Virasoro algebra

introduction

• Virasoro algebra could be pre-knowledge for the study of CFT
• important results on Virasoro algebra are
  • (i)Kac Determinant Formula
  • (ii) discrete series (A. J. Wassermann, Lecture Notes on the Kac-Moody and Virasoro algebras)
  • (iii) GKO construction of discrete series (this is added on 22nd of Oct, 2007)
• representation theory, see highest weight representation of Vir
  • full classification of all CFT's for c<1
  • no classification for c>1

Virasoro algebra

• Lie algebra of vector fields on the unit circle

\[f(z)\frac{d}{dz}\]

• commutator

\[[f(z)\frac{d}{dz},g(z)\frac{d}{dz}]=(fg'-f'g)\frac{d}{dz}\]

• Virasoro generators

\[L_n=-z^{n+1}\frac{d}{dz}\]

• they satisfy the following relation (Witt algebra)

\[[L_m,L_n]=(m-n)L_{m+n}\]

• Homological algebra tells that there is 1-dimensional central extension of Witt algebra
• taking a central extension of lie algebras, we get the Virasoro algebra \(L_n,n\in \mathbb{Z}\)

\[[c,L_n]=0\] \[[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n}\]

central charge and conformal weight

• highest weight representation
• \(c\) is called the central charge
• \(h\) is sometimes called a conformal dimension or conformal weights

Verma module

• highest weight representation of Vir

unitarity and ghost

• unitarity means the inner product in the space of states is positive definite (or semi-positive definite)
• A state with negative norm is called a ghost.
• If a ghost is found on any level the represetation cannot occur in a unitary theory

unitary irreducible representations

• highest weight representation of Vir

affine Lie algebras

• a highest weight representation V of an affine Kac-Moody algebra gives unitary representation of the Virasoro algebra
• This is because V is a unitary highest weight representation of the AKMA.
• Read chapter 4 of Kac-Raina on Wedge space
• unitary representations of affine Kac-Moody algebras

character of minimal models

• minimal models
• bosonic characters of minimal models

No-Ghost theorem

• refer to the No-ghost theorem and the construction of moonshine module and monster Lie algbera

history

• the central extension of the Virasoro algebras was not observed until 1974, and even then only in private communication from J. Weis quoted in [CT74]

related items

• Deformed Virasoro algebra
• vertex algebras
• BRST quantization and cohomology
• Virasoro singular vectors
• minimal models
• bosonic characters of Virasoro minimal models(Rocha-Caridi formula)

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxNHBISlk2T1E1cVU/edit
• http://ask.sagemath.org/question/3289/a-sage-implementation-of-the-virasoro-algebra-and

encyclopedia

• Virasoro algebra by V. Kac
• http://en.wikipedia.org/wiki/Virasoro_algebra

questions

• http://mathoverflow.net/questions/117287/motivation-of-virasoro-algebra

exposition

• Douglas Lundholm, The Virasoro algebra and its representations in physics , January 10, 2005

articles

• Hu, Haihong. “Quantum Group Structure of the Q-Deformed Virasoro Algebra.” Letters in Mathematical Physics 44, no. 2 (April 1, 1998): 99–103. doi:10.1023/A:1007475521529. http://www.springerlink.com/content/kn757431511020g2/
• Goddard, P., A. Kent, and D. Olive. “Unitary Representations of the Virasoro and Super-Virasoro Algebras.” Communications in Mathematical Physics 103, no. 1 (1986): 105–19. http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1104114626
• Friedan, Daniel, Zongan Qiu, and Stephen Shenker. “Conformal Invariance, Unitarity, and Critical Exponents in Two Dimensions.” Physical Review Letters 52, no. 18 (April 30, 1984): 1575–78. doi:10.1103/PhysRevLett.52.1575. http://prola.aps.org/abstract/PRL/v52/i18/p1575_1
• Feigin, B. L., and D. B. Fuchs. “Verma Modules over the Virasoro Algebra.” In Topology, edited by Ludwig D. Faddeev and Arkadii A. Mal’cev, 230–45. Lecture Notes in Mathematics 1060. Springer Berlin Heidelberg, 1984. http://link.springer.com/chapter/10.1007/BFb0099939.
• Belavin, A. A., A. M. Polyakov, and A. B. Zamolodchikov. “Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory.” Nuclear Physics B 241, no. 2 (July 23, 1984): 333–80. doi:10.1016/0550-3213(84)90052-X.
• [CT74] Chodos, Alan, and Charles B. Thorn. ‘Making the Massless String Massive’. Nuclear Physics B 72, no. 3 (18 April 1974): 509–22. doi:10.1016/0550-3213(74)90159-X.
• Virasoro, M. A. ‘Subsidiary Conditions and Ghosts in Dual-Resonance Models’. Physical Review D 1, no. 10 (15 May 1970): 2933–36. doi:10.1103/PhysRevD.1.2933.
• Fubini, S., and G. Veneziano. ‘Level Structure of Dual-Resonance Models’. Il Nuovo Cimento A 64, no. 4 (1 December 1969): 811–40. doi:10.1007/BF02758835.