4.2. Virasoro cocycle
In this section we assume that \bbk is a field of characteristic zero
and \witt is the Witt algebra over \bbk as in
Definition 4.1.1.
Virasoro cocycle.
The bilinear map {\gamma}_{\vir} \colon \witt \times \witt \to \bbk
given on basis elements of \witt by
\begin{aligned}
{\gamma}_{\vir}(\ell_n,\ell_m) = \frac{n^3 - n}{12} \, \delta_{n+m,0}
\end{aligned}
is called the Virasoro cocycle.
Lean code for Definition4.2.1●1 definition
Associated Lean declarations
-
defdefined in VirasoroProject/VirasoroCocycle.leancomplete
def VirasoroProject.WittAlgebra.virasoroCocycle.{u_1} (𝕜 : Type u_1) [Field 𝕜] [CharZero 𝕜] : VirasoroProject.LieTwoCocycle 𝕜 (VirasoroProject.WittAlgebra 𝕜) 𝕜
def VirasoroProject.WittAlgebra.virasoroCocycle.{u_1} (𝕜 : Type u_1) [Field 𝕜] [CharZero 𝕜] : VirasoroProject.LieTwoCocycle 𝕜 (VirasoroProject.WittAlgebra 𝕜) 𝕜
The Virasoro cocycle.
The Virasoro cocycle is a 2-cocycle.
The Virasoro cocycle is a 2-cocycle, {\gamma}_{\vir} \in \Coc^2(\witt,\bbk).
Lean code for Lemma4.2.2●1 definition
Associated Lean declarations
-
defdefined in VirasoroProject/VirasoroCocycle.leancomplete
def VirasoroProject.WittAlgebra.virasoroCocycle.{u_1} (𝕜 : Type u_1) [Field 𝕜] [CharZero 𝕜] : VirasoroProject.LieTwoCocycle 𝕜 (VirasoroProject.WittAlgebra 𝕜) 𝕜
def VirasoroProject.WittAlgebra.virasoroCocycle.{u_1} (𝕜 : Type u_1) [Field 𝕜] [CharZero 𝕜] : VirasoroProject.LieTwoCocycle 𝕜 (VirasoroProject.WittAlgebra 𝕜) 𝕜
The Virasoro cocycle.
By the construction if Definition 4.2.1,
{\gamma}_{\vir} \colon \witt \times \witt \to \bbk is bilinear.
It's antisymmetry on basis elements of the Witt algebra is easily checked,
so {\gamma}_{\vir} is antisymmetric. It remains to prove the Leibniz
rule for {\gamma}_{\vir}, i.e., that for X,Y,X \in \witt, we have
\begin{aligned}
{\gamma}_{\vir}(X,[Y,Z]) = \; & {\gamma}_{\vir}([X,Y],Z) + {\gamma}_{\vir}(Y,[X,Z]) .
\end{aligned}
This formula is trilinear in X,Y,Z, so it suffices to verify it for basis
vectors X=\ell_n, Y=\ell_m, Z=\ell_k. We calculate
\begin{aligned}
{\gamma}_{\vir}(\ell_n , [\ell_m , \ell_k])
= \; & {\gamma}_{\vir}\big( \ell_n , (m-k) \ell_{m+k} \big) \\
= \; & (m-k) \, \frac{n^3 - n}{12} \delta_{n+m+k,0} .
\end{aligned}
and
\begin{aligned}
& {\gamma}_{\vir}([\ell_n , \ell_m] , \ell_k)
+ {\gamma}_{\vir}(\ell_m , [\ell_n , \ell_k]) \\
= \; & {\gamma}_{\vir}((n-m) \ell_{n+m} , \ell_k)
+ {\gamma}_{\vir}(\ell_m , (n-k) \ell_{n+k}) \\
= \; & (n-m) \, \frac{(n+m)^3 - (n+m)}{12} \delta_{n+m+k,0}
+ (n-k) \, \frac{m^3-m}{12} \delta_{n+m+k,0} .
\end{aligned}
Both of the above results are nonzero only if k=-(n+m),
in which case m-k = 2m+n and n-k = 2n+m,
so it suffices to note that
\begin{aligned}
(2m+n) \, (n^3 - n)
= (n-m) \, \big( (n+m)^3 - (n+m) \big)
+ (2 n + m) (m^3 - m)
\end{aligned}
to verify the Leibniz rule for {\gamma}_{\vir}.
The Virasoro cocyle is nontrivial.
The cohomology class [{\gamma}_{\vir}] \in \Coh^2(\witt,\bbk)
of the Virasoro cocycle is nonzero.
Lean code for Lemma4.2.3●1 theorem
Associated Lean declarations
-
theoremdefined in VirasoroProject/VirasoroCocycle.leancomplete
theorem VirasoroProject.WittAlgebra.cohomologyClass_virasoroCocycle_ne_zero.{u_1} (𝕜 : Type u_1) [Field 𝕜] [CharZero 𝕜] : (VirasoroProject.WittAlgebra.virasoroCocycle 𝕜).cohomologyClass ≠ 0
theorem VirasoroProject.WittAlgebra.cohomologyClass_virasoroCocycle_ne_zero.{u_1} (𝕜 : Type u_1) [Field 𝕜] [CharZero 𝕜] : (VirasoroProject.WittAlgebra.virasoroCocycle 𝕜).cohomologyClass ≠ 0
The Virasoro cocycle is cohomologically nontrivial.
Assume, by way of contradiction, that {\gamma}_{\vir} \in \Cob^2(\witt,\bbk),
i.e., that {\gamma}_{\vir} = \partial \beta for some \beta \in \Coch^1(\witt,\bbk).
Then, in particular, for every n \in \bZ we would have
\begin{aligned}
{\gamma}_{\vir}(\ell_n,\ell_{-n})
= \beta \big( [\ell_n, \ell_{-n}] \big) = 2 n \, \beta ( \ell_0 ) .
\end{aligned}
By Definition 4.2.1, this would imply
\begin{aligned}
\frac{n^3-n}{12} = 2 n \, \beta ( \ell_0 )
\end{aligned}
for all n \in \bZ. Considering for example n=3 and n=6, we then get
\begin{aligned}
2 = 6 \, \beta (\ell_0)
\qquad \text{ and } \qquad
\frac{35}{2} = 12 \, \beta (\ell_0) ,
\end{aligned}
which obviously yield a contradiction.