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A [[Lie algebra|Lie algebra]], denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v0967101.png" />, over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v0967102.png" /> with basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v0967103.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v0967104.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v0967105.png" /> and the following commutation relations (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v0967106.png" />):
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v0967107.png" /></td> </tr></table>
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v0967108.png" /></td> </tr></table>
+
An infinite-dimensional [[Lie algebra|Lie algebra]], denoted by  $\mathop{\rm Vir} $,
 +
over  $  \mathbf C $ with basis $L_{n}$ (
 +
$n \in \mathbf Z $), $c$
 +
and the following commutation relations ($m , n \in \mathbf Z $):
  
Since the vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v0967109.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671010.png" />) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671011.png" /> satisfy the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671012.png" />, the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671013.png" /> is a central extension (which is, in fact, universal) of the Lie algebra of holomorphic vector fields on the punctured complex plane having finite Laurent series. For this reason the Virasoro algebra plays a key role in conformal field theory.
+
$$
 +
[ L _ {m} , L _ {n} ]  = \
 +
( m- n) L _ {m+} n + \delta _ {m, - n }
  
On the other hand, letting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671015.png" /> is the parameter on the unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671016.png" />, one gets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671017.png" />. Hence the Lie algebra of vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671018.png" /> with finite Fourier series is a real form of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671019.png" /> consisting of elements fixed under the anti-linear involution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671021.png" />. For this reason the Virasoro algebra is intimately related to the representation theory of the group of diffeomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671022.png" />, of the loop groups and to affine Kac–Moody algebras (see [[Kac–Moody algebra|Kac–Moody algebra]]).
+
\frac{m  ^ {3} - m }{12}
 +
c ,
 +
$$
  
The representation theory of the Virasoro algebra has numerous applications in mathematics and theoretical physics. The most interesting, positive-energy representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671023.png" /> in a complex vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671024.png" />, are defined by the property that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671025.png" /> acts as a scalar, denoted by the same letter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671026.png" /> (called the central charge), and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671027.png" /> (the energy operator) is diagonalizable with finite-dimensional eigenspaces and with real spectrum bounded below:
+
$$
 +
[ c , L _ {n} ]  = 0 .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671028.png" /></td> </tr></table>
+
Since the vector fields  $  d _ {n} = - z  ^ {n+} 1 ( d / dz) $(
 +
$  n \in \mathbf Z $)
 +
on  $  \mathbf C \setminus  \{ 0 \} $
 +
satisfy the relation  $  [ d _ {m} , d _ {n} ] = ( m- n) d _ {m+} n $,
 +
the Lie algebra  $  \mathop{\rm Vir} $
 +
is a central extension (which is, in fact, universal) of the Lie algebra of holomorphic vector fields on the punctured complex plane having finite Laurent series. For this reason the Virasoro algebra plays a key role in conformal field theory.
 +
 
 +
On the other hand, letting  $  z = \mathop{\rm exp}  i \theta $,
 +
where  $  \theta $
 +
is the parameter on the unit circle  $  S  ^ {1} $,
 +
one gets  $  d _ {n} = ie ^ {i n \theta } ( d / d \theta ) $.
 +
Hence the Lie algebra of vector fields on  $  S  ^ {1} $
 +
with finite Fourier series is a real form of the Lie algebra  $  \mathop{\rm Vir} / \mathbf C c $
 +
consisting of elements fixed under the anti-linear involution  $  L _ {n} \rightarrow L _ {-} n $,
 +
$  c \mapsto c $.
 +
For this reason the Virasoro algebra is intimately related to the representation theory of the group of diffeomorphisms of  $  S  ^ {1} $,
 +
of the loop groups and to affine Kac–Moody algebras (see [[Kac–Moody algebra|Kac–Moody algebra]]).
 +
 
 +
The representation theory of the Virasoro algebra has numerous applications in mathematics and theoretical physics. The most interesting, positive-energy representations of  $  \mathop{\rm Vir} $
 +
in a complex vector space  $  V $,
 +
are defined by the property that  $  c $
 +
acts as a scalar, denoted by the same letter  $  c $(
 +
called the central charge), and that  $  L _ {0} $(
 +
the energy operator) is diagonalizable with finite-dimensional eigenspaces and with real spectrum bounded below:
 +
 
