Bootstrapping Yang-Mills matrix integrals

TL;DR

The paper tackles nonperturbative control of the large-$N$ bosonic $D$-matrix Yang–Mills integrals by pairing positivity bootstrap with an analytic trajectory bootstrap. It develops loop equations, an $\mathrm{O}(D)$ singlet decomposition, and a large-$D$ expansion to produce rigorous bounds for moments such as $\langle \mathrm{tr}\:XX\rangle$ and $\langle \mathrm{tr}\:XXXX\rangle$, across length cutoffs up to $L_{\max}=12$. The positivity bounds form shrinking islands that converge toward large-$D$ predictions, with some $L_{\max}=12$ results rivaling Monte Carlo accuracy, while the analytic trajectory bootstrap uses explicit large-$D$ formulas to construct moment-trajectories and eigenvalue densities that yield accurate finite-$D$ results. Together, these methods yield a coherent picture of the matrix-model observables, offer explicit densities and resolvents, and point to promising directions for extending bootstrap techniques to supersymmetric cases and higher-dimensional settings.

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 cutoff $L_{\max}$. 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 $D$.

Figures (6)

• Figure 1: The positivity bounds at $L_{\max}=4$. From light to dark, the blue curves correspond to the lower bounds of $\left\langle\mathop{\mathrm{tr}}\nolimits XXXX\right\rangle$ for $\left\langle\mathop{\mathrm{tr}}\nolimits XX\right\rangle\geq 0$ with $D=3,4,\dots,10,\infty$. The black solid curve indicates the large $D$ limit of the lower bound. The leading prediction of the large $D$ expansion (orange point) is on the black curve. We insert some $D$-dependent factors so that the large $D$ results are associated with finite coordinates.
• Figure 2: The positivity bounds at $L_{\max}=6$. From light to dark, the green curves correspond to the lower bounds of $\left\langle\mathop{\mathrm{tr}}\nolimits XXXX\right\rangle$ for $D=3,4, 6,10,25, \infty$. The orange dot represents the leading prediction of the large $D$ expansion. The dashed line indicates that the left boundary violates the positive semi-definite condition except for the orange point. As in figure \ref{['fig:Lmax4']}, we insert some $D$-dependent factors for the clarity of the $D\rightarrow\infty$ results.
• Figure 3: Positivity bounds on $\left\langle\mathrm{tr}\,XX\right\rangle\,,\left\langle\mathrm{tr}\,XXXX\right\rangle$ for $D=3,...,10$. From light to dark, the shaded regions correspond to cutoffs $L_{\max}=8,10,12$. The allowed regions shrink rapidly as $L_{\max}$ increases, and become islands at $L_{\max}\ge8$ for $D\geq 4$, but at $L_{\max}=12$ for $D=3$. The red points indicate the Monte Carlo results, while the purple cross represents the $1/D$ expansion series to subleading order, i.e., $\left(\frac{1}{\sqrt{2D}}+\frac{7}{6\sqrt{2}D^{3/2}},\,\frac{1}{D}+\frac{5}{2D^2}\right)$.
• Figure 4: Positivity bounds at $L_{\max}=8$ with $D=20, 50,100, 200, 500$. In the first plot, the islands shrink to the point associated with the leading terms of the large $D$ limit. In the second plot, the islands converge as $D$ grows. The point associated with the subleading terms of the $1/D$ expansion is located on the boundary of the allowed region.
• Figure 5: Positivity bounds at $L_{\max}=10$ with $D=50,100,200,500$. In the first plot, as $D$ increases, the islands shrink to the leading large $D$ results more rapidly than the $L_\text{max}$ islands. In the second plot, the left lower tip of the islands converge to the subleading results of the $1/D$ expansion.