The spectral gluing theorem revisited

We strengthen the gluing theorem occurring on the spectral side of the geometric Langlands conjecture. While the latter embeds $IndCoh_N(LS_G)$ into a category glued out of 'Fourier coefficients' parametrized by standard parabolics, our refinement explicitly identifies the essential image of such embedding

Lifting vector bundles to Witt vector bundles

Let $p$ be a prime, and let $S$ be a scheme of characteristic $p$. Let $n \geq 2$ be an integer. Denote by $\mathbf{W}_n(S)$ the scheme of Witt vectors of length $n$, built out of $S$. The main objective of this paper concerns the question of extending (=lifting) vector bundles on $S$ to vector bundles on $\mathbf{W}_n(S)$. After introducing the formalism of Witt-Frobenius Modules and Witt vector bundles, we study two significant particular cases, for which the answer is positive: that of line bundles, and that of the tautological vector bundle of a projective space. We give several applications of our point of view to classical questions in deformation theory---see the Introduction for details. In particular, we show that the tautological vector bundle of the Grassmannian $Gr_{\mathbb{F}_p}(m,n)$ does not extend to $\mathbf{W}_2(Gr_{\mathbb{F}_p}(m,n))$, if $2 \leq m \leq n-2$. In the Appendix, we give algebraic details on our (new) approach to Witt vectors, using polynomial laws and divided powers. It is, we believe, very convenient to tackle lifting questions.Comment: Enriched version, with an appendi

Loop Spaces and Connections

We examine the geometry of loop spaces in derived algebraic geometry and extend in several directions the well known connection between rotation of loops and the de Rham differential. Our main result, a categorification of the geometric description of cyclic homology, relates S^1-equivariant quasicoherent sheaves on the loop space of a smooth scheme or geometric stack X in characteristic zero with sheaves on X with flat connection, or equivalently D_X-modules. By deducing the Hodge filtration on de Rham modules from the formality of cochains on the circle, we are able to recover D_X-modules precisely rather than a periodic version. More generally, we consider the rotated Hopf fibration Omega S^3 --> Omega S^2 --> S^1, and relate Omega S^2-equivariant sheaves on the loop space with sheaves on X with arbitrary connection, with curvature given by their Omega S^3-equivariance.Comment: Revised versio