Short Multiplets Research Articles

From free fields to interacting SCFTs via representation theory

We ask when it is possible to construct arbitrary unitary multiplets of the superconformal algebra with eight Poincaré supercharges that are compatible with locality from (continuous deformations of) representations in free field theories. We answer this question in two, three, and five dimensions. In four dimensions, we resort to an intricate but self-consistent web of conjectures. If correct, these conjectures imply various new non-perturbative constraints on short multiplets in any local unitary 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 superconformal field theory and on an unusual set of related vertex algebras. Throughout, we connect our results with properties of deformations in the space of theories.

Superconformal indices and localization in N = 2B quantum mechanics

Superconformal ‘type B’ quantum mechanical sigma models arise in a variety of interesting contexts, such as the description of D-brane bound states in an AdS2 decoupling limit. Focusing on N = 2B models, we study superconformal indices which count short multiplets and provide an alternative to the standard Witten index, as the latter suffers from infrared issues. We show that the basic index receives contributions from lowest Landau level states in an effective magnetic field and that, due to the noncompactness of the target space, it is typically divergent. Fortunately, the models of interest possess an additional target space isometry which allows for the definition of a well-behaved refined index. We compute this index using localization of the functional integral and find that the result agrees with a naive application of the Atiyah-Bott fixed point formula outside of it’s starting assumptions. In the simplest examples, this formula can also be directly verified by explicitly computing the short multiplet spectrum.

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.

6D (2,0) bootstrap with the soft-actor-critic algorithm

We study numerically the 6D (2,0) superconformal bootstrap using the soft-Actor-Critic (SAC) algorithm as a stochastic optimizer. We focus on the four-point functions of scalar superconformal primaries in the energy-momentum multiplet. Starting from the supergravity limit, we perform searches for adiabatically varied central charges and derive two curves for a collection of 80 CFT data (70 of these data correspond to unprotected long multiplets and 10 to protected short multiplets). We conjecture that the two curves capture the A- and D-series (2,0) theories. Our results are competitive when compared to the existing bounds coming from standard numerical bootstrap methods, and data obtained using the OPE inversion formula. With this paper we are also releasing our Python implementation of the SAC algorithm, BootSTOP. The paper discusses the main functionality features of this package.

From β to η: a new cohomology for deformed Sasaki-Einstein manifolds

We discuss in detail the different analogues of Dolbeault cohomology groups on Sasaki-Einstein manifolds and prove a new vanishing result for the transverse Dolbeault cohomology groups {H}_{overline{partial}}^{left(p,0right)}(k) graded by their charge under the Reeb vector. We then introduce a new cohomology, η-cohomology, which is defined by a CR structure and a holomorphic function f with non-vanishing η ≡ df. It is the natural cohomology associated to a class of supersymmetric type IIB flux backgrounds that generalise the notion of a Sasaki-Einstein manifold. These geometries are dual to finite deformations of the 4d mathcal{N} = 1 SCFTs described by conventional Sasaki-Einstein manifolds. As such, they are associated to Calabi-Yau algebras with a deformed superpotential. We show how to compute the η-cohomology in terms of the transverse Dolbeault cohomology of the undeformed Sasaki-Einstein space. The gauge-gravity correspondence implies a direct relation between the cyclic homologies of the Calabi-Yau algebra, or equivalently the counting of short multiplets in the deformed SCFT, and the η-cohomology groups. We verify that this relation is satisfied in the case of S5, and use it to predict the reduced cyclic homology groups in the case of deformations of regular Sasaki-Einstein spaces. The corresponding Calabi-Yau algebras describe non-commutative deformations of ℙ2, ℙ1× ℙ1 and the del Pezzo surfaces.

Mathcal{N} = 2 supersymmetric partially massless fields and other exotic non-unitary superconformal representations

We find and classify the simplest mathcal{N} = 2 SUSY multiplets on AdS4 which contain partially massless fields. We do this by studying representations of the mathcal{N} = 2, d = 3 superconformal algebra of the boundary, including new shortening conditions that arise in the non-unitary regime. Unlike the mathcal{N} = 1 case, the simplest mathcal{N} = 2 multiplet containing a partially massless spin-2 is short, containing several exotic fields. More generally, we argue that mathcal{N} = 2 supersymmetry allows for short multiplets that contain partially massless spin-s particles of depth t = s − 2.

