On the String Actions for the Generalized Two-dimensional Yang-Mills Theories

We study the structures of partition functions of the large N generalized twodimensional Yang-Mills theories (gYM2) by recasting the higher Casimirs. We clarify the appropriate interpretations of them and try to extend the Cordes-MooreRamgoolam’s topological string model describing the ordinary YM2 [4] to those describing gYM2. The concept of ”deformed gravitational descendants” will be introduced for this purpose. e-mail address: [email protected] 1. It is an old problem to understand relationships between Yang-Mills theory and string theory [1], especially for the two-dimensional case (YM2) [2]. In early years of 90’s one great progress was given by Gross and Taylor within the framework of YM2 on any compact Riemann surface M [3]. In these celebrated works they investigated in detail the large N expansion of the partition function by making use of some group theoretical techniques, and showed that the partition function is realized as an asymptotic series of which terms are all some homotopy invariants of ramified covering maps onto the considered Riemann surface M . This strongly supports the possibility of reformulating Y M2 as a theory of topological string with the space of maps f : Σ −→ M as the configuration space. Inspired with these Gross-Taylor’s studies, Cordes, Moore and Ramgoolam gave an elegant implication [4]. They firstly considered the 0-area limit (or equivalently, the weak coupling limit) of the large N YM2 and proved that each term of the Gross-Taylor’s asymptotic series is exactly equal to the Euler number of the moduli space of branched covers. Based on this observation they constructed the world sheet action of a topological string model corresponding to the large N YM2. Their topological string theory is formulated to calculate the Euler number of the moduli space and, to this aim, includes some extra degrees of freedom the ”co-fields” [4] (see also [9].) For the case of non-zero area, they gave a conjecture (and partially proved) that by adding the simple perturbation of the ”area operator” one can reproduce the result of the non-zero area case. They also presented a stimulating speculation; the appropriate perturbations of the gravitational descendants of area operators might correspond to the ”generalized 2-dim Yang-Mills theories” (gYM2) [7, 8], which are given by replacing the 2nd Casimir with some higher Casimirs in the heat Kernel Boltsmann weight of YM2 [5, 6, 7]. In this article we shall present a detailed study of gY M2 motivated with this speculation. Our goal is to exhibit the world-sheet actions for the topological string models describing gYM2, in other words, the suitable perturbation terms to the CMR’s string model reproducing the higher Casimirs. We must establish the rule to translate the algebraic data of higher Casimirs appearing in gYM2 into the geometric data of topological string theory. We will observe that the concept of deformed gravitational descendants need to be incorpolated. 2. We shall start with a short review for the work [4]. LetM be an arbitrary compact Riemann surface with genus p, and G be a compact group (gauge group). The two-dimensional