The tangent complex and Hochschild cohomology of $\mathcal {E}_n$-rings

By J. Francis. Compositio Mathematica, 2011

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Abstract In this work, we study the deformation theory of ${\mathcal {E}}_n$-rings and the ${\mathcal {E}}_n$ analogue of the tangent complex, or topological André–Quillen cohomology. We prove a generalization of a conjecture of Kontsevich, that there is a fiber sequence $A[n-1] \rightarrow T_A\rightarrow {\mathrm {HH}}^*_{{\mathcal {E}}_{n}}\!(A)[n]$, relating the ${\mathcal {E}}_n$-tangent complex and ${\mathcal {E}}_n$-Hochschild cohomology of an ${\mathcal {E}}_n$-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, $B^{n-1}A^\times \rightarrow {\mathrm {Aut}}_A\rightarrow {\mathrm {Aut}}_{{\mathfrak B}^n\!A}$. Here ${\mathfrak B}^n\!A$ is an enriched $(\infty ,n)$-category constructed from $A$, and ${\mathcal {E}}_n$-Hochschild cohomology is realized as the infinitesimal automorphisms of ${\mathfrak B}^n\!A$. These groups are associated to moduli problems in ${\mathcal {E}}_{n+1}$-geometry, a less commutative form of derived algebraic geometry, in the sense of the work of Toën and Vezzosi and the work of Lurie. Applying techniques of Koszul duality, this sequence consequently attains a nonunital ${\mathcal {E}}_{n+1}$-algebra structure; in particular, the shifted tangent complex $T_A[-n]$ is a nonunital ${\mathcal {E}}_{n+1}$-algebra. The ${\mathcal {E}}_{n+1}$-algebra structure of this sequence extends the previously known ${\mathcal {E}}_{n+1}$-algebra structure on ${\mathrm {HH}}^*_{{\mathcal {E}}_{n}}\!(A)$, 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 ${\mathcal {E}}_n$-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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bff5bc7985beb983e2b8045b6ead1d6bc7c37cc3

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The higher Morita category of n–algebras (2014, background): “introduced by Lurie [ 32, §5.5] and also independently by Andrade [1], and have since been extensively developed by a number of other authors, in particular Francis and collaborators; see for example [13, 5, 4]. An overview can also be found in Ginot’s lecture notes [18].) (iii) An En-algebra A is (”
Derived algebraic geometry (2014, background): “ral structure of a E n+2-algebra, and thus that HH E n+1(X)[n+1] has a natural dg-lie algebra structure, at last when X is a quotient stack of a derived quasi-projective scheme by a linear group (see [Fran] for the special case of ane derived schemes). The conjecture5.8follows from the main result o”