In mathematics, the Virasoro algebra is a complex Lie algebra and the unique nontrivial central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string theory. It is named after Miguel Ángel Virasoro.
Structure The Virasoro algebra is spanned by generators Ln for n ∈ Z {\displaystyle n\in \mathbb {Z} } and the central charge c. These generators satisfy [ c , L n ] = 0 {\displaystyle [c,L_{n}]=0} and
The factor of 1 12 {\displaystyle {\frac {1}{12}}} 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 or Schottenloher, Thm. 5.1, pp. 79. The Virasoro algebra has a presentation in terms of two generators (e.g. L3 and L−2) and six relations. The generators L n > 0 {\displaystyle L_{n>0}} are called annihilation modes, while L n < 0 {\displaystyle L_{n<0}} are creation modes. A basis of creation generators of the Virasoro algebra's universal enveloping algebra is the set
L = { L − n 1 L − n 2 ⋯ L − n k } k ∈ N 0 < n 1 ≤ n 2 ≤ ⋯ n k {\displaystyle {\mathcal {L}}={\Big \{}L_{-n_{1}}L_{-n_{2}}\cdots L_{-n_{k}}{\Big \}}_{\begin{array}{l}k\in \mathbb {N} \\0<n_{1}\leq n_{2}\leq \cdots n_{k}\end{array}}}
For L ∈ L {\displaystyle L\in {\mathcal {L}}} , let | L | = ∑ i = 1 k n i {\displaystyle |L|=\sum _{i=1}^{k}n_{i}} , then
[ L 0 , L ] = | L | L {\displaystyle [L_{0},L]=|L|L} .
Representation theory In any indecomposable representation of the Virasoro algebra, the central generator c {\displaystyle c} of the algebra takes a constant value, also denoted c {\displaystyle c} and called the representation's central charge. A vector v {\displaystyle v} in a representation of the Virasoro algebra has conformal dimension (or conformal weight) h {\displaystyle h} if it is an eigenvector of L 0 {\displaystyle L_{0}} with eigenvalue h {\displaystyle h} :
L 0 v = h v {\displaystyle L_{0}v=hv}
An L 0 {\displaystyle L_{0}} -eigenvector v {\displaystyle v} is called a primary state (of dimension h {\displaystyle h} ) if it is annihilated by the annihilation modes,
L n > 0 v = 0 {\displaystyle L_{n>0}v=0}
Highest weight representations A highest weight representation of the Virasoro algebra is a representation generated by a primary state v {\displaystyle v} . A highest weight representation is spanned by the L 0 {\displaystyle L_{0}} -eigenstates { L v } L ∈ L {\displaystyle \{Lv\}_{L\in {\mathcal {L}}}} . The conformal dimension of L v {\displaystyle Lv} is h + | L | {\displaystyle h+|L|} , where | L | ∈ N {\displaystyle |L|\in \mathbb {N} } is called the level of L v {\displaystyle Lv} . Any state whose level is not zero is called a descendant state of v {\displaystyle v} . For any h , c ∈ C {\displaystyle h,c\in \mathbb {C} } , the Verma module V c , h {\displaystyle {\mathcal {V}}_{c,h}} of central charge c {\displaystyle c} and conformal dimension h {\displaystyle h} is the representation whose basis is { L v } L ∈ L {\displaystyle \{Lv\}_{L\in {\mathcal {L}}}} , for v {\displaystyle v} a primary state of dimension h {\displaystyle h} . The Verma module is the largest possible highest weight representation. The Verma module is indecomposable, and for generic values of h , c ∈ C {\displaystyle h,c\in \mathbb {C} } it is also irreducible. When it is reducible, there exist other highest weight representations with these values of h , c ∈ C {\displaystyle h,c\in \mathbb {C} } , called degenerate representations, which are quotients of the Verma module. In particular, the unique irreducible highest weight representation with these values of h , c ∈ C {\displaystyle h,c\in \mathbb {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 vectors A singular vector or null vector of a highest weight representation is a state that is both descendant and primary. A sufficient condition for the Verma module V c , h {\displaystyle {\mathcal {V}}_{c,h}} to have a singular vector is h = h r , s ( c ) {\displaystyle h=h_{r,s}(c)} for some r , s ∈ N ∗ {\displaystyle r,s\in \mathbb {N} ^{*}} , where
h r , s ( c ) = 1 4 ( ( β r − β − 1 s ) 2 − ( β − β − 1 ) 2 ) , where c = 1 − 6 ( β − β − 1 ) 2 . {\displaystyle h_{r,s}(c)={\frac {1}{4}}{\Big (}(\beta r-\beta ^{-1}s)^{2}-(\beta -\beta ^{-1})^{2}{\Big )}\ ,\quad {\text{where}}\quad c=1-6(\beta -\beta ^{-1})^{2}\ .}
Then the singular vector has level r s {\displaystyle rs} and conformal dimension
h r , s + r s = h r , − s {\displaystyle h_{r,s}+rs=h_{r,-s}}
Here are the values of h r , s ( c ) {\displaystyle h_{r,s}(c)} for r s ≤ 4 {\displaystyle rs\leq 4} , together with the corresponding singular vectors, written as L r , s v {\displaystyle L_{r,s}v} for v {\displaystyle v} the primary state of V c , h r , s ( c ) {\displaystyle {\mathcal {V}}_{c,h_{r,s}(c)}} :
r , s h r , s L r , s 1 , 1 0 L − 1 2 , 1 − 1 2 + 3 4 β 2 L − 1 2 − β 2 L − 2 1 , 2 − 1 2 + 3 4 β − 2 L − 1 2 − β − 2 L − 2 3 , 1 − 1 + 2 β 2 L − 1 3 − 4 β 2 L − 1 L − 2 + 2 β 2 ( 2 β 2 + 1 ) L − 3 1 , 3 − 1 + 2 β − 2 L − 1 3 − 4 β − 2 L − 1 L − 2 + 2 β − 2 ( 2 β − 2 + 1 ) L − 3 4 , 1 − 3 2 + 15 4 β 2 L − 1 4 − 10 β 2 L − 1 2 L − 2 + 2 β 2 ( 12 β 2 + 5 ) L − 1 L − 3 + 9 β 4 L − 2 2 − 6 β 2 ( 6 β 4 + 4 β 2 + 1 ) L − 4 2 , 2 3 4 ( β − β − 1 ) 2 L − 1 4 − 2 ( β 2 + β − 2 ) L − 1 2 L − 2 + ( β 2 − β − 2 ) 2 L − 2 2 + 2 ( 1 + ( β + β − 1 ) 2 ) L − 1 L − 3 − 2 ( β + β − 1 ) 2 L − 4 1 , 4 − 3 2 + 15 4 β − 2 L − 1 4 − 10 β − 2 L − 1 2 L − 2 + 2 β − 2 ( 12 β − 2 + 5 ) L − 1 L − 3 + 9 β − 4 L − 2 2 − 6 β − 2 ( 6 β − 4 + 4 β − 2 + 1 ) L − 4 {\displaystyle {\begin{array}{|c|c|l|}\hline r,s&h_{r,s}&L_{r,s}\\\hline \hline 1,1&0&L_{-1}\\\hline 2,1&-{\frac {1}{2}}+{\frac {3}{4}}\beta ^{2}&L_{-1}^{2}-\beta ^{2}L_{-2}\\\hline 1,2&-{\frac {1}{2}}+{\frac {3}{4}}\beta ^{-2}&L_{-1}^{2}-\beta ^{-2}L_{-2}\\\hline 3,1&-1+2\beta ^{2}&L_{-1}^{3}-4\beta ^{2}L_{-1}L_{-2}+2\beta ^{2}(2\beta ^{2}+1)L_{-3}\\\hline 1,3&-1+2\beta ^{-2}&L_{-1}^{3}-4\beta ^{-2}L_{-1}L_{-2}+2\beta ^{-2}(2\beta ^{-2}+1)L_{-3}\\\hline 4,1&-{\frac {3}{2}}+{\frac {15}{4}}\beta ^{2}&{\begin{array}{r}L_{-1}^{4}-10\beta ^{2}L_{-1}^{2}L_{-2}+2\beta ^{2}\left(12\beta ^{2}+5\right)L_{-1}L_{-3}\\+9\beta ^{4}L_{-2}^{2}-6\beta ^{2}\left(6\beta ^{4}+4\beta ^{2}+1\right)L_{-4}\end{array}}\\\hline 2,2&{\frac {3}{4}}\left(\beta -\beta ^{-1}\right)^{2}&{\begin{array}{l}L_{-1}^{4}-2\left(\beta ^{2}+\beta ^{-2}\right)L_{-1}^{2}L_{-2}+\left(\beta ^{2}-\beta ^{-2}\right)^{2}L_{-2}^{2}\\+2\left(1+\left(\beta +\beta ^{-1}\right)^{2}\right)L_{-1}L_{-3}-2\left(\beta +\beta ^{-1}\right)^{2}L_{-4}\end{array}}\\\hline 1,4&-{\frac {3}{2}}+{\frac {15}{4}}\beta ^{-2}&{\begin{array}{r}L_{-1}^{4}-10\beta ^{-2}L_{-1}^{2}L_{-2}+2\beta ^{-2}\left(12\beta ^{-2}+5\right)L_{-1}L_{-3}\\+9\beta ^{-4}L_{-2}^{2}-6\beta ^{-2}\left(6\beta ^{-4}+4\beta ^{-2}+1\right)L_{-4}\end{array}}\\\hline \end{array}}}
Singular vectors for arbitrary r , s ∈ N ∗ {\displaystyle r,s\in \mathbb {N} ^{*}} may be computed using various algorithms, and their explicit expressions are known. If β 2 ∉ Q {\displaystyle \beta ^{2}\notin \mathbb {Q} } , then V c , h {\displaystyle {\mathcal {V}}_{c,h}} has a singular vector at level N {\displaystyle N} if and only if h = h r , s ( c ) {\displaystyle h=h_{r,s}(c)} with N = r s {\displaystyle N=rs} . If β 2 ∈ Q {\displaystyle \beta ^{2}\in \mathbb {Q} } , there can also exist a singular vector at level N {\displaystyle N} if N = r s + r ′ s ′ {\displaystyle N=rs+r's'} with h = h r , s ( c ) {\displaystyle h=h_{r,s}(c)} and h + r s = h r ′ , s ′ ( c ) {\displaystyle h+rs=h_{r',s'}(c)} . This singular vector is now a descendant of another singular vector at level r s {\displaystyle rs} . The integers r , s {\displaystyle r,s} that appear in h r , s ( c ) {\displaystyle h_{r,s}(c)} are called Kac indices. It can be useful to use non-integer Kac indices for parametrizing the conformal dimensions of Verma modules that do not have singular vectors, for example in the critical random cluster model.
Shapovalov form For any c , h ∈ C {\displaystyle c,h\in \mathbb {C} } , the involution L n ↦ L ∗ = L − n {\displaystyle L_{n}\mapsto L^{*}=L_{-n}} defines an automorphism of the Virasoro algebra and of its universal enveloping algebra. Then the Shapovalov form is the symmetric bilinear form on the Verma module V c , h {\displaystyle {\mathcal {V}}_{c,h}} such that ( L v , L ′ v ) = S L , L ′ ( c , h ) {\displaystyle (Lv,L'v)=S_{L,L'}(c,h)} , where the numbers S L , L ′ ( c , h ) {\displaystyle S_{L,L'}(c,h)} are defined by
L ∗ L ′ v = | L | = | L ′ | S L , L ′ ( c , h ) v {\displaystyle L^{*}L'v{\underset {|L|=|L'|}{=}}S_{L,L'}(c,h)v} and S L , L ′ ( c , h ) = | L | ≠ | L ′