Showing papers on "Topological string theory published in 2013"

Instanton effects in ABJM theory from Fermi gas approach

Abstract: 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 function.

3d TQFT from 6d SCFT

Abstract: We study the six-dimensional (2, 0) superconformal field theory on S 1 × S 2 × M via compactification to five dimensions, where M is a three-manifold. Twisted along M, the five-dimensional theory has a half of $ \mathcal{N}=\left( {2,2} \right) $ supersymmetry on S 2, the other half being broken by a superpotential. We show that in the limit where M is infinitely large, the twisted theory reduces to a three-dimensional topological quantum field theory which is closely related to Chern-Simons theory for the complexified gauge group.

Axion topological field theory of topological superconductors

Abstract: Topological superconductors are gapped superconductors with gapless and topologically robust quasiparticles propagating on the boundary. In this paper, we present a topological field theory description of three-dimensional time-reversal invariant topological superconductors. In our theory the topological superconductor is characterized by a topological coupling between the electromagnetic field and the superconducting phase fluctuation, which has the same form as the coupling of ``axions'' with an Abelian gauge field. As a physical consequence of our theory, we predict the level crossing induced by the crossing of special ``chiral'' vortex lines, which can be realized by considering $s$-wave superconductors in proximity with the topological superconductor. Our theory can also be generalized to the coupling with a gravitational field.

The toroidal block and the genus expansion

Abstract: We study the correspondence between four-dimensional supersymmetric gauge theories and two-dimensional conformal field theories in the case of $ \mathcal{N}={2^{*}} $ gauge theory. We emphasize the genus expansion on the gauge theory side, as obtained via geometric engineering from the topological string. This point of view uncovers modular properties of the one-point conformal block on a torus with complexified intermediate momenta: in the large intermediate weight limit, it is a power series whose coefficients are quasimodular forms. The all-genus viewpoint that the conformal field theory approach lends to the topological string yields insight into the analytic structure of the topological string partition function in the field theory limit.

Transformations of Spherical Blocks

Abstract: We further explore the correspondence between $ \mathcal{N} $ = 2 supersymmetric SU(2) gauge theory with four flavors on ϵ-deformed backgrounds and conformal field theory, with an emphasis on the ϵ-expansion of the partition function natural from a topological string theory point of view. Solving an appropriate null vector decoupling equation in the semi-classical limit allows us to express the instanton partition function as a series in quasi-modular forms of the group Γ(2), with the expected symmetry W(D 4) ⋊ S 3. In the presence of an elementary surface operator, this symmetry is enhanced to an action of $ W\left( {D_4^{(1) }} \right)\rtimes {S_4} $ on the instanton partition function, as we demonstrate via the link between the null vector decoupling equation and the quantum Painleve VI equation.

Simple models with non-Abelian moduli on topological defects

Abstract: I explain how conventional topological defects -- the Abrikosov-Nielsen-Olesen (ANO) string and domain walls -- can acquire non-Abelian moduli localized on their world sheets. The set-up is conceptually similar and generalizes that used by Witten for cosmic strings.

Bernoulli number identities from quantum field theory and topological string theory

Abstract: We present a new method for the derivation of convolution identities for finite sums of products of Bernoulli numbers. Our approach is motivated by the role of these identities in quantum field theory and string theory. We first show that the Miki identity and the Faber-Pandharipande-Zagier (FPZ) identity are closely related, and give simple unified proofs which naturally yield a new Bernoulli number convolution identity. We then generalize each of these three identities into new families of convolution identities depending on a continuous parameter. We rederive a cubic generalization of Miki's identity due to Gessel and obtain a new similar identity generalizing the FPZ identity. The generalization of the method to the derivation of convolution identities of arbitrary order is outlined. We also describe an extension to identities which relate convolutions of Euler and Bernoulli numbers.

Five-dimensional topologically twisted maximally supersymmetric Yang-Mills theory

Abstract: Herein, we consider a topologically twisted version of maximally supersymmetric Yang-Mills theory in five dimensions which was introduced by Witten in 2011. We consider this theory on a five manifold of the form M-4 x I for M-4 an oriented Riemannian four manifold. The complete and unique action of the theory in bulk is written down and is shown to be invariant under two scalar supersymmetries.

