Generalized geometry and the Hodge decomposition

By M. Gualtieri. 2004

72 citations recorded by the source corpus.

In this lecture, we review some of the concepts of generalized geometry, as introduced by Hitchin and developed in the speaker's thesis. We also prove a Hodge decomposition for the twisted cohomology of a compact generalized K\"ahler manifold, as well as a generalization of the $dd^c$-lemma of K\"ahler geometry.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/aaf5f3f028a4c57e476d41c568f40879f9a3c686

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Supergravity as generalised geometry I: type II theories (2011, background): “Finally the O(p, q) × O(q, p) structure provides two additional chirality operators Γ on Spin(d, d) × R+ spinors which one can define as [40, 43, 46] Γ = 1 d!ǫ Γa1 .”
Generalized complex geometry and T-duality (2011, unknown): “As a consequence, it is Spin0-invariant, in particular, for B ∈ Ω 2(M) we have (eB ∧ ρ1, e B ∧ ρ2) = (ρ1, ρ2).”
SU(3) × SU(3) compactification and mirror duals of magnetic fluxes (2006, background): “ht further representations. Denoting by U r,s the set of forms transforming in the (r,s) representation of SU(3)×SU(3) together these decompositions can be arranged in a diamond as given in Table 2.1 [53].5 U 1,¯1 U 1,¯1 U ¯3,¯1 U 1,3 U 3,¯1 U ¯3,3 U 1,¯3 U ¯1,¯1 U 3 ,U ¯3,¯3 U U ¯1,3 U 3,¯3 U ¯3,1 ”
From ten to four and back again: how to generalize the geometry (2007, background): “e as +i-eigenspaces the bundles L1 and L2 respectively. From the compatibility (2.6a) follows furthermore that we can construct the following generalized Hodge decomposition of diﬀerential forms, see [6, 56]: Λ• T⋆ M ⊗ C= M p,q Up,q , (A.9) where Up,q is the intersection of the ip-eigenspace of J1 a”
The decomposition of forms and cohomology of generalized complex manifolds (2006, unknown): “Recently, by finding a suitable generalization of Hodge theory for generalized Kähler manifolds [7], Gualtieri proved that generalized Kähler manifolds satisfy the generalized ∂∂-lemma with respect to both of the generalized complex structures involved.”
Generalized structures of N=1 vacua (2005, unknown): “r in a moment. Since theformalsums offormsarearepresentation ofO(6,6)ofwhich SU(3)×SU(3) is a subgroup, we can decompose the forms under SU(3)×SU(3). The decomposition is given by the following basis [2,32], analogous to the Hodge diamond: Φ+ γmΦ + Φ+γ m γmΦ¯ − γ mΦ +γ n Φ −γ m Φ¯ − γ mΦ¯ −γ n γmΦ −”
Generalized Kähler Geometry (2010, background): “This equivalence, first shown in [2], was followed by a number of results, such as those contained in [3, 4, 5, 6, 7, 8, 9, 10, 11, 12], which are not easily accessed purely from the bi-Hermitian point of view.”