Coulomb Branch Operators Research Articles

3d “chiral” Kutasov-Schwimmer duality

We propose a “chiral” version of Kutasov-Schwimmer duality in a 3d N=2SU(N) gauge theory with F fundamental matters, F¯ anti-fundamental matters and an adjoint matter X with a tree-level superpotential W=trXk+1. The theory exhibits a rich structure of the baryonic and (dressed) Coulomb branch operators. At first sight, the duality seems incorrect due to a mismatch of the anti-baryonic branch of the moduli space under the duality transformation. The duality well works by realizing that the anti-baryonic operators are identified with some of the dressed Coulomb branch corresponding to non-abelian monopoles. This generalizes the SU(N) “chiral” duality with (anti-)fundamental matters, which we previously proposed.

Coulomb branch quantization and abelianized monopole bubbling

We develop an approach to the study of Coulomb branch operators in 3D mathcal{N} = 4 gauge theories and the associated quantization structure of their Coulomb branches. This structure is encoded in a one-dimensional TQFT subsector of the full 3D theory, which we describe by combining several techniques and ideas. The answer takes the form of an associative and noncommutative star product algebra on the Coulomb branch. For “good” and “ugly” theories (according to the Gaiotto-Witten classification), we also exhibit a trace map on this algebra, which allows for the computation of correlation functions and, in particular, guarantees that the star product satisfies a truncation condition. This work extends previous work on abelian theories to the non-abelian case by quantifying the monopole bubbling that describes screening of GNO boundary conditions. In our approach, monopole bubbling is determined from the algebraic consistency of the OPE. This also yields a physical proof of the Bullimore-Dimofte-Gaiotto abelianization description of the Coulomb branch.

Compactifications of 6d mathcal{N} = (1, 0) SCFTs with non-trivial Stiefel-Whitney classes

We consider compactifications of very Higgsable 6d mathcal{N} =(1, 0) SCFTs on T2 with non-trivial Stiefel-Whitney classes (or equivalently ’t Hooft magnetic fluxes) introduced for their flavor symmetry groups. These systems can also be studied as twisted S1 compactifications of the corresponding 5d theories. We deduce various properties of the resulting 4d mathcal{N} =2 SCFTs by combining these two viewpoints. In particular, we find that all 4d rank-1 mathcal{N} =2 SCFTs with a dimension-6 Coulomb branch operator with flavor symmetry mathfrak{e} 8, mathfrak{u}mathfrak{s}mathfrak{p} (10), mathfrak{s}mathfrak{u} (4) and mathfrak{s}mathfrak{u} (3) can be uniformly obtained by starting from a single-tensor theory in 6d.We also have a mostly independent appendix where we propose a rule to determine the Coulomb branch dimensions of 4d mathcal{N} =2 theories obtained by T2 compactifications of 6d very Higgsable theories with and without Stiefel-Whitney twist.

3d Self-dualities

We investigate self-dualities in three-dimensional N=2 supersymmetric gauge theories. The electric and magnetic theories share the same gauge group. The examples include SU(2N), SO(7) and SO(8) gauge theories with various matter contents. In some examples, the duality exchanges the role of the baryon and Coulomb branch operators. In other examples, the Coulomb branch operator becomes an elementary field on the dual side. These self-dualities in turn teach us a correct quantum structure of the Coulomb moduli space of vacua. Some dualities exhibit symmetry enhancement.

Confinement in 3d mathcal{N} = 2 Spin(N) gauge theories with vector and spinor matters

We present various confinement phases in three-dimensional mathcal{N} = 2 Spin(N) gauge theories with vector and spinor matters. The quantum Coulomb branch in the moduli space of vacua is drastically modified when the rank of the gauge group and the matter contents are changed. In many examples, the Coulomb branch is one- or two-dimensional but its interpretation varies. In some examples, the Coulomb branch becomes three-dimensional and we need to introduce a “dressed” Coulomb branch operator.

