Superconformal Theories Research Articles

Tracy-Widom Distribution in Four-Dimensional Supersymmetric Yang-Mills Theories.

Various observables in different four-dimensional superconformal Yang-Mills theories can be computed exactly as Fredholm determinants of truncated Bessel operators. We exploit this relation to determine their dependence on the 't Hooft coupling constant. Unlike the weak coupling expansion, which has a finite radius of convergence, the strong coupling expansion is factorially divergent, necessitating the inclusion of nonperturbative, exponentially small corrections. We develop a method to systematically compute these corrections and discuss the resurgent properties of the resulting transseries.

Lightlike conformal reduction of 6d (1, 0) theories

We study 6d (1, 0) superconformal theories. These have a natural lightlike conformal Killing vector, the Dirac current. We perform a conformal dimensional reduction along the Dirac current down to five-dimensions in such a way that we always preserve at least two real supercharges.

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.

Supersymmetric Cardy formula and the Weak Gravity Conjecture in AdS/CFT

The Weak Gravity Conjecture (WGC) in anti-de Sitter spacetime (AdS) asserts the existence of an operator in the boundary conformal field theory (CFT) whose scaling dimension-to-charge ratio satisfies a certain upper bound. This bound is specified by the ratio of the conformal central charge c and the flavor central charge kF. We propose a modified bound in AdS5/CFT4, determined by a combination of two central charges 3c − 2a instead of c. This combination arises in the Cardy-like limit of the 4d superconformal index, which captures the Bekenstein-Hawking entropy of large BPS black holes in AdS5. Using the new bound, we find that certain superconformal field theories (SCFTs) that are previously thought to violate the AdS WGC, including SQCDs in the conformal window, do satisfy the WGC. We check this version of the WGC against all possible superconformal gauge theories with SU(N) gauge group admitting a large N limit when the superpotential is absent. We conjecture the modified version of the WGC is a generic property of any 4d SCFT, regardless of the existence of a weakly coupled gravity dual or a large N limit.

Wilson loop correlators at strong coupling 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} = 2 quiver gauge theories

We consider 4-dimensional 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 quiver theories with SU(N)M gauge group and bi-fundamental matter and we evaluate correlation functions of n coincident Wilson loops in the planar limit of the theory. Exploiting specific untwisted/twisted combinations of these operators and using supersymmetric localization, we are able to resum the whole perturbative expansion and find exact expressions for these correlators that are valid for all values of the ’t Hooft coupling. Moreover, we analytically derive the leading strong coupling behaviour of the correlators, showing that they obey a remarkable simple rule. Our analysis is complemented by numerical checks based on a Padé resummation of the perturbative series.

Marginal deformations of Calabi-Yau hypersurface hybrids with (2,2) supersymmetry

We study two-dimensional non-linear sigma models with (2,2) supersymmetry and a holomorphic superpotential that are believed to flow to unitary compact (2,2) superconformal theories with central charges cL = cR = 9. The SCFTs have a set of marginal deformations, and some of these can be realized as deformations of parameters of the UV theory, making it possible to apply techniques such as localization to probe the deformations of the SCFT in terms of a UV Lagrangian. In this work we describe the UV lifts of the remaining SCFT infinitesimal deformations, the so-called non-toric and non-polynomial deformations. Our UV theories naturally arise as geometric phases of gauged linear sigma models, and it may be possible to extend our results to find lifts of all SCFT deformations to the gauged linear sigma model.

Self-duality for [formula omitted]-extended superconformal gauge multiplets

We develop a general formalism of duality rotations for N-extended superconformal gauge multiplets in conformally flat backgrounds as an extension of the approach given in arXiv:2107.02001. Additionally, we construct U(1) duality-invariant models for the N=2 superconformal gravitino multiplet recently described in arXiv:2305.16029. Each of them is automatically self-dual with respect to a superfield Legendre transformation. A method is proposed to generate such self-dual models, including a family of superconformal theories.

Comments on Non-invertible Symmetries in Argyres-Douglas Theories

We demonstrate the presence of non-invertible symmetries in an infinite family of superconformal Argyres-Douglas theories. This class of theories arises from diagonal gauging of the flavor symmetry of a collection of multiple copies of Dp(SU(N)) theories. The same set of theories that we study can also be realized from 6d mathcal{N} = (1, 0) compactification on a torus. The main example in this class is the (A2, D4) theory. We show in detail that this specific theory bears the same structures of non-invertible duality and triality defects as those of mathcal{N} = 4 super Yang-Mills with gauge algebra \U0001d530\U0001d532(2). We extend this result to infinitely many other Argyres-Douglas theories in the same family, including those with central charges a = c whose conformal manifold is one dimensional, and those with a ≠ c whose conformal manifold has dimension larger than one. Our result is supported by examining certain special cases that can be realized in terms of theories of class \U0001d4ae.

