标题： 调和$1$-形式在卡拉比-丘流形的实轨道上 显示英文标题

标题： Harmonic $1$-forms on real loci of Calabi-Yau manifolds

摘要： 我们数值研究是否存在在某些精心构造的卡拉比-丘流形实轨上处处不为零的调和$1$形式，这将产生潜在的新$G_2$流形实例及其度量的显式描述。 我们分两步进行：首先，我们使用神经网络在每个流形上计算一个近似的卡拉比-丘度量。其次，我们使用另一个神经网络根据近似度量计算一个近似调和的$1$形式，然后检查找到的解。 在两个流形上，可以利用微分几何排除存在处处不为零的调和$1$形式的可能性。 第三个流形的实轨微分同胚于$S^1 \times S^2$，我们的数值结果表明，当卡拉比-丘度量接近奇异极限时，它会存在一个处处不为零的调和$1$形式。 我们解释了这样的近似解如何可能用于数值验证证明我们的示例流形必须存在一个处处不为零的调和$1$形式。 显示英文摘要

摘要： We numerically study whether there exist nowhere vanishing harmonic $1$-forms on the real locus of some carefully constructed examples of Calabi-Yau manifolds, which would then give rise to potentially new examples of $G_2$-manifolds and an explicit description of their metrics. We do this in two steps: first, we use a neural network to compute an approximate Calabi-Yau metric on each manifold. Second, we use another neural network to compute an approximately harmonic $1$-form with respect to the approximate metric, and then inspect the found solution. On two manifolds existence of a nowhere vanishing harmonic $1$-form can be ruled out using differential geometry. The real locus of a third manifold is diffeomorphic to $S^1 \times S^2$, and our numerics suggest that when the Calabi-Yau metric is close to a singular limit, then it admits a nowhere vanishing harmonic $1$-form. We explain how such an approximate solution could potentially be used in a numerically verified proof for the fact that our example manifold must admit a nowhere vanishing harmonic $1$-form.