We propose a construction of the quantum-corrected Coulomb branch of a general 3d gauge theory with $\mathcal{N}=4$ supersymmetry, in terms of local coordinates associated with an abelianized theory. In a fixed complex structure, the holomorphic functions on the Coulomb branch are given by expectation values of chiral monopole operators. We construct the chiral ring of such operators, using equivariant integration over BPS moduli spaces. We also quantize the chiral ring, which corresponds to placing the 3d theory in a 2d Omega background. Then, by unifying all complex structures in a twistor space, we encode the full hyperkähler metric on the Coulomb branch. We verify our proposals in a multitude of examples, including SQCD and linear quiver gauge theories, whose Coulomb branches have alternative descriptions as solutions to the Bogomolnyi and/or Nahm equations.
We introduce a variant of the $K$-theoretic quantized Coulomb branch constructed by Braverman--Finkelberg--Nakajima, by application of a new virtual intersection theory. In the abelian case, we define Verma modules for such virtual Coulomb branches, and relate them to the moduli spaces of quasimaps into the corresponding Higgs branches. The descendent vertex functions, defined by $K$-theoretic quasimap invariants of the Higgs branch, can be realized as the associated Whittaker functions. The quantum $q$-difference modules and Bethe algebras (analogue of quantum $K$-theory rings) can then be de…
We show how to exactly calculate the refined indices of N=4 U(1) times U(N) supersymmetric quiver quantum mechanics in the Coulomb branch by using the localization technique. The Coulomb branch localization is discussed from the viewpoint of both non-linear and gauged linear sigma models. A classification of fixed points in the Coulomb branch differs from one in the Higgs branch, but the derived indices completely agree with the results which were obtained by the localization in the Higgs branch. In the Coulomb branch localization, the refined indices can be written as a summation over differe…
The study of Coulomb branches of 3-dimensional N=4 gauge theories via the associated Hilbert series, the so-called monopole formula, has been proven useful not only for 3-dimensional theories, but also for Higgs branches of 5 and 6-dimensional gauge theories with 8 supercharges. Recently, a conjecture connected different phases of 6-dimensional Higgs branches via gauging of a discrete global $S_n$ symmetry. On the corresponding 3-dimensional Coulomb branch, this amounts to a geometric $S_n$-quotient. In this note, we prove the conjecture on Coulomb branches with unitary nodes and, moreover, ex…
The Coulomb branch of the potential between two static colored sources is calculated for the Yang-Mills theory using the electric Schroedinger representation.
We study Maldacena's conjecture and the AdS/SYM correspondence on the Coulomb branch. Several interesting aspects of this conjectured AdS/SYM correspondence on the Coulomb branch are pointed out and clarified.
We develop a new method for constructing $3d$ $\mathcal{N}=4$ Coulomb branch chiral rings in terms of gauge-invariant generators and relations while making the global symmetry manifest. Our examples generalise to all balanced quivers of type $A$ and $D$ whose Coulomb branches are closures of nilpotent orbits. This new approach is a synthesis of operator counting using Hilbert series and explicit algebraic construction introduced by Bullimore, Dimofte and Gaiotto with significant potential for further generalisation to other quivers, including non-simply laced. The method also identifies comple…
We study (4,4) supersymmetric field theories in two dimensions with a one dimensional Coulomb branch. These theories have applications in string theory. Our analysis explains the known relation between $A-D-E$ groups and modular invariants of affine SU(2).
