Superconformal indices, Sasaki-Einstein manifolds, and cyclic homologies

Abstract

The superconformal index of the quiver gauge theory dual to type IIB string theory on the product of an arbitrary smooth Sasaki-Einstein manifold with five-dimensional AdS space is calculated both from the gauge theory and gravity viewpoints. We find complete agreement. Along the way, we find that the index on the gravity side can be expressed in terms of the Kohn-Rossi cohomology of the Sasaki-Einstein manifold and that the index of a quiver gauge theory equals the Euler characteristic of the cyclic homology of the Ginzburg dg algebra associated to the quiver.

Is toric duality a Seiberg-like duality in (2 + 1)-d ?

“…We hope to figure out whether there is procedure analogous to symmetrization procedure which will help in resolving this discrepancy. For Fano C 3 and C 5 , our index computation upto order x 2 matched with the gravity index [48,49]. We are presently working on refined index beyond order x 2 for chiral quivers and comparing with the corresponding gravity index.…”

Laplace operators on Sasaki-Einstein manifolds

“…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].…”

“…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 .…”

“…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.…”

“…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, . .…”