Barel Skuratovsky: Chaos in gauge theories

Are pure gauge theories chaotic, and what are the right diagnostics? The only meaningful objects are gauge-invariant, so we probe pure $SU(N)$ Yang--Mills at large $N$ with the Wilson loop: the phase a heavy test charge picks up around a closed spatial curve. The loop's time evolution contains exactly two modes: rigid displacement, and splitting where it self-intersects, as a result of the chromomagnetic term $D_{j}F_{ji}$, which acts as a source insertion along the loop. Chaos stems from the non-abelian structure of the theory: the gauge field fails to commute with itself; the gluons carry the charge they couple to. An interesting exception appears in two dimensions, where $D_{j}F_{ji}=0$ because there is no transverse direction: indeed, two-dimensional Yang--Mills is integrable, and we solve the thermal loop correlator exactly. Where the source is present, each split adds two constituents, leading to operator growth: a negative-binomial size distribution of index $\tfrac12$ with exponentially growing mean, organized by the loop's geometric propagator, a sum over splitting histories. In a nutshell: chaos in pure gauge theories is a result of the non-abelian group structure, appears only in $d>2$, and is described geometrically, at large $N$, by the splitting of the initial loop. If time allows, we will discuss the calculation of the out-of-time-order correlator (OTOC) of these loops via ladder diagrams, giving $\lambda_{L}\propto\lambda T\ln\tfrac{1}{\lambda}$ at small 't~Hooft coupling $\lambda$, and outline a geometric derivation of the Bethe-Salpeter kernel.