Abstract. Keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with Verdier’s notion of quotient of a triangulated category modulo a triangulated subcategory. This work is an attempt to further develop his theory.

In this review we study BPS D-branes on Calabi–Yau threefolds. Such D-branes naturally divide into two sets called A-branes and B-branes which are most easily understood from topological field theory. The main aim of this paper is to provide a self-contained guide to the derived category approach to B-branes and the idea of Π-stability. We argue that this mathematical machinery is hard to avoid for a proper understanding of B-branes. A-branes and B-branes are related in a very complicated and interesting way which ties in with the “homological mirror symmetry ” conjecture of Kontsevich. We motivate and exploit this form of mirror symmetry. The examples of the quintic 3-fold, flops and orbifolds are discussed at some length. In the latter

This is a short nontechnical note summarizing the motivation and results of my recent work on D-brane categories. I also give a brief outline of how this framework can be applied to study the dynamics of topological D-branes and why this has a bearing on the homological mirror symmetry conjecture. This note can be read without any

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We construct an A-infinity structure on the tensor product of two A-infinity algebras by using the simplicial decomposition of the Stasheff polytope. The key point is the construction of an operad AA-infinity based on the simplicial Stasheff polytope. The operad AA-infinity admits a coassociative diagonal and the operad A-infinity is a retract by deformation of it. We compare these constructions with analogous constructions due to Saneblidze-Umble and Markl-Shnider based on the Boardman-Vogt cubical decomposition of the Stasheff polytope.

We discuss general properties of A∞-algebras and their applications to the theory of open strings. The properties of cyclicity for A∞-algebras are examined in detail. We prove the decomposition theorem, which is a stronger version of the minimal model theorem, for A∞algebras and cyclic A∞-algebras and discuss various consequences of it. In particular it is applied to classical open string field theories and it is shown that all classical open string field theories on a fixed conformal background are cyclic A∞-isomorphic to each other. The same results hold for classical closed string field theories, whose algebraic structure is governed by cyclic L∞-algebras. Contents 1 Introduction and Summary 2 1.1 A∞-space and A∞-algebras.............................. 2 1.2 A∞-structure and classical open string field theory................. 6 1.3 Dual description; formal noncommutative supermanifold.............. 13

We study deformations of Landau-Ginzburg D-branes corresponding to obstructed rational curves on Calabi-Yau threefolds. We determine D-brane moduli spaces and D-brane superpotentials by evaluating higher products up to homotopy in the Landau-Ginzburg orbifold category. For concreteness we work out the details for lines on a perturbed Fermat quintic. In this case we show that our results reproduce the local analytic structure of the Hilbert scheme of curves on the threefold.

Abstract. We consider exact Lagrangian submanifolds in cotangent bundles. Under certain additional restrictions (triviality of the fundamental group of the cotangent bundle, and of the Maslov class and second Stiefel-Whitney class of the Lagrangian submanifold) we prove such submanifolds are Floer-cohomologically indistinguishable from the zero-section. This implies strong restrictions on their topology. An essentially equivalent result was recently proved independently by Nadler [16], using a different approach. 1.

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. Let A be a connected graded algebra and let E denote its Extalgebra i Exti A (kA, kA). There is a natural A∞-structure on E, and we prove that this structure is mainly determined by the relations of A. In particular, the coefficients of the A∞-products mn restricted to the tensor powers of Ext1 A (kA, kA) give the coefficients of the relations of A. We also relate the mn’s to Massey products.