Generalized contact geometry and T-duality

Abstract

We study generalized almost contact structures on odd-dimensional manifolds. We introduce a notion of integrability and show that the class of these structures is closed under symmetries of the Courant-Dorfman bracket, including T-duality. We define a notion of geometric type for generalized almost contact structures, and study its behavior under T-duality.

AN ABSTRACT MORIMOTO THEOREM FOR GENERALIZEDF-STRUCTURES

“…The generalized complex structures constructed with our method come in families and thus, even in the generalized contact case, they are more general than those of [6]. For instance, we show that the Morimoto product of two copies of the normal generalized almost contact structures on S 3 introduced in [1] yields holomorphic Poisson deformations of the Calabi-Eckmann complex structures on S 3 × S 3 for every choice of complex structure on the T 2 fiber.…”

“…Let J be the split generalized F -structure on E ⊥ corresponding to a generalized almost contact structure (E, L) and let Φ be the extension of J to TM by 0. Given an isotropic frame {e 1 , e 2 } of E such that 2 e 1 , e 2 = 1 then (Φ, e 1 , e 2 ) is a generalized almost contact triple as defined in [1]. Therefore, the set of generalized contact triples up to a change of frame of E can be identified with the union of all SGF(E ⊥ ), as E ranges over all rank 2 split structures on M that are trivial subbundles of TM.…”

“…In a different direction, we are able to extend Sekiya's characterization of invariant generalized (almost) complex structures ( [16], [1]) from products of the form M × R to products of M with an arbitrary finite dimensional Lie group.…”

“…For instance, we are able to describe two constructions of generalized CRFstructures on flat principal bundles, one of which extends previous work [2] on normal contact pairs. A second class of examples of Morimoto data beyond the global product case comes from a generalized version of a classical theorem of Blair, Ludden and Yano [3] which states that Hermitian bicontact manifolds with bicontact forms (η 1 , η 2 ) of bidegree (1,1) are locally the product of normal contact manifolds. In this paper we prove an Abstract Blair-Ludden-Yano Theorem at the level of Hermitian bicontact data, a notion that we introduce in order to isolate the features of classical Hermitian bicontact structures of bidegree (1, 1) that we need.…”

Generalized contact bundles

“…Several approaches to odd-dimensional analogues of generalized complex structures can be found in the literature [13,22,18,19,1]. They are often named generalized contact structures and all of them include contact structures globally defined by a contact 1-form.…”