative case; moreover, dispersion relations can be written on the same basis as in usual QFT. Finally, based on the unitarity constraint on the partial-wave amplitudes in NC QFT, we can conclude that, for theories with space-space noncommutativity (&i = 0), the total cross-section is subject to an upper bound (23) identical to the Froissart-Martin bound in its high-energy behaviour, when the incoming particle momentum p’ is orthogonal to the NC plane. This is the first example of a nonlocal theory, in which cross-sections do have an upper high-energy bound.

256

Acknowledgments T h e financial support of the Academy of Finland under the Project no. 54023 is acknowledged.

References 1. N. Seiberg and E. Witten, JHEP 9909,032 (1999), hep-th/9908142. 2. Y. Liao and K. Sibold, Phys. Lett. B549,352 (2002), hepth/0209221. 3. M. Chaichian, M. N. Mnatsakanova, A. Tureanu and Yu. S. Vernov, Nucl. Phys. B673,476 (2003), hep-th/0306158. 4. L. Alvarez-GaumB and M. A. Vkquez-Mozo, Nucl. Phys. B668,293 (2003), h e p t h/0305093. 5. M. Chaichian, M. N. Mnatsakanova, K. Nishijima, A. Tureanu and Yu. S. Vernov, hep- th/0402212. 6. M. Chaichian, K. Nishijima and A. Tureanu, Phys. Lett. B568, 146 (2003), h e p t h/0209008. 7. M. M. Sheikh-Jabbari, Phys. Rev. Lett. 84,5265 (2000), hep-th/0001167. 8. S. M. Carroll, J. A. Harvey, V. A. Kostelecky, C. D. Lane, T. Okamoto, Phys. Rev. Lett. 87,141601 (2001), hep-th/0105082. 9. P. Aschieri, B. JurEo, P. Schupp and J. Wess, Nucl. Phys. B651,45 (2003), hepth/0205214. 10. M. Froissart, Phys. Rev. 123,1053 (1961). 11. A. Martin,Phys. Rev. 129,1432 (1963); Nuovo Cim. 42,901 (1966). 12. M. Chaichian and A. Tureanu, hep-th/0403032. 13. M. R. Douglas and N. A. Nekrasov, Rev. Mod. Phys. 73,977 (2001), hepth/O 106048. 14. R. J. Szabo, Phys. Rept. 378,207 (2003), hep-th/0109162. 15. J. Gomis and T. Mehen, Nucl. Phys. B591,265 (2000), hep-th/0005129. 16. N. Seiberg, L. Susskind and N. Toumbas, JHEP 0006,044 (2000), h e p th/0005015. 17. L. Alvarez-GaumB and J. L. F. Barbon, Int. J. Mod. Phys. A16,1123 (2001), h e p t h/0006209. 18. S. Mandelstam, Phys. Rev. 112,1344 (1958). 19. H. Lehmann, Nuovo Cimento 10,579 (1958). 20. R. Jost and H. Lehmann, Nuovo Cimento 5, 1598 (1957). 21. F. Dyson, Phys. Rev. 110,1460 (1958). 22. S. S. Schweber, An Introduction to Relativistic Quantum Field Theory, Row, Peterson and Company (1961). 23. A. Martin, Scattering Theory: Unitarity, Analyticity and Crossing, Springer Verlag (1969). 24. N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields, Wiley, New York, 3rd ed. (1980). 25. L. Alvarez-GaumB, J. L. F. Barbon and R. Zwicky, JHEP 0105,057 (2001), hepth/0103069. 26. M. Chaichian, C. Montonen and A. Tureanu, Phys. Lett. B566,263 (2003),

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hepth/0305243. 27. A. S. Wightman, Matematika 6:4,96 (1962). 28. V. S. Vladimirov, Sou. Math. Dokl. 1, 1039 (1960); Methods of the Theory of Functions of Several Complex Variables, Cambridge, Massachusetts, MIT Press (1966). 29. D. Ya. Petrina, Ukr. Mat. Zh. 13,No. 4, 109 (1961) (in Russian). 30. A. S. Wightman, J. Indian Math. SOC.24, 625 (1960-61). 31. N. N. Bogoliubov, A. A. Logunov and I. T. Todorov, Introduction t o Aziomatic Quantum Field Theory, W. A. Benjamin, Inc., New York (1975).

MANY FACES OF D-BRANES: FROM FLAT SPACE, VIA ADS TO PP-WAVES

M. ZAMAKLAR MPI fur Gravitationsphysik A m Muhlenberg 1 14476 Golm, Germany E-mail: [email protected] We review recent studies of branes in Ads x S and pp-wave spaces using effective action methods based on probe branes and supergravity. We also summarise results on an algebraic study of D-branes in these spaces, using extensions of the superisometry algebras which include brane charges.

