Superconformal Index Research Articles

Superconformal indices for [formula omitted] theories with multiple duals

Following a recent work of Dolan and Osborn, we consider superconformal indices of four-dimensional N = 1 supersymmetric field theories related by an electric–magnetic duality with the SP ( 2 N ) gauge group and fixed rank flavour groups. For the SP ( 2 ) (or SU ( 2 ) ) case with 8 flavours, the electric theory has index described by an elliptic analogue of the Gauss hypergeometric function constructed earlier by the first author. Using the E 7 -root system Weyl group transformations for this function, we build a number of dual magnetic theories. One of them was originally discovered by Seiberg, the second model was built by Intriligator and Pouliot, the third one was found by Csáki et al. We argue that there should be in total 72 theories dual to each other through the action of the coset group W ( E 7 ) / S 8 . For the general SP ( 2 N ) , N > 1 , gauge group, a similar multiple duality takes place for slightly more complicated flavour symmetry groups. Superconformal indices of the corresponding theories coincide due to the Rains identity for a multidimensional elliptic hypergeometric integral associated with the B C N -root system.

Nonrelativistic superconformal M2-brane theory

We investigate the low energy physics of particles in the symmetric phase of the = 6 mass-deformed ABJM theory in terms of the superconformal nonrelativistic field theory with 14 supercharges. They describe a certain kind of excitations on M2 branes in the background of external four-form flux. We study the nonrelativistic superconformal algebra and their representations by using the operator-state correspondence with the related harmonic oscillator Hamiltonian. We find the unitarity bounds on the scaling dimension and particle number of any local operator, and comment on subtleties in computing the superconformal Witten index that counts the chiral operators.

The complete superconformal index for [formula omitted] Chern–Simons theory

We calculate the superconformal index for N = 6 Chern–Simons-matter theory with gauge group U ( N ) k × U ( N ) − k at arbitrary allowed value of the Chern–Simons level k. The calculation is based on localization of the path integral for the index. Our index counts supersymmetric gauge invariant operators containing inclusions of magnetic monopole operators, where latter operators create magnetic fluxes on 2-sphere. Through analytic and numerical calculations in various sectors, we show that our result perfectly agrees with the index over supersymmetric gravitons in AdS 4 × S 7 / Z k in the large N limit. Monopole operators in nontrivial representations of U ( N ) × U ( N ) play important roles. We also comment on possible applications of our methods to other superconformal Chern–Simons theories.

Superconformal indices for orbifold Chern-Simons theories

We calculate the superconformal indices of recently discovered three-dimensional N=4,5 Chern-Simons-matter theories and compare them with the corresponding indices of supergravity on AdS4 times orbifolds of S7. We find perfect agreement in the large N and large k limit, provided that the twisted sector contributions at the fixed loci of the orbifolds are properly taken into account. We also discuss the index for the so-called dual ABJM proposal.

Applications of the superconformal index for protected operators and q-hypergeometric identities to [formula omitted] dual theories

The results of Römelsberger for an N = 1 superconformal index counting protected operators, satisfying a BPS condition and which cannot be combined to form long multiplets, are analysed further. The index is expressible in terms of single particle superconformal characters for N = 1 scalar and vector multiplets. For SQCD, involving SU ( N c ) gauge groups and appropriate numbers of flavours N f , the formula used to construct the index may be proved to give identical results for theories linked by Seiberg duality using recently proved theorems for q-series elliptic hypergeometric integrals. The discussion is also extended to Kutasov–Schwimmer dual theories in the large N c , N f limit and to dual theories with Sp ( N ) and SO ( N ) gauge groups. For the former, a transformation identity for elliptic hypergeometric integrals directly verifies that the index is the same for the electric and magnetic theory. For SO ( N ) theories the corresponding result may also be obtained from the same basic identity. An expansion of the index to several orders is also obtained in a form where the detailed protected operator content may be read off. Relevant mathematical results are reviewed.

Superconformal indices for 𝒩 = 6 Chern Simons theories

Aharony, Bergman, Jafferis and Maldacena have recently proposed a dual gravitational description for a family of superconformal Chern Simons theories in three spacetime dimensions. In this note we perform the one loop computation that determines the field theory superconformal index of this theory and compare with the index computed over the Fock space of dual supersymmetric gravitons. In the appropriate limit (large N and large k) we find a perfect match.