Fermionic Localization of the Schwarzian Theory

TL;DR

The authors show that the Schwarzian theory, describing the low-energy, reparametrization-driven dynamics of SYK and certain dilaton gravities, has a path integral that is one-loop exact via fermionic localization. By modeling the integration space as the symplectic quotient diff(S^1)/SL(2,R) and applying the Duistermaat-Heckman framework, they derive the exact partition function Z(g) and analyze the associated density of states, including a careful discussion of the measure and gauge zero-modes. They then extend the framework to Virasoro coadjoint orbits, Virasoro-Kac-Moody, and super-Virasoro generalizations, showing that one-loop exactness persists across these families with distinct saddle structures and zero-mode counting. The results connect to SYK's triple-scaling limit and provide predictions for spectra and densities of states that align with numerical SYK studies, while clarifying the role of coadjoint-orbit geometry in the low-energy dynamics of these systems.

Abstract

The SYK model is a quantum mechanical model that has been proposed to be holographically dual to a $1+1$-dimensional model of a quantum black hole. An emergent "gravitational" mode of this model is governed by an unusual action that that has been called the Schwarzian action. It governs a reparametrization of a circle. We show that the path integral of the Schwarzian theory is one-loop exact. The argument uses a method of fermionic localization, even though the model itself is purely bosonic.

Figures (2)

• Figure 1: Top: Exact density of states of the Schwarzian theory near the ground state, for different amounts of supersymmetry. For the two $\mathcal{N} =2$ cases we chose $\hat{q} = 3$ and we omitted the $\delta(E)$ contribution to the spectrum. Bottom: The numerical density of states from exact diagonalization of (super) SYK with a four-fermion Hamiltonian. The Hilbert space dimensions used are respectively $2^{16},2^{16},2^{18},2^{19}$. The $\mathcal{N} = 0$ case is an average using data from Cotler:2016fpe. The other curves are for a single realization.
• Figure 2: Typical configurations of the discretized $\phi$ varible that contribute to $Z(g)$ for the discretized Schwarzian theory. The configurations were generated using the Metropolis algorithm, with a lattice of $n = 300$. The figure at left contains only small fluctuations about the saddle point, which would be a straight line $\phi_j = \frac{2\pi j}{n}$.