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Fundamental weight systems are quantum states

Corfield, David, Sati, Hisham, Schreiber, Urs (2023) Fundamental weight systems are quantum states. Letters in Mathematical Physics, . (doi:10.1007/s11005-023-01725-4) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:101929)

The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided.
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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.

Item Type: Article
DOI/Identification number: 10.1007/s11005-023-01725-4
Uncontrolled keywords: Distance kernel
Subjects: Q Science > QC Physics > QC20 Mathematical Physics
Divisions: Divisions > Division of Arts and Humanities > Department of Philosophy
Funders: University of Kent (
Depositing User: David Corfield
Date Deposited: 03 Jul 2023 13:58 UTC
Last Modified: 14 Nov 2023 14:21 UTC
Resource URI: (The current URI for this page, for reference purposes)

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