Z k parafermion theory
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introduction
- parafermionic Hilbert space
- defined by the algebra of parafermionic fields \(\psi_1\) and \(\psi _1^{\dagger }\) of dimension 1-1/k and central charge
\[c=\frac{k \dim \mathfrak{g}}{k+h^{\vee}}-\operatorname{rank}\mathfrak{g}=\frac{3k}{k+2}-1=\frac{2(k-1)}{(k+2)}\] where \(\mathfrak{g}=\mathfrak{sl}_2\) and \(k=2\)
- the highest-weight modules are parametrized by an integer (Dynkin label) \(l\) with \(0\leq l < k\)
- \(\mathbb{Z}_k\) parafermion theory is known to be equivalent to the coset \(\hat{\text{su}}(2)_k/\hat{u}(1)_k\)
- Kac and Peterson (1984) obtained expression for the parafermion characters
- Lepowsky-Primc (1985) expression in fermionic form
- third expression
examples
- \(k=1\), Ising models
- \(k=2\), 3-states Potts model
\(\mathbb{Z}_{n+1}\) theory
- central charge\(\frac{2n}{n+3}\)
history
- String functions and branching functions
- CFT on torus and modular invariant partition functions
- Graded parafermion theory
computational resource
expositions
- Gepner, Level Two String Functions and Rogers Ramanujan Type Identities
- http://physics.stackexchange.com/questions/76617/what-is-parafermion-in-condensed-matter-physics
articles
- Bianchini, Davide, Elisa Ercolessi, Paul A. Pearce, and Francesco Ravanini. ‘RSOS Quantum Chains Associated with Off-Critical Minimal Models and \(\mathbb{Z}_n\) Parafermions’. arXiv:1412.4942 [cond-Mat, Physics:hep-Th, Physics:math-Ph], 16 December 2014. http://arxiv.org/abs/1412.4942.
- Fateev, V. A., and Al. B. Zamolodchikov. “Integrable Perturbations of ZN Parafermion Models and the O(3) Sigma Model.” Physics Letters B 271, no. 1–2 (November 14, 1991): 91–100. doi:10.1016/0370-2693(91)91283-2.
- Bilal, Adel. “Bosonization of ZN Parafermions and su(2)N KAČ-Moody Algebra.” Physics Letters B 226, no. 3–4 (August 10, 1989): 272–78. doi:10.1016/0370-2693(89)91194-5.