[Paper Review] BPS Chaos

Yiming Chen, Henry W. Lin+1 more|arXiv (Cornell University)|Jul 28, 2024

Complex Systems and Decision MakingDecision Sciences3 citations

TL;DR

This paper investigates quantum chaos in BPS black hole microstates using the LMRS diagnostic for chaos in the boundary CFT. It finds that 1/2- and 1/4-BPS states in $$\mathcal{N}=4$ SYM and 2-charge D1-D5 fuzzballs exhibit only weak chaos, with Thouless time scaling as a power of $N$, in contrast to finite-horizon BPS black holes that are strongly chaotic with order-one Thouless time. The results resolve a tension between smooth horizonless geometries and strong chaos in holography.

ABSTRACT

Black holes are chaotic quantum systems that are expected to exhibit random matrix statistics in their finite energy spectrum. Lin, Maldacena, Rozenberg and Shan (LMRS) have proposed a related characterization of chaos for the ground states of BPS black holes with finite area horizons. On a separate front, the "fuzzball program" has uncovered large families of horizon-free geometries that account for the entropy of holographic BPS systems, but only in situations with sufficient supersymmetry to exclude finite area horizons. The highly structured, non-random nature of these solutions seems in tension with strong chaos. We verify this intuition by performing analytic and numerical calculations of the LMRS diagnostic in the corresponding boundary quantum system. In particular we examine the 1/2 and 1/4-BPS sectors of $\mathcal{N}=4$ SYM, and the two charge sector of the D1-D5 CFT. We find evidence that these systems are only weakly chaotic, with a Thouless time determining the onset of chaos that grows as a power of $N$. In contrast, finite horizon area BPS black holes should be strongly chaotic, with a Thouless time of order one. In this case, finite energy chaotic states become BPS as $N$ is decreased through the recently discovered "fortuity" mechanism. Hence they can plausibly retain their strongly chaotic character.

Motivation & Objective

• To resolve the tension between the highly structured, non-random nature of horizonless BPS geometries (fuzzballs) and the expectation of strong quantum chaos in black holes.
• To test whether BPS states in supersymmetric systems—specifically 1/2- and 1/4-BPS states in $$\mathcal{N}=4$ SYM and 2-charge D1-D5 CFT—exhibit strong chaos as predicted by random matrix theory.
• To quantify the degree of chaos using the LMRS diagnostic, particularly the Thouless time, in the boundary quantum systems dual to fuzzball geometries.
• To explore the role of the 'fortuity' mechanism in enabling strongly chaotic behavior in 1/16-BPS states as $N$ is reduced.

Proposed method

• Analytically and numerically compute the spectral form factor and level spacing statistics in the BPS subspace of $$\mathcal{N}=4$ SYM and D1-D5 CFT using the one-loop dilatation operator and projected twist operators.
• Use the LMRS diagnostic to assess chaos by measuring the Thouless time, defined as the onset of the linear ramp in the spectral form factor.
• Construct a basis of the Hilbert space for 1/4-BPS states using fermionic Fock space and compute the action of the dilatation operator via numerical diagonalization.
• Apply group-theoretic techniques to analyze the spectrum of the projected operator $\widehat{\Sigma}_2$, leveraging the symmetric group $S_N$ and character theory to compute eigenvalues and degeneracies.
• Perform randomized simulations by sampling Gaussian-distributed coupling constants $C$ to test whether introducing disorder enhances chaos, comparing Thouless time scaling.
• Use the convolution theorem on the Cayley graph of $S_N$ to interpret $\widehat{\Sigma}_2$ as a discrete kinetic term and derive its eigenvalues via Fourier modes (characters) of irreducible representations.

Experimental results

Research questions

• RQ1 Do 1/2-BPS states in $$\mathcal{N}=4$ SYM exhibit strong quantum chaos as predicted by random matrix theory?
• RQ2 What is the scaling of the Thouless time in 1/4-BPS states of $$\mathcal{N}=4$ SYM, and does it indicate weak or strong chaos?
• RQ3 How does the Thouless time scale in the 2-charge D1-D5 CFT fuzzball system, and does it support the presence of weak chaos?
• RQ4 Can the 'fortuity' mechanism explain the transition to strong chaos in 1/16-BPS states as $N$ is reduced?
• RQ5 Does randomizing the coupling constants $C$ in the projected operator $\widehat{\Sigma}_2$ lead to a significant reduction in the Thouless time, indicating enhanced chaos?

Key findings

• The 1/2-BPS sector of $$\mathcal{N}=4$ SYM exhibits weak chaos, with the Thouless time scaling as a power of $N$, indicating a slow onset of chaos.
• In the 1/4-BPS sector of $$\mathcal{N}=4$ SYM, numerical diagonalization of the one-loop dilatation operator shows level spacing statistics consistent with weak chaos and a Thouless time growing with $N$.
• For the 2-charge D1-D5 CFT, the projected twist operator's eigenvalue statistics show no evidence of level repulsion, confirming weak chaos with a Thouless time scaling as a power of $N$.
• The spectrum of the projected operator $\widehat{\Sigma}_2$ in the $C=1$ case exhibits a large number of degeneracies, with the number of distinct eigenvalues scaling as $N^{2.4}$ for $N \geq 10$, suggesting underlying group structure.
• Randomizing the coupling constants $C$ in $\widehat{\Sigma}_2$ leads to a faster onset of the linear ramp in the spectral form factor, but the Thouless time still scales with $N$, indicating persistent weak chaos.
• The 'fortuity' mechanism allows finite-energy chaotic states to become BPS as $N$ decreases, enabling them to retain strongly chaotic behavior, consistent with finite-horizon BPS black holes.