Z k parafermion theory

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

• http://www.google.com/search?hl=en&tbs=tl:1&q=

related items

• String functions and branching functions
• CFT on torus and modular invariant partition functions
• Graded parafermion theory

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxZzRnSkZJZ0kyZlE/edit

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

• 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.