Abstract
We present a formula for the Hilbert series that counts gauge invariant chiral operators in a large class of 3d ${\cal N} \ge 2$ Yang-Mills-Chern-Simons theories. The formula counts 't Hooft monopole operators dressed by gauge invariants of a residual gauge theory of massless fields in the monopole background. We provide a general formula for the case of abelian theories, where nonperturbative corrections are absent, and consider a few examples of nonabelian theories where nonperturbative corrections are well understood. We also analyze in detail nonabelian ABJ(M) theories as well as worldvolume theories of M2-branes probing Calabi-Yau fourfold and hyperK\"ahler twofold singularities with ${\cal N} = 2$ and ${\cal N} = 3$ supersymmetry.
Highlights
Three-dimensional gauge theories have several features that are absent in their fourdimensional counterparts, such as the non-trivial dynamics of abelian gauge groups, the presence of Chern-Simons couplings and interesting effects of real masses and FayetIliopoulos parameters
We present a formula for the Hilbert series that counts gauge invariant chiral operators in a large class of 3d N ≥ 2 Yang-Mills-Chern-Simons theories
In light of the correspondence between real scalars in vector multiplets and magnetic charges of monopole operators, in order to study the moduli spaces of vacua of gauge theories perturbed by real masses and FI parameters it is natural to consider a generalization of the Hilbert series (2.1) and (2.10) where we include background monopole operators for the global non-R symmetry group
Summary
Three-dimensional gauge theories have several features that are absent in their fourdimensional counterparts, such as the non-trivial dynamics of abelian gauge groups, the presence of Chern-Simons couplings and interesting effects of real masses and FayetIliopoulos parameters. Where we introduced fugacities xi for the U(1)r gauge group in addition to t for the U(1)R symmetry and xi for the flavor symmetry Taking all these facts into account, we conclude that in the absence of a superpotential, the Hilbert series that counts gauge invariant dressed chiral monopole operators of a U(1)r gauge theory takes the general form: r. Our Hilbert series formalism is closely related to the old semiclassical analysis of the moduli space [1, 2, 22] (see the more recent [4]), but with the added benefit of providing a general formula to count gauge invariant chiral operators. We will consider non-generic theories, such as M2-brane theories, that flow to CFT’s with interesting conical moduli spaces parametrized by dressed monopole operators in sections 4, 5 and 6
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