Supersymmetry and large-N limit in a zero-dimensional two-matrix model

Volume 222, number 3,4

PHYSICSLETTERSB

25 May 1989

SUPERSYMMETRY AND LARGE-N L I M I T IN A Z E R O - D I M E N S I O N A L TWO-MATRIX M O D E L Jorge ALFARO and Juan C. RETAMAL Facultad de Fisica, Pontificia Universidad Catolica de Chile, Casilla 6177, Santiago 22, Chile

Received 10 February 1989

We study the zero-dimensionaltwo-hermitean-matrixmodel, by using a new method to obtain the large-Nlimit of a quantum field theory. This method predicts a closed systemof integral equations that givesthe solution in a closed form.

1. Introduction

The large-N expansion of a quantum field theory (QFT) remains as an interesting possibility to understand the non-perturbative region of a real system like QCD [ 1 ]. In this context many efforts have been made in order to clarify some qualitative aspects of a theory, for example, the large-N reduction and the studies of the phase structure of vector theories [ 2,3 ]. We know that in the large-N limit of a QFT, the perturbative expansion is reduced to the so-called planar Feynman graphs. It is very important to find a general method to sum this remaining planar series. The answer to this problem is known for matrix models in zero and one dimension [ 4 ]. These models contain a large parameter N which corresponds to the order of the invariance group. In the zero-dimensional case the existence of this parameter suggests the application of the saddle point method in the path integral of the model. But this method can only be applied after reducing the number of degrees of freedom in the action. In the one-hermitean-matrix model such a reduction is achieved by means of a change of variables to U (N) invariants plus angles and one obtains a closed solution [4]. In the two-hermiteanmatrix model it is not easy to achieve this reduction by this change of variables and the solution obtained is not closed [5 ]. The two-hermitean-matrix model has been studied by the author of ref. [ 6 ] using the theory of diffusion equations and of orthogonal polynomials with a non-local weight. 0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )

The aim of this work is to apply an alternative method to obtain the large-N limit of a QFT. This method has been developed in previous papers [ 7,8 ] and is based on a supersymmetric (SUSY) extension of the theory. Originally this approach has been introduced in the context of a stochastic formulation of a model [ 8 ] that contains the so-called hidden SUSY [9]. But it is possible to define a SUSY extended model in the standard formulation of QFT [ 10 ]. In this paper we use this idea to study the zero-dimensional two-hermitean-matrix model. The master field equations determining the U (N) invariants of the model are obtained from a completely general set of Ward identities associated with this SUSY. The lowest order identities allow to find a closed system of integral equations. We show that this system of equations contains the previously known solution to the two-hermitean-matrix model. The paper is divided as follows: In section 2 we define the model and its SUSY extension; in section 3 we obtain the Ward identities and a general relation between the U ( N ) invariants; in section 4 we derive the system of integral equations and we find their solution in a particular case. Moreover, we show that this system contains the solution given in ref. [ 5 ]. Finally, in section 5 we discuss our results. 2. The model and its SUSY extension

The action for the zero-dimensional two-hermitean-matrix model is given by [ 5,6 ] 429

Volume 222, number 3,4

PHYSICS LETTERS B 3. Ward identities

S= ½tr M { + ½tr M 2 + (g/4N) tr M 4 + (g/4N) t r M 4 -ctrM~M2.

(2.1)

This action is invariant under M-+ UMU*, UeU (N). The SUSY extension of this model is defined as [ 10] S'-~S-~- E Itla,ik~'la,ki, a,ik

25 May 1989

a=l,2,

(2.2)

In this section we obtain the Ward identities associated to SUSY invariance and the master field equation implied by these. For the two-hermitean-matrix model in d= 0 we have the following general Ward identity:

(StrF(M~,M2)~a)=O, where ~a and ~P~ are NX N matrices whose entries are Grassman variables. The new action S' is invariant under SUSY transformations 8M~ = e gJ~ + ~ ,

a=l,2,

(3.1)

where F(MI, M2) is an arbitrary product of M~ and M> All possible expressions like (3.1) can be generated by (8 tr exp(iklM~ ) exp(illM2)

8gJa= 8SISMa(,

8~'la = -SSISM,,E,

a=l,2,

× exp (ik2Ml) exp (il2M2)...

