spectral triple

A spectral triple $\left(A, H, D\right)$ is the core analytic structure in Noncommutative Geometry, generalizing Riemannian spin manifolds by encoding geometric and metric data via an algebra, a Hilbert space, and a Dirac-type operator.

Spectral triples, $\left(A, H, D\right)$, are the foundational analytic structure in Noncommutative Geometry (NCG), a concept pioneered by Alain Connes. The triple comprises a $\ast$-algebra $A$, a separable Hilbert space $H$, and an unbounded self-adjoint Dirac-type operator $D$. This framework replaces a classical Riemannian spin manifold's smooth functions and Dirac operator with operator-algebraic data. The key constraint is the bounded commutator condition, $\left[a, D\right] < \infty$ for all $a \in A$. This analytic data is sufficient to recover the manifold's metric structure and allows for the generalization of the Atiyah-Singer index theorem. A critical application is the almost-commutative model, which successfully incorporates the Standard Model of particle physics within this geometric framework.

https://en.wikipedia.org/wiki/Spectral_triple