Nonperturbative effects and nonperturbative definitions in matrix models and topological strings

We develop techniques to compute multi-instanton corrections to the 1/N expansion in matrix models described by orthogonal polynomials. These techniques are based on finding trans-series solutions, i.e. formal solutions with exponentially small corrections, to the recursion relations characterizing the free energy. We illustrate this method in the Hermitian, quartic matrix model, and we provide a detailed description of the instanton corrections in the Gross-Witten-Wadia (GWW) unitary matrix model. Moreover, we use Borel resummation techniques and results from the theory of resurgent functions to relate the formal multi-instanton series to the nonperturbative definition of the matrix model. We study this relation in the case of the GWW model and its double-scaling limit, providing in this way a nice illustration of various mechanisms connecting the resummation of perturbative series to nonperturbative results, like the cancellation of nonperturbative ambiguities. Finally, we argue that trans-series solutions are also relevant in the context of topological string theory. In particular, we point out that in topological string models with both a matrix model and a large N gauge theory description, the nonperturbative, holographic definition involves a sum over the multi-instanton sectors of the matrix modelComment: 50 pages, 12 figures, comments and references added, small correction

Nonperturbative aspects of ABJM theory

Using the matrix model which calculates the exact free energy of ABJM theory on S^3 we study non-perturbative effects in the large N expansion of this model, i.e., in the genus expansion of type IIA string theory on AdS4xCP^3. We propose a general prescription to extract spacetime instanton actions from general matrix models, in terms of period integrals of the spectral curve, and we use it to determine them explicitly in the ABJM matrix model, as exact functions of the 't Hooft coupling. We confirm numerically that these instantons control the asymptotic growth of the genus expansion. Furthermore, we find that the dominant instanton action at strong coupling determined in this way exactly matches the action of an Euclidean D2-brane instanton wrapping RP^3.Comment: 26 pages, 14 figures. v2: small corrections, final version published in JHE

Unquenched flavor and tropical geometry in strongly coupled Chern-Simons-matter theories

We study various aspects of the matrix models calculating free energies and Wilson loop observables in supersymmetric Chern-Simons-matter theories on the three-sphere. We first develop techniques to extract strong coupling results directly from the spectral curve describing the large N master field. We show that the strong coupling limit of the gauge theory corresponds to the so-called tropical limit of the spectral curve. In this limit, the curve degenerates to a planar graph, and matrix model calculations reduce to elementary line integrals along the graph. As an important physical application of these tropical techniques, we study N=3 theories with fundamental matter, both in the quenched and in the unquenched regimes. We calculate the exact spectral curve in the Veneziano limit, and we evaluate the planar free energy and Wilson loop observables at strong coupling by using tropical geometry. The results are in agreement with the predictions of the AdS duals involving tri-Sasakian manifoldsComment: 32 pages, 7 figures. v2: small corrections, added an Appendix on the relation with the approach of 1011.5487. v3: further corrections and clarifications, final version to appear in JHE

Exact Results in ABJM Theory from Topological Strings

Recently, Kapustin, Willett and Yaakov have found, by using localization techniques, that vacuum expectation values of Wilson loops in ABJM theory can be calculated with a matrix model. We show that this matrix model is closely related to Chern-Simons theory on a lens space with a gauge supergroup. This theory has a topological string large N dual, and this makes possible to solve the matrix model exactly in the large N expansion. In particular, we find the exact expression for the vacuum expectation value of a 1/6 BPS Wilson loop in the ABJM theory, as a function of the 't Hooft parameters, and in the planar limit. This expression gives an exact interpolating function between the weak and the strong coupling regimes. The behavior at strong coupling is in precise agreement with the prediction of the AdS string dual. We also give explicit results for the 1/2 BPS Wilson loop recently constructed by Drukker and TrancanelliComment: 18 pages, two figures, small misprints corrected and references added, final version to appear in JHE

