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.

arXiv comment41 pages. v2: additional references

References in corpus (91)

• Monopole Condensation, And Confinement In N=2 Supersymmetric Yang-Mills Theory
• Monopoles, Duality and Chiral Symmetry Breaking in N=2 Supersymmetric QCD
• Liouville Correlation Functions from Four-dimensional Gauge Theories
• Structure Constants and Conformal Bootstrap in Liouville Field Theory
• Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes
• Two and three-point functions in Liouville theory
• Holomorphic Anomalies in Topological Field Theories
• Topological Amplitudes in String Theory
• Nekrasov Functions and Exact Bohr-Sommerfeld Integrals
• Bits and Pieces in Logarithmic Conformal Field Theory
• Symplectic Fermions
• Multi-Instanton Calculus and Equivariant Cohomology
• On combinatorial expansion of the conformal blocks arising from AGT conjecture
• An Algorithm for the Microscopic Evaluation of the Coefficients of the Seiberg-Witten Prepotential
• Polymers and percolation in two dimensions and twisted N=2 supersymmetry
• A Local Logarithmic Conformal Field Theory
• Gravitational corrections in supersymmetric gauge theory and matrix models
• ABCD of instantons
• On Modular Invariant Partition Functions of Conformal Field Theories with Logarithmic Operators
• A lecture on the Liouville vertex operators
• Recursion relations in CFT and N=2 SYM theory
• On AGT conjecture
• Conformal blocks as Dotsenko-Fateev Integral Discriminants
• The Omega deformed B-model for rigid N=2 theories
• Recursive representation of the torus 1-point conformal block
• Nekrasov Functions from Exact BS Periods: the Case of SU(N)
• Search for the ${\Lambda }_b\to \Lambda {\eta }^{\prime }$ and ${\Lambda }_b\to \Lambda \eta$ decays with the LHCb detector
• Extended Holomorphic Anomaly in Gauge Theory
• Matone's Relation in the Presence of Gravitational Couplings
• Method of Generating q-Expansion Coefficients for Conformal Block and N=2 Nekrasov Function by beta-Deformed Matrix Model
• On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles
• Virasoro conformal blocks in closed form
• A direct proof of AGT conjecture at beta = 1
• Isomonodromic tau-functions from Liouville conformal blocks
• A non-rational CFT with c=1 as a limit of minimal models
• Conformal field theory of Painlevé VI
• String theory of the Omega deformation
• A note on instanton counting for N=2 gauge theories with classical gauge groups
• Deformed Topological Partition Function and Nekrasov Backgrounds
• The Omega Deformation From String and M-Theory
• Finalizing the proof of AGT relations with the help of the generalized Jack polynomials
• An A_r threesome: Matrix models, 2d CFTs and 4d N=2 gauge theories
• Deformed N=2 theories, generalized recursion relations and S-duality
• Holomorphicity and Modularity in Seiberg-Witten Theories with Matter
• Exotic instanton counting and heterotic/type I' duality
• Differential equation for four-point correlation function in Liouville field theory and elliptic four-point conformal blocks
• Modular anomaly equation, heat kernel and S-duality in N=2 theories
• Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals
• Bilinear equations on Painleve tau functions from CFT
• Worldsheet Realization of the Refined Topological String
• Instanton Expansions for Mass Deformed N=4 Super Yang-Mills Theories
• Virasoro vacuum block at next-to-leading order in the heavy-light limit
• On "Dotsenko-Fateev" representation of the toric conformal blocks
• Holographic interpretation of 1-point toroidal block in the semiclassical limit
• Classical torus conformal block, N=2* twisted superpotential and the accessory parameter of Lame equation
• Q-deformed Painleve tau function and q-deformed conformal blocks
• On Gauge Theory and Topological String in Nekrasov-Shatashvili Limit
• The toroidal block and the genus expansion
• F-theoretic vs microscopic description of a conformal N=2 SYM theory
• Exotic prepotentials from D(-1)D7 dynamics
• Recursion representation of the Neveu-Schwarz superconformal block
• Transformations of Spherical Blocks
• Magnetic expansion of Nekrasov theory: the SU(2) pure gauge theory
• Non-Perturbative Nekrasov Partition Function from String Theory
• N=1 SUSY Conformal Block Recursive Relations
• Modular anomaly equations in N=2* theories and their large-N limit
• Stringy instanton corrections to N=2 gauge couplings
• Mathieu equation and Elliptic curve
• On AGT description of N=2 SCFT with N_f=4
• Deformed supersymmetric gauge theories from the fluxtrap background
• Elliptic recurrence representation of the N=1 Neveu-Schwarz blocks
• SU(2)_0 and OSp(2|2)_{-2} WZNW models : Two current algebras, one Logarithmic CFT
• Elliptic recurrence representation of the N=1 superconformal blocks in the Ramond sector
• Quantum geometry from the toroidal block
• Riemann-Hilbert treatment of Liouville theory on the torus
• The exact 8d chiral ring from 4d recursion relations
• Elliptic recursion for 4-point superconformal blocks and bootstrap in N=1 SLFT
• Quasimodular instanton partition function and the elliptic solution of Korteweg-de Vries equations
• Exact results in N=2 gauge theories
• $\mathcal{N}=2^{\star }$ from Topological Amplitudes in String Theory
• Matching branches of non-perturbative conformal block at its singularity divisor
• On the large $\Omega$-deformations in the Nekrasov-Shatashvili limit of $\mathcal{N}=2^{\ast }$ SYM
• BPS States in the Duality Web of the Omega deformation
• 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
• Resumming instantons in N=2* theories with arbitrary gauge groups
• Recurrence relations for toric N=1 superconformal blocks
• Exact Partition Functions for the $q$-State Potts Model with a Generalized Magnetic Field on Lattice Strip Graphs
• Conformal blocks related to the R-R states in the \hat c =1 SCFT
• On new exact conformal blocks and Nekrasov functions
• Discrete matrix models for partial sums of conformal blocks associated to Painlevé transcendents

Cited by in corpus (4)

• A slow review of the AGT correspondence
• Modular and duality properties of surface operators in N=2* gauge theories
• Geodesic description of Heavy-Light Virasoro blocks
• Chiral trace relations in $\Omega$-deformed $\mathcal{N}=2$ theories