A new $S{L}_2$-action on the rank-two $\beta \gamma$-system

We construct an explicit rational ${\mathrm{S}\mathrm{L}}_2(\mathbb{C})$-action on the underlying vector space of the rank-two $\beta \gamma$-system. The construction is motivated by Creutzig’s branching decomposition with respect to $L_{-1}({\mathfrak{s}\mathfrak{l}}_2)$ and a rank-one Heisenberg vertex algebra. We realize the raising and lowering operators as residues of lattice-type fields, prove that they preserve the original $\beta \gamma$-system despite being defined on localizations, and establish their commutation relations and integrability. The resulting action commutes with the affine $L_{-1}({\mathfrak{s}\mathfrak{l}}_2)$-action and all nonzero Heisenberg modes, while the Heisenberg zero mode serves as its Cartan operator. Each affine multiplicity space is thereby identified with $V_n\otimes {\mathcal{F}}_{\mathrm{o}\mathrm{s}\mathrm{c}}$, where $V_n$ is the irreducible ${\mathrm{S}\mathrm{L}}_2(\mathbb{C})$-module of highest weight $n$ and ${\mathcal{F}}_{\mathrm{o}\mathrm{s}\mathrm{c}}$ is the Heisenberg oscillator Fock space. Thus the Heisenberg charge decomposition is realized as the weight decomposition of $V_n$. The infinitesimal action is not by vertex algebra derivations and does not preserve the symmetric conformal grading.