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 $$c=\frac{k \dim \mathfrak{g}}{k+h^{\vee}}=\frac{3k}{k+2}-1=\frac{2(k-1)}{(k+2)}$$

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

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

• Fendley, Paul. “Parafermionic Edge Zero Modes in Z_n-Invariant Spin Chains.” arXiv:1209.0472 [cond-Mat, Physics:hep-Th], September 3, 2012. http://arxiv.org/abs/1209.0472.
• Mathieu, Pierre. “Paths and Partitions: Combinatorial Descriptions of the Parafermionic States.” Journal of Mathematical Physics 50, no. 9 (September 1, 2009): 095210. doi:10.1063/1.3157921.
• Cabra, D. C. 1998. “Spinons and Parafermions in Fermion Cosets.” In Supersymmetry and Quantum Field Theory, edited by Julius Wess and Vladimir P. Akulov, 220–229. Lecture Notes in Physics 509. Springer Berlin Heidelberg. http://link.springer.com/chapter/10.1007/BFb0105250.
• 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.
• Gepner, Doron, and Zongan Qiu. 1987. “Modular Invariant Partition Functions for Parafermionic Field Theories.” Nuclear Physics B 285: 423–453. doi:10.1016/0550-3213(87)90348-8.
• Gepner, Doron. 1987. “New Conformal Field Theories Associated with Lie Algebras and Their Partition Functions.” Nuclear Physics B 290: 10–24. doi:10.1016/0550-3213(87)90176-3.

character formula

• Gepner, Doron. “On The Characters of Parafermionic Field Theories.” arXiv:1410.1778 [hep-Th, Physics:math-Ph], October 7, 2014. http://arxiv.org/abs/1410.1778.
• Belavin, Alexander A., and Doron R. Gepner. “Generalized Rogers Ramanujan Identities Motivated by AGT Correspondence.” Letters in Mathematical Physics 103, no. 12 (December 1, 2013): 1399–1407. doi:10.1007/s11005-013-0653-2.
• Genish, A., and D. Gepner. “Level Two String Functions and Rogers Ramanujan Type Identities.” arXiv:1405.1387 [hep-Th, Physics:math-Ph], May 6, 2014. http://arxiv.org/abs/1405.1387.
• Keegan, Sinéad, and Werner Nahm. 2011. “Nahm’s conjecture and coset models.” 1103.4986 (March 25). http://arxiv.org/abs/1103.4986
• Fortin, J.-F., P. Mathieu, and S. O. Warnaar. 2006. “Characters of Graded Parafermion Conformal Field Theory”. ArXiv e-print hep-th/0602248. http://arxiv.org/abs/hep-th/0602248.
• Schilling, Anne, and S. Ole Warnaar. “Conjugate Bailey Pairs.” arXiv:math/9906092, June 14, 1999. http://arxiv.org/abs/math/9906092.
• Georgiev, Galin. ‘Combinatorial Constructions of Modules for Infinite-Dimensional Lie Algebras, II. Parafermionic Space’. arXiv:q-alg/9504024, 25 April 1995. http://arxiv.org/abs/q-alg/9504024.
• Georgiev, Galin. ‘Combinatorial Constructions of Modules for Infinite-Dimensional Lie Algebras, I. Principal Subspace’. arXiv:hep-th/9412054, 6 December 1994. http://arxiv.org/abs/hep-th/9412054.
• B. Feigin and A. Stoyanovsky, Functional models for representations of current algebras and semi-infinite Schubert cells cf. also the short version . Func. Anal. Appl. 28 (1994), p. 15.
• B. Feigin and A. Stoyanovsky, Quasi-particles models for the representations of Lie algebras and geometry of flag manifold. preprint RIMS-942 (1993) hep-th/9308079 http://arxiv.org/abs/hep-th/9308079
• Kedem, R., T. R. Klassen, B. M. McCoy, and E. Melzer. 1993. “Fermionic Sum Representations for Conformal Field Theory Characters.” Physics Letters. B 307 (1-2): 68–76. doi:10.1016/0370-2693(93)90194-M.
• Dasmahapatra, Srinandan. 1993. “String Hypothesis and Characters of Coset CFTs”. ArXiv e-print hep-th/9305024. http://arxiv.org/abs/hep-th/9305024.’
• [KNS93] Kuniba, A., T. Nakanishi, and J. Suzuki. 1993. “Characters in Conformal Field Theories from Thermodynamic Bethe Ansatz.” Mod. Phys. Lett. A8 (1993) 1649-1660 arXiv:hep-th/9301018 (January 7). doi:10.1142/S0217732393001392. http://arxiv.org/abs/hep-th/9301018.
• Gepner, Doron. 1987. “New Conformal Field Theories Associated with Lie Algebras and Their Partition Functions.” Nuclear Physics B 290: 10–24. doi:10.1016/0550-3213(87)90176-3.
• [LP1985] Lepowsky, James, and Mirko Primc. 1985. Structure of the Standard Modules for the Affine Lie Algebra $A^{(1)}_1$. Vol. 46. Contemporary Mathematics. American Mathematical Society, Providence, RI. http://www.ams.org/mathscinet-getitem?mr=814303.
• Kac, Victor G, and Dale H Peterson. 1984. “Infinite-dimensional Lie Algebras, Theta Functions and Modular Forms.” Advances in Mathematics 53 (2) (August): 125–264. doi:10.1016/0001-8708(84)90032-X.