Coulomb Branch Operators Research Articles

BCFT One-point Functions of Coulomb Branch Operators

We show that supersymmetry can be used to compute the BCFT one-point function coefficients for chiral primary operators, in 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SCFTs with 12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{1}{2} $$\\end{document}-BPS boundary conditions. The main ingredient is the hemisphere partition function, with the boundary condition on the equatorial S3. A supersymmetric Ward identity relates derivatives with respect to the chiral coupling constants to the insertion of the primaries at the pole of the hemisphere. Exact results for the one-point functions can be then obtained in terms of the localization matrix model. We discuss in detail the example of the super Maxwell theory in the bulk, interacting with 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SCFTs on the boundary. In particular we derive the action of the SL(2,ℤ) duality on the one-point functions.

Rigorous Holographic Bound on AdS Scale Separation.

We give an elementary proof of the following property of unitary, interacting four-dimensional N=2 superconformal field theories: At large central charge c, there exist at least sqrt[c] single-trace, scalar superconformal primary operators with dimensions Δ≲sqrt[c] (suppressing multiplicative logarithmic corrections). This follows from a stronger, more refined bound on the spectral density in terms of the asymptotic growth rate of the central charge. The proof employs known results on the structure of Coulomb branch operators. Interpreted holographically, this bounds the possible degree of scale separation in semiclassical AdS_{5} half-maximal supergravity. In particular, the bulk must contain an infinite tower of charged scalar states of energies parametrically below the large black hole threshold E_{BH}∼c. We address the extreme case of AdS_{5} pure supergravity, ruling it out as the asymptotic limit of certain sequences in theory space, though the general question remains open.

A matrix-model approach to integrated correlators in a N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SYM theory

In a N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 superconformal gauge theory with matter hypermultiplets transforming in the symmetric and anti-symmetric representations of SU(N), we study the integrated correlators of two Coulomb-branch operators and two moment-map operators using localization. In the corresponding matrix model we identify the operator associated with the integrated insertions of moment-map operators and provide for it an exact expression valid for all values of the coupling constant in the planar limit. This allows us to study the integrated correlators at strong-coupling where we show that they behave as the 2-point functions of the Coulomb-branch operators, up to an overall constant dependent only on the conformal dimensions of the latter. The strong-coupling relation between integrated correlators and 2-point functions turns out to be the same as in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM at large N, despite the reduced amount of supersymmetry in our theory.

A study of mathcal{N} = 1 SCFT derived from mathcal{N} = 2 SCFT: index and chiral ring

One can derive a large class of new mathcal{N} = 1 SCFTs by turning on mathcal{N} = 1 preserving deformations for mathcal{N} = 2 Argyres-Dougals theories. In this work, we use mathcal{N} = 2 superconformal indices to get indices of mathcal{N} = 1 SCFTs, then use these indices to derive chiral rings of mathcal{N} = 1 SCFTs. For a large class of mathcal{N} = 2 theories, we find that the IR theory contains only free chirals if we deform the parent mathcal{N} = 2 theory using the Coulomb branch operator with smallest scaling dimension. Our results provide interesting lessons on studies of mathcal{N} = 1 theories, such as a-maximization, accidental symmetries, chiral ring, etc.

Vanishing short multiplets in rank one 4d/5d SCFTs

We study the short multiplet spectrum in 4d mathcal{N} = 2 superconformal theories of low rank using the full superconformal indices and the selection rules from the superconformal representation theory. We find a universal expression for the leading terms for the superconformal index of rank one H0, H1, H2, D4, E6, E7 theories. From this result, we argue that certain short multiplets appear in the operator product expansions involving stress-tensor, conserved current, and Coulomb branch operator vanish. We also apply the same procedure to 5d superconformal theories and find that E1 theory has vanishing short multiplets analogous to that of the H1 theory.

