Title: Local mirror symmetry and the sunset Feynman integral Show Chinese title

Title: 局部镜像对称性和太阳环费曼积分

Abstract: We study the sunset Feynman integral defined as the scalar two-point self-energy at two-loop order in a two dimensional space-time. We firstly compute the Feynman integral, for arbitrary internal masses, in terms of the regulator of a class in the motivic cohomology of a 1-parameter family of open elliptic curves. Using an Hodge theoretic (B-model) approach, we show that the integral is given by a sum of elliptic dilogarithms evaluated at the divisors determined by the punctures. Secondly we associate to the sunset elliptic curve a local non-compact Calabi-Yau 3-fold, obtained as a limit of elliptically fibered compact Calabi-Yau 3-folds. By considering the limiting mixed Hodge structure of the Batyrev dual A-model, we arrive at an expression for the sunset Feynman integral in terms of the local Gromov-Witten prepotential of the del Pezzo surface of degree 6. This expression is obtained by proving a strong form of local mirror symmetry which identifies this prepotential with the second regulator period of the motivic cohomology class. Show Chinese abstract

Abstract: 我们研究了二维时空中两环阶的标量两点自能定义的日落费曼积分。 我们首先在任意内部质量的情况下，通过一个1参数开椭圆曲线族的动机上同调类的正则化来计算费曼积分。 使用Hodge理论（B模型）方法，我们证明该积分由在由穿孔决定的除子上求值的椭圆双对数之和给出。 其次，我们将日落椭圆曲线与一个局部非紧致的卡拉比-丘3流形相关联，该流形是椭圆纤维化紧致卡拉比-丘3流形的极限。 通过考虑Batyrev对偶A模型的极限混合Hodge结构，我们得到了日落费曼积分在6度的del Pezzo曲面的局部Gromov-Witten预势中的表达式。 该表达式是通过证明局部镜像对称性的强形式得出的，该形式将该预势与动机上同调类的第二个正则化周期相联系。