The Batalin–Vilkovisky Algebra in the String Topology of Classifying Spaces

Kuribayashi, Katsuhiko ; Menichi, Luc

Canadian journal of mathematics, Tome 71 (2019) no. 4, pp. 843-889

Résumé

For almost any compact connected Lie group $G$ and any field $\mathbb{F}_{p}$, we compute the Batalin–Vilkovisky algebra $H^{\star +\text{dim}\,G}(\text{LBG};\mathbb{F}_{p})$ on the loop cohomology of the classifying space introduced by Chataur and the second author. In particular, if $p$ is odd or $p=0$, this Batalin–Vilkovisky algebra is isomorphic to the Hochschild cohomology $HH^{\star }(H_{\star }(G),H_{\star }(G))$. Over $\mathbb{F}_{2}$ , such an isomorphism of Batalin–Vilkovisky algebras does not hold when $G=\text{SO}(3)$ or $G=G_{2}$. Our elaborate considerations on the signs in string topology of the classifying spaces give rise to a general theorem on graded homological conformal field theory.

Détail

DOI : 10.4153/CJM-2018-021-9

Mots-clés : string topology, Batalin–Vilkovisky algebra, classifying space