The Characteristic Dimension of Four-Dimensional {mathcal {N}} = 2 SCFTs

In this paper we introduce the characteristic dimension of a four dimensional {{mathcal {N}}}=2 superconformal field theory, which is an extraordinary simple invariant determined by the scaling dimensions of its Coulomb branch operators. We prove that only nine values of the characteristic dimension are allowed, -infty , 1 ,6/5, 4/3, 3/2, 2, 3, 4, and 6, thus giving a new organizing principle to the vast landscape of 4d {mathcal {N}}=2 SCFTs. Whenever the characteristic dimension differs from 1 or 2, only very constrained special Kähler geometries (i.e. isotrivial, diagonal and rigid) are compatible with the corresponding set of Coulomb branch dimensions and extremely special, maximally strongly coupled, BPS spectra are allowed for the theories which realize them. Our discussion applies to superconformal field theories of arbitrary rank, i.e. with Coulomb branches of any complex dimension. Along the way, we predict the existence of new {{mathcal {N}}}=3 theories of rank two with non-trivial one-form symmetries.

On the Nekrasov partition function of gauged Argyres-Douglas theories

We study SU(2) gauge theories coupled to (A1, DN) theories with or without a fundamental hypermultiplet. For even N, a formula for the contribution of (A1, DN) to the Nekrasov partition function was recently obtained by us with Y. Sugawara and T. Uetoko. In this paper, we generalize it to the case of odd N in the classical limit, under the condition that the relevant couplings and vacuum expectation values of Coulomb branch operators of (A1, DN) are all turned off. We apply our formula to the (A2, A5) theory to find that its prepotential is related to that of the SU(2) gauge theory with four fundamental flavors by a simple change of variables.

SUSY Localization for Coulomb Branch Operators in Omega-Deformed 3d $$\mathcal {N}$$ = 4 Gauge Theories

We perform SUSY localization for Coulomb branch operators of 3d $$\mathcal {N}=4$$ gauge theories in $$\mathbb {R}^3$$ with $$\Omega $$ -deformation. Our study provides a path integral foundation to the so-called abelianization procedure that has been used to study the Coulomb branch. For the dressed monopole operators whose expectation values do not involve non-perturbative corrections, our computations reproduce the results of abelianization. For the expectation values of other operators and the correlation functions of multiple operators in U(N) gauge theories, we compute the non-perturbative corrections due to monopole bubbling using matrix models obtained by string theory (brane) construction. We relate the results of localization to algebraic structures discussed in the mathematical literature, and also uncover a similar relation for line operators in 4d $$\mathcal {N}=2$$ gauge theories. For U(N) (quiver) gauge theories in 3d we demonstrate a direct correspondence between wall-crossing in matrix models and the ordering of operators.

Probing 7-branes on orbifolds

D3 branes in the vicinity of an E6, E7, or E8 stack of 7-branes in flat space are known to host at low energy a famous class of strongly-coupled mathcal{N} = 2 superconformal field theories featuring exceptional global symmetry. What if, instead, the 7-branes wrap an orbifold? We give a systematic characterization of such theories in the case of ℂ2/ℤn, and determine their main properties, like global symmetries, spectra of Coulomb-branch operators, and patterns of Higgs-branch flows. We put forward a set of rules to construct their magnetic quivers, which directly generalize what happens in the perturbative case, and later derive them using the duality with M5 branes probing M9 walls.

Fermi-gas correlators of ADHM theory and triality symmetry

We analytically study the Fermi-gas formulation of sphere correlation functions of the Coulomb branch operators for 3d \mathcal{N}=4𝒩=4 ADHM theory with a gauge group U(N)U(N), an adjoint hypermultiplet and ll hypermultiplets which can describe a stack of NN M2-branes at A_{l-1}Al−1 singularities. We find that the leading coefficients of the perturbative grand canonical correlation functions are invariant under a hidden triality symmetry conjectured from the twisted M-theory. The triality symmetry also helps us to fix the next-to-leading corrections analytically.

