The tangent complex and Hochschild cohomology of E_n-rings

Abstract

In this work, we study the deformation theory of $\cE_n$-rings and the $\cE_n$ analogue of the tangent complex, or topological Andr\'e-Quillen cohomology. We prove a generalization of a conjecture of Kontsevich, that there is a fiber sequence $A[n-1] \ra T_A\ra \hh^*_{\cE_{n}}(A)[n]$, relating the $\cE_n$-tangent complex and $\cE_n$-Hochschild cohomology of an $\cE_n$-ring $A$. We give two proofs: The first is direct, reducing the problem to certain stable splittings of configuration spaces of punctured Euclidean spaces; the second is more conceptual, where we identify the sequence as the Lie algebras of a fiber sequence of derived algebraic groups, $B^{n-1}A^\times\ra \Aut_A\ra \Aut_{\fB^nA}$. Here $\fB^nA$ is an enriched $(\oo,n)$-category constructed from $A$, and $\cE_n$-Hochschild cohomology is realized as the infinitesimal automorphisms of $\fB^nA$. These groups are associated to moduli problems in $\cE_{n+1}$-geometry, a {\it less} commutative form of derived algebraic geometry, in the sense of To\"en-Vezzosi and Lurie. Applying techniques of Koszul duality, this sequence consequently attains a nonunital $\cE_{n+1}$-algebra structure; in particular, the shifted tangent complex $T_A[-n]$ is a nonunital $\cE_{n+1}$-algebra. The $\cE_{n+1}$-algebra structure of this sequence extends the previously known $\cE_{n+1}$-algebra structure on $\hh^*_{\cE_{n}}(A)$, given in the higher Deligne conjecture. In order to establish this moduli-theoretic interpretation, we make extensive use of factorization homology, a homology theory for framed $n$-manifolds with coefficients given by $\cE_n$-algebras, constructed as a topological analogue of Beilinson-Drinfeld's chiral homology. We give a separate exposition of this theory, developing the necessary results used in our proofs. This work is an elaboration of a chapter of the author's 2008 PhD thesis, \cite{thez}.