Fundamental weight systems are quantum states

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 \({\mathfrak {g}}{\mathfrak {l}}(n)\)-weight systems on horizontal chord diagrams with N strands may be identified with the Cayley distance kernel at inverse temperature \(\beta = \textrm{ln}(n)\) 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 \(\beta = \text {ln}(n)\) for all \(n = 1,2,3, \ldots \). In particular, this proves that all fundamental \({\mathfrak {g}}{\mathfrak {l}}(n)\)-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.