Quantum geometry of refined topological strings

Abstract

We consider branes in refined topological strings. We argue that their wave-functions satisfy a Schrödinger equation depending on multiple times and prove this in the case where the topological string has a dual matrix model description. Furthermore, in the limit where one of the equivariant rotations approaches zero, the brane partition function satisfies a time-independent Schrödinger equation. We use this observation, as well as the back reaction of the brane on the closed string geometry, to offer an explanation of the connection between integrable systems and N = 2 gauge systems in four dimensions observed by Nekrasov and Shatashvili.

Exact quantization conditions for the relativistic Toda lattice

“…Another important ingredient in our calculation is the quantum mirror map, which was introduced in [49]. This promotes the non-trivial Kähler parameters T i , i = 1, · · · , N − 1, to functions of the z i , i = 1, · · · , N , and of .…”

“…We will denote the resulting functions by JHEP05(2016)133 T i (z 1 , · · · , z N ; ) or simply by T i ( ). As explained in [49], the quantum mirror map can be computed as follows. Let us denote the equation for the mirror curve (2.46) by H(x, p) = 0.…”

Quantum geometry of resurgent perturbative/nonperturbative relations

“…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]. There are surprisingly close parallels between the resurgent structures found in such theories for the partition function (or free energy) as a function of (at least) two parameters, g s and N , and the resurgent structure of the Schrödinger energy eigenvalue u( , N ), as a function of and the perturbative level number N .…”

Torus Knots and Mirror Symmetry

“…The topological recursion of [19], which computes open and closed topological string amplitudes in toric Calabi-Yau manifolds, might have a generalization which gives the mirror of the refined topological vertex [25] (recent work in this direction can be found in [1]). If the only data entering in this generalization turn out to be the same ones appearing in the original recursion (i.e., if the refinement only requires the knowledge of the spectral curve and of the natural differential on it, as it happens for example in the β deformation [12]), then one should be able to use our spectral curve (3.25) to refine the colored HOMFLY polynomial of torus knots.…”