1. Introduction

Since their discovery in 1995, D-branes have become the center of intensive research of the string theory community. In this process a lot has been learnt about various manifestations of these objects. It has become clear that, depending on the regime in which one works, D-branes can be described using a variety of techniques. In situations with a small number of branes and weak string coupling, methods of open string conformal field theory are appropriate. Open string CFT techniques have, however, been applied mainly to the study of D-branes in very restricted classes of backgrounds. The main obstacle in the study of branes using these techniques is the lack of knowledge concerning the quantisation of strings in arbitrary backgrounds. The problem can be simplified partially by restricting one’s interest t o low-energy processes. In this case a description of branes using effective actions becomes viable. The effective actions of the closed string CFTs are various supergravity actions. These actions are supersymmetric generalisations of the Einstein-Hilbert action. The open string CFT leads to the Dirac-Born-Infeld action (DBI), which (as the name suggests) is a generalisation of the Dirac action for the relativistic membrane to higher dimensional objects, modified to include the brane’s “inner” degrees of freedom, i.e. gauge fields. The latter are included in a fashion proposed a long time 258

259

ago by Born and Infeld in order to regularise the infinite electromagnetic energy of a classical electron. The total (bulk plus brane) effective action is quite complicated due to the non-trivial coupling between the brane and supergravity fields. Hence in order to use this action, one is often forced to simplify the problem further. Increasing the number of branes, for example, leads to the regime where gravitational back-reaction of the branes cannot be neglected, while the gauge theory on the worldvolume of the branes becomes strongly coupled. In this regime branes can be described as purely gravitational solutions, using only the bulk effective action. On the other side, when the number of branes is very small, the probe brane approach is appropriate: only the worldvolume excitations (i.e. scalars and gauge fields) are dynamical fields, while the background is “frozen”. In the first two parts of this report we will partially survey our recent study of Dbranes in Ads and plane-wave (pp-wave) geometries using the supergravity and Dirac-Born-Infeld effective actions. Finally, a very powerful method for classifying possible brane configurations in arbitrary backgrounds is the so-called algebraic method. The full information about the non-perturbative spectrum of string theory (in a given background) is encoded in the “central” extensions of the appropriate superisometry algebra of the background. Unfortunately, the explicit forms of the algebras are essentially not known beyond flat space. Recently, however, we have made an important step in understanding the construction of central extensions of Ads and pp-wave superisometry algebras. In the last part of this survey we report on these results. 2. The Probe Brane Approach 2.1. h m Flat Space to A d s

The problem of understanding the full brane-background system is simplified dramatically by making a restriction to the effective actions, and then further restricting to the probe brane approach. However, this still leaves one with a generically complicated action which one has to solve in order to find the exact embedding of the brane surface in the target space. In generic backgrounds, and in cases of supersymmetric brane configurations, one often uses kappa-symmetry or calibration methods in order to replace the problem of solving second-order differential equations to the problem of solving first order, BPS-like equations. Both of these methods, however, are in a certain sense ad hoc, since they both require good intuition about the ansatze for the brane embeddings.

260

On the other hand, the situation is simpler in special backgrounds such as Ads or pp-waves. All supersymmetric brane configurations in these spaces are inherited from flat space and can be ‘Lderived”by starting from flat space. Given a brane configuration in flat space one first replaces some of the branes in the configuration by their supergravity solutions and subsequently one focuses on their corresponding near-horizon geometries, while keeping the remaining branes as probes. The actual equations for the embedding of the probe can usually be deduced directly from the flat space equations, using Poincark coordinates for the Ads background. This is due to the fact that in this coordinate system, the relation to the flat space Cartesian coordinates is direct. Moreover, it turns out that the same equations that describe the embedding of the brane in flat space also solve the D B I action in the near horizon geometry in Poincark coordinates. The essential reason why the inheritance property holds is the fact that the brane configurations are supersymmetric. This situation should be contrasted to non-BPS configurations, for example a circular 0 1 string in the space transverse to a 0 3 brane. In flat space one can easily derive the solution that describes shrinking of this string, and one can show that this solution (in Cartesian coordinates), when interpreted in Poincarh coordinates, does not solve the DBI equations of motion of the 0 1 string in the Ads5 x S5space. Instead of listing all possible brane configurations which one can derive using this method, let us illustrate it on a very simple example of two 0 3 branes intersecting over a string, 0301 2 3 ----- D301--4 5----.

(1)

In flat space the embedding equations of the second brane are given by

-

2%. -- cz-const., .

