"Virasoro algebra"의 두 판 사이의 차이

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* start with given c and h
 
* start with given c and h
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* construct <math>M(c,h)</math>
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* construct <math>M(c,h)</math><br>
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** quotients from the Universal enveloping algebra
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** tensor product from the one dimensional Borel subalgebra representations
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* there exists a unique contravariant hermitian form
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* a natural grading given by the <math>L_0</math>-eigenvalues
 
* contains a unique maximal submodule, and its quotient is the unique (up to isomorphism) irreducible representation with highest weight
 
* contains a unique maximal submodule, and its quotient is the unique (up to isomorphism) irreducible representation with highest weight
 
* When is <math>M(c,h)</math> unitary? 
 
* When is <math>M(c,h)</math> unitary? 
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<h5>Discrete series unitary representations</h5>
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<h5>discrete series unitary representations</h5>
  
 
* c<1 case
 
* c<1 case
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<math>s= 1, 2, 3,\cdots, r</math>
 
<math>s= 1, 2, 3,\cdots, r</math>
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* constructed by GKO construction
  
 
 
 
 
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<h5>affine Lie algebras</h5>
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* the highest weight representation V of an affine Kac-Moody algebra gives unitary representation of the Virasoro algebra
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* This is because V is unitary highest weigh representation of the AKMA. 
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* Read chapter 4 of Kac-Raina on Wedge space
  
 
 
 
 

2009년 7월 28일 (화) 15:16 판

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

 

 

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
  • 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? 

 

unitary representations
  • They are classified by c>1 and c<1 case.
    • \(c> 1, h > 0\)
    • \(c\geq 1, h \geq 0\)
    • \(c>0, h >0\) with Kac determinant condition
      • called the discrete series representations

 

discrete series unitary representations
  • c<1 case

 

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

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

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

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

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

  • constructed by GKO construction

 

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

 

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