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imported>Pythagoras0 |
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| 1번째 줄: | 1번째 줄: | ||
| − | + | ==introduction== | |
* Virasoro algebra could be pre-knowledge for the study of CFT | * Virasoro algebra could be pre-knowledge for the study of CFT | ||
| 11번째 줄: | 11번째 줄: | ||
** no classification for c>1 | ** no classification for c>1 | ||
| − | + | ||
| − | + | ||
| − | + | ==Virasoro algebra== | |
| − | * Lie algebra | + | * Lie algebra of vector fields on the unit circle<br><math>f(z)\frac{d}{dz}</math><br> |
* commutator<br><math>[f(z)\frac{d}{dz},g(z)\frac{d}{dz}]=(fg'-f'g)\frac{d}{dz}</math><br> | * commutator<br><math>[f(z)\frac{d}{dz},g(z)\frac{d}{dz}]=(fg'-f'g)\frac{d}{dz}</math><br> | ||
| 25번째 줄: | 25번째 줄: | ||
* 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> | ||
| − | + | ||
| − | + | ||
| − | + | ==central charge and conformal weight== | |
* highest weight representation | * highest weight representation | ||
| − | * <math>c</math> | + | * <math>c</math> is called the central charge |
| − | * <math>h</math> | + | * <math>h</math> is sometimes called a conformal dimension or conformal weights |
| − | + | ||
| − | + | ||
| − | + | ==Verma module== | |
* [[highest weight representation of Vir]] | * [[highest weight representation of Vir]] | ||
| − | + | ||
| − | + | ||
| − | + | ==unitarity and ghost== | |
| − | * unitarity | + | * 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. | * 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 | * If a ghost is found on any level the represetation cannot occur in a unitary theory | ||
| − | + | ||
| − | + | ||
| − | + | ==unitary irreducible representations== | |
* [[highest weight representation of Vir]] | * [[highest weight representation of Vir]] | ||
| − | + | ||
| − | + | ||
| − | + | ==affine Lie algebras== | |
| − | * a highest weight representation | + | * a highest weight representation V of an [[affine Kac-Moody algebra]] gives unitary representation of the Virasoro algebra |
| − | * This is because V is | + | * This is because V is a unitary highest weight representation of the AKMA. |
* Read chapter 4 of Kac-Raina on Wedge space | * Read chapter 4 of Kac-Raina on Wedge space | ||
* [[unitary representations of affine Kac-Moody algebras]] | * [[unitary representations of affine Kac-Moody algebras]] | ||
| − | + | ||
| − | + | ||
| − | + | ==character of minimal models== | |
* [[minimal models]] | * [[minimal models]] | ||
* [[bosonic characters of Virasoro minimal models(Rocha-Caridi formula)|bosonic characters of minimal models]] | * [[bosonic characters of Virasoro minimal models(Rocha-Caridi formula)|bosonic characters of minimal models]] | ||
| − | + | ||
| − | + | ||
| − | + | ==No-Ghost theorem== | |
| − | * refer to | + | * refer to the [[3917551|No Ghost theorem]] |
| − | + | ||
| − | + | ||
| − | + | ==관련된 항목들== | |
* [[vertex algebras|Vertex Algebras]] | * [[vertex algebras|Vertex Algebras]] | ||
| 100번째 줄: | 100번째 줄: | ||
* [[bosonic characters of Virasoro minimal models(Rocha-Caridi formula)|bosonic characters of minimal models]] | * [[bosonic characters of Virasoro minimal models(Rocha-Caridi formula)|bosonic characters of minimal models]] | ||
| − | + | ||
| − | + | ||
| − | + | ==encyclopedia== | |
| − | * [http://eom.springer.de/v/v096710.htm Virasoro algebra] | + | * [http://eom.springer.de/v/v096710.htm Virasoro algebra] by V. Kac |
* http://en.wikipedia.org/wiki/Virasoro_algebra | * http://en.wikipedia.org/wiki/Virasoro_algebra | ||
| − | + | ||
| − | + | ||
| − | + | ==exposition== | |
* Douglas Lundholm, [http://www.math.kth.se/%7Edogge/files/virasoro.pdf The Virasoro algebra and its representations in physics] , January 10, 2005 | * Douglas Lundholm, [http://www.math.kth.se/%7Edogge/files/virasoro.pdf The Virasoro algebra and its representations in physics] , January 10, 2005 | ||
| − | + | ||
| − | + | ||
| − | + | ||
| − | + | ||
| − | + | ==articles== | |
* [http://www.springerlink.com/content/kn757431511020g2/ Quantum Group Structure of the q-Deformed Virasoro Algebra]<br> | * [http://www.springerlink.com/content/kn757431511020g2/ Quantum Group Structure of the q-Deformed Virasoro Algebra]<br> | ||
** Haihong Hu | ** Haihong Hu | ||
* [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1104114626 Unitary representations of the Virasoro and super-Virasoro algebras]<br> | * [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1104114626 Unitary representations of the Virasoro and super-Virasoro algebras]<br> | ||
| − | ** P. Goddard, A. Kent and D. Olive, | + | ** P. Goddard, A. Kent and D. Olive, Comm. Math. Phys. 103, no. 1 (1986), 105–119. |
* [http://prola.aps.org/abstract/PRL/v52/i18/p1575_1 Conformal invariance, unitarity and critical exponents in two dimensions]<br> | * [http://prola.aps.org/abstract/PRL/v52/i18/p1575_1 Conformal invariance, unitarity and critical exponents in two dimensions]<br> | ||
| − | ** Friedan, D., Qiu, Z. and Shenker, S., | + | ** Friedan, D., Qiu, Z. and Shenker, S., Phys. Rev. Lett. 52 (1984) 1575-1578 |
* [http://www.springerlink.com/content/122636vk15g86472/ Verma modules over the Virasoro algebra]<br> | * [http://www.springerlink.com/content/122636vk15g86472/ Verma modules over the Virasoro algebra]<br> | ||
2012년 10월 27일 (토) 15:30 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
- refer to the No Ghost theorem
관련된 항목들
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