N=2 Super Lie Conformal Algebra#

The \(N=2\) super Lie conformal algebra is an extension of the Virasoro Lie conformal algebra (with generators \(L,C\)) by an even generator \(J\) which is primary of conformal weight \(1\) and two odd generators \(G_1,G_2\) which are primary of conformal weight \(3/2\). The remaining \(\lambda\)-brackets are given by:

\[\begin{split}[J_\lambda J] &= \frac{\lambda}{3} C, \\ [J_\lambda G_1] &= G_1, \\ [J_\lambda G_2] &= -G_2, \\ [{G_1}_\lambda G_1] &= [{G_2}_\lambda G_2 ] = 0, \\ [{G_1}_\lambda G_2] &= L + \frac{1}{2} TJ + \lambda J + \frac{\lambda^2}{6}C.\end{split}\]


  • Reimundo Heluani (2020-06-03): Initial implementation.

class sage.algebras.lie_conformal_algebras.n2_lie_conformal_algebra.N2LieConformalAlgebra(R)#

Bases: GradedLieConformalAlgebra

The N=2 super Lie conformal algebra.


  • R – a commutative ring; the base ring of this super Lie conformal algebra.


sage: x = polygen(ZZ, 'x')
sage: F.<x> = NumberField(x^2 - 2)
sage: R = lie_conformal_algebras.N2(F); R
The N=2 super Lie conformal algebra over Number Field in x with defining polynomial x^2 - 2
sage: R.inject_variables()
Defining L, J, G1, G2, C
sage: G1.bracket(G2)
{0: L + 1/2*TJ, 1: J, 2: 1/3*C}
sage: G2.bracket(G1)
{0: L - 1/2*TJ, 1: -J, 2: 1/3*C}
sage: G1.degree()
sage: J.degree()

The topological twist is a Virasoro vector with central charge 0:

sage: L2 = L - 1/2*J.T()
sage: L2.bracket(L2) == {0: L2.T(), 1: 2*L2}

The sum of the fermions is a generator of the Neveu-Schwarz Lie conformal algebra:

sage: G = (G1 + G2)
sage: G.bracket(G)
{0: 2*L, 2: 2/3*C}