Eur. Phys. J. C (2011) 71: 1592
https://doi.org/10.1140/epjc/s10052-011-1592-7

Regular Article - Theoretical Physics

AdS solutions in gauge supergravities and the global anomaly for the product of complex two-cycles

¹ Departamento de Física, Universidade Estadual de Londrina, Caixa Postal 6001, Londrina-Paraná, Brazil
² Consejo Superior de Investigaciones Científicas (ICE/CSIC) and Institut d’Estudis Espacials de Catalunya (IEEC), Facultat de Ciències, Universitat Autònoma de Barcelona, Campus UAB, Torre C5-Par-2a pl, 08193, Bellaterra (Barcelona), Spain

^* e-mail: elizalde@ieec.uab.es
^** e-mail: elizalde@math.mit.edu

Received: 9 November 2010
Revised: 13 January 2011
Published online: 22 March 2011

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 H ⁿ /Γ (where H ⁿ denotes the product of n upper half-planes, H ², equipped with the co-compact action of Γ⊂SL(2,ℝ) ⁿ ) and (H ⁿ )^∗/Γ (where (H ²)^∗=H ²∪{cusp of Γ} and Γ is a congruence subgroup of SL(2,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 the 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.