The space of unital algebra homomorphisms $C^\infty(S)\toΛ_2$: a singular determinantal model
Year: 2025
Type: preprint
Abstract: Let $Λ_2$ be the real Grassmann algebra on two odd generators. Define $\mathcal{J}(S)$ as the set of unital algebra homomorphisms $C^\infty(S)\to Λ_2$ that are not required to preserve parity. This space is motivated by replacing the commutative local test algebra appearing in Weil's near-point construction with $Λ_2$. Each homomorphism expands to give a base point and three tangent vectors. The homomorphism condition, together with the noncommutativity of $Λ_2$, forces two of these tangent vectors to be linearly dependent. The resulting parameter space is $\mathcal{C}(TS) \times_S TS$, where $\mathcal{C}(TS)$ is the fibrewise determinantal cone of pairs of linearly dependent tangent vectors. When $\dim S \ge 2$, $\mathcal{J}(S)$ carries fibrewise conical singularities along its zero section; when $\dim S = 1$, $\mathcal{J}(S)$ is a smooth manifold.
Source: arXiv (Cornell University)
Author: Yan, Zhiwei
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4416101293
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