Quantization condition from exact WKB for difference equations

Abstract

A well-motivated conjecture states that the open topological string partition function on toric geometries in the Nekrasov-Shatashvili limit is annihilated by a difference operator called the quantum mirror curve. Recently, the complex structure variables parameterizing the curve, which play the role of eigenvalues for related operators, were conjectured to satisfy a quantization condition non-perturbative in the NS parameter . Here, we argue that this quantization condition arises from requiring single-valuedness of the partition function, combined with the requirement of smoothness in the parameter . To determine the monodromy of the partition function, we study the underlying difference equation in the framework of exact WKB.

Resurgence matches quantization

“…Based upon earlier constructions [19][20][21][22][23][24][25][26][27][28][29][30], it was proposed in [31] that there is a precise correspondence between the spectral theory of operators obtained by quantizing the mirror curve, and topological string theory on the toric Calabi-Yau geometry. This proposal has since led to many recent developments, e.g., [32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51] (see [52] for an introduction). In addition, the topological string/spectral theory correspondence of [31] provides a nonperturbative definition of the topological-string partition function on toric Calabi-Yau geometries.…”

Hofstadter’s butterfly in quantum geometry

“…The complete eigenvalues are determined by an exact version of the Bohr-Sommerfeld quantization condition [13], based on earlier attempts [14,15]. This quantization condition has not yet been rigorously proven, but passes extensive analytical and numerical tests 9 [16][17][18][19][20][21][22][23][24][25][26][27][28]. We should note that there is a parallel development purely in the 5d gauge theoretic framework [30][31][32].…”

“…Moreover, the free energy -( ) F t t R q , log ; NS 4 is an expansion in terms ofe t , and thus 11 In this paper, we focus only on the case that Î  x and Î  . In principle, one can consider the problem for Î  x or Î  , as studied in [27,41] for instance. Though we do not yet see a visible structure in the general case, it might give a clue to unify the two spectral problems in the relativistic Toda lattice and in the Hofstadter model.…”

Exact quantization conditions for the elliptic Ruijsenaars-Schneider model

“…Schematically, these two functions can be written as [68,69]. 18 This was also suggested in [71,72] in the context of open topological strings in the NS limit.…”