Localization and duality for ABJM latitude Wilson loops

We investigate several aspects of BPS latitude Wilson loops in gauge theories in three dimensions with mathcal{N} ≥ 4 supersymmetry. We derive a matrix model for the bosonic latitude Wilson loop in ABJM using supersymmetric localization, and show how to extend the computation to more general Chern-Simons-matter theories. We then define latitude type Wilson and vortex loop operators in theories without Chern-Simons terms, and explore a connection to the recently derived superalgebra defining local Higgs and Coulomb branch operators in these theories. Finally, we identify a BPS loop operator dual to the bosonic latitude Wilson loop which is a novel bound state of Wilson and vortex loops, defined using a worldvolume supersymmetric quantum mechanics.

Schur correlation functions from q -deformed Yang-Mills theory

We construct the wave functions in the $q$-deformed 2D Yang-Mills theory that compute torus correlation functions of affine currents in the vertex operator algebra associated to a class of 4D $\mathcal{N}=2$ superconformal field theory. These wave functions are then shown to reduce to the topological correlators of a set of Coulomb branch operators in the $T[SU(N)]$ theory, from which those correlators in the 3D mirror dual of the 4D ${T}_{N}$ theories can be computed.

Extremal correlators and random matrix theory

We study the correlation functions of Coulomb branch operators of four-dimensional mathcal{N} = 2 Superconformal Field Theories (SCFTs). We focus on rank-one theories, such as the SU(2) gauge theory with four fundamental hypermultiplets. “Extremal” correlation functions, involving exactly one anti-chiral operator, are perhaps the simplest nontrivial correlation functions in four-dimensional Quantum Field Theory. We show that the large charge limit of extremal correlators is captured by a “dual” description which is a chiral random matrix model of the Wishart-Laguerre type. This gives an analytic handle on the physics in some particular excited states. In the limit of large random matrices we find the physics of a non-relativistic axion-dilaton effective theory. The random matrix model also admits a ’t Hooft expansion in which the matrix is taken to be large and simultaneously the coupling is taken to zero. This explains why the extremal correlators of SU(2) gauge theory obey a nontrivial double scaling limit in states of large charge. We give an exact solution for the first two orders in the ’t Hooft expansion of the random matrix model and compare with expectations from effective field theory, previous weak coupling results, and we analyze the non-perturbative terms in the strong ’t Hooft coupling limit. Finally, we apply the random matrix theory techniques to study extremal correlators in rank-1 Argyres-Douglas theories. We compare our results with effective field theory and with some available numerical bootstrap bounds.

S-duality and correlation functions at large R-charge

We study the ratio of pairs of adjacent correlators of Coulomb-branch operators in SU(2) mathcal{N} = 2 sqcd with four flavors within the framework of the large quantum number expansion. Capitalizing on the order-by-order S-duality invariance of the large-R-charge expansion, we compute ab initio the dependence of the leading large- behavior on the marginal coupling τ and we find excellent agreement with numerical estimates from localization.

Bootstrapping Coulomb and Higgs branch operators

We apply the numerical conformal bootstrap to correlators of Coulomb and Higgs branch operators in 4d mathcal{N} = 2 superconformal theories. We start by revisiting previous results on single correlators of Coulomb branch operators. In particular, we present improved bounds on OPE coefficients for some selected Argyres-Douglas models, and compare them to recent work where the same cofficients were obtained in the limit of large r charge. There is solid agreement between all the approaches. The improved bounds can be used to extract an approximate spectrum of the Argyres-Douglas models, which can then be used as a guide in order to corner these theories to numerical islands in the space of conformal dimensions. When there is a flavor symmetry present, we complement the analysis by including mixed correlators of Coulomb branch operators and the moment map, a Higgs branch operator which sits in the same multiplet as the flavor current. After calculating the relevant superconformal blocks we apply the numerical machinery to the mixed system. We put general constraints on CFT data appearing in the new channels, with particular emphasis on the simplest Argyres-Douglas model with non-trivial flavor symmetry.

