Superconformal Indices, Sasaki-Einstein Manifolds, and Cyclic Homologies

TL;DR

The paper derives and matches the single-trace superconformal index on both gauge-theory and gravity sides for D3-branes at a Calabi–Yau cone with Sasaki–Einstein base, demonstrating exact equivalence. The gauge-theory computation expresses the index via holomorphic data on the Calabi–Yau cone: $1+\mathcal{I}_{s.t.}=\sum_{0\le p-q\le 2}(-1)^{p-q}\mathrm{Tr}\,t^{3R}|H^{q}(X,\wedge^p\Omega'_X)$, while the gravity computation recasts it in terms of Kohn–Rossi cohomology on the base: $1+\mathcal{I}_{s.t.}=\sum_{0\le p-q\le 2}\mathrm{Tr}\,t^{3R}|H^{p,q}_{\bar{\partial}_B}(Y)$. A key bridge is Ginzburg’s DG algebra, whose reduced cyclic homology $\overline{HC}_*(\mathfrak{D})$ reproduces the same single-trace index, linking open-string quiver data to closed-string cohomology via Hochschild/Cyclic homology. The KK analysis on Sasaki–Einstein manifolds further yields a precise organization of bosonic KK modes into 4d supermultiplets, with shortening controlled by holomorphic data, leading to a complete gauge/gravity match and a geometric understanding of the index. These results illuminate deep connections between quiver gauge theories, Calabi–Yau geometry, and holographic spectra, and suggest avenues for extending to orbifolds and baryonic sectors.

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.

Figures (1)

• Figure 1: The quiver for $Y=S^5/\mathbb{Z}_3$ (left) and for the circle bundle over $dP_5$ (right).