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 $\rm\, U(1)_R$ symmetry. A set of novel topological and geometric results are developed which promise to be useful for the study and classification of Coulomb bran…
Even though for generic $\mathcal{N}=1$ theories it is not possible to separate distinct branches of supersymmetric vacua, in this paper we study a special class of $\mathcal{N}=1$ SCFTs, these of Class $\mathcal{S}_k$ for which it is possible to define Coulomb and Higgs branches precisely as for the $\mathcal{N}=2$ theories of Class $\mathcal{S}$ from which they descend. We study the BPS operators that parameterise these branches of vacua using the different limits of the superconformal index as well as the Coulomb and Higgs branch Hilbert Series. Finally, with the tools we have developed, we…
To date, the best effort made to simply determine the Coulomb branch global symmetry of a theory from a $3d$ $\mathcal{N}=4$ quiver is by applying an algorithm based on its balanced gauge nodes. This often gives the full global symmetry, but there have been many cases seen where it instead gives only a subgroup. This paper presents a method for constructing several families of $3d$ $\mathcal{N}=4$ unitary quivers where the true global symmetry is enhanced from that predicted by the balance algorithm, motivated by the study of Coulomb branch Hasse diagrams. This provides a rich list of examples…
We construct 4d superconformal field theories (SCFTs) whose Coulomb branches have singular complex structures. This implies, in particular, that their Coulomb branch coordinate rings are not freely generated. Our construction also gives examples of distinct SCFTs which have identical moduli space (Coulomb, Higgs, and mixed branch) geometries. These SCFTs thus provide an interesting arena in which to test the relationship between moduli space geometries and conformal field theory data. We construct these SCFTs by gauging certain discrete global symmetries of $\mathcal N=4$ superYang-Mills (sYM)…
Three-dimensional Coulomb branches have a prominent role in the study of moduli spaces of supersymmetric gauge theories with $8$ supercharges in $3,4,5$, and $6$ dimensions. Inspired by simply laced $3$d $\mathcal{N}=4$ supersymmetric quiver gauge theories, we consider Coulomb branches constructed from non-simply laced quivers with edge multiplicity $k$ and no flavor nodes. In a computation of the Coulomb branch as the space of dressed monopole operators, a center-of-mass $U(1)$ symmetry needs to be ungauged. Typically, for a simply laced theory, all choices of the ungauged $U(1)$ (i.e. all ch…
We discuss planar scattering amplitudes on the Coulomb branch of N=4 super Yang-Mills. The vacuum expectation values on the Coulomb branch can be used to regulate infrared divergences. We argue that this has a number of conceptual as well as practical advantages over dimensional regularisation.
We compute holographically the vevs of all chiral primary operators for supergravity solutions corresponding to the Coulomb branch of N=4 SYM and find exact agreement with the corresponding field theory computation. Using the dictionary between 10d geometries and field theory developed to extract these vevs, we propose a gravity dual of a half supersymmetric deformation of N=4 SYM by certain irrelevant operators.
We compute certain two-point functions in D=4, ${\cal N}=4$, SU(N) SYM theory on the Coulomb branch using SUGRA/SYM duality and find an infinite set of first order poles at masses of order $({\rm Higgs~scale})/(g_{YM} \sqrt{N})$.
We study Coulomb branch moduli spaces of a class of three dimensional $\mathcal{N}=4$ gauge theories whose quiver satisfies the balance condition. The Coulomb branch is described by dressed monopole operators which can be counted using the Monopole formula. We mainly focus on A-type quivers in this paper, using Hilbert Series to study their moduli spaces, and present the interesting pattern which emerges. All of these balanced A-type quiver gauge theories can be realized on brane intervals in Type IIB string theory, where mirror symmetry acts by exchanging the five branes and induces an equiva…
University of Toronto（多伦多大学）；Perimeter Institute of Theoretical Physics（理论物理前沿研究所）；National Research University Higher School of Economics（国立高等经济学院）；Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究所）；Institute for Information Transmission Problems（信息传输问题研究所）；Research Institute for Mathematical Sciences, Kyoto University（京都大学数学科学研究所）；Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo（东京大学宇宙物理与数学 Kavli 研究所（WPI））
We establish and develop a correspondence between certain crystal bases (Kashiwara crystals) and the Coulomb branch of three-dimensional $ \mathcal{N} =4 $ gauge theories. The result holds for simply-laced, non-simply laced and affine quivers. Two equivalent derivations are given in the non-simply laced case, either by application of the axiomatic rules or by folding a simply-laced quiver. We also study the effect of turning on real masses and the ensuing simplification of the crystal. We present a multitude of explicit examples of the equivalence. Finally, we put forward a correspondence betw…
This is the third in a series of three papers on the systematic analysis of rank 1 four dimensional $\mathcal{N}=2$ SCFTs. In the first two papers we developed and carried out a strategy for classifying and constructing physical planar rank-1 Coulomb branch geometries of $\mathcal{N}=2$ SCFTs. Here we describe general features of the Higgs and mixed branch geometries of the moduli space of these SCFTs, and use this, along with their Coulomb branch geometry, to compute their conformal and flavor central charges. We conclude with a summary of the state of the art for rank-1 $\mathcal{N}=2$ SCFTs…
We compute the contribution of discrete Coulomb vacua to A-Model correlators in toric Gauged Linear Sigma Models. For models corresponding to a compact variety, this determines the correlators at arbitrary genus. For non-compact examples, our results imply the surprising conclusion that the quantum cohomology relations break down for a subset of the correlators.
