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

관련된 항목들

• Vertex Algebras
• BRST Cohomology
• minimal models
• bosonic characters of minimal models

매스매티카 파일 및 계산 리소스

• 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

exposition

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

articles

• Quantum Group Structure of the q-Deformed Virasoro Algebra
  
  • Haihong Hu
• Unitary representations of the Virasoro and super-Virasoro algebras
  
  • P. Goddard, A. Kent and D. Olive, Comm. Math. Phys. 103, no. 1 (1986), 105–119.

• Conformal invariance, unitarity and critical exponents in two dimensions
  
  • Friedan, D., Qiu, Z. and Shenker, S., Phys. Rev. Lett. 52 (1984) 1575-1578

• Verma modules over the Virasoro algebra
  
  • B.L. Feigin and D.B. Fuchs, Lecture Notes in Math. 1060 (1984), pp. 230–245.

• Infinite conformal symmetry in two-dimensional quantum field theory
  
  • Alexander Belavin, Alexander Polyakov and Alexander Zamolodchikov, Nucl. Phys. B241 (1984) 333–380