Duality and confinement in 3d [formula omitted] “chiral” SU(N) gauge theories

We study low-energy dynamics of three-dimensional N=2SU(N) “chiral” gauge theories with F fundamental and F¯ anti-fundamental matters without a Chern–Simons term. Compared to a naive semi-classical analysis of the Coulomb branch, its quantum structure is highly richer than expected due to so-called “dressed” Coulomb branch (monopole) operators. We propose dualities and confinement phases for the “chiral” SU(N) theories. The theories with N>F>F¯ exhibit spontaneous supersymmetry breaking. The very many Coulomb branch operators generally remain exactly massless and are non-trivially mapped under the dualities. Some dualities lead to a novel duality between SU(N) and USp(2N˜) theories. For the 3d N=2SU(2) gauge theory with 2F doublets, there are generally F+2 “chiral” and “non-chiral” dual descriptions.

The singularity structure of scale-invariant rank-2 Coulomb branches

We compute the spectrum of scaling dimensions of Coulomb branch operators in 4d rank-2 mathcal{N}=2 superconformal field theories. Only a finite rational set of scaling dimensions is allowed. It is determined by using information about the global topology of the locus of metric singularities on the Coulomb branch, the special Kähler geometry near those singularities, and electric-magnetic duality monodromies along orbits of the U(1)R symmetry. A set of novel topological and geometric results are developed which promise to be useful for the study and classification of Coulomb branch geometries at all ranks.

Coulomb branch operators and mirror symmetry in three dimensions

We develop new techniques for computing exact correlation functions of a class of local operators, including certain monopole operators, in three-dimensional mathcal{N}=4 abelian gauge theories that have superconformal infrared limits. These operators are position-dependent linear combinations of Coulomb branch operators. They form a one-dimensional topological sector that encodes a deformation quantization of the Coulomb branch chiral ring, and their correlation functions completely fix the (n ≤ 3)-point functions of all half-BPS Coulomb branch operators. Using these results, we provide new derivations of the conformal dimension of half-BPS monopole operators as well as new and detailed tests of mirror symmetry. Our main approach involves supersymmetric localization on a hemisphere HS3 with half-BPS boundary conditions, where operator insertions within the hemisphere are represented by certain shift operators acting on the HS3 wavefunction. By gluing a pair of such wavefunctions, we obtain correlators on S3 with an arbitrary number of operator insertions. Finally, we show that our results can be recovered by dimensionally reducing the Schur index of 4D mathcal{N}=2 theories decorated by BPS ’t Hooft-Wilson loops.

Argyres-Douglas theories, chiral algebras and wild Hitchin characters

We use Coulomb branch indices of Argyres-Douglas theories on S1 × L(k, 1) to quantize moduli spaces {mathrm{mathcal{M}}}_H of wild/irregular Hitchin systems. In particular, we obtain formulae for the “wild Hitchin characters” — the graded dimensions of the Hilbert spaces from quantization — for four infinite families of {mathrm{mathcal{M}}}_H , giving access to many interesting geometric and topological data of these moduli spaces. We observe that the wild Hitchin characters can always be written as a sum over fixed points in {mathrm{mathcal{M}}}_H under the U(1) Hitchin action, and a limit of them can be identified with matrix elements of the modular transform STkS in certain two-dimensional chiral algebras. Although naturally fitting into the geometric Langlands program, the appearance of chiral algebras, which was known previously to be associated with Schur operators but not Coulomb branch operators, is somewhat surprising.

A-twisted correlators and Hori dualities

The Hori-Tong and Hori dualities are infrared dualities between two-dimensional gauge theories with mathcal{N} = (2, 2) supersymmetry, which are reminiscent of four-dimensional Seiberg dualities. We provide additional evidence for those dualities with U(Nc), USp(2Nc), SO(N ) and O(N ) gauge groups, by matching correlation functions of Coulomb branch operators on a Riemann surface Σg , in the presence of the topological A-twist. The O(N ) theories studied, denoted by O+(N ) and O_(N ), can be understood as {mathbb{Z}}_2 orbifolds of an SO(N ) theory. The correlators of these theories on Σg with g > 0 are obtained by computing correlators with {mathbb{Z}}_2 -twisted boundary conditions and summing them up with weights determined by the orbifold projection.

