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==Virasoro algebra== | ==Virasoro algebra== | ||
| − | * Lie algebra of vector fields on the unit circle | + | * Lie algebra of vector fields on the unit circle |
| − | * commutator | + | :<math>f(z)\frac{d}{dz}</math><br> |
| − | + | * commutator | |
| − | * Virasoro generators | + | :<math>[f(z)\frac{d}{dz},g(z)\frac{d}{dz}]=(fg'-f'g)\frac{d}{dz}</math><br> |
| − | * they satisfy the following relation (Witt algebra) | + | * Virasoro generators |
| + | :<math>L_n=-z^{n+1}\frac{d}{dz}</math><br> | ||
| + | * they satisfy the following relation (Witt algebra) | ||
| + | :<math>[L_m,L_n]=(m-n)L_{m+n}</math><br> | ||
* Homological algebra tells that there is 1-dimensional central extension of Witt algebra | * Homological algebra tells that there is 1-dimensional central extension of Witt algebra | ||
| − | * taking a [[central extension of groups and Lie algebras|central extension of lie algebras]], we get the Virasoro algebra | + | * taking a [[central extension of groups and Lie algebras|central extension of lie algebras]], we get the Virasoro algebra <math>L_n,n\in \mathbb{Z}</math> |
| + | :<math>[c,L_n]=0</math> | ||
| + | :<math>[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n}</math><br> | ||
2013년 2월 26일 (화) 13:46 판
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
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
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
No-Ghost theorem
관련된 항목들
매스매티카 파일 및 계산 리소스
encyclopedia
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