Topological Strings from Quantum Mechanics (1410.3382v3)

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

• The paper introduces an explicit spectral determinant formula that connects the non-perturbative quantum mechanics of toric Calabi–Yau manifolds with topological string theory.
• It employs mirror symmetry to derive quantum operators and exact quantization conditions featuring non-perturbative corrections via a M-theoretic framework.
• The work is supported by strong numerical and analytical evidence, underscoring its implications for background independence and advancements in quantum geometry.

An Analysis of "Topological Strings from Quantum Mechanics"

The paper "Topological Strings from Quantum Mechanics" by Alba Grassi, Yasuyuki Hatsuda, and Marcos Mariño offers a significant proposal in the interaction between mathematical physics and quantum mechanics. The authors introduce a correspondence relating a non-perturbative quantum-mechanical operator with toric Calabi–Yau manifolds, presenting a conjecture for an explicit formula for its spectral determinant. They further investigate the implications of this conjecture in terms of both enumerative geometry and quantum mechanics, sharing results consistent with existing numerical data.

This paper intricately combines various domains, introducing a framework that potentially advances our understanding of topological string theory and its non-perturbative effects. The proposed correspondence provides a novel perspective, comparing the function of the topological string with quantum mechanics, particularly through quantum-mechanical operators associated with toric Calabi-Yau manifolds. One of the central conjectures is that the spectral determinant of these operators is explained via the M-theoretic version of the topological string free energy.

Key Insights and Claims

1. Spectral Determinant Formula: The authors conjecture an explicit formula for the spectral determinant associated with a toric Calabi-Yau manifold. This determinant is expressed in terms of the M-theoretic analogue of the topological string free energy.
2. Quantum Operators and Mirror Symmetry: The paper investigates quantum operators derived from mirror symmetry principles. This approach links the spectral properties of these operators to quantum mechanics, with particular focus on difference operators.
3. Exact Quantization Conditions: From the proposed spectral determinant, exact quantization conditions are derived. These quantization conditions involve non-perturbative corrections captured in the theta function's behavior—a notable departure from traditional perturbative quantization reliant on the Nekrasov–Shatashvili limit.
4. Numerical and Analytical Validation: Strong numerical results supporting this theory come from analyzing the cases of local $^2$, local $^1 \times ^1$, and local $_1$. The predicted spectra closely match empirical data, reinforcing the validity of the conjecture.
5. Background Independence: The spectral determinant is shown as an entire function of the moduli space, emphasizing the background independence of the theory—a significant theoretical advancement.

Implications and Future Directions

This research lays groundwork for deeper exploration of the intersections between M-theory and topological strings, laying a path toward a more profound understanding of quantum geometry and enumerative geometry in the context of Calabi-Yau spaces. The conjecture not only advances the non-perturbative analysis of these theories but also integrates findings from refined topological strings and functional difference operators.

The potential impact extends into mathematical physics, where the correspondence could influence methods for solving quantum integrable systems and other areas concerned with the spectral theory of operators. Future work may explore extending these results to other geometric configurations or higher genus cases, further enriching the theoretical landscape established in this paper.

In conclusion, "Topological Strings from Quantum Mechanics" contributes an innovative conjecture with substantial theoretical and numerical foundations, opening new pathways in understanding and applying the principles of non-perturbative quantum mechanics and topological string theory.