(i=2,6,7,8,9).

(2)

We now we replace the first brane with its near horizon geometry, i.e. with the AdSS x S5 space, which in Poincarh coordinates yields ds2 = R2u2(- dt2 +dx: +dx; +dxz) u2 = ~42

+ ... + X: +

dx:

+ $(dx: +. . .

+ . . . + dx:

= du2 -I-u2dRi.

When all coefficients ci in Eq. (2) are zero, we see that the Eq. (2) define a maximal-curvature Ads2 x S1 submanifold.1*2In the case when some of the ci # 0, the Eq. (2) solve the 0 3 DBI action in the (full) 0 3 brane

261

supergravity background. However, when taking the near horizon limit, one in addition needs to rescale the parameters q to zero in order for the solution to survive this limit. As we focus on the region near the 0 3 brane that becomes the background, we simultaneously have to bring the probe 0 3 brane closer and closer to the horizon. The resulting geometry of the 0 3 brane probe describes a brane which starts at the Ads boundary, extends in the u-direction up to some point and then folds back to the b~undary.~ We have recently extended4 the analysis described above to the cases of supersymmetric brane configuration intersecting under an angle in flat space. Due to supersymmetry the inheritance property goes through, as in the previous situation. The resulting geometries of the AdS branes are, however, different, since they non-trivially mix the Ads and sphere submanifolds, i.e. unlike before, the worldvolume surfaces are not factorisable into a product of Ads and sphere submanifolds. Another interesting type of brane that has appeared in Ref. 4 is one whose worldvolume surfaces axe direct products of Ads and sphere submanifolds, but where mixing is achieved via a mixed worldvolume flux which has one index in the sphere and one index in the Ads part. These branes wrap non-supersymmetric target space cycles which are stabilised only after the mixed worldvolume flux is turned on. To construct them, one starts from the flat space configuration of branes intersecting under an angle and performs T-duality in such a way that branes which will be replaced with the background do not carry any worldvolume flux, while the brane which will become a probe carries flu. Then, as before, one takes the near horizon limit of this configuration.

2.2. From AdS to pp-waves

It was realised a long time.ago by Penrose that an infinitely boosted observer in an arbitrary spacetime, in a neighborhood of its geodesic, sees a very simplified background geometry: the geometry of a gravitational wave. This dramatic simplification has recently been used extensively for a direct check of the gauge-gravity (AdS/CFT) corre~pondence,~ one which avoids the standard strong-weak coupling problems. On the gravity side, the Penrose limit amounts to a suitable rescaling of the coordinates and parameters characterising the (super)gravity solution, in such a way that one focuses on the region close to an arbitrary null geodesic. In the same way in which the background undergoes simplifica-

262

tion, so do different objects present in the initial space. The geometry of the resulting branes can easily be derived by rescaling the embedding equations of the branes in the same way as the target space coordinates. Since the pp-wave space is homogeneous but not isotropic, there are three basic families of D-branes which appear in the limit, depending on the relative orientation of the brane and the wave?’ longitudinal D-branes for which the pp-wave propagates along the worldvolume of the D-brane, transuersal D-branes for which the pp-wave propagates in a direction transverse to the D-brane but the timelike direction is along its worldvolume, and instantonic D-branes for which both the direction in which the pp-wave propagates and the timelike direction are transverse to the D-brane. The first class originates from Ads branes where the geodesic along which the boost was performed belonged to the worldvolume of the brane (before the limit), while for the second case, the brane was co-moving with the observer along the the geodesic (i.e. it was infinitely boosted). The third class of branes can be obtained from the first class by a formal T-duality in the timelike direction of the wave. The pp-wave coordinates split into three groups: the “lightcone coordinates” u and w , and two four-dimensional subspaces with SO(4) x SO(4) isometry group. The split of the transverse coordinates is due to the nonvanishing 5-form flux. In the case of longitudinal branes, the worldvolume coordinates split accordingly into three sets: the “lightcone coordinates” u and v, m coordinates along the first SO(4) subspace and n along the second SO(4). A Dp brane ( m n = p - 1) with such orientation is denoted with (+,-, m, n). The number of preserved supersymmetries depends on the values for (n,m)?

+

0

0

0

+

1/2-BPS D-branes with embedding (+, -, m 2, m ) , for m = 1,.. . ,4, 1/4-BPS D-string with embedding (+, -, 0, 0), non-supersymmetric D-branes with embedding (+,-, m, m ) , for m = 1,2,3.