A modified melting crystal model and the Ablowitz–Ladik hierarchy

Abstract: This paper addresses the issue of integrable structure in a modified melting crystal model of topological string theory on the resolved conifold. The partition function can be expressed as the vacuum expectation value of an operator on the Fock space of 2D complex free fermion fields. The quantum torus algebra of fermion bilinears behind this expression is shown to have an extended set of ?shift symmetries?. They are used to prove that the partition function (deformed by external potentials) is essentially a tau function of the 2D Toda hierarchy. This special solution of the 2D Toda hierarchy can also be characterized by a factorization problem of matrices. The associated Lax operators turn out to be quotients of first-order difference operators. This implies that the solution of the 2D Toda hierarchy in question is actually a solution of the Ablowitz?Ladik (equivalently, the relativistic Toda) hierarchy. As a byproduct, the shift symmetries are shown to be related to matrix-valued quantum dilogarithmic functions.

Resurgent Transseries and the Holomorphic Anomaly

Abstract: 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.

Glueball and meson spectrum in large-N massless QCD

Abstract: We provide outstanding numerical evidence that in large-N massless QCD the joint spectrum of the masses squared, for fixed integer spin s and unspecified parity and charge conjugation, obeys exactly the following laws: m_k^2 = (k+s/2) Lambda_QCD^2 for s even, m_k^2 = 2(k+s/2) Lambda_QCD^2 for s odd, k = 1,2,... for glueballs, and m_n^2 = 1/2 (n+s/2) Lambda_QCD^2, n = 0,1,... for mesons. One of the striking features of these laws is that they imply that the glueball and meson masses squared form exactly-linear Regge trajectories in the large-N limit of massless QCD, all the way down to the low-lying states: A fact unsuspected so far. The numerical evidence is based on lattice computations by Meyer-Teper in SU(8) YM for glueballs, and by Bali et al. in SU(17) quenched massless QCD for mesons, that we analyze systematically. The aforementioned spectrum for spin-0 glueballs is implied by a Topological Field Theory underlying the large-N limit of YM, whose glueball propagators satisfy as well fundamental universal constraints arising from the asymptotic freedom and the renormalization group. No other presently existing model meets both the infrared spectrum and the ultraviolet constraints. We argue that some features of the aforementioned spectrum of glueballs and mesons of any spin could be explained by the existence of a Topological String Theory dual to the Topological Field Theory.

Non-holomorphic deformations of special geometry and their applications

Abstract: The aim of these lecture notes is to give a pedagogical introduction to the subject of non-holomorphic deformations of special geometry. This subject was first introduced in the context of \(N=2\) BPS black holes, but has a wider range of applicability. A theorem is presented according to which an arbitrary point-particle Lagrangian can be formulated in terms of a complex function \(F\), whose features are analogous to those of the holomorphic function of special geometry. A crucial role is played by a symplectic vector that represents a complexification of the canonical variables, i.e. the coordinates and canonical momenta. We illustrate the characteristic features of the theorem in the context of field theory models with duality invariances. The function \(F\) may depend on a number of external parameters that are not subject to duality transformations. We introduce duality covariant complex variables whose transformation rules under duality are independent of these parameters. We express the real Hesse potential of \(N=2\) supergravity in terms of the new variables and expand it in powers of the external parameters. Then we relate this expansion to the one encountered in topological string theory. These lecture notes include exercises which are meant as a guidance to the reader.

Refined Chern-Simons theory and (q, t)-deformed Yang-Mills theory: Semi-classical expansion and planar limit

Abstract: We study the relationship between refined Chern-Simons theory on lens spaces S 3/ $ \mathbb{Z} $ p and (q,t)-deformed Yang-Mills theory on the sphere S 2. We derive the instanton partition function of (q, t)-deformed U(N) Yang-Mills theory and describe it explicitly as an analytical continuation of the semi-classical expansion of refined Chern-Simons theory. The derivations are based on a generalization of the Weyl character formula to Macdonald polynomials. The expansion is used to formulate q-generalizations of β-deformed matrix models for refined Chern-Simons theory, as well as conjectural formulas for the χ y -genus of the moduli space of U(N) instantons on the surface $ \mathcal{O} $ (−p) → $ \mathbb{P} $ 1 for all p ≥ 1 which enumerate black hole microstates in refined topological string theory. We study the large N phase structures of the refined gauge theories, and match them with refined topological string theory on the resolved conifold.

