Superalgebras, symplectic bosons and the Sugawara construction
Introduce 2n real boson fields \(\zeta^{\alpha}(z)=\sum_{s}\zeta_ s^{\alpha}z^{-s}\), where the summation is taken over \(s\in {\mathbb{Z}}\) (Ramond case) or \(s\in {\mathbb{Z}}+1/2\) (Neveu-Schwarz case), subject to the hermiticity condition \(\zeta_ s^{\alpha \dag}=\zeta^{\alpha}_{- s}\). They satisfy the commutation relations \([\zeta_ r^{\alpha},\zeta_ s^{\beta}]=iJ^{\alpha \beta}\delta_{r,-s}\), where \(J=diag(\begin{pmatrix} 0&1 \\ -1&0 \end{pmatrix},...,\begin{pmatrix} 0&1 \\ -1&0 \end{pmatrix})\) (n-copies). Take the fields to act in a space with vacuum satisfying \(\zeta_ r^{\alpha}| 0>=0\) \((r>0)\). One can form representations of \(\hat{\mathfrak g}\) for suitable, possible noncompact, \({\mathfrak g}\) out of real symplectic representations using boson fields. Having such a representation of \(\hat{\mathfrak g}\), one can perform the Sugawara construction \({\mathcal L}(z)\) of the Virasoro algebra. On the other hand, one can form the free Virasoro algebra with the central charge \[ c=-n: L^{\zeta}(z)=\sum_{n}L_ n^{\zeta}z^{-n}=i/2\quad J_{\alpha \beta}: zd\zeta^{\alpha}(z)/dz\quad \zeta^{\beta}(z): - \epsilon n/8, \] where \(\epsilon =0\) (NS case) or \(\epsilon =1\) (R case), \(J_{\alpha \beta}=J^{-1}.\) One of the main results of the first half is the following ``superalgebra theorem'', which is the bosonic version of the ``symmetric space theorem'': \({\mathcal L}(z)=L^{\zeta}(z)\) if and only if the representation is that given by the fermionic generators \(\{s^{\alpha}\}\) of a superalgebra \({\mathfrak a}\), containing \({\mathfrak g}\) as its bosonic part and possessing a quadratic Casimir operator, \(\eta_{ab}t\) at \(b+iJ_{\alpha \beta}s^{\alpha}s^{\beta}\), where \(\eta_{ab}\) is the symmetric invariant form on \({\mathfrak g}\), used to construct \({\mathcal L}(z).\) After giving some examples of the superalgebra theorem, the authors extend the Sugawara construction to affine superalgebras \(\hat{\mathfrak a}\), and discuss the construction of representations of such affine superalgebras using orthosymplectic representations of \({\mathfrak a}\) together with fermion and symplectic boson fields. Finally the ``super- symmetric space theorem'' is given, which contains the symmetric space theorem and the superalgebra theorem as special cases.
scientific article; zbMATH DE number 4033961