We provide a proof for the conjectured equality of the generating function of integrated Higgs and Coulomb branch topological operators in 3d $\mathcal{N}\ge 4$ gauge theories and the three sphere partition function deformed by mass or FI parameters. The equality is the result of cohomological equivalence and applies to all theories in this class, including ABJM and other generalized Gaiotto-Witten models, and those without an explicit supersymmetric Lagrangian.
We consider N=1 SU(N_c) gauge theory with an adjoint matter field $Φ$, N_f flavors of fundamentals Q and antifundamentals \tQ, and tree-level superpotential of the form $\tQΦ^l Q$. This superpotential is relevant or marginal for $lN_f\leq 2N_c$. The theory has a Coulomb branch which is not lifted by quantum corrections. We find the exact effective gauge coupling on the Coulomb branch in terms of a family of hyperelliptic curves, thus providing a generalization of known results about N=2 SUSY QCD to N=1 context. The Coulomb branch has singular points at which mutually nonlocal dyons become mass…
The Higgs branch of minimally supersymmetric five dimensional SQCD theories increases in a significant way at the UV fixed point when the inverse gauge coupling is tuned to zero. It has been a long standing problem to figure out how, and to find an exact description of this Higgs branch. This paper solves this problem in an elegant way by proposing that the Coulomb branches of three dimensional ${\cal N}=4$ supersymmetric quiver gauge theories, named "Exceptional Sequences", provide the solution to the problem. Thus, once again, 3d ${\cal N}=4$ Coulomb branches prove to be useful tools in solv…
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…
The Coulomb branches of certain 3-dimensional N=4 quiver gauge theories are closures of nilpotent orbits of classical or exceptional algebras. The monopole formula, as Hilbert series of the associated Coulomb branch chiral ring, has been successful in describing the singular hyper-Kähler structure. By means of the monopole formula with background charges for flavour symmetries, which realises real mass deformations, we study the resolution properties of all (characteristic) height two nilpotent orbits. As a result, the monopole formula correctly reproduces (i) the existence of a symplectic res…
This is a companion paper of arXiv:1601.03586. We study Coulomb branches of unframed and framed quiver gauge theories of type $ADE$. In the unframed case they are isomorphic to the moduli space of based rational maps from ${\mathbb C}P^1$ to the flag variety. In the framed case they are slices in the affine Grassmannian and their generalization. In the appendix, written jointly with Joel Kamnitzer, Ryosuke Kodera, Ben Webster, and Alex Weekes, we identify the quantized Coulomb branch with the truncated shifted Yangian.
