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)

 

 

Virasoro algebra

• Lie algebra of vector fields on the unit circle
  \(f(z)\frac{d}{dz}\)
  \(L_n=-z^{n+1}\frac{d}{dz}\)
  
• they satisfy the following relation
  \([L_m,L_n]=(m-n)L_{m+n}\)
  
• 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

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

 

 

Verma module

• start with given c and h
• construct \(M(c,h)\)
  
  • quotients from the Universal enveloping algebra
  • tensor product from the one dimensional Borel subalgebra representations
• there exists a unique contravariant hermitian form
• http://mathworld.wolfram.com/HermitianForm.html
• http://en.wikipedia.org/wiki/Hermitian_adjoint
• http://en.wikipedia.org/wiki/Hermitian_form#Hermitian_form
• contravariance means
  
  • \(L_n\) and \(L_{-n}\) act as adjoints to each other, i.e.
    \(<{L_n}v,w>=<w,L_{-n}w>\)
    
• a natural grading given by the \(L_0\)-eigenvalues
• contains a unique maximal submodule, and its quotient is the unique (up to isomorphism) irreducible representation with highest weight
• When is \(M(c,h)\) unitary? 
• to understand the submodules of the Verma module, we refer to Feigin and Fuks.

 

 

 

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

• They are classified by c>1 and c<1 case.
  
  • \(c> 1, h > 0\) positive definite
  • \(c\geq 1, h \geq 0\) positive semi-definite
  • \(0<c<1, h> 0\) with Kac determinant condition
    
    • called the discrete series representations or minimal models

 

 

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

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• Virasoro algebra, Dedekind eta-function and Specialized Macdonald's identities

 

 

No-Ghost theorem

• refer to the No Ghost theorem

 

 

관련된 항목들

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

 

 

encyclopedia

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

 

 

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