We propose several different types of construction principles for new classes of Toda field theories based on root systems defined on Lorentzian lattices. In analogy to conformal and affine Toda theories based on root systems of semi-simple Lie algebras, also their Lorentzian extensions come about in conformal and massive variants. We carry out the Painlevé integrability test for the proposed theories, finding in general only one integer valued resonance corresponding to the energy-momentum tensor. Thus most of the Lorentzian Toda field theories are not integrable, as the remaining resonances, that grade the spins of the W-algebras in the semi-simple cases, are either non-integer or complex valued. We analyze in detail the classical mass spectra of several massive variants. Lorentzian Toda field theories may be viewed as perturbed versions of integrable theories equipped with an algebraic framework.

中文翻译：

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洛伦兹户田场论

我们为基于洛伦兹格上定义的根系统的新型户田场论提出了几种不同类型的构造原则。类似于基于半简单李代数根系统的共形和仿射 Toda 理论，它们的洛伦兹扩展也出现在共形和大量变体中。我们对所提出的理论进行了 Painlevé 可积性检验，通常只发现一个与能量-动量张量相对应的整数值共振。因此，大多数 Lorentzian Toda 场论是不可积的，因为在半简单情况下对 W 代数的自旋进行分级的剩余共振要么是非整数值，要么是复值。我们详细分析了几种大规模变体的经典质谱。