Refined Chern-Simons theory and (q,t)-deformed Yang-Mills theory: Semi-classical expansion and planar limit

Abstract: We study the relationship between refined Chern-Simons theory on lens spaces S^3/Z_p and (q,t)-deformed Yang-Mills theory on the sphere S^2 We derive the instanton partition function of (q,t)-deformed U(N) Yang-Mills theory and describe it explicitly as an analytical continuation of the semi-classical expansion of refined Chern-Simons theory The derivations are based on a generalization of the Weyl character formula to Macdonald polynomials The expansion is used to formulate q-generalizations of beta-deformed matrix models for refined Chern-Simons theory, as well as conjectural formulas for the chi_y-genus of the moduli space of U(N) instantons on the surface O(-p)--->P^1 for all p which enumerate black hole microstates in refined topological string theory We study the large N phase structures of the refined gauge theories, and match them with refined topological string theory on the resolved conifold

Torus Knots in Lens Spaces & Topological Strings

Abstract: We study the invariant of knots in lens spaces defined from quantum Chern-Simons theory. By means of the knot operator formalism, we derive a generalization of the Rosso-Jones formula for torus knots in L(p,1). In the second part of the paper, we propose a B-model topological string theory description of torus knots in L(2,1).

A matrix model for the topological string II: The spectral curve and mirror geometry

Abstract: In a previous paper, we presented a matrix model reproducing the topological string partition function on an arbitrary given toric Calabi-Yau manifold. Here, we compute the spectral curve of our matrix model and thus provide a matrix model derivation of the large volume limit of the BKMP "remodeling the B-model" conjecture, the claim that Gromov-Witten invariants of any toric Calabi-Yau threefold coincide with the spectral invariants of its mirror curve.

Integrable lattice models from four-dimensional field theories

Abstract: This note gives a general construction of an integrable lattice model (and a solution of the Yang-Baxter equation with spectral parameter) from a four-dimensional field theory which is a mixture of topological and holomorphic. Spin-chain models arise in this way from a twisted, deformed version of N=1 gauge theory.

Remarks on partition functions of topological string theory on generalized conifolds

Abstract: The method of topological vertex for topological string theory on toric Calabi-Yau 3-folds is reviewed. Implications of an explicit formula of partition functions in the “on-strip” case, typically the generalized conifolds, are considered. Generating functions of part of the partition functions are shown to be tau functions of the KP hierarchy. The associated Baker-Akhiezer functions play the role of wave functions, and satisfy q-difference equations. These q-difference equations represent the quantum mirror curves conjectured by Gukov and Su lkowski.

Modified melting crystal model and Ablowitz-Ladik hierarchy

Abstract: This paper addresses the issue of integrable structure in a modified melting crystal model of topological string theory on the resolved conifold. The partition function can be expressed as the vacuum expectation value of an operator on the Fock space of 2D complex free fermion fields. The quantum torus algebra of fermion bilinears behind this expression is shown to have an extended set of "shift symmetries". They are used to prove that the partition function (deformed by external potentials) is essentially a tau function of the 2D Toda hierarchy. This special solution of the 2D Toda hierarchy can be characterized by a factorization problem of $\ZZ\times\ZZ$ matrices as well. The associated Lax operators turn out to be quotients of first order difference operators. This implies that the solution of the 2D Toda hierarchy in question is actually a solution of the Ablowitz-Ladik (equivalently, relativistic Toda) hierarchy. As a byproduct, the shift symmetries are shown to be related to matrix-valued quantum dilogarithmic functions.

