The following are brief notes on a variety of subjects related to open/closed string duality that weren't substantial enough to make it into their own papers.

Relation of c=1 to A model on CP^1 via Hurwitz numbers

In Complex matrix model duality I fleshed out an equality noticed earlier in the literature between certain correlation functions of half-BPS operators in N=4 SYM, encapsulated by a complex matrix model, and amplitudes of tachyons in the non-critical c=1 string compactified at the self-dual radius. I also showed how to express these amplitudes as sums over Hurwitz numbers that count holomorphic maps from the worldsheet to the sphere CP^1 with three branch points. Each bipartite Feynman diagram of the complex matrix model is a dessin d'enfant for a Belyi map.

One natural question to ask is: Is the c=1 string then related to the A model on CP^1?

Another natural question: If the c=1 string is some kind of half-BPS reduction of (free) N=4, can we embed it in some bigger model with full PSU(2,2|4) symmetry that might correspond to the zero radius limit of AdS_5 x S^5? One could for example try to write the c=1, R=1 string with its tachyons in terms of some coset model with PSU(1|1) symmetry and then extend it.

Discrete gauge/string duality from localization

Open-closed-open duality

In a talk in 2010 Open-Closed-Open String Duality at the Second Jo'burg Workshop on String Theory, Gopakumar suggested that there might exist dualities between field theories which correspond to graph dualities of their Feynman diagrams. He called this open-open duality, in contrast to open-closed string duality. He explored an example for Hermitian matrix models. In Complex matrix model duality I explored implications for a matrix model with a single complex matrix and its relations to the c=1, R=1 string.

The reason this is interesting is that it might be easier to see how open-closed duality works from the open-open dual of a field theory.

The following notes extend the complex matrix model duality to a multiple-matrix model and one with spacetime dependence.

Lorentzian open-closed duality

Mathematics problems