Showing papers on "Topological string theory published in 2008"

Open string amplitudes and large order behavior in topological string theory

Abstract: We propose a formalism inspired by matrix models to compute open and closed topological string amplitudes in the B-model on toric Calabi-Yau manifolds. We find closed expressions for various open string amplitudes beyond the disk, and in particular we write down the annulus amplitude in terms of theta functions on a Riemann surface. We test these ideas on local curves and local surfaces, providing in this way generating functionals for open Gromov-Witten invariants in the spirit of mirror symmetry. In the case of local curves, we study the open string sector near the critical point which leads to 2d gravity, and we show that toric D-branes become FZZT branes in a double-scaling limit. We use this connection to compute non-perturbative instanton effects due to D-branes that control the large order behavior of topological string theory on these backgrounds.

Topological Strings and (Almost) Modular Forms

Abstract: The B-model topological string theory on a Calabi-Yau threefold X has a symmetry group Γ, generated by monodromies of the periods of X. This acts on the topological string wave function in a natural way, governed by the quantum mechanics of the phase space H3(X). We show that, depending on the choice of polarization, the genus g topological string amplitude is either a holomorphic quasi-modular form or an almost holomorphic modular form of weight 0 under Γ. Moreover, at each genus, certain combinations of genus g amplitudes are both modular and holomorphic. We illustrate this for the local Calabi-Yau manifolds giving rise to Seiberg-Witten gauge theories in four dimensions and local IP2 and IP1 × IP1. As a byproduct, we also obtain a simple way of relating the topological string amplitudes near different points in the moduli space, which we use to give predictions for Gromov-Witten invariants of the orbifold \({{\mathbb {C}^3} / {\mathbb {Z}_3}}\).

Topological String Theory on Compact Calabi–Yau: Modularity and Boundary Conditions

Abstract: The topological string partition function Z(λ,t,t) =exp(λ2 g-2 Fg(t, t)) is calculated on a compact Calabi–Yau M. The Fg(t, t) fulfil the holomorphic anomaly equations, which imply that ψ=Z transforms as a wave function on the symplectic space H3(M, Z). This defines it everywhere in the moduli space M(M) along with preferred local coordinates. Modular properties of the sections Fg as well as local constraints from the 4d effective action allow us to fix Z to a large extent. Currently with a newly found gap condition at the conifold, regularity at the orbifold and the most naive bounds from Castelnuovo’s theory, we can provide the boundary data, which specify Z, e.g. up to genus 51 for the quintic.

Nonperturbative effects and nonperturbative definitions in matrix models and topological strings

Abstract: 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 model.

Topological Strings And (Almost) Modular Forms

Abstract: hep-th/0607100 arXiv:hep-th/0607100v2 4 May 2007 Topological Strings and (Almost) Modular Forms Mina Aganagic, 1 Vincent Bouchard, 2 Albrecht Klemm, 3 University of California, Berkeley, CA 94720, USA Mathematical Sciences Research Institute, Berkeley, CA 94720, USA University of Wisconsin, Madison, WI 53706, USA Abstract The B-model topological string theory on a Calabi-Yau threefold X has a symmetry group Γ, generated by monodromies of the periods of X. This acts on the topological string wave function in a natural way, governed by the quantum mechanics of the phase space H 3 (X). We show that, depending on the choice of polarization, the genus g topological string amplitude is either a holomorphic quasi-modular form or an almost holomorphic modular form of weight 0 under Γ. Moreover, at each genus, certain combinations of genus g amplitudes are both modular and holomorphic. We illustrate this for the local Calabi- Yau manifolds giving rise to Seiberg-Witten gauge theories in four dimensions and local IP 2 amplitudes near different points in the moduli space, which we use to give predictions for Z Gromov-Witten invariants of the orbifold C 3 /Z 3 . July 2006 and IP 1 ×IP 1 . As a byproduct, we also obtain a simple way of relating the topological string

Supersymmetric Gauge Theories, Intersecting Branes and Free Fermions

Abstract: We show that various holomorphic quantities in supersymmetric gauge theories can be conveniently computed by configurations of D4-branes and D6-branes. These D-branes intersect along a Riemann surface that is described by a holomorphic curve in a complex surface. The resulting I-brane carries two-dimensional chiral fermions on its world-volume. This system can be mapped directly to the topological string on a large class of non-compact Calabi-Yau manifolds. Inclusion of the string coupling constant corresponds to turning on a constant B-field on the complex surface, which makes this space non-commutative. Including all string loop corrections the free fermion theory is elegantly formulated in terms of holonomic D-modules that replace the classical holomorphic curve in the quantum case.