All these results are valid for the brane placed at the “origin” of the ppwave. If we rigidly move the first or second type of brane outside the origin (without turning worldvolume fluxes), supersymmetry is always reduced t o 1/4. However, the previous three classes do not capture all the branes which can appear in p p - w a v e ~ . In ~ ?the ~ process of Penrose rescaling, not all objects of the initial space will be inherited by the final wave geometry. It

263

is usually said that in order to have a nontrivial Penrose limit of a brane in some background, one needs to take the limit along a geodesic which belongs to the brane. This statement is intuitively understandable: in the Penrose limit an infinitesimal region around the geodesic gets zoomed out. Hence, those parts of the brane which are placed at some nonzero distance from the geodesic get pushed off to infinity. However, this reasoning can be circumvented if the distance between the geodesic and the brane is determined by free parameters of the s ~ l u t i o n In . ~ that case one can take the Penrose limit along a geodesic that does not belong to the brane, as long as the parameter labeling the brane in a family of solutions is appropriately scaled. For example, let us consider the family of solutions corresponding t o two intersecting D-branes and let us take the Penrose geodesic to lie on one of the two branes. Then the Penrose limit of the other brane can be nontrivial if, while taking the Penrose limit of the target space metric, we simultaneously scale the angle between the two branes to zero. It should be emphasized that the final configuration obtained in this way is different from the one which is obtained by first sending the angle to zero and then taking the Penrose limit of the metric. Namely, if we first set the angIe to zero (and hence reduce the problem to taking a limit of orthogonally intersecting branes) the resulting brane will be “flat” (i.e. placed at zi = const. in Brinkman coordinates). However, if we follow the procedure outlined above, the resulting D-brane is a brane with a relativistic pulse propagating on its worldvolume (i.e. with some of xi = const. getting replaced by zi(z+)).The precise form of these worldvolume waves carries information about the position of the brane with respect to the geodesic before the limit was been taken.

3. The Supergravity Approach When the number of branes in some space becomes very large, the probe brane approach is inadequate and a supergravity description takes over. Finding a supergravity solutions for D-branes in AdS spaces is, however, still an open problem; the explicit constructions have been carried out only in a few specific cases. One of the reasons for this is that fully localised supergravity solutions for D-brane intersections in flat space are generically not known. Hence in order to construct the brane solutions in asymptotically Ads and pp-wave spaces, one has to start from scratch. We will present here the construction of the (extremal) D-brane solutions in asymp

264

totically pp-wave spaces. The main difficulty in constructing these solutions consists of identifying a coordinate system where the description of the D-brane is the simplest. This is similar to the problem that one would face if one would only know about Minkowski space in spherical coordinates and would try t o describe flat D-branes in these coordinates. Cartesian coordinates are the natural coordinates to describe infinitely extending D-branes in flat space. So the question that one should first ask is what are the analogues of the Cartesian coordinates for D-branes in pp-wave backgrounds? The answer to this question is more complicated than in flat space, as it depends very much on what kind of D-branes one considers. It was shown8i9 that Brinkman coordinates are the natural coordinates for a description of 1/4-BPS and nonsupersymmetric D-branes, while the natural coordinates for the 1/2BPS D-branes are the Rosen coordinates. For the metric part of the ansatz, one writes a simple standard metric for a superposition of D-branes with waves, ds2 = H(y, y')-i (2du(dv

+ S(z, z', y, y')du)

- dZ2 - d?")

- H(y,y')i(dq2 + d g 2 ) . (4) The metric is given in the string frame, and the D-brane worldvolume co. . ,z", x" = d', . . . ,P ) ,while the directions ordinates are (u, v,zi= d,. transverse to the D-brane are (y" = y', . . . ,Y(~-"), yrA = y'l, . . . ,Y ' ( ~ - ~ ) ) . The function H characterising the D-brane is at this stage allowed to depend on all transverse coordinates y, y'. The ansatz for the RR field strength and the dilaton reads

where '*' in F[S]denotes Hodge duality with respect to the metric (4) and W ( z )is an undetermined function which can depend on all directions transverse to the pp-wave. Also, in the case of the 0 3 brane, one has to add to the form ( 5 ) its Hodge dual. One of the main characteristics and perhaps limitations of this ansatz is that the metric is diagonal in Brinkman coordinates. This property forces

