View source for Virasoro algebra

{{Group theory sidebar}}

In [[mathematics]], the '''Virasoro algebra''' (named after the physicist [[Miguel Ángel Virasoro (physicist)|Miguel Angel Virasoro]])<ref>{{cite journal|author=M. A. Virasoro |year=1970 |title=Subsidiary conditions and ghosts in dual-resonance models |journal=[[Physical Review D]] |volume=1 |issue=10 |pages=2933–2936|doi=10.1103/PhysRevD.1.2933|bibcode = 1970PhRvD...1.2933V }}</ref> is a complex [[Lie algebra]], the unique [[Lie algebra extension#Central|central extension]] of the [[Witt algebra]]. It is widely used in [[two-dimensional conformal field theory]] and in [[string theory]].

== Definition ==
The '''Virasoro algebra''' is [[linear span|spanned]] by '''generators''' {{math|''L<sub>n</sub>''}} for {{math|''n'' ∈ ℤ}} and the '''[[central charge]]''' {{mvar|c}}.
These generators satisfy <math>[c,L_n]=0</math> and 
{{Equation box 1
|indent =::
|equation = <math>[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n,0}</math> .
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7}}
The factor of   {{Fraction|1|12}}  is merely a matter of convention. For a derivation of the algebra as the unique central extension of the [[Witt algebra]], see [[Lie algebra extension#Virasoro algebra|derivation of the Virasoro algebra]].

The Virasoro algebra has a [[Presentation of a group|presentation]] in terms of 2 generators (e.g. {{mvar|L}}<sub>3</sub> and {{mvar|L}}<sub>−2</sub>) and 6 relations.<ref>{{Cite journal | doi = 10.1007/BF01218387| title = A presentation for the Virasoro and super-Virasoro algebras| journal = Communications in Mathematical Physics| volume = 117| issue = 4| pages = 595| year = 1988| last1 = Fairlie | first1 = D. B.| last2 = Nuyts | first2 = J.| last3 = Zachos | first3 = C. K.|bibcode = 1988CMaPh.117..595F }}</ref><ref>{{Cite journal | doi = 10.1007/BF01221412| title = Redundancy of conditions for a Virasoro algebra| journal = Communications in Mathematical Physics| volume = 122| issue = 1| pages = 171–173| year = 1989| last1 = Uretsky | first1 = J. L. |bibcode = 1989CMaPh.122..171U }}</ref>

==Representation theory==

===Highest weight representations===
A '''highest weight representation''' of the Virasoro algebra is a representation generated by a '''primary state''': a vector <math>v</math> such that 
:<math> L_{n>0} v = 0 \quad , \quad  L_0 v = hv \ ,</math> 
where the number {{mvar|h}} called the '''conformal dimension''' or '''conformal weight''' of <math>v</math>.<ref name="BYB">P. Di Francesco, P. Mathieu, and D. Sénéchal, ''Conformal Field Theory'', 1997, {{ISBN|0-387-94785-X}}</ref>

A highest weight representation is spanned by eigenstates of <math>L_0</math>. The eigenvalues take the form <math>h+N</math>, where the integer <math>N\geq 0</math> is called the '''level''' of the corresponding eigenstate. 

More precisely, a highest weight representation is spanned by <math>L_0</math>-eigenstates of the type <math>L_{-n_1}L_{-n_2}\cdots L_{-n_k}v</math> with <math>0<n_1\leq n_2\leq \cdots n_k</math> and <math>k\geq 0</math>, whose levels are <math>N=\sum_{i=1}^k n_i</math>. Any state whose level is not zero is called a '''descendant state''' of <math>v</math>.

For any pair of complex numbers {{mvar|h}} and {{mvar|c}}, the [[Verma module]] <math>\mathcal V_{c,h}</math> is 
the largest possible highest weight representation. (The same letter, {{mvar|c}}, is used for both the element {{mvar|c}} of the Virasoro algebra and its eigenvalue in a representation.) 

The states <math>L_{-n_1}L_{-n_2}\cdots L_{-n_k}v</math> with <math>0<n_1\leq n_2\leq \cdots n_k</math> and <math>k\geq 0</math> form a basis of the Verma module. The Verma module is indecomposable, and for generic values of {{mvar|h}} and {{mvar|c}} it is also irreducible. When it is reducible, there exist other highest weight representations with these values of {{mvar|h}} and {{mvar|c}}, called '''degenerate representations''', which are cosets of the Verma module. In particular,
the unique irreducible highest weight representation with these values of {{mvar|h}} and {{mvar|c}} is the quotient of the Verma module by its maximal submodule.

A Verma module is irreducible if and only if it has no singular vectors.