A cubic deformation of ABJM: the squashed, stretched, warped, and perturbed gets invaded

A superpotential deformation that is cubic in one of the chiral superfields of ABJM makes the latter theory flow into a new mathcal{N} = 2 superconformal phase. This is holographically dual to a warped AdS4×wS7 solution of M-theory equipped with a squashed and stretched metric on S7. We determine the spectrum of spin-2 operators of the cubic deformation at low energies by computing the spectrum of Kaluza-Klein (KK) gravitons over the dual AdS4 solution. We calculate, numerically, the complete graviton spectrum and, analytically, the spectrum of gravitons that belong to short multiplets. We also use group theory to assess the structure of the full KK spectrum, and conclude that mathcal{N} = 2 supermultiplets cannot be allocated KK level by KK level. This phenomenon, usually referred to as “space invaders scenario”, is also known to occur for another AdS4 solution based on a different squashed S7.

Polology of Superconformal Blocks

We systematically classify all possible poles of superconformal blocks as a function of the scaling dimension of intermediate operators, for all superconformal algebras in dimensions three and higher. This is done by working out the recently-proven irreducibility criterion for parabolic Verma modules for classical basic Lie superalgebras. The result applies to correlators for external operators of arbitrary spin, and indicates presence of infinitely many short multiplets of superconformal algebras, most of which are non-unitary. We find a set of poles whose positions are shifted by linear in $\mathcal{N}$ for $\mathcal{N}$-extended supersymmetry. We find an interesting subtlety for 3d $\mathcal{N}$-extended superconformal algebra with $\mathcal{N}$ odd associated with odd non-isotropic roots. We also comment on further applications to superconformal blocks.

Vanishing OPE coefficients in 4d mathcal{N}=2 SCFTs

We compute the superconformal characters of various short multiplets in 4d mathcal{N}=2 superconformal algebra, from which selection rules for operator products are obtained. Combining with the superconformal index, we show that a particular short multiplet appearing in the n-fold product of stress-tensor multiplet is absent in the (A1, A2n) Argyres-Douglas (AD) theory. This implies that certain operator product expansion (OPE) coefficients involving this multiplet vanish whenever the central charge c is identical to that of the AD theory. Similarly, by considering the n-th power of the current multiplet, we show that a particular short multiplet and OPE coefficients vanish for a class of AD theories with ADE flavor symmetry. We also consider the generalized AD theory of type (Ak−1, An−1) for coprime k, n and compute its Macdonald index using the associated W -algebra under a mild assumption. This allows us to show that a number of short multiplets and OPE coefficients vanish in this theory. We also provide a Mathematica file along with this paper, where we implement the algorithm by Cordova-Dumitrescu-Intriligator to compute the spectrum of 4d mathcal{N}=2 superconformal multiplets as well as their superconformal character.

Spin 2 operators in holographic 4d mathcal{N}=2 SCFTs

A broad class of holographic duals for 4d mathcal{N}=2 SCFTs is based on the general half-BPS AdS5 solutions to M-theory constructed by Lin, Lunin and Maldacena, and their Kaluza-Klein reductions to Type IIA. We derive the equations governing spin 2 fluctuations around these solutions. The resulting partial differential equations admit families of universal solutions which are constructed entirely out of the generic data characterizing the background. This implies the existence of families of spin 2 operators in the dual SCFTs, in short superconformal multiplets which we identify.

OPE selection rules for Schur multiplets in 4D mathcal{N}=2 superconformal field theories

We compute general expressions for two types of three-point functions of (semi-)short multiplets in four-dimensional mathcal{N}=2 superconformal field theories. These (semi-)short multiplets are called “Schur multiplets” and play an important role in the study of associated chiral algebras. The first type of the three-point functions we compute involves two half-BPS Schur multiplets and an arbitrary Schur multiplet, while the second type involves one stress tensor multiplet and two arbitrary Schur multiplets. From these three-point functions, we read off the corresponding OPE selection rules for the Schur multiplets. Our results particularly imply that there are non-trivial selection rules on the quantum numbers of Schur operators in these multiplets. We also give a conjecture on the selection rules for general Schur multiplets.

