Resurgence in topological string theory

One of the main results of this theory describes, in a quantitative way, the relation between the perturbative and nonperturbative information of a system. Encoded in the asymptotic growth of the series coefficients of perturbation theory is the information necessary to reconstruct nonperturbative sectors. All these sectors can be put together in a formal object called the transseries, whose different coefficients are related to each other by resurgence relations. The resurgent approach has been applied succesfully to problems in mathematics, on differential and difference equations, and in physics, on quantum mechanics and even quantum field theory. It is currently a very active area of research merging the efforts of both physicists and mathematicians. This thesis performs a resurgent analysis of the perturbative topological string theory. Using the holomorphic anomaly equations it is possible to compute coefficients of the perturbative free energy to very high order and analyze their asymptotic growth. In agreement with resurgence, it is found that nonperturbative sectors coming from a transseries control this growth. It is shown that this transseries can be computed as a solution of a natural extension of the holomorphic anomaly equations. The first half of this thesis is concerned with the main properties of the theory of resurgence and with the computation of the perturbative topological string free energy. These results are then applied to a concrete topological string example. A careful study of the asymptotic growth of the perturbative free energies is performed and various resurgence relations are uncovered. These relations involve elements of the transseries describing the full nonperturbative free energy. General properties of the transseries satisfying the holomorphic anomaly equations are described, including the role of the instanton actions, the presence of holomorphic ambiguities and the possibility of resonance. The numerical results are found to match, to high precision, the elements of the computed transseries. The asymptotic nature of the higher instanton sectors is also studied and a complicated net of resurgence relations is found

Resurgent Transseries and the Holomorphic Anomaly: Nonperturbative Closed Strings in Local CP2

The holomorphic anomaly equations describe B-model closed topological strings in Calabi-Yau geometries. Having been used to construct perturbative expansions, it was recently shown that they can also be extended past perturbation theory by making use of resurgent transseries. These yield formal nonperturbative solutions, showing integrability of the holomorphic anomaly equations at the nonperturbative level. This paper takes such constructions one step further by working out in great detail the specific example of topological strings in the mirror of the local CP2 toric Calabi-Yau background, and by addressing the associated (resurgent) large-order analysis of both perturbative and multi-instanton sectors. In particular, analyzing the asymptotic growth of the perturbative free energies, one finds contributions from three different instanton actions related by Z_3 symmetry, alongside another action related to the Kahler parameter. Resurgent transseries methods then compute, from the extended holomorphic anomaly equations, higher instanton sectors and it is shown that these precisely control the asymptotic behavior of the perturbative free energies, as dictated by resurgence. The asymptotic large-order growth of the one-instanton sector unveils the presence of resonance, i.e., each instanton action is necessarily joined by its symmetric contribution. The structure of different resurgence relations is extensively checked at the numerical level, both in the holomorphic limit and in the general nonholomorphic case, always showing excellent agreement with transseries data computed out of the nonperturbative holomorphic anomaly equations. The resurgence relations further imply that the string free energy displays an intricate multi-branched Borel structure, and that resonance must be properly taken into account in order to describe the full transseries solution.Comment: 63 pages, 54 images in 24 figures, jheppub-nosort.sty; v2: corrected figure, minor changes, final version for CM

Resurgent Transseries and the Holomorphic Anomaly

The gauge theoretic large N expansion yields an asymptotic series which requires a nonperturbative completion in order to be well defined. Recently, within the context of random matrix models, it was shown how to build resurgent transseries solutions encoding the full nonperturbative information beyond the 't Hooft genus expansion. On the other hand, via large N duality, random matrix models may be holographically described by B-model closed topological strings in local Calabi-Yau geometries. This raises the question of constructing the corresponding holographically dual resurgent transseries, tantamount to nonperturbative topological string theory. This paper addresses this point by showing how to construct resurgent transseries solutions to the holomorphic anomaly equations. These solutions are built upon (generalized) multi-instanton sectors, where the instanton actions are holomorphic. The asymptotic expansions around the multi-instanton sectors have both holomorphic and anti-holomorphic dependence, may allow for resonance, and their structure is completely fixed by the holomorphic anomaly equations in terms of specific polynomials multiplied by exponential factors and up to the holomorphic ambiguities -- which generalizes the known perturbative structure to the full transseries. In particular, the anti-holomorphic dependence has a somewhat universal character. Furthermore, in the nonperturbative sectors, holomorphic ambiguities may be fixed at conifold points. This construction shows the nonperturbative integrability of the holomorphic anomaly equations, and sets the ground to start addressing large-order analysis and resurgent nonperturbative completions within closed topological string theory.Comment: 59 pages, jheppub-nosort.sty; v2: small additions, minor changes, refs updated; v3: more minor corrections, final version for AH

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

On Asymptotics and Resurgent Structures of Enumerative Gromov-Witten Invariants

Making use of large-order techniques in asymptotics and resurgent analysis, this work addresses the growth of enumerative Gromov-Witten invariants---in their dependence upon genus and degree of the embedded curve---for several different threefold Calabi-Yau toric-varieties. In particular, while the leading asymptotics of these invariants at large genus or at large degree is exponential, at combined large genus and degree it turns out to be factorial. This factorial growth has a resurgent nature, originating via mirror symmetry from the resurgent-transseries description of the B-model free energy. This implies the existence of nonperturbative sectors controlling the asymptotics of the Gromov-Witten invariants, which could themselves have an enumerative-geometry interpretation. The examples addressed include: the resolved conifold; the local surfaces local P^2 and local P^1 x P^1; the local curves and Hurwitz theory; and the compact quintic. All examples suggest very rich interplays between resurgent asymptotics and enumerative problems in algebraic geometry

Addition of torsion to chiral gravity

Three-dimensional gravity in anti-de Sitter space is considered, including torsion. The derivation of the central charges of the algebra that generates the asymptotic isometry group of the theory is reviewed, and a special point of the theory, at which one of the central charges vanishes, is compared with the chiral point of topologically massive gravity. This special point corresponds to a singular point in the Chern-Simons theory, where one of the two coupling constants of the SL(2,R) actions vanishes. A prescription to approach this point in the space of parameters is discussed, and the canonical structure of the theory is analyzed. © 2011 American Physical Society.Fil: Santamaría, Ricardo Couso. Universidad de Santiago de Compostela; EspañaFil: Edelstein, Jose Daniel. Centro de Estudios Cientificos; Chile. Universidad de Santiago de Compostela; España. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; ArgentinaFil: Garbarz, Alan Nicolás. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; ArgentinaFil: Giribet, Gaston Enrique. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; Argentin