Superstring limit of Yang–Mills theories

Abstract

It was pointed out by Shifman and Yung that the critical superstring on $X^{10}={\mathbb R}^4\times Y^6$, where $Y^6$ is the resolved conifold, appears as an effective theory for a U(2) Yang-Mills-Higgs system with four fundamental Higgs scalars defined on $\Sigma_2\times{\mathbb R}^2$, where $\Sigma_2$ is a two-dimensional Lorentzian manifold. Their Yang-Mills model supports semilocal vortices on ${\mathbb R}^2\subset\Sigma_2\times{\mathbb R}^2$ with a moduli space $X^{10}$. When the moduli of slowly moving thin vortices depend on the coordinates of $\Sigma_2$, the vortex strings can be identified with critical fundamental strings. We show that similar results can be obtained for the low-energy limit of pure Yang-Mills theory on $\Sigma_2\times T^2_p$, where $T^2_p$ is a two-dimensional torus with a puncture $p$. The solitonic vortices of Shifman and Yung then get replaced by flat connections. Various ten-dimensional superstring target spaces can be obtained as moduli spaces of flat connections on $T^2_p$, depending on the choice of the gauge group. The full Green-Schwarz sigma model requires extending the gauge group to a supergroup and augmenting the action with a topological term.