Superconformal Index Research Articles

Superconformal index of higher derivative mathcal{N}=1 multiplets in four dimensions

Supersymmetric partition function of mathcal{N}=1 superconformal theories on Sβ1 × S3 is related to the superconformal index receiving contributions from short representations. The leading coefficients in the small β (high “temperature”) expansion of the index were previously related to the conformal anomaly coefficients of the theory. Assumptions underlying universality of these relations were tested only for simplest low-spin unitary multiplets. Here we consider examples of higher derivative non-unitary mathcal{N}=1 multiplets that naturally appear in the context of extended conformal supergravities and compute their superconformal index. We compare the coefficients in the small β expansion of the index with those proposed earlier for unitary multiplets and suggest some modifications that should apply universally to all types of theories. We also comment on the structure of subleading terms and the case of mathcal{N}=4 conformal supergravity.

Superconformal partition functions and non-perturbative topological strings

We propose a non-perturbative definition for refined topological strings. This can be used to compute the partition function of superconformal theories in 5 dimensions on squashed S5 and the superconformal index of a large number of 6 dimensional (2, 0) and (1, 0) theories, including that of N coincident M5 branes. The result can be expressed as an integral over the product of three combinations of topological string amplitudes. SL(3,Z) modular transformations acting by inverting the coupling constants of the refined topological string play a key role.

4d mathcal{N}=3 indices via discrete gauging

A class of 4d mathcal{N}=3 SCFTs can be obtained from gauging a discrete subgroup of the global symmetry group of mathcal{N}=4 Super Yang-Mills theory. This discrete subgroup contains elements of both the SU(4) R-symmetry group and the SL(2, ℤ) S-duality group of mathcal{N}=4 SYM. We give a prescription for how to perform the discrete gauging at the level of the superconformal index and Higgs branch Hilbert series. We interpret and match the information encoded in these indices to known results for rank one mathcal{N}=3 theories. Our prescription is easily generalised for the Coulomb branch and the Higgs branch indices of higher rank theories, allowing us to make new predictions for these theories. Most strikingly we find that the Coulomb branches of higher rank theories are generically not-freely generated.

Vertex operator algebras, Higgs branches, and modular differential equations

Every four-dimensional mathcal{N}=2 superconformal field theory comes equipped with an intricate algebraic invariant, the associated vertex operator algebra. The relationships between this invariant and more conventional protected quantities in the same theories have yet to be completely understood. In this work, we aim to characterize the connection between the Higgs branch of the moduli space of vacua (as an algebraic geometric entity) and the associated vertex operator algebra. Ultimately our proposal is simple, but its correctness requires the existence of a number of nontrivial null vectors in the vacuum Verma module of the vertex operator algebra. Of particular interest is one such null vector whose presence suggests that the Schur index of any mathcal{N}=2 SCFT should obey a finite order modular differential equation. By way of the “high temperature” limit of the superconformal index, this allows the Weyl anomaly coefficient a to be reinterpreted in terms of the representation theory of the associated vertex operator algebra. We illustrate these ideas in a number of examples including a series of rank-one theories associated with the “Deligne-Cvitanović exceptional series” of simple Lie algebras, several families of Argyres-Douglas theories, an assortment of class mathcal{S} theories, and mathcal{N}=2 super Yang-Mills with mathfrak{s}mathfrak{u}(n) gauge group for small-to-moderate values of n.

Index and duality of minimal mathcal{N} = 4 Chern-Simons-matter theories

We perform a first step analysis toward generalization of the classification of mathcal{N} = 4 linear quiver gauge theories by Gaiotto and Witten including Chern-Simons interaction. For this we investigate minimal mathcal{N}=4mathrm{U}{left({N}_1right)}_ktimes mathrm{U}{left({N}_2right)}_{-k} Chern-Simons theories and their superconformal indices. In the previous publication we analyzed the three-sphere partition function of the theories, which implies that the theory is good/ugly/bad if k−N1 −N2 is greater than/equal to/smaller than −1. In this paper we verify that this classification is consistent with the behavior of the superconformal index. We compare the superconformal indices for several pairs of non-bad theories connected by the Hanany-Witten transition and confirm their coincidence up to the contribution of one hypermultiplet.

A new duality between mathcal{N} = 8 superconformal field theories in three dimensions

We propose a new duality between two 3d mathcal{N} = 8 superconformal Chern-Simons-matter theories: the U(3)1 × U(3)−1 ABJM theory and a theory consisting of the product between the (SU(2)3 × SU(2)− 3)/ℤ2 BLG theory and a free mathcal{N} = 8 theory of eight real scalars and eight Majorana fermions. As evidence supporting this duality, we show that the moduli spaces, superconformal indices, S3 partition functions, and certain OPE coefficients of BPS operators in the two theories agree.

