Generalized Kähler Geometry

Abstract

Generalized Kähler geometry is the natural analogue of Kähler geometry, in the context of generalized complex geometry. Just as we may require a complex structure to be compatible with a Riemannian metric in a way which gives rise to a symplectic form, we may require a generalized complex structure to be compatible with a metric so that it defines a second generalized complex structure. We explore the fundamental aspects of this geometry, including its equivalence with the bi-Hermitian geometry on the target of a 2-dimensional sigma model with (2, 2) supersymmetry, as well as the relation to holomorphic Dirac geometry and the resulting derived deformation theory. We also explore the analogy between pre-quantum line bundles and gerbes in the context of generalized Kähler geometry.A striking feature of the Courant bracket is that it has symmetries fixing the underlying space M. Any closed 2-form B ∈ Ω 2,cl (M) acts on TM, preserving the Courant bracket, via the bundle map:(1.2)Because of this symmetry, we may "twist" or modify the global structure of TM as an orthogonal bundle with a Courant bracket, keeping its local structure unchanged. These twisted structures are therefore classified by a characteristic class in H 1 (Ω 2,cl (M)) and are called exact Courant algebroids [13,18,19].

Ricci flow, Courant algebroids, and renormalization of Poisson–Lie T-duality

“…A generalized metric on a Courant algebroid E → M , as defined in [7], is a vector subbundle V + ⊂ E such that , is positive definite on V + and negative definite on…”

“…if E = (T ⊕ T * )M with the structure given by (3), then a generalized metric V + ⊂ E is the graph of a bilinear form e = g + b on T M such that the symmetric part g of e is a Riemannian metric on M . The skew-symmetric part b of e depends on the splitting (g does not), and for a given V + there is a unique splitting of E such that b = 0 (see [7]); the closed 3-form corresponding to this splitting will be denoted H. An exact CA with a generalized metric is thus equivalent to a pair (g, H).…”

Blow-Ups in Generalized Kähler Geometry

“…One can prove this directly in local coordinates using the integrability conditions (see [15]), or in the following more abstract way (see [14]). If L 1 is a Dirac structure on (M, H 1 ) and L 2 a Dirac structure on (M, H 2 ), we can form their Baer-sum on (M,…”

“…Noteworthy examples of the latter include even dimensional compact Lie groups (Gualtieri [14]) and some specific solvmanifolds (Fino,Tomassini [9]). …”

“…2, we lay out the basic ingredients from generalized complex and generalized Kähler geometry. For a more detailed explanation including proofs we refer to [13] and [14]. In Sect.…”