[Paper Review] Two Categories of Dirac Manifolds

Brett Milburn|arXiv (Cornell University)|Dec 17, 2007

Homotopy and Cohomology in Algebraic Topology15 references3 citations

TL;DR

This paper introduces two distinct categories of Dirac manifolds—Dirac maps and dual-Dirac maps—unifying Poisson, complex, symplectic, and holomorphic structures under generalized complex geometry. The key contribution is the characterization of Dirac groups and generalized complex groups as Lie groups with compatible Dirac structures, generalizing Poisson groups and providing a framework for quantization via Courant algebroids and B-transforms.

ABSTRACT

We define two categories of Dirac manifolds, i.e. manifolds with complex Dirac structures. The first notion of maps I call \emph{Dirac maps}, and the category of Dirac manifolds is seen to contain the categories of Poisson and complex manifolds as full subcategories. The second notion, \emph{dual-Dirac maps}, defines a \emph{dual-Dirac category} which contains presymplectic and complex manifolds as full subcategories. The dual-Dirac maps are stable under B-transformations. In particular we get two structures of a category on Hitchin'sgeneralized complex manifolds, i.e., two reasonable notions of generalized complex maps. We also generalize further to get categories of Dirac manifolds for which the Dirac structures lie in arbitrary exact Courant algebroids. As an example, we consider the case of a Lie group with a complex Dirac structure and establish conditions for which multiplication is a Dirac map.

Motivation & Objective

• To define two new categories of Dirac manifolds using Dirac maps and dual-Dirac maps, unifying Poisson and complex geometry.
• To generalize Poisson groups by introducing Dirac groups and generalized complex groups as Lie groups with compatible Dirac structures.
• To establish conditions under which group multiplication is a Dirac or dual-Dirac map, particularly in the context of exact Courant algebroids.
• To extend the theory to twisted Dirac structures using B-fields and H-fluxes, generalizing twisted Poisson groups.
• To explore connections to representation theory of affine Lie algebras through exact Courant algebroids and vertex algebroids.

Proposed method

• Define Dirac maps as morphisms preserving generalized complex structures, generalizing Poisson and holomorphic maps.
• Introduce dual-Dirac maps that unify symplectic and complex maps, stable under B-transforms.
• Use the Courant bracket and generalized almost complex structures on $\mathcal{V}_M = TM \oplus T^*M$ to define integrability conditions.
• Characterize Dirac groups via a $G$-invariant ideal $E \subset \mathfrak{g}_{\mathbb{C}}$ and a section $\varepsilon \in \Gamma(G, \wedge^2 \tilde{E})$.
• Apply B-transforms to relate Dirac structures to twisted Courant algebroids and derive conditions for $\mu^*H = H \oplus H + dB$.
• Use left-invariant trivializations of $\mathcal{V}_G$ to reduce the problem to Lie algebra data, including $d_E\varepsilon = -H|_E$.

Experimental results

Research questions

• RQ1 What conditions define a morphism between Dirac manifolds that unifies Poisson and complex geometry?
• RQ2 How can Dirac groups be characterized as generalizations of Poisson groups within generalized complex geometry?
• RQ3 What is the role of B-transforms in stabilizing the dual-Dirac category and extending the theory to exact Courant algebroids?
• RQ4 Under what conditions is the multiplication map of a Lie group a dual-Dirac map in the presence of an H-flux?
• RQ5 How do generalized complex groups relate to holomorphic Poisson groups, and what is their significance in quantization?

Key findings

• Dirac maps form a category that fully contains both Poisson and complex manifolds, providing a unifying framework for these structures.
• Dual-Dirac maps form a category that fully contains both symplectic and complex manifolds, with stability under B-transforms.
• A Lie group $G$ is a Dirac group if and only if it admits a $G$-invariant ideal $E \subset \mathfrak{g}_{\mathbb{C}}$, a section $\varepsilon \in \Gamma(G, \wedge^2 \tilde{E})$, and a 2-form $B$ satisfying specific transformation and integrability conditions.
• For generalized complex groups, the structure $D = L(E, \varepsilon)$ is integrable if $d_E\varepsilon = -H|_E$, and such groups are equivalent to holomorphic Poisson groups.
• When $G$ is semisimple, the only possible $\varepsilon$ is zero, and $B$ must be of the form $\pi_1^*\omega \oplus \pi_2^*\omega$ with $\omega|_E = 0$.
• The condition $\mu^*H = H \oplus H + dB$ ensures that multiplication is a morphism in the dual-Dirac category, and integrability requires $[\beta, \beta] = \beta_\sharp H$.