## 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] T-A {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× {mathrm {Aut}-A {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.

Original language | English (US) |
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Pages (from-to) | 430-480 |

Number of pages | 51 |

Journal | Compositio Mathematica |

Volume | 149 |

Issue number | 3 |

DOIs | |

State | Published - Mar 1 2013 |

## Keywords

- Hochschild cohomology
- Koszul duality
- algebras
- categories
- deformation theory
- factorization homology
- operads
- the tangent complex
- topological chiral homology

## ASJC Scopus subject areas

- Algebra and Number Theory