"Virasoro algebra"의 두 판 사이의 차이

2010년 3월 23일 (화) 13:36 판

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)

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

Lie algebra of vector fields on the unit circle
\(f(z)\frac{d}{dz}\)
\(L_n=-z^{n+1}\frac{d}{dz}\)

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.

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

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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

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 <h5>Virasoro algebra</h5>
-*  Lie algebra of vector fields on the unit circle<br><math>f(z)\frac{d}{dz}</math><br>
+*  Lie algebra of vector fields on the unit circle<br><math>f(z)\frac{d}{dz}</math><br><math>L_n=-z^{n+1}\frac{d}{dz}</math><br>
-* <math>L_n=-z^{n+1}\frac{d}{dz}</math>
-*
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-<h5>discrete series unitary representations</h5>
-*  c<1 case<br><math>m= 2, 3, 4.\cdots</math><br><math>c = 1-{6\over m(m+1)} = 0,\quad 1/2,\quad 7/10,\quad 4/5,\quad 6/7,\quad 25/28, \ldots</math><br><math>h_{r,s}(c) = {((m+1)r-ms)^2-1 \over 4m(m+1)}</math><br><math>r = 1, 2, 3,\cdots,m-1</math><br><math>s= 1, 2, 3,\cdots, r</math><br>
-* 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
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 * [[vertex algebras|Vertex Algebras]]
 * [[BRST quantization and cohomology|BRST Cohomology]]
+* [[#]]