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

• according to Virasoro
• When I went back to Wisconsin I began to look carefully at the low lying resonances of the model by calculating by brute force their couplings to n-ground states and checking whether there were cancellations.
• My luck then was a direct consequence of my laziness.
• I knew that calculations were much simpler if α(0) = 1 (this as a by-product of an earlier work on an alternative dual, crossing-symmetric amplitude), so I was routinely working at that value.
• Thus when I found that at the first level the ghost decoupled I could continue to the second level and there find that there were some additional decouplings.
• At that exact moment I heard from Gabriele the unwelcome news that at least two groups had derived the first level results. Fortunately, by then I was used to this kind of frustration and did not rush to publish but continued trying to simplify the calculation. The paper still shows how messy the original calculation was, but how it simplifies considerably once the Fubini–Veneziano [FV69] creation–annihilation operators were used.
• Furthermore, written in that way, the generalization to the mth level became trivial.
• The operator turns out to create a resonance uncoupled to any number of ground states
• Thus there are as many uncoupled resonances as there are ghosts and therefore I assumed that all the ghosts had been killed. I was worried that I was trading ghosts for a tachyon.
• On the other hand, I happily dismissed the possibility that I could be killing good resonances and leaving ghosts alive.
• I have checked that this was not the case for the first two levels but it was Thorn, Brower and Goddard that took this issue seriously.
• the operators \(L_m\)'s showed up in 1971 paper by Veneziano and Fubini
• 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

• Virasoro, M. A. “The Little Story of an Algebra.” In String Theory and Fundamental Interactions, edited by Maurizio Gasperini and Jnan Maharana, 137–44. Lecture Notes in Physics 737. Springer Berlin Heidelberg, 2008. http://link.springer.com/chapter/10.1007/978-3-540-74233-3_6.
• Douglas Lundholm, The Virasoro algebra and its representations in physics , January 10, 2005
• Goddard, Peter, and David Olive. “Kac-Moody and Virasoro Algebras in Relation to Quantum Physics.” International Journal of Modern Physics A 01, no. 02 (July 1, 1986): 303–414. doi:10.1142/S0217751X86000149.
• Roger, Claude. ‘Sur Les Origines Du Cocycle de Virasoro’. arXiv:1006.0641 [math], 2 June 2010. http://arxiv.org/abs/1006.0641.

articles

• Nieri, Fabrizio. “An Elliptic Virasoro Symmetry in 6d.” arXiv:1511.00574 [hep-Th, Physics:math-Ph], November 2, 2015. http://arxiv.org/abs/1511.00574.
• Dykes, Kathlyn. “Length Two Extensions of Modules for the Witt Algebra.” arXiv:1510.06792 [math], October 22, 2015. http://arxiv.org/abs/1510.06792.
• 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.
• Fubini, S, and G Veneziano. “Algebraic Treatment of Subsidiary Conditions in Dual Resonance Models.” Annals of Physics 63, no. 1 (March 1971): 12–27. doi:10.1016/0003-4916(71)90295-8.
• 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.
• [FV69] 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.
• Chiu, Charles B., Satoshi Matsuda, and Claudio Rebbi. “Factorization Properties of the Dual Resonance Model: A General Treatment of Linear Dependences.” Physical Review Letters 23, no. 26 (December 29, 1969): 1526–30. doi:10.1103/PhysRevLett.23.1526.
• Gliozzi, F. “Ward-like Identities and Twisting Operator in Dual Resonance Models.” Lettere Al Nuovo Cimento 2, no. 18 (December 1, 1969): 846–50. doi:10.1007/BF02755080.

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