Justification of the D1-string ansatz on the IKKT cosmological background

Establish that the matrix configuration used to represent two parallel D1 strings superposed on the nontrivial emergent cosmological background of the IKKT matrix model is valid. Specifically, prove that the ansatz A_μ^b = A_μ^q + A_μ^{D1} (and the block-diagonal variant combining A_μ^q with A_μ^{1,2}) correctly and consistently describes two parallel D1 strings on the quantum background A_μ^q, thereby justifying its use in the one-loop effective action analysis.

Background

The paper proposes a dynamical mechanism within the IKKT matrix model to explain the SO(9) → SO(3) × SO(6) symmetry breaking, relying on D1-strings as excitations and the absence of forces between parallel D1-strings. To compute the one-loop effective action and argue for vanishing forces on a cosmological background, the authors introduce an ansatz that superposes D1-string matrices onto the quantum background matrices.

This ansatz underpins the core physical argument: only three spatial dimensions can grow large because D1-strings winding those dimensions must annihilate, which requires worldsheet intersections; without forces, intersections are suppressed for more than three large dimensions. A rigorous validation of the ansatz is therefore essential to the paper’s mechanism.