The tangent complex and Hochschild cohomology of E_n-rings

In this work, we study the deformation theory of ${\text{cE}}_n$-rings and the ${\text{cE}}_n$ 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 $A[n-1]\text{ra}{T}_A\text{ra}{\text{hh}}_{{\text{cE}}_n}^{\ast }(A)[n]$, relating the ${\text{cE}}_n$-tangent complex and ${\text{cE}}_n$-Hochschild cohomology of an ${\text{cE}}_n$-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, $B^{n-1}A^\times\ra \Aut_A\ra \Aut_{\fB^nA}$\mathrm{.}Here$\fB^nA$isanenriched$(\oo,n)$-category constructed from $A$, and ${\text{cE}}_n$-Hochschild cohomology is realized as the infinitesimal automorphisms of ${\text{fB}}^{n}A$. These groups are associated to moduli problems in ${\text{cE}}_{n+1}$-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 ${\text{cE}}_{n+1}$-algebra structure; in particular, the shifted tangent complex $T_A[-n]$ is a nonunital ${\text{cE}}_{n+1}$-algebra. The ${\text{cE}}_{n+1}$-algebra structure of this sequence extends the previously known ${\text{cE}}_{n+1}$-algebra structure on ${\text{hh}}_{{\text{cE}}_n}^{\ast }(A)$, 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 ${\text{cE}}_n$-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}.