Dirac structure

Definition

A Dirac structure on a Riemannian manifold $(M,g)$ is a quadruple $(E,c,{\mathrm{\nabla }}^E,{\mathrm{\nabla }}^M)$ where:

• $E$ is a Hermitian vector bundle on $M$
• $c:T^{*}(M)\to End(E)$ is a self-adjoint Clifford structure on $E$
• ${\mathrm{\nabla }}^M$ is a metric connection on $M$
• ${\mathrm{\nabla }}^E$ is a Hermitian connection on $E$ compatible with Clifford multiplication

A geometric Dirac structure is a Dirac structure where the metric connection ${\mathrm{\nabla }}^M$ is the Levi-Civita connection.