Showing papers on "Topological string theory published in 2011"

Quantum black holes, localization, and the topological string

Abstract: We use localization to evaluate the functional integral of string field theory on AdS 2 × S 2 background corresponding to the near horizon geometry of supersymmetric black holes in 4d compactifications with $ \mathcal{N} = 2 $ supersymmetry. In particular, for a theory containing n v + 1 vector multiplets, we show that the functional integral localizes exactly onto an ordinary integral over a finite-dimensional submanifold in the field space labeling a continuous family of instanton solutions in which auxiliary fields in the vector multiplets are excited with nontrivial dependence on AdS 2 coordinates. These localizing solutions are universal in that they follow from the off-shell supersymmetry transformations and do not depend on the choice of the action. They are parametrized by n v + 1 real parameters {C I; I = 0,…, n v } that correspond to the values of the auxiliary fields at the center of AdS 2. In the Type-IIA frame, assuming D-terms evaluate to zero on the solutions for reasons of supersymmetry, the classical part of the integrand equals the absolute square of the partition function of the topological string as conjectured by Ooguri, Strominger, and Vafa; however evaluated at the off-shell values of scalar fields at the center of AdS 2. In addition, there are contributions from one-loop determinants, brane-instantons, and nonperturbative orbifolds that are in principle computable. These results thus provide a concrete method to compute exact quantum entropy of these black holes including all perturbative and nonperturbative corrections and can be used to establish a precise relation between the quantum degeneracies of black holes and the topological string.

The Laplace transform of the cut-and-join equation and the Bouchard-Marino conjecture on Hurwitz numbers

Abstract: Author(s): Eynard, Bertrand; Mulase, Motohico; Safnuk, Brad | Abstract: We calculate the Laplace transform of the cut-and-join equation of Goulden, Jackson and Vakil. The result is a polynomial equation that has the topological structure identical to the Mirzakhani recursion formula for the Weil-Petersson volume of the moduli space of bordered hyperbolic surfaces. We find that the direct image of this Laplace transformed equation via the inverse of the Lambert W-function is the topological recursion formula for Hurwitz numbers conjectured by Bouchard and Marino using topological string theory.

Quantization of Integrable Systems and a 2d/4d Duality

Abstract: We present a new duality between the F-terms of supersymmetric field theories defined in two- and four-dimensions respectively. The duality relates N=2 supersymmetric gauge theories in four dimensions, deformed by an Omega-background in one plane, to N=(2,2) gauged linear sigma-models in two dimensions. On the four dimensional side, our main example is N=2 SQCD with gauge group SU(L) and 2L fundamental flavours. Using ideas of Nekrasov and Shatashvili, we argue that the Coulomb branch of this theory provides a quantization of the classical Heisenberg SL(2) spin chain. Agreement with the standard quantization via the Algebraic Bethe Ansatz implies the existence of an isomorphism between the chiral ring of the 4d theory and that of a certain two-dimensional theory. The latter can be understood as the worldvolume theory on a surface operator/vortex string probing the Higgs branch of the same 4d theory. We check the proposed duality by explicit calculation at low orders in the instanton expansion. One striking consequence is that the Seiberg-Witten solution of the 4d theory is captured by a one-loop computation in two dimensions. The duality also has interesting connections with the AGT conjecture, matrix models and topological string theory where it corresponds to a refined version of the geometric transition.

Extended Holomorphic Anomaly in Gauge Theory

Abstract: The partition function of an $${\mathcal {N}=2}$$ gauge theory in the Ω-background satisfies, for generic value of the parameter $${\beta=-{\epsilon_1}/{\epsilon_2}}$$ , the, in general extended, but otherwise β-independent, holomorphic anomaly equation of special geometry. Modularity together with the (β-dependent) gap structure at the various singular loci in the moduli space completely fixes the holomorphic ambiguity, also when the extension is non-trivial. In some cases, the theory at the orbifold radius, corresponding to β = 2, can be identified with an “orientifold” of the theory at β = 1. The various connections give hints for embedding the structure into the topological string.