265

one to delocalise the supersymmetric solutions along some directions transverse t o the brane when solving the equations of motion.a The smearing procedure physically means that one is constructing an array of D-branes of the same type with an infinitesimally small spacing. However, as we have seen before, the probe brane results tell us that, unless we turn on additional bulk fluxes (sourced by the worldvolume fluxes of the I/ZBPS D-branes), a periodic array of rigid D-branes in Brinkman coordinates with orientation (+, -,n 2 , n ) is only one quarter supersymmetric. Hence the supersymmetric solutions that we find due to the smearing procedure are only 1/4 BPS. However, these restrictions have to be imposed only on the harmonic function characterising the D-brane, and not on the function characterising the pp-wave. Hence, all our solutions asymptotically tend to the unmodified Hpp-wave. Also, despite the simplicity of the ansatz, the nonsupersymmetric solutions, describing branes with (+,-, m, m) orientation, are fully localised. Pluging the ansatz (4)-(7) into the equations of motion and the Bianchi identities one obtains, depending on the orientation of the branes, solutions with the following characteristics. The presence of the D-brane modifies the function S which characterises the pp-wave, while the function H (which specifies the D-brane) is completely unmodified by the presence of the wave. Therefore, this ansatz does not catch the back-reaction of the pp-wave on the D-brane. For a generic embedding of the D-brane, one expects that the (fully localised) D-brane is modified by the wave. However, as our fully localised, nonsupersymmetric solution demonstrates, this does not have to hold for specific embeddings. By examining the behavior of the radially infalling geodesics, one discovers that if the pure pp-wave was focusing the geodesics, this attractive behaviour is strengthened in the presence of a supersymmetric brane, as one would expect. Surprisingly, however, the non-supersymmetric geometries exhibit repulsion behavior.

+

4. The Algebraic Approach

Rather surprisingly, a modification of the superalgebra of anti-de-Sitter backgrounds which accounts for the presence of D-branes in the string spectrum is still unknown. At an algebraic level, D-branes manifest themselves through non-zero expectation values of bosonic tensorial charges. There exists a widespread, but incorrect, belief that the inclusion of these "This is the same type of restriction that one faces when constructing supergravity solutions for intersecting D-branes, with a simple diagonal ansatz.

266

brane charges into the anti-de-Sitter superalgebras follows the well-known flat-space pattern. In flat space, the inclusion of brane charges leads to a rather minimal modification of the super-Poincar6 algebra: the bosonic tensorial charges appear on the right-hand side of the anti-commutator of supercharges, transform as tensors under the Lorentz boosts and rotations, while they commute with all other generators. The brane charges are therefore often loosely called “central”, and the resulting algebra is referred to as the maximal bosonic “central” extension of the super-Poincar6 algebra. However, despite several attempts to construct a similar modification of anti-de-Sitter superalgebras, a physically satisfactory solution is as of yet unknown. There are two basic physical requirements which have to be satisfied by an anti-de-Sitter algebra which is modified to include brane charges. The algebra has to include at least the brane charges which correspond to all D-branes that are already known to exist, and it also has to admit at least the supergraviton multiplet in its spectrum. Mathematically consistent modifications of anti-de-Sitter superalgebras can be constructed, but all existing proposals fail to satisfy one or both of these physical criteria.1° In Ref. 11 we have identified a simple reason why previous attempts to extend anti-de-Sitter superalgebras with brane charges have failed: such extensions are only physically acceptable when one adds new fermionic brane charges as well. The necessity of including new fermionic brane charges into the modified algebra can be understood from a very simple argument based on Jacobi identities, in combination with the two physical requirements just mentioned.ll Let Consider an anti-de-Sitter superisometry algebra, or a p p wave contraction of it. The bracket of supercharges can, very symbolically, be written in the form

where Q and M are the supercharges and rotation generators respectively (we have grouped together momentum and rotation generators by using a notation in the embedding space). Suppose now that we add a bosonic tensorial brane charge 2 on the right-hand side of this bracket. This extension has to be made consistently with the Jacobi identities. Consider the (Q, Q, 2 ) identity, which takes the symbolic form

267

As the brane charge 2 is a tensor charge, it will transform non-trivially under the rotation generators. This implies that the first term of (9) will not vanish. The Jacobi identity can then only hold if 2 also transforms nontrivially under the action of the supersymmetry generators! (In flat space, only the vanishing bracket [P,Z]appears in the first term of the Jacobi identity (9), because in that case the { Q , Q } anti-commutator closes on the translation generators). The simplest option is to assume that no new fermionic charges should be introduced, and that therefore symbolically

21 = Qa .

(10) Although it is possible to construct an algebra based on (lo), which satisfies all Jacobi identities, it is physically unsatisfactory.1° The essential reason is that brackets like (10) are incompatible with multiplets on which the brane charge is zero (the left-hand side would vanish on all states in the multiplet, while the right-hand side is not zero). In other words, one cannot “turn off” the brane charges. The only other way out is to add new fermionic charges Qb, to the algebra, such that (10) is replaced with [Qa,

[Qa,21 = Q&.

