Spinor module - Knowlpedia

Definition

Let $(V,q)$ be a finite-dimensional nondegenerate real or complex quadratic space. A spinor module $\Delta$ is a specified module for the corresponding real, complex, or complexified Clifford algebra, usually chosen irreducible in the relevant module category. Since the spin group lies in the group of units of the even Clifford algebra, the Clifford action restricts to a representation

\rho:\operatorname{Spin}(V,q)\longrightarrow\operatorname{GL}(\Delta),

called a spin representation.

Irreducibility and scalar field

The phrase “a spinor module” does not specify a unique module until the scalar field, signature, grading convention, and irreducibility condition have been fixed. Over $\mathbb C$ , the possibilities are described by the classification of complex Clifford modules. Over $\mathbb R$ , the answer depends on the signature modulo $8$ .

These choices also determine which additional spinor conditions exist. A Dirac spinor is complex, a Weyl spinor has a specified chirality, and Majorana and Majorana–Weyl spinors use compatible real structures. These are distinct definitions rather than interchangeable names for elements of $\Delta$ .

From modules to bundles

Given a spin structure on a manifold, a chosen spinor module produces the spinor bundle by the associated-bundle construction. Clifford multiplication then acts fiberwise, and a lifted metric connection yields the corresponding Dirac operator.

References

H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin Geometry, Princeton University Press, 1989. DOI record. Relevant: Chapter I, Sections 4–5.
Daniel S. Freed, Five Lectures on Supersymmetry, American Mathematical Society, 1999. Relevant: Lectures 1–2.
Pierre Deligne, “Notes on spinors,” in Quantum Fields and Strings: A Course for Mathematicians, Volume 1, American Mathematical Society, 1999, pp. 99–135. Relevant: signature-dependent real and complex spinors.