2d small N=4 Long-multiplet superconformal block

We study 2d N=4 superconformal field theories, focusing on its application on numerical bootstrap study. We derive the superconformal block by utilizing the global part of the super Virasoro algebra and set up the crossing equations for the non-BPS long-multiplet 4-point function. Along the way, we build global N=4 superconformal short and long multiplets and compute all possible 2,3-point functions of long-multiplets that are needed to construct the superconformal blocks and the crossing equations. Since we consider a long-multiplet 4-point function, the number of crossing equations is huge, and we expect it to give a strong constraint than the usual superconformal bootstrap analysis, which relies on BPS 4-point functions. In addition, we present an alternative way to derive crossing equations using N=4 superspace and comment on a puzzle.

Mathcal{N}=2 moduli of AdS4 vacua: a fine-print study

We analyze the moduli spaces near mathcal{N}=4 supersymmetric AdS4 vacua of string theory by combining and comparing various approaches: (a) the known exact solutions of Type IIB string theory with localized 5-brane sourcees; (b) the holographically dual 3d quiver gauge theories; (c) gauged supergravity; and (d) the representations of the superconformal algebra mathfrak{o}mathfrak{s}mathfrak{p}left(4left|4right|right) . Short multiplets containing the marginal mathcal{N}=2 deformations transform in the (2; 0), (0; 2) or (1; 1) representations of the R-symmetry group SU(2)H× SU(2)C. The first two are classified by the chiral rings of the Higgs and Coulomb branches, while the latter contain mixed-branch operators. We identify the origin of these moduli in string theory, matching in particular the operators of the chiral rings with open strings on the magnetized 5-brane sources. Our results provide new evidence for the underlying holographic duality. The existence of a large number of bound-state moduli highlights the limitations of effective supergravity.

Perturbing AdS6 \xd7wS4: linearised equations and spin-2 spectrum

We initiate the analysis of the Kaluza-Klein mass spectrum of massive IIA supergravity on the warped AdS6 ×wS4 background, by deriving the linearised equations of motion of bosonic and fermionic fluctuations, and determining the mass spectrum of those of spin-2. The spin-2 modes are given in terms of hypergeometric functions and a careful analysis of their boundary conditions uncovers the existence of two branches of mass spectra, bounded from below. The modes that saturate the bounds belong to short multiplets which we identify in the representation theory of the mathfrak{f}(4) symmetry superalgebra of the AdS6 ×wS4 solution.

Spin-2 spectrum of six-dimensional field theories

We analyze the mass spectrum of spin-2 excitations around the gravity duals of "linear quiver" supersymmetric conformal field theories (SCFT's) in six dimensions. We show that for the entire family of gravity solutions it satisfies a bound which corresponds to a unitarity bound for the scaling dimension of the dual field theory operators. We determine the masses of excitations which belong to short multiplets, and for certain gravity solutions we obtain the Kaluza-Klein modes and the corresponding mass spectrum, fully and explicitly. Finally, we discuss an intuitive picture of the dual operators in terms of the effective descriptions of the SCFT's.

Deformations of superconformal theories

We classify possible supersymmetry-preserving relevant, marginal, and irrelevant deformations of unitary superconformal theories in $d \geq 3$ dimensions. Our method only relies on symmetries and unitarity. Hence, the results are model independent and do not require a Lagrangian description. Two unifying themes emerge: first, many theories admit deformations that reside in multiplets together with conserved currents. Such deformations can lead to modifications of the supersymmetry algebra by central and non-central charges. Second, many theories with a sufficient amount of supersymmetry do not admit relevant or marginal deformations, and some admit neither. The classification is complicated by the fact that short superconformal multiplets display a rich variety of sporadic phenomena, including supersymmetric deformations that reside in the middle of a multiplet. We illustrate our results with examples in diverse dimensions. In particular, we explain how the classification of irrelevant supersymmetric deformations can be used to derive known and new constraints on moduli-space effective actions.

