Mathematical Physics

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.

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.

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.

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.

I will explain how to axiomatize the notion of a chiral WZW model using the formalism of VOAs (vertex operator algebras). This class of models is in almost bijective correspondence with pairs (G,k), where G is a connected (not necessarily simply connected) Lie group and k in H^4(BG,Z) is a degree four cohomology class subject to a certain positivity condition. To my surprise, I have found a couple extra models which satisfy all the defining properties of chiral WZW models, but which don't come from pairs (G,k) as above. The simplest such model is the simple current extension of the affine VOA E8xE8 at level (2,2) by the group Z/2. Thus from a quantum perspective, the Lie group E8 x E8 behaves as if it had a non-trivial center.

 

It is interesting to note that, unlike Chern–Simons theories whose gauge group can be disconnected, the gauge group of a chiral WZW model is necessarily connected. This raises the question of what is the chiral WZW model associated to a disconnected gauge group? Whatever the answer turns out to be, mathematicians will probably need to enlarge the class of objects that they agree to call ‘chiral conformal field theories’ in order to accommodate these yet to be defined models.

I will present some speculations on the matter.

This is a joint work with A.Kuznetsov and L.Rybnikov.

We study a moduli problem on a nodal curve of arithmetic genus 1, whose solution is an open subscheme in the zastava space for projective line. This moduli space is equipped with a natural Poisson structure, and we compute it in a natural coordinate system. We compare this Poisson structure with the trigonometric Poisson structure on the transversal slices in an affine flag variety.

We conjecture that certain generalized minors give rise to a cluster structure on the trigonometric zastava.

We will discuss a (conjectural) explicit presentation for the equivariant cohomology of Nakajima quiver varieties of type ADE.  This presentation arises as a shadow of the expected symplectic duality between slices to Schubert varieties in the affine Grassmannian and Nakajima quiver varieties (a.k.a. the expected Coulomb and Higgs branches for a quiver gauge theory). 

 

Categorical symplectic geometry studies an invariant of symplectic manifolds called the "Fukaya (A-infinity) category", which consists of the Lagrangian submanifolds and a symplectically-robust intersection theory of these Lagrangians.  Over the last two decades the Fukaya category has emerged as a powerful tool: for instance, it has produced inroads to Arnol'd's Nearby Lagrangians Conjecture, and it allowed Kontsevich to formulate the the Homological Mirror Symmetry conjecture.

In this talk I will describe a project, joint with Satyan Devadoss, Stefan Forcey, and Katrin Wehrheim, which attempts to relate the Fukaya categories of different symplectic manifolds via a notion of functoriality.  After mentioning some analytic results about the singular quilts necessary for this construction, I will describe the combinatorial component: with Devadoss and Forcey we are constructing a family of polytopes that specialize to the associahedra in two different ways, and can be thought of as the 2-categorical version of associahedra.