A String Theory for Two-Dimensional Yang-Mills Theory II

In earlier work we proposed a string theory dual to two dimensional Yang-Mills theory at zero coupling (which can also be thought of as a $BF$ theory), given by a Polyakov-like generalization of Ho\v rava's topological rigid string theory, and we showed that it correctly reproduces (in the $1/N$ expansion) several partition functions of $SU(N)$ Yang-Mills theory. In the present paper, we generalise this to Wilson loop expectation values by adding boundaries with one Dirichlet and one Neumann boundary condition to our string worldsheets. We discuss in detail several examples, including examples where the worldsheet has branch points or orientation-reversing tubes, or where the Wilson loop has one or more self-intersections, and we show that in all of them the string theory reproduces the known Yang-Mills expectation values. We argue that examples with orientation-reversing tubes or self-intersecting Wilson loops cannot be brought to the conformal gauge, so we analyse them in a different gauge.