Showing papers on "Topological string theory published in 2005"

The Topological Vertex

Abstract: We construct a cubic field theory which provides all genus amplitudes of the topological A-model for all non-compact toric Calabi-Yau threefolds. The topology of a given Feynman diagram encodes the topology of a fixed Calabi-Yau, with Schwinger parameters playing the role of Kahler classes of the threefold. We interpret this result as an operatorial computation of the amplitudes in the B-model mirror which is the quantum Kodaira-Spencer theory. The only degree of freedom of this theory is an unconventional chiral scalar on a Riemann surface. In this setup we identify the B-branes on the mirror Riemann surface as fermions related to the chiral boson by bosonization.

Pure spinor formalism as an N=2 topological string

Abstract: Following suggestions of Nekrasov and Siegel, a non-minimal set of fields are added to the pure spinor formalism for the superstring Twisted ĉ = 3 N = 2 generators are then constructed where the pure spinor BRST operator is the fermionic spin-one generator, and the formalism is interpreted as a critical topological string Three applications of this topological string theory include the super-Poincare covariant computation of multiloop superstring amplitudes without picture-changing operators, the construction of a cubic open superstring field theory without contact-term problems, and a new four-dimensional version of the pure spinor formalism which computes F-terms in the spacetime action

Chern-Simons theory and topological strings

Abstract: A review of the relation between Chern-Simons gauge theory and topological string theory on noncompact Calabi-Yau spaces is given. This relation has made it possible to give an exact solution of topological string theory on these spaces to all orders in the string coupling constant. Here the focus is on the construction of this solution, which is encoded in the topological vertex, and the implications of the physics of string/gauge theory duality for knot theory and for the geometry of Calabi-Yau manifolds.

The Existence of Supersymmetric String Theory with Torsion

Abstract: We derived an existence criterion to the Supersymmetric String Theory with Torsion proposed by Strominger and proved the existence of such theory for a class of Calabi-Yau threefolds.

Chern-Simons Theory, Matrix Models, and Topological Strings

Abstract: PART I: MATRIX MODELS, CHERN-SIMONS THEORY, AND THE LARGE N EXPANSION PART II: TOPOLOGICAL STRINGS PART III: THE TOPOLOGICAL STRING / GAUGE THEORY CORRESPONDENCE

Topological M-theory as unification of form theories of gravity

Abstract: We introduce a notion of topological M-theory and argue that it provides a unification of form theories of gravity in various dimensions. Its classical solutions involve G_2 holonomy metrics on 7-manifolds, obtained from a topological action for a 3-form gauge field introduced by Hitchin. We show that by reductions of this 7-dimensional theory, one can classically obtain 6-dimensional topological A and B models, the self-dual sector of loop quantum gravity in four dimensions, and Chern–Simons gravity in 3 dimensions. We also find that the 7-dimensional M-theory perspective sheds some light on the fact that the topological string partition function is a wavefunction, as well as on S-duality between the A and B models. The degrees of freedom of the A and B models appear as conjugate variables in the 7-dimensional theory. Finally, from the topological M-theory perspective, we find hints of an intriguing holographic link between non-supersymmetric Yang–Mills in four dimensions and A model topological strings on twistor space.

A paradigm of open/closed duality Liouville D-branes and the Kontsevich model

Abstract: We argue that topological matrix models (matrix models of the Kontsevich type) are examples of exact open/closed duality. The duality works at finite N and for generic 't Hooft couplings. We consider in detail the paradigm of the Kontsevich matrix integral for two-dimensional topological gravity. We demonstrate that the Kontsevich model arises by topological localization of cubic open string field theory on N stable branes. Our analysis is based on standard worldsheet methods in the context of non-critical bosonic string theory. The stable branes have Neumann (FZZT) boundary conditions in the Liouville direction. Several generalizations are possible.

A mini-course on topological strings

Abstract: These are the lecture notes for a short course in topological string theory that I gave at Uppsala University in the fall of 2004. The notes are aimed at PhD students who have studied quantum field theory and general relativity, and who have some general knowledge of ordinary string theory. The main purpose of the course is to cover the basics: after a review of the necessary mathematical tools, a thorough discussion of the construction of the A- and B-model topological strings from twisted N = (2,2) supersymmetric field theories is given. The notes end with a brief discussion on some selected applications.

On supertwistors, the Penrose-Ward transform and N=4 super Yang-Mills theory

Abstract: It was recently shown by Witten that B-type open topological string theory with the supertwistor space $\mathbb{C}P^{3|4}$ as a target space is equivalent to holomorphic Chern--Simons (hCS) theory on the same space. This hCS theory in turn is equivalent to self-dual $\mathcal{N}=4$ super-Yang–Mills (SYM) theory in four dimensions. We review the supertwistor description of self-dual and anti-self-dual $\mathcal{N}$-extended SYM theory as the integrability of SYM fields on complex $(2|\mathcal{N})$-dimensional superplanes and demonstrate the equivalence of this description to Witten's formulation. The equivalence of the field equations of hCS theory on an open subset of $\mathbb{C}P^{3|\mathcal{N}}$ to the field equations of self-dual $\mathcal{N}$-extended SYM theory in four dimensions is made explicit. Furthermore, we extend the picture to the full $\mathcal{N}=4$ SYM theory and, by using the known supertwistor description of this case, we show that the corresponding constraint equations are (gauge) equivalent to the field equations of hCS theory on a quadric in $\mathbb{C}P^{3|3}\times \mathbb{C}P^{3|3}$.

