Two-Dimensional Chiral Matrix Models and String Theories

CERN-TH/96-311

hep-th/9611011

Two-Dimensional Chiral Matrix Mo dels

and String Theories

Ivan K. Kostov

C.E.A. - Saclay, Service de Physique Theorique

F-91191 Gif-Sur-Yvette, France

Matthias Staudacher

CERN, Theory Division

CH-1211 Geneva 23, Switzerland

We formulate and solve a class of two-dimensional matrix gauge mo d-

els describing ensembles of non-folding surfaces covering an oriented, dis-

cretized, two-dimensional manifold. We interpret the mo dels as string the-

ories characterized by a set of coupling constants asso ciated to worldsheet

rami cation p oints of various orders. Our approach is closely related to,

but simpler than, the string theory describing two-dimensional Yang-Mills

theory. Using recently develop ed character expansion metho ds we exactly

solve the mo dels for target space lattices of arbitrary internal connectivity

and top ology.

SPhT-96/123

CERN-TH/96-311

November 1996

member of CNRS

[email protected] a.f r

[email protected]. ch

1. Intro duction

The idea that the strong interactions are describ ed by a string theory

which is in some sense dual to p erturbative QCD is a ma jor challenge for

high energy theory. More generally, a D -dimensional con ning Yang-Mills

theory is exp ected to de ne a string theory with D -dimensional target space

stable in the interval 2 D 4.

This \YM string" has b een constructed so far only for D = 2 [1],[2].

It has b een shown that the partition function of pure U (N ) gauge theory

de ned on a two-dimensional manifold of given genus and area can be rep-

resented in terms of a weighted sum over maps from a worldsheet to

spacetime . The allowed worldsheet con gurations represent minimal

area maps ! . The latter condition is equivalent to the condition

W T

that the emb edded surfaces are not allowed to have folds. Rami cation

p oints are however allowed, as it has b een suggested in earlier studies [3].

In other words, the path integral of the \YM string" is over all branched

covers of the target manifold [1], [2].

Unfortunately, this construction is highly involved and do es not easily

reduce to a system of simple geometrical principles . One can nevertheless

sp eculate, using the analogy with the random-walk representation of the

O (N ) mo del, that there exists an underlying 2D string theory with clear

geometrical interpretation. The YM string is obtained from the latter by

tuning the interactions due to rami cation p oints and adding new contact

interactions via microscopic tub es, etc.

One is thus led to lo ok for the most general 2D string theory whose

path integral is given by branched covers of the target space. Any such

string theory is invariant under area-preserving di eomorphisms of the tar-

get space . Its partition function dep ends on only through its genus

T T

and area. In addition to the top ological coupling constant N there is

a set of couplings t asso ciated with the interactions due to rami cation

p oints of order n.

The random surface representation of YM on a lattice found in [4] is geometrically

clear but has the inconvenience of b eing highly redundant. For example, the folds are not

forbidden but their contribution vanishes as a result of cancellations.

In this letter we prop ose a discretization of the path integral for such

string theories, in which the target space represents a two-dimensional

simplicial complex with given top ology. The branch p oints (the images

of the rami cation p oints) are thus lo cated at the vertices of the target

lattice . We will not try to establish a worldsheet description but instead

construct and solve a class of equivalent matrix mo dels. Our approach is

thus analogous to quantizing Polyakov string theory by employing randomly

triangulated surfaces.

The matrix mo del asso ciated with the target space will be formu-

lated as a 2D lattice gauge theory whose lo cal link variables are complex

N N matrices. However, the matrices are no longer unitary: the Haar

measure on U (N ) is replaced by a Gaussian measure. The mo del resembles

very much the 2D Weingarten mo del [5] with the imp ortant di erence that

the lattice action represents a sum over the positively oriented cells only.

The latter restriction eliminates from the string path integral all surfaces

containing folds (which are b elieved to cause the trivial critical b ehavior of

the standard Weingarten mo del [6]). We will demonstrate that these 2D

matrix mo dels are exactly solvable for any target space lattice and any N

by employing the character expansion metho ds recently develop ed in [7],

[8], [9], [10]. We pay sp ecial attention to the cases of spherical and toroidal

top ology. In the rst case we observe that the string theory exists only for

a target space with suciently large area. At the critical area a third-order

transition takes place due to the entropy of the rami cation p oints. In the

second case we nd the same partition function as the chirally p erturb ed

conformal eld theory considered recently by R. Dijkgraaf [11]. In this letter

we restrict our attention to the discrete case since it involves combinatorial

problems interesting on their own. The continuum limit of an in nitely

dense target lattice will be considered in detail elsewhere [12].

