Counting paths in Bratteli diagrams for SU(2)_k
Toufik Mansour, Simone Severini
TL;DR
This paper provides an explicit formula for counting paths in Bratteli diagrams related to SU(2)_k Chern-Simons-Witten theory, connecting it with Dyck paths and Chebyshev polynomials to determine Hilbert space dimensions.
Contribution
It introduces a novel explicit formula for path counting in Bratteli diagrams for any level k, based on a new relation with Dyck paths and Chebyshev polynomials.
Findings
Derived an explicit formula for path counts in Bratteli diagrams
Established a connection with Dyck paths and Chebyshev polynomials
Enabled calculation of Hilbert space dimensions for SU(2)_k theories
Abstract
It is known that the Hilbert space dimensionality for quasiparticles in an SU(2)_k Chern-Simons-Witten theory is given by the number of directed paths in certain Bratteli diagrams. We present an explicit formula for these numbers for arbitrary k. This is on the basis of a relation with Dyck paths and Chebyshev polynomials.
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