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 $⟨\text{tr}XX⟩$ and $⟨\text{tr}XXXX⟩$ 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 $D$.

v2: 49 pages, 6 figures, discussions improved, typos corrected, references added