Correlation between Polyakov loops oriented in two different directions in SU(N) gauge theory on a two dimensional torus

We consider SU(N) gauge theories on a two dimensional torus with finite area, $A$. Let $T_μ(A)$ denote the Polyakov loop operator in the $μ$ direction. Starting from the lattice gauge theory on the torus, we derive a formula for the continuum limit of $\langle g_1(T_1(A)) g_2(T_2(A)) \rangle$ as a function of the area of the torus where $g_1$ and $g_2$ are class functions. We show that there exists a class function $ξ_0$ for SU(2) such that $\langle ξ_0(T_1(A)) ξ_0(T_2(A))\rangle > 1$ for all finite area of the torus with the limit being unity as the area of the torus goes to infinity. Only the trivial representation contributes to $ξ_0$ as $A\to\infty$ whereas all representations become equally important as $A\to 0$.

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