Superconformal indices and M2-branes

Abstract

Abstract: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 KaluzaKlein multiplets on generic Sasaki-Einstein seven-manifolds.

Laplace operators on Sasaki-Einstein manifolds

“…4 As long as we restrict to one of these cases, (1) takes the form of a bound on the spectrum of ∆. This was conjectured and partially shown in the context of the calculations of the superconformal index [23,24] in [19,20]. Here, the spectrum was constructed from primitive elements of Ω p,q .…”

“…The conformal energy, R-charge, and spin of any SCFT operator have to satisfy the unitarity bounds [18], which should be reflected on the supergravity side in the spectrum of ∆. We will argue shortly that equation (1) allows us to re-derive the unitarity bounds from supergravity when considered in conjunction with the calculations in [19,20].…”

“…In the case where the Sasaki-Einstein manifold has a coset structure, this has been done using harmonic analysis [21]. [19,20] obtained the structure of the Kaluza-Klein spectrum of generic Sasaki-Einstein manifolds using a construction similar to that in [22], which can be nicely summarized in terms of the identities in appendix A: given any eigen k-form ω of ∆, one diagonalizes the action of ∆ on the k + 1-forms {∂ B ω,∂ B ω, L η ω, Lω, ∂ B∂B ω, . .…”

“…Assuming that every element of H p,q ∂ B (S) has a representative closed under∂ * B , the bound (2.1) is saturated on the elements of H p,q ∂ B (S). These are the forms that correspond to the short multiplets in the SCFT, and (2.1) together with the expressions for the derived eigenmodes of ∆ given in [19,20] allows to recover the unitarity bounds from supergravity. Note that (2.1) and a precursor to (1) were already conjectured in those references.…”

“…Note that (2.1) and a precursor to (1) were already conjectured in those references. Furthermore, the appendix of [20] contains an argument that every element of H p,q ∂ B (S) is either primitive, carrying zero charge, or both. For the cases of interest in the context of that paper it turned out that all elements are primitive.…”