Computing the spectral action for fuzzy geometries: from random noncommutative geometry to bi-tracial multimatrix models

Abstract

A fuzzy geometry is a certain type of spectral triple whose Dirac operator crucially turns out to be a finite matrix. This notion was introduced in [J. Barrett, J. Math. Phys. 56, 082301 (2015)] and accommodates familiar fuzzy spaces like spheres and tori. In the framework of random noncommutative geometry, we use Barrett's characterization of Dirac operators of fuzzy geometries in order to systematically compute the spectral action S(D) = Trf (D) for 2ndimensional fuzzy geometries. In contrast to the original Chamseddine-Connes spectral action, we take a polynomial f with f (x) → ∞ as |x| → ∞ in order to obtain a well-defined path integral that can be stated as a random matrix model with action of the type S(D) = N • tr F + i tr Ai • tr Bi, being F, Ai and Bi noncommutative polynomials in 2 2n−1 complex N × N matrices that parametrize the Dirac operator D. For arbitrary signature-thus for any admissible KO-dimension-formulas for 2-dimensional fuzzy geometries are given up to a sextic polynomial, and up to a quartic polynomial for 4-dimensional ones, with focus on the octo-matrix models for Lorentzian and Riemannian signatures. The noncommutative polynomials F, Ai and Bi are obtained via chord diagrams and satisfy: independence of N ; self-adjointness of the main polynomial F (modulo cyclic reordering of each monomial); also up to cyclicity, either self-adjointness or anti-self-adjointness of Ai and Bi simultaneously, for fixed i. Collectively, this favors a free probabilistic perspective for the large-N limit we elaborate on.

Phase Transition in Random Noncommutative Geometries

“…Another method to gain insight into these models might be through blobbed topological recursion [5,2]. Additionally one might be able to apply results from Free Probability theory as suggested in [19]. Finally, in a different but related direction, it would be interesting to apply the random matrix theory techniques we used in this paper to finite spectral triples based on operator systems developed recently in [10].…”

“…In [1,2], formal aspects of these models and their generalizations is studied through topological recursion techniques. In [19] the algebraic structure of the action functional of these models is further analyzed.…”

Spectral Statistics of Dirac Ensembles

“…Each term of D 2 consists of two linearly independent matrices tensored with some commutator or anticommutator of some skew-Hermitian or Hermitian random matrix. Using Proposition 3.5 of [21] we know…”

“…Proof of this can be seen from using the explicit formulas given in section 4 of [21] and knowing that all odd trace matrix powers contribute nothing in the large N limit, see Appendix A. This result may very well be true for higher powers but since no general formula for trace of powers of D for this class of models is known, it is difficult to prove such a result.…”

Riemannian Geometry of a Discretized Circle and Torus

“…Yet there is currently no method to recover all geometric objects like Ricci tensor or the torsion through the spectral methods. There exists a huge discrepancy between the usual methods of recovering the geometric objects like the scalar of curvature for the manifolds and their deformations and the attemt to use of spectral methods [3,4,21,27] in the finite-dimensional case.…”