Infinity inner products on A-infinity algebras
By Thomas Tadler. 2008
55 citations recorded by the source corpus.
We give a self contained introduction to A∞-algebras, A∞-bimodules and maps between them. The case of A∞-bimodule-map between A and its dual space A∗, which we call ∞-inner-product, will be investigated in detail. In particular, we describe the graph complex associated to ∞-inner-product. In a later paper, we show how ∞-inner-products can be used to model the string topology BV-algebra on the free loop space of a Poincaré duality space. A∞-algebras were first introduced by J. Stasheff in [S1] in the study of the homotopy associativity of H-spaces. Since then, the concept has found numerous applications in many fields of mathematics and physics. Our interest in A∞-algebras stems from applying the concept to the chain level of a Poincaré duality space, and to ultimately obtain a model for string topology operations defined by M. Chas and D. Sullivan in [CS]. To this end, it is necessary to develop an appropriate algebraic notion of Poincaré duality. A detailed introduction to such a concept will be presented in these notes. In order to describe Poincaré duality for a topological space X , note that its cohomology H is an algebra under the cup product, and thus both H and its dual H are bimodules over H . With this, Poincaré duality is given by a bimodule equivalence H(X) → H(X) between the homology and cohomology of X . A suitable chain level version of this may be obtained by considering the bimodule concept in a homotopy invariant way. Our approach for Poincaré duality consists of examining the A∞-algebra structure on the cochains A of X . With this, one can see, that both A and its dual A are in fact A∞-bimodules in an appropriate sense, described below. Finally, we may complete the analogy by taking the chain level Poincaré duality to be a map between the A∞-bimodules A and A , which we call an ∞-inner-product. We will review these concepts and will investigate a useful graph complex that is associated to A∞-algebras with ∞-inner-products. We show how the graph complex gives rise to a sequence of polyhedra, which include as a special case Stasheff’s associahedra coming from A∞-algebras. This paper is the first in a series, setting the foundation for the algebraic notation necessary to describe Poincaré duality at the chain level, and with this, modeling string topology algebraically. The next step in this direction is taken in [TZS] in collaboration with M. Zeinalian, where the following theorem is proved. Theorem ([TZS] Theorem 3.1.4). Let X be a compact triangulated Poincaré duality space, in which the closure of every simplex is contractible. Then there exists a symmetric ∞-inner-product on the cochains A of X, which on the lowest level induces Poincaré duality on homology H → H. It is interesting to note, that the lowest level of this theorem consists of capping with the fundamental cycle of the space, which does not give a bimodule map at the chain level, but which requires the notion of A∞-bimodule maps. 1
Subfields: Algebra and Number Theory, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/eb97a47835241a52f1becc0ed39eeaf56cb234e4
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Citations in the corpus
- Fukaya A∞-structures Associated to Lefschetz Fibrations. II (2014, background): “For more background material concerning A∞bimodules, see [50, 24, 22, 25, 41].”
- Homotopy Algebras Inspired by Classical Open-Closed String Field Theory (2004, background): “It would be worth investigating a strongly homotopy invariant analog in the sense of [55, 56], which we do not discuss in this paper.”
- Fukaya A∞-structures associated to Lefschetz fibrations. I (2009, background): “For generalities about A∞-bimodules, see for instance [37, 30].”
- Cyclic symmetry and adic convergence in Lagrangian Floer theory (2009, background): “We follow [18] in various places (see also [3], [20]).”
- Symplectic homology as Hochschild homology (2006, background): “Now B, being part of the Fukaya category of M, has a weak version of cyclic symmetry [ 20 ], and this implies that Q ∼= A∨[1 − d] canonically.”
- Suspending Lefschetz Fibrations, with an Application to Local Mirror Symmetry (2009, background): “For a more thorough exposition of the theory of A∞-bimodules and their morphisms, see [18, 16] (our sign conventions are related to those in [16] by reversing the ordering of the entries in (2.”