===Singular vectors===
A '''singular vector''' or '''null vector''' of a highest weight representation is a state that is both descendent and primary. 

A sufficient condition for the Verma module <math>\mathcal V_{c,h}</math> to have a singular vector at the level <math>N</math> is <math>h=h_{r,s}(c)</math> for some positive integers <math>r,s</math> such that <math>N=rs</math>, with
:<math> h_{r,s}(c) = \frac14\Big((b+b^{-1})^2 - (br + b^{-1}s)^2\Big)\ ,\quad \text{where} \quad c=1+6(b+b^{-1})^2\ . </math>
In particular, <math> h_{1,1}(c) = 0</math>, and the reducible Verma module <math>\mathcal V_{c,0}</math> has a singular vector <math> L_{-1}v </math> at the level <math>N=1</math>. Then <math> h_{2,1}(c) = -\frac12 - \frac34 b^2 </math>, and the corresponding reducible Verma module has a singular vector <math> (L_{-1}^2 + b^2 L_{-2})v </math> at the level <math>N=2</math>.

This condition for the existence of a singular vector at the level <math>N</math> is not necessary. In particular, there is a singular vector at the level <math>N</math> if <math> N = rs + r's'</math> with <math> h=h_{r,s}(c)</math> and <math> h+rs = h_{r',s'}(c)</math>. This singular vector is now a descendent of another singular vector at the level <math>rs</math>. This type of singular vectors can however only exist if the central charge is of the type <math> c = 1-6\frac{(p-q)^2}{pq}</math> with <math>p,q\in\mathbb{Z}</math>. (For <math>p>q\geq 2</math> coprime, these are the central charges of the [[minimal models]].)<ref name="BYB"/>

===Hermitian form and unitarity===
A highest weight representation with a real value of <math>c</math> has a unique [[Hermitian form]] such that the adjoint of <math>L_n</math> is <math>L_{-n}</math>, and the norm of the primary state is one.
The representation is called '''unitary''' if that Hermitian form is positive definite. 
Since any singular vector has zero norm, all unitary highest weight representations are irreducible. 

The [[Gram determinant]] of a basis of the level <math>N</math> is given by the '''Kac determinant formula''', 
:<math>  A_N\prod_{1\le r,s\le N}(h-h_{r,s}(c))^{p(N-rs)} ~,</math>
where the function ''p''(''N'') is the [[partition function (number theory)|partition function]], and ''A''<sub>''N''</sub> is a positive constant that does not depend on <math>h</math> or <math>c</math>. 
The Kac determinant formula was stated by [[V. Kac]] (1978), and its first published proof was given by Feigin and Fuks (1984).

The irreducible highest weight representation with values  {{mvar|h}} and {{mvar|c}} is unitary  if and only if either {{mvar|c}} ≥1 and  {{mvar|h}}≥0, or 
:<math> c \in\left\{ 1-{6\over m(m+1)}\right\}_{m=2,3,4,\ldots} = \left\{ 0,\frac12,\frac{7}{10},\frac45,\frac67,\frac{25}{28}, \ldots\right\}</math>
for ''m'' = 2, 3, 4, ...,  and ''h'' is one of the values 
:<math> h = h_{r,s}(c) = {((m+1)r-ms)^2-1 \over 4m(m+1)}</math>
for ''r'' = 1, 2, 3, ..., ''m''&minus;1 and ''s''= 1, 2, 3, ..., ''r''.

[[Daniel Friedan]], Zongan Qiu, and [[Stephen Shenker]] (1984) showed that these conditions are necessary, and [[Peter Goddard (physicist)|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 algebra]]s) to show that they are sufficient.

==Applications==

===Conformal field theory===
In two dimensions, the algebra of local [[conformal map|conformal transformations]] is made of two copies of the [[Witt algebra]].
It follows that the symmetry algebra of [[two-dimensional conformal field theory]] is the Virasoro algebra. Technically, the [[conformal bootstrap]] approach to two-dimensional CFT relies on [[Virasoro conformal block|Virasoro conformal blocks]], special functions that include and generalize the characters of representations of the Virasoro algebra.

===String theory===
Since the Virasoro algebra comprises the generators of the conformal group of the [[worldsheet]], the [[Stress–energy tensor|stress tensor]] in [[string theory]] obeys the commutation relations of (two copies of) the Virasoro algebra.  This is because the conformal group decomposes into separate diffeomorphisms of the forward and back lightcones.  Diffeomorphism invariance of the worldsheet implies additionally that the stress tensor vanishes.  This is known as the [[Virasoro constraint]], and in the [[Quantum mechanics|quantum theory]], cannot be applied to all the states in the theory, but rather only on the physical states (compare [[Gupta–Bleuler formalism]]).

