(Topological) modular forms with level structures: decompositions and duality

Topological modular forms with level structure: decompositions and duality

Topological modular forms with level structure were introduced in full generality by Hill and Lawson. We show that these decompose additively in many cases into a few simple pieces and give an application to equivariant $TMF$. Furthermore, we show which $Tmf_1(n)$ are self-Anderson dual up to a shift, both with and without their natural $C_2$-action.

拓扑模形式的对偶

Duality for Topological Modular Forms

已经观察到拓扑模形式的谱的某些局部化是自对偶的（Mahowald-Rezk，Gross-Hopkins）。我们提供了一个完整的解释，这些结果是内部的几何（紧化）模堆栈的椭圆曲线$ \M $，但只有在派生的设置是真实的。当$ 2 $被反转时，椭圆曲线的水平$ 2 $结构的选择提供了$ \M $的几何上良好表现的覆盖，这允许人们将$ Tmf $视为在$ GL_2(\Z/2) $的自然作用下的$ Tmf(2) $的同伦不动点，具有水平$ 2 $结构的拓扑模形式。利用Grothendieck-Serre对偶性，得到了$ Tmf(2) $是自对偶的.伴随的泰特谱的消失使$ Tmf $自身成为安德森自对偶。

The local cohomology spectral sequence for topological modular forms

We discuss proofs of local cohomology theorems for topological modular forms, based on Mahowald-Rezk duality and on Gorenstein duality, and then make the associated local cohomology spectral sequences explicit, including their differential patterns and hidden extensions.

拓扑模形式的Anderson对偶性及其微分几何体现

Anderson duality of topological modular forms and its differential-geometric manifestations

本文构造了实现拓扑模形式（TMF）Anderson对偶性的谱态射及其微分版本，利用RO(G)分次TMF理论识别了负度同伦群元素，并探讨了与顶点算子代数相关的猜想。

The equivalence between Feynman transform and Verdier duality

The equivalence between dg duality and Verdier duality has been established for cyclic operads earlier. We propose a generalization of this correspondence from cyclic operads and dg duality to twisted modular operads and Feynman transform. Specifically, for each twisted modular operad $\mathcal{P}$ (taking values in dg-vector spaces over a field $k$ of characteristic 0), there is a certain sheaf $\mathcal{F}$ associated with it on the moduli space of stable metric graphs such that the Verdier dual sheaf $D\mathcal{F}$ is associated with the Feynman transform $F\mathcal{P}$ of $\mathcal{P}$. In…

Hodge分解与Poincaré对偶模型

Hodge decompositions and Poincare duality models

本文通过扩展CDGA结构，构造具有Hodge分解的CDGA，并证明其自然诱导出V的微分Poincaré对偶模型，消除了原有假设中的额外条件。