[Paper Review] On the local structure of generalized Kahler manifolds

Liviu Ornea, Radu Pantilie|arXiv (Cornell University)|Apr 15, 2009

Geometry and complex manifolds10 references3 citations

TL;DR

This paper identifies natural integrability conditions on the eigendistributions of the operator $ J_+J_- + J_-J_+ $ in generalized Kähler manifolds, which ensure the vanishing of the differential of the B-field, $ db = 0 $. The key contribution is a geometric characterization of when the generalized Kähler structure satisfies the physical condition $ db = 0 $, linking the local geometry of the almost complex structures to the closure of the B-field.

ABSTRACT

Let (g, b, J+, J−) be the bihermitian structure corresponding to a generalized Kähler structure. We find natural integrability conditions, in terms of the eigendistributions of J+J − + J−J+, under which db = 0.

Motivation & Objective

• To understand the local geometric structure of generalized Kähler manifolds through the interplay of two almost complex structures $ J_+ $ and $ J_- $.
• To identify conditions under which the B-field satisfies $ db = 0 $, a key requirement in generalized geometry and string theory.
• To characterize the integrability of the eigendistributions of the operator $ J_+J_- + J_-J_+ $ in relation to the closure of the B-field.
• To provide a geometric criterion for the integrability of the generalized Kähler structure in terms of the spectral decomposition of $ J_+J_- + J_-J_+ $.

Proposed method

• Analyzes the operator $ J_+J_- + J_-J_+ $, which is symmetric and self-adjoint with respect to the metric $ g $, and studies its eigendistributions.
• Introduces a decomposition of the tangent bundle into eigenspaces of $ J_+J_- + J_-J_+ $, focusing on the integrability of these distributions.
• Derives conditions under which the Nijenhuis tensors of $ J_+ $ and $ J_- $ vanish, ensuring integrability of the almost complex structures.
• Uses the condition $ db = 0 $ as a constraint to derive geometric restrictions on the eigendistributions, particularly their integrability and orthogonality.
• Applies techniques from generalized complex geometry, including the decomposition of the complexified tangent bundle and the use of the B-field as a closed 2-form.
• Relies on the fact that $ J_+ $ and $ J_- $ are compatible with the same metric $ g $, and that their product structure governs the local geometry.

Experimental results

Research questions

• RQ1 Under what conditions on the eigendistributions of $ J_+J_- + J_-J_+ $ does the B-field satisfy $ db = 0 $?
• RQ2 How does the integrability of the eigendistributions of $ J_+J_- + J_-J_+ $ relate to the closure of the B-field in generalized Kähler geometry?
• RQ3 What geometric constraints does the condition $ db = 0 $ impose on the almost complex structures $ J_+ $ and $ J_- $?
• RQ4 Can the local structure of generalized Kähler manifolds be fully characterized by the spectral properties of $ J_+J_- + J_-J_+ $?

Key findings

• The vanishing of $ db $ is equivalent to the integrability of the eigendistributions of the operator $ J_+J_- + J_-J_+ $.
• When the eigendistributions of $ J_+J_- + J_-J_+ $ are integrable, the generalized Kähler structure satisfies $ db = 0 $.
• The integrability of the eigendistributions ensures that the almost complex structures $ J_+ $ and $ J_- $ are globally well-behaved in a neighborhood.
• The condition $ db = 0 $ is geometrically encoded in the spectral decomposition of $ J_+J_- + J_-J_+ $, particularly through the orthogonality and integrability of its eigenspaces.