Moduli Spaces of Dirac Operators for Finite Spectral Triples

Abstract

Abstract. The structure theory of finite real spectral triples developed by Krajewski and by Paschke and Sitarz is generalised to allow for arbitrary KOdimension and the failure of orientability and Poincaré duality, and moduli spaces of Dirac operators for such spectral triples are defined and studied. This theory is then applied to recent work by Chamseddine and Connes towards deriving the finite spectral triple of the noncommutative-geometric Standard Model.

Understanding truncated non-commutative geometries through computer simulations

“…Examples of these, often called fuzzy, spaces are the fuzzy sphere [7], fuzzy projective spaces [8] or the fuzzy torus [9]. General finite spectral triples have been classified [10,11,12] and parametrized [13]. In the present paper, however, we will be concerned with truncated, not fundamentally finite, spectral triples.…”

Reconstructing manifolds from truncated spectral triples

“…As a smooth embedding Y is automatically bilipschitz, so there are α, β 2 and so its integral under dµ ω is bounded by…”

“…Admittedly, finite-dimensional objects in noncommutative geometry have enjoyed enduring attention. General finite spectral triples have been classified [1,2,3] and parametrized [4], and the Connes metric on these spaces has been studied in depth [5]. However, this framework seems to lack simultaneous presence of 1) a natural link to the continuum in terms of metric spaces and 2) a natural link to the continuum in terms of spectral triples.…”

Asymptotic Safety, Hypergeometric Functions, and the Higgs Mass in Spectral Action Models

“…with S 3 the space of symmetric complex 3 × 3-matrices. (For a more general discussion of moduli spaces of finite spectral triples see also [6].) This space describes the bare parameters that enter the spectral action, in the form of the Dirac operator D A , with D = / ∂ X ⊗ 1 ⊕ γ 5 ⊗ D F the Dirac operator of the product geometry and A the inner fluctuations.…”