Vanishing theorems and string backgrounds. (English) Zbl 0990.53078

Ivanov, S.; Papadopoulos, G.

Classical Quantum Gravity 18, No. 6, 1089-1110 (2001).

Authors’ summary: We show various vanishing theorems for the cohomology groups of compact Hermitian manifolds for which the Bismut connection has a (restricted) holonomy contained in \(SU(n)\) and classify all such manifolds of dimension four. In this way we provide necessary conditions for the existence of such structures on compact Hermitian manifolds with vanishing first Chern class of non-Kähler type. Then we apply our results to solutions of the string equations and show that such solutions admit various cohomological restrictions such as, for example, that under certain natural assumptions the plurigenera vanish. We also find that under some assumptions the string equations are equivalent to the condition that a certain vector is parallel with respect to the Bismut connection.

Reviewer: A.D.Osborne (Keele)

MSC:

53C80	Applications of global differential geometry to the sciences
81T30	String and superstring theories; other extended objects (e.g., branes) in quantum field theory
83E30	String and superstring theories in gravitational theory

Keywords:

vanishing theorems; cohomology groups; compact Hermitian manifolds; Bismut connection; vanishing first Chern class; string equations