http://cds.cern.ch/record/994225/files/0610327.pdf

Four-dimensional string compactifications with D-branes, orientifolds and fluxes

Double field theory (DFT) is a proposal to incorporate T-duality, a distinctive symmetry of string theory, as a symmetry of a field theory defined on a double configuration space. The aim of this review is to provide a pedagogical presentation of DFT and its applications. We first introduce some basic ideas on T-duality and supergravity in order to proceed to the construction of generalized diffeomorphisms and an invariant action on the double space. Steps towards the construction of a geometry on the double space are discussed. We then address generalized Scherk–Schwarz compactifications of DFT and their connection to gauged supergravity and flux compactifications. We also discuss U-duality extensions and present a brief parcours on worldsheet approaches to DFT. Finally, we provide a summary of other developments and applications that are not discussed in detail in the review.

https://iopscience.iop.org/article/10.1088/0264-9381/30/16/163001/pdf

Double field theory: a pedagogical review

We establish a relation between disk amplitudes involving N_o open and N_c closed strings and disk amplitudes with only N_o+2N_c open strings. This map, which represents a sort of generalized KLT relation on the disk, reveals important structures between open & closed and pure open string disk amplitudes: it relates couplings of brane and bulk string states to pure brane couplings. On the string world-sheet this becomes a non-trivial monodromy problem, which reduces the disk amplitude of N_o open and N_c closed strings to a sum of many color ordered partial subamplitudes of N_o+2N_c open strings. This sum can be further reduced to a sum over (N_o+2N_c-3)! subamplitudes of N=N_o+2N_c open strings only. Hence, the computation of disk amplitudes involving open and closed strings is reduced to computing these subamplitudes in the open string sector. In this sector we find a string theory generalization and proof of the Kleiss-Kuijf and Bern-Carrasco-Johanson relations: All order alpha' identities between open string subamplitudes are derived, which reproduce these field-theory relations in the limit alpha'->0. These identities allow to reduce the number of independent subamplitudes of an open string N-point amplitude to (N-3)!. This number is identical to the dimension of a minimal basis of generalized Gaussian hypergeometric functions describing the full N-point open string amplitude.

Open & Closed vs. Pure Open String Disk Amplitudes

We conjecture a relation between the sl(N) knot homology, recently introduced by Khovanov and Rozansky, and the spectrum of BPS states captured by open topological strings. This conjecture leads to new regularities among the sl(N) knot homology groups and suggests that they can be interpreted directly in topological string theory. We use this approach in various examples to predict the sl(N) knot homology groups for all values of N. We verify that our predictions pass some non-trivial checks.

/pdf/khovanov-rozansky-homology-and-topological-strings-2ex4ul34pg.pdf

Khovanov-Rozansky homology and topological strings

Cohen‐macaulay modules over cohen‐macaulay rings

We study B-type D-branes in linear sigma models with Abelian gauge groups. The most important finding is the grade restriction rule. It classifies representations of the gauge group on the Chan-Paton factor, which can be used to define a family of D-branes over a region of the K\"ahler moduli space that connects special points of different character. As an application, we find a precise, transparent relation between D-branes in various geometric phases as well as free orbifold and Landau-Ginzburg points. The result reproduces and unifies many of the earlier mathematical results on equivalences of D-brane categories, including the McKay correspondence and Orlov's construction.

Phases Of N=2 Theories In 1+1 Dimensions With Boundary

We investigate B-type topological Landau-Ginzburg theory with one variable, with D2brane boundary conditions. The allowed brane configurations are determined in terms of the possible factorizations of the superpotential, and compute the corresponding open string chiral rings. These are characterized by bosonic and fermionic generators that satisfy certain relations. Moreover we show that the disk correlators, being continuous functions of deformation parameters, satisfy the topological sewing constraints, thereby proving consistency of the theory. In addition we show that the open string LG model is, in its content, equivalent to a certain triangulated category introduced by Kontsevich, and thus may be viewed as a concrete physical realization of it.

/pdf/landau-ginzburg-realization-of-open-string-tft-4lb8p6uyix.pdf

Landau-Ginzburg Realization of Open String TFT

We reconsider the issue of localization in open-closed B-twisted Lan- dau-Ginzburg models with arbitrary Calabi-Yau target. Through careful analsysis of zero-mode reduction, we show that the closed model allows for a one-parameter fam- ily of localization pictures, which generalize the standard residue representation. The parameter λ which indexes these pictures measures the area of worldsheets with S 2 topology, with the residue representation obtained in the limit of small area. In the boundary sector, we find a double family of such pictures, depending on parameters λ andwhich measure the area and boundary length of worldsheets with disk topology. We show that setting � = 0 and varying λ interpolates between the localization picture of the B-model with a noncompact target space and a certain residue representation proposed recently. This gives a complete derivation of the boundary residue formula, starting from the explicit construction of the boundary coupling. We also show that the various localization pictures are related by a semigroup of homotopy equivalences.

/pdf/localization-and-traces-in-open-closed-topological-landau-4wfp29yfz7.pdf

Localization and traces in open-closed topological Landau-Ginzburg models

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.

/pdf/superpotentials-a-relations-and-wdvv-equations-for-open-41isunsnww.pdf

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

We give a systematic derivation of the consistency conditions which constrain open-closed disk amplitudes of topological strings. They include the A-infinity 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.

Manfred Herbst

Papers

Phases Of N=2 Theories In 1+1 Dimensions With Boundary

Landau-Ginzburg Realization of Open String TFT

Localization and traces in open-closed topological Landau-Ginzburg models

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

Superpotentials, A-infinity Relations and WDVV Equations for Open Topological Strings