Superpotentials, A ∞ Relations and WDVV Equations for Open Topological Strings

Abstract: We give a systematic derivation of the consistency conditions which constrain open-closed disk amplitudes of topological strings. They include the A∞ relations (which generalize associativity of the boundary product of topological field theory), as well as certain homotopy versions of bulk-boundary crossing symmetry and Cardy constraint. We discuss integrability of amplitudes with respect to bulk and boundary deformations, and write down the analogs of WDVV equations for the space-time superpotential. We also study the structure of these equations from a string field theory point of view. As an application, we determine the effective superpotential for certain families of D-branes in B-twisted topological minimal models, as a function of both closed and open string moduli. This provides an exact description of tachyon condensation in such models, which allows one to determine the truncation of the open string spectrum in a simple manner.

The Hitchin functionals and the topological B-model at one loop

Abstract: The quantization in quadratic order of the Hitchin functional, which defines by critical points a Calabi–Yau structure on a six-dimensional manifold, is performed. The conjectured relation between the topological B-model and the Hitchin functional is studied at one loop. It is found that the genus one free energy of the topological B-model disagrees with the one-loop free energy of the minimal Hitchin functional. However, the topological B-model does agree at one-loop order with the extended Hitchin functional, which also defines by critical points a generalized Calabi–Yau structure. The dependence of the one-loop result on a background metric is studied, and a gravitational anomaly is found for both the B-model and the extended Hitchin model. The anomaly reduces to a volume-dependent factor if one computes for only Ricci-flat Kahler metrics

Topological mirror symmetry with fluxes

Abstract: Motivated by SU(3) structure compactifications, we show explicitly how to construct half-flat topological mirrors to Calabi-Yau manifolds with NS fluxes. Units of flux are exchanged with torsion factors in the cohomology of the mirror; this is the topological complement of previous differential-geometric mirror rules. The construction modifies explicit SYZ fibrations for compact Calabi-Yaus. The results are of independent interest for SU(3) compactifications. For example one can exhibit explicitly which massive forms should be used for Kaluza-Klein reduction, proving previous conjectures. Formality shows that these forms carry no topological information; this is also confirmed by infrared limits and old classification theorems.

Topological open string amplitudes on orientifolds

Abstract: We study topological open string amplitudes on orientifolds without fixed planes. We determine the contributions of the untwisted and twisted sectors as well as the BPS structure of the amplitudes. We illustrate our general results in various examples involving D-branes in toric orientifolds. We perform the computations by using both the topological vertex and unoriented localization. We also present an application of our results to the BPS structure of the coloured Kauffman polynomial of knots.

N=1/2 quiver gauge theories from open strings with R-R fluxes

Abstract: We consider a four dimensional N=1 gauge theory with bifundamental matter and a superpotential, defined on stacks of fractional branes. By turning on a flux for the R-R graviphoton field strength and computing open string amplitudes with insertions of R-R closed string vertices, we introduce a non-anticommutative deformation and obtain the N=1/2 version of the theory. We also comment on the appearance of a new structure in the effective Lagrangian.

Deformed PP-waves from the Lunin-Maldacena background

Abstract: In this article we study a pp-wave limit of the Lunin-Maldacena background. We show that the relevant string theory background is a homogeneous pp-wave. We obtain the string spectrum. The dual field theory is a deformation of = 4 super Yang-Mills theory. We have shown that, for a class of operators, at O(gYM2) and at leading order in N, all contributions to the anomalous dimension come from F-terms. We are able to identify the operator in the deformed super Yang-Mills which is dual to the lowest string mode. By studying the undeformed theory we are able to provide some evidence, directly in the field theory, that a small set of nearly protected operators decouple. We make some comments on operators in the Yang-Mills theory that are dual to excited string modes.

Extending the Picard-Fuchs system of local mirror symmetry

Abstract: We propose an extended set of differential operators for local mirror symmetry. If X is Calabi-Yau such that dimH4(X,Z)=0, then we show that our operators fully describe mirror symmetry. In the process, a conjecture for intersection theory for such X is uncovered. We also find operators on several examples of type X=KS through similar techniques. In addition, open string Picard-Fuchs systems are considered.