 +
$$
 +
V  =  \oplus _ {j \geq  j _ {0} } V _ {j} .
 +
$$
  
 
The character of such a representation is the (formal) series
 
The character of such a representation is the (formal) series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671029.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm ch}  V  = \sum _ { j } (  \mathop{\rm dim}  V _ {j} ) q  ^ {j} .
 +
$$
  
The first positive-energy representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671030.png" /> were implicitly constructed by M.A. Virasoro [[#References|[a1]]] in 1970, using an Abelian version of the Sugawara construction (see [[Kac–Moody algebra|Kac–Moody algebra]]) in the framework of string theory. Since that time, and especially since the proof of the no-ghost theorem [[#References|[a2]]], the representation theory of the Virasoro algebra has become a key ingredient of string theory (see [[#References|[a3]]]). The Virasoro central extension itself was previously discovered by mathematicians [[#References|[a4]]], [[#References|[a5]]]; paper [[#References|[a2]]] is one of the earliest references in the physics literature containing a correct formula for the central term.
+
The first positive-energy representations of $  \mathop{\rm Vir} $
 +
were implicitly constructed by M.A. Virasoro [[#References|[a1]]] in 1970, using an Abelian version of the Sugawara construction (see [[Kac–Moody algebra|Kac–Moody algebra]]) in the framework of string theory. Since that time, and especially since the proof of the no-ghost theorem [[#References|[a2]]], the representation theory of the Virasoro algebra has become a key ingredient of string theory (see [[#References|[a3]]]). The Virasoro central extension itself was previously discovered by mathematicians [[#References|[a4]]], [[#References|[a5]]]; paper [[#References|[a2]]] is one of the earliest references in the physics literature containing a correct formula for the central term.
  
An irreducible positive-energy representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671031.png" /> in a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671032.png" /> admits a non-zero vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671033.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671034.png" />, such that
+
An irreducible positive-energy representation of $  \mathop{\rm Vir} $
 +
in a vector space $  V $
 +
admits a non-zero vector v _ {h} \in V $,  
 +
where $  h \in \mathbf R $,  
 +
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671035.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$ \tag{a1 }
 +
L _ {n} ( v _ {h} )  = \delta _ {n, 0 }  h v _ {h} \ \
 +
\textrm{ for }  n \geq  0 ,\ \
 +
c( v _ {h} ) =  cv _ {h} .
 +
$$
  
 
Then one has:
 
Then one has:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671036.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
$$ \tag{a2 }
 +
= \sum _ {0 < j _ {1} \leq  j _ {2} \leq  \dots }
 +
\mathbf C L _ {- j _ {s}  } \dots L _ {- j _ {1}  } v _ {h} .
 +
$$
  
This representation is determined uniquely by the two real constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671037.png" />, the central charge, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671038.png" />, the conformal dimension, and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671039.png" />. It is called degenerate if (a2) is not a direct sum decomposition.
+
This representation is determined uniquely by the two real constants $  c $,  
 +
the central charge, and $  h $,  
 +
the conformal dimension, and is denoted by $  L ^ {c , h } $.  
 +
It is called degenerate if (a2) is not a direct sum decomposition.
  
The first basic result of the representation theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671040.png" /> [[#References|[a6]]] (see [[#References|[a7]]] for a proof) states that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671041.png" /> is degenerate if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671043.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671044.png" /> and some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671045.png" /> where
+
The first basic result of the representation theory of $  \mathop{\rm Vir} $[[#References|[a6]]] (see [[#References|[a7]]] for a proof) states that $  L ^ {c, h } $
 +
is degenerate if and only if $  c = c( m) $,  
 +
$  h = h( m) $
 +
for some $  m \in \mathbf R \setminus  \{ 0, 1 \} $
 +
and some $  r , s = 1 , 2 \dots $
 +
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671046.png" /></td> </tr></table>
+
$$
 +
c( m)  = 1 -
 +
\frac{6}{m(}
 +
m+ 1) ,\ \
 +
h _ {r , s }  ( m)  =
 +
\frac{(( m+ 1) r- ms)  ^ {2} - 1 }{4m(}
 +
m+ 1) .
 +
$$
  
