Classical irregular block, N=2 pure gauge theory and Mathieu equation

Abstract

Combining the semiclassical/Nekrasov-Shatashvili limit of the AGT conjecture and the Bethe/gauge correspondence results in a triple correspondence which identifies classical conformal blocks with twisted superpotentials and then with Yang-Yang functions. In this paper the triple correspondence is studied in the simplest, yet not completely understood case of pure SU(2) super-Yang-Mills gauge theory. A missing element of that correspondence is identified with the classical irregular block. Explicit tests provide a convincing evidence that such a function exists. In particular, it has been shown that the classical irregular block can be recovered from classical blocks on the torus and sphere in suitably defined decoupling limits of classical external conformal weights. These limits are "classical analogues" of known decoupling limits for corresponding quantum blocks. An exact correspondence between the classical irregular block and the SU(2) gauge theory twisted superpotential has been obtained as a result of another consistency check. The latter determines the spectrum of the 2-particle periodic Toda (sin-Gordon) Hamiltonian in accord with the Bethe/gauge correspondence. An analogue of this statement is found entirely within 2d CFT. Namely, considering the classical limit of the null vector decoupling equation for the degenerate irregular block a celebrated Mathieu's equation is obtained with an eigenvalue determined by the classical irregular block. As it has been checked this result reproduces a well known weak coupling expansion of Mathieu's eigenvalue. Finally, yet another new formulae for Mathieu's eigenvalue relating the latter to a solution of certain Bethe-like equation are found.

arXiv comment47 pages, 3 figures

References in corpus (40)

• Monopole Condensation, And Confinement In N=2 Supersymmetric Yang-Mills Theory
• Liouville Correlation Functions from Four-dimensional Gauge Theories
• Structure Constants and Conformal Bootstrap in Liouville Field Theory
• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• Integrability and Seiberg-Witten Exact Solution
• Liouville theory revisited
• Supersymmetric vacua and Bethe ansatz
• Analytic Continuation of Liouville Theory
• Nekrasov Functions and Exact Bohr-Sommerfeld Integrals
• Quantization of Integrable Systems and Four Dimensional Gauge Theories
• Quantum integrability and supersymmetric vacua
• Entanglement Entropy at Large Central Charge
• Instanton counting on blowup. I. 4-dimensional pure gauge theory
• An Algorithm for the Microscopic Evaluation of the Coefficients of the Seiberg-Witten Prepotential
• Integrability and Seiberg-Witten Theory: Curves and Periods
• On non-conformal limit of the AGT relations
• Asymptotically free N=2 theories and irregular conformal blocks
• On AGT conjecture
• Deformed Prepotential, Quantum Integrable System and Liouville Field Theory
• Nekrasov Functions from Exact BS Periods: the Case of SU(N)
• Classical Conformal Blocks and Painleve VI
• Prepotential and the Seiberg-Witten Theory
• Deforming SW curve
• Exact absorption probabilities for the D3-brane
• Proving the AGT relation for N_f = 0,1,2 antifundamentals
• Comments on Renyi entropy in AdS${}_3$/CFT${}_2$
• Non-conformal limit of AGT relation from the 1-point torus conformal block
• Classical torus conformal block, N=2* twisted superpotential and the accessory parameter of Lame equation
• The toroidal block and the genus expansion
• Integrable structure of Quantum Field Theory: Classical flat connections versus quantum stationary states
• Liouville theory and uniformization of four-punctured sphere
• M-Theoretic Derivations of 4d-2d Dualities: From a Geometric Langlands Duality for Surfaces, to the AGT Correspondence, to Integrable Systems
• Liouville theory, N=2 gauge theories and accessory parameters
• On the monodromy problem for the four-punctured sphere
• Absorption Cross Section of Scalar Field in Supergravity Background
• Riemann-Hilbert treatment of Liouville theory on the torus
• Accessory parameters for Liouville theory on the torus
• D-Branes and their Absorptivity in Born-Infeld Theory
• Hyperbolic deformation of the strip-equation and the accessory parameters for the torus
• Compactification in deconstructed gauge theory with topologically non-trivial link fields

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