图积分、费曼周期与单值多重zeta值
Graph integrals, Feynman periods, and single-valued multiple zeta values
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中文总结 AI 辅助
研究图积分是什么数的问题，通过证明原始规范积分与变形量子化中的复位置空间积分一致，得出图的规范积分值为单值多重zeta值，且每个单值多重zeta值是特定图费曼周期的有理线性组合，还表明在交换图复形中相关上闭链一致。
中文摘要 AI 辅助
生成一般线性群稳定上同调的博雷尔类可用不变微分形式表示。沿热带托雷利映射拉回这些形式会产生与图相关的规范收敛积分，它们与\(\mathrm{GL}_n\)和图复形的上同调紧密相连。一个自然问题是这些图积分是什么数。我们表明原始规范积分与变形量子化中出现的一族复位置空间积分一致，从而回答了这个问题。结果是图的规范积分值为单值多重zeta值。我们还推断每个单值多重zeta值都是具有无质量传播子的图的费曼周期的有理线性组合。最后，在交换图复形中，我们的结果意味着两个相关上闭链一致。
英文摘要
The Borel classes generating the stable cohomology of the general linear group can be represented by invariant differential forms. It is known that pulling these forms back along a tropical Torelli map yields canonical convergent integrals associated to graphs, which are closely connected to the cohomology of $\mathrm{GL}_n$ and of graph complexes. A natural question is what numbers these graph integrals are. We answer this for primitive canonical integrals by showing that they coincide with a family of complex position-space integrals arising in deformation quantisation. As a consequence, canonical integrals of graphs evaluate to single-valued multiple zeta values. We further deduce that every single-valued multiple zeta value occurs as a rational linear combination of Feynman periods of graphs with massless propagators. Finally, in the commutative graph complex, our result implies that the two associated cocycles agree.
