Classical limit of irregular blocks and Mathieu functions

Abstract

The Nekrasov-Shatashvili limit of the N=2 SU(2) pure gauge (Omega-deformed) super Yang-Mills theory encodes the information about the spectrum of the Mathieu operator. On the other hand, the Mathieu equation emerges entirely within the frame of two-dimensional conformal field theory (2d CFT) as the classical limit of the null vector decoupling equation for some degenerate irregular block. Therefore, it seems to be possible to investigate the spectrum of the Mathieu operator employing the techniques of 2d CFT. To exploit this strategy, a full correspondence between the Mathieu equation and its realization within 2d CFT has to be established. In our previous paper [1], we have found that the expression of the Mathieu eigenvalue given in terms of the classical irregular block exactly coincides with the well known weak coupling expansion of this eigenvalue in the case in which the auxiliary parameter is the noninteger Floquet exponent. In the present work we verify that the formula for the corresponding eigenfunction obtained from the irregular block reproduces the so-called Mathieu exponent from which the noninteger order elliptic cosine and sine functions may be constructed. The derivation of the Mathieu equation within the formalism of 2d CFT is based on conjectures concerning the asymptotic behaviour of irregular blocks in the classical limit. A proof of these hypotheses is sketched. Finally, we speculate on how it could be possible to use the methods of 2d CFT in order to get from the irregular block the eigenvalues of the Mathieu operator in other regions of the coupling constant.

41 pages, 1 figure

References in corpus (16)

• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• On AGT relation in the case of U(3)
• Asymptotically free N=2 theories and irregular conformal blocks
• Resurgence and the Nekrasov-Shatashvili Limit: Connecting Weak and Strong Coupling in the Mathieu and Lam'e Systems
• Deforming SW curve
• Gauge theories on Omega-backgrounds from non commutative Seiberg-Witten curves
• Whittaker vectors of the Virasoro algebra in terms of Jack symmetric polynomial
• AGT conjecture and Integrable structure of Conformal field theory for c=1
• Classical irregular block, N=2 pure gauge theory and Mathieu equation
• Quantum Hitchin Systems via beta-deformed Matrix Models
• Classical conformal blocks from TBA for the elliptic Calogero-Moser system
• Classical Virasoro irregular conformal block
• Uniformization, Calogero-Moser/Heun duality and Sutherland/bubbling pants
• Classical Virasoro irregular conformal block II
• Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models
• Analytic continuation formulae for the BPZ conformal block

Cited by in corpus (10)

• Quantum Geometry of Resurgent Perturbative/Nonperturbative Relations
• A slow review of the AGT correspondence
• Bands and gaps in Nekrasov partition function
• Bethe/Gauge correspondence in odd dimension: modular double, non-perturbative corrections and open topological strings
• Resurgence, Painleve Equations and Conformal Blocks
• Classical irregular blocks, Hill's equation and PT-symmetric periodic complex potentials
• Solving Heun's equation using conformal blocks
• Classical conformal blocks, Coulomb gas integrals and Richardson-Gaudin models
• The Toda system and solution to the $\mathcal{N}$=2 SUSY Yang-Mills theory
• Solvable spectral problems from 2d CFT and N=2 gauge theories