11 research outputs found
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: SantamarÃa, 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 CientÃficas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de FÃsica de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de FÃsica de Buenos Aires; ArgentinaFil: Garbarz, Alan Nicolás. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de FÃsica de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de FÃsica de Buenos Aires; ArgentinaFil: Giribet, Gaston Enrique. Consejo Nacional de Investigaciones CientÃficas y Técnicas; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de FÃsica de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de FÃsica de Buenos Aires; Argentin