Counting holomorphic disks using bounding chains

Sun, 10/06/2018 - 12:15


In 2003, Welschinger defined invariants of real symplectic
manifolds of complex dimensions 2 and 3, which are related to counts
of pseudo-holomorphic disks with boundary and interior point
constraints (Solomon, 2006). The problem of extending the definition
to higher dimensions remained open until recently (Georgieva, 2013,
and Solomon-Tukachinsky, 2016-17).

  In the talk I will give some background on the problem, and describe
a generalization of Welschinger's invariants to higher dimensions,
with boundary and interior constraints, a.k.a. open Gromov-Witten
invariants. This generalization is constructed in the language of
$A_\infty$-algebras and bounding chains, where bounding chains play
the role of boundary point constraints. If time permits, we will
describe equations, a version of the open WDVV equations, which the
resulting invariants satisfy. These equations give rise to recursive
formulae that allow the computation of all invariants of
$\mathbb{C}P^n$ for odd $n$.

  This is joint work with Jake Solomon.

  No previous knowledge of any of the objects mentioned above will be assumed.