(2.3) where ~(+ (e = ee = ~(= 0. The existence of this SUSY invariance allows to apply the following theorem valid for a QFT with U (N) invariance, proved in ref. [ 8 ]. Let F be a U ( N ) invariant function of the quantum field and 8 the SUSY variation then we have the following theorem.

×exp(ik~Ml ) exp(ilnM2) 7"a ) =0,

where kt, li and n are arbitrary numbers. If we decompose each matrix in its invariant subspaces and take into account the theorem enunciated above, we have at leading order in large N exp (ik~ x,~, ) exp(il~y~,)... O/l~l--.~nfln

Theorem. The U ( N ) invariant function of the

× exp ( i k . x J exp (il.yp.)

quantum field are SUSY invariants in leading order of large N (e.g.)

× 8 tr T~ 1T~ 1...T?" T~" ~a =0,

( S F ) = 8F=O.

(2.4)

(3.2)

(3.3)

where x and y denote the eigenvalues of Ml and M2, respectively. This expression is valid for all k, and l~ so that we can write

It is suitable to decompose each matrix in its invariant subspaces as follows:

g tr T?' T~' ...T~"T~" g~a=0.

Ma=~m~T~,

From this last identity with a = 1 and using (2.3) and (2.5) we get the following equation:

a=l,2,...,

a=l,2,...,

(2.5)

ot

where the T~ are the orthogonal projectors to the invariant subspaces and m a represents the eigenvalues of the matrices. The eigenvalues of an hermitean matrix are U (N) invariants, so by the theorem they are SUSY invariants. This argument allows to determine the SUSY variations of the projectors [8 ] 1

ot Y 7 8T~= E - ~r (T~fM~Ta+TagM~T~a)" e+c, ma - m a

(2.6)

430

(

(3.4)

1)

A(xm)-cYP°-- ~,,~Z x,~,--xr T'~'pl.....P"

k=2

7 ~ o t l Xotl - - X 7

× T~a~,k+1ak+,..... ~. ( 1 -- 5,~ke) +

1

E - T}'fl113~2~2. . . . . ~¢~, x m - x ~

~"'

where we have introduced the notation

(

3.5 )

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that this system can be decoupled by transforming the second equation in (4.1) as follows:

3(x,~) =x~ + (g/N)x 3, T~,p...... a, = t r To(~Tgt...T?"Tg".

(3.6)

By taking a = 2 in (3.4) we get an expression analogous to (3.5). Eq. (3.6) for n arbitrary relates the coupling of order n with all couplings of lowest order in a recursive way. In this paper we will restrict ourselves to the study of the case n = 1 in (3.4) which gives us equations that contain couplings of two projectors only. In this case we obtain the following system of equations:

1

3(x~)-2 ~ - -

p

~,~" T~px,~ - x~

#

x,~--x~

A(yp)T,~p=cx~,.

4. Integral equations In this section we derive the system of integral equations for the density of eigenvalues of each matrix. By taking the continuum limit in (3.7) as made in ref. [ 4 ] we obtain b2

al

a2

bl

u(x) Pn(X' ) ~', --P.(X) ~ ~u(x') dx' -- ~ X_--U~ al

al

(4.2)

where b2 ¢1

P n ( x ) = j ynv(Y) F(x,y) dy.

