Local mirror symmetry and the sunset Feynman integral
Local mirror symmetry and the sunset Feynman integral
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DOI：
10.4310/atmp.2017.v21.n6.a1
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发表时间：
2016-01
期刊：
影响因子：
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通讯作者：
S. Bloch;M. Kerr;P. Vanhove
中科院分区：
文献类型：
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作者：
S. Bloch;M. Kerr;P. Vanhove
研究了在二维时空中定义为双环阶标量两点自能的日落费曼积分。首先，我们计算了任意内质量的Feynman积分，该积分是基于1参数开椭圆曲线族的动力上同调中一类的调节器。利用Hodge理论（b -模型）方法，我们证明了积分是由在由点确定的除数处求值的椭圆二对数和给出的。其次，我们将一个局部非紧化的Calabi-Yau 3褶作为椭圆纤维紧化Calabi-Yau 3褶的极限与日落椭圆曲线关联。通过考虑Batyrev对偶a模型的极限混合Hodge结构，我们得到了6次del Pezzo曲面局部Gromov-Witten预势的日落Feynman积分表达式。这个表达式是通过证明一个局部镜像对称的强形式得到的，该局部镜像对称与动机上同调类的第二个调节周期相一致。
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.