Correlation functions of Coulomb branch operators

We consider the correlation functions of Coulomb branch operators in four-dimensional mathcal{N} = 2 Superconformal Field Theories (SCFTs) involving exactly one antichiral operator. These extremal correlators are the “minimal” non-holomorphic local observables in the theory. We show that they can be expressed in terms of certain determinants of derivatives of the four-sphere partition function of an appropriate deformation of the SCFT. This relation between the extremal correlators and the deformed four-sphere partition function is non-trivial due to the presence of conformal anomalies, which lead to operator mixing on the sphere. Evaluating the deformed four-sphere partition function using supersymmetric localization, we compute the extremal correlators explicitly in many interesting examples. Additionally, the representation of the extremal correlators mentioned above leads to a system of integrable differential equations. We compare our exact results with previous perturbative computations and with the four-dimensional tt∗ equations. We also use our results to study some of the asymptotic properties of the perturbative series expansions we obtain in mathcal{N} = 2 SQCD.

Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory

We introduce several families of $\mathcal{N}=(2,2)$ UV boundary conditions in 3d $\mathcal N=4$ gauge theories and study their IR images in sigma-models to the Higgs and Coulomb branches. In the presence of Omega deformations, a UV boundary condition defines a pair of modules for quantized algebras of chiral Higgs- and Coulomb-branch operators, respectively, whose structure we derive. In the case of abelian theories, we use the formalism of hyperplane arrangements to make our constructions very explicit, and construct a half-BPS interface that implements the action of 3d mirror symmetry on gauge theories and boundary conditions. Finally, by studying two-dimensional compactifications of 3d $\mathcal{N}=4$ gauge theories and their boundary conditions, we propose a physical origin for symplectic duality - an equivalence of categories of modules associated to families of Higgs and Coulomb branches that has recently appeared in the mathematics literature, and generalizes classic results on Koszul duality in geometric representation theory. We make several predictions about the structure of symplectic duality, and identify Koszul duality as a special case of wall crossing.

Argyres-Douglas theories and S-duality

We generalize S-duality to N=2 superconformal field theories (SCFTs) with Coulomb branch operators of non-integer scaling dimension. As simple examples, we find minimal generalizations of the S-dualities discovered in SU(2) gauge theory with four fundamental flavors by Seiberg and Witten and in SU(3) gauge theory with six fundamental flavors by Argyres and Seiberg. Our constructions start by weakly gauging diagonal SU(2) and SU(3) flavor symmetry subgroups of two copies of a particular rank-one Argyres-Douglas theory (along with sufficient numbers of hypermultiplets to guarantee conformality of the gauging). As we explore the resulting conformal manifold of the SU(2) SCFT, we find an action of S-duality on the parameters of the theory that is reminiscent of Spin(8) triality. On the other hand, as we explore the conformal manifold of the SU(3) theory, we find that an exotic rank-two SCFT emerges in a dual SU(2) description.

NILPOTENT ORBITS AND CODIMENSION-2 DEFECTS OF 6d${\mathcal N} = (2, 0)$ THEORIES

We study the local properties of a class of codimension-2 defects of the 6d [Formula: see text] theories of type J = A, D, E labeled by nilpotent orbits of a Lie algebra [Formula: see text], where [Formula: see text] is determined by J and the outer-automorphism twist around the defect. This class is a natural generalization of the defects of the six-dimensional (6d) theory of type SU (N) labeled by a Young diagram with N boxes. For any of these defects, we determine its contribution to the dimension of the Higgs branch, to the Coulomb branch operators and their scaling dimensions, to the four-dimensional (4d) central charges a and c and to the flavor central charge k.

Gastrojejunostomy

We propose a “chiral” version of Kutasov-Schwimmer duality in a 3d N=2 SU(N) gauge theory with F fundamental matters, F¯ anti-fundamental matters and an adjoint matter X with a tree-level superpotential W=trXk+1. The theory exhibits a rich structure of the baryonic and (dressed) Coulomb branch operators. At first sight, the duality seems incorrect due to a mismatch of the anti-baryonic branch of the moduli space under the duality transformation. The duality well works by realizing that the anti-baryonic operators are identified with some of the dressed Coulomb branch corresponding to non-abelian monopoles. This generalizes the SU(N) “chiral” duality with (anti-)fundamental matters, which we previously proposed.