The tangent complex and Hochschild cohomology of E_n-rings

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}.Comment: May vary slightly from the published versio

Derived Algebraic Geometry

This text is a survey of derived algebraic geometry. It covers a variety of general notions and results from the subject with a view on the recent developments at the interface with deformation quantization.Comment: Final version. To appear in EMS Surveys in Mathematical Science

Deriving Deligne-Mumford Stacks with Perfect Obstruction Theories

We give conditions for a n-connective quasicoherent obstruction theory on a Deligne-Mumford stack to come from the structure of a connective spectral Deligne-Mumford stack on the underlying topos.Comment: 25 pages; v.4: Assumptions added to correct Lemma; Main result now gives necessary and sufficient conditions for an obstruction theory to be induced by a derived structure; Erratum to published version is to appea