2. De nition of the mo dels

By target space we will understand an oriented triangulated surface

(two-dimensional simplicial complex) of genus G containing N p oints, N

0 1

links and N two-dimensional cells. These numb ers are related by the Euler

formula

N N +N = 2 2G: (2:1)

0 1 2

In addition, each cell c is characterized by its area A .

We will rst consider the simplest string theory with this target space,

in which all rami cation p oints have Boltzmann weight one, and the corre-

sp onding matrix mo del. The generalized mo del, to be considered in sect. 6,

will dep end on a set of external eld variables asso ciated with the p oints of

At each link ` is de ned a eld variable representing an N N

matrix with complex elements. The partition function of the matrix mo del

is de ned as

Y Y

Z = [D ] exp (N Tr ); (2:2)

` c c

of link variables where = e , denotes the ordered pro duct

` c c

`2@c

along the oriented b oundary @c of the cell c, and the integration over the

link variables is p erformed with the Gaussian measure

N N Tr

: (2:3) [D ] = (N= ) d d e

` ij ij

Note that we omit the complex conjugate from the plaquette action,

eliminating thereby orientation reversing plaquettes. This is why we call

the mo del with partition function (2.2) a chiral matrix mo del. The mo del is

invariant under complex conjugation of the matrix variables and reversing

the orientation of the target space.

The p erturbative expansion of (2.2) results in a representation of the

free energy F = ln Z in terms of connected lattice surfaces emb edded

in the target surface . Each of these surfaces is obtained by placing

plaquettes on the faces of the lattice and gluing any two together along

the edges. All plaquettes should have the same orientation, which means

that the surfaces cannot have folds. It is geometrically evident that the

number of plaquettes, say n, covering each cell is constant throughout .

Thus the surface is wrapping n times and its area is A = nA

W T W T

where A = A is the total area of the target space. The surface

T c W c

may have rami cation p oints whose images are p oints of . The map

! de nes at each p oint p 2 a branching number B . The

W T T p

branch p oints are the p oints with B 6= 0. By the Riemann-Hurwitz formula

2g 2 = n(2G 2) + B where g is the genus of the surface . The

p W

free energy F = ln Z of our chiral mo del can be written as

1 1

X X

nA 22g

F = e N F (n; g jG; N ) (2:4)

n=1

g =G

where F (n; g jG; N ) is the number of no-fold surfaces of genus g covering n

times . This number dep ends on only through its genus G and the

T T

number of the N of allowed lo cations of the branched p oints.

3. Exact solution by the character expansion metho d

Applying to (2.2) the same strategy as in ref.[10], we expand the ex-

p onential of the action for each cell c as a sum over the characters

of the p olynomial representations of GL(N ). We use the shifted weights

h = fh ;h ;::: ;h g where h are related to the lengths m ; :::; m of the

1 2 N i 1 N

rows of the Young tableau by h = N i + m and are therefore sub jected

i i

to the constraint h > h > ::: > h 0. We will denote by jhj = m

1 2 N i i

the total number of boxes of the Young tableau and by

h h

i j

: (3:1) =

i j

the dimension of the representation h. The character expansion of the

exp onential then reads

N Tr jhj

c c

( ); (3:2) e =

where we have de ned

h !

jhj

= N : (3:3)

(N i)!

i=1

Now the mo del can be diagonalized in h-space by using the following

two simple but powerful facts. Firstly, one easily proves [7],[10]

[D ] ( ) = ; (3:4)

h h

where is given as ab ove by (3.3). Secondly, one uses that complex

matrices may b e diagonalized by a bi-unitary transformation U V , where

U; V are unitary and is a p ositive de nite and diagonal matrix. Then

one uses the formula (3.4) together with the ssion and fusion rules for

unitary matrices

(A) (B ); D U (AU B U ) =

h h h

(3:5)

D U (AU ) (U B ) = (AB );

h;h

h h

to derive ssion and fusion rules for complex matrices corresp onding to the

measure (2.3):

[D ] (AB ) = (A) (B );

h h h

(3:6)

(AB ): [D ] (A) ( B ) =

h;h

h h

Therefore, complex matrices b ehave exactly like unitary matrices apart from

the extra lo cal factor for each integration .