Exact results in 3d mathcal{N} = 2 Spin(7) gauge theories with vector and spinor matters

We study three-dimensional mathcal{N} = 2 Spin(7) gauge theories with NS spinorial matters and with Nf vectorial matters. The quantum Coulomb branch on the moduli space of vacua is one- or two-dimensional depending on the matter contents. For particular values of (Nf, NS ), we find s-confinement phases and derive exact superpotentials. The 3d dynamics of Spin(7) is connected to the 4d dynamics via KK-monopoles. Along the Higgs branch of the Spin(7) theories, we obtain 3d mathcal{N} = 2 G2 or SU(4) theories and some of them lead to new s-confinement phases. As a check of our analysis we compute superconformal indices for these theories.

A \u201cLagrangian\u201d for the E7 superconformal theory

We find an mathcal{N}=1 gauge theory that flows to the rank-one mathcal{N}=2 superconformal field theory with E7 flavor symmetry. We first obtain a Lagrangian description for the R0,N theory, which appears in the S-dual description of the SU(N) gauge theory with 2N fundamental hypermultiplets. This is a straightforward generalization of the proposed Lagrangian description for the E6 theory. The E7 theory is then obtained via partial Higgsing of the R0,4 theory. From this Lagrangian description, we compute the full superconformal index. We also consider twisted dimensional reduction on S2 to obtain mathcal{N}=left(0,4right) theory for the E7 one instanton string and compute its elliptic genus.

Studying superconformal symmetry enhancement through indices

In this note we classify the necessary and the sufficient conditions that an index of a superconformal theory in 3 ≤ d ≤ 6 must obey for the theory to have enhanced supersymmetry. We do that by noting that the index distinguishes a superconformal multiplet contribution to the index only up to a certain equivalence class it lies in. We classify the equivalence classes in d = 4 and build a correspondence between mathcal{N}=1 and mathcal{N}>1 equivalence classes. Using this correspondence, we find a set of necessary conditions and a sufficient condition on the d = 4 mathcal{N}=1 index for the theory to have mathcal{N}>1 SUSY. We also find a necessary and sufficient condition on a d = 4 mathcal{N}>1 index to correspond to a theory with mathcal{N}>2 . We then use our results to study some of the d = 4 theories described by Agarwal, Maruyoshi and Song, and find that the theories in question have only mathcal{N}=1 SUSY despite having rational central charges. In d = 3 we classify the equivalence classes, and build a correspondence between mathcal{N}>2 and mathcal{N}>2 equivalence classes. Using this correspondence, we classify all necessary or sufficient conditions on an 1le mathcal{N}le 3 superconformal index in d = 3 to correspond to a theory with higher SUSY, and find a necessary and sufficient condition on an mathcal{N}=4 index to correspond to an mathcal{N}=4 theory. Finally, in d = 6 we find a necessary and sufficient condition for an mathcal{N}=1 index to correspond to an mathcal{N}>2 theory.

4d mathcal{N} = 1 quiver gauge theories and the An Bailey lemma

We study the integral Bailey lemma associated with the An-root system and identities for elliptic hypergeometric integrals generated thereby. Interpreting integrals as superconformal indices of four-dimensional mathcal{N} = 1 quiver gauge theories with the gauge groups being products of SU(n + 1), we provide evidence for various new dualities. Further confirmation is achieved by explicitly checking that the ‘t Hooft anomaly matching conditions holds. We discuss a flavour symmetry breaking phenomenon for supersymmetric quantum chromodynamics (SQCD), and by making use of the Bailey lemma we indicate its manifestation in a web of linear quivers dual to SQCD that exhibits full s-confinement.

Low-energy dynamics of 3d mathcal{N} = 2 G2 supersymmetric gauge theory

We study a three-dimensional mathcal{N} = 2 supersymmetric G2 gauge theory with and without fundamental matters. We find that a classical Coulomb branch of the moduli space of vacua is partly lifted by monopole-instantons and the quantum Coulomb moduli space would be described by a complex one-dimensional space. Depending on the number of the matters in a fundamental representation, the low-energy dynamics of the theory shows various phases like s-confinement or quantum merging of the Coulomb and the Higgs branches. We also investigate superconformal indices as an independent check of our analysis.

Wall Crossing Invariants from Spectral Networks

A new construction of BPS monodromies for 4d {mathcal {N}}=2 theories of class {mathcal {S}} is introduced. A novel feature of this construction is its manifest invariance under Kontsevich–Soibelman wall crossing, in the sense that no information on the 4d BPS spectrum is employed. The BPS monodromy is encoded by topological data of a finite graph, embedded into the UV curve C of the theory. The graph arises from a degenerate limit of spectral networks, constructed at maximal intersections of walls of marginal stability in the Coulomb branch of the gauge theory. The topology of the graph, together with a notion of framing, encode equations that determine the monodromy. We develop an algorithmic technique for solving the equations and compute the monodromy in several examples. The graph manifestly encodes the symmetries of the monodromy, providing some support for conjectural relations to specializations of the superconformal index. For A_1-type theories, the graphs encoding the monodromy are “dessins d’enfants” on C, the corresponding Strebel differentials coincide with the quadratic differentials that characterize the Seiberg–Witten curve.

