The tangent complex and Hochschild cohomology of εn-rings

In this work, we study the deformation theory of epsilon(n)-rings and the epsilon(n) analogue of the tangent complex, or topological Andre-Quillen cohomology. We prove a generalization of a conjecture of Kontsevich, that there is a fiber sequence A[n - 1] -> T-A -> HH*epsilon(n)(A)[n], relating the epsilon(n)-tangent complex and epsilon(n)-Hochschild cohomology of an epsilon(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, Bn-1 A(x) -> Aut(A) -> Aut(BnA). Here B(n)A is an enriched (infinity, n)-category constructed from A, and epsilon(n)-Hochschild cohomology is realized as the infinitesimal automorphisms of B(n)A. These groups are associated to moduli problems in epsilon(n+1)-geometry, a less commutative form of derived algebraic geometry, in the sense of the work of Toen and Vezzosi and the work of Lurie. Applying techniques of Koszul duality, this sequence consequently attains a nonunital epsilon(n+1)-algebra structure; in particular, the shifted tangent complex T-A[-n] is a nonunital epsilon(n+1)-algebra. The epsilon(n+1)-algebra structure of this sequence extends the previously known epsilon(n+1)-algebra structure on HH*epsilon(n)(A), given in the higher Deligne conjecture. In order to establish this modulitheoretic interpretation, we make extensive use of factorization homology, a homology theory for framed n-manifolds with coefficients given by epsilon(n)-algebras, constructed as a topological analogue of Beilinson and Drinfeld's chiral homology. We give a separate exposition of this theory, developing the necessary results used in our proofs.