Global anomaly and a family of structures on fold product of complex two-cycles

TL;DR

This paper constructs new supergravity solutions based on products of hyperbolic spaces, analyzes global anomalies related to torsion cohomology, and studies deformations of complex structures with stability results.

Contribution

It introduces a novel class of supergravity solutions involving hyperbolic space products and examines their global anomaly conditions and deformation stability.

Findings

Global anomaly conditions involve torsion in cuspidal cohomology.

Stability of generalized complex structures verified for certain discrete subgroups.

New supergravity solutions constructed on hyperbolic space products.

Abstract

We propose a new set of IIB type and eleven-dimensional supergravity solutions which consists of the $n$-fold product of 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 Freed-Witten global anomaly condition have been analyzed. We argue that the torsion part of the cuspidal cohomology involves in the global anomaly condition. Infinitisimal deformations of generalized complex (and K\"ahler) structures also has been analyzed and stability theorem in the case of a discrete subgroup of $SL(2, {\mathbb R})^n$ with the compact quotient ${\bf H}^n/\Gamma$ was verified.