[Paper Review] The spinor bundle of Riemannian products

Frank Klinker|ArXiv.org|Dec 4, 2002

Advanced Differential Geometry Research6 references3 citations

TL;DR

This paper establishes a canonical isomorphism between the spinor bundle of a Riemannian product manifold $M = M_1 imes \\-cdots \times M_N$ and the tensor product of the spinor bundles of its factors, under a general class of metrics that are not necessarily Riemannian products. The construction uses Clifford multiplication and $\mathbb{Z}_2$-grading to define a spinor bundle on $M$ via tensoring the spinor bundles of the factors, proving that $M$ is spin and the spinor bundle is isomorphic to a subbundle of the tensor product bundle, even without holonomy or local product assumptions.

ABSTRACT

In this note we compare the spinor bundle of a Riemannian manifold $(M=M_1 imes... imes M_N,g)$ with the spinor bundles of the Riemannian factors $(M_i,g_i)$. We show, that - without any holonomy conditions - the spinor bundle of $(M,g)$ for a special class of metrics is isomorphic to a bundle obtained by tensoring the spinor bundles of $(M_i,g_i)$ in an appropriate way.

Motivation & Objective

• To establish a rigorous construction of the spinor bundle on a Riemannian product manifold $M = M_1 \times \cdots \times M_N$ from the spinor bundles of its factors.
• To resolve the lack of a general proof for the isomorphism between the spinor bundle of $M$ and the tensor product of the spinor bundles of the $M_i$, a fact widely used in physics but previously unproven.
• To clarify the conditions under which such an isomorphism holds, particularly for metrics that are not necessarily Riemannian products, but satisfy a specific metric decomposition (2).
• To extend the known result for $N=2$ with a 1-dimensional factor (Baum, 1989a) to arbitrary dimensions and general metrics of the form (2).

Proposed method

• The paper constructs a bundle $W = \bigotimes_{i=1}^N (S_i^+ \oplus S_i^-)$ from the spinor bundles $S_i$ of the factors $M_i$, equipped with a $\mathbb{Z}_2$-grading.
• It defines a Clifford multiplication on $W$ via a modified action using the linear maps $\delta_k$, which track the grading sign when inserting an operator on the $k$-th factor.
• The metric on $M$ is assumed to be of the form $g_{ab}|_{TM_i} = A_i^c_a g_i^{cd} A_i^d_b$, ensuring the tangent bundle splits orthogonally and the frame bundle reduces to $SO(D_1) \times \cdots \times SO(D_N)$.
• The spin structure on $M$ is constructed via reduction of the $Spin(D)$-principal bundle to $Spin(D_1) \times \cdots \times Spin(D_N)$, using the fiber product construction $(P_G \times P_{\widetilde{H}})/P_H$.
• The spinor bundle $S$ on $M$ is identified as a subbundle of $W$ via the isomorphism $S \simeq (P_G \times P_{\widetilde{H}})/P_H \times_{\widetilde{G}} \hat{S}$, where $\hat{S}$ is the standard spinor representation.
• The key technical step is proving the Clifford relation $(XY + YX)\Xi = -2g(X,Y)\Xi$ holds on $W$ for all $X,Y \in TM$ and $\Xi \in W$, which confirms the bundle carries a valid Clifford module structure.

Experimental results

Research questions

• RQ1 Under what conditions is the spinor bundle of a Riemannian product manifold $M = M_1 \times \cdots \times M_N$ isomorphic to the tensor product of the spinor bundles of its factors?
• RQ2 Can the isomorphism between the spinor bundle of $M$ and the tensor product of the factor spinor bundles be established without assuming the metric is a Riemannian product or that the holonomy splits?
• RQ3 How does the Clifford multiplication on the total spinor bundle $W$ extend from the individual factors $M_i$ to the product manifold $M$?
• RQ4 What role does the $\mathbb{Z}_2$-grading play in ensuring the correct transformation properties of spinors under Clifford multiplication on $M$?
• RQ5 Is the spin structure on $M$ uniquely determined by the spin structures on the $M_i$ under the given metric condition (2), and how is this reflected in the bundle reduction?

Key findings

• The spinor bundle of $M = M_1 \times \cdots \times M_N$ is isomorphic to a subbundle $S$ of the tensor product bundle $W = \bigotimes_{i=1}^N (S_i^+ \oplus S_i^-)$, even when the metric $g$ is not a Riemannian product, provided it satisfies the metric condition (2).
• The Clifford relation $(XY + YX)\Xi = -2g(X,Y)\Xi$ holds on $W$ for all $X,Y \in TM$ and $\Xi \in W$, confirming that $W$ carries a valid Clifford module structure.
• The spin structure on $M$ is induced from the product of the spin structures on the $M_i$, and the $Spin(D)$-principal bundle on $M$ reduces to $Spin(D_1) \times \cdots \times Spin(D_N)$, ensuring $M$ is spin.
• For the special case where $A_i = \text{id}$, the construction recovers the standard result that the spinor bundle of a Riemannian product is the tensor product of the factor spinor bundles.
• The construction generalizes the known result for $N=2$ with a 1-dimensional factor (Baum, 1989a) to arbitrary $N$ and general metrics of the form (2), filling a gap in the mathematical literature.
• The result holds in the pseudo-Riemannian setting as well, as noted in Remark 6(1), extending its applicability beyond Riemannian geometry.