Instanton effects and quantum spectral curves

TL;DR

This work analyzes a spectral problem from quantizing the local mirror curve of ${\mathbb P}^1\times{\mathbb P}^1$ in the NS limit, showing that perturbative quantum periods do not fix the spectrum and that an infinite series of non-perturbative, in $hbar$, instanton corrections—controlled by the unrefined topological string—are needed. The authors propose an exact WKB quantization condition combining perturbative quantum periods with these instanton terms and test it against precise numerical spectra, finding excellent agreement. They derive how these non-perturbative contributions connect to the ABJM grand potential, linking membrane and worldsheet instantons to GV invariants, and argue for a non-perturbative definition of topological strings via spectral problems. The results reveal a deep, non-perturbative relationship between NS-refined and conventional topological strings and suggest broad applicability to local CY geometries and related gauge theories.

Abstract

We study a spectral problem associated to the quantization of a spectral curve arising in local mirror symmetry. The perturbative WKB quantization condition is determined by the quantum periods, or equivalently by the refined topological string in the Nekrasov-Shatashvili (NS) limit. We show that the information encoded in the quantum periods is radically insufficient to determine the spectrum: there is an infinite series of instanton corrections, which are non-perturbative in \hbar, and lead to an exact WKB quantization condition. Moreover, we conjecture the precise form of the instanton corrections: they are determined by the standard or un-refined topological string free energy, and we test our conjecture successfully against numerical calculations of the spectrum. This suggests that the non-perturbative sector of the NS refined topological string contains information about the standard topological string. As an application of the WKB quantization condition, we explain some recent observations relating membrane instanton corrections in ABJM theory to the refined topological string.

Figures (2)

• Figure 1: For a given energy $E$, we have a real periodic trajectory given by (\ref{['real-t']}), which is shown in the figure on the left for $E=3$. The horizontal axis represents the time $t$ and it runs through a full period, from $-8 K(k)$ to $8 K(k)$. There is also an imaginary periodic trajectory for $\theta= {\rm Im}(q)$, given by (\ref{['im-t']}), which is shown in the figure on the right, also for $E=3$. Here, the imaginary time $\tau$ runs from $-4 K(k')$ to $4 K(k')$.
• Figure 2: The trajectory (\ref{['real-t']}) describes a closed orbit in the phase space $(x,p)$, represented schematically in the figure on the left. After complexifying the exponentiated variable ${\rm e}^{x/2}$, this closed orbit becomes a torus, as shown in the figure on the right. The imaginary trajectory (\ref{['im-t']}) is a closed orbit around the $A$-cycle of the torus.