Abstract
Using numerical methods for finding Ricci-flat metrics, we explore the spectrum of local operators in two-dimensional conformal field theories defined by sigma models on Calabi-Yau targets at large volume. Focusing on the examples of K3 and the quintic, we show that the spectrum, averaged over a region in complex structure moduli space, possesses the same statistical properties as the Gaussian orthogonal ensemble of random matrix theory.
Highlights
Random matrices are a hallmark of quantum chaos
Focusing on the examples of K3 and the quintic, we show that the spectrum, averaged over a region in complex structure moduli space, possesses the same statistical properties as the Gaussian orthogonal ensemble of random matrix theory
Some of the most prominent features of quantum chaos are the repulsion of nearby energy levels and the long-range rigidity of the spectrum, both of which contrast sharply with the statistics of a Poisson random process. (See the nearest-neighbor level spacings (NNS) and Σ2 plots in figure 1.)
Summary
Random matrices are a hallmark of quantum chaos (see [1–4] and references therein for a review). RMT-like statistics for the energy spectrum is often taken to be a defining property of quantum chaos and a signature of the kind of classical dynamics underlying the quantum system.2 Despite this long history, we note that there are few universal or rigorous results on chaos and ergodicity, other than for 2d billiards [13], compact manifolds with negative curvature [14–17], and related systems. A particular diagnostic, the spectral form factor (SFF), has been used to investigate spectral correlations in quantum many-body systems such as the SYK model [26–29] as well as in the context of black hole physics and gravitational systems [30–34] In this situation, the essential result is that the black hole energy levels are discrete, non-degenerate and chaotic. This behavior, suitably averaged, is precisely captured by random matrix statistics and leads to features known colloquially as the ramp and the plateau. (See the SFF plot in figure 1.)
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