Fortuity in SYK Models

TL;DR

This work investigates fortuity in ${\cal N}=2$ SUSY SYK models, where generic $Q$-cohomology concentrates BPS states in a single ${R}$-charge sector, revealing a deep link between fortuity, Schwarzian dynamics, and near-BPS black holes. The authors formulate the universal "supercharge chaos" conjecture, arguing that near fortuitous states the generic ${q}$-local supercharge is well described by the ${\rm TW}$ random matrix ensemble, with strong chaos and random-matrix statistics observed in BPS-projected operators. They construct and analyze two-flavor SYK toy models that host both fortuitous and monotonous states, showing fortuitous states are highly chaotic (random-like), whereas monotonous states are far more orderly; this is supported by large-${N}$ ${G-}\Sigma$ analyses and LMRS-type numerics. The paper also studies how following ${N}$ (via analytic continuation) leads to a chaos-invasion picture where an exponential number of fortuitous states enter the BPS subspace simultaneously, and it examines how sparsity and higher-supersymmetry theories (e.g., ${\cal N}=4$) affect concentration and bulk interpretations. Overall, the results suggest a universal boundary mechanism—the ${R}$-charge concentration governed by a generic supercharge—that links microscopic cohomology, random matrices, and holographic gravity, with potential implications for the nonlinear charge constraints of SUSY black holes and their bulk duals.

Abstract

We study the fortuity phenomenon in supersymmetric Sachdev-Ye-Kitaev (SYK) models. For generic choices of couplings, all the BPS states in the $\mathcal{N}=2$ SUSY SYK model are fortuitous. The SYK models reveal an intimate connection between fortuity and the Schwarzian description of supersymmetric black holes, reflected in a sharp feature of $R$-charge concentration - microscopically, all the fortuitous states are concentrated in particular charge sectors. We propose that both $R$-charge concentration and the random matrix behavior near the BPS states are key properties of a generic $q$-local supercharge and formulate these as a supercharge chaos conjecture. We expect supercharge chaos to hold universally for supercharges in holographic CFTs near their fortuitous states, potentially providing a microscopic interpretation for the charge constraints of supersymmetric black holes. We also construct SYK models that contain both fortuitous states and monotonous states and contrast their properties, providing further evidence that monotonous states are less chaotic than fortuitous states.

Figures (12)

• Figure 1: If we fix $p$ and take $N\rightarrow\infty$, then generically the map from $\alpha_p$ to $C\wedge \alpha_p$ is full rank apart from the obvious kernel being the $Q$-exact states. In the drawing, we illustrate the case of $q=3$ and the red arrow denotes the $Q$-map from $\mathcal{H}^1$ to $\mathcal{H}^4$.
• Figure 2: (a) When $p \geq \frac{N}{2} - 1$, $Q \ket{\alpha_p} = 0$ has a large number of solutions since we are mapping from a larger space to a smaller one. (b) The BPS states are concentrated in the sectors $p = \frac{N}{2} , \frac{N}{2} \pm 1$. Here we consider $N$ even and $q = 3$.
• Figure 3: (a) In red, we display the one-dimensional line where superconformal solutions exist in the large $N$ limit. The same line can be derived either from the index or the Schwinger-Dyson equations. The dashed line, which labels a cochain complex with fixed $J$, crosses the red line at one point. (b) We show a table containing the number of BPS states in various charge sectors, computed by exact diagonalization of the $N=9$ theory. $\textbf{N}_\psi$ ($\textbf{N}_\chi$) increases from $0$ to $9$ along the horizontal (vertical) direction. We highlight the fortuitous states in red. The dashed line denotes an example of a cochain complex with $J = 10$, along which we find BPS spectra $\{0,0,3231,0,0\}$, which indeed exhibits $R$-charge concentration.
• Figure 4: We compare the location of the BPS states (in red) with the location of the maximal dimension spaces (in green). We see the two lines are close to each other but do not overlap apart from some crossings.
• Figure 5: (a) With additional structure in the coupling $C_{ijk}$, we introduce a family of monotonous states with $\textbf{N}_{\psi} = \textbf{N}_{\chi}$, denoted by the blue line. The location of the fortuitous states, denoted by the red line, is not modified compared to the generic two-flavor model. The two lines are cleanly separated in the large $N$ limit, only crossing in the middle. (b) We show the BPS spectra of the model (\ref{['Qexample']}) at $N=9$, which can be compared with Figure \ref{['fig:twofconcentration']} (b). At $\textbf{N}_{\psi} = \textbf{N}_{\chi}=1$, we find a single monotonous state $\ket{V}$, shown in blue. As opposed to (a), for $N=9$, the fortuitous and monotonous states are not yet cleanly separated. In the sectors that are in purple, we have a mixture of fortuitous and monotonous states built from $V$.

Theorems & Definitions (3)

• Conjecture 1: Supercharge Chaos
• Definition 1: Monotonous cohomology
• Definition 2: Monotonous and fortuitous state