The Yang-mills String as the A-model on the Twistor Space of the Complex Two-dimensional Projective Space with Fluxes and Wilson Loops: the Beta Function

We argue that the string theory dual to a certain sector of the four-dimensional Yang-Mills theory at large-N is the A-model wrapping N Lagrangian Dbranes on the twistor space of the complex two-dimensional projective space, with a certain flux and Wilson loop background, by finding that the targetspace beta function of this A-model coincides with the large-N beta function of the Yang-Mills theory. The beta function is obtained by the quantization of the A-model Chern-Simons effective action in target space, provided a certain B-field and Wilson loop background is coupled to the A-model world-sheet sigma-model. The B-field provides the embedding of the four-dimensional permanent address space-time into the two-dimensional base of the Lagrangian-twistorial ChernSimons by the non-commutative large-N Eguchi-Kawai reduction. In fact, in presence of the Wilson loop and B-field background, the large-N loop equation of the twistorial Chern-Simons theory implies the non-commutative vortex equation of (anti-)self-dual type of the pure Yang-Mills theory restricted to a Lagrangian submanifold in space-time, that in turn have been shown to occur, on the Yang-Mills side, in the localization of the loop equation of the full Yang-Mills theory in the (anti-)self-dual variables for a certain diagonal embedding of quasi BPS Wilson loops. Remarkably, the Wilsonean large-N beta function of full Yang-Mills and the large-N A-model target-space beta function coincide.