Having at hand (3.6), we can pro ceed with the exact solution of (2.2),

just as in the gauge theory case [13],[14]. Using the fusion rule one pro-

gressively eliminates links between adjoining cells until one is left with a

single cell. The remaining links along that plaquette are integrated out by

applying the ssion rule. One arrives at the formula

N +2G2

22G

jhjA

e : (3:7) Z =

h h

The ssion and fusion rules actually hold for any U (N ) invariant measure [D ], if

we de ne the factor by the integral (3.4). In particular one could consider p otentials

higher than linear in in (2.3).

We observe that, in accord with the surface representation (2.4), the solu-

tion's only dep endence on the original target space lattice data is through

the genus G, the number of vertices N , and the total area A of the target

0 T

space. It is therefore clear that a given character expansion (3.7) corre-

sp onds to many di erent complex multi matrix mo dels (2.2) with the same

partition function: As many, as we can form simplicial complexes di er-

ing in their internal connectivity but constrained by the global data G,

N and A . (This is a lattice version of the invariance with resp ect to

0 T

area-preserving automorphisms.) In particular we can represent the target

space by a one-cell complex and write down a matrix mo del with a minimal

number N = N 1+2G of link variables giving the expansion (3.7):

1 0 `

N 1 N 1

G G

0 0

h i

Y Y Y Y

y y

A y

Z = [d ][d ] [D ] exp Ne Tr :

s s~ k s s~ k

s=1 s=1

k =1 k =1

(3:8)

Note that the trace in the exp onent of (3.8) corresp onds to the element

of the homotopy group of a surface with G handles and N punctures which

is equivalent to the identity (Fig. 1).

...... 1 N -1 2 0

Fig.1: The target space of the e ective one-plaquette mo del.

4. Spherical target space

In the case of a spherical target space (G = 0) the partition function

(3.7) b ecomes:

N 2

2 jhjA

Z = e : (4:1)

G=0

h h

The simplest example of N = 2 is describ ed by a Gaussian integral. There

is only one surface covering n times a sphere with two punctures and it

1 1

contributes e coming from the cyclic to the free energy (the factor

n n

symmetry). No higher-genus surfaces are contributing. Summing on n gives

therefore

2 A

F = N ln(1 e ); (4:2)

which is also the result of the Gaussian integration.

If we have N 3 the situation b ecomes non-trivial. Now there are

contributions from non-spherical (g > 0) world sheets as well. Concentrat-

ing on the leading (g = 0, i.e. N = 1) term, we can pro ceed by using

the saddlep oint techniques of [7]. Here the sum (4.1) describ es a \gas" of

mutually repulsive weights at thermal equilibrium and is dominated by the

contribution of a \classical" con guration fhg. One intro duces continuum

and a density (h) = (h h ). The saddlep oint variables h =

N N

density is found by functionally varying (4.1) with resp ect to h; this gives

(h ) 2 N 1 h

dh = log h + A log : (4:3)

h h 2 2 h b

It is straightforward to explicitly solve this equation and calculate the free

energy. Here we will merely write out the equation determining the critical

b ehavior; one nds, for the variable u = ( a + b) , the algebraic equation

A N 1

T 0

(N 1) e u = u 1: (4:4)

This equation predicts for N 3 a critical area A = A > 0: The sum

0 T

. over surfaces diverges if the area of the target manifold falls b elow A

A quite similar transition has already observed in YM by Douglas and

Kazakov [15]. One nds for the free energy near the critical area

2 c

F N (A A ) (4:5)

i.e., the same critical b ehavior as the Polyakov string in zero dimensions.

We can understand the critical b ehavior by considering e.g. N = 3

y y

qN Tr

1 2

Z = [D ][D ] e : (4:6)

1 2

where q = e . Eq.(4.6) generates planar diagrams which can be inter-

preted as abstract plaquettes without emb edding. Its critical b ehavior is in

the university class of pure gravity.

For N 4 the mo dels (3.8) are intractable without our character

expansion techniques. The qualitative b ehavior remains the same: As seen

from (4.4), we can always nd the b ehavior of pure 2D gravity. The critical

size of the target sphere is given by

N 1

: (4:7) A = 2 ln(N 1) + (N 2) ln

0 0

N 2

and tends to in nity in the continuum limit N ! 1. It is therefore evident

that the theory should be mo di ed in order to have a sensible continuum

limit.

