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.

The partition function of ABJM theory on the three-sphere has nonperturbative corrections due to membrane instantons in the M-theory dual. We show that the full series of membrane instanton corrections is completely determined by the refined topological string on the Calabi-Yau manifold known as local P 1 × P 1 , in the NekrasovShatashvili limit. Our result can be interpreted as a first-principles derivation of the full series of non-perturbative effects for the closed topological string on this Calabi-Yau background. Based on this, we make a proposal for the non-perturbative free energy of topological strings on general, local Calabi-Yau manifolds.

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We study the instanton effects of the ABJM partition function using the Fermi gas formalism. We compute the exact values of the partition function at the Chern-Simons levels k = 1, 2, 3, 4, 6 up to N = 44, 20, 18, 16, 14 respectively, and extract non-perturbative corrections from these exact results. Fitting the resulting non-perturbative corrections by their expected forms from the Fermi gas, we determine unknown parameters in them. After separating the oscillating behavior of the grand potential, which originates in the periodicity of the grand partition function, and the worldsheet instanton contribution, which is computed from the topological string theory, we succeed in proposing an analytical expression for the leading D2-instanton correction. Just as the perturbative result, the instanton corrections to the partition function are expressed in terms of the Airy

The partition function of the ABJM theory receives non-perturbative corrections due to instanton effects. We study these non-perturbative corrections, including bound states of worldsheet instantons and membrane instantons, in the Fermi-gas approach. We require that the total non-perturbative correction should be always finite for arbitrary Chern-Simons level. This finiteness is realized quite non-trivially because each bound state contribution naively diverges at some levels. The poles of each contribution should be canceled out in total. We use this pole cancellation mechanism to find unknown bound state corrections from known ones. We conjecture a general expression of the bound state contribution. Summing up all the bound state contributions, we find that the effect of bound states is simply incorporated into the worldsheet instanton correction by a redefinition of the chemical potential in the Fermi-gas system. Analytic expressions of the 3-and 4-membrane instanton corrections are also proposed.