Owen Gwilliam

One-dimensional Chern–Simons theory and theÂgenus

We construct a Chern‐Simons gauge theory for dg Lie and L‐infinity algebras on any one-dimensional manifold and quantize this theory using the Batalin‐Vilkovisky formalism and Costello’s renormalization techniques. Koszul duality and derived geometry allow us to encode topological quantum mechanics, a nonlinear sigma model of maps from a 1‐manifold into a cotangent bundle T X , as such a Chern‐ Simons theory. Our main result is that the effective action of this theory is naturally identified with the y A class of X . From the perspective of derived geometry, our quantization constructs a projective volume form on the derived loop space LX that can be identified with the y A class. 57R56; 18G55, 58J20

Enhancing the filtered derived category

Abstract The filtered derived category of an abelian category has played a useful role in subjects including geometric representation theory, mixed Hodge modules, and the theory of motives. We develop a natural generalization using current methods of homotopical algebra, in the formalisms of stable ∞-categories, stable model categories, and pretriangulated, idempotent-complete dg categories. We characterize the filtered stable ∞-category Fil ( C ) of a stable ∞-category C as the left exact localization of sequences in C along the ∞-categorical version of completion (and prove analogous model and dg category statements). We also spell out how these constructions interact with spectral sequences and monoidal structures. As examples of this machinery, we construct a stable model category of filtered D -modules and develop the rudiments of a theory of filtered operads and filtered algebras over operads. This paper is also available at arXiv:1602.01515v3 .