On the worldsheet theories of strings dual to free large N gauge theories

TL;DR

This work analyzes the worldsheet theories dual to free large $N$ gauge theories using Gopakumar's Strebel-differential prescription to map space-time Feynman diagrams into worldsheet correlators. It provides explicit constructions for sphere and torus diagrams, derives edge-length–to–Schwinger-parameter dictionaries, and computes worldsheet OPEs, illustrating both consistency with local worldsheet CFT expectations and striking localization of correlators to subspaces of moduli space. The study reveals that, even for conformal space-times, the mapping preserves Poincaré and scaling symmetries but can break special conformal invariance on the worldsheet, and it documents cases where the worldsheet theory must be nonlocal or supplemented by global contributions. Overall, the results support the feasibility of a Gopakumar-type worldsheet dual for free gauge theories while highlighting puzzling features—such as moduli-space localization and symmetry realizations—that invite further refinement of the dual construction and interpretation.

Abstract

We analyze in detail some properties of the worldsheet of the closed string theories suggested by Gopakumar to be dual to free large N SU(N) gauge theories (with adjoint matter fields). We use Gopakumar's prescription to translate the computation of space-time correlation functions to worldsheet correlation functions for several classes of Feynman diagrams, by explicit computations of Strebel differentials. We compute the worldsheet operator product expansion in several cases and find that it is consistent with general worldsheet conformal field theory expectations. A peculiar property of the construction is that in several cases the resulting worldsheet correlation functions are non-vanishing only on a sub-space of the moduli space (say, for specific relations between vertex positions). Another strange property we find is that for a conformally invariant space-time theory, the mapping to the worldsheet does not preserve the special conformal symmetries, so that the full conformal group is not realized as a global symmetry on the worldsheet (even though it is, by construction, a symmetry of all integrated correlation functions).

Figures (16)

• Figure 1: The horizontal curves for a four-punctured sphere. Each domain is conformally a disk. The critical graph is easily seen. This figure was produced by a numerical analysis which is described in Moeller:2004yy.
• Figure 2: (borrowed from Gopakumars) A characteristic ring domain in the vicinity of a double pole (marked with a dot). The non-closed horizontal trajectories are shown by thick lines. These begin and end at zeros marked by a cross.
• Figure 3: (borrowed from Gopakumars) A specific (planar) 6-point diagram is reduced to a skeleton graph by collapsing color loops which are homotopic to a line.
• Figure 4: The three possibilities for the topology of the critical curves of the Strebel differential : (a)$\Delta>0$, (b)$\Delta=0$, (c)$\Delta<0$.
• Figure 5: The critical graphs in the different regions in the decorated moduli space. We exhibit a slice with constant $p_\infty$. On the lines $p_0+p_1=p_\infty$ and $p_0=p_1\pm p_\infty$ we have the interpolating degenerate diagram, such as Figure \ref{['regs3Point']}(b).