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quick note at spin structure on the characterization over Kähler manifolds
Added to spin structure a quick section Examples – On the 2-sphere computing the unique spin structure on the 2-sphere explicitly as a square root of the canonical bundle. (Still needs polishing…)
have expanded the Definition-section at spin structure a bit more, highlighting the obstructing 2-bundles / bundle gerbes a bit more.
added pointer to
for spin structures on orbifolds.
and to
Added:
Just like an orientation of a real vector space $V$ equipped with an inner product is an isometry between the top exterior power of $V$ and real numbers, a spin structure on a real vector space $V$ equipped with an inner product is an isomorphism in the bicategory of algebras, bimodules, and intertwiners from the Clifford algebra of $V$ to the Clifford algebra of the real vector space $\mathbf{R}^n$ of the same dimension $n=\dim V$ with the canonical inner product.
Spin structures naturally form a category, with morphisms being (isometric) isomorphisms of bimodules as described above.
I have moved the algebraic definition out of the Idea-section into the Definition-section, now here.
Also added hyperlinks to a bunch of technical terms.
Incidentally, since the rest of the entry discusses spin-structures in the generality of bundles, it would be good to add a comment on that to the algebraic definition, too.
have expanded out the publication data for the references that discuss spin-structure as anomaly-cancellation for the worldline theory of the spinning particle:
{#Witten85} Edward Witten, p. 65-68 in: Global anomalies in string theory, in: W. Bardeen and A. White (eds.) Symposium on Anomalies, Geometry, Topology, World Scientific (1985) 61-99 [WittenGlobalAnomaliesInStringTheory.pdf:file, spire:214913]
Luis Alvarez-Gaumé, p. 165 of: Supersymmetry and the Atiyah-Singer index theorem, Comm. Math. Phys. 90 2 (1983) 161-173 [doi:10.1007/BF01205500, euclid]
Daniel Friedan, P. Windey, Supersymmetric derivation of the Atiyah-Singer index and the chiral anomaly, Nucl. Phys. B 235 (1984) 395 [doi:10.1016/0550-3213(84)90506-6]
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