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

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.