Non-perturbative Phenomena and Background (In)dependence in Field and String Theory

场论描述了广泛的物理现象。许多场论表现出交换强耦合和弱耦合以及交换基本自由度和非微扰自由度的对偶性。理解对偶性使场论和弦理论都有了一个巨大的飞跃，它们进一步把引力耦合纳入其中。在弦理论中，同一理论的对偶公式可以被赋予几何意义。此外，还可以仔细研究理论对耦合和参数的依赖性，即它的背景依赖性。获得的见解允许物理量的精确计算。此外，解决背景依赖性问题带来了有趣的新见解，允许连接和重新思考一些最基本的物理概念。该研究计划的目标是扩展和深化对弦理论背景依赖性的理解，以及赋予物理量背景独立意义的可能性。这里的重点是与弦理论的微扰环展开有关的现象，而弦理论的微扰环展开是由弦耦合常数决定的。与这种耦合相关的非微扰效应是D-膜，它是弦理论的边界条件。而且，这种扩展对应于把场论与引力耦合起来。四维物理理论的这项研究的前景是获得一个精确的处理的有效耦合的物理理论工程的D膜设置，并获得洞察到一个有效的量子引力理论。此外，对弦理论的非微扰完备性和S-对偶性的见解也是值得期待的。在数学方面，这些概念是镜像对称领域的研究主题，在镜像对称中，物理对偶被转化为数学中不同领域的令人惊讶的关系，并允许计算数学不变量。

Field theories describe a broad range of physical phenomena. Many field theories exhibit dualities which exchange strong and weak coupling together with an exchange of fundamental and non-perturbative degrees of freedom. Understanding dualities has led to a great leap forward for both field and string theories which furthermore incorporate the coupling to gravity. Within string theory, a geometrical meaning can be given to dual formulations of the same theory. Furthermore it is possible to scrutinize the dependence of the theory on its couplings and parameters, i.e. its background dependence. The gained insights allow exact computations of physical quantities. Moreover addressing the question of background dependence has led to intriguing new insights allowing to connect and rethink some of the most fundamental physical notions. The goal of this research program is to extend and deepen the understanding of background dependence of string theories and the possibility to give a background independent meaning to physical quantities. The focus hereby is on phenomena which are related to the perturbative loop expansion of string theory which is governed by the string coupling constant. Non-perturbative effects with respect to this coupling are the D-branes, which are the boundary conditions of string theory. This expansion moreover corresponds to coupling field theories to gravity. Prospects of this research for four-dimensional physical theories are to obtain a precise handle on the effective couplings of physical theories engineered by D-brane setups and to obtain insights into an effective theory of quantum gravity. Furthermore, insights into the non-perturbative completion and S-duality of string theory are expected. On the mathematical side these very concepts are subject of research in the field of mirror symmetry where physical dualities are translated into surprising relations of different areas in mathematics and allow to compute mathematical invariants.