Showing papers on "Topological string theory published in 1998"

M theory and topological strings. 2.

Abstract: It is shown how the topological string amplitudes encode the BPS structure of wrapped M2 branes in M-theory compactification on Calabi-Yau threefolds. This in turn is related to a twisted supersymmetric index in 5 dimensions which receives contribution only from BPS states. The spin dependence of BPS states in 5 dimensions is captured by the string coupling constant dependence of topological string amplitudes.

Yang-Mills theory as a deformation of topological field theory, dimensional reduction, and quark confinement

Abstract: We propose a reformulation of Yang-Mills theory as a perturbative deformation of a novel topological (quantum) field theory. We prove that this reformulation of four-dimensional QCD leads to quark confinement in the sense of an area law of the Wilson loop. First, Yang-Mills theory with a non-Abelian gauge group G is reformulated as a deformation of a novel topological field theory. Next, a special class of topological field theories is defined by both Becchi-Rouet-Stora-Tyupin (BRST) and anti-BRST exact actions corresponding to the maximal Abelian gauge leaving the maximal torus group Hof G invariant. Then we find topological field theory $(Dg2)$ has a hidden supersymmetry for a choice of maximal Abelian gauge. As a result, the D-dimensional topological field theory is equivalent to the $(D\ensuremath{-}2)$-dimensional coset $G/H$ nonlinear sigma model in the sense of the Parisi-Sourlas dimensional reduction. After maximal Abelian gauge fixing, the topological property of the magnetic monopole and antimonopole of four-dimensional Yang-Mills theory is translated into that of an instanton and anti-instanton in a two-dimensional equivalent model. It is shown that the linear static potential in four dimensions follows from the instanton--anti-instanton gas in the equivalent two-dimensional nonlinear sigma model obtained from the four-dimensional topological field theory by dimensional reduction, while the remaining Coulomb potential comes from the perturbative part in four-dimensional Yang-Mills theory. The dimensional reduction opens a path for applying various exact methods developed in two-dimensional quantum field theory to study the nonperturbative problem in low-energy physics of four-dimensional quantum field theories.

Timelike T-Duality, de Sitter Space, Large $N$ Gauge Theories and Topological Field Theory

Abstract: T-Duality on a timelike circle does not interchange IIA and IIB string theories, but takes the IIA theory to a type $IIB^*$ theory and the IIB theory to a type $IIA^*$ theory. The type $II^*$ theories admit E-branes, which are the images of the type II D-branes under timelike T-duality and correspond to imposing Dirichlet boundary conditions in time as well as some of the spatial directions. The effective action describing an E$n$-brane is the $n$-dimensional Euclidean super-Yang-Mills theory obtained by dimensionally reducing 9+1 dimensional super-Yang-Mills on $9-n$ spatial dimensions and one time dimension. The $IIB^*$ theory has a solution which is the product of 5-dimensional de Sitter space and a 5-hyperboloid, and the E4-brane corresponds to anon-singular complete solution which interpolates between this solution and flat space. This leads to a duality between the large $N$ limit of the Euclidean 4-dimensional U(N) super-Yang-Mills theory and the $IIB^*$ string theory in de Sitter space, and both are invariant under the same de Sitter supergroup. This theory can be twisted to obtain a large $N$ topological gauge theory and its topological string theory dual. Flat space-time may be an unstable vacuum for the type $II^*$ theories, but they have supersymmetric cosmological solutions.

Abelian Chern–Simons theory. I. A topological quantum field theory

Abstract: We give a construction of the Abelian Chern–Simons gauge theory from the point of view of a 2+1-dimensional topological quantum field theory. The definition of the quantum theory relies on geometric quantization ideas that have been previously explored in connection to the non-Abelian Chern–Simons theory [J. Diff. Geom. 33, 787–902 (1991); Topology 32, 509–529 (1993)]. We formulate the topological quantum field theory in terms of the category of extended 2- and 3-manifolds introduced in a preprint by Walker in 1991 and prove that it satisfies the axioms of unitary topological quantum field theories formulated by Atiyah [Publ. Math. Inst. Hautes Etudes Sci. Pans 68, 175–186 (1989)].

Duality in twisted N=4 supersymmetric gauge theories in four dimensions

Abstract: We consider a twisted version of the four-dimensional N=4 supersymmetric Yang-Mills theory with gauge groups SU(2) and SO(3), and bare masses for two of its chiral multiplets, thereby breaking N=4 down to N=2. Using the wall-crossing technique introduced by Moore and Witten within the u-plane approach to twisted topological field theories, we compute the partition function and all the topological correlation functions for the case of simply-connected spin four-manifolds of simple type. By including 't Hooft fluxes, we analyse the properties of the resulting formulae under duality transformations. The partition function transforms in the same way as the one first presented by Vafa and Witten for another twist of the N=4 supersymmetric theory in their strong coupling test of S-duality. Both partition functions coincide on K3. The topological correlation functions turn out to transform covariantly under duality, following a simple pattern which seems to be inherent in a general type of topological quantum field theories.

