Annales Henri Poincaré

, Volume 17, Issue 11, pp 3177–3235 | Cite as

Topological Strings from Quantum Mechanics

  • Alba GrassiEmail author
  • Yasuyuki Hatsuda
  • Marcos Mariño


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 \({{\mathbb{P}}^2}\), local \({{\mathbb{P}}^1 \times {\mathbb{P}}^1}\) and local \({{\mathbb{F}}_1}\). 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.


Theta Function Topological String ABJM Theory Canonical Partition Function Orbifold Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Département de Physique Théorique et Section de MathématiquesUniversité de GenèveGenevaSwitzerland
  2. 2.DESY Theory Group, DESY HamburgHamburgGermany

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