Since the work of Mikhail Kapranov, it is known that the shifted tangent complex T[−1] of a smooth algebraic variety X is endowed with a weak Lie structure, the bracket being given by the Atiyah class. Moreover any complex of quasi-coherent sheaves on X is endowed with a weak Lie action of this tangent Lie algebra. We will generalize this result to (finite enough) derived Artin stacks, without any smoothness assumption. This in particular applies to (finite enough) singular schemes. This work uses tools of both derived algebraic geometry and ∞-category theory.

If M is a symplectic manifold then the space of smooth loops C(S¹,M) inherits of a quasi-symplectic form. We will focus in this work on an algebraic analogue of that result. Kapranov and Vasserot introduced and studied the formal loop space of a scheme X. It is an algebraic version of the space of smooth loops in a differentiable manifold. We generalize their construction to higher dimensional loops. To any scheme X ‒ not necessarily smooth ‒ we associate L^d(X), the space of loops of dimension d. We prove it has a structure of (derived) Tate scheme ‒ ie its tangent is a Tate module: it is infinite dimensional but behaves nicely enough regarding duality. We also define the bubble space B^d(X), a variation of the loop space. We prove that B^d(X) is endowed with a natural symplectic form as soon as X has one.

Tate objects have been studied by many authors. They allow us to deal with infinite dimensional spaces by identifying some more structure. In this article, we set up the theory of Tate objects in stable (∞,1)-categories, while the literature only treats with exact categories. We will prove the main properties expected from Tate objects. This new setting includes several useful examples: Tate objects in the category of spectra for instance, or in the derived category of a derived algebraic object.

We provide a generalization to the higher dimensional case of the construction of the current algebra g((z)), its Kac–Moody extension g̃ and of the classical results relating them to the theory of G-bundles over a curve. For a reductive algebraic group G with Lie algebra g, we define a dg-Lie algebra g_n of n-dimensional currents in g. For any symmetric G-invariant polynomial P on g of degree n+1, we get a higher Kac-Moody algebra g̃_n,P as a central extension of g_n by the base field k. Further, for a smooth, projective variety X of dimension n≥2, we show that g_n acts infinitesimally on the derived moduli space RBun^rig_G(X,x) of G-bundles over X trivialized at the neighborhood of a point x ∈ X. Finally, for a representation φ: G → GL_r we construct an associated determinantal line bundle on RBun^rig_G(X,x) and prove that the action of g_n extends to an action of g̃_n,P on such bundle for P the (n+1)^th Chern character of φ.

We show that the relative algebraic K-theory functor fully determines the absolute cyclic homology over any field k of characteristic 0. More precisely, we prove that the tangent complex of K-theory, in terms of (abelian) deformation problems over k, is cyclic homology. We also show that the Loday-Quillen-Tsygan generalized trace comes as the tangent morphism of the canonical map BGL_∞ → K mapping a vector bundle to its class in K-theory. The proof builds on results of Goodwillie, using Wodzicki’s excision for cyclic homology and formal deformation theory à la Lurie-Pridham.

Let X be a smooth affine variety over a field k of characteristic 0 and T(X) be the Lie algebra of regular vector fields on X. We compute the Lie algebra cohomology of T(X) with coefficients in k. The answer is given in topological terms relative to any embedding of k into complex numbers and is analogous to the classical Gelfand-Fuks computation for smooth vector fields on a C-infinity manifold. Unlike the C-infinity case, our setup is purely algebraic: no topology on T(X) is present. The proof is based on the techniques of factorization algebras, both in algebro-geometric and topological contexts.

In this paper we define and study a generalization of the Belinson-Drinfeld Grassmannian to the case where the curve is replaced by a smooth projective surface X, and the trivialization data are given on loci suitably associated to a nonlinear flag of closed subschemes. In order to do this, we first establish some general formal gluing results for moduli of almost perfect complexes, perfect complexes and torsors. We then construct a simplicial object Fl_X of flags of closed subschemes of a smooth projective surface X, naturally associated to the operation of taking union of flags. We prove that this simplicial object has the 2-Segal property. For an affine complex algebraic group G, we finally define a derived, flag analog Gr_X of the Beilinson-Drinfeld Grassmannian of G-bundles on the surface X, and show that most of the properties of the Beilinson-Drinfeld Grassmannian for curves can be extended to our flag generalization. In particular, we prove a factorization formula, the existence of a canonical flat connection, and define a chiral product on suitable sheaves on Fl_X and on Gr_X. We also sketch the construction of actions of flags analogs of the loop group and of the positive loop group on Gr_X. To fixed ``large'' flags on X, we associate "exotic" derived structures on the classical stack of G-bundles on X. Analogs of the flag Grassmannian for other Perf-local stacks (replacing the stack of G-bundles) are briefly considered, and flag factorization is proved for them, too.

Let X be a (-1)-shifted symplectic derived Deligne‒Mumford stack. In this paper we introduce the Darboux stack of X, parametrizing local presentations of X as a derived critical locus of a function f on a smooth formal scheme U. Local invariants such as the Milnor number μ_f, the perverse sheaf of vanishing cycles P_U,f and the category of matrix factorizations MF(U,f) are naturally defined on the Darboux stack, without ambiguity. The stack of non-degenerate flat quadratic bundles acts on the Darboux stack and our main theorem is the contractibility of the quotient stack when taking a further homotopy quotient identifying isotopic automorphisms. As a corollary we recover the gluing results for vanishing cycles by Brav‒Bussi‒Dupont‒Joyce‒Szendrői. In a second part (to appear), we will apply this general mechanism to glue the motives of the locally defined categories of matrix factorizations MF(U,f) under the prescription of additional orientation data, thus answering positively conjectures by Kontsevich‒Soibelman and Toda in motivic Donaldson‒Thomas theory.

In this paper we prove formal glueing, along an arbitrary closed subscheme Z of a scheme X, for the stack of pseudo-coherent, perfect complexes, and G-bundles on X (for G a smooth affine algebraic group). By iterating this result, we get a decomposition of these stacks along an arbitrary nonlinear flag of subschemes in X. By taking points over the base field, we deduce from this both a formal glueing, and a flag-related decomposition formula for the corresponding derived ∞-categories of pseudo-coherent and perfect complexes. We finish the paper by highlighting some expected progress in the subject matter of this paper, that might be related to a Geometric Langlands program for higher dimensional varieties. In the Appendix we also prove a localization theorem for the stack of pseudo-coherent complexes, which parallels Thomason’s localization results for perfect complexes.