Topological Strings from Quantum Mechanics

Abstract

We propose a general correspondence which associates a non-perturbative quantummechanical operator to a toric Calabi-Yau manifold, and we conjecture an explicit formula for its spectral determinant in terms of an M-theoretic version of the topological string free energy. As a consequence, we derive an exact quantization condition for the operator spectrum, in terms of the vanishing of a generalized theta function. The perturbative part of this quantization condition is given by the Nekrasov-Shatashvili limit of the refined topological string, but there are non-perturbative corrections determined by the conventional topological string. We analyze in detail the cases of local P 2 , local P 1 × P 1 and local F 1 . In all these cases, the predictions for the spectrum agree with the existing numerical results. We also show explicitly that our conjectured spectral determinant leads to the correct spectral traces of the corresponding operators. Physically, our results provide a non-perturbative formulation of topological strings on toric Calabi-Yau manifolds, in which the genus expansion emerges as a 't Hooft limit of the spectral traces. Since the spectral determinant is an entire function on moduli space, it leads to a background independent formulation of the theory. Mathematically, our results lead to precise, surprising conjectures relating the spectral theory of functional difference operators to enumerative geometry.

Exact quantization conditions for the relativistic Toda lattice

“…Inspired by the HMO mechanism in ABJM theory [23], it was proposed in [22] that these poles should be cancelled by non-perturbative contributions in . In the case of one-dimensional Hamiltonians associated to quantized mirror curves, the precise form of these non-perturbative effects was conjectured in [22,[24][25][26][27][28][29].…”

“…However, to do this, one has to specify very carefully the boundary conditions satisfied by ψ(µ), as it happens for example in the simpler case of the standard Toda lattice. In this paper, instead of solving this equation analytically, we will propose an exact quantization condition for the eigenvalues H 1 , · · · , H N −1 , based on insights from [8] and on the recent progress in the quantization of mirror curves [22,24,25].…”

“…These poles are not physical, and they should be cancelled by non-perturbative contributions. In order to find these contributions, we will use recent results on the quantization of mirror curves from [22,24,25]. It turns out that the quantum Baxter equation (2.22) in the case N = 2 is identical to the quantization of the mirror curve of local P 1 × P 1 .…”

“…It turns out that the quantum Baxter equation (2.22) in the case N = 2 is identical to the quantization of the mirror curve of local P 1 × P 1 . A conjectural, analytic solution to the corresponding eigenvalue problem was presented in [22] for the so-called "maximally supersymmetric case" = 2π, and then extended to arbitrary in [25]. This solution involves the resummation of the all-orders WKB result, but it incorporates in addition explicit non-perturbative contributions that cancel the poles.…”

“…In order to see how this works, and to open the way for generalizations, let us consider a general, toric CY manifold X and let us introduce, as in [25,29,51], a B-field B satisfying the following requirement: for all d, j L and j R such that the BPS invariant…”