Topological resolution of gauge theory singularities

Some gauge theories with Coulomb branches exhibit singularities in perturbation theory, which are usually resolved by nonperturbative physics. In string theory this corresponds to the resolution of timelike singularities near the core of orientifold planes by effects from F or M theory. We propose a new mechanism for resolving Coulomb branch singularities in three-dimensional gauge theories, based on Chern-Simons interactions. This is illustrated in a supersymmetric $SU(2)$ Yang-Mills-Chern-Simons theory. We calculate the one-loop corrections to the Coulomb branch of this theory and find a result that interpolates smoothly between the high-energy metric (that would exhibit the singularity) and a regular singularity-free low-energy result. We suggest possible applications to singularity resolution in string theory and speculate a relationship to a similar phenomenon for the orientifold six-plane in massive IIA supergravity.

Figure 1

Plot of the function $G(|\varphi |)$ in the one-loop Coulomb branch metric in YM-CS (namely, $S\supset \int G(|\varphi |){\partial }_{\mu }{\varphi }_3{\partial }^{\mu }{\overline{\varphi }}_3$), for different values of $k$. The four curves correspond to $k=0$, 6, 10, 14 (in red, violet, magenta, and blue, respectively), with $g=1$ in all cases. For $k=0$, the metric vanishes at $\varphi =\frac{g}{2\sqrt{2}\pi }$, while it stays regular everywhere along the Coulomb branch for $k$ large enough that our perturbative computations can be trusted.