Andr\'e-Quillen cohomology in the context of curved algebras

Abstract

The Andr\'e-Quillen cohomology of an algebra with coefficients in a module is defined by deriving a functor based on K\"ahler differential forms. It can be computed using a cofibrant resolution of the algebra in a model category structure where weak equivalences are quasi-isomorphisms. This construction works for algebras over an operad, providing a cohomology theory tailored for each type of algebra. For curved algebras however, the notion of quasi-isomorphism is meaningless. The occurrence and importance of curved structures in various research topics (symplectic topology, deformation theory, derived geometry, mathematical physics) motivate the development of their homotopy theory and Andr\'e-Quillen cohomology theory. To get a homotopical context with an appropriate notion of weak equivalence, we consider filtered complete modules with a predifferential inducing a differential on the associated graded. Curved algebras in such modules are algebras over a curved operad. In this article, we consider curved operads which are not necessarily augmented. Bar and cobar constructions adapted to these curved operads are developed, as well as Koszul duality theory. Consequently, we obtain homotopy versions of our curved algebras and make it explicit for interesting cases. Two main examples are the curved operads encoding curved unital associative algebras and curved complex Lie algebras. In particular, homotopy curved unital associative algebras describe the structure of Floer complexes of lagrangian submanifolds and Fukaya categories in symplectic topology. Bar and cobar constructions for curved algebras are also developed, and we obtain resolutions from which we compute their Andr\'e-Quillen cohomology with module coefficients. Our computations in the case of curved complex Lie algebras reveal an interesting link between their Andr\'e-Quillen cohomology and derived complex analytic geometry.