Superconformal index, BPS monodromy and chiral algebras

We show that specializations of the 4d mathcal{N}=2 superconformal index labeled by an integer N is given by Tr ℳN where ℳ is the Kontsevich-Soibelman monodromy operator for BPS states on the Coulomb branch. We provide evidence that the states enumerated by these limits of the index lead to a family of 2d chiral algebras {mathcal{A}}_N . This generalizes the recent results for the N = −1 case which corresponds to the Schur limit of the superconformal index. We show that this specialization of the index leads to the same integrand as that of the elliptic genus of compactification of the superconformal theory on S2× T2 where we turn on frac{1}{2}N units of U(1)r flux on S2.

Rahman\u2019s Biorthogonal Rational Functions and Superconformal Indices

We study biorthogonal functions related to basic hypergeometric integrals with coupled continuous and discrete components. Such integrals appear as superconformal indices for three-dimensional quantum field theories and also in the context of solvable lattice models. We obtain explicit biorthogonal systems given by products of two of Rahman’s biorthogonal rational {}_{10}W_9-functions or their degenerate cases. We also give new bilateral extensions of the Jackson and q-Saalschütz summation formulas and new continuous and discrete biorthogonality measures for Rahman’s functions.

Monopole quivers and new 3D [formula omitted] dualities

We present a new family of dualities for three-dimensional gauge theories, motivated by the brane realization of the reduction of four-dimensional dualities on a circle. This family can be understood as a generalization of Aharony duality to quiver gauge theories whose nodes interact via monopole terms in the superpotential. We refer to this family of theories as monopole quivers. We corroborate the new dualities by checking the equivalence of the three-sphere partition functions, obtained from the standard circle reduction of the four-dimensional superconformal index. As a special case, we recover some dualities recently discussed in the literature.

On three-dimensional quiver gauge theories of type B

We study three-dimensional supersymmetric quiver gauge theories with a nonsimply laced global symmetry primarily focusing on framed affine BN quiver theories. Using a supersymmetric partition function on a three sphere, and its transformation under S-duality, we study the three-dimensional ADHM quiver for SO(2N + 1) instantons with a half-integer Chern-Simons coupling. The theory after S-duality has no Lagrangian, and can not be represented by a single quiver, however its partition function can be conveniently described by a collection of framed affine BN quivers. This correspondence can be conjectured to generalize three-dimensional mirror symmetry to theories with nontrivial Chern-Simons terms. In addition, we propose a formula for the superconformal index of a theory described by a framed affine BN quiver.

Supersymmetric Casimir energy and mathrm{S}mathrm{L}left(3,mathrm{mathbb{Z}}right) transformations

We provide a recipe to extract the supersymmetric Casimir energy of theories defined on primary Hopf surfaces directly from the superconformal index. It involves an mathrm{S}mathrm{L}left(3,mathrm{mathbb{Z}}right) transformation acting on the complex structure moduli of the background geometry. In particular, the known relation between Casimir energy, index and partition function emerges naturally from this framework, allowing rewriting of the latter as a modified elliptic hypergeometric integral. We show this explicitly for mathcal{N}=1 SQCD and mathcal{N}=4 supersymmetric Yang-Mills theory for all classical gauge groups, and conjecture that it holds more generally. We also use our method to derive an expression for the Casimir energy of the nonlagrangian mathcal{N}=2 SCFT with E6 flavour symmetry. Furthermore, we predict an expression for Casimir energy of the mathcal{N}=1 SP(2N) theory with SU(8) × U(1) flavour symmetry that is part of a multiple duality network, and for the doubled mathcal{N}=1 theory with enhanced E7 flavour symmetry.

Exact deconstruction of the 6D (2,0) theory

The dimensional-deconstruction prescription of Arkani-Hamed, Cohen, Kaplan, Karch and Motl provides a mechanism for recovering the A-type (2,0) theories on T2, starting from a four-dimensional mathcal{N}=2 circular-quiver theory. We put this conjecture to the test using two exact-counting arguments: in the decompactification limit, we compare the Higgs-branch Hilbert series of the 4D mathcal{N}=2 quiver to the “half-BPS” limit of the (2,0) superconformal index. We also compare the full partition function for the 4D quiver on S4 to the (2,0) partition function on S4× T2. In both cases we find exact agreement. The partition function calculation sets up a dictionary between exact results in 4D and 6D.

Single-flavor Abelian mirror symmetry on ℝℙ2 × S1

The superconformal index on ℝℙ2 × S1 can be derived exactly by the localization technique and applied to the direct proof of Abelianmirror symmetry. We find two sets of parity conditions compatible with the unorientable property of ℝℙ2 and then rigorously show two kinds of Abelian mirror symmetry via the index on ℝℙ2 × S1.

Enhancement of Supersymmetry via Renormalization Group Flow and the Superconformal Index.

We find a four-dimensional N=1 gauge theory which flows to the minimal interacting N=2 superconformal field theory, the Argyres-Douglas theory, in the infrared up to the extra free chiral multiplets. The gauge theory is obtained from a certain N=1 preserving deformation of the N=2 SU(2) gauge theory with four fundamental hypermultiplets. From this description, we compute the full superconformal index and find agreements with the known results in special limits.