Deformation Quantization of Shifted Poisson Structures

A crucial ingredient in the theory of shifted Poisson structures on general derived Artin stacks is the method of formal localization. Formal localization is interesting in its own right as a new, very power ful tool that will prove useful in order to globalize tricky constructions and results, whose extension from the local case presents obstructions that only vanish formally locally.

After the pioneering work of J. Lurie in [DAG-IX], the possibility of a derived version of analytic geometry drew the attention of several mathematicians. The goal of this talk is to provide an overview of the state of art of derived analytic geometry, addressing both the complex and the non-archimedean setting. After providing a series of motivations for derived analytic geometry, I will survey the main results obtained in my PhD thesis: derived versions of GAGA theorems, the existence of the analytic cotangent complex and an analytic version of Lurie's representability theorem. If time will permit, I will conclude the talk by discussing the possible future directions. Parts of the results I will talk about have been obtained in collaboration with T. Y. Yu.

Let (X,w) be a -1-shifted symplectic derived scheme or stack over C in the sense of Pantev-Toen-Vaquie-Vezzosi with an "orientation" (square root of det L_X). We explain how to construct a perverse sheaf P on the classical truncation X=t_0(X), over a base ring A. The hypercohomology H*(P) is regarded as a categorification of X. Now suppose i : L --> X is a Lagrangian in (X,w) in the sense of PTVV, with a "relative orientation". We outline a programme (work in progress) to construct a natural morphism \mu : A_L[vdim L] --> i^!(P) of constructible complexes on L=t_0(L). If i is proper this is equivalent to a hypercohomology in H^{-vdim L}(P). These natural morphisms / hypercohomology classes \mu satisfy various identities under products, composition of Lagrangian correspondences, etc. This programme will have interesting applications. In particular: (a) Take (X,w) to be the derived moduli stack of coherent sheaves on a Calabi-Yau 3-fold Y, so that the orientation is essentially "orientation data" in the sense of Kontsevich-Soibelman 2008. Then we regard H*(P) as being the Cohomological Hall Algebra of Y (cf Kontsevich and Soibelman 2010 for quivers). Consider i : Exact --> (X,w) x (X,-w) x (X,w) the moduli stack of exact sequences of coherent sheaves on Y, with projections to first, second and third factors. This is a Lagrangian in -1-shifted symplectic. Suppose we have a relative orientation. Then the hypercohomology element \mu associated to Exact should give the COHA multiplication on H*(P), and identities on \mu should imply associativity of multiplication. (b) Let (S,w) be a classical symplectic C-scheme, or complex symplectic manifold, of dimension 2n, and L --> S, M --> S be algebraic / complex Lagrangians (or derived Lagrangians in the PTVV sense), proper over S. Suppose we are given "orientations" on L,M, i.e. square roots of the canonical bundles K_L,K_M. Then the derived intersection X = L x_S M is -1-shifted symplectic and oriented, so we get a perverse sheaf P on X. We regard the shifted hypercohomology H^{*-n}(P) as being a version of the "Lagrangian Floer cohomology" HF*(L,M), and the morphisms L --> M in a "Fukaya category" of (S,w). If L,M,N are oriented Lagrangians in (S,w), then the triple intersection L x_S M x_S N is Lagrangian in the triple product (L x_S M) x (M x_S N) x (N x_S L). The associated hypercohomology element should correspond to the product HF*(L,M) x HF*(M,N) --> HF*(L,N) which is composition of morphisms in the "Fukaya category". Using these techniques we intend to define "Fukaya categories" of algebraic symplectic / complex symplectic manifolds, with many nice properties. Different parts of this programme are joint work with subsets of Lino Amorim, Oren Ben-Bassat, Chris Brav, Vittoria Bussi, Delphine Dupont, Pavel Safronov, and Balazs Szendroi.

