The mass term in non-Abelian gauge field dynamics on matrix D-branes and T-duality in the $\sigma$-model approach

TL;DR

This paper explores how T-duality extends to non-Abelian gauge fields on matrix D-branes, revealing a mass term in gauge dynamics through RG analysis, contrasting with direct calculations under different boundary conditions.

Contribution

It demonstrates the emergence of a mass term in non-Abelian gauge field dynamics on matrix D-branes via RG analysis, highlighting the role of boundary conditions in direct calculations.

Findings

Mass term appears in RG beta-function analysis.

Direct calculations with fixed boundary conditions do not produce the mass term.

Dynamically realized boundary conditions lead to the mass term.

Abstract

The formal extension of the T-duality rules for open strings from Abelian to non-Abelian gauge field background leads in a well known manner to the notion of matrix valued D-brane position. The application of this concept to the non-Abelian gauge field RG $\beta $-function of the corresponding $\sigma $-model yields a mass term in the gauge field dynamics on the matrix D-brane. The direct calculation in a corresponding D-brane model does $not$ yield such a mass term, if the Dirichlet boundary condition is implemented as a constraint on the integrand in the defining functional integral. However, the mass term arises in the direct calculation for a D-brane model with dynamically realized boundary condition.