Blocking up D-branes : Matrix renormalization ?

TL;DR

This paper explores a matrix renormalization approach to coarse-grain D-brane configurations in string theory, revealing self-similar structures and a tendency for matrices to become more diagonal, with implications for understanding noncommutative geometries.

Contribution

It introduces a novel blocking-up method for D-brane matrices, demonstrating self-similarity in certain geometries and analyzing the behavior of matrix elements under coarse-graining.

Findings

Noncommutative geometries exhibit self-similarity under blocking-up.

Matrices tend to become less off-diagonal through the process.

Blocking-up does not show chaotic behavior, indicating stable physics.

Abstract

Drawing analogies with block spin techniques used to study continuum limits in critical phenomena, we attempt to block up D-branes by averaging over near neighbour elements of their (in general noncommuting) matrix coordinates, i.e.\ in a low energy description. We show that various D-brane (noncommutative) geometries arising in string theory appear to behave sensibly under blocking up, given certain key assumptions in particular involving gauge invariance. In particular, the (gauge-fixed) noncommutative plane, fuzzy sphere and torus exhibit a self-similar structure under blocking up, if some ``counterterm'' matrices are added to the resulting block-algebras. Applying these techniques to matrix representations of more general D-brane configurations, we find that blocking up averages over far-off-diagonal matrix elements and brings them in towards the diagonal, so that the matrices become ``less off-diagonal'' under this process. We describe heuristic scaling relations for the matrix elements under this process. Further, we show that blocking up does not appear to exhibit any ``chaotic'' behaviour, suggesting that there is sensible physics underlying such a matrix coarse-graining. We also discuss briefly interrelations of these ideas with B-fields and noncommutativity.