On the relation between Stokes multipliers and the T-Q systems of conformal field theory

Abstract

The vacuum expectation values of the so-called Q-operators of certain integrable quantum field theories have recently been identified with spectral determinants of particular Schrödinger operators. In this paper we extend the correspondence to the T-operators, finding that their vacuum expectation values also have an interpretation as spectral determinants. As byproducts we give a simple proof of an earlier conjecture of ours, proved by another route by Suzuki, and generalise a problem in PT symmetric quantum mechanics studied by Bender and Boettcher. We also stress that the mapping between Q-operators and Schrödinger equations means that certain problems in integrable quantum field theory are related to the study of Regge poles in non-relativistic potential scattering.

Quantum geometry of resurgent perturbative/nonperturbative relations

“…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. Further motivation along these lines comes from the ODE/Integrable Model correspondence [66][67][68][69][70], which provides explicit mappings between monodromy operators in certain Schrödinger systems and Yang-Baxter operators in integrable models. We are also strongly motivated by the geometric relation between supersymmetric gauge theories, matrix models and topological strings [71][72][73][74][75], for which a rich web of resurgent structures has been comprehensively established both analytically and numerically [76][77][78][79][80][81][82][83][84][85][86].…”

Boundary RG flow associated with the AKNS soliton hierarchy

“…For the equation (138) with q = 0 and n < 0 the corresponding difference equations were derived in the work [46] (see also [47,48]). The case q = 0, n > 0 was studied in the unpublished paper [35].…”

Finite lattice Bethe ansatz systems and the Heun equation

“…Some years ago, an unexpected connection was found between certain 0 + 1 dimensional quantum-mechanical problems and 1 + 1 dimensional conformal field theories [1,2,3,4,5,6,7,8]. The simplest example involves the Schrödinger equation 1) and the fact that (1.1) has unique solution y(x, E), entire in x and E, which decays along the positive real axis as x → ∞.…”

“…The simplest example involves the Schrödinger equation 1) and the fact that (1.1) has unique solution y(x, E), entire in x and E, which decays along the positive real axis as x → ∞. The function y(x, E) can be shown [5] to satisfy a Stokes relation…”