Fundamental weight systems are quantum states
arXiv:2105.02871 · doi:10.1007/s11005-023-01725-4
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.
arXiv comment21 pages; 1 figure; v2: bound sharpened
References in corpus (20)
- Kernel methods in machine learning
- A survey of graphical languages for monoidal categories
- Towards a full solution of the large N double-scaled SYK model
- Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction
- Holographic tensor network models and quantum error correction: A topical review
- Vassiliev knot invariants and Chern-Simons perturbation theory to all orders
- Algebraic Approach to Entanglement and Entropy
- Equivariant Cohomotopy implies orientifold tadpole cancellation
- Twisted Cohomotopy implies M-theory anomaly cancellation on 8-manifolds
- Framed M-branes, corners, and topological invariants
- Resolving brane collapse with 1/N corrections in non-Abelian DBI
- Majorana dimers and holographic quantum error-correcting codes
- The Fundamental Theorem of Vassiliev Invariants
- Twisted Cohomotopy implies level quantization of the full 6d Wess-Zumino term of the M5-brane
- Differential Cohomotopy implies intersecting brane observables via configuration spaces and chord diagrams
- Twisted Cohomotopy implies M5-brane anomaly cancellation
- M/F-Theory as Mf-Theory
- Finite N effects on the collapse of fuzzy spheres
- Vassiliev and Quantum Invariants of Braids
- Geodesic string condensation from symmetric tensor gauge theory: a unifying framework of holographic toy models