Coulomb branches of star-shaped quivers

Abstract

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. The algebraic techniques supply an abelianization map, which embeds the Coulomb-branch chiral ring into a vastly simpler abelian algebra A. Relations among chiral-ring operators, and their deformation quantization, are canonically induced from the embedding into A. In the case of star-shaped quivers -whose Coulomb branches are related to Higgs branches of 4d N = 2 theories of Class 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. arXiv:1808.05226v2 [hep-th] 7 Dec 2018In the initial work [12], the precise image of the embedding (1.4) was only identified in a handful of examples; however, at least in principle, a complete combinatorial construction of the image has since been described by Webster [18].In the case of T N,k theories, we will identify the putative generators of C[M C ] proposed by [21][22][23][24][25] (from a 4d Higgs-branch perspective) as elements of A. We will show how to explicitly verify and then quantize the conjectured relations among them.An important insight in the derivation of C[M 4d H ] chiral-ring relations in [24] was that various generators could be "diagonalized," as tensors for the SU (N ) k flavor symmetry. We find that the abelian algebra A plays a surprisingly important role in this diagonalization. In particular, the eigenvalues of the generators, which are complicated algebraic functions on the actual moduli space M C ≈ M 4d H , 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[M 4d H ] admits a deformation quantization may not be obvious. However, this extra structure is completely natural (and physical) in 3d Coulomb branches. Indeed, in the recent mathematical/TQFT constructions of Coulomb branches [12,13,[15][16][17][18], one typically works with quantized algebras from the very beginning. In physical terms, the Poisson structure in the chiral ring of 3d N = 4 theories arises from topological descent in the Rozansky-Witten twist [36][37][38], and quantization comes from turning on an Omega background [39,40]. (See also [41,42]. An analogous quantization arising from an Omega background in four dimensions is familiar from [43][44][45][46][47].)We note that when k = 1 or k = 2, the expected relation between the Coulomb branch of T N,k and the Higgs branch of T N [Σ 0,k ] breaks down. Neither the 3d nor the 4d theories are CFT's in this case. Nevertheless, the Coulomb branch of T N,k is still a well-defined hyperkähler manifold, in fact a smooth manifold. We will see explicitly that the Coulombbranch chiral rings of T N,k are consistent with k = 1 : M C T * SL(N, C...

Chiral algebra, localization, modularity, surface defects, and all that

“…Further, there is a known connection, on the one hand, between the three-sphere partition function and the quantized Coulomb and Higgs branches [49,58,59]. On the other hand, there is a similar story relating three-dimensional N = 4 theories on a space of the form R × R 2 , where the latter factor is the Ω-background [60,62,[110][111][112]. Roughly speaking, the latter construction corresponds to a "local" version of the former, akin to the logic in Section 6.…”

Mirror symmetry and line operators

“…These are "abelianized" monopole operators discussed in [96,104], slight renormalizations of the abelianized monopole operators of [74,103]. They act on states as u + a |n, k; σ = P −m σ(a) − (k σ(a) + 1)ε |n + 1, k + e σ(a) ; σ .…”

Coulomb branch quantization and abelianized monopole bubbling

“…Existing approaches to deriving the Coulomb branch chiral rings of 3D N = 4 quiver gauge theories or their quantizations include the Hilbert series [82,88], abelianization [59,70], combinations of the aforementioned techniques [89], and incorporating half-BPS local operators into the type IIB brane/S-duality realization [90] of 3D mirror symmetry [91]. In particular, the Hilbert series can be used to infer the quantum numbers of the generators and their relations (such as for U , U Sp, and SO gauge theories, for which the Coulomb branch is a complete intersection [82]), but it does not specify numerical coefficients.…”