Matrix models, gauge theory and emergent geometry

Abstract

We present, theoretical predictions and Monte Carlo simulations, for a simple three matrix model that exhibits an exotic phase transition. The nature of the transition is very different if approached from the high or low temperature side. The high temperature phase is described by three self interacting random matrices with no background spacetime geometry. As the system cools there is a phase transition in which a classical two-sphere condenses to form the background geometry. The transition has an entropy jump or latent heat, yet the specific heat diverges as the transition is approached from low temperatures. We find no divergence or evidence of critical fluctuations when the transition is approached from the high temperature phase. At sufficiently low temperatures the system is described by small fluctuations, on a background classical two-sphere, of a U (1) gauge field coupled to a massive scalar field. The critical temperature is pushed upwards as the scalar field mass is increased. Once the geometrical phase is well established the specific heat takes the value 1 with the gauge and scalar fields each contributing 1/2.

Lectures on Matrix Field Theory

“…The model S + V was studied in [20] and [21] where the instability of the sphere was interpreted along the lines of an emergent geometry phenomena. For vanishing potential m 2 , µ −→ 0 the transition from/to the fuzzy sphere phase was found to have a discontinuity in the internal energy, i.e.…”

2D fuzzy anti-de Sitter space from matrix models

“…This leads to mathematically wellcontrolled toy models for noncommutative field theory and geometry, see e.g. [1][2][3][4][5][6][7][8][9][10] and references therein. However, most of the work so far has been for spaces with Euclidean signature, and it would be desirable to know more about fuzzy spaces with Minkowski signature.…”

Low-dimensional Yang-Mills theories: Matrix models and emergent geometry

“…Monte Carlo simulations are a powerful tool for understanding the nature of the transition and investigating the physics of the transition in more detail. It was found using such simulations that the behavior described by effective potential (2) is a surprisingly good approximation [9], [10]. The results of simulations are shown in Fig.…”