5. Toroidal target space

If the target space is a torus (G = 1) the Vandermonde determinants

in (3.7) cancel, the free energy is now order O (N ) and the saddlep oint

metho d is no longer applicable. However, we can do even b etter in this

case and immediately give a result to all orders in . Each term in the

sum (3.7) factorizes into a pro duct over the rows of the Young tableau; the

sum remains nevertheless non-trivial due to the ordering constraint on the

weights. Using (3.3), rewrite Z as

X Y

i j

jhjA

Z = e 1 : (5:1)

G=1

fi;j g2h

Here fi; j g denotes the a box in row i and column j and the pro duct go es

over all boxes in the tableau h. Slicing the tableau through the diagonal

and counting the fraction of boxes in the rows of the upp er half and the

columns of the lower half we can elegantly express (5.1) as

1 n n

1 1

Y Y Y

N N

dz k k

0 0

n+ n+

2 2

Z = [1 + zq 1+ 1 ][ 1 + z q ]

G=1

2iz N N

n=0

k=1 k =1

(5:2)

where we have denoted q = e . The contour integral ensures that an

equal number of rows and columns emanate from the diagonal; it eliminates

con gurations that cannot be interpreted as a tableau . The leading order

of the free energy gives the partition function of the noninteracting string.

It is immediately extracted from either (5.1) or (5.2) and is identical to

what one nds for the continuum Yang-Mills theory on a torus [1]:

n 2

F = ln(1 q )+O(N ): (5:3)

n=1

The di erences start with the terms. These are in principle computable

in terms of quasi mo dular forms as in [16],[18],[17],[11].

Similarly to the YM theory, the matrix mo dels with toroidal target

spaces can be interpreted as chiral deformations of a two-dimensional free

eld theory de ned by the action [16],[17],[11]

Z I

s d z

n+1

S = @'@' + (@') : (5:4)

2 n +1

The partition function (5.2) is equal to the zero U (1) charge sector of the

partition function of (5.4) computed on a torus, with an appropriate choice

of the couplings fs g.

Let us mention an interesting interpretation of the mo del with N = 1

whose target space is a torus with one puncture. Explicitly, we have

y y

qN Tr

1 ~

Z = [D ][D ] e : (5:5)

1 ~

The mo del is sup er cially very similar to (4.6) of the last section. Again

a network of square plaquettes (corresp onding to the vertices of (5.5)) is

generated. But here a lo cal rule suppresses all positive curvature: The co or-

dination numb ers at the vertices of such a surface can takevalues 4; 8; 12; :::,

which corresp ond to zero or negative curvature. No lo cal curvature uctu-

ations, leading to pure quantum gravity b ehavior, are p ossible: For xed

genus g the surfaces contain only a nite number of quantized, negative

curvature defects of total curvature 4 (g 1). The mo del is therefore

similar in spirit to the \almost at planar graphs" of [8], where discrete

surfaces of vanishing or p ositive lo cal curvature were considered.

It is interesting to p oint out that a very similar result was rst found by Douglas

[16] in the context of continuum 2D Yang-Mills theory on the torus. The combinatorial

derivation of (5.2) is due to Dijkgraaf [17].

6. The general case

Let us now consider the discretized string theory in the general case

when a rami cation p oint of order n = 2; 3; 4; ::: asso ciated with the p oint

(p)

p 2 is weighted by a factor t . It is straightforward to appropriately

mo dify the original matrix theory and solve it by rep eating the same steps

explained in sect. 3.

(p)

We will parametrize the couplings t by an N N external matrix

eld

n (p)

Tr(H ) (6:1) t =

and insert the matrix H into the r.h.s. of (2.2). This should be done in

the following way: For each p oint p we cho ose one of the cells c such that

p 2 @c and insert the matrix H into the corresp onding trace. For example,

the one-plaquette theory (3.8) b ecomes

N 1 N 1

G G

0 0

h i

Y Y Y Y

y y

Z = [d ][d ] [D ] exp qN Tr H ( ) ( H ) :

s s~ k N s s~ k k

0 s

s=1 s=1

k =1 k =1

(6:2)

Then one can pro ceed as in sect. 3, arriving at the following nal ex-

pression replacing (3.7):

22G

X Y

h h

jhjA

Z = e (H ) : (6:3)

h h

h k =1

The mo del (6.3) can be solved in the spherical limit using the metho ds

develop ed in [7]-[10]. Let us note that in the case of a sphere with three

punctures (G = 0; N = 3) one repro duces the mo del of the dually weighted

planar graphs in the most general formulation presented in [10]:

