Abstract not found

hep-th/yymmnnn

A two-dimensional topological sigma-model on a generalized Calabi-Yau target space X is defined. The model is constructed in Batalin-Vilkovisky formalism using only a generalized complex structure J and a pure spinor ρ on X. In the present construction the algebra of Q-transformations automatically closes off-shell, the model transparently depends only on J, the algebra of observables and correlation functions for topologically trivial maps in genus zero are easily defined. The extended moduli space appears naturally. The familiar action of the twisted N = 2 CFT can be recovered after a gauge fixing. In the open case, we consider an example of generalized deformation of complex structure by a holomorphic Poisson bivector β and recover holomorphic noncommutative Kontsevich ∗-product. 1

The topological string partition function Z(λ,t, ¯t) = exp(λ 2g−2 Fg(t, ¯t)) is calculated on a compact Calabi-Yau M. The Fg(t, ¯t) fulfill the holomorphic anomaly equations, which imply that Ψ = Z transforms as a wave function on the symplectic space H 3 (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.

Abstract. Stability conditions are a mathematical way to understand Π-stability for D-branes in string theory. Spaces of stability conditions seem to be related to moduli spaces of conformal field theories. This is a survey article describing what is currently known about spaces of stability conditions, and giving some pointers for future research. 1.

This expository paper describes sewing conditions in two-dimensional open/closed topological field theory. We include a description of the G-equivariant case, where G is a finite group. We determine the category of boundary conditions in the case that the closed string algebra is semisimple. In this case we find that sewing constraints – the most primitive form of worldsheet locality – already imply that D-branes are (G-twisted) vector bundles on spacetime. We comment on extensions to cochain-valued theories and various applications. Finally, we give uniform proofs of all relevant sewing theorems using Morse theory. August

In the wake of Donaldson’s pioneering work [6], Picard-Lefschetz theory has been extended from its original context in algebraic geometry to (a very large class of) symplectic manifolds. Informally speaking, one can view the theory as analogous to Kirby calculus: one of its basic insights is that one can give a (non-unique)

Abstract. This is the second of two papers in which we prove that a cell model of the moduli space of curves with marked points and tangent vectors at the marked points acts on the Hochschild co–chains of a Frobenius algebra. We also prove that a there is dg–PROP action of a version of Sullivan Chord diagrams which acts on the normalized Hochschild co-chains of a Frobenius algebra. These actions lift to operadic correlation functions on the co–cycles. In particular, the PROP action gives an action on the homology of a loop space of a compact simply–connected manifold. In this second part, we discretize the operadic and PROPic structures of the first part. We also introduce the notion of operadic correlation functions and use them in conjunction with operadic maps from the cell

“One imagines trying to push the input circles through levels of a harmonic function on the surface. As critical levels are passed the circles are cut and reconnected near the critical points. The Poincaré dual cocycle creates the possibility of positioning the surface inside the target manifold.” Abstract. (from [12]) 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 assumptions. It is assumed the punctures are labeled and divided into nonempty sets of inputs and outputs. The inputs are marked by a tangent direction and the outputs are weighted by nonnegative real numbers adding to unity. It is assumed the gluing of inputs to outputs lands on the pseudomanifold boundary of the cell decomposition and the entire pseudomanifold boundary is decomposed into pieces by all such factorings. It is further assumed that the action is equivariant with respect to the toroidal action of rotating the markings. A main result of compactified string topology is the Theorem 1. (closed strings) Each oriented smooth manifold has a 2D field theory with a closed string compactification on the equivariant chains of its free loop space mod constant loops. The sum over all surface types of the top pseudomanifold chain yields a chain X satisfying the master equation dX + X ∗ X = 0 where ∗ is the sum over all gluings. This structure is well defined up to homotopy*. The genus zero parts yields an infinity Lie bialgebra on the equivariant chains of the free loop space mod constant loops. The higher genus terms provide further elements of algebraic structure * called a “quantum Lie bialgebra ” partially resolving the involutive identity. There is also a compactified discussion and a Theorem 2 for open strings as the first step to a more complete theory. We note a second step for knots. *See the Appendix “Homotopy theory of the master equation ” for more explanation.

Abstract. This paper gives a construction, using heat kernels, of differential forms on the moduli space of metrised ribbon graphs, or equivalently on the moduli space of Riemann surfaces with boundary. The construction depends on a manifold with a bundle of Frobenius algebras, satisfying various conditions. These forms satisfy gluing conditions which mean they form an open topological conformal field theory, i.e. a kind of open string theory. If the integral of these forms converged, it would yield the purely quantum part of the partition function of a Chern-Simons type gauge theory. Yang-Mills theory on