M2 Odd-Dimensional Spinor And Dirac Data

Let $M$ be a closed oriented Riemannian three-manifold with a specified Spin lift of its oriented orthonormal frame bundle and a metric-compatible principal Spin connection. Choose one of the two irreducible complex modules of the negative-square complex Clifford algebra in dimension three. Its restriction to Spin is the unique irreducible two-dimensional complex Spin module. The associated bundle and its formal geometric Dirac differential expression are defined below; no self-adjoint operator domain or family regularity is implied by this definition.

Use the smooth structure on the Spin lift induced by its equivariant map $\lambda :P\to P_{SO}(TM)$. Explicitly, in a continuous principal chart it is $(x,g)\mapsto (x,a(x)\rho (g))$ for continuous $a:U\to SO(3)$. Near any point lift $a$ through a local inverse of the double covering $\rho$ and absorb that lift into the principal chart. Thus $\lambda$ is a local homeomorphism. Pulling back the smooth charts of $P_{SO}(TM)$ gives a smooth atlas on $P$; chart changes are restrictions of smooth frame-bundle chart changes. The right Spin action is smooth in these charts because $\rho$ is a smooth covering homomorphism. Here a principal Spin connection means a smooth $\mathfrak{s}\mathfrak{p}\mathfrak{i}\mathfrak{n}(3)$-valued one-form $\omega$ on $P$ with $R_g^{\ast }\omega ={\mathrm{Ad}}_{g^{-1}}\omega$ and $\omega (X^{\mathrm{\#}})=X$, where $X_p^{\mathrm{\#}}={\frac{d}{dt}\mid }_{0}p\exp (tX)$. Compatibility in this item means $d\rho \circ \omega ={\lambda }^{\ast }{\omega }_{LC}$, with ${\omega }_{LC}$ the Levi-Civita connection form. Since $d\rho$ is an isomorphism, this specifies $\omega$ uniquely. In a local smooth section $s:U\to P$, the associated connection is $d+d\sigma (s^{\ast }\omega )$.

Let $\sigma$ be the selected two-dimensional Spin representation and let $S:=P{\times }_{\sigma }{\mathbb{C}}^2$ be its associated Hermitian spinor bundle. Let $c:T^{\ast }M\to {\mathrm{End}}_{\mathbb{C}}(S)$ be the associated negative-square Clifford action. The compatible principal Spin connection induces a unitary connection ${\mathrm{\nabla }}^S$ satisfying

For any local orthonormal coframe $(e^1,e^2,e^3)$ with dual frame $(e_1,e_2,e_3)$, define the formal Dirac expression $D_S\psi ={\sum }_{j=1}^{3}c(e^j){\mathrm{\nabla }}_{e_j}^S\psi \mathrm{.}$ The formal expression is independent of the chosen local orthonormal frame. Its self-adjoint closure, Sobolev domain, compact resolvent, smooth dependence in families, and eta regularization are not conclusions of this item; they must be proved by an applicable analytic supplier or carried as explicit inputs in a later data definition.

• Wernli, Lecture Notes on Spin Geometry, §§1.5.1–1.5.3 and 2.3.3–2.3.5: Spin representations, odd-dimensional spinor bundles, and the Dirac expression; the frame-independence calculation is supplied above.