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Register a new userA Study of U(N) Lattice Gauge Theory in 2-dimensions
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Spenta R. Wadia
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This is an edited version of an unpublished 1979 EFI (U. Chicago) preprint: The U(N) lattice gauge theory in 2-dimensions can be considered as the statistical mechanics of a Coulomb gas on a circle in a constant electric field. The large N limit of this system is discussed and compared with exact answers for finite N. Near the fixed points of the renormalization group and especially in the critical region where one can define a continuum theory, computations in the thermodynamic limit $(N rightarrow infty)$ are in remarkable agreement with those for finite and small N. However, in the intermediate coupling region the thermodynamic computation, unlike the one for finite N, shows a continuous phase transition. This transition seems to be a pathology of the infinite N limit and in this simple model has no bearing on the physical continuum limit.
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We study the U(2) lattice gauge theory in the pure gauge sector using the simplest action, with determinant and fundamental terms, having the naive continuum limit of SU(2)$times$U(1). We determine part of the phase diagram of the model and find a first-order critical line which goes through the U(1) critical point. We show how to deduce both the order parameter of the first-order transition and the U(2) renormalization group flow from the lattice potential in the determinant and fundamental representations. We give evidence that a Monte-Carlo simulation of the model is indeed consistent with the symmetric SU(2)$times$U(1) continuum limit in the weak coupling pertubative regime.
Motivated by applications to soft supersymmetry breaking, we revisit the expansion of the Seiberg-Witten solution around the multi-monopole point on the Coulomb branch of pure $SU(N)$ $mathcal{N}=2$ gauge theory in four dimensions. At this point $N-1$ mutually local magnetic monopoles become massless simultaneously, and in a suitable duality frame the gauge couplings logarithmically run to zero. We explicitly calculate the leading threshold corrections to this logarithmic running from the Seiberg-Witten solution by adapting a method previously introduced by DHoker and Phong. We compare our computation to existing results in the literature; this includes results specific to $SU(2)$ and $SU(3)$ gauge theories, the large-$N$ results of Douglas and Shenker, as well as results obtained by appealing to integrable systems or topological strings. We find broad agreement, while also clarifying some lingering inconsistencies. Finally, we explicitly extend the results of Douglas and Shenker to finite $N$, finding exact agreement with our first calculation.
Recently it was shown that the scaling dimension of the operator $phi^n$ in $lambda(barphiphi)^2$ theory may be computed semiclassically at the Wilson-Fisher fixed point in $d=4-epsilon$, for generic values of $lambda n$, and this was verified to two loop order in perturbation theory at leading and subleading $n$. This result was subsequently generalised to operators of fixed charge $Q$ in $O(N)$ theory and verified up to four loops in perturbation theory at leading and subleading $Q$. More recently, similar semiclassical calculations have been performed for the classically scale-invariant $U(N)times U(N)$ theory in four dimensions, and verified up to two loops, once again at leading and subleading $Q$. Here we extend this verification to four loops. We also consider the corresponding classically scale-invariant theory in three dimensions, similarly verifying the leading and subleading semiclassical results up to four loops in perturbation theory.
We consider general aspects of N=2 gauge theories in three dimensions, including their multiplet structure, anomalies and non-renormalization theorems. For U(1) gauge theories, we discuss the quantum corrections to the moduli space, and their relation to ``mirror symmetries of 3d N=4 theories. Mirror symmetry is given an interpretation in terms of vortices. For SU(N_c) gauge groups with N_f fundamental flavors, we show that, depending on the number of flavors, there are quantum moduli spaces of vacua with various phenomena near the origin.
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.
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