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

  • Katsuhiko Kuribayashi (a1) and Luc Menichi (a2)

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.

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The first author was partially supported by JSPS KAKENHI Grant Number 25287008.

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