Nonperturbative Stabilization of D-Instantons in the Bosonic IIB Matrix Model

Abstract

We study the bosonic type IIB (IKKT) matrix model and the fate of the D-instanton positions $p_{\mu }^{(i)}$, the diagonal components of the $d$ Hermitian matrices, whose one-loop effective potential infamously drives them to a single point. We show that this collapse is an artifact of the leading (one-loop) truncation, while the actual non-collapse of the $p_{\mu }^{(i)}$ is a nonperturbative effect: it is invisible at one loop but already present in the exact (all-loop) two-body interaction. Since the two-body sector of the $N\times N$ model factorizes into copies of $N=2$, this interaction is captured exactly by the $\mathrm{U}(2)$ model, and we find that the two D-instantons do not collapse onto each other. To set up the computation, we gauge-fix the $\text{U}(N)$ symmetry in a way that keeps the diagonal and off-diagonal sectors distinct, and we handle the residual $\mathrm{U}(1{)}^N$ symmetry with an auxiliary-ghost BRST construction. This construction generates a new ghost four-leg vertex; the resulting Faddeev-Popov determinant admits a systematic large-separation expansion that organizes the effective potential into a many-body decomposition, separating the interaction into two-body, three-body, and higher-body contributions. The exact $N=2$ partition function is finite at finite separation; its naive Lorenz-gauge form develops a negative region at separations of order one, which we trace to a Gribov ambiguity of the Lorenz gauge and resolve with the maximal diagonal gauge--the classical frame containing the perturbative vacuum--where the short-distance force is finite and repulsive, so that the two-body potential develops a stable minimum at finite separation. These results are consistent with a stable, non-collapsed distribution of the $p_{\mu }^{(i)}$; establishing the detailed distribution and full $N$-body non-collapse requires the higher-body potentials and is left to future work.

41 pages, 5 figures; v2: minor modifications, references added