New Lagrangian Structures from String Theory – NeLaSt2

The aim of this project is to define and study new lagrangian objects, in the framework of derived symplectic geometry, and deduce results in traditional representation theory or string theory via topological field theories. We split this project in two axe which can be treated independently, but are motivated by common questions arising from physics.
The first axis has several tasks particularly suitable for a postdoctoral researcher, while the second will be carried out in close collaboration with Damien Calaque and Julien Grivaux.

Symplectic geometry is a natural setting for the hamiltonian formulation of classical mechanics, as most phase spaces appear to be symplectic.
The cotangent bundle of a manifold is an example of a symplectic manifold, and actually, symplectic manifolds do not have local invariants: it follows from a theorem of Darboux that every symplectic manifold is locally the cotangent to R^n.
Lagrangian submanifolds play a crucial role in symplectic geometry. Generalizing Darboux’s theorem, Weinstein proved that in the neighborhood of a lagrangian submanifold L, every symplectic manifold is a neighborhood of the zero section of T*L. Thus lagrangian submanifolds can naturally be interpreted as generalized configurations of a classical mechanical system. These submanifolds pop up everywhere: graphs of closed 1-forms, graphs of symplectomorphisms (an example of which is the time t=1 flow of a hamiltonian vector field), conormal bundles, zero loci of moment maps, etc. The need to deal with singular spaces becomes clear, which is the purpose of derived geometry.
A leitmotiv of derived geometry is to replace geometric perturbations with homological perturbations in order to compute fiber products. Homological perturbations can be made functorial (in a higher categorical sense), and make sense in the algebro-geometric context, resolving pathological behaviours of many moduli spaces appearing in classical physics (spaces of solutions of equations of motion - as opposed to quantum physics).

The first Axis of this project consists in studying new cohomological Hall algebras (COHAs) built on lagrangian subvarieties constructed in the framework of derived geometry by Bozec, Calaque and Scherotzke. These COHAs form the quintessence of geometric representation theory, consisting in studying quantum groups by geometric means. It led to the definition of algebraic structures on cohomology groups of lagrangian subvarieties, yielding the recent resolution of crucial conjectures.

The second axis aims at solving a conjecture of Moore and Tachikawa regarding topological field theories (TFTs). The TFT they’re aiming at is a functor from the category of oriented cobordisms X of dimension 2 to a category of symplectic holomorphic hamiltonian varieties. In physics, classical field theories typically associate to a manifold a space of fields, where the precise notion of space required might be some derived stack. In many examples the space of fields is a mapping space Map(X,T) for some fixed target T, endowed with a symplectic structure which plays an important role in quantizing the classical theory. This involves the so-called AKSZ construction which has been promoted to the context of derived geometry, where constructions require a small number of assumption.