Resumming the string perturbation series

We use the AdS/CFT correspondence to study the resummation of a perturbative genus expansion appearing in the type II superstring dual of ABJM theory. Although the series is Borel summable, its Borel resummation does not agree with the exact non-perturbative answer due to the presence of complex instantons. The same type of behavior appears in the WKB quantization of the quartic oscillator in Quantum Mechanics, which we analyze in detail as a toy model for the string perturbation series. We conclude that, in these examples, Borel summability is not enough for extracting non-perturbative information, due to non-perturbative effects associated to complex instantons. We also analyze the resummation of the genus expansion for topological string theory on local $\mathbb P^1 \times \mathbb P^1$, which is closely related to ABJM theory. In this case, the non-perturbative answer involves membrane instantons computed by the refined topological string, which are crucial to produce a well-defined result. We give evidence that the Borel resummation of the perturbative series requires such a non-perturbative sector.Comment: 31 pages, 9 figures; v3 : clarifications added and misprints correcte

A One-Loop Test of Quantum Supergravity

The partition function on the three-sphere of ABJM theory and its generalizations has, at large N, a universal, subleading logarithmic term. Inspired by the success of one-loop quantum gravity for computing the logarithmic corrections to black hole entropy, we try to reproduce this universal term by a one-loop calculation in Euclidean eleven-dimensional supergravity on AdS_4 \times X_7. We find perfect agreement between the results of ABJM theory and the eleven dimensional supergravity.Comment: 12 pages, accepted for publication in Classical and Quantum Gravit

Painlev\'e kernels and surface defects at strong coupling

It is well established that the spectral analysis of canonically quantized four-dimensional Seiberg-Witten curves can be systematically addressed via the Nekrasov-Shatashvili functions. In this paper we explore another aspect of the relation between $\mathcal{N}=2$ supersymmetric gauge theories in four dimensions and operator theory. Specifically, we study an example of an integral operator connected to Painlev\'e equations and whose spectral traces compute correlation functions of the 2d Ising model. This operator does not correspond to a canonically quantized Seiberg-Witten curve, but its kernel can nevertheless be interpreted as the density matrix of an ideal Fermi gas. Adopting the approach of Tracy and Widom, we provide an explicit expression for its eigenfunctions via an $\mathrm{O}(2)$ matrix model. We then show that these eigenfunctions are computed by surface defects in $\mathrm{SU}(2)$ super Yang-Mills in the self-dual phase of the $\Omega$-background. Our result also yields a strong coupling expression for such defects which resums the instanton expansion. Even though we focus on one concrete example, we expect these results to hold for a larger class of operators arising in the context of isomonodromic deformation equations.Comment: 49 pages, 1 figur

Argyres-Douglas theories, Painlev\'e II and quantum mechanics

We show in details that the all-orders genus expansion of the two-cut Hermitian cubic matrix model reproduces the perturbative expansion of the $H_1$ Argyres-Douglas theory coupled to the $\Omega$ background. In the self-dual limit we use the Painlev\'e/gauge correspondence and we show that, after summing over all instanton sectors, the two-cut cubic matrix model computes the tau function of Painlev\'e II without taking any double scaling limit or adding any external fields. We decode such solution within the context of trans-series. Finally in the Nekrasov-Shatashvili limit we connect the $H_1$ and the $H_0$ Argyres-Douglas theories to the quantum mechanical models with cubic and double well potentials.Comment: 33 pages, 3 figures, minor corrections, some references adde

Quantization conditions and functional equations in ABJ(M) theories

The partition function of ABJ(M) theories on the three-sphere can be regarded as the canonical partition function of an ideal Fermi gas with a non-trivial Hamiltonian. We propose an exact expression for the spectral determinant of this Hamiltonian, which generalizes recent results obtained in the maximally supersymmetric case. As a consequence, we find an exact WKB quantization condition determining the spectrum which is in agreement with numerical results. In addition, we investigate the factorization properties and functional equations for our conjectured spectral determinants. These functional equations relate the spectral determinants of ABJ theories with consecutive ranks of gauge groups but the same Chern-Simons coupling.Comment: 29 pages, 3 figures. v2: minor corrections and clarifications adde