$4D\text{−}2D$ equivalence for large-$N$ Yang-Mills theory

General string-theoretic considerations suggest that four-dimensional large-$N$ gauge theories should have dual descriptions in terms of two-dimensional conformal field theories. However, for nonsupersymmetric confining theories such as pure Yang-Mills theory, a long-standing challenge has been to explicitly show that any such dual descriptions actually exist. In this paper, we consider the large-$N$ limit of four-dimensional pure Yang-Mills theory compactified on a three-sphere in the solvable limit where the sphere radius is small compared to the strong length scale, and demonstrate that the confined-phase spectrum of this gauge theory coincides with the spectrum of an irrational two-dimensional conformal field theory.

Figure 1

A conjectured phase diagram for large-$N$ YM theory on $S^3\times S^1$. In the analytically tractable regime $\mathrm{\Lambda }R\ll 1$, the deconfinement transition occurs at $\beta \sim R$, while for $\mathrm{\Lambda }R\gg 1$, lattice studies have shown that it occurs at $\beta \sim 1/\mathrm{\Lambda }$. This sketch illustrates the natural conjecture that these two limiting cases are smoothly connected. The results of this paper apply in the $\mathrm{\Lambda }R\to 0$ region indicated by the blue line.

Figure 2

The numerical values of Eq. (16) with $|m|$, $|n|\le 10$, plotted within the unit-$q$ disk.