Exact partition functions for deformed $\mathcal{N}=2$ theories with $N_{f}=4$ flavours

TL;DR

This paper derives exact formulas for the partition functions of certain deformed $ =2$ supersymmetric gauge theories with four flavors, revealing special fixed ratios of parameters and their relation to conformal blocks via AGT correspondence.

Contribution

It introduces a condition on multi-instanton contributions that yields finite pole partition functions with explicit modular properties, extending understanding of $ =2$ theories with four flavors.

Findings

Finite pole partition functions expressed via Eisenstein series and theta functions.

Exact recursive relations for conformal blocks confirming closed-form solutions.

Analytic proofs of conjectured expressions for $ ext{Pi}_1$ and $ ext{Pi}_2$ cases.

Abstract

We consider the $\Omega$-deformed $\mathcal{N}=2$ $SU(2)$ gauge theory in four dimensions with $N_{f}=4$ massive fundamental hypermultiplets. The low energy effective action depends on the deformation parameters $\varepsilon_{1}, \varepsilon_{2}$, the scalar field expectation value $a$, and the hypermultiplet masses ${\bf m}=(m_{1}, m_{2}, m_{3}, m_{4})$. Motivated by recent findings in the $\mathcal{N}=2^{*}$ theory, we explore the theories that are characterized by special fixed ratios $\varepsilon_{2}/\varepsilon_{1}$ and ${\bf m}/\varepsilon_{1}$ and propose a simple condition on the structure of the multi-instanton contributions to the prepotential determining the effective action. This condition determines a finite set $\Pi_{N}$ of special points such that the prepotential has $N$ poles at fixed positions independent on the instanton number. In analogy with what happens in the $\mathcal{N}=2^{*}$ gauge theory, the full prepotential of the $\Pi_{N}$ theories may be given in closed form as an explicit function of $a$ and the modular parameter $q$ appearing in special combinations of Eisenstein series and Jacobi theta functions with well defined modular properties. The resulting finite pole partition functions are related by AGT correspondence to special 4-point spherical conformal blocks of the Virasoro algebra. We examine in full details special cases where the closed expression of the block is known and confirms our Ansatz. We systematically study the special features of Zamolodchikov's recursion for the $\Pi_{N}$ conformal blocks. As a result, we provide a novel effective recursion relation that can be exactly solved and allows to prove the conjectured closed expressions analytically in the case of the $\Pi_{1}$ and $\Pi_{2}$ conformal blocks.