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In mathematics, the Virasoro algebra (named after the physicist Miguel Angel Virasoro)[1] is a complex Lie algebra, the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string theory.

Contents

DefinitionEdit

The Virasoro algebra is spanned by generators Ln for n ∈ ℤ and the central charge c. These generators satisfy   and

  .

The factor of ​112 is merely a matter of convention. For a derivation of the algebra as the unique central extension of the Witt algebra, see derivation of the Virasoro algebra.

The Virasoro algebra has a presentation in terms of 2 generators (e.g. L3 and L−2) and 6 relations.[2][3]

Representation theoryEdit

Highest weight representationsEdit

A highest weight representation of the Virasoro algebra is a representation generated by a primary state: a vector   such that

 

where the number h called the conformal dimension or conformal weight of  .[4]

A highest weight representation is spanned by eigenstates of  . The eigenvalues take the form  , where the integer   is called the level of the corresponding eigenstate.

More precisely, a highest weight representation is spanned by  -eigenstates of the type   with   and  , whose levels are  . Any state whose level is not zero is called a descendant state of  .

For any pair of complex numbers h and c, the Verma module   is the largest possible highest weight representation. (The same letter, c, is used for both the element c of the Virasoro algebra and its eigenvalue in a representation.)

The states   with   and   form a basis of the Verma module. The Verma module is indecomposable, and for generic values of h and c it is also irreducible. When it is reducible, there exist other highest weight representations with these values of h and c, called degenerate representations, which are cosets of the Verma module. In particular, the unique irreducible highest weight representation with these values of h and 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 vectorsEdit

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   to have a singular vector at the level N is   for some positive integers r,s such that N=rs, with

 

In particular,  , and the reducible Verma module   has a singular vector   at the level N=1. Then  , and the corresponding reducible Verma module has a singular vector   at the level N=2.

This condition for the existence of a singular vector at the level   is not necessary. In particular, there is a singular vector at the level   if   with   and  . This singular vector is now a descendent of another singular vector at the level  . This type of singular vectors can however only exist if the central charge is of the type   with  . (For   coprime, these are the central charges of the minimal models.)[4]

Hermitian form and unitarityEdit

A highest weight representation with a real value of   has a unique Hermitian form such that the adjoint of   is  , 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   is given by the Kac determinant formula,

 

where the function p(N) is the partition function, and AN is a positive constant that does not depend on   or  . 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 h and c is unitary if and only if either c ≥1 and h≥0, or

 

for m = 2, 3, 4, ..., and h is one of the values

 

for r = 1, 2, 3, ..., m−1 and s= 1, 2, 3, ..., r.

Daniel Friedan, Zongan Qiu, and Stephen Shenker (1984) showed that these conditions are necessary, and 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 algebras) to show that they are sufficient.

ApplicationsEdit

Conformal field theoryEdit

In two dimensions, the algebra of local 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 blocks, special functions that include and generalize the characters of representations of the Virasoro algebra.

String theoryEdit

Since the Virasoro algebra comprises the generators of the conformal group of the worldsheet, the 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 theory, cannot be applied to all the states in the theory, but rather only on the physical states (compare Gupta–Bleuler formalism).

GeneralizationsEdit

Super Virasoro algebrasEdit

There are two 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 numbers. There are further extensions of these algebras with more supersymmetry, such as the N = 2 superconformal algebra.

W-algebrasEdit

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 algebrasEdit

The Virasoro algebra is a subalgebra of the universal enveloping algebra of any affine Lie algebra, as shown by the Sugawara construction. In this sense, affine Lie algebras are extensions of the Virasoro algebra.

Meromorphic vector fields on Riemann surfacesEdit

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.[5] This can be further generalized to supermanifolds.[6]

Vertex Virasoro algebra and conformal Virasoro algebraEdit

The Virasoro algebra also has vertex algebraic and conformal algebraic counterparts, which basically come from arranging all the basis elements into generating series and working with single objects.

HistoryEdit

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 > 0) by R. E. Block (1966, page 381) and independently rediscovered (in characteristic 0) by I. M. Gelfand and D. B. Fuchs [de] (1968). Virasoro (1970) wrote down some operators generating the Virasoso algebra (later known as the Virasoro operators) while studying dual resonance models, 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 alsoEdit

NotesEdit

  1. ^ M. A. Virasoro (1970). "Subsidiary conditions and ghosts in dual-resonance models". Physical Review D. 1 (10): 2933–2936. Bibcode:1970PhRvD...1.2933V. doi:10.1103/PhysRevD.1.2933.
  2. ^ Fairlie, D. B.; Nuyts, J.; Zachos, C. K. (1988). "A presentation for the Virasoro and super-Virasoro algebras". Communications in Mathematical Physics. 117 (4): 595. Bibcode:1988CMaPh.117..595F. doi:10.1007/BF01218387.
  3. ^ Uretsky, J. L. (1989). "Redundancy of conditions for a Virasoro algebra". Communications in Mathematical Physics. 122 (1): 171–173. Bibcode:1989CMaPh.122..171U. doi:10.1007/BF01221412.
  4. ^ a b P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, 1997, ISBN 0-387-94785-X
  5. ^ Krichever, I. M.; Novikov, S.P. (1987). "Algebras of Virasoro type, Riemann surfaces and structures of the theory of solitons". Funkts. Anal. Appl. 21 (2): 46–63. doi:10.1007/BF01078026.
  6. ^ Rabin, J. M. (1995). "Super elliptic curves". Journal of Geometry and Physics. 15 (3): 252–280. arXiv:hep-th/9302105. Bibcode:1995JGP....15..252R. doi:10.1016/0393-0440(94)00012-S.

ReferencesEdit