Classical irregular block, N $$ \mathcal{N} $$ = 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 p2q 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 p2q 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. 1

Quantum geometry of resurgent perturbative/nonperturbative relations

“…We adopt the semiclassical path integral approach of Balian and Bloch [34,35] and the geometric Stokes diagram and monodromy picture [6][7][8]36], to express the P/NP relations in terms of all-orders WKB ("exact WKB") actions and dual actions. In so doing, it proves useful to adopt some of the language and ideas from supersymmetric gauge theories, integrability, conformal field theory and wall-crossing [37][38][39][40][41][42][43][44][45][46][47][48], given the close connection of such theories with exact WKB [49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65]. This is similar to the philosophy of [27], where the refined holomorphic anomaly equation of topological string theory is shown to describe these P/NP relations for some 1d quantum oscillator systems.…”

“…For the Mathieu system, which is associated with N = 2 supersymmetric gauge theory [49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65], the P/NP relation is:…”

“…If one knows the expansion of one of them to some order in , the other can be deduced. In the context of gauge theory (N = 2 * , SU(2) theory) (5.44) is the quantum Matone relation where the modular parameter t, and are identified with the complex gauge coupling and the Omega deformation parameter 1 (in the limit 2 = 0), respectively [42][43][44][55][56][57][58][59][60][61][62][63][64][65].…”

Classical Virasoro irregular conformal block

“…The simplest case is obtained using the Bohr-Sommerfeld periods of onedimensional sine-Gordon model [13]. Similarly, degenerate conformal block is used to obtain the information on the classical limit of the ICB in [10,14] In this paper, we will provide NS limit of the inner product of irregular modules for arbitrary rank using the IMM and evaluate the partition function in a systematic way.…”

“…One may find more rigorous proof for this dominant behavior. Let us first consider the simplest case (the rank 1/2) given in [14]. (This corresponds to N f = 0, SU(2) and is obtained using some appropriate limit from the rank 1 case.…”