[Paper Review] Topological Strings from Quantum Mechanics

Alba Grassi, Yasuyuki Hatsuda+1 more|arXiv (Cornell University)|Oct 13, 2014

Black Holes and Theoretical Physics65 references18 citations

TL;DR

This paper proposes a non-perturbative duality between quantum mechanics and topological strings on toric Calabi–Yau manifolds, conjecturing that the spectral determinant of a quantum operator associated with the mirror curve is determined by an M-theoretic generalization of the topological string free energy. The exact quantization condition arises from the zeros of a generalized theta function constructed from this free energy, unifying perturbative Nekrasov–Shatashvili and non-perturbative conventional topological string contributions, with full agreement to numerical spectra in local P², local F₁, and local P¹×P¹ geometries.

ABSTRACT

We propose a general correspondence which associates a non-perturbative quantum-mechanical 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 P2, local P1xP1 and local F1. 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

Motivation & Objective

• To establish a non-perturbative correspondence between quantum mechanical operators on toric Calabi–Yau manifolds and topological string theory.
• To resolve the gap in the perturbative quantization condition for spectral operators by including non-perturbative corrections from the conventional topological string.
• To provide an exact, background-independent formulation of topological strings via the spectral determinant of a quantum operator.
• To unify the Nekrasov–Shatashvili limit of the refined topological string with the conventional topological string in a single non-perturbative framework.

Proposed method

• Associate a non-perturbative quantum mechanical operator ˆρX to each toric Calabi–Yau manifold via quantization of its mirror curve WX(ex, ep) = 0.
• Conjecture that the spectral determinant of ˆρX is encoded in a modified grand potential JX, derived from an M-theoretic version of the topological string free energy.
• Construct a generalized theta function from JX, whose zeros yield the exact quantization condition for the operator spectrum.
• Use the spectral traces of ˆρX to recover the genus expansion of the topological string in the 't Hooft limit.
• Apply the formalism to local P², local F₁, and local P¹×P¹, verifying agreement with numerical spectra.
• Derive semiclassical corrections to the grand potential using WKB methods and integral representations.

Experimental results

Research questions

• RQ1 How can the full non-perturbative spectrum of a quantum operator on a toric Calabi–Yau manifold be determined from topological string theory?
• RQ2 What is the precise role of the Nekrasov–Shatashvili limit and the conventional topological string in constructing a complete quantization condition?
• RQ3 Can the spectral determinant of the quantum operator be expressed as a holomorphic, entire function on moduli space, leading to a background-independent formulation?
• RQ4 Why does the conjectured quantization condition reproduce the correct spectrum in cases where previous perturbative conditions fail?
• RQ5 How do the spectral traces of the operator relate to the topological string partition function near the orbifold point?

Key findings

• The spectral determinant of the quantum operator ˆρX is conjectured to be an entire function on moduli space, constructed from a generalized theta function derived from the modified grand potential JX.
• The exact quantization condition for the spectrum is given by the vanishing of this generalized theta function, which combines perturbative Nekrasov–Shatashvili and non-perturbative conventional topological string contributions.
• For local P², local F₁, and local P¹×P¹, the proposed quantization condition reproduces all existing numerical results for the spectrum with perfect agreement.
• The spectral traces of ˆρX are shown to be computable from the behavior of the topological string near the orbifold point, confirming the duality at the level of traces.
• The semiclassical correction to the grand potential is derived analytically and matches the numerical results in the local P² case.
• The proposal provides a non-perturbative completion of the topological string, where the genus expansion emerges as a 't Hooft limit of spectral traces, and the full free energy is Borel-summable via the M-theoretic framework.