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Register a new userSuperconformal Indices for ${cal N}=6$ Chern Simons Theories
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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.
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We show that the N=8 superconformal Bagger-Lambert theory based on the Lorentzian 3-algebra can be derived by taking a certain scaling limit of the recently proposed N=6 superconformal U(N)xU(N) Chern-Simons-matter theories at level (k, -k). The scaling limit (and Inonu-Wigner contraction) is to scale the trace part of the bifundamental fields as X_0 -> lambda^{-1} X_0 and an axial combination of the two gauge fields as B_{mu} -> lambda B_mu. Simultaneously we scale the level as k -> lambda^{-1} k and then take lambda -> 0 limit. Interestingly the same constraint equation partial^2 X_0=0 is derived by imposing finiteness of the action. In this scaling limit, M2-branes are located far from the origin of C^4/Z_k compared to their fluctuations and Z_k identification becomes a circle identification. Hence the scaled theory describes N=8 supersymmetric theory of 2-branes with dynamical coupling. The coupling constant is promoted to a space-time dependent SO(8) vector X_0^I and we show that the scaled theory has a generalized conformal symmetry as well as manifest SO(8) with the transformation of the background fields X_0^I.
We investigate phases of 3d ${cal N}=2$ Chern-Simons-matter theories, extending to three dimensions the celebrated correspondence between 2d gauged Wess-Zumino-Witten (GWZW) models and non-linear sigma models (NLSMs) with geometric targets. We find that although the correspondence in 3d and 2d are closely related by circle compactification, an important subtlety arises in this process, changing the phase structure of the 3d theory. Namely, the effective theory obtained from the circle compactification of a phase of a 3d ${cal N}=2$ gauge theory is, in general, different from the phase of the 3d ${cal N}=2$ theory on ${mathbb R}^2times S^{1}$, which means taking phases of a 3d gauge theory does not necessarily commute with compactification. We compute the Witten index of each effective theory to check this observation. Furthermore, when the matter fields have the same non-minimal charges, the 3d ${cal N}=2$ Chern-Simons-matter theory with a proper Chern-Simons level will decompose into several identical 2d gauged linear sigma models (GLSMs) for the same target upon reduction to 2d. To illustrate this phenomenon, we investigate how vacua of the 3d gauge theory for a weighted projective space $Wmathbb{P}_{[l,cdots,l]}$ move on the field space when we change the radius of $S^{1}$.
In $mathcal N geq 2$ superconformal Chern-Simons-matter theories we construct the infinite family of Bogomolnyi-Prasad-Sommerfield (BPS) Wilson loops featured by constant parametric couplings to scalar and fermion matter, including both line Wilson loops in Minkowski spacetime and circle Wilson loops in Euclidean space. We find that the connection of the most general BPS Wilson loop cannot be decomposed in terms of double-node connections. Moreover, if the quiver contains triangles, it cannot be interpreted as a supermatrix inside a superalgebra. However, for particular choices of the parameters it reduces to the well-known connections of 1/6 BPS Wilson loops in Aharony-Bergman-Jafferis-Maldacena (ABJM) theory and 1/4 BPS Wilson loops in $mathcal N = 4$ orbifold ABJM theory. In the particular case of $mathcal N = 2$ orbifold ABJM theory we identify the gravity duals of a subset of operators. We investigate the cohomological equivalence of fermionic and bosonic BPS Wilson loops at quantum level by studying their expectation values, and find strong evidence that the cohomological equivalence holds quantum mechanically, at framing one. Finally, we discuss a stronger formulation of the cohomological equivalence, which implies non-trivial identities for correlation functions of composite operators in the defect CFT defined on the Wilson contour and allows to make novel predictions on the corresponding unknown integrals that call for a confirmation.
We present a trace formula for a Witten type Index for superconformal field theories in d=3,5 and 6 dimensions, generalizing a similar recent construction in d=4. We perform a detailed study of the decomposition of long representations into sums of short representations at the unitarity bound to demonstrate that our trace formula yields the most general index (i.e. quantity that is guaranteed to be protected by superconformal symmetry alone) for the corresponding superalgebras. Using the dual gravitational description, we compute our index for the theory on the world volume of N M2 and M5 branes in the large N limit. We also compute our index for recently constructed Chern Simons theories in three dimensions in the large N limit, and find that, in certain cases, this index undergoes a large N phase transition as a function of chemical potentials.
Following a recent work of Dolan and Osborn, we consider superconformal indices of four dimensional ${mathcal N}=1$ supersymmetric field theories related by an electric-magnetic duality with the SP(2N) 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 Csaki 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 $SP(2N), N>1,$ 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 $BC_N$-root system.
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