Non-perturbative quantum geometry III

Abstract

The Nekrasov-Shatashvili limit of the refined topological string on toric CalabiYau manifolds and the resulting quantum geometry is studied from a non-perturbative perspective. The quantum differential and thus the quantum periods exhibit Stokes phenomena over the combined string coupling and quantized Kähler moduli space. We outline that the underlying formalism of exact quantization is generally applicable to points in moduli space featuring massless hypermultiplets, leading to non-perturbative band splitting. Our prime example is local P 1 × P 1 near a conifold point in moduli space. In particular, we will present numerical evidence that in a Stokes chamber of interest the string based quantum geometry reproduces the non-perturbative corrections for the Nekrasov-Shatashvili limit of 4d supersymmetric SU(2) gauge theory at strong coupling found in the previous part of this series. A preliminary discussion of local P 2 near the conifold point in moduli space is also provided.

Hofstadter’s butterfly in quantum geometry

“…In this case, one needs a trans-series expansion to explain the spectrum for finite f. Roughly speaking, the non-perturbative order f e 1 is also related to the width of the bands, and thus it is extremely narrow in the weak flux limit. Recently, the non-perturbative band splitting in the very similar (but different) setup was also confirmed in 16 [41].…”

“…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.…”

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].…”