String theories as the adiabatic limit of Yang-Mills theory

We consider Yang-Mills theory with a matrix gauge group G on a direct product manifold M=Σ2×H2, where Σ2 is a two-dimensional Lorentzian manifold and H2 is a two-dimensional open disc with the boundary S1=∂H2. The Euler-Lagrange equations for the metric on Σ2 yield constraint equations for the Yang-Mills energy-momentum tensor. We show that in the adiabatic limit, when the metric on H2 is scaled down, the Yang-Mills equations plus constraints on the energy-momentum tensor become the equations describing strings with a world sheet Σ2 moving in the based loop group ΩG=C∞(S1,G)/G, where S1 is the boundary of H2. By choosing G=Rd−1,1 and putting to zero all parameters in ΩRd−1,1 besides Rd−1,1, we get a string moving in Rd−1,1. In another paper of the author, it was described how one can obtain the Green-Schwarz superstring action from Yang-Mills theory on Σ2×H2 while H2 shrinks to a point. Here we also consider Yang-Mills theory on a three-dimensional manifold Σ2×S1 and show that in the limit when the radius of S1 tends to zero, the Yang-Mills action functional supplemented by a Wess-Zumino-type term becomes the Green-Schwarz superstring action. © 2015 American Physical Society