Tyler Schulz | Searcharxiv

arXiv2025

Supercritical phase transition on the Toeplitz algebra of $\mathbb N^\times \ltimes \mathbb Z$

We study the high-temperature equilibrium for the C*-algebra $\mathcal T (\mathbb N^\times \ltimes \mathbb N)$ recently considered by an Huef, Laca and Raeburn. We show that the simplex of KMS$_\beta$ states at each inverse temperature $\beta$ in the critical interval $(0,1]$ is a Bauer simplex whose space of extreme points is homeomorphic to $\mathbb N \sqcup\{\infty\}$. This is in contrast to the uniqueness of equilibrium at high temperature observed in previously considered systems arising from number theory. We also show that quotients of our system exhibit spontaneous symmetry-breaking by finite cyclotomic Galois groups and establish their connection to the Bost-Connes phase transition.

math.OA↗

arXiv2020

Zeta functions and topology of Heisenberg cycles for linear ergodic flows

Placing a Dirac-Schr\"odinger operator along the orbit of a flow on a compact manifold $M$ defines an $\R$-equivariant spectral triple over the algebra of smooth functions on $M$. We study some of the properties of these triples, especially their zeta functions, which have the form $\trace (fH^{-s})$ with $f$ the restriction to $\R$ of a function on $M$ and $H = -\frac{\partial^2}{\partial x^2} + x^2$ the harmonic oscillator. The meromorphic continuation property and pole structure of these zeta functions is related to ergodic time averages in dynamics. The construction reproduces the `Heisenberg cycles' of Lesch and Moscovici, in the case of the periodic flow on the circle, where it produces a spectral triple over the smooth irrational torus in the irrational rotation algebra $A_\h$. We strengthen a result of these authors, showing that the zeta function $\trace (aH^{-s})$ extend mermomorphically for any element $a$ of the C*-algebra $A_\h$. Another variant of the construction produces a spectral cycle for $A_\h\otimes A_{1/\h}$ and a spectral triple over a suitable subalgebra with the meromorphic continuation property if $\h$ satisfies a Diophantine condition. The class of this cycle defines a fundamental class in the sense that it determines a KK-duality. We employ the Local Index Theorem of Connes and Moscovici in order to elaborate an index theorem of Connes for certain classes of differential operators on the line and compute the intersection form on K-theory induced by the fundamental class.

math.KT↗