Batalin-Vilkovisky algebra structures on Hochschild cohomology

Luc Menichi

Bulletin de la Société Mathématique de France (2009)

  • Volume: 137, Issue: 2, page 277-295
  • ISSN: 0037-9484

Abstract

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Let M be any compact simply-connected oriented d -dimensional smooth manifold and let 𝔽 be any field. We show that the Gerstenhaber algebra structure on the Hochschild cohomology on the singular cochains of M , H H * ( S * ( M ) , S * ( M ) ) , extends to a Batalin-Vilkovisky algebra. Such Batalin-Vilkovisky algebra was conjectured to exist and is expected to be isomorphic to the Batalin-Vilkovisky algebra on the free loop space homology on M , H * + d ( L M ) introduced by Chas and Sullivan. We also show that the negative cyclic cohomology H C - * ( S * ( M ) ) has a Lie bracket. Such Lie bracket is expected to coincide with the Chas-Sullivan string bracket on the equivariant homology H * S 1 ( L M ) .

How to cite

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Menichi, Luc. "Batalin-Vilkovisky algebra structures on Hochschild cohomology." Bulletin de la Société Mathématique de France 137.2 (2009): 277-295. <http://eudml.org/doc/272441>.

@article{Menichi2009,
abstract = {Let $M$ be any compact simply-connected oriented $d$-dimensional smooth manifold and let $\mathbb \{F\}$ be any field. We show that the Gerstenhaber algebra structure on the Hochschild cohomology on the singular cochains of $M$, $HH^*(S^*(M),S^*(M))$, extends to a Batalin-Vilkovisky algebra. Such Batalin-Vilkovisky algebra was conjectured to exist and is expected to be isomorphic to the Batalin-Vilkovisky algebra on the free loop space homology on $M$, $H_\{*+d\}(LM)$ introduced by Chas and Sullivan. We also show that the negative cyclic cohomology $HC^*_-(S^*(M))$ has a Lie bracket. Such Lie bracket is expected to coincide with the Chas-Sullivan string bracket on the equivariant homology $H_*^\{S^1\}(LM)$.},
author = {Menichi, Luc},
journal = {Bulletin de la Société Mathématique de France},
keywords = {string topology; Batalin-Vilkovisky algebra; Gerstenhaber algebra; Hochschild cohomology; free loop space},
language = {eng},
number = {2},
pages = {277-295},
publisher = {Société mathématique de France},
title = {Batalin-Vilkovisky algebra structures on Hochschild cohomology},
url = {http://eudml.org/doc/272441},
volume = {137},
year = {2009},
}

TY - JOUR
AU - Menichi, Luc
TI - Batalin-Vilkovisky algebra structures on Hochschild cohomology
JO - Bulletin de la Société Mathématique de France
PY - 2009
PB - Société mathématique de France
VL - 137
IS - 2
SP - 277
EP - 295
AB - Let $M$ be any compact simply-connected oriented $d$-dimensional smooth manifold and let $\mathbb {F}$ be any field. We show that the Gerstenhaber algebra structure on the Hochschild cohomology on the singular cochains of $M$, $HH^*(S^*(M),S^*(M))$, extends to a Batalin-Vilkovisky algebra. Such Batalin-Vilkovisky algebra was conjectured to exist and is expected to be isomorphic to the Batalin-Vilkovisky algebra on the free loop space homology on $M$, $H_{*+d}(LM)$ introduced by Chas and Sullivan. We also show that the negative cyclic cohomology $HC^*_-(S^*(M))$ has a Lie bracket. Such Lie bracket is expected to coincide with the Chas-Sullivan string bracket on the equivariant homology $H_*^{S^1}(LM)$.
LA - eng
KW - string topology; Batalin-Vilkovisky algebra; Gerstenhaber algebra; Hochschild cohomology; free loop space
UR - http://eudml.org/doc/272441
ER -

References

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