Remarks on worldsheet theories dual to free large $N$ gauge theories

We continue to investigate properties of the worldsheet conformal field theories (CFTs) which are conjectured to be dual to free large $N$ gauge theories, using the mapping of Feynman diagrams to the worldsheet suggested in [R. Gopakumar, Phys. Rev. D 70, 025009 (2004); R. GopakumarPhys. Rev. D70, 025010 (2004); R. GopakumarC. R. Physique 5, 1111 (2004); R. GopakumarPhys. Rev. D 72, 066008 (2005)]. The modular invariance of these CFTs is shown to be built into the formalism. We show that correlation functions in these CFTs which are localized on subspaces of the moduli space may be interpreted as delta-function distributions, and that this can be consistent with a local worldsheet description given some constraints on the operator product expansion coefficients. We illustrate these features by a detailed analysis of a specific four-point function diagram. To reliably compute this correlator, we use a novel perturbation scheme which involves an expansion in the large dimension of some operators.

Figure 1

The skeleton graph which we refer to as the “Broom diagram.” We depict here what happens if we make the naive choice of branches, in which we do not change branches when we rotate $\eta$. We see that for this choice the rotation in the $\eta$ plane changes the position of one of the edges, and thus changes the correlator we are computing.

Figure 2

The skeleton graph of the Broom diagram. The $x_i$ label space-time points where operators are inserted. Bold lines represent many homotopic contractions. On the other hand, the line connecting $x_2$ and $x_4$ has a small number of contractions.

Figure 3

The Broom diagram in field theory (the straight lines) and its dual graph.

Figure 4

The ${\Pi }_1$ diagram and its dual graph.

Figure 5

The $Y$ diagram and its dual graph.

Figure 6

The ${\Pi }_2$ diagram and its dual graph.

Figure 7

The critical (and the dual) graph for the Broom expanded around the $Y$ diagram.

Figure 8

The skeleton (and the dual graph) of the field theory graph we consider. The line connecting 0 and $\eta$ is taken to have few contractions compared to the bold lines.

Figure 9

Three of the 6 possible blowups of the fourfold degenerate zero, the other 3 are similar. The corresponding (dual) field theory graph contains a self-contraction for the figure on the left. The insertions are at 0, 1, $\eta$, $\infty$, and are exactly in this order also for the faces in the figures.

Figure 10

A typical behavior of the integrand. Depicted here is the integrand as a function of $\mathrm{Im}[\eta ]$ for some given $\mathrm{Re}[\eta ]$. The vertical axis is proportional to the amplitude (denoted by $S$). As $m_{0\eta }\to 0$, the integrand approaches $\delta (\mathrm{Im}[\eta ])$.

Figure 11

The Square diagram.

Figure 12

The Whale diagram, which satisfies ${\gamma }_{\eta }+1={\gamma }_0+{\gamma }_1$, like the Square diagram.

Figure 13

The curve on the complex $\eta$ plane for which the Strebel condition is satisfied. On the left ${\gamma }_0={\gamma }_1=70$ and on the right ${\gamma }_0=40$, ${\gamma }_1=70$. In general, one can see numerically that, as ${\gamma }_0$ is taken to 1 the curve around $\eta =1$ shrinks to zero size, and as ${\gamma }_1$ is taken to 1 the curve around $\eta =0$ shrinks to zero size. This implies that the diagrams corresponding to these curves cover the full region corresponding to the OPE limits $\eta \to 0$, 1.

Figure 14

The (disconnected) critical graphs and the leaves of the two possible topologies. The arrows indicate directions of increasing $C$. On the left we have shape I, and on the right shape II.

Figure 15

On the left we draw the critical graph of the Square diagram for a typical real solution. On the right we have the critical curves when slightly moving to the complex plane. We see how the graph smoothly becomes disconnected, and has topology I.

Figure 16

On the left, we draw the critical graph of a complex solution for $\eta$. It is easy to see that this critical graph is dual to the Whale diagram. In the next two graphs we again slightly move away from the point where the Strebel conditions are satisfied. We keep the same real part of $\eta$ and first move up and then down along the imaginary axis. The critical graph has topology II when lowering $\mathrm{Im}(\eta )$, and topology I when raising $\mathrm{Im}(\eta )$. The last graph is a generic complex $\eta$ not satisfying the Strebel conditions. The critical graph is disconnected and we do not have a cell decomposition.