We study ultraviolet (UV) incarnations of deep infrared (IR) physics on the Coulomb branch of the simplest interacting 4D $\mathcal{N}=2$ superconformal field theory: the minimal Argyres-Douglas (MAD) theory. One of the most basic properties of the Coulomb branch is an emergent infinite-dimensional higher-spin symmetry. While the MAD theory is interacting and therefore does not have such a symmetry, we find UV operators that encode the emergent complex higher-spin symmetry on the Coulomb branch. Moreover, we show that cousins of these UV operators give rise to cousins of the IR higher-spin mul…
We review the relation of certain integrals over the Coulomb phase of $d=4$, N=2 SO(3) supersymmetric Yang-Mills theory with Donaldson-Witten theory. We describe a new way to write an important contact term in the theory and show how the integrals generalize to higher rank gauge groups.
The moduli space of instantons on C^2 for any simple gauge group is studied using the Coulomb branch of N=4 gauge theories in three dimensions. For a given simple group G, the Hilbert series of such an instanton moduli space is computed from the Coulomb branch of the quiver given by the over-extended Dynkin diagram of G. The computation includes the cases of non-simply-laced gauge groups G, complementing the ADHM constructions which are not available for exceptional gauge groups. Even though the Lagrangian description for non-simply laced Dynkin diagrams is not currently known, the prescriptio…
Quantized Coulomb branch of 4d $\mathcal{N}=2$ $Sp(N)$ gauge theory and spherical DAHA of $(C_N^{\vee}, C_N)$-type
Department of Current Legal Study, Faculty of Law, Meiji Gakuin University（明治学院大学法学部现行法学科）；Institute for Mathematical Informatics, Meiji Gakuin University（明治学院大学数学信息学研究所）
The Dijkgraaf-Vafa approach is used in order to study the Coulomb branch of the Leigh-Strassler massive deformation of N=4 SYM with gauge group U(N). The theory has N=1 SUSY and an N-dimensional Coulomb branch of vacua, which can be described by a family of ``generalized'' Seiberg-Witten curves. The matrix model analysis is performed by adding a tree level potential that selects particular vacua. The family of curves is found: it consists of order N branched coverings of a base torus, and it is described by multi-valued functions on the latter. The relation between the potential and the vacuum…
Under reasonable assumptions about the complex structure of the set of singularities on the Coulomb branch of $\mathcal N=2$ superconformal field theories, we present a relatively simple and elementary argument showing that the scaling dimension, $Δ$, of a Coulomb branch operator of a rank $r$ theory is allowed to take values in a finite set of rational numbers$Δ\in \big[\frac{n}{m}\big|n,m\in\mathbb N, 0<m\le n, gcd(n,m)=1,\ φ(n)\le2r\big]$ where $φ(n)$ is the Euler totient function. The maximal dimension grows superlinearly with rank as $Δ_\text{max} \sim r \ln\ln r$. This agrees with the re…
We investigate the structure of the five-point W boson scattering amplitude in N=4 SYM on the Coulomb branch in a small mass limit. We show that up to two loops the IR divergences exponentiate and are controlled by the $Γ_{oct}$ anomalous dimension similar to the four-point amplitude case considered recently in the literature. We also make a conjecture regarding the all-loop structure of the five-point amplitude
There has been a recent progress in understanding the chiral ring of 3d $\mathcal{N}=4$ superconformal gauge theories by explicitly constructing an exact generating function (Hilbert series) counting BPS operators on the Coulomb branch. In this paper we introduce Coulomb branch Hilbert series in the presence of background magnetic charges for flavor symmetries, which are useful for computing the Hilbert series of more general theories through gluing techniques. We find a simple formula of the Hilbert series with background magnetic charges for $T_ρ(G)$ theories in terms of Hall-Littlewood poly…
This paper addresses a long standing problem - to identify the chiral ring and moduli space (i.e. as an algebraic variety) on the Coulomb branch of an N = 4 superconformal field theory in 2+1 dimensions. Previous techniques involved a computation of the metric on the moduli space and/or mirror symmetry. These methods are limited to sufficiently small moduli spaces, with enough symmetry, or to Higgs branches of sufficiently small gauge theories. We introduce a simple formula for the Hilbert series of the Coulomb branch, which applies to any good or ugly three-dimensional N = 4 gauge theory. The…
The constraining mathematical structure of the Coulomb branch of four dimensional $\mathcal{N}=2$ supersymmetric theories is discussed. The presentation follows a somewhat different route from other excellent reviews on the subject and it is geared towards using this tool to classify four dimensional $\mathcal{N}=2$ superconformal field theories. This is the writeup of the lectures given at the Winter School "YRISW 2020" to appear in a special issue of JPhysA.
