arXiv research

Analyzes string topology operations using Chen's integrals and homotopy transfer.

String topology coproduct and Turaev cobracket computed for surfaces.

Let M be a closed, oriented, n -manifold, and LM its free loop space. Chas and Sullivan defined a commutative algebra structure in the homology of LM, and a Lie algebra structure in its equivariant homology. These structures are known as the string topology loop product and string bracket, respectively. In this paper w…

The data of a "2D field theory with a closed string compactification" is an equivariant chain level action of a cell decomposition of the union of all moduli spaces of punctured Riemann surfaces with each component compactified as a pseudomanifold with boundary. The axioms on the data are contained in the following ass…

The string bracket introduced by Chas and Sullivan [math.GT/9911159] is reinterpreted from the point of view of topological field theories in the Batalin-Vilkovisky or BRST formalisms. Namely, topological action functionals for gauge fields (generalizing Chern-Simons and BF theories) are considered together with genera…

From an operad C with an action of a group G, we construct new operads using the homotopy fixed point and orbit spectra. These new operads are shown to be equivalent when the generalized G-Tate cohomology of C is trivial. Applying this theory to the little disk operad C_2 (which is an S^1 operad) we obtain variations o…

We develop the theory of simplicial extensions for bundle gerbes and their characteristic classes with a view towards studying descent problems and equivariance for bundle gerbes. Equivariant bundle gerbes are important in the study of orbifold sigma models. We consider in detail two examples: the basic bundle gerbe on…

We present an explicit formula for the topology and H-flux of the T-dual of a general type II compactification, significantly generalizing earlier results. Our results apply to T-dualities with respect to any circle action on spacetime. As before, T-duality exchanges type IIA and type IIB string theories. A new consequ…

We refine the intersection product in homology to an equivariant setting, which unifies several known constructions. As an application, we give a common generalisation of the Chas-Sullivan string product on a manifold and the Chataur-Menichi string product on the classifying space by defining a string product on the Bo…

Unified classification of equivariant principal bundles using higher homotopy theory.

We construct equivariant, string and leading order characteristic classes and Chern-Simons classes for certain infinite rank bundles associated to fibrations occurring in loop spaces, Gromov-Witten theory and gauge theory. Results include a restatement of the S^1 index theorem using equivariant classes on the tangent b…

Perturbative string amplitudes are correctly derived from the string geometry theory, which is one of the candidates of a non-perturbative formulation of string theory. In order to derive non-perturbative effects rather easily, we formulate topological string geometry theory. We derive the perturbative partition functi…

By using the equivariant localization formula of toric varieties. We prove the vanishing of the Witten genus of some string complete intersections in smooth toric varieties.

We give a classifying theory for $LG$-bundles, where $LG$ is the loop group of a compact Lie group $G$, and present a calculation for the string class of the universal $LG$-bundle. We show that this class is in fact an equivariant cohomology class and give an equivariant differential form representing it. We then use t…

We establish the family rigidity and vanishing theorems on the equivariant $K$-theory level for the Witten type operators on String$^c$ manifolds introduced by Chen-Han-Zhang.

New K-theory framework reveals exotic brane charges and conformal blocks.

New smooth models for string groups defined in ∞-categories.

Study topological G₂ and Spin(7) strings at 1-loop using double complexes.

Constructs 2-representations and 2-vector bundles for Lie 2-groups.

This paper is an exposition of the new subject of String Topology. We present an introduction to this exciting new area, as well as a survey of some of the latest developments, and our views about future directions of research. We begin with reviewing the seminal paper of Chas and Sullivan, which started String Topolog…

We begin by presenting a symmetric version of the circle equivariant T-duality result in a joint work of the second author with Siye Wu, thereby generalising the results there. We then initiate the study of twisted equivariant Courant algebroids and equivariant generalised geometry and apply it to our context. As befor…

There is an interpretation of open string field theory in algebraic topology. An interpretation of closed string field theory can be deduced from this open string theory to obtain as well the interpretation of open and closed string field theory combined.

We study complex Chern-Simons theory on a Seifert manifold $M_3$ by embedding it into string theory. We show that complex Chern-Simons theory on $M_3$ is equivalent to a topologically twisted supersymmetric theory and its partition function can be naturally regularized by turning on a mass parameter. We find that the d…

Study string topology on symmetric spaces, showing non-triviality and nilpotency results.