Refined Topological Vertex and Instanton Counting

Abstract: It has been proposed recently that topological A-model string amplitudes for toric Calabi-Yau 3-folds in non self-dual graviphoton background can be caluculated by a diagrammatic method that is called the ``refined topological vertex''. We compute the extended A-model amplitudes for SU(N)-geometries using the proposed vertex. If the refined topological vertex is valid, these computations should give rise to the Nekrasov's partition functions of = 2 SU(N) gauge theories via the geometric engineering. In this article, we verify the proposal by confirming the equivalence between the refined A-model amplitude and the K-theoretic version of the Nekrasov's partition function by explicit computation.

The Topological G 2 String

Abstract: We construct new topological theories related to sigma models whose target space is a seven-dimensional manifold of G2 holonomy. We define a new type of topological twist and identify the BRST operator and the physical states. Unlike the more familiar six-dimensional case, our topological model is defined in terms of conformal blocks and not in terms of local operators of the original theory. We also present evidence that one can extend this definition to all genera and construct a seven-dimensional topological string theory. We compute genus zero correlation functions and relate these to Hitchin’s functional for three-forms in seven dimensions. Along the way we develop the analogue of special geometry for G2 manifolds. When the seven-dimensional topological twist is applied to the product of a Calabi–Yau manifold and a circle, the result is an interesting combination of the six-dimensional A and B models.

A String theoretic model of gauge mediated supersymmetry beaking

Abstract: We propose a robust supergravity model of dynamical supersymmetry breaking and gauge mediation, and a natural embedding in non-perturbative string theory with D-branes. A chiral field (and its mirror) charged under "anomalous" U(1)'s acts as a Polonyi field whose hierarchical Polonyi-term can be generated by string instantons. Further quartic superpotential terms arise naturally as a tree-level decoupling effect of massive string states. A robust supersymmetry breaking minimum allows for gauge mediation with soft masses at the TeV scale, which we realise for a globally consistent SU(5) GUT model of Type I string theory, with a D1-instanton inducing the Polonyi term.

Mutual Chern-Simons theory for Z 2 topological order

Abstract: We study several different ${Z}_{2}$ topological ordered states in frustrated spin systems. The effective theories for those different ${Z}_{2}$ topological orders all have the same form---a ${Z}_{2}$ gauge theory which can also be written as a mutual $U(1)\ifmmode\times\else\texttimes\fi{}U(1)$ Chern-Simons theory. However, we find that the different ${Z}_{2}$ topological orders are reflected in different projective realizations of lattice symmetry in the same effective mutual Chern-Simons theory. This result is obtained by comparing the ground-state degeneracy, the ground-state quantum numbers, the gapless edge state, and the projective symmetry group of quasiparticles calculated from the slave-particle theory and from the effective mutual Chern-Simons theories. Our study reveals intricate relations between topological order and symmetry.

Nonperturbative effects and nonperturbative definitions in matrix models and topological strings

Abstract: 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 model

Matching gauge theory and string theory in a decoupling limit of AdS/CFT

Abstract: We identify a regime of the AdS/CFT correspondence in which we can quantitatively match N=4 super Yang-Mills (SYM) for small 't Hooft coupling with weakly coupled type IIB string theory on AdS_5 x S^5. We approach this regime by taking the same decoupling limit on both sides of the correspondence. On the gauge theory side only the states in the SU(2) sector survive, and in the planar limit the Hamiltonian is given by the XXX_{1/2} Heisenberg spin chain. On the string theory side we show that the decoupling limit corresponds to a non-relativistic limit. In this limit some of the bosonic modes and all of the fermionic modes of the string become infinitely heavy and decouple. We first take the decoupling limit of the string sigma-model classically. This enables us to identify a semi-classical regime with semi-classical string states even though we are in a regime corresponding to small 't Hooft coupling. We furthermore analyze the quantum corrections that enter in taking the limit. From this we infer that gauge theory and string theory match, both in terms of the action and the spectrum, for the leading part and the first correction away from the semi-classical regime. Finally we consider the implications for the hitherto unexplained matching of the one-loop contribution to the energy of certain gauge theory and string theory states, and we explain how our results give a firm basis for the matching of the Hagedorn temperature in hep-th/0608115.