The Todd class enters algebraic geometry in two places, in the Hirzebruch-Riemann-Roch formula and in the correction of the HKR isomorphism needed to make the Hochschild cohomology isomorphic to polyvector field cohomology (Kontsevich’s claim, proved by Calaque and van den Bergh). In the case of orbifolds the Riemann-Roch formula is known, but not the analogue of Kontsevich’s result. However, we can try to use the former as a guide towards a conjectural formulation for the latter. The problem with this approach is that in the case of an orbifold it is not obvious what the Todd class actually is. This happens because the Riemann-Roch formula mixes the Todd class with the Chern character and it is difficult to separate one from the other. In my talk I shall discuss what the study of loop groups of orbifolds predicts the correct Todd class to be, and then I shall explain how the orbifold Riemann-Roch formula can be rewritten to make this prediction consistent.

In many situations, geometric objects on a space have some kind of singular support, which refines the usual support. For instance, for smooth X, the singular support of a D-module (or a perverse sheaf) on X is as a conical subset of the cotangent bundle; similarly, for quasi-smooth X, the singular support of a coherent sheaf on X is a conical subset of the cohomologically shifted cotangent bundle. I would like to describe a higher categorical version of this notion. Let X be a smooth variety, and let Z be a closed conical isotropic subset of the cotangent bundle of X. I will define a 2-category associated with Z; its objects may be viewed as `categories over X with singular support in Z'. In particular, if Z is the zero section, we simply consider categories over Z in the usual sense. This talk is based on a joint project with D.Gaitsgory. The project is motivated by the local geometric Langlands correspondence; I plan to sketch the relation with the Langlands correspondence in the talk.

An important result in shifted symplectic geometry is the existence of shifted symplectic forms on mapping spaces with symplectic target and oriented source. I provide several examples of more complicated situations where stacks of maps shifted symplectic structures, or maps between them have Lagrangian structures. These include spaces of framed maps, pushforwards of perfect complexes, and perfect complexes on open varieties.

Let L be an exact Lagrangian submanifold of a cotangent bundle T^* M. If a topological obstruction vanishes, a local system of R-modules on L determines a constructible sheaf of R-modules on M -- this is the Nadler-Zaslow construction. I will discuss a variant of this construction that avoids Floer theory, and that allows R to be a ring spectrum. The talk is based on joint work with Xin Jin.

We give a definition of relative Calabi-Yau structure on a dg functor f: A --> B, discussing a examples coming from algebraic geometry, homotopy theory, and representation theory. When A=0, this returns the usual definition of Calabi-Yau structure on a smooth dg category B. When A itself is endowed with a Calabi-Yau structure and relative Calabi-Yau structure on f is compatible with the absolute structure on A, then we sketch the construction of a shifted symplectic structure on the derived moduli space M_A of pseudo-perfect A-modules, as well as the construction of a Lagrangian structure on the induced map f* : M_B --> M_A of derived moduli. This is joint work with Tobias Dyckerhoff.

Thanks to a result of Arinkin and Cāldāru, the derived self-intersection of a closed smooth subscheme of an ambiant scheme (over a field of characteristic zero) is a formal object if and only if the conormal bundle of the subscheme extends to a locally free sheaf at the first order. In this talk, we will explain a program as well as new results in order to describe these derived self-intersections in the non-formal case.

In this talk we will review various point-of-views on classical Chern-Simons theory and moduli of flat connections. We will explain how derived symplectic geomletry (after Pantev-Toën-Vaquié-Vezzosi) somehow reconciles all of these. If time permits, we will discuss a bit the quantization problem.

In this talk, we'll discuss what it means to be a cohomology theory for topological stacks, using a notion of local symmetric monoidal inversion of objects in families. While the general setup is abstract, it specializes to many cases of interest, including Schwede's global spectra. We will then go on to discuss various examples with particular emphasis on elliptic cohomology. It turns out that TMF sees more objects as dualizable (or even invertible) than one might naively expect.