On the shape of a D-brane bound state and its topology change

TL;DR

This paper introduces a method to extract the shape and topology of D-brane bound states from matrix models, revealing phase transitions and topology changes consistent with gauge/gravity duality, supported by numerical simulations.

Contribution

It generalizes a prescription for identifying D-brane positions from matrices, enabling direct analysis of shape and topology changes in D-brane bound states.

Findings

Detected topology change during a phase transition in matrix models.

Validated the prescription through Monte-Carlo simulations.

Proposed a criterion for the validity of D-brane position definitions.

Abstract

As is well known, coordinates of D-branes are described by NxN matrices. From generic non-commuting matrices, it is difficult to extract physics, for example, the shape of the distribution of positions of D-branes. To overcome this problem, we generalize and elaborate on a simple prescription, first introduced by Hotta, Nishimura and Tsuchiya, which determines the most appropriate gauge to make the separation between diagonal components (D-brane positions) and off-diagonal components. This prescription makes it possible to extract the distribution of D-branes directly from matrices. We verify the power of it by applying it to Monte-Carlo simulations for various lower dimensional Yang-Mills matrix models. In particular, we detect the topology change of the D-brane bound state for a phase transition of a matrix model; the existence of this phase transition is expected from the gauge/gravity duality, and the pattern of the topology change is strikingly similar to the counterpart in the gravity side, the black hole/black string transition. We also propose a criterion, based on the behavior of the off-diagonal components, which determines when our prescription gives a sensible definition of D-brane positions. We provide numerical evidence that our criterion is satisfied for the typical distance between D-branes. For a supersymmetric model, positions of D-branes can be defined even at a shorter distance scale. The behavior of off-diagonal elements found in this analysis gives some support for previous studies of D-brane bound states.