ABSTRACT:

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.

Go figure.

(This is based on joint work with Linhui Shen and David Treumann.)