Confining ensemble of dyons

We construct the integration measure over the moduli space of an arbitrary number of $N$ kinds of dyons of the pure $SU(N)$ gauge theory at finite temperatures. The ensemble of dyons governed by the measure is mathematically described by a (supersymmetric) quantum field theory that is exactly solvable and is remarkable for a number of striking features: (i) The free energy has the minimum corresponding to the zero average Polyakov line, as expected in the confining phase; (ii) the correlation function of two Polyakov lines exhibits a linear potential between static quarks in any $N$-ality nonzero representation, with a calculable string tension roughly independent of temperature; (iii) the average spatial Wilson loop falls off exponentially with its area and the same string tension; (iv) at a critical temperature, the ensemble of dyons rearranges and deconfines; and (v) the estimated ratio of the critical temperature to the square root of the string tension is in excellent agreement with the lattice data.

Figure 1

A bunch of profile functions $w_{12}$ (left), $w_{23}$ (middle), and $w_{31}$ (right) inside the $SU(3)$ string for five values of the parameter $\gamma$: $\arg (\gamma )=(4,5,6,7,8)\times (\pi /9)$. The solid red curves display imaginary parts, and the dashed blue curves display real parts of $w_{12}$, $w_{23}$, and $w_{31}$, respectively, as functions of the distance $z$ from the Wilson loop plane. The string tension (the action) is identical for all five curves.