Open/closed string duality for topological gravity with matter

Abstract: The exact FZZT brane partition function for topological gravity with matter is computed using the dual two-matrix model. We show how the effective theory of open strings on a stack of FZZT branes is described by the generalized Kontsevich matrix integral, extending the earlier result for pure topological gravity. Using the well-known relation between the Kontsevich integral and a certain shift in the closed-string background, we conclude that these models exhibit open/closed string duality explicitly. Just as in pure topological gravity, the unphysical sheets of the classical FZZT moduli space are eliminated in the exact answer. Instead, they contribute small, nonperturbative corrections to the exact answer through Stokes' phenomenon.

Anatomy of zero-norm states in string theory

Abstract: We calculate and identify the counterparts of zero-norm states in the old covariant first quantized (OCFQ) spectrum of open bosonic string in two other quantization schemes of string theory, namely, the light-cone Del Giudice--Di Vecchia--Fubine zero-norm states and the off-shell Becchi-Rouet-Stora-Tyutin (BRST) zero-norm states (with ghost) in the Witten string field theory (WSFT). In particular, special attention is paid to the interparticle zero-norm states in all quantization schemes. For the case of the off-shell BRST zero-norm states, we impose the no-ghost conditions and recover exactly two types of on-shell zero-norm states in the OCFQ string spectrum for the first few low-lying mass levels. We then show that off-shell gauge transformations of WSFT are identical to the on-shell stringy gauge symmetries generated by two types of zero-norm states in the generalized massive $\ensuremath{\sigma}$-model approach of string theory. The high-energy limit of these stringy gauge symmetries was recently used to calculate the proportionality constants, conjectured by Gross, among high-energy scattering amplitudes of different string states. Based on these zero-norm state calculations, we have thus related gauge symmetry of WSFT to the high-energy stringy symmetry of Gross.

Topological quantum field theory and four manifolds

Abstract: I review some recent results on four-manifold invariants which have been obtained in the context of topological quantum field theory. I focus on three different aspects: (a) the computation of correlation functions, which give explicit results for the Donaldson invariants of non-simply connected manifolds, and for generalizations of these invariants to the gauge groupSU(N);(b) compactifications to lower dimensions, and connections to three-manifold topology and to intersection theory on the moduli space of flat connections on Riemann surfaces; (c) four-dimensional theories with critical behaviour, which give some remarkable constraints on Seiberg-Witten invariants and new results on the geography of four-manifolds.

Comments on BRST quantization of strings

Abstract: The BRST quantization of strings is revisited and the derivation of the path integral measure for scattering amplitudes is streamlined. Gauge invariances due to zero modes in the ghost sector are taken into account by using the Batalin-Vilkovisky formalism. This involves promoting the moduli of Riemann surfaces to quantum mechanical variables on which BRST transformations act. The familiar ghost and antighost zero mode insertions are recovered upon integrating out auxiliary fields. In contrast to the usual treatment, the gauge-fixed action including all zero mode insertions is BRST invariant. Possible anomalous contributions to BRST Ward identities due to boundaries of moduli space are reproduced in a novel way. Two models are discussed explicitly: bosonic string theory and topological gravity coupled to the topological A-model.

The Topological G2 String

Abstract: We construct new topological theories related to sigma models whose target space is a seven dimensional manifold of G_2 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 G_2 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.

On topological F-theory

Abstract: We consider the construction of a topological version of F-theory on a particular Spin(7) 8-manifold which is a Calabi-Yau 3-fold times a 2-torus. We write an action for this theory in eight dimensions and reduce it to lower dimensions using Hitchin's gradient flow method. A symmetry of the eight-dimensional theory which follows from modular transformations of the torus induces duality transformations of the variables of the topological A- and B-models. We also consider target space form actions in the presence of background fluxes in six dimensions.

RTN lectures on the non AdS/non CFT Correspondence

Abstract: Recently, Witten proposed a topological string theory in twistor space that is dual to a weakly coupled gauge theory. In this lectures we will discuss aspects of the twistor string theory. Along the way we will learn new things about Yang-Mills scattering amplitudes. The string theory sheds light on Yang-Mills perturbation theory and leads to new methods for computing Yang-Mills scattering amplitudes 2 .

Exact Black Hole Degeneracies and the Topological String

Abstract: Motivated by the recent conjecture of Ooguri, Strominger and Vafa, we compute the semi-canonical partition function of BPS black holes in N=4 and N=8 string theories, to all orders in perturbation theory. Not only are the black hole partition functions surprisingly simple; they capture the full topological string amplitudes, as expected from the OSV conjecture. The agreement is not perfect, however, as there are differences between the black hole and topological string partition functions even at the perturbative level. We propose a minimal modification of the OSV conjecture, in which these differences are understood as a nontrivial measure factor for the topological string.

c < 1 string from two dimensional black holes

Abstract: We study a topological string description of the c < 1 non-critical string whose matter part is defined by the time-like linear dilaton CFT. We show that the topologically twisted N=2 SL(2,R)/U(1) model (or supersymmetric 2D black hole) is equivalent to the c < 1 non-critical string compactified at a specific radius by comparing their physical spectra and correlation functions. We examine another equivalent description in the topological Landau-Ginzburg model and check that it reproduces the same scattering amplitudes. We also discuss its matrix model dual description.