Generalized Ablowitz-Ladik hierarchy in topological string theory

Abstract: This paper addresses the issue of integrable structures in topological string theory on generalized conifolds. Open string amplitudes of this theory can be expressed as the matrix elements of an operator on the Fock space of 2D charged free fermion fields. The generating function of these amplitudes with respect to the product of two independent Schur functions becomes a tau function of the 2D Toda hierarchy. The associated Lax operators turn out to have a particular factorized form. This factorized form of the Lax operators characterizes a generalization of the Ablowitz-Ladik hierarchy embedded in the 2D Toda hierarchy. The generalized Ablowitz-Ladik hierarchy is thus identified as a fundamental integrable structure of topological string theory on the generalized conifolds.

Notes on reductions of superstring theory to bosonic string theory

Abstract: It is in general very subtle to integrate over the odd moduli of super Riemann surfaces in perturbative superstring computations. We study how these subtleties go away in favorable cases, including the embedding of $ \mathcal{N}=0 $ string to $ \mathcal{N}=1 $ string by Berkovits and Vafa, and the relation of the graviphoton amplitude and the topological string amplitude by Antoniadis, Gava, Narain and Taylor and Bershadsky, Cecotti, Ooguri and Vafa. The Poincare dual of the moduli space of Riemann surfaces in the moduli space of super Riemann surfaces plays an important role.

Framing the Di-Logarithm (over Z)

Abstract: Motivated by their role for integrality and integrability in topological string theory, we introduce the general mathematical notion of "s-functions" as integral linear combinations of poly-logarithms. 2-functions arise as disk amplitudes in Calabi-Yau D-brane backgrounds and form the simplest and most important special class. We describe s-functions in terms of the action of the Frobenius endomorphism on formal power series and use this description to characterize 2-functions in terms of algebraic K-theory of the completed power series ring. This characterization leads to a general proof of integrality of the framing transformation, via a certain orthogonality relation in K-theory. We comment on a variety of possible applications. We here consider only power series with rational coefficients; the general situation when the coefficients belong to an arbitrary algebraic number field is treated in a companion paper.

Notes on topological strings and knot contact homology.

Abstract: We give an introduction to the physics and mathematics involved in the recently observed relation between topological string theory and knot contact homology and then discuss this relation. This note is based on two lectures given at the 2013 Gokova Geometry and Topology Conference, and reports on joint work by Aganagic, Ng, Vafa, and the author [1].

Dijkgraaf-Vafa conjecture and β-deformed matrix models

Abstract: We study the β-deformed matrix models using the method of refined topological string theory. The refined holomorphic anomaly equation and boundary conditions near the singular divisors of the underlying geometry fix the refined amplitudes recursively. We provide exact test of the quantum integrality conjecture in the Nekrasov-Shatashvili limit. We check the higher genus exact formulae with perturbative matrix model calculations.

Dijkgraaf-Vafa conjecture and beta-deformed matrix models

Abstract: We study the beta-deformed matrix models using the method of refined topological string theory. The refined holomorphic anomaly equation and boundary conditions near the singular divisors of the underlying geometry fix the refined amplitudes recursively. We provide exact test of the quantum integrality conjecture in the Nekrasov-Shatashvili limit. We check the higher genus exact formulae with perturbative matrix model calculations.

M5-branes, toric diagrams and gauge theory duality

Abstract: We explore a duality between five-dimensional supersymmetric linear quiver gauge theories compactified on a circle, which are the five-dimensional uplifts of four-dimensional superconformal linear quiver theories. We find a correspondence between the gauge theory parameters of two dual theories, under which identical infrared effective coupling constants are obtained on the Coulomb branch. Two independent approaches using M-theory and the topological string theory give a consistent result.

Semiclassical partition functions for gravity with cosmic strings

Abstract: In this paper we describe an approach to construct semiclassical partition functions in gravity which are complete in the sense that they contain a complete description of the differentiable structures of the underlying 4-manifold. In addition, we find our construction naturally includes cosmic strings. We prove that the mass density of these strings uniquely specifies the topology of the leaves of a dimension 2 foliation, and conjecture that spacetime topology emerges as a result of the symmetry breaking of the fundamental fields. We discuss some possible applications of the partition functions in the fields of both quantum gravity and topological string theory.