This paper considers the higher derivative terms in the effective action of type II string theory and in particular the behaviour of the automorphic forms they contain in all the different possible limits of the string parameters. The automorphic forms are thought to obey Poisson equations which contain the Laplacian defined on the coset space to which the scalars fields belong and we compute this Laplacian in all the possible string theory limits. We also consider these Poisson equations in the decompactification limit of a single dimension and by making two assumptions, one on the generic form of this equation and the other on the behaviour of the automorphic forms in this limit, we find strong constraints on the allowed form of this differential equation. We show that these constraints allow one to recover much of what was previously known about the automorphic forms corresponding to terms in the effective action that have fourteen or fewer space-time derivatives in a simple way.
Automorphic properties of low energy string amplitudes in various dimensions
本文研究了d-环面上紧化的最大超对称弦论的四引力子振幅的低能展开中出现的高阶导数相互作用的模依赖系数。这些自守函数是通过施加各种相容性条件来确定的，对于D ^6 R ^4阶以下的项和不同的d值。它们满足拉普拉斯本征值方程，有或没有源项，其解决方案给出的爱森斯坦级数，或更一般的自守函数，某些抛物子群的相关U-对偶群。尽管弦论的表达式是有限的，但相应的超引力场论极限的紫外发散被编码在不同的符号中。这个分析包括有趣的SL（d）和SO（d，d）爱森斯坦级数的超环面紧化的单圈和双圈弦和超重力振幅的表示。
Unlike the $\mathcal{R}^4$ and $\nabla^4\mathcal{R}^4$ couplings, whose coefficients are Langlands-Eisenstein series of the U-duality group, the coefficient $\mathcal{E}_{(0,1)}^{(d)}$ of the $\nabla^6\mathcal{R}^4$ interaction in the low-energy effective action of type II strings compactified on a torus $T^d$ belongs to a more general class of automorphic functions, which satisfy Poisson rather than Laplace-type equations. In earlier work, it was proposed that the exact coefficient is given by a two-loop integral in exceptional field theory, with the full spectrum of mutually 1/2-BPS states r…
Generalized dualities had an intriguing incursion into Double Field Theory (DFT) in terms of local $O(d,d)$ transformations. We review this idea and use the higher derivative formulation of DFT to compute the first order corrections to generalized dualities. Our main result is a unified expression that can be easily specified to any generalized T-duality (Abelian, non-Abelian, Poisson-Lie, etc.) or deformations such as Yang-Baxter, in any of the theories captured by the bi-parametric deformation (bosonic, heterotic strings and HSZ theory), in any supergravity scheme related by field redefiniti…
We introduce the Poisson bracket operator which is an alternative quantum counterpart of the Poisson bracket. This operator is defined using the operator derivative formulated in quantum analysis and is equivalent to the Poisson bracket in the classical limit. Using this, we derive the quantum canonical equation which describes the time evolution of operators. In the standard applications of quantum mechanics, the quantum canonical equation is equivalent to the Heisenberg equation. At the same time, this equation is applicable to c-number canonical variables and then coincides with the canonic…