(11) In this case it becomes possible to find representations in which both 2 and the new charge Qb, are realised trivially, as expected for e.g. the supergraviton multiplet, while still allowing for multiplets with non-zero brane charges. This formal argument based on Jacobi identities may come as a surprise, and one would perhaps find it more convincing to see new fermionic brane charges appear in eqcplicit models. In Ref. 11 we have shown that such charges indeed do appear. In order to show this, we have analysed the world-volume superalgebras of the supermatrix model and the supermembrane in a pp-wave limit of the anti-de-Sitter background. These models exhibit, in the absence of brane charges, a world-volume version of the superisometry algebra of the background geometry. When bosonic winding charges are included, the algebra automatically exhibits fermionic winding charges as well. Moreover, configurations on which these charges are non-zero can be found explicitly, or can alternatively be generated from configurations on which the fermionic winding charges are zero. On the basis of these results we have briefly discussed a D-brane extension of the osp* (814) superisometry algebra with bosonic as well as fermionic brane charges, which avoids the problems with purely bosonic modifications as first observed in Ref. 10. A partial construction of this algebra has been carried out in Ref. 12.

268

References 1. A. Bilal and C.-S. Chu, Nucl. Phys. B547,179 (1999), hepth/9810195. 2. J. Gutowski, G. Papadopoulos, and P. K. Townsend, Phys. Rev. D60,106006 (1999), hepth/9905156. 3. A. Karch and E. Katz, JHEP 06, 043 (2002), hep-th/0205236. 4. G. Sarkissian and M. Zamaklar, JHEP 03, 005 (2004), hep-thf0308174. 5. D. Berenstein, J. M. Maldacena, and H. Nastase, JHEP 04, 013 (2002), hepth/0202021. 6. K . Skenderis and M. Taylor, JHEP 06, 025 (2002), hepth/0204054. 7. G. Sarkissian and M. Zamaklar, Symmetry breaking, permutation D-branes o n group manifolds: Boundary states and geometric description, h e p t h/03 12215. 8. P. Bain, P. Meessen, and M. Zamaklar, Class. Quant. G m v . 20, 913 (2003), h e p t h/0205 106. 9. P. Bain, K . Peeters, and M. Zamaklar, Phys. Rev. D67, 066001 (2003), hepth/0208038. 10. P. Meessen, K. Peeters, and M. Zamaklar, O n central eztensions of anti-deSitter alge bras, hep-t h/0302 198. 11. K. Peeters and M. Zamaklar, Phys. Rev. D69, 066009 (2004), hepth/0311110. 12. S. Lee and J.-H. Park, Noncentral extension of the Ads5 x S5 superalgebm: Supermultiplet of brane charges, hep-th/0404051.

ABSTRACTS AND TITLES OF REPORTS NOT INCLUDED IN THE VOLUME

Perturbation of Spectra of Operator Matrices D. Djordjevik Department of Mathematics, Faculty of Sciences, University of Nis, Serbia E-mail: [email protected]. yu

If A , B , C are bounded operators on Banach or Hilbert spaces and

we determined the sets

n

uT(~c),

C

where u7 denotes any of the following part of the spectrum: left (right) spectrum, left (right) essential spectrum, Browder or Weil spectrum.

Boundary Liouville Theory and 2D Quantum Gravity I. Kostov Saclay , France E-mail: [email protected] We study the boundary correlation functions in Liouville theory and

2D quantum gravity. We find that all the fundamental Liouville structure constants obey functional equations similar to the one obtained for the two-point function by Fateev, Zamolodchikov and Zamolodchikov. These take the form of finite-difference equations with respect to the boundary parameters. Then we show how these equations can be derived in a discrete model of 2D quantum gravity and give them a geometrical meaning.

269

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Manifestation of TeV Gravity A. Nicolaidis Theoretical Physics Department, University of Thessaloniki, Greece E-mail: [email protected] In recent unification models, gravity propagates in 4+d dimensions, while standard model fields are confined to a four dimensional brane. As a consequence, gravity becomes strong not at the Planck scale, but at the TeV scale. We searched for signatures of TeV gravity and extra dimensions in the cosmic rays. We have interpreted the cosmic ray spectrum "knee" (the steepening of the cosmic ray spectrum at energy 10'5.5eV), as due to missing energy from graviton bremsstrahlung. We estimated the graviton production in pp collisions, in the soft graviton approximation. By reproducing the cosmic ray spectrum in the "knee" region, we deduced that the fundamental scale of gravity is M = 8 TeV and the number of extra dimensions is d = 4 (for details see Gen. Rel. Grav. 35, 1117 (2003), hep-ph/0109247). We studied also gravitational scattering in the background of a massive black hole, living in 4+d dimensions. Two regimes appear. For large impact parameters, the deflection angle follows a power-law behavior, reminiscent of the Rutherford-type scattering. For small impact parameters, the deflection angle develops a logarithmic singularity and becomes infinite at a critical b value. This singularity is reflected into a strong enhancement of the backward scattering. We suggest then, as distinctive signature of black hole formation in particle collisions at TeV energies, the observation of the backward scattering events (for details see hep-ph/0307321).