Topological strings, two-dimensional Yang-Mills theory and Chern-Simons theory on torus bundles

Abstract: We study the relations between two-dimensional Yang-Mills theory on the torus, topological string theory on a Calabi-Yau threefold whose local geometry is the sum of two line bundles over the torus, and Chern-Simons theory on torus bundles. The chiral partition function of the Yang-Mills gauge theory in the large N limit is shown to coincide with the topological string amplitude computed by topological vertex techniques. We use Yang-Mills theory as an efficient tool for the computation of Gromov-Witten invariants and derive explicitly their relation with Hurwitz numbers of the torus. We calculate the Gopakumar-Vafa invariants, whose integrality gives a non-trivial confirmation of the conjectured nonperturbative relation between two-dimensional Yang-Mills theory and topological string theory. We also demonstrate how the gauge theory leads to a simple combinatorial solution for the Donaldson-Thomas theory of the Calabi-Yau background. We match the instanton representation of Yang-Mills theory on the torus with the nonabelian localization of Chern-Simons gauge theory on torus bundles over the circle. We also comment on how these results can be applied to the computation of exact degeneracies of BPS black holes in the local Calabi-Yau background.

Flop Invariance of Refined Topological Vertex and Link Homologies

Abstract: It has been proposed recently that the topological A-model string theory on local toric Calabi-Yau manifolds has a two parameter extension. Amplitudes of the two parameter topological strings can be computed using a diagrammatic method called the refined topological vertex. In this paper we study properties of the refined amplitudes under the flop transition of toric Calabi-Yau three-folds. We also discuss that the slicing invariance and the flop transition imply a simple formula for the homological sl(N) invariants of the Hopf link. The new expression for the invariants gives a simple refinement of the Hopf link invariant of Chern-Simons theory.

Bubbling Calabi-Yau geometry from matrix models

Abstract: We study bubbling geometry in topological string theory. Specifically, we analyse Chern-Simons theory on both the 3-sphere and lens spaces in the presence of a Wilson loop of an arbitrary representation. For each three manifold, we formulate a multi-matrix model whose partition function is the Wilson loop vev and compute the spectral curve. This spectral curve is closely related to the Calabi-Yau threefold which is the gravitational dual of the Wilson loop. Namely, it is the reduction to two dimensions of the mirror to the Calabi-Yau. For lens spaces the dual geometries are new. We comment on a similar matrix model relevant for Wilson loops in AdS/CFT.

Selfduality and Chern-Simons Theory

Abstract: We propose a relation between the operator of S-duality (of N=4 super Yang-Mills theory in 3+1D) and a topological theory in one dimension lower. We construct the topological theory by compactifying N=4 super Yang-Mills on a circle with an S-duality and R-symmetry twist. The S-duality twist requires a selfdual coupling constant. We argue that for a sufficiently low rank of the gauge group the three-dimensional low-energy description is a topological theory, which we conjecture to be a pure Chern-Simons theory. This conjecture implies a connection between the action of mirror symmetry on the sigma-model with Hitchin's moduli space as target space and geometric quantization of the moduli space of flat connections on a Riemann surface.

D-branes and Closed String Field Theory

Abstract: We construct solitonic states in the OSp invariant string field theory, which are BRST invariant in the leading order of regularization parameter. One can show that these solitonic states describe D-branes and ghost D-branes, by calculating the scattering amplitudes.

D-brane instantons and the effective field theory of flux compactifications

Abstract: We provide a description of the effects of fluxes on euclidean D-brane instantons purely in terms of the 4d effective action. The effect corresponds to the dressing of the effective non-perturbative 4d effective vertex with 4d flux superpotential interactions, generated when the moduli fields made massive by the flux are integrated out. The description in terms of effective field theory allows a unified description of non-perturbative effects in all flux compactifications of a given underlying fluxless model, globally in the moduli space of the latter. It also allows us to describe explicitly the effects on D-brane instantons of fluxes with no microscopic description, like non-geometric fluxes. At the more formal level, the description has interesting connections with the bulk-boundary map of open-closed two-dimensional topological string theory, and with the $\NN=1$ special geometry.

