Question

How to understand the non-abelian DBI action?

Answer 1

String Theory and M-theory, A Modern Introduction" by Becker, Becker and Schwarz. It says that the understanding of the square root of the determinant is to compute the determinant of each of the N2N2 matrix elements. (See below) However, I think the gauge invariant is violated if we compute the trace of the square root in this way. How can I understand it and is there any reference about this way of understanding?

(Actually, I know that in many literature, the non-abelian case is understood as an expansion where one should use the symmetric trace.)

Answer 2

We study the way Lorentz covariance can be reconstructed from Matrix Theory as a IMF description of M-theory.

The problem is actually related to the interplay between a non abelian Dirac-Born-Infeld action and Super-Yang-Mills as its generalized non-relativistic approximation.

All this physics shows up by means of an analysis of the asymptotic expansion of the Bessel functions $K_\nu$ that profusely appear in the computations of amplitudes at finite temperature and solitonic calculations.

We hope this might help to better understand the issue of getting a Lorentz covariant formulation in relation with the $N\to +\infty$ limit.

There are also some computations that could be of some interest in Relativistic Statistical Mechanics.

Old section 3 suppressed, the end of old section 4 is now an appendix. For the obssesed reader, we also stress that the work has nothing to do with any proposal of modification for the DBI action in the non abelian case.