Quantization of integrable systems and a 2d/4d duality

Abstract: We present a new duality between the F-terms of supersymmetric field theories defined in two-and four-dimensions respectively. The duality relates $ \mathcal{N} = 2 $ super-symmetric gauge theories in four dimensions, deformed by an Ω-background in one plane, to $ \mathcal{N} = \left( {2,2} \right) $ gauged linear σ-models in two dimensions. On the four dimensional side, our main example is $ \mathcal{N} = 2 $ SQCD with gauge group G = SU(L) and N F = 2 L fundamental flavours. Using ideas of Nekrasov and Shatashvili, we argue that the Coulomb branch of this theory provides a quantization of the classical Heisenberg SL(2) spin chain. Agreement with the standard quantization via the Algebraic Bethe Ansatz implies the existence of an isomorphism between the chiral ring of the 4 d theory and that of a certain two-dimensional theory. The latter can be understood as the worldvolume theory on a surface operator/vortex string probing the Higgs branch of the same 4 d theory. We check the proposed duality by explicit calculation at low orders in the instanton expansion. One striking consequence is that the Seiberg-Witten solution of the 4 d theory is captured by a one-loop computation in two dimensions. The duality also has interesting connections with the AGT conjecture, matrix models and topological string theory where it corresponds to a refined version of the geometric transition.

Surface operator, bubbling Calabi-Yau and AGT relation

Abstract: Surface operators in $ \mathcal{N} = 2 $ four-dimensional gauge theories are interesting half-BPS objects. These operators inherit the connection of gauge theory with the Liouville conformal field theory, which was discovered by Alday, Gaiotto and Tachikawa. Moreover it has been proposed that toric branes in the A-model topological strings lead to surface operators via the geometric engineering. We analyze the surface operators by making good use of topological string theory. Starting from this point of view, we propose that the wave-function behavior of the topological open string amplitudes geometrically engineers the surface operator partition functions and the Gaiotto curves of corresponding gauge theories. We then study a peculiar feature that the surface operator corresponds to the insertion of the degenerate fields in the conformal field theory side. We show that this aspect can be realized as the geometric transition in topological string theory, and the insertion of a surface operator leads to the bubbling of the toric Calabi-Yau geometry.

SU(N) quantum Racah coefficients & non-torus links

Abstract: It is well-known that the SU(2) quantum Racah coefficients or the Wigner $6j$ symbols have a closed form expression which enables the evaluation of any knot or link polynomials in SU(2) Chern-Simons field theory. Using isotopy equivalence of SU(N) Chern-Simons functional integrals over three balls with one or more $S^2$ boundaries with punctures, we obtain identities to be satisfied by the SU(N) quantum Racah coefficients. This enables evaluation of the coefficients for a class of SU(N) representations. Using these coefficients, we can compute the polynomials for some non-torus knots and two-component links. These results are useful for verifying conjectures in topological string theory.

The uses of the refined matrix model recursion

Abstract: We study matrix models in the β-ensemble by building on the refined recursion relation proposed by Chekhov and Eynard. We present explicit results for the first β-deformed corrections in the one-cut and the two-cut cases, as well as two applications to supersymmetric gauge theories: the calculation of superpotentials in N=1 gauge theories, and the calculation of vevs of surface operators in superconformal N=2 theories and their Liouville duals. Finally, we study the β-deformation of the Chern–Simons matrix model. Our results indicate that this model does not provide an appropriate description of the Ω-deformed topological string on the resolved conifold, and therefore that the β-deformation might provide a different generalization of topological string theory in toric Calabi–Yau backgrounds.

Non-perturbative topological strings and conformal blocks

Abstract: We give a non-perturbative completion of a class of closed topological string theories in terms of building blocks of dual open strings. In the specific case where the open string is given by a matrix model these blocks correspond to a choice of integration contour. We then apply this definition to the AGT setup where the dual matrix model has logarithmic potential and is conjecturally equivalent to Liouville conformal field theory. By studying the natural contours of these matrix integrals and their monodromy properties, we propose a precise map between topological string blocks and Liouville conformal blocks. Remarkably, this description makes use of the light-cone diagrams of closed string field theory, where the critical points of the matrix potential correspond to string interaction points.

