Fundamental weight systems are quantum states

TL;DR

This paper demonstrates that fundamental weight systems in knot theory are positive quantum states by linking them to the Cayley distance kernel on symmetric groups, revealing their positivity properties across different parameters.

Contribution

It establishes that fundamental gl(n)-weight systems are quantum states by analyzing the positivity of the Cayley distance kernel, a novel connection in the context of weight systems and quantum states.

Findings

Cayley distance kernel is positive at beta=ln(n) for all n.

Fundamental gl(n)-weight systems are quantum states.

Positivity phases depend on inverse temperature beta.

Abstract

Weight systems on chord diagrams play a central role in knot theory and Chern-Simons theory; and more recently in stringy quantum gravity. We highlight that the noncommutative algebra of horizontal chord diagrams is canonically a star-algebra, and ask which weight systems are positive with respect to this structure; hence we ask: Which weight systems are quantum states, if horizontal chord diagrams are quantum observables? We observe that the fundamental gl(n)-weight systems on horizontal chord diagrams with N strands may be identified with the Cayley distance kernel at inverse temperature beta=ln(n) on the symmetric group on N elements. In contrast to related kernels like the Mallows kernel, the positivity of the Cayley distance kernel had remained open. We characterize its phases of indefinite, semi-definite and definite positivity, in dependence of the inverse temperature beta; and we prove that the Cayley distance kernel is positive (semi-)definite at beta=ln(n) for all n=1,2,3,... In particular, this proves that all fundamental gl(n)-weight systems are quantum states, and hence so are all their convex combinations. We close with briefly recalling how, under our "Hypothesis H", this result impacts on the identification of bound states of multiple M5-branes.