==Generalizations==
===Super Virasoro algebras===
{{main|Super Virasoro algebra}}
There are two [[Lie superalgebra|supersymmetric N=1 extensions]] of the Virasoro algebra, called the [[Neveu-Schwarz algebra]] and the [[Ramond algebra]]. Their theory is similar to that of the Virasoro algebra, now involving [[Grassmann number]]s. There are further extensions of these algebras with more supersymmetry, such as the [[N = 2 superconformal algebra]].

===W-algebras===
{{main|W-algebra}}
W-algebras are associative algebras which contain the Virasoro algebra, and which play an important role in [[two-dimensional conformal field theory]]. Among W-algebras, the Virasoro algebra has the particularity of being a Lie algebra. 
 
===Affine Lie algebras===
{{main|affine Lie algebra}}
The Virasoro algebra is a subalgebra of the universal enveloping algebra of any affine Lie algebra, as shown by the [[Wess-Zumino-Witten_model#Sugawara_construction|Sugawara construction]]. In this sense, affine Lie algebras are extensions of the Virasoro algebra.

===Meromorphic vector fields on Riemann surfaces===
The Virasoro algebra is a central extension of the Lie algebra of meromorphic vector fields with two poles on a genus 0 Riemann surface.
On a higher genus compact Riemann surface, the Lie algebra of meromorphic vector fields with two poles also has a central extension, which is a generalization of the Virasoro algebra.<ref>{{cite journal | last1 = Krichever | first1 = I. M. | last2 = Novikov | first2 = S.P. | year = 1987 | title = Algebras of Virasoro type, Riemann surfaces and structures of the theory of solitons | url = | journal = Funkts. Anal. Appl. | volume = 21 | issue = 2| pages = 46–63 | doi = 10.1007/BF01078026 }}</ref> This can be further generalized to supermanifolds.<ref>{{Cite journal | doi = 10.1016/0393-0440(94)00012-S| title = Super elliptic curves| journal = Journal of Geometry and Physics| volume = 15| issue = 3| pages = 252–280| year = 1995| last1 = Rabin | first1 = J. M. |arxiv = hep-th/9302105 |bibcode = 1995JGP....15..252R }}</ref>

===Vertex Virasoro algebra and conformal Virasoro algebra===
The Virasoro algebra also has [[Vertex Operator Algebra|vertex algebraic]] and [[Lie conformal algebra|conformal algebraic]] counterparts, which basically come from arranging all the basis elements into generating series and working with single objects.

==History<!--'Virasoro operator' and 'Virasoro operators' redirect here-->==
The Witt algebra (the Virasoro algebra without the central extension) was discovered by [[É. Cartan]] (1909). Its analogues over finite fields were studied by [[E. Witt]] in about the 1930s. The central extension of the Witt algebra that gives the Virasoro algebra was first found (in characteristic ''p'' &gt; 0) by [[R. E. Block]] (1966, page 381) and independently rediscovered (in characteristic 0) by [[I. M. Gelfand]] and {{Interlanguage link multi|Dmitry Fuchs{{!}}D. B. Fuchs|de|3=Dmitry Borisovich Fuchs}} (1968). Virasoro (1970) wrote down some operators generating the Virasoso algebra (later known as the '''Virasoro operators'''<!--boldface per WP:R#PLA-->) while studying [[dual resonance model]]s, though  he did not find the central extension.  The central extension giving the Virasoro algebra was rediscovered in physics shortly after by J. H. Weis, according to Brower and Thorn (1971, footnote on page 167).

==See also==
*[[Witt algebra]]
*[[Heisenberg algebra]]
*[[WZW model]]
*[[Conformal field theory]]
*[[Goddard–Thorn theorem]]
*[[Lie conformal algebra]]
*[[Super Virasoro algebra]]
*[[W-algebra]]