(H ) (H ) (H ): (6:4) Z =

1 2 3

(A =0;G=0;N =3)

h h h

T 0

7. Concluding remarks

1. We have constructed a discretization of the most general string eld

theory of nonfolding surfaces immersed in a two-dimensional compact space-

time. In the continuum limit the sum over the p ositions of a branch p oint

should be replaced by an integral with resp ect to the area. This means that

the couplings t should scale as

t (7:1) t =

n n

where t are the interaction constants of the continuum theory. The prop-

erties of the expansion (6.3) in this limit will be studied elsewhere [12].

2. The YM-string should b e obtained by a sp ecial tuning of the coupling

constants and by intro ducing contact (tub e-like) interactions. These inter-

actions can be implemented by considering the H-matrices as dynamical

elds.

3. We p oint out an interesting equivalence between ensembles of cover-

ings of a sphere with xed numb er of punctures and an ensemble of abstract

(i.e., without emb edding) planar graphs. It is therefore not a miracle that

we nd a third order phase transition as in the case of pure 2D gravity. In

our case the transition is due to the entropy of surfaces with a large num-

ber of rami cation p oints. A very similar transition has already observed

in YM by Douglas and Kazakov [15].

5. The chiral matrix mo dels de ned on a torus give a nice geometri-

cal interpretation of the general chiral deformation of top ological theory

studied by R. Dijkgraaf [11]. The sum over surfaces can again be inter-

preted in terms of abstract planar graphs describing random surfaces with

non-p ositive lo cal curvature.

Acknowledgements

One of us (I.K.) thanks the CERN Theory Division for hospitality.

References

[1] D. Gross, Nucl. Phys. B B 400 (1993) 161; D. Gross and W. Taylor,

Nucl. Phys. B 400 (1993) 181.

[2] S. Cordes, G. Mo ore and S. Ramgo olam, Lectures on 2D Yang-Mil ls

Theory, Equivariant Cohomology and Topological Field Theories, hep-

th/9411210, 1993 Les Houches and Trieste Pro ceedings.

[3] V.A. Kazakov and I. Kostov, Nucl. Phys. B 176 (1980) 199; V.A. Kaza-

kov, Nucl. Phys. B 179 (1981) 283.

[4] V. Kazakov, Phys. Lett. B 128 (1983); K. O'Brien and J.-B. Zub er,

Phys. Lett. B 144 (1984) 407; I. Kostov, Nucl. Phys. B 265 (1986), 223,

B 415 (1994) 29.

[5] D. Weingarten, Phys. Lett. B 90 (1980) 280.

[6] B. Durhuus, J. Frohlich and T. Jonsson, Nucl. Phys. B 240 FS[12] (84)

453.

[7] V.A. Kazakov, M. Staudacher and T. Wynter, Comm. Math. Phys. 177

(1996) 451.

[8] V.A. Kazakov, M. Staudacher and T. Wynter, Comm. Math. Phys. 179

(1996) 235.

[9] V.A. Kazakov, M. Staudacher and T. Wynter, Nucl. Phys. B 471 (1996)

309.

[10] V.A. Kazakov, M. Staudacher and T. Wynter, Advances in Large N

Group Theory and the Solution of Two-Dimensional R Gravity, hep-

th/9601153, 1995 Cargese Pro ceedings.

[11] R. Dijkgraaf, Chiral Deformations of Conformal Field Theories, hep-

th/9609022.

[12] I. Kostov, M. Staudacher and T. Wynter, in preparation.

[13] A.A. Migdal, Zh. Eksp. Teor. Fiz. 69 (1975) 810 (Sov. Phys. JETP 42

(413)).

[14] B. Rusakov, Mo d. Phys. Lett. A5 (1990) 693.

[15] M.R. Douglas and V.A. Kazakov, Phys. Lett. B 312 (1993) 219.

[16] M.R. Douglas, Conformal Field Theory Techniques in Large N Yang-

Mil ls Theory, hep-th/9311130, 1993 Cargese Pro ceedings.

[17] R. Dijkgraaf, Mirror Symmetry and El liptic Curves, in The Moduli

Space of Curves, Progress in Mathematics 129 (Birkhauser, 1995), 149.

[18] R. Rudd, The String Partition Function for QCD on the Torus, hep- th/9407176.