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

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 ${\epsilon }_1,{\epsilon }_2$, the scalar field expectation value $a$, and the hypermultiplet masses $\mathbf{m}=(m_1,m_2,m_3,m_4)$. Motivated by recent findings in the $\mathcal{N}=2^{\ast }$ theory, we explore the theories that are characterized by special fixed ratios ${\epsilon }_2/{\epsilon }_1$ and $\mathbf{m}/{\epsilon }_1$ 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 $\mathcal{N}=2^{\ast }$ 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.

41 pages. v2: additional references

References in corpus (17)

• Extended Holomorphic Anomaly in Gauge Theory
• On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles
• A direct proof of AGT conjecture at beta = 1
• Differential equation for four-point correlation function in Liouville field theory and elliptic four-point conformal blocks
• Exotic instanton counting and heterotic/type I' duality
• Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals
• On "Dotsenko-Fateev" representation of the toric conformal blocks
• F-theoretic vs microscopic description of a conformal N=2 SYM theory
• N=1 SUSY Conformal Block Recursive Relations
• Modular anomaly equations in N=2* theories and their large-N limit
• Elliptic recurrence representation of the N=1 Neveu-Schwarz blocks
• Quantum geometry from the toroidal block
• Riemann-Hilbert treatment of Liouville theory on the torus
• Critical dense polymers with Robin boundary conditions, half-integer Kac labels and ${\mathbb{Z}}_4$ fermions
• Exact partition functions for the $\Omega$-deformed $\mathcal{N}=2^{\ast }$ $SU(2)$ gauge theory
• Conformal blocks related to the R-R states in the \hat c =1 SCFT
• Discrete matrix models for partial sums of conformal blocks associated to Painlevé transcendents

Cited by in corpus (1)

• Geodesic description of Heavy-Light Virasoro blocks