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Two-Dimensional Chiral Matrix Models and String Theories

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Two-Dimensional Chiral Matrix Models and String Theories CERN-TH/96-311 hep-th/9611011 Two-Dimensional Chiral Matrix Mo dels and String Theories y 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 y [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 W spacetime . The allowed worldsheet con gurations represent minimal T 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 2 covers of the target manifold [1], [2]. Unfortunately, this construction is highly involved and do es not easily 1 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 2 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 1 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 n p oints of order n. 1 The random surface representation of YM on a lattice found in [4] is geometrically 2 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 T 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 T 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- T 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 T (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 2 formula N N +N = 2 2G: (2:1) 0 1 2 In addition, each cell c is characterized by its area A . c 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 . T 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 Z Y Y Z = [D ] exp (N Tr ); (2:2) ` c c c ` Q A 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 y 2 N N Tr ` ` : (2:3) [D ] = (N= ) d d e ` ij ij ij y Note that we omit the complex conjugate from the plaquette action, c 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 W in the target surface . Each of these surfaces is obtained by placing T plaquettes on the faces of the lattice and gluing any two together along T 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 . T Thus the surface is wrapping n times and its area is A = nA W T W T P 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 T ! 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 p P 2g 2 = n(2G 2) + B where g is the genus of the surface . The p W p free energy F = ln Z of our chiral mo del can be written as 1 1 X X nA 22g T F = e N F (n; g jG; N ) (2:4) 0 n=1 g =G where F (n; g jG; N ) is the number of no-fold surfaces of genus g covering n 0 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. 0 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 h 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 Y h h i j : (3:1) = h i j i<j the dimension of the representation h.
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