Abstract

We use tools from generalized complex geometry to develop the theory of SKT (a.k.a. pluriclosed Hermitian) manifolds and more generally manifolds with special holonomy with respect to a metric connection with closed skew-symmetric torsion. We develop Hodge theory on such manifolds showing how the reduction of the holonomy group causes a decomposition of the twisted cohomology. For SKT manifolds this decomposition is accompanied by an identity between different Laplacian operators and equates different cohomologies defined in terms of the SKT structure. We illustrate our theory with examples based on Calabi–Eckmann manifolds, instantons, Hopf surfaces and Lie groups.

Highlights

  • Looking beyond the Levi–Civita connection in Riemannian geometry, one finds a number of other metric connections with interesting properties

  • The “strong” torsion condition refers to the latter: a strong torsion connection on a Riemannian manifold is a metric connection whose torsion is skew symmetric and closed

  • A weaker notion is that of a parallel Hermitian structure which for us means a connection with closed, skew symmetric torsion for which the holonomy is U (n)

Read more

Summary

Introduction

Looking beyond the Levi–Civita connection in Riemannian geometry, one finds a number of other metric connections with interesting properties. Requiring less supersymmetry without giving up on the idea altogether leads one to consider models where there is more left than right supersymmetry or models where the right side is absent These conditions lead to (2, 1) or (2, 0) supersymmetric sigma models and supersymmetry holds if and only if the target space has an SKT structure [20]. The opposite seemed to be the case: Since SKT manifolds are, in particular, complex, their space of forms inherits a natural bi-grading, but a simple check in concrete examples shows that there is no corresponding decomposition of their cohomology.

Linear algebra
Intrinsic torsion of generalized Hermitian structures
The Nijenhuis tensor
The intrinsic torsion and the road to integrability
Parallel Hermitian and bi-Hermitian structures
SKT structures
Hodge theory
Signature and Euler characteristic of almost Hermitian manifolds
Hodge theory on parallel Hermitian manifolds
Relation to Dolbeault cohomology
Hermitian symplectic structures
Integrability
Instantons over complex surfaces
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call