[Paper Review] Pure Spinors on Lie groups

Anton Alekseev, Henrique Bursztyn+1 more|Archive ouverte UNIGE (University of Geneva)|Sep 10, 2007

Homotopy and Cohomology in Algebraic Topology27 references9 citations

TL;DR

This paper develops a unified framework for pure spinors and Dirac structures on Lie groups with bi-invariant metrics, using Clifford algebra and spinor representations to characterize generalized complex structures. It establishes that Lagrangian subalgebras in 𝔤 ⊕ ḡ define Dirac structures, and shows that pure spinors on G correspond to elements in Cl(𝔤), with integrability equivalent under the differential d + η. The key contribution is a new Dirac-geometric approach to quasi-Hamiltonian G-spaces, valid beyond compact groups, enabling new constructions of volume forms and extending the theory to complex semi-simple groups.

ABSTRACT

For any manifold M, the direct sum TM \oplus T*M carries a natural inner product given by the pairing of vectors and covectors. Differential forms on M may be viewed as spinors for the corresponding Clifford bundle, and in particular there is a notion of \emph{pure spinor}. In this paper, we study pure spinors and Dirac structures in the case when M=G is a Lie group with a bi-invariant pseudo-Riemannian metric, e.g. G semi-simple. The applications of our theory include the construction of distinguished volume forms on conjugacy classes in G, and a new approach to the theory of quasi-Hamiltonian G-spaces.

Motivation & Objective

• To develop a systematic theory of pure spinors and Dirac structures on Lie groups G equipped with a bi-invariant pseudo-Riemannian metric.
• To generalize the theory of quasi-Hamiltonian G-spaces beyond compact groups by using Dirac geometry and spinor modules.
• To construct canonical volume forms on conjugacy classes and the big cell of complex semi-simple groups via pure spinor constructions.
• To establish a correspondence between different moment map formulations (K*, P, 𝔰𝔭*) using Dirac morphisms and isomorphisms.
• To provide a unified framework for generalized complex structures on G using the trivialization 𝕋G ≅ G × (𝔤 ⊕ ḡ) and spinor bundle isomorphisms.

Proposed method

• Utilize the trivialization 𝕋G ≅ G × (𝔤 ⊕ ḡ) to identify the generalized tangent bundle with the direct sum of the Lie algebra and its opposite.
• Construct a spinor bundle isomorphism 𝒫: G × Cl(𝔤) → ∧T*G that maps the standard Clifford action on Cl(𝔤) to the natural action on differential forms.
• Define pure spinors on G as images of pure spinors in Cl(𝔤) under the isomorphism 𝒫, with integrability equivalent under the differential d + η.
• Use the Clifford differential d_Cl on Cl(𝔤), given by commutator with a cubic element, to relate to the twisted de Rham differential d + η on Ω(G).
• Characterize Dirac structures via Lagrangian subalgebras 𝔰 ⊂ 𝔤 ⊕ ḡ, which induce generalized foliations on G.
• Establish Dirac isomorphisms between different moment map models (K*, P, 𝔰𝔭*) using diffeomorphisms and closed 2-forms, preserving the Dirac structure.

Experimental results

Research questions

• RQ1 How can pure spinors on a Lie group G with a bi-invariant metric be characterized using Clifford algebra and spinor modules?
• RQ2 What is the relationship between integrability of pure spinors on G and the twisted de Rham differential d + η?
• RQ3 How do Dirac structures on G arise from Lagrangian subalgebras of 𝔤 ⊕ ḡ, and what are their geometric consequences?
• RQ4 Can the theory of quasi-Hamiltonian G-spaces be extended beyond compact groups using Dirac geometry and spinor methods?
• RQ5 How are different moment map formulations (K*, P, 𝔰𝔭*) related via Dirac morphisms and volume form constructions?

Key findings

• The generalized tangent bundle 𝕋G is trivialized as G × (𝔤 ⊕ ḡ), allowing a global description of Dirac structures via Lagrangian subalgebras of 𝔤 ⊕ ḡ.
• The spinor bundle ∧T*G is isomorphic to G × Cl(𝔤), under which the Clifford action on Cl(𝔤) corresponds to the natural action on differential forms.
• The differential d + η on Ω(G) corresponds to the Clifford differential d_Cl on Cl(𝔤), so integrability of a pure spinor φ ∈ Ω(G) is equivalent to integrability of its preimage in Cl(𝔤).
• The Cartan-Dirac structure (corresponding to the pure spinor 1 ∈ Cl(𝔤)) induces the foliation of G by conjugacy classes.
• The Gauss-Dirac structure on complex semi-simple G has a dense open leaf corresponding to the big cell in the Gauss decomposition.
• Volume forms on quasi-Hamiltonian G-spaces are constructed via the top-degree part of φ^top ∧ ψ, and are invariant under the group action, with explicit formulas involving (Φ^A)^{2ρ} and exp(ω) for K*-valued moment maps.