The basic idea of the foundational work [[#References|[a8]]] on conformal field theory is to use degenerate representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671047.png" /> to write down differential equations for correlation functions. The most complete results have been obtained [[#References|[a9]]] for the  "most degenerate"  representations, called the minimal series representations. These correspond to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671048.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671050.png" /> are relatively prime positive integers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671052.png" /> [[#References|[a8]]].
+
The basic idea of the foundational work [[#References|[a8]]] on conformal field theory is to use degenerate representations of $  \mathop{\rm Vir} $
 +
to write down differential equations for correlation functions. The most complete results have been obtained [[#References|[a9]]] for the  "most degenerate"  representations, called the minimal series representations. These correspond to $  m = ( p  ^  \prime  ) / ( p - p  ^  \prime  ) $,  
 +
where $  p $
 +
and $  p  ^  \prime  $
 +
are relatively prime positive integers and $  r < p  ^  \prime  - 1 $,  
 +
$  s \leq  p - 1 $[[#References|[a8]]].
  
The characters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671053.png" /> were computed in [[#References|[a10]]]. It follows that after letting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671054.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671055.png" /> becomes a modular function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671056.png" /> on the upper half-plane for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671057.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671058.png" /> is a representation of minimal series (then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671059.png" />) [[#References|[a11]]].
+
The characters $  \mathop{\rm ch}  L ^ {c, h } $
 +
were computed in [[#References|[a10]]]. It follows that after letting $  q = e ^ {2 \pi i \tau } $,  
 +
the function $  q  ^ {a}  \mathop{\rm ch}  L ^ {c, h } $
 +
becomes a modular function in $  \tau $
 +
on the upper half-plane for some $  a \in \mathbf R $
 +
if and only if $  L ^ {c, h } $
 +
is a representation of minimal series (then $  a = - c( m)/24 $)  
 +
[[#References|[a11]]].
  
The representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671060.png" /> carries a unique Hermitian form such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671061.png" /> has norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671062.png" /> and the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671064.png" /> are adjoint. Another important class of representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671065.png" /> are the unitary ones, i.e. those for which the Hermitian form is positive definite. The complete list of unitary representations is (see [[#References|[a12]]], [[#References|[a13]]], [[#References|[a7]]]):
+
The representation $  L ^ {c, h } $
 +
carries a unique Hermitian form such that v _ {h} $
 +
has norm $  1 $
 +
and the operators $  L _ {n} $
 +
and $  L _ {-} n $
 +
are adjoint. Another important class of representations $  L ^ {c, h } $
 +
are the unitary ones, i.e. those for which the Hermitian form is positive definite. The complete list of unitary representations is (see [[#References|[a12]]], [[#References|[a13]]], [[#References|[a7]]]):
  
a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671067.png" />;
+
a) $  c \geq  1 $,  
 +
$  h \geq  0 $;
  
b) minimal series with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671068.png" />.
+
b) minimal series with $  m = 2, 3,\dots $.
  
The minimal series representations (especially the unitary ones) are intimately related to statistical lattice models (see [[#References|[a14]]]). For example, the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671069.png" /> is identified with the Ising model, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096710/v09671070.png" /> with the Potts model, etc.
+
The minimal series representations (especially the unitary ones) are intimately related to statistical lattice models (see [[#References|[a14]]]). For example, the case $  m = 3 $
 +
is identified with the Ising model, $  m = 4 $
 +
with the Potts model, etc.
  
 
Among other areas of applications of the Virasoro algebra one should mention the theory of moduli spaces of curves (see [[#References|[a15]]]–[[#References|[a18]]]).
 
