上半平面上的手性德拉姆复形与模形式
Chiral de Rham complex on the upper half plane and modular forms
AI总结：
该数学研究针对同余子群构造上半平面手性德拉姆复形的顶点算子代数，引入不变滤过证明其与模形式同构，给出提升与特征公式，还修正了Rankin-Cohen括号并证明其非零性。
摘要：
针对任一同余子群$\Gamma$，我们研究由上半平面上的手性德拉姆复形的$\Gamma$-不变整体截面构造的顶点算子代数$\Omega^{ch}(\mathbb H,\Gamma)$，这些截面在所有尖点处全纯。我们在整体截面上引入一种$SL(2,\mathbb R)$-不变滤过，并证明分次代数上的$\Gamma$-不变量与若干份模形式同构。我们还给出了模形式提升至$\Omega^{ch}(\mathbb H,\Gamma)$的显式公式，并计算了$\Omega^{ch}(\mathbb H,\Gamma)$的特征公式。此外，我们证明顶点代数结构修正了Rankin-Cohen括号，且修正后的括号在涉及Eisenstein级数的常模形式之间非零。
英文摘要：
For any congruence subgroup $Γ$, we study the vertex operator algebra $Ω^{ch}(\mathbb H,Γ)$ constructed from the $Γ$-invariant global sections of the chiral de Rham complex on the upper half plane, which are holomorphic at all the cusps. We introduce an $SL(2,\mathbb R)$-invariant filtration on the global sections and show that the $Γ$-invariants on the graded algebra is isomorphic to certain copies of modular forms. We also give an explicit formula for the lifting of modular forms to $Ω^{ch}(\mathbb H,Γ)$ and compute the character formula of $Ω^{ch}(\mathbb H,Γ)$. Furthermore, we show that the vertex algebra structure modifies the Rankin-Cohen bracket, and the modified bracket becomes non-zero between constant modular forms involving the Eisenstein series.
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