On some symplectic aspects of moduli stack of Chen connections

Mahmoud Zeinalian, LIU and CUNY
Fine Hall 322

The study of the Poisson geometry of the Teichmuller space and the moduli space of local systems gave rise to the discovery of the Goldman bracket of curves on an oriented surface which in turn led Chas and Sullivan to discover string topology operations on chains on the free loop space of an arbitrary oriented manifold. Their string topology operations also generalized the Turaev cobracket which did not come from a Poisson geometric origin, and the search for the geometric meaning of all string topology operations continues. In this direction, I will discuss some Poisson geometric aspects of the moduli stack of Z-graded Chen connections and how in the large N-limit an additional relevant structure appears (N=dimension of the fibre). Unlike the Z-graded case, the somewhat conceptually different Z/2 graded case, studied several years ago with Hossein Abbaspour, did not require the use of derived geometry. The simple reason is that there are very few maps (e.g. traces) from a Z-graded object to a ground ring concentrated in degree zero, whereas in the Z/2 graded setting, viable maps exist. In the derived setting the single ground ring is replaced by the class of all non-positively graded differential graded algebras, with the differential going up towards the origin, as the test objects (i.e. a deformation functor). I plan to review the necessary background material before discussing recent work. This is part of a joint work in progress with Gregory Ginot and Owen Gwilliam.