Unified theory of orbifolds and cohomology.

Cartan calculus applied to string topology homology.

We prove that the open topological string partition function on a D-brane configuration in a Calabi-Yau manifold X takes the form of a closed topological string partition function on a different Calabi-Yau manifold X_b. This identification shows that the physics of D-branes in an arbitrary background X of topological s…

A De Rham model for string topology based on the theory of iterated integrals is presented.

We construct a space of string diagrams, which are a type of fatgraph with some additional data, and show that there are string topology operations on the chains of the free loop space of a closed Riemannian manifold which are parameterized by the chains on the space of string diagrams. These operations are shown to re…

Motivated by the Moore-Segal axioms for an open-closed topological field theory, we consider planar open string topological field theories. We rigorously define a category 2Thick whose objects and morphisms can be thought of as open strings and diffeomorphism classes of planar open string worldsheets. Just as the categ…

We demonstrate the equivalence of all loop closed topological string amplitudes on toric local Calabi-Yau threefolds with computations of certain knot invariants for Chern-Simons theory. We use this equivalence to compute the topological string amplitudes in certain cases to very high degree and to all genera. In parti…

We study refined topological string theory in the presence of orientifolds by counting second-quantized BPS states in M-theory. This leads us to propose a new integrality condition for both refined and unrefined topological strings when orientifolds are present. We define the SO(2N) refined Chern-Simons theory which co…

In this paper, we discuss Hochschild chain models for some of the string topology operations. We use these models to simplify the proofs and computations of some of the results in string topology. Along the way we also make some new observations. We further discuss how nonnilpotent local level homology classes with res…

The most general gauge-invariant marginal deformation of four-dimensional abelian BF-type topological field theory is studied. It is shown that the deformed quantum field theory is topological and that its observables compute, in addition to the usual linking numbers, smooth intersection indices of immersed surfaces wh…

This paper explains the conjectured algebraic duality between genus zero Gromov-Witten theory and genus zero "Closed String topology". This duality in another perspective is discussed on page 87 of the book "Frobenius manifold, quantum cohomology, and moduli spaces" (by Yuri Manin). This paper also discusses Fulton Mac…

Homflypt skein theory and string topology linked via 2-groupoids.

We make a precision test of a recently proposed conjecture relating Chern-Simons gauge theory to topological string theory on the resolution of the conifold. First, we develop a systematic procedure to extract string amplitudes from vacuum expectation values (vevs) of Wilson loops in Chern-Simons gauge theory, and then…

Proves every equivariant vector bundle over toric manifolds is a Klyachko bundle.

In these lecture notes we discuss a body of work in which Morse theory is used to construct various homology and cohomology operations. In the classical setting of algebraic topology this is done by constructing a moduli space of graph flows, using homotopy theoretic methods to construct a virtual fundamental class, an…

Proves non-solvability of concordance groups using Milnor invariants.

Upper bounds for Lagrangian capacities of Liouville domains

E-string theory reveals modular properties of 4-manifold invariants.

New smooth 2-group extensions from bundle gerbes on manifolds.

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 st…

Study uses equivariant topology to measure distances between G metric spaces.

There are fundamental open problems in the precise global nature of RR-field tadpole cancellation conditions in string theory. Moreover, the non-perturbative lift as M5/MO5-anomaly cancellation in M-theory had been based on indirect plausibility arguments,lacking a microscopic underpinning in M-brane charge quantizatio…

Let $M$ be a closed, oriented manifold of dimension $d$. Let $LM$ be the space of smooth loops in $M$. Chas and Sullivan recently defined a product on the homology $H_*(LM)$ of degree $-d$. They then investigated other structure that this product induces, including a Batalin -Vilkovisky structure, and a Lie algebra str…

Given a smooth closed manifold M with a family {L_i} of closed submanifolds, we consider the free loop space LM and the spaces PM(L_i,L_j) of open strings (paths g:[0,1]->M with g(0) in L_i, and g(1) in L_j). We construct string topology operations resulting in an open-closed TQFT on the family (h_*(LM),h_*(PM(L_i,L_j)…