Fortuity with a single matrix

TL;DR

This work analyzes a single $U(N)$ adjoint fermionic matrix quantum mechanics that is exactly solvable and rich in fortuitous, finite-$N$ BPS states. The authors solve the spectrum via ${\rm SU}(N)$ representation theory, identifying the maximal representation ${\mathbf{r}}_*$ as hosting zero-energy states with an exponentially large degeneracy, while fortuity is shown to hinge on the finite-$N$ structure of the theory. They then study the fortuitous sector in the large-$N$ limit using a unitary matrix integral with the ${\mathbf{r}}_*$ character, uncovering delicate saddle-point structures and a mechanism to “follow $N$” by decoupling the last row/column of fermions; at special fugacities, multiple saddle configurations cancel as $N$ is reduced, causing the fortuitous states to disappear. The paper also discusses generalizations to follow $N$ in other unitary integrals, singlet fortuity, and a minimal chaotic matrix model with $R$-charge concentration, linking exact solvability to broader questions about fortuity in holography and black-hole microstate counting.

Abstract

We construct and study a supersymmetric quantum mechanical model with a single $U(N)$ adjoint fermionic matrix. The model is exactly solvable yet contains a large number of fortuitous states. We investigate these states exactly at finite $N$ and, in the large $N$ limit, via a unitary matrix model. In particular, we develop a way to "follow $N$" in the unitary matrix integral and study how the answer of the integral depends sensitively on the value of $N$.

Figures (8)

• Figure 1: In our model, as well as in models such as the SUSY SYK model, the fortuitous states depend on $N$ in an extremely sensitive way. After an exponential number of states become BPS at $N= N_*$, they immediately become "null" at $N = N_* - 1$.
• Figure 2: We can form a highest weight state $\ket{\lambda}$ by occupying all the fermion modes in the lower triangle. The figure illustrates the case when $N=4$. The arrow shows the action of a positive-root generator $J_{E^{23}}$ on $\Psi_{43}$. It transforms $\Psi_{43}$ into $\Psi_{42}$, which is already occupied, and thus annihilating the state.
• Figure 3: The Young diagram of the maximal representation ${\textbf{r}}_*$. The $i$-th row has $2N-2i$ boxes and it has the maximum number of $N-1$ rows.
• Figure 4: We fix the number of rows of the staircase Young diagram as $N_*$ and compare it with the value of $N$. See main text for discussion.
• Figure 5: At special values $q =e^{\pm {\mathrm i} \frac{\pi}{3}}$, one can find a family of saddle points. This includes (a) a uniform distribution spanning an interval of length $2\pi/3$, as in (\ref{['rhouniform']}), but also includes multicut as well as highly fragmented saddle points shown in (b), arising by shifting blocks of eigenvalues by multiples of $2\pi/3$.