Bands and gaps in Nekrasov partition function

Abstract

Abstract:We discuss the effective twisted superpotentials of 2d N = (2, 2) theories arising upon the reduction of 4d N = 2 gauge theories on the Ω-deformed cigar-like geometry. We explain field-theoretic origins of the gaps in the spectrum in the corresponding quantum mechanical (QM) systems. We find local 2d descriptions of the physics near these gaps by resumming the non-perturbative part of the twisted superpotential and discuss arising wall-crossing phenomena. The interpretation of the associated phenomena in the classical Liouville theory and in the scattering of two heavy states in AdS 3 gravity is suggested. Some comments concerning a possible interpretation of the band structure in QM in terms of the Schwinger monopole-pair production in 4d are presented.

Toward the pole

“…k with k > 1 is given in the work [5]. Square roots 1 + 4q 2n /ζ 2 n (n±ν) 2 , entering functions g (n) k with k ∈ N, have branching points at ν = n, n±2iq n /ζ n .…”

Tying Up Instantons with Anti-Instantons

“…where q N +1 = q 1 , is famously dual to the pure N = 2 super-Yang-Mills with gauge group SU (N ). It was observed in the classical limit in [35,56], in [15,16] in the formal quantization, in [74] in the actual L 2 -quantization on the real line q i ∈ R. It is also possible to study the eigenvalue problem (126) where q i ∈ iR/2πZ, however new phenomena arise in this case, notably the non-perturbative splitting of the levels for which our formalism is being developed (see [45] for the current status of the problem for N = 2, 3, and [34] for new developments). We expect to see our non-linear superpositions of instantons and antiinstantons in the quantum mechanical model in the limit where all Gibbs potentials β k → ∞.…”

Resurgence, Painlevé equations and conformal blocks

“…In the Mathieu system, the band splitting is due to the effect of real instantons, while the gap splitting is due to complex instantons[45,46]. A similar interpretation for PVI is consistent with the quantum geometry and exact WKB explanations of the P/NP relations for all genus 1 systems, in terms of all-orders actions and dual actions[55,57,70,[72][73][74][75][76][77][78][79].3. A similar, but not identical, relation between spectral problems and Painlevé equations arisesfor the PI equation, whose tritronquée poles have been associated with a leading order WKB analysis of the cubic QM oscillator[69,80,81].4.…”

“…The monodromy parameter ν is identified with the label N of the spectral gap. The spectral interpretation of the remaining boundary condition parameter, κ(ν, r), was not identified in [60,69], but by comparison with the full non-perturbative energy spectrum, including the all-orders gap splitting, we see that κ(ν, r) is directly related to the non-perturbative width of the N th gap, with ν = N [46,54,77]. Indeed, with the above identifications, we can re-express (2.29) as…”