Bootstrapping Yang-Mills matrix integrals

Abstract

We revisit the large $N$ limit of bosonic $D$-matrix Yang-Mills integrals using two complementary bootstrap methods. In the positivity bootstrap, we obtain bounds for $\langle \text{tr} XX \rangle$ and $\langle \text{tr} XXXX \rangle$ at various length cutoffs $L_{\max}$. For $D=3$, we do not find an isolated region until $L_{\max}=12$. For larger $D$, the allowed regions become islands at $L_{\max}=8$ and shrink rapidly as $L_{\max}$ increases. The precision of some $L_{\max}=12$ islands is comparable to that of Monte Carlo estimates. For a fixed $L_{\max}$, the allowed region also shrinks with $D$ and converges to the large $D$ expansion results. We further deduce the analytic expressions of various types of trajectories and eigenvalue distributions at large $D$. Based on these explicit formulas, we propose some ansatz for the analytic trajectory bootstrap and obtain accurate results for finite $D$.