Among other areas of applications of the Virasoro algebra one should mention the theory of moduli spaces of curves (see [[#References|[a15]]]–[[#References|[a18]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.M. Virasoro,  "Subsidary conditions and ghosts in dual-resonance models"  ''Phys. Rev.'' , '''D1'''  (1970)  pp. 2933–2936</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Goddard,  C.B. Thorn,  "Compatibility of the dual Pomeron with unitarity and the absence of ghosts in the dual resonance model"  ''Phys. Lett.'' , '''4'''  (1972)  pp. 235–238</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.B. Green,  J.H. Schwarz,  E. Witten,  "Superstring theory" , Cambridge Univ. Press  (1987)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.E. Block,  "On the Mills–Seligman axioms for Lie algebras of classical type"  ''Trans. Amer. Math. Soc.'' , '''121'''  (1966)  pp. 378–392</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  I.M. Gel'fand,  D.B. Fuks,  "The cohomology of the Lie algebra of vector fields in a circle"  ''Funct. Anal. Appl.'' , '''2'''  (1968)  pp. 342–343  ''Funkts. Anal. i Prilozh.'' , '''2''' :  4  (1968)  pp. 92–93</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  V.G. Kac,  "Highest weight representations of infinite dimensional Lie algebras" , ''Proc. Internat. Congress Mathematicians (Helsinki, 1978)'' , '''1''' , Acad. Sci. Fennicae  (1980)  pp. 299–304</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  V.G. Kac,  A.K. Raina,  "Bombay lectures on highest weight representations" , World Sci.  (1987)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A.A. Belavin,  A.M. Polyakov,  A.B. Zamolodchikov,  "Infinite conformal symmetry in two-dimensional quantum field theory"  ''Nuclear Phys.'' , '''B241'''  (1984)  pp. 333–380</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  G. Felder,  "BRST approach to minimal models"  ''Nuclear Phys.'' , '''B317'''  (1989)  pp. 215–236</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  B.L. Feigin,  D.B. [D.B. Fuks] Fuchs,  "Verma models 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. 230–245</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  V.G. Kac,  M. Wakimoto,  "Modular invariant representations of infinite-dimensional Lie algebras and superalgebras"  ''Proc. Nat. Acad. Sci. USA'' , '''85'''  (1988)  pp. 4956–4960</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  D. Friedan,  Z. Qui,  S. Shenker,  "Conformal invariance, unitary and two dimensional critical exponents"  ''Publ. MSRI'' , '''3'''  (1985)  pp. 419–449</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  P. Goddard,  A. Kent,  D. Olive,  "Unitary representations of the Virasoro and super-Virasoro algebras"  ''Comm. Math. Phys.'' , '''103'''  (1986)  pp. 105–119</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  C. Itzykson,  J.-M. Dronfree,  "Statistical field theory" , Cambridge Univ. Press  (1989)</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  E. Arbarello,  C. De Concini,  V.G. Kac,  C. Processi,  "Moduli spaces of curves and representation theory"  ''Comm. Math. Phys.'' , '''117'''  (1988)  pp. 1–36</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  M.L. Kontzevich,  "Virasoro algebra and Teichmüller spaces"  ''Funct. Anal. Appl.'' , '''21''' :  2  (1987)  pp. 156–157  ''Funkts. Anal. i Prilozh.'' , '''21''' :  2  (1987)  pp. 78–79</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  A.A. Beilinson,  V.V. Schechtman,  "Determinant bundles and Virasoro algebras"  ''Comm. Math. Phys.'' , '''118'''  (1988)  pp. 651–701</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  N. Kawamoto,  Y. Namikawa,  A. Tsuchiya,  Y. Yamada,  "Geometric realization of conformal field theory on Riemann surfaces"  ''Comm. Math. Phys.'' , '''116'''  (1988)  pp. 