The Rational Topology of Gauge Groups and of Spaces of Connections S. Terzic University of Montenegro, Faculty of Sciences, Cetinjski put bb, 81 000 Podgorica, Serbia Montenegro E-mail: [email protected]. y u In this talk we are going to present general approach which completely solves the problem on computation of the rational homotopy groups and the rational cohomology of the gauge groups and of the space of connections modulo gauge transformations for principal bundles over four-manifolds. The latter are assumed to be equipped with appropriate Sobolev topolo-

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gies and we also assume that G is a semisimple compact simply connected Lie group and M is compact and simply connected. Note that in some particular cases some of these computations have already been done. Namely, Donaldson computed the cohomology structure of the quotients of spaces of connections for SU(2)-principal bundles over compact simply connected four-manifolds, but the proof essentially uses the fact that the structure group is SU(2). We propose here a general approach which appeals to Sullivan’s minimal model theory. We proceed as follows. First we compute the rational homotopy groups of the gauge group using the result of Singer characterising the weak homotopy type of base point preserving gauge groups, and the result of Milnor giving the homotopy type of a simply connected four-manifold M . Having computed the rational homotopy groups of the quotients of spaces of connections, the nilpotency of the space of connections modulo based point gauge transformations group will make it possible to apply Sullivan’s minimal model theory for the cohomology computation.

String Theory and Matrix Models A . Morozov

Strong Interactions and Stability in Quasi Localized Gravity R. Rattazzi

Strings and Branes in Non-compact Backgrounds V. Schomerus

Two Dimensional Gravity and Liouville Field Theory A . B. Zamolodchikov

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Acknowledgment There are many people and institutions we would like to acknowledge at the end of this “scientific adventure”. Many thanks are due to all authors and participants for their cooperation and to the members of the International Advisory Committee, including those who could not attend. The crucial financial support came from UNESCO-ROSTE Venice, and it was a great privilege to work together with Mr. Vladimir Kouzminov, deputy director of this office. Kind assistance of Mrs. Iulia Nechifor from the same office and Ms. Vesna Filipovic-Nikolic from the Permanent Delegation of Serbia and Montenegro in UNESCO-Paris is also acknowledged. The participation of the researchers from developing countries, i.e. their travel expenses, was substantially supported by International Centre for Theoretical Physics (ICTP) - Trieste. We are especially indebted to Prof. Goran Senjanovic, Prof. Seif Randjbar-Daemi and Prof. Faheem Hussain for this support. Optimal conditions for the workshop were provided through joint applications of Institute of Physics, Humboldt University, Berlin and Department of Physics NiS, to Deutscher Akademischer Austauschdienst (DAAD) coordinated by Prof. Dieter Luest, as well as of Department of Physics, LMU-Munich with Physics Department NiS, coordinated by Prof. Julius Wess, to Deutsche Forschungsgemeinschaft (DFG). The support of Serbian Ministry of Science, Technology and Development was greatly appreciated. It was an encouraging sign for the future support of fundamental research. Many thanks are due t o the Assistant Minister, Aleksandar Belic. Last but not least, our gratitude goes out to the International Association of Mathematical Physics (IAMP) and its President, Prof. David Brydges, as well as to JAT Airways company and Faculty of Science, NiS, for their donation in the early stages when the status of the whole conference was questionable. Finally, this Balkan Workshop could not have been realised without great devotion and efforts of the students of Department of Physics, NiS, D. Dimitrijevic, J. Stankovic and G. Stanojevic. For the moral support and kind attendance to the Conference, we are most grateful to Prof. Martin Huber, President of the European Physical Society (EPS), Prof. Ilija Savic, President of the Serbian Physical Society and academician Zvonko Maric, Institute of Physics, Belgrade. For the Organizing Committee Goran DjordjeviC NiS, September 2004

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Sponsors We would like to thank the following for support: UNESCO - ROSTE (Regional Bureau for Science in Europe, Venice) ICTP (The Abdus Salam International Centre for Theoretical Physics), Italy DFG Deutsche Forschungsgemeinschaft, Germany Ministry of Science, Technologies and Development, Republic of Serbia DAAD (Deutscher Akademischer Austausch Dienst), Germany IAMP (International Association of Mathematical Physics) JAT (Yugoslav Airlines)