Gravity as the Square of Gauge Theory

Abstract: The BCJ squaring relations provide a simple prescription for the computation of gravity amplitudes in terms of gauge theory ingredients. Unlike the KLT relations, the squaring relations are directly applicable both at tree and loop level. We review the derivation of these relations from on-shell recursion relations, and discuss an off-shell approach to these relations in which the interactions of the gravity Lagrangian arise as the square of the gaugetheory interactions. This article is based on work with Zvi Bern, Tristan Dennen and Yu-tin Huang [Z. Bern, T. Dennen, Y.-t. Huang and M. Kiermaier, Phys. Rev. D 82 (2010), 065003, arXiv:1004.0693 (Ref. 1))] which was presented at String Field Theory and Related Aspects 2010.

What is the Simplest Gauge-String Duality?

Abstract: We make a proposal for the string dual to the simplest large N theory, the Gaussian matrix integral in the 'tHooft limit, and how this dual description emerges from double line graphs. This is a specic realisation of the general approach to gauge-string duality which associates worldsheet riemann surfaces to the Feynman- 'tHooft diagrams of a large N gauge theory. We show that a particular version (proposed by Razamat) of this connection, involving integer Strebel dierentials, naturally explains the combinatorics of Gaussian matrix correlators. We nd that the correlators can be explicitly realised as a sum over a special class of holomorphic maps (Belyi maps) from the worldsheet to a target space P 1 . We are led to identify this target space with the riemann surface associated with the (eigenvalues of the) matrix model. In the process, an AdS/CFT like dictionary, for arbitrary correlators of single trace operators, also emerges in which the holomorphic maps play the role of stringy Witten diagrams. Finally, we provide some evidence that the above string dual is the conventional A-model topological string theory on P 1 .

M5-branes, toric diagrams and gauge theory duality

Abstract: In this article we explore the duality between the low energy effective theory of five-dimensional N=1 SU(N)^{M-1} and SU(M)^{N-1} linear quiver gauge theories compactified on S^1. The theories we study are the five-dimensional uplifts of four-dimensional superconformal linear quivers. We study this duality by comparing the Seiberg-Witten curves and the Nekrasov partition functions of the two dual theories. The Seiberg-Witten curves are obtained by minimizing the worldvolume of an M5-brane with nontrivial geometry. Nekrasov partition functions are computed using topological string theory. The result of our study is a map between the gauge theory parameters, i.e., Coulomb moduli, masses and UV coupling constants, of the two dual theories. Apart from the obvious physical interest, this duality also leads to compelling mathematical identities. Through the AGTW conjecture these five-dimentional gauge theories are related to q-deformed Liouville and Toda SCFTs in two-dimensions. The duality we study implies the relations between Liouville and Toda correlation functions through the map we derive.

On the topology of the hypermultiplet moduli space in type II/CY string vacua

Abstract: By analyzing qualitative aspects of NS5-brane instanton corrections, we determine the topology of the hypermultiplet moduli space M_H in Calabi-Yau compactifications of type II string theories at fixed value of the dilaton and of the Calabi-Yau metric. Specifically, we show that for fivebrane instanton couplings to be well-defined, translations along the intermediate Jacobian must induce non-trivial shifts of the Neveu-Schwarz axion which had thus far been overlooked. As a result, the Neveu-Schwarz axion parametrizes the fiber of a circle bundle, isomorphic to the one in which the fivebrane partition function is valued. In the companion paper arXiv:1010.5792, we go beyond the present analysis and take steps towards a quantitative description of fivebrane instanton corrections, using a combination of mirror symmetry, S-duality, topological string theory and twistor techniques.

