Exact Results in Quantum Theory

I will present the relationship between a wide class of amplitudes in topological string theory and quiver representation theory. For a given string theory model, we find its dual description in terms of effective spacetime theory, or supersymmetric quantum mechanics, whose interactions are succinctly encoded in a quiver graph. This relation, in various more specific situations, leads to very interesting mathematical and physics statements. For example, for topological string configurations related to knots, the quiver point of view enables one to prove some old conjectures about knot invariants and leads to new knot invariants. On the other hand, we show that amplitudes in topological strings on toric Calabi-Yau manifolds without compact four-cycles, which take the form of q-hypergeometric functions, can be also encoded in some quiver. This leads to interesting new statements about hypergeometric functions and their q-analogs. In this talk, I will summarize these relations and discuss prospects for future work.