String theories as the adiabatic limit of Yang-Mills theory

Abstract

We consider Yang-Mills theory with a matrix gauge group $G$ on a direct product manifold $M={\Sigma }_2\times H^2$, where ${\Sigma }_2$ is a two-dimensional Lorentzian manifold and $H^2$ is a two-dimensional open disc with the boundary $S^1=\partial H^2$. The Euler-Lagrange equations for the metric on ${\Sigma }_2$ yield constraint equations for the Yang-Mills energy-momentum tensor. We show that in the adiabatic limit, when the metric on $H^2$ is scaled down, the Yang-Mills equations plus constraints on the energy-momentum tensor become the equations describing strings with a worldsheet ${\Sigma }_2$ moving in the based loop group $\Omega G=C^{\infty }(S^1,G)/G$, where $S^1$ is the boundary of $H^2$. By choosing $G=R^{d-1,1}$ and putting to zero all parameters in $\Omega R^{d-1,1}$ besides $R^{d-1,1}$, we get a string moving in $R^{d-1,1}$. In arXiv:1506.02175 it was described how one can obtain the Green-Schwarz superstring action from Yang-Mills theory on ${\Sigma }_2\times H^2$ while $H^2$ shrinks to a point. Here we also consider Yang-Mills theory on a three-dimensional manifold ${\Sigma }_2\times S^1$ and show that in the limit when the radius of $S^1$ tends to zero, the Yang-Mills action functional supplemented by a Wess-Zumino-type term becomes the Green-Schwarz superstring action.

11 pages, v3: clarifying remarks added, new section on embedding of the Green-Schwarz superstring into d=3 Yang-Mills theory included

References in corpus (2)

• Magnetic Monopole Dynamics, Supersymmetry, and Duality
• Green-Schwarz superstring as subsector of Yang-Mills theory

Cited by in corpus (4)

• Green-Schwarz superstring as subsector of Yang-Mills theory
• Supermembrane limit of Yang-Mills theory
• Superstring limit of Yang-Mills theories
• Superstring theories as low-energy limit of supergroup gauge theories