247–308</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  M.M. Virasoro,  "Subsidary conditions and ghosts in dual-resonance models"  ''Phys. Rev.'' , '''D1'''  (1970)  pp. 2933–2936</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Goddard,  C.B. Thorn,  "Compatibility of the dual Pomeron with unitarity and the absence of ghosts in the dual resonance model"  ''Phys. Lett.'' , '''4'''  (1972)  pp. 235–238</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  M.B. Green,  J.H. Schwarz,  E. Witten,  "Superstring theory" , Cambridge Univ. Press  (1987)</TD></TR>
 +
<TR><TD valign="top">[a4]</TD> <TD valign="top">  R.E. Block,  "On the Mills–Seligman axioms for Lie algebras of classical type"  ''Trans. Amer. Math. Soc.'' , '''121'''  (1966)  pp. 378–392 {{ZBL|0136.30402}}</TD></TR>
 +
<TR><TD valign="top">[a5]</TD> <TD valign="top">  I.M. Gel'fand,  D.B. Fuks,  "The cohomology of the Lie algebra of vector fields in a circle"  ''Funct. Anal. Appl.'' , '''2'''  (1968)  pp. 342–343  ''Funkts. Anal. i Prilozh.'' , '''2''' :  4  (1968)  pp. 92–93</TD></TR>
 +
<TR><TD valign="top">[a6]</TD> <TD valign="top">  V.G. Kac,  "Highest weight representations of infinite dimensional Lie algebras" , ''Proc. Internat. Congress Mathematicians (Helsinki, 1978)'' , '''1''' , Acad. Sci. Fennicae  (1980)  pp. 299–304</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  V.G. Kac,  A.K. Raina,  "Bombay lectures on highest weight representations" , World Sci.  (1987)</TD></TR>
 +
<TR><TD valign="top">[a8]</TD> <TD valign="top">  A.A. Belavin,  A.M. Polyakov,  A.B. Zamolodchikov,  "Infinite conformal symmetry in two-dimensional quantum field theory"  ''Nuclear Phys.'' , '''B241'''  (1984)  pp. 333–380</TD></TR>
 +
<TR><TD valign="top">[a9]</TD> <TD valign="top">  G. Felder,  "BRST approach to minimal models"  ''Nuclear Phys.'' , '''B317'''  (1989)  pp. 215–236</TD></TR>
 +
<TR><TD valign="top">[a10]</TD> <TD valign="top">  B.L. Feigin,  D.B. [D.B. Fuks] Fuchs,  "Verma models 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. 230–245</TD></TR>
 +
<TR><TD valign="top">[a11]</TD> <TD valign="top">  V.G. Kac,  M. Wakimoto,  "Modular invariant representations of infinite-dimensional Lie algebras and superalgebras"  ''Proc. Nat. Acad. Sci. USA'' , '''85'''  (1988)  pp. 4956–4960</TD></TR>
 +
<TR><TD valign="top">[a12]</TD> <TD valign="top">  D. Friedan,  Z. Qui,  S. Shenker,  "Conformal invariance, unitary and two dimensional critical exponents"  ''Publ. MSRI'' , '''3'''  (1985)  pp. 419–449</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  P. Goddard,  A. Kent,  D. Olive,  "Unitary representations of the Virasoro and super-Virasoro algebras"  ''Comm. Math. Phys.'' , '''103'''  (1986)  pp. 105–119</TD></TR>
 +
<TR><TD valign="top">[a14]</TD> <TD valign="top">  C. Itzykson,  J.-M. Dronfree,  "Statistical field theory" , Cambridge Univ. Press  (1989)</TD></TR>
 +
<TR><TD valign="top">[a15]</TD> <TD valign="top">  E. Arbarello,  C. De Concini,  V.G. Kac,  C. Processi,  "Moduli spaces of curves and representation theory"  ''Comm. Math. Phys.'' , '''117'''  (1988)  pp. 1–36</TD></TR>
 +
<TR><TD valign="top">[a16]</TD> <TD valign="top">  M.L. Kontzevich,  "Virasoro algebra and Teichmüller spaces"  ''Funct. Anal. Appl.'' , '''21''' :  2  (1987)  pp. 156–157  ''Funkts. Anal. i Prilozh.'' , '''21''' :  2  (1987)  pp. 78–79</TD></TR>
 +
<TR><TD valign="top">[a17]</TD> <TD valign="top">  A.A. Beilinson,  V.V. Schechtman,  "Determinant bundles and Virasoro algebras"  ''Comm. Math. Phys.'' , '''118'''  (1988)  pp. 651–701</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  N. Kawamoto,  Y. Namikawa,  A. Tsuchiya,  Y. Yamada,  "Geometric realization of conformal field theory on Riemann surfaces"  ''Comm. Math. Phys.'' , '''116'''  (1988)  pp. 247–308</TD></TR>
 +
</table>