We study the low energy effective dynamics of four-dimensional $\mathcal{N}=1$ supersymmetric gauge theories of class $\mathcal{S}_k$ on the generalized Coulomb branch. The low energy effective gauge couplings are naturally encoded in algebraic curves $\mathcal{X}$, which we derive for general values of the couplings and mass deformations. We then recast these IR curves $\mathcal{X}$ to the UV or M-theory form $\mathcal{C}$: the punctured Riemann surfaces on which the six-dimensional $\mathcal{N}=(1,0)$ ${A_{k-1}}$ SCFTs are compactified giving the class $\mathcal{S}_k$ theories. We find that …
This paper tests a conjecture on discrete non-Abelian gauging of 3d $\mathcal{N} = 4$ supersymmetric quiver gauge theories. Given a parent quiver with a bouquet of $n$ nodes of rank $1$, invariant under a discrete $S_n$ global symmetry, one can construct a daughter quiver where the bouquet is substituted by a single adjoint $n$ node. Based on the main conjecture in this paper, the daughter quiver corresponds to a theory where the $S_n$ discrete global symmetry is gauged and the new Coulomb branch is a non-Abelian orbifold of the parent Coulomb branch. We demonstrate and test the conjecture for…
The absence of exotics is a conjectural property of the spectrum of BPS states of four--dimensional $\mathcal{N}=2$ supersymmetric QFT's. In this letter we revisit the precise statement of this conjecture, and develop a general strategy that, if applicable, entails the absence of exotic BPS states. Our method is based on the Coulomb branch formula and on quiver mutations. In particular, we obtain the absence of exotic BPS states for all pure SYM theories with simple simply--laced gauge group $G$, and, as a corollary, of infinitely many other lagrangian $\mathcal{N}=2$ theories.
Certain star shaped quivers exhibit a pattern of symmetry enhancement on the Coulomb branch of $3d$ $\mathcal{N}=4$ supersymmetric gauge theories. This paper studies a subclass of theories where such global symmetry enhancement occurs through a computation of the Highest Weight Generating Function (HWG) and of the corresponding Hilbert Series (HS), providing a further test of the Coulomb branch formula. This special subclass has a feature in which the HWG takes a particularly simple form, as a simple rational function which is either a product of simple poles (termed freely generated) or a sim…
Coulomb branch chiral rings of $\mathcal N=2$ SCFTs are conjectured to be freely generated. While no counter-example is known, no direct evidence for the conjecture is known either. We initiate a systematic study of SCFTs with Coulomb branch chiral rings satisfying non-trivial relations, restricting our analysis to rank 1. The main result of our study is that (rank-1) SCFTs with non-freely generated CB chiral rings when deformed by relevant deformations, always flow to theories with non-freely generated CB rings. This implies that if they exist, they must thus form a distinct subset under RG f…
We compute correlators of chiral operators in half-twisted (0,2) supersymmetric gauged linear sigma models. Our results give simple algebraic formulas for a (0,2) generalization of genus zero Gromov-Witten invariants of compact toric varieties. We derive compact expressions for deformed quantum cohomology relations and apply our general method to several examples.
We study scattering amplitudes of massive BPS states on the Coulomb branch of $4d$ $\mathcal{N}=4$ super-Yang-Mills, utilising a little group covariant on-shell superspace for massive particles. Super-BCFW recursion for massive amplitudes is constructed and its validity is proven for all Coulomb branch superamplitudes. We then determine the exact three-particle superamplitudes for massive states. These ingredients allow us to explicitly compute the four- and five-particle superamplitudes, which is the first non-trivial usage of BCFW recursion for amplitudes with entirely massive external state…
