Abstract
We consider gauge theories of non-Abelian $finite$ groups, and discuss the 1+1 dimensional lattice gauge theory of the permutation group $S_N$ as an illustrative example. The partition function at finite $N$ can be written explicitly in a compact form using properties of $S_N$ conjugacy classes. A natural large-$N$ limit exists with a new 't Hooft coupling, $\lambda=g^2 \log N$. We identify a Gross-Witten-Wadia-like phase transition at infinite $N$, at $\lambda=2$. It is first order. An analogue of the string tension can be computed from the Wilson loop expectation value, and it jumps from zero to a finite value. We view this as a type of large-$N$ (de-)confinement transition. Our holographic motivations for considering such theories are briefly discussed.
Highlights
AND CONCLUSIONIn this paper, we will discuss lattice gauge theory of the finite group SN with gauge coupling g in two dimensions, and consider its large-N limit
Even though the string tension is not a directly meaningful quantity to define in these theories, we will find that there is a closely related quantity one can define via the Wilson loop expectation value
This “string tension” undergoes a jump from zero to a finite value at the transition, suggesting that it may be useful to view this as an analog of a large-Nconfinement transition
Summary
We will discuss lattice gauge theory of the finite group SN with gauge coupling g in two dimensions, and consider its large-N limit. One of our motivations for considering theories with finite gauge groups is the possibility that they may provide a simple setting for the introduction of boundary gauge invariance in discussions of holographic tensor networks and quantum error correcting codes. Our present interest in these theories stems from the fact that in the SN example we consider, (a) the finite-N partition function is explicitly calculable and can be written in a simple enough form, (b) the analog of a large-N first order (de)confinement transition is present, and (c) the ’t Hooft coupling has an interesting new scaling. Various nice properties of this group will enable us to obtain an exact, analytical expression for the partition function which will prove to be very convenient for our investigations on phase transition
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