发表机构
Okinawa Institute of Science and Technology; Nanjing University; University of Science and Technology of China（冲绳科学技术大学院大学; 南京大学; 中国科学技术大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在秩二βγ系统的底空间上构造显式有理SL2(C)作用，受Creutzig分支分解启发，将升降算子实现为格点型场留数，证明其保持系统并满足对易与可积性，与仿射L-1(sl2)及Heisenberg模对易，实现Heisenberg荷分解为权分解。
AI 中文摘要
我们在秩二 $\beta\gamma$-系统的底层向量空间上构造了一个显式的有理 $\mathrm{SL}_2(\mathbb C)$-作用。该构造受 Creutzig 关于 $L_{-1}(\mathfrak{sl}_2)$ 和秩一 Heisenberg 顶点代数的分支分解的启发。我们将升算子和降算子实现为格点型场的留数，证明尽管它们定义在局部化上，仍保持原始的 $\beta\gamma$-系统，并确立它们的对易关系和可积性。所得作用与仿射 $L_{-1}(\mathfrak{sl}_2)$-作用以及所有非零 Heisenberg 模对易，而 Heisenberg 零模充当其 Cartan 算子。每个仿射重数空间因此被等同于 $V_n\otimes\mathcal F_{\mathrm{osc}}$，其中 $V_n$ 是最高权为 $n$ 的不可约 $\mathrm{SL}_2(\mathbb C)$-模，$\mathcal F_{\mathrm{osc}}$ 是 Heisenberg 振荡 Fock 空间。因此，Heisenberg 荷分解被实现为 $V_n$ 的权分解。该无穷小作用不是由顶点代数导子给出的，也不保持对称共形分次。
英文摘要
We construct an explicit rational \(\mathrm{SL}_2(\mathbb C)\)-action on the underlying vector space of the rank-two \(βγ\)-system. The construction is motivated by Creutzig's branching decomposition with respect to \(L_{-1}(\mathfrak{sl}_2)\) 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 \(βγ\)-system despite being defined on localizations, and establish their commutation relations and integrability. The resulting action commutes with the affine \(L_{-1}(\mathfrak{sl}_2)\)-action and all nonzero Heisenberg modes, while the Heisenberg zero mode serves as its Cartan operator. Each affine multiplicity space is thereby identified with \(V_n\otimes\mathcal F_{\mathrm{osc}}\), where \(V_n\) is the irreducible \(\mathrm{SL}_2(\mathbb C)\)-module of highest weight \(n\) and \(\mathcal F_{\mathrm{osc}}\) 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.
Comments12 pages. Comments are welcome!
