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.

Localisation on Sasaki-Einstein manifolds from holomorphic functions on the cone

“…Building on the work of KQZ we will argue that the one-loop super determinant can be expressed in terms of the so-called Kohn-Rossi cohomology groups H p,q ∂b (Y ) and the Lie-derivative along the Reeb vector ξ. Previously, the H p,q ∂b (Y ) have appeared in the context of holographic calculations of superconformal indices of three-and four-dimensional SCFTs [15,16,17]. Together with the isomorphism H 0,0 ∂b (Y ) ∼ = H 0 (O C(Y ) ), our result allows for a easy evaluation of the perturbative partition function, as the whole calculation reduces to the counting of holomorphic functions on the Calabi-Yau cone C(Y ), weighted by their charge along the Reeb.…”

Laplace operators on Sasaki-Einstein manifolds

“…This was conjectured and partially shown in the context of the calculations of the superconformal index [22,23] in [18,19]. 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 [17], 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 [18,19].…”

“…In the case where the Sasaki-Einstein manifold has a coset structure, this has been done using harmonic analysis [20]. [18,19] obtained the structure of the Kaluza-Klein spectrum of generic Sasaki-Einstein manifolds using a construction similar to that in [21], 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 ( 13) 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 (13) together with the expressions for the derived eigenmodes of ∆ given in [18,19] allows to recover the unitarity bounds from supergravity. Note that (13) and a precursor to (1) were already conjectured in those references.…”