On Multimatrix Models Motivated by Random Noncommutative Geometry I: The Functional Renormalization Group as a Flow in the Free Algebra
Abstract
Random noncommutative geometry can be seen as a Euclidean pathintegral approach to the quantization of the theory defined by the Spectral Action in noncommutative geometry (NCG). With the aim of investigating phase transitions in random NCG of arbitrary dimension, we study the nonperturbative Functional Renormalization Group for multimatrix models whose action consists of noncommutative polynomials in Hermitian and antiHermitian matrices. Such structure is dictated by the Spectral Action for the Dirac operator in Barrett's spectral triple formulation of fuzzy spaces.The present mathematically rigorous treatment puts forward "coordinatefree" language that might be useful also elsewhere, all the more so because our approach holds for general multimatrix models. The toolkit is a noncommutative calculus on the free algebra that allows to describe the generator of the renormalization group flow  a noncommutative Laplacian introduced here  in terms of Voiculescu's cyclic gradient and RotaSaganStein noncommutative derivative. We explore the algebraic structure of the Functional Renormalization Group Equation and, as an application of this formalism, we find the $\beta$functions, identify the fixed points in the large$N$ limit and obtain the critical exponents of $2$dimensional geometries in two different signatures.
 Publication:

Annales Henri Poincaré
 Pub Date:
 September 2021
 DOI:
 10.1007/s00023021010254
 arXiv:
 arXiv:2007.10914
 Bibcode:
 2021AnHP...22.3095P
 Keywords:

 Mathematical Physics;
 High Energy Physics  Theory;
 Mathematics  Operator Algebras;
 58B34;
 81XX (Primary);
 15B52;
 46L54 (Secondary)
 EPrint:
 50 pages + glossary and four appendices. Four figures and some tables. v5: Conform with the Annales Henri Poincar\'e version (except, that the "supplementary material" there is part of the appendices here)