N=4拓扑振幅和Lorentzian Borcherds代数中的增强规范群
Enhanced Gauge Groups in N=4 Topological Amplitudes and Lorentzian Borcherds Algebras
摘要：
我们继续研究在T^2上紧化的杂化弦论中N=4拓扑振幅的代数性质，这一研究始于arXiv：1102.1821。在这项工作中，我们评估一个特定的单圈振幅为任何增强规范群h \subset e_8 + e_8，即任意选择的威尔逊线模量。我们证明了结果的某个解析部分具有无穷乘积表示，其中乘积取在Lorentz卡茨-穆迪代数g^{++}的正根上.后者是通过补数g=（e_8 + e_8）/h的双重扩张得到的。无穷积关于全T-对偶群SO（2，18;Z）的有限指数子群是自同构的，并且通过Borcherds-Gritsenko-Nikulin的哲学，这定义了广义卡茨-穆迪代数G（g^{++}）的分母公式，它是g^{++}的“自同构修正”。我们明确地给出了一些例子G（g^{++}）的根重数。
英文摘要：
We continue our study of algebraic properties of N=4 topological amplitudes in heterotic string theory compactified on T^2, initiated in arXiv:1102.1821. In this work we evaluate a particular one-loop amplitude for any enhanced gauge group h \subset e_8 + e_8, i.e. for arbitrary choice of Wilson line moduli. We show that a certain analytic part of the result has an infinite product representation, where the product is taken over the positive roots of a Lorentzian Kac-Moody algebra g^{++}. The latter is obtained through double extension of the complement g= (e_8 + e_8)/h. The infinite product is automorphic with respect to a finite index subgroup of the full T-duality group SO(2,18;Z) and, through the philosophy of Borcherds-Gritsenko-Nikulin, this defines the denominator formula of a generalized Kac-Moody algebra G(g^{++}), which is an 'automorphic correction' of g^{++}. We explicitly give the root multiplicities of G(g^{++}) for a number of examples.
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