Crystal structure of N-(2-hydroxy-5-methylphenyl)benzamide

In the title compound, C14H13NO2, the mean plane of the non-H atoms of the central amide fragment C-N-C(=O)-C (r.m.s. deviation = 0.029 Å) forms dihedral angles of 5.63 (6) and 10.20 (5)° with the phenyl and hy­droxy­phenyl rings, respectively. A short intra­molecular N-H...O contact is present. In the crystal, the mol­ecules are linked by O-H...O hydrogen bonds to generate C(7) chains along [100]. The chains are reinforced by weak C-H...O contacts, which together with the O-H...O bonds lead to R22(7) loops. Very weak N-H...O inter­actions link the mol­ecules into inversion dimers

Interacting fermions and N=2 Chern-Simons-matter theories

The partition function on the three-sphere of N=3 Chern-Simons-matter theories can be formulated in terms of an ideal Fermi gas. In this paper we show that, in theories with N=2 supersymmetry, the partition function corresponds to a gas of interacting fermions in one dimension. The large N limit is the thermodynamic limit of the gas and it can be analyzed with the Hartree and Thomas-Fermi approximations, which lead to the known large N solutions of these models. We use this interacting fermion picture to analyze in detail N=2 theories with one single node. In the case of theories with no long-range forces we incorporate exchange effects and argue that the partition function is given by an Airy function, as in N=3 theories. For the theory with g adjoint superfields and long-range forces, the Thomas-Fermi approximation leads to an integral equation which determines the large N, strongly coupled R-charge.Comment: 29 pages, 4 figure

The large N limit of M2-branes on Lens spaces

We study the matrix model for N M2-branes wrapping a Lens space L(p,1) = S^3/Z_p. This arises from localization of the partition function of the ABJM theory, and has some novel features compared with the case of a three-sphere, including a sum over flat connections and a potential that depends non-trivially on p. We study the matrix model both numerically and analytically in the large N limit, finding that a certain family of p flat connections give an equal dominant contribution. At large N we find the same eigenvalue distribution for all p, and show that the free energy is simply 1/p times the free energy on a three-sphere, in agreement with gravity dual expectations.Comment: 28 pages, 4 figure

Non-perturbative effects and the refined topological string

The partition function of ABJM theory on the three-sphere has non-perturbative 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 P1xP1, in the Nekrasov-Shatashvili 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.Comment: 38 pages, 5 figure

Towards Vacuum Superstring Field Theory: The Supersliver

We extend some aspects of vacuum string field theory to superstring field theory in Berkovits' formulation, and we study the star algebra in the fermionic matter sector. After clarifying the structure of the interaction vertex in the operator formalism of Gross and Jevicki, we provide an algebraic construction of the supersliver state in terms of infinite-dimensional matrices. This state is an idempotent string field and solves the matter part of the equation of motion of superstring field theory with a pure ghost BRST operator. We determine the spectrum of eigenvalues and eigenvectors of the infinite-dimensional matrices of Neumann coefficients in the fermionic matter sector. We then analyze coherent states based on the supersliver and use them in order to construct higher-rank projector solutions, as well as to construct closed subalgebras of the star algebra in the fermionic matter sector. Finally, we show that the geometric supersliver is a solution to the superstring field theory equations of motion, including the (super)ghost sector, with the canonical choice of vacuum BRST operator recently proposed by Gaiotto, Rastelli, Sen and Zwiebach.Comment: 45 pages, JHEP styl

Numerical studies of the ABJM theory for arbitrary N at arbitrary coupling constant

We show that the ABJM theory, which is an N=6 superconformal U(N)*U(N) Chern-Simons gauge theory, can be studied for arbitrary N at arbitrary coupling constant by applying a simple Monte Carlo method to the matrix model that can be derived from the theory by using the localization technique. This opens up the possibility of probing the quantum aspects of M-theory and testing the AdS_4/CFT_3 duality at the quantum level. Here we calculate the free energy, and confirm the N^{3/2} scaling in the M-theory limit predicted from the gravity side. We also find that our results nicely interpolate the analytical formulae proposed previously in the M-theory and type IIA regimes. Furthermore, we show that some results obtained by the Fermi gas approach can be clearly understood from the constant map contribution obtained by the genus expansion. The method can be easily generalized to the calculations of BPS operators and to other theories that reduce to matrix models.Comment: 35 pages, 20 figures; reference added. The simulation code is available upon request to [email protected]