Long multiplets in supersymmetric mechanics

The ``long'' indecomposable $\mathcal{N}=2$, $d=1$ multiplet $(\mathbf{2},\mathbf{4},\mathbf{2})$ defined in B. Assel et al. [J. High Energy Phys. 07 (2015) 043] as a deformation of the pair of chiral multiplets $(\mathbf{2},\mathbf{2},\mathbf{0})$ and $(\mathbf{0},\mathbf{2},\mathbf{2})$ by a number of the mass-dimension parameters is described in the superfield approach. We present its most general superfield and component actions as well as a generalization to the case with the superfields of the opposite Grassmann parities and dimensionless deformation parameter. We show that the long $\mathcal{N}=2$, $d=1$ multiplets are naturally embedded into the chiral $SU(2|1)$, $d=1$ superfields having nonzero external spins with respect to $SU(2)\ensuremath{\subset}SU(2|1)$. A superfield with spin $s$ contains $2s$ long multiplets and two short multiplets $(\mathbf{2},\mathbf{2},\mathbf{0})$ and $(\mathbf{0},\mathbf{2},\mathbf{2})$. Two possible $\mathcal{N}=4$, $d=1$ generalizations of the $\mathcal{N}=2$ long multiplet in the superfield approach are also proposed.

Evidence for the holographic dual ofN=3solution in massive type IIA supergravity

We calculate the Kaluza-Klein spectrum of spin-2 fluctuations around the ${\cal N}=3$ warped ${\rm AdS}_4\times M_6$ solution in massive IIA supergravity. This solution was conjectured to be dual to the $D=3$ ${\cal N}=3$ superconformal ${\rm SU}(N)$ Chern-Simons matter theory with level $k$ and 2 adjoint chiral multiplets. The ${\rm SO}(3)_R\times{\rm SO}(3)_D$ isometry of the ${\cal N}=3$ solution is identified with the ${\rm SU}(2)_F\times {\rm SU}(2)_{\cal R}$ global symmetry of the dual ${\cal N}=3$ SCFT. We show that the ${\rm SO}(3)_R\times{\rm SO}(3)_D$ quantum numbers and the ${\rm AdS}$ energies carried by the BPS spin-2 modes match precisely with those of the spin-2 gauge invariant operators in the short multiplets of operators in the ${\cal N}=3$ SCFT. We also compute the Euclidean action of the ${\cal N}=3$ solution and the free energy of the ${\cal N}=3$ SCFT on $S^3$, in the limit $N\gg k$. Remarkably, the results show a complete agreement.

Conformal anomaly c-coefficients of superconformal 6d theories

We propose general relations between the conformal anomaly and the chiral (R-symmetry and gravitational) anomaly coefficients in 6d (1,0) superconformal theories. The suggested expressions for the three type B conformal anomaly c-coefficients complement the expression for the type A anomaly a-coefficient found in arXiv:1506.03807. We check them on several examples -- the standard (1,0) hyper and tensor multiplets as well as some higher derivative short multiplets containing vector fields that generalize the superconformal 6d vector multiplet discussed in arXiv:1506.08727. We also consider a family of higher derivative superconformal (2,0) 6d multiplets associated to 7d multiplets in the KK spectrum of 11d supergravity compactified on S^4. In particular, we prove that (2,0) 6d conformal supergravity coupled to 26 tensor multiplets is free of all chiral and conformal anomalies. We discuss some interacting (1,0) superconformal theories, predicting the c-coefficients for the "E-string" theory on multiple M5-branes at E_8 9-brane and for the theory describing M5-branes at an orbifold singularity. Finally, we elaborate on holographic computation of subleading corrections to conformal anomaly coefficients coming from R^2+R^3 terms in 7d effective action, revisiting, in particular, the (2,0) theory case.

Superconformal indices and M2-branes

We derive the superconformal index of the world-volume theory on M2-branes probing the cone over an arbitrary Sasaki-Einstein seven-manifold. The index is expressed in terms of the cohomology groups of the cone. We match our supergravity results with known results from gauge theory. Along the way we derive the spectrum of short Kaluza-Klein multiplets on generic Sasaki-Einstein seven-manifolds.