Hodge Strings and Elements of K. Saito’s Theory of Primitive Form

Abstract: The Hodge strings construction of solutions to associativity equations is proposed. From the topological string theory point of view, this construction formalizes the integration over the position of the marked point procedure for computation of amplitudes. From the mathematical point of view the Hodge strings construction is just a composition of elements of harmonic theory (known among physicists as a t-part of t − t* equations [CV, D2]) and the K. Saito construction of flat coordinates (starting from flat connection with a spectral parameter).

Relationship between the comma theory and Witten's string field theory

Abstract: The comma representation of interacting string field theory is further elucidated. The proof that Witten's vertex solves the comma overlap equations is established. In this representation, the associativity of the star algebra is seen to hold. The relationship of the symmetry K in the standard formulation of Witten's string field theory to that in the comma theory is discussed.

BRST-anti-BRST symmetry and observables for topological gravity

Abstract: We consider topological four-dimensional (4D) gravity with an independent spin connection by using a superspace formalism This gives rise to the basic fields of the quantized theory as well as to two pairs of extra fields, which are needed to close the BRST--anti-BRST algebra off shell Therefore we build a gauge-fixing action written in BRST--anti-BRST exact form leading to an effective one, which allows us to fix all of the symmetries at once In particular, the topological symmetries are fixed as in the model of topological 4D self-dual gravity We construct the observables related to both BRST symmetry and anti-BRST symmetry We find that the anti-BRST invariant observables are not fundamentally different from the BRST invariant ones, since there is a complete mirror symmetry between them The obtained observables extend those constructed within the equivariant method in the context of topological 4D self-dual gravity

Topological Gravity as Large N Topological Gauge Theory

Abstract: We consider topological closed string theories on Calabi-Yau manifolds which compute superpotential terms in the corresponding compactified type II effective action. In particular, near certain singularities we compare the partition function of this topological theory (the Kodaira-Spencer theory) to $SU(\infty)$ Chern-Simons theory on the vanishing 3-cycle. We find agreement between these theories, which we check explicitly for the case of shrinking $S^3$ and Lens spaces, at the perturbative level. Moreover, the gauge theory has non-perturbative contributions which have a natural interpretation in the Type IIB picture. We provide a heuristic explanation for this agreement as well as suggest further equivalences in other topological gravity/gauge systems.

Summary of the Theory of Primitive Forms

Abstract: In this article we review the theory of primitive forms associated to the semi- universal deformation of an isolated critical point of a holomorphic function basically following the original paper [18] as elementary as possible.

Topological term in the string representation of the Wilson loop in the dilute instanton gas approximation

Abstract: A topological term related to the number of self-intersections of the string world-sheet is shown to emerge in the string representation of the Wilson loop in the dilute instanton gas. The coupling constant of this term occurs to be proportional to the topological charge of the instanton gas under consideration.

'Hodge strings' and elements of K. Saito's theory of the primitive form

Abstract: The "Hodge strings" construction of solutions to associativity equations is proposed. From the topological string theory point of view this construction formalizes the "integration over the position of the marked point" procedure for computation of amplitudes. From the mathematical point of view the "Hodge strings" construction is just a composition of elements of harmonic theory (known among physicists as a $t$-part of $t-t^*$ equations) and the K.Saito construction of flat coordinates (starting from flat connection with a spectral parameter). We also show how elements of K.Saito theory of primitive form appear naturally in the "Landau-Ginzburg" version of harmonic theory if we consider the holomorphic pieces of germs of harmonic forms at the singularity.

Space-time Uncertainty Principle from Breakdown of Topological Symmetry

Abstract: Starting from topological quantum field theory, we derive space–time uncertainty relation with respect to the time interval and the spatial length proposed by Yoneya through breakdown of topological symmetry in the large-N matrix model. This work suggests that the topological symmetry might be an underlying higher symmetry behind the space–time uncertainty principle of string theory.

"Hodge strings" and elements of K.Saito's theory of the Primitive form

Abstract: The "Hodge strings" construction of solutions to associativity equations is proposed. From the topological string theory point of view this construction formalizes the "integration over the position of the marked point" procedure for computation of amplitudes. From the mathematical point of view the "Hodge strings" construction is just a composition of elements of harmonic theory (known among physicists as a $t$-part of $t-t^*$ equations) and the K.Saito construction of flat coordinates (starting from flat connection with a spectral parameter). We also show how elements of K.Saito theory of primitive form appear naturally in the "Landau-Ginzburg" version of harmonic theory if we consider the holomorphic pieces of germs of harmonic forms at the singularity.

Non-Perturbative Results in Global SUSY and Topological Field Theories

Abstract: In this lecture we review some non-perturbative results obtained in globally supersymmetric theories and show how they can be obtained in the framework of topological theories.