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The Batalin–Vilkovisky Algebra in the String Topology of Classifying Spaces
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 71 / Issue 4 / August 2019
- Published online by Cambridge University Press:
- 09 January 2019, pp. 843-889
- Print publication:
- August 2019
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- Article
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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.
