Rational BV-algebra in string topology. (English) Zbl 1160.55006

Félix, Yves; Thomas, Jean-Claude

Bull. Soc. Math. Fr. 136, No. 2, 311-327 (2008).

For a smooth orientable manifold \(M\) of dimension \(m\), let \(LM\) denote the space of free loops on \(M\), and let \(HH^*(A,R)\) denote the Hochschild cohomology of a differential graded algebra \(A\) with coefficients in the differential graded \(A\)-module \(R\). Recall that Chas-Sullivan showed that the shifted homology \({\mathbb H}_*(LM,{\mathbf k}):=H_{*+m}(LM,{\mathbf k})\) has the structure of a BV-algebra. If \({\mathbf k}\) is a field, it is also known (due to a result of J. Jones) that there is an isomorphism of graded vector spaces \({\mathbb H}_*(LM,{\mathbf k})\cong HH^*(C^*(M);C^*(M))\). In this paper, the authors prove that if \(M\) is \(1\)-connected and \(ch({\mathbf k})=0\), then Poincaré duality induces a \(BV\)-algebra structure on \(HH^*(C^*(M);C^*(M))\) and that there is an isomorphism of BV-algebras, \({\mathbb H}_*(LM,{\mathbf k})\cong HH^*(C^*(M);C^*(M))\). Moreover, they prove that the Chas-Sullivan product and the BV-operator behave well with respect to a Hodge decomposition of \({\mathbb H}_*(LM)\).

Reviewer: Kohhei Yamaguchi (Tokyo)

MSC:

55P35	Loop spaces
55N33	Intersection homology and cohomology in algebraic topology
81T30	String and superstring theories; other extended objects (e.g., branes) in quantum field theory
13D03	(Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)

Keywords:

string homology; rational homotopy; Hochschild cohomology; free loop space homology; BV-algebra; Gerstenhaber algebra