Latest revision as of 11:16, 23 March 2023


An infinite-dimensional Lie algebra, denoted by $\mathop{\rm Vir} $, over $ \mathbf C $ with basis $L_{n}$ ( $n \in \mathbf Z $), $c$ and the following commutation relations ($m , n \in \mathbf Z $):

$$ [ L _ {m} , L _ {n} ] = \ ( m- n) L _ {m+} n + \delta _ {m, - n } \frac{m ^ {3} - m }{12} c , $$

$$ [ c , L _ {n} ] = 0 . $$

Since the vector fields $ d _ {n} = - z ^ {n+} 1 ( d / dz) $( $ n \in \mathbf Z $) on $ \mathbf C \setminus \{ 0 \} $ satisfy the relation $ [ d _ {m} , d _ {n} ] = ( m- n) d _ {m+} n $, the Lie algebra $ \mathop{\rm Vir} $ is a central extension (which is, in fact, universal) of the Lie algebra of holomorphic vector fields on the punctured complex plane having finite Laurent series. For this reason the Virasoro algebra plays a key role in conformal field theory.

On the other hand, letting $ z = \mathop{\rm exp} i \theta $, where $ \theta $ is the parameter on the unit circle $ S ^ {1} $, one gets $ d _ {n} = ie ^ {i n \theta } ( d / d \theta ) $. Hence the Lie algebra of vector fields on $ S ^ {1} $ with finite Fourier series is a real form of the Lie algebra $ \mathop{\rm Vir} / \mathbf C c $ consisting of elements fixed under the anti-linear involution $ L _ {n} \rightarrow L _ {-} n $, $ c \mapsto c $. For this reason the Virasoro algebra is intimately related to the representation theory of the group of diffeomorphisms of $ S ^ {1} $, of the loop groups and to affine Kac–Moody algebras (see Kac–Moody algebra).

The representation theory of the Virasoro algebra has numerous applications in mathematics and theoretical physics. The most interesting, positive-energy representations of $ \mathop{\rm Vir} $ in a complex vector space $ V $, are defined by the property that $ c $ acts as a scalar, denoted by the same letter $ c $( called the central charge), and that $ L _ {0} $( the energy operator) is diagonalizable with finite-dimensional eigenspaces and with real spectrum bounded below:

$$ V = \oplus _ {j \geq j _ {0} } V _ {j} . $$

The character of such a representation is the (formal) series

$$ \mathop{\rm ch} V = \sum _ { j } ( \mathop{\rm dim} V _ {j} ) q ^ {j} . $$

The first positive-energy representations of $ \mathop{\rm Vir} $ were implicitly constructed by M.A. Virasoro [a1] in 1970, using an Abelian version of the Sugawara construction (see Kac–Moody algebra) in the framework of string theory. Since that time, and especially since the proof of the no-ghost theorem [a2], the representation theory of the Virasoro algebra has become a key ingredient of string theory (see [a3]). The Virasoro central extension itself was previously discovered by mathematicians [a4], [a5]; paper [a2] is one of the earliest references in the physics literature containing a correct formula for the central term.

An irreducible positive-energy representation of $ \mathop{\rm Vir} $ in a vector space $ V $ admits a non-zero vector $ v _ {h} \in V $, where $ h \in \mathbf R $, such that

$$ \tag{a1 } L _ {n} ( v _ {h} ) = \delta _ {n, 0 } h v _ {h} \ \ \textrm{ for } n \geq 0 ,\ \ c( v _ {h} ) = cv _ {h} . $$

Then one has:

$$ \tag{a2 } V = \sum _ {0 < j _ {1} \leq j _ {2} \leq \dots } \mathbf C L _ {- j _ {s} } \dots L _ {- j _ {1} } v _ {h} . $$

This representation is determined uniquely by the two real constants $ c $, the central charge, and $ h $, the conformal dimension, and is denoted by $ L ^ {c , h } $. It is called degenerate if (a2) is not a direct sum decomposition.

The first basic result of the representation theory of $ \mathop{\rm Vir} $[a6] (see [a7] for a proof) states that $ L ^ {c, h } $ is degenerate if and only if $ c = c( m) $, $ h = h( m) $ for some $ m \in \mathbf R \setminus \{ 0, 1 \} $ and some $ r , s = 1 , 2 \dots $ where

$$ c( m) = 1 - \frac{6}{m(} m+ 1) ,\ \ h _ {r , s } ( m) = \frac{(( m+ 1) r- ms) ^ {2} - 1 }{4m(} m+ 1) . $$

The basic idea of the foundational work [a8] on conformal field theory is to use degenerate representations of $ \mathop{\rm Vir} $ to write down differential equations for correlation functions. The most complete results have been obtained [a9] for the "most degenerate" representations, called the minimal series representations. These correspond to $ m = ( p ^ \prime ) / ( p - p ^ \prime ) $, where $ p $ and $ p ^ \prime $ are relatively prime positive integers and $ r < p ^ \prime - 1 $, $ s \leq p - 1 $[a8].

The characters $ \mathop{\rm ch} L ^ {c, h } $ were computed in [a10]. It follows that after letting $ q = e ^ {2 \pi i \tau } $, the function $ q ^ {a} \mathop{\rm ch} L ^ {c, h } $ becomes a modular function in $ \tau $ on the upper half-plane for some $ a \in \mathbf R $ if and only if $ L ^ {c, h } $ is a representation of minimal series (then $ a = - c( m)/24 $) [a11].

