Using Arakelov geometry, we compute the partition function of the noncompact free boson at genus two. We begin by compiling a list of modular invariants which appear in the Arakelov theory of Riemann surfaces. Using these quantities, we express the genus two partition function as a product of modular forms, as in the well-known genus one case. We check that our result has the expected obstruction to holomorphic factorization and behavior under degeneration.

中文翻译：

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Arakelov几何中的两个自由玻色子

使用Arakelov几何，我们可以计算第二类非紧实自由玻色子的分区函数。我们首先编译出现在Riemann曲面的Arakelov理论中的模不变量列表。使用这些量，我们将两个分区函数表示为模块化形式的乘积，就像在众所周知的一个案例中一样。我们检查我们的结果对变性下的全同因式分解和行为具有预期的阻碍。