Anomalies of E8 gauge theory on String manifolds

Abstract: In this note we revisit the subject of anomaly cancelation in string theory and M-theory on manifolds with String structure and give three observations. First, that on String manifolds there is no E8 x E8 global anomaly in heterotic string theory. Second, that the description of the anomaly in the phase of the M-theory partition function of Diaconescu-Moore-Witten extends from the Spin case to the String case. Third, that the cubic refinement law of Diaconescu-Freed-Moore for the phase of the M-theory partition function extends to String manifolds. The analysis relies on extending from invariants which depend on the Spin structure to invariants which instead depend on the String structure. Along the way, the one-loop term is refined via the Witten genus.

Crystal Melting and Toric Calabi-Yau Manifolds

Abstract: We construct a statistical model of crystal melting to count BPS bound states of D0 and D2 branes on a single D6 brane wrapping an arbitrary toric Calabi-Yau threefold. The three-dimensional crystalline structure is determined by the quiver diagram and the brane tiling which characterize the low energy effective theory of D branes. The crystal is composed of atoms of different colors, each of which corresponds to a node of the quiver diagram, and the chemical bond is dictated by the arrows of the quiver diagram. BPS states are constructed by removing atoms from the crystal. This generalizes the earlier results on the BPS state counting to an arbitrary non-compact toric Calabi-Yau manifold. We point out that a proper understanding of the relation between the topological string theory and the crystal melting involves the wall crossing in the Donaldson-Thomas theory.

G(2) Hitchin functionals at one loop

Abstract: We consider the quantization of the effective target space description of topological M-theory in terms of the Hitchin functional whose critical points describe seven manifolds with a G2 structure. The one-loop partition function for this theory is calculated and an extended version of it, that is related to the generalized G2 geometry, is compared with the topological G2 string. We relate the reduction of the effective action for the extended G2 theory to the Hitchin functional description of the topological string in six dimensions. The dependence of the partition functions on the choice of background G2 metric is also determined.

A Note on topological M5-branes and string-fivebrane duality

Abstract: We derive the stability conditions for the M5-brane in topological M-theory using k-symmetry. The non-linearly self-dual 3-form on the world-volume is necessarily non-vanishing, as is the case also for the 2-form field strengths on coisotropic branes in topological string theory. It is demonstrated that the self-duality is consistent with the stability conditions, which are solved locally in terms of a tensor in the representation 6 of SU(3) subset of G(2). The double dimensional reduction of the M5-brane is the D4-brane, and its direct reduction is an NS5-brane. We show that the equation of motion for the 3-form on the NS5-brane wrapping a Calabi-Yau space is exactly the Kodaira-Spencer equation, providing support for a string-fivebrane duality in topological string theory.

Mirror symmetry formulae for the elliptic genus of complete intersections

Abstract: In this paper we calculate the elliptic genus of certain complete inter- sections in products of projective spaces. We show that it is equal to the elliptic genus of the Landau-Ginzburg models that are, according to Hori and Vafa, mirror partners of these complete intersections. This provides additional evidence of the validity of their construction. Understanding the mathematics behind Quantum and in particular Conformal Field Theories has been a challenge for more then twenty five years. The usual ground for mathematical interpretations of Quantum Field Theory predictions has been the Topological Quantum Field Theory, which is a certain reduction of the genuine Quan- tum Field Theory. There have been significant mathematical advances in this area. Mirror symmetry is among the major motivations behind these advances. Vaguely stated, mirror symmetry is a duality between complex and symplectic geometry. As was originally discovered by physicists (6), among its specific manifestations is a strik- ing connection between the"number of curves of a given genus" in a symplectic man- ifold and the periods of a holomorphic form of a different complex manifold. Ever since, the"explicit" construction of the mirror partner for a given manifold has be- come the central task for the mathematicians. There is a vast amount of work done in this direction. In this paper, we follow the line pioneered by Gepner (8) and developed by Vafa (25) who discovered that the Conformal Field Theory defined by a manifold, the so-called the sigma model, can be identical to the Conformal Field Theory of an a different type, the so-called Landau-Ginsburg model. This point of view was further developed by Witten (28). As far as the application of this idea to mirror symmetry is concerned, the major reference for the purposes of this paper is the work of Hori and Vafa (15). They showed that the mirror partner of a large class of manifolds turns out to be a Landau-Ginsburg model of some kind, or its orbifold. The"proof" that these models form a mirror pair would consist of picking an invariant which is known to be identical for mirror partners and by a calculation showing that it is indeed the same for given hypothetical mirror partners. In (15) such an invariant is given by the periods of a holomorphic form on a manifold and the so called BPS masses in the Landau-Ginsburg model. Of course the identities implied by mirror symmetry in Topological Quantum Field Theory are a reduction of stronger identities in the

THE TOPOLOGICAL THEORY OF THE MILNOR INVARIANT $\bar\mu(1,2,3)$

Abstract: We study a topological Abelian gauge theory that generalizes the Abelian Chern–Simons one, and that leads in a natural way to the Milnor's link invariant when the classical action on-shell is calculated.

