Virasoro algebra

A 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
[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