发表机构
Google DeepMind; Institut de Physique Théorique, Université Paris Saclay, CNRS, CEA; Laboratoire d’Annecy-le-Vieux de Physique Théorique (LAPTh), CNRS and Université Savoie Mont-Blanc（谷歌DeepMind; 巴黎萨克雷大学理论物理研究所，法国国家科学研究中心，法国原子能和替代能源委员会; 阿讷西勒沃理论物理实验室，法国国家科学研究中心和萨瓦蒙特布朗克大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究将等质量的三圈和四圈日落费曼积分与局部Calabi-Yau四流形、五流形的零亏格格罗莫夫-威滕理论关联，推导了积分的显式表达式、归一化及相关性质，还证明了多覆盖变换的整性并给出端点值等结果。
AI 中文摘要
我们利用局部Calabi–Yau四流形和五流形$X_L=\operatorname{Tot}(K_{F_L})$的零亏格格罗莫夫-威滕理论，确定了大欧几里得动量下等质量二维三圈和四圈日落费曼积分$I^{(3)}_\circleddash$与$I^{(4)}_\circleddash$，其中$F_L\subset(\mathbb P^1)^{L+1}$、$L=3,4$是多重次数为$(1,\ldots,1)$的光滑超曲面。两圈日落积分由局部Calabi–Yau三流形的预势描述；对于$L\geq3$，无标记稳定映射的模空间具有正虚维且无预势，因此该费曼积分转而由带插入项$\mathcal F_{\gamma_L}$的单点势$\gamma_L=H^{L-1}/(L-1)!$、其引力后代以及来自Gamma类的$\zeta(3)$边界项描述。在两个圈阶下，镜像坐标$R$的弗罗贝尼乌斯展开中$R^{L-1}$的系数为含诺维科夫变量$Q=\exp(R)$的$-\theta_Q\mathcal F_{\gamma_L}$。三圈时，整个费曼积分是仅关于$\mathcal F_{\gamma_3}$的显式表达式，因为零亏格递推可由此确定所有后代级数；四圈时，两条对角级数已足够，且$\zeta(3)$项本身受$\theta_Q\mathcal F_{\gamma_4}$修正。三圈时，我们还独立从皮卡-富克斯方程和梅林变换的极部推导了解析归一化，证明了任意整数插入下多覆盖变换的整性并给出显式量子$K$公式，计算了精确周期联络多项式。最后，我们给出欧几里得轴上的端点值、诺维科夫变量$Q$中泰勒半径的界、四圈端点非解析性的计算机辅助证明，以及两、三、四圈归一化的对比。
英文摘要
We determine the equal-mass two-dimensional three- and four-loop sunset Feynman integrals $I^{(3)}_\circleddash$ and $I^{(4)}_\circleddash$ at large Euclidean momentum in terms of the genus-zero Gromov--Witten theory of the local Calabi--Yau fourfold and fivefold $X_L=\operatorname{Tot}(K_{F_L})$, where $F_L\subset(\mathbb P^1)^{L+1}$, $L=3,4$, is a smooth hypersurface of multidegree $(1,\ldots,1)$. At two loops the sunset integral is governed by the prepotential of a local Calabi--Yau threefold. For $L\geq3$ the moduli spaces of unmarked stable maps have positive virtual dimension and there is no prepotential, therefore the Feynman integral is governed instead by the one-point potential $\mathcal F_{γ_L}$ with insertion $γ_L=H^{L-1}/(L-1)!$, by its gravitational descendants, and by $ζ(3)$ boundary terms coming from the Gamma class. At both loop orders the coefficient of $R^{L-1}$ in the Frobenius expansion in the mirror coordinate $R$ is $-θ_Q\mathcal F_{γ_L}$ with the Novikov variable $Q=\exp(R)$. At three loops the whole Feynman integral is an explicit expression in $\mathcal F_{γ_3}$ alone, because genus-zero recursion determines all descendant series from it; at four loops two diagonal series suffice, and the $ζ(3)$ term is itself corrected by $θ_Q\mathcal F_{γ_4}$. At three loops we also derive the analytic normalization independently, from the Picard--Fuchs equation and the polar part of a Mellin transform, prove integrality of the multiple-cover transforms for arbitrary integral insertions together with an explicit quantum-$K$ formula, and compute the exact period connection polynomials. Finally we give endpoint values along the Euclidean axis, bounds on the Taylor radii in the Novikov variable $Q$, a computer-assisted proof of the four-loop endpoint nonanalyticity, and a comparison of the normalizations at two, three and four loops.
Comments70 pages. Implementation codes available at https://github.com/pierrevanhove/Sunset-Gromov-Witten