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Southeastern European Network in Theoretical and Mat hematical Physics Statement of Intention Participants of BALKAN WORKSHOP BW2003 Mathematical, Theoretical and Phenomenological Challenges Beyond Standard Model: Perspectives of Balkans Collaboration (VrnjaEka Banja, Serbia, August 29 September 2, 2003.) from Bulgaria, Greece, Moldova, Romania and Serbia and Montenegro, initiate the mobility-teaching-research NETWORK IN THEORETICAL AND MATHEMATICAL PHYSICS of Southeast part of Europe. Our goal is to establish closer relations between science faculties and research institutes and individual scientists in the region of Southeastern Europe. We are going to improve the teaching in mathematical and theoretical physics on undergraduate and postgraduate levels, as well as, joint scientifical work. The initiators propose establishing a few different working groups of particular interest in selected topics (quantum field theory, gravity, statistical physics, dynamical systems etc.). This initiative is and will be open to all institutions and individual scientists from the region who share the ideas of the Network. We expect that scientists and scientific institutions from the region (of Southeast Europe, in particular: Albania, Bosnia and Herzegovina, Croatia, Former Yugoslav Republic of Macedonia, Turkey) will join this initiative in the near future. Scientists from all over the world are welcome to join the Network activities. The proposed collaboration will be realized in the following ways: through the exchange of professors with maximal duration of one semester (3 months); 0 exchange of students (preferably PhD students, or for preparation of a diploma work/thesis, or Master degree); 0 joint organization of various meetings (workshops, conferences and schools). The main joint event could be the “Summer Institute in Theoretical and Mathematical Physics”. We are expecting to organize it every year in different countries, based on the rotational scheme. The duration of the institute could be 2-4 weeks. The institute should consist of different events (seminars, workshops, schools and so on); 0 short visits: from a couple of days to a couple of weeks, including seminars and research; 0 support of the publications of

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proceedings, books and other issues and other means for the exchange of scientific information. The participants agree to support the foundation of the Initiative Committee to oversee the establishment and initial development of the Network with the following composition: Boyka Aneva (INRNE and University of Sofia, Bulgaria), Goran DjordjeviC (University of NiS, SCG), Argyris Nicolaidis (University of Thessaloniki, Greece), Corneliu Sochichiu (IAP, Chisinau, Moldova, INFN F'rascati, Italy), Mihai Visinescu (Bucharest, Romania). The membership of the Committee will be extended to assure the participation of all countries which join the Network. One of the prime objectives of this Committee will be the elaboration of the terms of the reference of the Network. The meeting of the Initiative Committee will be held during next 1 2 months. It was agreed that for this initial period, the Faculty of Science and Mathematics of the University of NiS would take responsibility for the coordination the Network activities. A Steering Committee consisting of outstanding and internationally leading researchers from the region and from all over the world is to be established. Based on previous contacts it is expected that financial support will be provided by: 0 local institutions, ministries and other foundations from our countries; 0 European Union, different European foundations, large national foundations in Western Europe; 0 international Institutions, like UNESCO, ICTP, CERN, IAE, JINR, Central European Initiative; private foundations.

Vrnjaeka Banja, Serbia and Montenegro, September 2, 2003.

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Epilogue This book has almost bean read and is about t o be closed. The last participants of the 2003 conference in VrnjaEka, Banja have left this beautiful resort a long time ago. In the meantime, many new findings and equations have been brought to light, many new papers have been published, and many new relations have been established. However, physicists all around the world, in particular, young ones from the Balkan region (or Southeastern Europe, as others may prefer to call it) are still in need of scientific contact, books, equipment, exchange of ideas and new knowledge. We do hope that Balkan Workshop series and the Southeastern European Network in Mathematical and Theoretical Physics (SEENET-MTP, http://seenet-mtp.pmf.ni.ac.yu)will be proved helpful and valuable in this respect, and it should be our imperative that their scope of impact be broadened. In the course of preparatory activities for the next in the series of the Balkan workshops - BW2005 (http://www.pmf.ni.ac.yu/bw2OO5) we learnt about the Nordic Network Discovery Physics at the LHC (http://www.hep.lu.se/nlhc). The example of the Nordic countries should be kept in mind when tracing further paths of our cooperation. As always, time will be the most crucial judge of all our efforts and endeavors. The people in the Initiative and Steering/Advisory Committee, together with all their friends, supporters, physicists and mathematicians of “good will” will try to offer their own contribution to this long and difficult process of making our beautiful region a respectable part of the world community or a t least, by asking, like billions before us did, what the universe in fact is. Welcome to the SEENET-MTP!