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Modular groups

Infinite dimensional Lie group

In mathematics, the Virasoro algebra (named after the physicist Miguel Angel Virasoro) is a complex Lie algebra, given as a central extension of the complex polynomial vector fields on the circle, and is widely used in conformal field theory and string theory.
Definition
The Virasoro algebra is spanned by elements L_n for n\in\mathbb{Z} and c with L_n + L_{n}, \quad i(L_n L_{n}) and c being real elements. Here the central element c is the central charge.
The algebra satisfies [c,L_n]=0 and [L_m,L_n]=(mn)L_{m+n}+\frac{c}{12}(m^3m)\delta_{m+n,0}. The factor of 1/12 is merely a matter of convention.
The Virasoro algebra is a central extension of the (complex) Witt algebra of complex polynomial vector fields on the circle. The Lie algebra of real polynomial vector fields on the circle is a dense subalgebra of the Lie algebra of diffeomorphisms of the circle.
The Virasoro algebra is obeyed by the stress tensor in string theory, since it comprises the generators of the conformal group of the worldsheet, 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 GuptaBleuler quantization).
Representation theory
A lowest weight representation of the Virasoro algebra is a representation generated by a vector v that is killed by L_i for i\geq 1, and is an eigenvector of L_0 and c. The letters h and c are usually used for the eigenvalues of L_0 and c on v. (The same letter c is used for both the element c of the Virasoro algebra and its eigenvalue.) For every pair of complex numbers h and c there is a unique irreducible lowest weight representation with these eigenvalues.
A lowest weight representation is called unitary if it has a positive definite inner product such that the adjoint of L_n is L_{n}. The irreducible lowest weight representation with eigenvalues h and c is unitary if and only if either c≥1 and h≥0, or c is one of the values
 c = 1{6\over m(m+1)} = 0,\quad 1/2,\quad 7/10,\quad 4/5,\quad 6/7,\quad 25/28, \ldots
for m = 2, 3, 4, .... and h is one of the values
 h = h_{r,s}(c) = {((m+1)rms)^21 \over 4m(m+1)}
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. The unitary irreducible lowest weight representations with c < 1 are called the discrete series representations of the Virasoro algebra. These are special cases of the representations with m = q/(p−q), 0<r<q, 0< s<p for p and q coprime integers and r and s integers, called the minimal models and first studied in Belavin et al. (1984).
The first few discrete series representations are given by:
 m = 2: c = 0, h = 0. The trivial representation.
 m = 3: c = 1/2, h = 0, 1/16, 1/2. These 3 representations are related to the Ising model
 m = 4: c = 7/10. h = 0, 3/80, 1/10, 7/16, 3/5, 3/2. These 6 representations are related to the tri critical Ising model.
 m = 5: c = 4/5. There are 10 representations, which are related to the 3state Potts model.
 m = 6: c = 6/7. There are 15 representations, which are related to the tri critical 3state Potts model.
The lowest weight representations that are not irreducible can be read off from the Kac determinant formula, which states that the determinant of the invariant inner product on the degree h+N piece of the lowest weight module with eigenvalues c and h is given by
 A_N\prod_{1\le r,s\le N}(hh_{r,s}(c))^{p(Nrs)}
which was stated by V. Kac (1978), (see also Kac and Raina 1987) and whose first published proof was given by Feigin and Fuks (1984). (The function p(N) is the partition function, and A_{N} is some constant.) The reducible highest weight representations are the representations with h and c given in terms of m, c, and h by the formulas above, except that m is not restricted to be an integer ≥ 2 and may be any number other than 0 and 1, and r and s may be any positive integers. This result was used by Feigin and Fuks to find the characters of all irreducible lowest weight representations.
Generalizations
There are two supersymmetric N=1 extensions of the Virasoro algebra, called the NeveuSchwarz algebra and the Ramond algebra. Their theory is similar to that of the Virasoro algebra. There are further extensions of these algebras with more supersymmetry, such as the N = 2 superconformal algebra.
The Virasoro algebra is a central extension of the Lie algebra of meromorphic vector fields on a genus 0 Riemann surface that are holomorphic except at two fixed points. I.V. Krichever and S.P. Novikov (1987) found a central extension of the Lie algebra of meromorphic vector fields on a higher genus compact Riemann surface that are holomorphic except at two fixed points, and M. Schlichenmaier (1993) extended this to the case of more than two points.
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. Unsurprisingly these are called the vertex Virasoro and conformal Virasoro algebras respectively.
History
The Witt algebra (the Virasoro algebra without the central extension) was discovered by E. 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 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 also
 Witt algebra
 Heisenberg algebra
 WZW model
 conformal field theory
 Goddard–Thorn theorem
 Lie conformal algebra
References
 R. E. Block (1966). "On the Mills–Seligman axioms for Lie algebras of classical type".
 R. C. Brower, C. B. Thorn (1971). "Eliminating spurious states from the dual resonance model". .
 E. Cartan (1909). "Les groupes de transformations continus, infinis, simples". Annals of Sci Ecole Normale Supérieur 26: 93–161.
 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. 230–245
 Friedan, D., Qiu, Z. and Shenker, S. (1984). "Conformal invariance, unitarity and critical exponents in two dimensions". .
 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. 342–343 Funkts. Anal. i Prilozh., 2 : 4 (1968) pp. 92–93
 P. Goddard, A. Kent and D. Olive (1986). "Unitary representations of the Virasoro and superVirasoro algebras". .
 Iohara, Kenji; Koga, Yoshiyuki (2011), Representation theory of the Virasoro algebra, Springer Monographs in Mathematics, London: SpringerVerlag London Ltd.,
 A. Kent (1991). "Singular vectors of the Virasoro algebra".
 Victor Kac (2001), "Virasoro algebra", in Hazewinkel, Michiel,
 V. G. Kac, Highest weight representations of infinite dimensional Lie algebras, Proc. Internat. Congress Mathematicians (Helsinki, 1978),
 V. G. Kac, A. K. Raina, Bombay lectures on highest weight representations, World Sci. (1987) ISBN 9971503956.
 V. K. Dobrev, Multiplet classification of the indecomposable highest weight modules over the NeveuSchwarz and Ramond superalgebras, Lett. Math. Phys. 11 (1986) 225234 & correction: ibid. 13 (1987) 260.
 I. M. Krichever, S.P. Novikov, Algebras of Virasoro type, Riemann surfaces and structures of the theory of solitons, Funkts. Anal. Appl., 21:2 (1987) p. 46–63.
 V. K. Dobrev, Characters of the irreducible highest weight modules over the Virasoro and superVirasoro algebras, Suppl. Rendiconti del Circolo Matematico di Palermo, Serie II, Numero 14 (1987) 2542.
 M. Schlichenmaier, Differential operator algebras on compact Riemann surfaces H.D. Doebner (ed.) V.K. Dobrev (ed.) A.G Ushveridze (ed.), Generalized Symmetries in Physics, Clausthal 1993, World Sci. (1994) p. 425–435
 M. A. Virasoro (1970). "Subsidiary conditions and ghosts in dualresonance models".
 A. J. Wassermann (2010). "Lecture notes on KacMoody and Virasoro algebras". arXiv:1004.1287 [math.RT].
 Wassermann, A. J. (2010). "Direct proofs of the FeiginFuchs character formula for unitary representations of the Virasoro algebra". arXiv:1012.6003 [math.RT].