An mathcal{N} = 1 Lagrangian for an mathcal{N} = 3 SCFT

We propose that a certain 4d mathcal{N} = 1 SU(2) × SU(2) gauge theory flows in the IR to an mathcal{N} = 3 SCFT plus a single free chiral field. The specific mathcal{N} = 3 SCFT has rank 1 and a dimension three Coulomb branch operator. The flow is generically expected to land at the mathcal{N} = 3 SCFT deformed by the marginal deformation associated with said Coulomb branch operator. We also present a discussion about the properties expected of various RG invariant quantities from mathcal{N} = 3 superconformal symmetry, and use these to test our proposal. Finally, we discuss a generalization to another mathcal{N} = 1 model that we propose is related to a certain rank 3 mathcal{N} = 3 SCFT through the turning of certain marginal deformations.

(Mis-)matching type-B anomalies on the Higgs branch

Building on [1], we uncover new properties of type-B conformal anomalies for Coulomb-branch operators in continuous families of 4D mathcal{N} = 2 SCFTs. We study a large class of such anomalies on the Higgs branch, where conformal symmetry is spontaneously broken, and compare them with their counterpart in the CFT phase. In Lagrangian the- ories, the non-perturbative matching of the anomalies can be determined with a weak coupling Feynman diagram computation involving massive multi-loop banana integrals. We extract the part corresponding to the anomalies of interest. Our calculations support the general conjecture that the Coulomb-branch type-B conformal anomalies always match on the Higgs branch when the IR Coulomb-branch chiral ring is empty. In the opposite case, there are anomalies that do not match. An intriguing implication of the mismatch is the existence of a second covariantly constant metric on the conformal manifold (other than the Zamolodchikov metric), which imposes previously unknown restrictions on its holonomy group.

Type-B anomaly matching and the 6D (2,0) theory

We study type-B conformal anomalies associated with frac{1}{2} -BPS Coulomb-branch operators in 4D mathcal{N} = 2 superconformal field theories. When the vacuum preserves the conformal symmetry these anomalies coincide with the two-point function coefficients in the Coulomb-branch chiral ring. They are non-trivial functions of exactly-marginal couplings that can be determined from the S4 partition function. In this paper, we examine the fate of these anomalies in vacua of the Higgs-branch moduli space, where conformal symmetry is spontaneously broken. We argue non-perturbatively that these anomalies are covariantly constant on conformal manifolds. In some cases, this can be used to show that they match in the broken and unbroken phases. Thus, we uncover a new class of data on the Higgs branch of 4D mathcal{N} = 2 conformal field theories that are exactly computable. An interesting application of this matching occurs in mathcal{N} = 2 circular quivers that deconstruct the 6D (2,0) theory on a torus. In that context, we argue that 4D supersymmetric localisation can be used to calculate non-trivial data involving frac{1}{2} -BPS operators of the 6D theory as exact functions of the complex structure of the torus.

Sphere correlation functions and Verma modules

We propose a universal IR formula for the protected three-sphere correlation functions of Higgs and Coulomb branch operators of N = 4 supersymmetric quantum field theories with massive, topologically trivial vacua.

The geometry of SUSY enhancement

We provide a precise geometric picture that demystifies the phenomenon of supersymmetry enhancement along certain RG flows of four-dimensional field theories, recently discovered by Maruyoshi and Song. It applies to theories of arbitrary rank and it is based on a hyperkähler-structure restoration on the moduli space of solutions of (twisted) Hitchin systems, which underly the class- mathcal{S} construction we use as an engineering tool. Along the way, we formulate a necessary algebraic condition for supersymmetry enhancement, and, when enhancement occurs, we are able to derive the Seiberg-Witten geometry and all conformal dimensions of Coulomb-branch operators for the infrared theory, without using a-maximization.