==Notes==
{{reflist}}

==References==
*{{cite journal |author=[[Alexander Belavin]], [[Alexander Markovich Polyakov|Alexander Polyakov]] and [[Alexander Zamolodchikov]] |year=1984 |title=Infinite conformal symmetry in two-dimensional quantum field theory |journal=[[Nuclear Physics B]] |volume=241 |pages=333–380 |doi=10.1016/0550-3213(84)90052-X |issue=2|bibcode = 1984NuPhB.241..333B }}
*{{cite journal |author=R. E. Block |year=1966  |title=On the Mills–Seligman axioms for Lie algebras of classical type |journal=[[Transactions of the American Mathematical Society]] |volume=121 |pages=378–392 |jstor=1994485 |doi=10.1090/S0002-9947-1966-0188356-3 |issue=2}}
*{{cite journal |author1=R. C. Brower |author2=C. B. Thorn |year=1971  |title=Eliminating spurious states from the dual resonance model |journal=[[Nuclear Physics B]] |volume=31 |issue=1 |pages=163–182 |doi=10.1016/0550-3213(71)90452-4|bibcode = 1971NuPhB..31..163B }}.
*{{cite journal |author=E. Cartan |year=1909 |title=Les groupes de transformations continus, infinis, simples |journal=Annals of Sci Ecole Normale Supérieur |volume=26 |pages=93–161 |jfm=40.0193.02 |doi=10.24033/asens.603}}
*B. L. Feigin, D. B. Fuchs, ''Verma modules over the Virasoro algebra''  L. D. Faddeev (ed.)  A. A. Mal'tsev (ed.), Topology. Proc. Internat. Topol. Conf. Leningrad 1982, Lect. notes in math., 1060, Springer  (1984)  pp.&nbsp;230–245
*{{cite journal |author=Friedan, D., Qiu, Z. and Shenker, S. |year=1984 |title=Conformal invariance, unitarity and critical exponents in two dimensions |journal=[[Physical Review Letters]] |volume=52 |pages=1575–1578 |doi=10.1103/PhysRevLett.52.1575 |issue=18|bibcode = 1984PhRvL..52.1575F }}.
*[[I. M. Gelfand|I.M. Gel'fand]], D. B. Fuchs,   ''The cohomology of the Lie algebra of vector fields in a circle''  Funct. Anal. Appl., 2  (1968)  pp.&nbsp;342–343  Funkts. Anal. i Prilozh., 2 : 4  (1968)  pp.&nbsp;92–93
*{{cite journal |author1=P. Goddard, A. Kent  |author2=D. Olive  |lastauthoramp=yes |year=1986 |title=Unitary representations of the Virasoro and super-Virasoro algebras |journal=[[Communications in Mathematical Physics]] |volume=103 |issue=1 |pages=105–119 |mr=0826859 |zbl=0588.17014 |doi=10.1007/BF01464283|bibcode = 1986CMaPh.103..105G }}.
*{{Citation | last1=Iohara | first1=Kenji | last2=Koga | first2=Yoshiyuki | title=Representation theory of the Virasoro algebra | publisher=Springer-Verlag London Ltd. | location=London | series=Springer Monographs in Mathematics | isbn=978-0-85729-159-2 | doi=10.1007/978-0-85729-160-8 | mr=2744610 | year=2011}}
*{{cite journal |author=A. Kent |year=1991 |title=Singular vectors of the Virasoro algebra |journal=[[Physics Letters B]] |volume=273 |issue=1–2 |pages=56–62 |doi=10.1016/0370-2693(91)90553-3|arxiv = hep-th/9204097 |bibcode = 1991PhLB..273...56K }}
* {{springer|author=Victor Kac|title=Virasoro algebra|id=v/v096710}}
*V. G. Kac,   ''Highest weight representations of infinite dimensional Lie algebras'', Proc. Internat. Congress Mathematicians (Helsinki, 1978), [http://www.mathunion.org/ICM/ICM1978.1/Main/icm1978.1.0299.0304.ocr.pdf pp.299-304]
*V. G. Kac, A. K. Raina,   ''Bombay lectures on highest weight representations'', World Sci.  (1987) {{isbn|9971-5-0395-6}}.
*{{cite journal | last1 = Dobrev | first1 = V. K. | year = 1986 | title = Multiplet classification of the indecomposable highest weight modules over the Neveu-Schwarz and Ramond superalgebras | url = | journal = Lett. Math. Phys. | volume = 11 | issue = 3| pages = 225–234 | doi=10.1007/bf00400220 | bibcode=1986LMaPh..11..225D}} & correction: ibid. '''13''' (1987) 260.
*V. K. Dobrev, ''Characters of the irreducible highest weight modules over the Virasoro and super-Virasoro algebras'', Suppl. [[Rendiconti del Circolo Matematico di Palermo]], Serie II, Numero 14 (1987) 25-42.
*{{cite arXiv|author=[[Antony Wassermann]]|title=Lecture notes on Kac-Moody and Virasoro algebras |eprint=1004.1287|class=math.RT|year=2010}}
*{{cite arXiv|author=[[Antony Wassermann]]|title=Direct proofs of the Feigin-Fuchs character formula for unitary representations of the Virasoro algebra|eprint=1012.6003|year=2010|class=math.RT}}

{{authority control}}
[[Category:Conformal field theory]]
[[Category:Lie algebras]]
[[Category:Mathematical physics]]