An Action for higher spin gauge theory in four dimensions

Abstract: An action principle is presented for Vasiliev's Bosonic higher spin gauge theory in four spacetime dimensions. The action is of the form of a broken topological field theory, and arises by an extension of the MacDowell-Mansouri formulation of general relativity. In the latter theory the local degrees of freedom of general relativity arise by breaking the gauge invariance of a topological theory from $sp(4)$ to the Lorentz algebra. In Vasiliev's theory the infinite number of degrees of freedom with higher spins similarly arise by the breaking of a topological theory with an infinite dimensional gauge symmetry extending $sp(4)$ to the Lorentz algebra. The Hamiltonian formulation of Vasilev's theory is then derived from our action, and it is shown that the Hamiltonian is a linear combination of constraints, as expected for a diffeomorphism invariant theory. The constraint algebra is computed and found to be first class.

Closed string cohomology in open string field theory

Abstract: We show that closed string states in bosonic string field theory are encoded in the cyclic cohomology of cubic open string field theory (OSFT) which, in turn, classifies the deformations of OSFT. This cohomology is then shown to be independent of the open string background. Exact elements correspond to closed string gauge transformations, generic boundary deformations of Witten’s 3-vertex and infinitesimal shifts of the open string background. Finally it is argued that the closed string cohomology and the cyclic cohomology of OSFT are isomorphic to each other.

Beta Deformation and Superpolynomials of (n,m) Torus Knots

Abstract: Recent studies in several interrelated areas -- from combinatorics and representation theory in mathematics to quantum field theory and topological string theory in physics -- have independently revealed that many classical objects in these fields admit a relatively novel one-parameter deformation. This deformation, known in different contexts under the names of Omega-background, refinement, or beta-deformation, has a number of interesting mathematical implications. In particular, in Chern-Simons theory beta-deformation transforms the classical HOMFLY invariants into Dunfield-Gukov-Rasmussen superpolynomials -- Poincare polynomials of a triply graded knot homology theory. As shown in arXiv:1106.4305, these superpolynomials are particular linear combinations of rational Macdonald dimensions, distinguished by the polynomiality, integrality and positivity properties. We show that these properties alone do not fix the superpolynomials uniquely, by giving an example of a combination of Macdonald dimensions, that is always a positive integer polynomial but generally is not a superpolynomial.

Bulk deformations of open topological string theory

Abstract: We present a general method to construct bulk-deformed open topological string theories from Landau-Ginzburg models. To this end we obtain a weak version of deformation quantisation, and we show how this together with the technique of homological perturbation allows to explicitly compute all bulk-deformed open topological string amplitudes at tree-level before tadpole-cancellation. Our approach is based on a coherent treatment of the problem in terms of the fundamental A-infinity- and L-infinity-structures involved.

A note on computations of the D-brane superpotential

Abstract: We develop some computational methods for the integrals over the 3-chains on the compact Calabi–Yau 3-folds that play a prominent role in the analysis of the topological B-model in the context of the open mirror symmetry. We discuss such 3-chain integrals in two approaches. In the first approach, we provide an algorithm to obtain the inhomogeneous Picard–Fuchs equations. In the second approach, we discuss the analytic continuation of the period integral to compute the 3-chain integral directly. The latter direct integration method is applicable for both on-shell and off-shell formalisms.

Bell's Inequalities, Superquantum Correlations, and String Theory

Abstract: We offer an interpretation of superquantum correlations in terms of a “doubly” quantum theory. We argue that string theory, viewed as a quantum theory with two deformation parameters, the string tension α', and the string coupling constant gs, is such a superquantum theory that transgresses the usual quantum violations of Bell's inequalities. We also discuss the ℏ→∞ limit of quantum mechanics in this context. As a superquantum theory, string theory should display distinct experimentally observable supercorrelations of entangled stringy states.