Seiberg-Witten Geometry of Four-Dimensional $\mathcal N=2$ Quiver Gauge Theories

Seiberg-Witten geometry of mass deformed $\mathcal N=2$ superconformal ADE quiver gauge theories in four dimensions is determined. We solve the limit shape equations derived from the gauge theory and identify the space $\mathfrak M$ of vacua of the theory with the moduli space of the genus zero holomorphic (quasi)maps to the moduli space ${\rm Bun}_{\mathbf G} (\mathcal E)$ of holomorphic $G^{\mathbb C}$-bundles on a (possibly degenerate) elliptic curve $\mathcal E$ defined in terms of the microscopic gauge couplings, for the corresponding simple ADE Lie group $G$. The integrable systems $\mathfrak P$ underlying the special geometry of $\mathfrak M$ are identified. The moduli spaces of framed $G$-instantons on ${\mathbb R}^{2} \times {\mathbb T}^{2}$, of $G$-monopoles with singularities on ${\mathbb R}^{2} \times {\mathbb S}^{1}$, the Hitchin systems on curves with punctures, as well as various spin chains play an important role in our story. We also comment on the higher-dimensional theories.

Non-planar corrections in orbifold/orientifold mathcal{N} = 2 superconformal theories from localization

We study non-planar corrections in two special mathcal{N} = 2 superconformal SU(N) gauge theories that are planar-equivalent to mathcal{N} = 4 SYM theory: two-nodes quiver model with equal couplings and mathcal{N} = 2 vector multiplet coupled to two hypermultiplets in rank-2 symmetric and antisymmetric representations. We focus on two observables in these theories that admit representation in terms of localization matrix model: free energy on 4-sphere and the expectation value of half-BPS circular Wilson loop. We extend the methods developed in arXiv:2207.11475 to derive a systematical expansion of non-planar corrections to these observables at strong ’t Hooft coupling constant λ. We show that the leading non- planar corrections are given by a power series in λ3/2/N2 with rational coefficients. Sending N and the coupling constant λ to infinity with λ3/2/N2 kept fixed corresponds to the familiar double scaling limit in matrix models. We find that in this limit the observables in the two models are related in a remarkably simple way: the free energies differ by the factor of 2, whereas the Wilson loop expectation values coincide. Surprisingly, these relations hold only at strong coupling, they are not valid in the weak coupling regime. We also discuss a dual string theory interpretation of the leading corrections to the free energy in the double scaling limit suggesting their relation to curvature corrections in type IIB string effective action.

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.

Argyres-Douglas avatars of Coulomb branch physics

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 multiplets. In terms of superconformal representation theory, we are led to a conjecture on the exact spectrum of {overline{mathcal{C}}}_{R,rleft(j,overline{j}right)} multiplets in the MAD theory for all R, r, j, and overline{j} satisfying R+overline{j}-j+1=0 , thereby making progress towards a full characterization of the protected spectrum. Along the way, we give a geometrical interpretation of these operators and include them in an extension of the Coulomb branch / mathcal{N} = 2 chiral operator correspondence.

Induced action for superconformal higher-spin multiplets using SCFT techniques

Recently, the interacting N=1 superconformal higher-spin theory in four dimensions has been proposed within the induced action approach. In this paper we initiate a program of computing perturbative corrections to the corresponding action and explicitly evaluate all quadratic terms. This is achieved by employing standard techniques from superconformal field theory.

B-RNS-GSS heterotic string in curved backgrounds

The recently established B-RNS-GSS formalism is extended for the description of the heterotic superstring in curved backgrounds. We propose a generalized action and BRST charge defined in the small Hilbert space with the standard form of an \U0001d4a9 = (1, 0) worldsheet superconformal theory with superconformal generator G and stress tensor T. We show that {G, G} = −2T implies the D=10 N=1 supergravity and super-Yang-Mills equations of motion, as well as holomorphicity of the BRST charge.

D4-branes wrapped on four-dimensional orbifolds through consistent truncation

We construct a consistent truncation of six-dimensional matter coupled F(4) gauged supergravity on a cornucopia of two-dimensional surfaces including a spindle, disc, domain wall and other novel backgrounds to four-dimensional minimal gauged supergravity. Using our consistent truncation we uplift known AdS2× Σ1 solutions giving rise to four-dimensional orbifold solutions, AdS2× Σ1 ⋉ Σ2. We further uplift our solutions to massive type IIA supergravity by constructing the full uplift formulae for six-dimensional U(1)2-gauged supergravity including all fields and arbitrary Romans mass and gauge coupling. The solutions we construct are naturally interpreted as the near-horizon geometries of asymptotically AdS6 black holes with a four-dimensional orbifold horizon. Alternatively, one may view them as the holographic duals of superconformal quantum mechanical theories constructed by compactifying five-dimensional USp(2N) theory living on a stack of D4-D8 branes on the four-dimensional orbifolds. As a first step to identifying these quantum mechanical theories we compute the Bekenstein-Hawking entropy holographically.

