Holomorphic anomaly of 2d Yang-Mills theory on a torus revisited

Abstract

We study the large N 't Hooft expansion of the chiral partition function of 2d U (N ) Yang-Mills theory on a torus. There is a long-standing puzzle that no explicit holomorphic anomaly equation is known for the partition function, although it admits a topological string interpretation. Based on the chiral boson interpretation we clarify how holomorphic anomaly arises and propose a natural anti-holomorphic deformation of the partition function. Our deformed partition function obeys a fairly traditional holomorphic anomaly equation. Moreover, we find a closed analytic expression for the deformed partition function. We also study the behavior of the deformed partition function both in the strong coupling/large area limit and in the weak coupling/small area limit. In particular, we observe that drastic simplification occurs in the weak coupling/small area limit, giving another nontrivial support for our anti-holomorphic deformation.

Boson-fermion correspondence and holomorphic anomaly equation in 2d Yang-Mills theory on torus

“…Although the partition function is not modular with this deformation, a nice BCOV-like JHEP07(2021)144 holomorphic anomaly equation can be obtained. In this paper, we derive the corresponding formula in fermionic formulation, which was also already conjectured in [14]. It is then easy to prove the conjectured holomorphic anomaly equation.…”

“…The partition function would be entirely holomorphic if one omits the S-dependence in the propagator. Here we follow the prescription in [14] of keeping the S-dependence, so that a BCOV-like holomorphic anomaly equation can be obtained.…”

“…As an example, the genus two free energy has contributions from two Feynman diagrams. The result is computed in [14] as…”

“…Okuyama and Sakai proposed an anholomorphic deformation of the propagators in the Feynman diagrams of the bosonic formulation, based on some physical arguments [14]. Although the partition function is not modular with this deformation, a nice BCOV-like JHEP07(2021)144 holomorphic anomaly equation can be obtained.…”

“…The paper is organized as the followings. In section 2 we review the formulation of holomorphic anomaly equation in [14]. In section 3, after briefly reviewing the boson-fermion correspondence, we derive the formula for the deformed partition function in the fermionic formulation.…”