In this talk, I will explain and generalize the following curious observation.
First, two things:
1. The quiver Q with one node and g arrows has invariants which are
conjecturally dimensions of parts of the middle dimensional
cohomology of the twisted character variety of a genus-g surface.
2. There is a famous special-Lagrangian submanifold of six-space
defined by Harvey and Lawson, and used by Aganagic, Klemm and Vafa in the
context of string theory. The open Gromov-Witten invariants of this
submanifold depend on an integer-valued “framing” at infinity.
Their generating function (superpotential) can be written as a
power series in dilogarithms with integer coefficients, the BPS numbers.
If we choose framing g, then the BPS numbers are the quiver invariants of Q.
Note that g does *not* represent any kind of genus in the physics story!?
In general, the invariants of *all* symmetric quivers with n nodes are
controlled by a single moduli space associated to a genus-n surface.
(This is based on joint work with Linhui Shen and David Treumann.)