Finite-$N$ Bootstrap Constraints in Matrix and Tensor Models

Abstract

We explore how matrix bootstrap techniques can be used to constrain matrix and tensor models at finite $N$, where $N$ is the dimension of the matrix/tensor, taking a Gaussian model with a quartic interaction as example. For matrix models, we find further evidence that bounds do not depend explicitly on $N$, but rather on properties of multi-trace expectation values. For tensor models, the structure of the Schwinger-Dyson equations allow for bounds that vary as a function of $N$, admitting a broader scan of the parameter space of the theory. In the latter case, we find novel bounds on the two-point function as a function of the quartic coupling of the theory.

Figures (6)

• Figure 1: Bounds for the two-point function $m_2$ as a function of the coupling $g$. The size of the Gram matrix is varied from $4 \times 4$ (top left), to $6 \times 6$ (top right), $8 \times 8$ (bottom left), and finally $12 \times 12$ (bottom right).
• Figure 2: Bounds on $m_2$ vs $g$ at finite $N$ for a $6 \times 6$ Gram matrix. The figure on the left shows bounds for $g < 0$ and the figure on the right shows the bounds for $g > 0$.
• Figure 3: Bounds on $m_2$ vs $g$ at finite $N$ for a $6 \times 6$ Gram matrix. The figure on the left shows bounds when imposing the large $N$ factorization properties and the figure on the right shows bounds when imposing the $N = 1$ factorization properties.
• Figure 4: Bounds for the two-point function of an order 3 tensor as a function of the coupling $g$ for $g \geq 0$. $N$ is chosen to be 1 (left), 4 (center), and 100 (right) respectively. The size of the Gram matrix is $4 \times 4$.
• Figure 5: Bounds for the two-point function of the order 3 tensor model as a function of negative values of the coupling $g$. $N$ is chosen to be 1 (left), 10 (center), and 100 (right) respectively. The size of the Gram matrix is $4 \times 4$.