The representation $ L ^ {c, h } $ carries a unique Hermitian form such that $ v _ {h} $ has norm $ 1 $ and the operators $ L _ {n} $ and $ L _ {-} n $ are adjoint. Another important class of representations $ L ^ {c, h } $ are the unitary ones, i.e. those for which the Hermitian form is positive definite. The complete list of unitary representations is (see [a12], [a13], [a7]):

a) $ c \geq 1 $, $ h \geq 0 $;

b) minimal series with $ m = 2, 3,\dots $.

The minimal series representations (especially the unitary ones) are intimately related to statistical lattice models (see [a14]). For example, the case $ m = 3 $ is identified with the Ising model, $ m = 4 $ with the Potts model, etc.

Among other areas of applications of the Virasoro algebra one should mention the theory of moduli spaces of curves (see [a15][a18]).

References

[a1] M.M. Virasoro, "Subsidary conditions and ghosts in dual-resonance models" Phys. Rev. , D1 (1970) pp. 2933–2936
[a2] P. Goddard, C.B. Thorn, "Compatibility of the dual Pomeron with unitarity and the absence of ghosts in the dual resonance model" Phys. Lett. , 4 (1972) pp. 235–238
[a3] M.B. Green, J.H. Schwarz, E. Witten, "Superstring theory" , Cambridge Univ. Press (1987)
[a4] R.E. Block, "On the Mills–Seligman axioms for Lie algebras of classical type" Trans. Amer. Math. Soc. , 121 (1966) pp. 378–392 Zbl 0136.30402
[a5] I.M. Gel'fand, D.B. Fuks, "The cohomology of the Lie algebra of vector fields in a circle" Funct. Anal. Appl. , 2 (1968) pp. 342–343 Funkts. Anal. i Prilozh. , 2 : 4 (1968) pp. 92–93
[a6] V.G. Kac, "Highest weight representations of infinite dimensional Lie algebras" , Proc. Internat. Congress Mathematicians (Helsinki, 1978) , 1 , Acad. Sci. Fennicae (1980) pp. 299–304
[a7] V.G. Kac, A.K. Raina, "Bombay lectures on highest weight representations" , World Sci. (1987)
[a8] A.A. Belavin, A.M. Polyakov, A.B. Zamolodchikov, "Infinite conformal symmetry in two-dimensional quantum field theory" Nuclear Phys. , B241 (1984) pp. 333–380
[a9] G. Felder, "BRST approach to minimal models" Nuclear Phys. , B317 (1989) pp. 215–236
[a10] B.L. Feigin, D.B. [D.B. Fuks] Fuchs, "Verma models 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. 230–245
[a11] V.G. Kac, M. Wakimoto, "Modular invariant representations of infinite-dimensional Lie algebras and superalgebras" Proc. Nat. Acad. Sci. USA , 85 (1988) pp. 4956–4960
[a12] D. Friedan, Z. Qui, S. Shenker, "Conformal invariance, unitary and two dimensional critical exponents" Publ. MSRI , 3 (1985) pp. 419–449
[a13] P. Goddard, A. Kent, D. Olive, "Unitary representations of the Virasoro and super-Virasoro algebras" Comm. Math. Phys. , 103 (1986) pp. 105–119
[a14] C. Itzykson, J.-M. Dronfree, "Statistical field theory" , Cambridge Univ. Press (1989)
[a15] E. Arbarello, C. De Concini, V.G. Kac, C. Processi, "Moduli spaces of curves and representation theory" Comm. Math. Phys. , 117 (1988) pp. 1–36
[a16] M.L. Kontzevich, "Virasoro algebra and Teichmüller spaces" Funct. Anal. Appl. , 21 : 2 (1987) pp. 156–157 Funkts. Anal. i Prilozh. , 21 : 2 (1987) pp. 78–79
[a17] A.A. Beilinson, V.V. Schechtman, "Determinant bundles and Virasoro algebras" Comm. Math. Phys. , 118 (1988) pp. 651–701
[a18] N. Kawamoto, Y. Namikawa, A. Tsuchiya, Y. Yamada, "Geometric realization of conformal field theory on Riemann surfaces" Comm. Math. Phys. , 116 (1988) pp. 247–308
How to Cite This Entry:
Virasoro algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Virasoro_algebra&oldid=15975
This article was adapted from an original article by Victor Kac (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article