Integrated correlators in mathcal{N} = 4 SYM via SL(2, ℤ) spectral theory

We perform a systematic study of integrated four-point functions of half-BPS operators in four-dimensional mathcal{N} = 4 super Yang-Mills theory with gauge group SU(N). These observables, defined by a certain spacetime integral of leftlangle {mathcal{O}}_2{mathcal{O}}_2{mathcal{O}}_p{mathcal{O}}_prightrangle where {mathcal{O}}_p is a superconformal primary of charge p, are known to be computable by supersymmetric localization, yet are non-trivial functions of the complexified gauge coupling τ. We find explicit and remarkably simple results for several classes of these observables, exactly as a function of N and τ. Their physical and formal properties are greatly illuminated upon employing the SL(2, ℤ) spectral decomposition: in this S-duality-invariant eigenbasis, the integrated correlators are fixed simply by polynomials in the spectral parameter. These polynomials are determined recursively by linear algebraic equations relating different N and p, such that all integrated correlators are ultimately fixed in terms of the integrated stress tensor multiplets in the SU(2) theory. Our computations include the full matrix of integrated correlators at low values of p, and a certain infinite class involving operators of arbitrary p. The latter satisfy an open lattice chain equation for all N, reminiscent of the Toda equation obeyed by extremal correlators in mathcal{N} = 2 superconformal theories. We compute ensemble averages of these observables and analyze our solutions at large N, confirming and predicting features of semiclassical AdS5× S5 supergravity amplitudes.

Strong coupling expansions in mathcal{N} = 2 quiver gauge theories

We study the 3-point functions of gauge-invariant scalar operators in four dimensional mathcal{N} = 2 superconformal quiver theories using supersymmetric localization in the planar limit of a large number of colors. By exploiting a web of nontrivial relations, we show that the 3-point functions can be expressed in terms of the 2-point functions through exact Ward-like identities that are valid for all values of the coupling constant. In this way, using recent results about the 2-point functions, we are able to obtain the asymptotic strong-coupling expansion of the 3-point functions and of the corresponding structure constants in the planar limit. Our results extend to sub-leading orders what has been recently found at leading order, where a precise match with calculations within the AdS/CFT correspondence at the supergravity level is possible. Therefore, our findings can be interpreted also as a prediction for the sub-leading string corrections to these holographic calculations.

Exact strong coupling results in mathcal{N} = 2 Sp(2N) superconformal gauge theory from localization

We apply the localization technique to compute the free energy on four-sphere and the circular BPS Wilson loop in the four-dimensional mathcal{N} = ∈ superconformal Sp(2N) gauge theory containing vector multiplet coupled to four hypermultiplets in fundamental representation and one hypermultiplet in rank-2 antisymmetric representation. This theory is unique among similar mathcal{N} = ∈ superconformal models that are planar-equivalent to mathcal{N} = ∆ SYM in that the corresponding localization matrix model has the interaction potential containing single-trace terms only. We exploit this property to show that, to any order in large N expansion and an arbitrary ’t Hooft coupling λ, the free energy and the Wilson loop satisfy Toda lattice equations. Solving these equations at strong coupling, we find remarkably simple expressions for these observables which include all corrections in 1/N and 1/ sqrt{lambda } . We also compute the leading exponentially suppressed mathcal{O}left({e}^{-sqrt{lambda }}right) corrections and consider a generalization to the case when the fundamental hypermultiplets have a non-zero mass. The string theory dual of this mathcal{N} = ∈ gauge theory should be a particular orientifold of AdS5 × S5 type IIB string and we discuss the string theory interpretation of the obtained strong-coupling results.

Yangian Symmetry in Five Dimensions.

Quantum gravity in AdS_{7}×S^{4} is dual to a six-dimensional (6D) superconformal field theory, known as the 6D (2,0) theory, which is very challenging to describe because it lacks a conventional Lagrangian description. On the other hand, certain null reductions of the 6D (2,0) theory give rise to 5D Lagrangian theories with SU(1,3) spacetime symmetry, SO(5) R symmetry, and 24 supercharges. This appears to be closely related to the superconformal group of a 3D superconformal Chern-Simons theory known as the Aharony-Bergman-Jafferis-Maldacena theory, which is believed to be integrable in the planar limit, if one exchanges the role of conformal and R symmetry. In this Letter, we construct a representation of the 5D superconformal group using 6D supertwistors and show that it admits an infinite dimensional extension known as Yangian symmetry, which opens up the possibility that these 5D theories are exactly solvable in the planar limit.

Worldsheet dual of free mathcal{N} = 2 quiver gauge theories

For a special family of 4d mathcal{N} = 2 superconformal quiver theories, the worldsheet dual corresponding to the free theory is identified. This result is obtained from the recently proposed worldsheet dual of free mathcal{N} = 4 SYM by applying a ℤk orbifold. In support of our proposal we show that the spectrum of both the untwisted as well as the twisted sectors coincides between the two descriptions.