Abstract
We study the Coulomb branches of 3d mathcal{N}=4 “star-shaped” quiver gauge theories and their deformation quantizations, by applying algebraic techniques that have been developed in the mathematics and physics literature over the last few years. The algebraic techniques supply an abelianization map, which embeds the Coulomb-branch chiral ring into a vastly simpler abelian algebra mathcal{A} . Relations among chiral-ring operators, and their deformation quantization, are canonically induced from the embedding into mathcal{A} . In the case of star-shaped quivers — whose Coulomb branches are related to Higgs branches of 4d mathcal{N}=2 theories of Class mathcal{S} — this allows us to systematically verify known relations, to generalize them, and to quantize them. In the quantized setting, we find several new families of relations.
Highlights
The Coulomb branches of 3d N = 4 gauge theories have long been an object of physical and mathematical interest
We study the Coulomb branches of 3d N = 4 “star-shaped” quiver gauge theories and their deformation quantizations, by applying algebraic techniques that have been developed in the mathematics and physics literature over the last few years
Physical studies [1, 2] led to the discovery of 3d mirror symmetry [3,4,5], and related the Coulomb branch of ADE quiver gauge theories to moduli spaces of monopoles and instantons [6, 7]
Summary
The 4d theories TN [Σ0,k] — and in particular the “trinion” theory at k = 3, which was called TN in [19] — are principal building blocks in the gluing construction of Class S theories Their Higgs branches M4Hd were conjecturally used to define a “2d TQFT valued in holomorphic symplectic varieties” in [20], fully constructed by [28] and [29]. The eigenvalues of the generators, which are complicated algebraic functions on the actual moduli space MC ≈ M4Hd, turn out to be extremely simple monomials in the algebra A This allows the entire diagonalization procedure to be deformation-quantized. From the perspective of 4d Higgs branches, the fact that the chiral ring C[M4Hd] admits a deformation quantization may not be obvious This extra structure is completely natural (and physical) in 3d Coulomb branches.
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