(4.3)

P, (x) + gP3(x) = cx (3.7)

bl

bl

P~ (x) is determined by the first equation in (4.1). Pn(x) for n > 1 can be expressed in terms ofP~ (x) as we see in (4.2), so that the third equation in (4.1) written as

1

= A(x,~)-cyp- r

cP,+~ (x) =A(x)Pn(x)

a2

=c ~ ypT,~p,

~ ~ o~ X a - - X ~

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(4.4)

determines a closed integral equation for u (x). Eq. (3.5) and the system of integral equations (4.1) are the main result of this paper. We explore below the system (4.1) in various simple cases. In the case g = 0 we obtain the following integral equation: bl

A ( x ) - 2 ~ u(x'--~)

=c2x'

(4.5)

al

where the solution is given by

u ( x ) = ( 1 / 2 n ) ( 1 - c 2 ) x / 4 / ( 1 - c 2 ) - x 2,

(4.6)

bl

and the

interval of definition

is a ~ = - b ~ = 2

× (1-cZ) -~/2. It is easy to verify that this result is

al bt

f

al b2

f 3(y) F(x, y) v(y) dy=cx,

(4.1)

a2

where u(x) and v(y) denote the densities of eigenvalues for M~ and Me respectively, with [a~, b~ ] and [a2, b2 ] being their intervals of definition and F(x, y) the continuum expression for T,~a. It is easy to see

in agreement with the perturbative calculation of the invariants tr (M~)" and tr (M~M2). On the other hand, we must notice that if we have an expression for Tap the first equation in (4.1) gives us an equation decoupled for u(x). We think in this way because this equation is similar to the one found by the authors in ref. [ 5 ]. One possibility is to consider a perturbative expansion for Tap in the coupling constant c,

T,~p= ~', T ~ c k,

(4.7)

k=O

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if we suppose that the eigenvalues are m o d i f i e d by the coupling so that

x,~=x~°)+cx~l)...,

y,~=y~°)+cy~) ....

(4.8)

25 May 1989

it is interesting to test it in m o r e realistic models. O f these, the simplest example is the o n e - m a t r i x m o d e l in m o r e dimensions. Work in that direction is in progress now.

Replacing (4.7) and (4.8) in (3.7) we get

F(x,y)=l +c(x-(x)

)(y-(y)

Acknowledgement

)

+c2[ (x-
....

(4.9)

where a c o n t i n u u m limit as N--. ~ is understood. Introducing this result in the first equation o f (4.1) we obtain bl

3 ( x ) - 2 "J[ u ( x ' ) dx' X--X'

J.A. wants to thank the m e m b e r s o f The Niels Bohr Institute for their w a r m hospitality. Especially he wants to thank Dr. J. A m b j o r n for m a n y interesting conversations. He also acknowledges support from D I U C 201-86 and F O N D E C Y T ~0394-88. J.R. wants to t h a n k L. Huerta a n d L. Vergara for various useful discussions and R. Rojas for reading the manuscript. J.R. acknowledges a C O N I C Y T doctoral fellowship.

al

=c
References

+C3( ( y - - ( y ) ) 3) X [ ( X - - ( X ) ) 2 - - ( ( X - - ( X ) ) 2 ) ] + ....

(4.10)

This is an integral equation for the eigenvalues ofM~. A similar expression for the eigenvalues o f M2 can be obtained. Eq. (4.10) is in agreement with eq. (4.6) in ref. [5 ] to the order o f c 3.

5. Discussion The m e t h o d e m p l o y e d here predicts a closed system o f integral equations not k n o w n previously. The system o f equations contains the right limit when the coupling constant c goes to zero corresponding to the p r o b l e m o f the o n e - h e r m i t e a n - m a t r i x m o d e l [ 4 ]. We have solved the case g = 0 which is in agreement with the perturbative calculation to this order; work in the general case is in progress. A n o t h e r interesting p r e d i c t i o n is eq. (3.5). It is a recurrence relation for the correlatiOns o f projectors. Higher o r d e r correlations o f projectors can be obt a i n e d in terms o f the coupling o f two projectors det e r m i n e d by the solution o f (4.1). Since the s u p e r s y m m e t r i c m e t h o d has led us to a quick a n d elegant solution o f the two-matrix model,

432

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