"Virasoro algebra"의 두 판 사이의 차이
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| 13번째 줄: | 13번째 줄: | ||
<h5>Virasoro algebra</h5> | <h5>Virasoro algebra</h5> | ||
| − | * 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> | + | * Lie algebra of vector fields on the unit circle<br><math>f(z)\frac{d}{dz}</math><br> |
| − | * they satisfy the following relation<br><math>[L_m,L_n]=(m-n)L_{m+n}</math><br> | + | * commutator<br><math>f(z)\frac{d}{dz}</math><br> |
| + | |||
| + | * Virasoro generators<br><math>L_n=-z^{n+1}\frac{d}{dz}</math><br> | ||
| + | * they satisfy the following relation (Witt algebra)<br><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 | ||
* taking a [[central extension of groups and Lie algebras|central extension of lie algebras]], we get the Virasoro algebra<br><math>L_n,n\in \mathbb{Z}</math><br><math>[c,L_n]=0</math><br><math>[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n}</math><br> | * taking a [[central extension of groups and Lie algebras|central extension of lie algebras]], we get the Virasoro algebra<br><math>L_n,n\in \mathbb{Z}</math><br><math>[c,L_n]=0</math><br><math>[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n}</math><br> | ||
| 23번째 줄: | 27번째 줄: | ||
<h5>central charge and conformal weight</h5> | <h5>central charge and conformal weight</h5> | ||
| + | * highest weight representation | ||
* <math>c</math> is called the central charge | * <math>c</math> is called the central charge | ||
* <math>h</math> is sometimes called a conformal dimension or conformal weights | * <math>h</math> is sometimes called a conformal dimension or conformal weights | ||
| 46번째 줄: | 51번째 줄: | ||
* When is <math>M(c,h)</math> unitary? | * When is <math>M(c,h)</math> unitary? | ||
* to understand the submodules of the Verma module, we refer to Feigin and Fuks. | * to understand the submodules of the Verma module, we refer to Feigin and Fuks. | ||
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2010년 9월 23일 (목) 23:20 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)
Virasoro algebra
- Lie algebra of vector fields on the unit circle
\(f(z)\frac{d}{dz}\) - commutator
\(f(z)\frac{d}{dz}\)
- 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
- 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.
\(<{L_n}v,w>=<w,L_{-n}w>\)
- \(L_n\) and \(L_{-n}\) act as adjoints to each other, i.e.
- 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
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
[/pages/3003682/attachments/1973999 전체화면_캡처_2009-08-08_오전_60339.jpg]
No-Ghost theorem
- refer to the No Ghost theorem
관련된 항목들
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