Crystals, instantons and quantum toric geometry

Abstract: We describe the statistical mechanics of a melting crystal in three dimensions and its relation to a diverse range of models arising in combinatorics, algebraic geometry, integrable systems, low-dimensional gauge theories, topological string theory and quantum gravity. Its partition function can be computed by enumerating the contributions from noncommutative instantons to a six-dimensional cohomological gauge theory, which yields a dynamical realization of the crystal as a discretization of spacetime at the Planck scale. We describe analogous relations between a melting crystal model in two dimensions and N=4 supersymmetric Yang-Mills theory in four dimensions. We elaborate on some mathematical details of the construction of the quantum geometry which combines methods from toric geometry, isospectral deformation theory and noncommutative geometry in braided monoidal categories. In particular, we relate the construction of noncommutative instantons to deformed ADHM data, torsion-free modules and a noncommutative twistor correspondence.

Aspects géométriques et intégrables des modèles de matrices aléatoires

Abstract: Travail realise en cotutelle avec l'universite Paris-Diderot et le Commissariat a l'Energie Atomique sous la direction de John Harnad et Bertrand Eynard.

SU(2) gauge theory of gravity with topological invariants

Abstract: The most general gravity Lagrangian in four dimensions contains three topological densities, namely Nieh-Yan, Pontryagin and Euler, in addition to the Hilbert-Palatini term. We set up a Hamiltonian formulation based on this Lagrangian. The resulting canonical theory depends on three parameters which are coefficients of these terms and is shown to admit a real SU(2) gauge theoretic interpretation with a set of seven first-class constraints. Thus, in addition to the Newton's constant, the theory of gravity contains three (topological) coupling constants, which might have non-trivial imports in the quantum theory.

Quelques probl\`emes de g\'eom\'etrie \'enum\'erative, de matrices al\'eatoires, d'int\'egrabilit\'e, \'etudi\'es via la g\'eometrie des surfaces de Riemann

Abstract: Complex analysis is a powerful tool to study classical integrable systems, statistical physics on the random lattice, random matrix theory, topological string theory,... All these topics share certain relations, called "loop equations" or "Virasoro constraints". In the simplest case, the complete solution of those equations was found recently: it can be expressed in the framework of differential geometry over a certain Riemann surface which depends on the problem : the "spectral curve". This thesis is a contribution to the development of these techniques, and to their applications. Keywords: random matrices, random maps, integrable systems, algebraic geometry, loop equations, topological recursion, Hurwitz numbers, Gromov-Witten invariants, Tracy-Widom laws, beta ensemble, large N asymptotics in random matrix theory.

Emergence of Supersymmetry, Gauge Theory and String in Condensed Matter Systems

Abstract: This is a set of lecture notes on the following topics : 1. Emergent supersymmetry[1] 2. Emergent gauge theory[2] 3. Critical spin liquid with Fermi surface[3] 4. Holographic description of quantum field theory[4]

Application of KBc subalgebra in string field theory

Abstract: Recently, a classical solution of open cubic string field theory (CSFT) which corresponds to the closed string vacuum is found by Erler and Schnabl. In their work, a very simple subalgebra of open string star algebra — called KBc subalgebra — plays a crucial role. In this talk, we demonstrate two applications of the KBc subalgebra. One is evaluation of classical and effective tachyon potential. It turns out that the level expansion in the KBc subalgebra terminates at a certain level, so that analytic evaluation of effective potential is available. The other application is regularization of the identity based solutions. It is demonstrated that the Okawa-Erler-Schnabl type solution naturally includes gauge invariant regularization of identity based solutions.

Application of KBc Subalgebra in String Field Theory(String Field Theory and Related Aspects (SFT2010))

Abstract: Recently, a classical solution of open cubic string field theory (CSFT) which corresponds to the closed string vacuum is found by Erler and Schnabl. In their work, a very simple subalgebra of open string star algebra — called KBc subalgebra — plays a crucial role. In this talk, we demonstrate two applications of the KBc subalgebra. One is evaluation of classical and effective tachyon potential. It turns out that the level expansion in the KBc subalgebra terminates at a certain level, so that analytic evaluation of effective potential is available. The other application is regularization of the identity based solutions. It is demonstrated that the Okawa-Erler-Schnabl type solution naturally includes gauge invariant regularization of identity based solutions.