Renormalization footprints in the phase diagram of the Grosse-Wulkenhaar model

Abstract

We construct and analyze a phase diagram of a self-interacting matrix field coupled to curvature of the non-commutative truncated Heisenberg space. In the infinite size limit, the model reduces to the renormalizable Grosse-Wulkenhaar's. The curvature term is crucial to renormalization. When turned off, the triple point collapses into the origin as matrices grow larger. When turned on, the triple point shifts away proportionally to the coupling strength and matrix size. Coupling attenuation that renormalizes the Grosse-Wulkenhaar model cannot contain the shifting, and the translational symmetry-breaking stripe phase escapes to infinity, taking away the problematic UV/IR mixing.

Second moment fuzzy-field-theory-like matrix models

“…The numerical analysis of these models led to most of the results for the phase structure of the fuzzy field theories. Moreover, recent simulations of the matrix model describing a version of noncommutative field theory that is free of the UV/IR mixing showed signs of retreat of the non-uniform order phase [22,23], further confirming the connection between the existence of this phase and the UV/IR mixing.…”

Fuzzy field theories in the string modes formalism

“…One setting where the consequences of such a discrete structure can be studied is the case of fuzzy spaces -finite mode approximations to compact manifolds, which retain the symmetries of the original space. Field theories on such spaces are described in terms of matrices and can be studied analytically [4][5][6][7][8][9][10][11][12][13][14][15] or numerically [16][17][18][19][20][21][22][23][24][25][26][27][28]. The standard approach using the momentum eigenstates leads to the most important consequence of the fuzzy structure -the UV/IR mixing.…”