Legendrian skein algebras and Hall algebras

I’ve posted a new preprint to the ArXiv titled “Legendrian skein algebras and Hall algebras”. From the abstract:

We compare two associative algebras which encode the “quantum topology” of Legendrian curves in contact threefolds of product type S×R. The first is the skein algebra of graded Legendrian links and the second is the Hall algebra of the Fukaya category of S. We construct a natural homomorphism from the former to the latter, which we show is an isomorphism if S is a disk with marked points and injective if S is the annulus.

The heart of the paper is a statement about the relation between Maurer-Cartan elements and smoothing of self-intersection points of Legendrian curves, which is actually something I have been thinking about on and off since my days as a grad student and which found a satisfying application here.

 

New preprint

I’ve posted a new preprint to the arXiv:

An extension of the Siegel space of complex abelian varieties and
conjectures on stability structures

Abstract: We study semi-algebraic domains associated with symplectic tori and
conjecturally identified with spaces of stability conditions on the Fukaya
categories of these tori. Our motivation is to test which results from the
theory of flat surfaces could hold for more general spaces of stability
conditions. The main results concern systolic bounds and volume of the moduli
space.