Semiclassical chaos in the Witten model

Abstract

Recently there has been an extremely large amount of interest regarding the SYK model, in
particular regarding its saturation of the chaos bound conjectured by Maldacena, Shenker
and Stanford. A related model, that shares many properties with SYK without the need
for quenched disorder, is Witten’s tensor model. In this thesis we will present the first evidence of
classically chaotic dynamics in a representation of the fermionic Witten model given in terms of
commutative variables. This evidence is obtained by first enacting a transformation that allows
us to express the Majorana fermions that appear in the Witten model in terms of complex
numbers, thus opening the door to the possibility of numerical investigations. The classical
dynamics of this representation are then extracted by way of a stationary phase approximation.
We then examine these dynamics for signs of classical chaos using two numerical methods:
estimation of the largest Lyapunov exponent and the 0-1 test for chaos. We find that the results
of the former are inconclusive, but for the latter we obtain definitive evidence of chaotic
dynamics. The 0-1 test also allows us to spread some light on the possible presence of conserved
quantities in this representation. We end with the derivation of an alternative representation of
the Witten Hamiltonian in terms so called dualised variables, which are also commutative.
Throughout we point out the successes and discuss drawbacks of both representations and
speculate as to how they both may be improved or extended.

Date of Award	30 Sept 2025
Original language	English
Awarding Institution	University of Bristol
Supervisor	Viveka Erlandsson (Supervisor) & Sebastian Muller (Supervisor)