Virasoro algebra

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

 

 

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
• 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.

 

unitary 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
      

 

discrete series unitary representations

• c<1 case

 

\(m= 2, 3, 4.\cdots\)

\(c = 1-{6\over m(m+1)} = 0,\quad 1/2,\quad 7/10,\quad 4/5,\quad 6/7,\quad 25/28, \ldots\)

\(h_{r,s}(c) = {((m+1)r-ms)^2-1 \over 4m(m+1)}\)

\(r = 1, 2, 3,\cdots,m-1\)

\(s= 1, 2, 3,\cdots, r\)

• Daniel Friedan, Zongan Qiu, and Stephen Shenker (1984) showed that these conditions are necessary
• Peter Goddard, Adrian Kent and David Olive (1986) used the coset construction or GKO construction (identifying unitary representations of the Virasoro algebra within tensor products of unitary representations of affine Kac-Moody algebras) to show that they are sufficient.

• constructed by GKO construction which uses the representation theory of affine Kac-Moody algebras

 

 

character of minimal models

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

 

 

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

 

 

No-Ghost theorem

• refer to the No Ghost theorem.

 

관련된 다른 주제들

• Vertex Algebras
• BRST Cohomology

 

참고할만한 자료

• Quantum Group Structure of the q-Deformed Virasoro Algebra
  
  • Haihong Hu
• Conformal invariance, unitarity and critical exponents in two dimensions
  
  • Friedan, D., Qiu, Z. and Shenker, S.
  • Phys. Rev. Lett. 52 (1984) 1575-1578
• 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.
• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Virasoro_algebra
• http://en.wikipedia.org/wiki/Coset_construction