Virasoro algebra

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

\( [L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n,0}. \)

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 Gupta-Bleuler 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

\( h = h_{r,s}(c) = {((m+1)r-ms)^2-1 \over 4m(m+1)} \)

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 3-state Potts model.
m = 6: c = 6/7. There are 15 representations, which are related to the tri critical 3-state 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

\( A_N\prod_{1\le r,s\le N}(h-h_{r,s}(c))^{p(N-rs)} \)

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 AN 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 Neveu-Schwarz 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 (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
WZW model
conformal field theory
No-ghost theorem
Lie conformal algebra

References

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