AdS Solutions in Gauge Supergravities and the Global Anomaly for the Product of Complex Two-Cycles

TL;DR

This paper explores new solutions in gauge supergravities using cohomological methods, focusing on Hilbert modular varieties and their cohomology, revealing their role in global anomalies for products of complex two-cycles.

Contribution

It introduces a novel class of solutions called Hilbert modular varieties and analyzes their cohomology, linking torsion cohomology to global anomaly conditions in supergravity.

Findings

Hilbert modular varieties are proposed as new solutions in gauge supergravities.

Cohomology analysis shows torsion parts influence global anomaly conditions.

Cuspidal cohomology can be mapped to the boundary, affecting anomaly considerations.

Abstract

Cohomological methods are applied for the special set of solutions corresponding to rotating branes in arbitrary dimensions, AdS black holes (which can be embedded in ten or eleven dimensions), and gauge supergravities. A new class of solutions is proposed, the Hilbert modular varieties, which consist of the $2n$-fold product of the two-spaces ${\bf H}^n/\Gamma$ (where ${\bf H}^n$ denotes the product of $n$ upper half-planes, $H^2$, equipped with the co-compact action of $\Gamma \subset SL(2, {\mathbb R})^n$) and $({\bf H}^n)^*/\Gamma$ (where $(H^2)^* = H^2\cup \{{\rm cusp\,\, of}\,\,\Gamma\}$ and $\Gamma$ is a congruence subgroup of $SL(2, {\mathbb R})^n$). The cohomology groups of the Hilbert variety, which inherit a Hodge structure (in the sense of Deligne), are analyzed, as well as bifiltered sequences, weight and Hodge filtrations, and it is argued that the torsion part of the cuspidal cohomology is involved in the global anomaly condition. Indeed, in presence of the cuspidal part, all cohomology classes can be mapped to the boundary of the space and the cuspidal contribution can be involved in the global anomaly condition.