"Z k parafermion theory"의 두 판 사이의 차이
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| 1번째 줄: | 1번째 줄: | ||
==introduction== | ==introduction== | ||
* parafermionic Hilbert space | * parafermionic Hilbert space | ||
| − | * defined by the algebra of parafermionic fields <math>\psi_1</math> and <math>\psi _1^{\dagger }</math> of dimension 1-1/k and central charge $$c=\frac{k \dim \mathfrak{g}}{k+h^{\vee}}=\frac{3k}{k+2}-1=\frac{2(k-1)}{(k+2)}$$ | + | * defined by the algebra of parafermionic fields <math>\psi_1</math> and <math>\psi _1^{\dagger }</math> 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 $k=2$ | where $\mathfrak{g}=\mathfrak{sl}_2$ and $k=2$ | ||
* the highest-weight modules are parametrized by an integer (Dynkin label) $l$ with <math>0\leq l < k</math> | * the highest-weight modules are parametrized by an integer (Dynkin label) $l$ with <math>0\leq l < k</math> | ||
| 8번째 줄: | 9번째 줄: | ||
* Lepowsky-Primc (1985) expression in fermionic form | * Lepowsky-Primc (1985) expression in fermionic form | ||
* third expression | * third expression | ||
| − | |||
==examples== | ==examples== | ||
2014년 10월 20일 (월) 00:39 판
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
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.