Lectures on Black Holes, Topological Strings, and Quantum Attractors (2.0)

Abstract: In these lecture notes, we review some recent developments on the relation between the macroscopic entropy of four-dimensional BPS black holes and the microscopic counting of states beyond the thermodynamical, large charge limit. After a brief overview of charged black holes in supergravity and string theory, we give an extensive introduction to special and very special geometry, attractor flows and topological string theory, including holomorphic anomalies. We then expose the Ooguri-Strominger-Vafa (OSV) conjecture which relates microscopic degeneracies to the topological string amplitude, and review precision tests of this formula on “small” black holes. Finally, motivated by a holographic interpretation of the OSV conjecture, we give a systematic approach to the radial quantization of BPS black holes (i.e. quantum attractors). This suggests the existence of a one-parameter generalization of the topological string amplitude and provides a general framework for constructing automorphic partition functions for black hole degeneracies in theories with sufficient degree of symmetry.

New Anomalies in Topological String Theory

Abstract: We show that the topological string partition function with D-branes on a compact Calabi-Yau manifold has new anomalies that spoil the recursive structure of the holomorphic anomaly equation and introduce dependence on wrong moduli (such as complex structure moduli in the A-model), unless the disk one-point functions vanish. This provides a microscopic explanation for the recent result of Walcher in arXiv:0712.2775 on counting of BPS states in M-theory using the topological string partition function. The relevance of vanishing disk one-point functions to large $N$ duality for compact Calabi-Yau manifolds is noted.

Aspects of Supersymmetric Gauge Theory and String Theory

Abstract: This thesis consists of two parts. In the first part we study some topics in $\mathcal{N}=1$ supersymmeric gauge theory and the relation to matrix models. We review the relevant non-perturbative techniques for computing effective superpotential, such as Seiberg-Witten curve. Then we review the proposal of Dijkgraaf and Vafa that relates the glueball superpotentials to the computation in matrix models. We then consider a case of multi-trace superpotential. We perform the perturbative computation of glueball superpotential in this case and explain the subtlety in identifying the glueball superfield. We also use these techniques to study phases of $\mathcal{N}=1$ gauge theory with flavors. In the second part we study topics in AdS/CFT correspondence and its plane wave limit. We review the plane wave geometry and BMN operators that corresponding to string modes. Then we study string interactions in the case of a highly curved plane wave background, and demonstrate the agreements between calculations of string interaction amplitudes in the two dual theories. Finally we study D3-brane giant gravitons and open string attached to them. Giant gravitons are non-perturbative objects that have very large R-charge.

Phase space polarization and the topological string: a case study

Abstract: We review and elaborate on our discussion in hep-th/0606112 on the interplay between the target space and the worldsheet description of the open topological string partition function, for the example of the conifold We discuss the appropriate phase space and canonical form for the system We find a map between choices of polarization and the worldsheet description, based on which we study the behavior of the partition function under canonical transformations

Composite strings in (2+1)-dimensional anisotropic weakly coupled Yang-Mills theory

Abstract: The small-scale structure of a string connecting a pair of static sources is explored for the weakly-coupled anisotropic SU(2) Yang-Mills theory in (2+1) dimensions A crucial ingredient in the formulation of the string Hamiltonian is the phenomenon of color smearing of the string constituents The quark-anti-quark potential is determined We close with some discussion of the standard, fully Lorentz-invariant Yang-Mills theory

Orbifold Gromov-Witten Invariants and Topological Strings

Abstract: In this contribution I propose a (hopefully) pedagogical ap- proach to the computation of orbifold Gromov-Witten invariants using mirror symmetry and topological string theory, focusing on the orbifold C 3 /Z3. Recent B-model developments on the mirror side, which led to predictions for "open orbifold Gromov-Witten invariants" of C 3 /Z3, are also addressed. This contribution is based on the results of (1) and (9).