Niclai maps in super gauge theory and gravity

1980年，赫尔曼·尼古拉发现，整体超对称理论可以在没有费米子自由度的情况下刻画：在任何耦合g上积分所有费米子(和势鬼)后得到的非局域纯玻色子理论允许对自由理论(在g=0时)进行非线性场变换，即所谓的Nicolai映射。或者，该映射可以通过将任何场泛函的相互作用关联函数等同于逆变换的场泛函的自由关联函数来定义。20世纪80年代初，在一维、二维和四维时空中，Wess-Zumino模型和超杨-Mills理论的Nicolai映象被提出了一种微扰构造方案，其方法是将引起耦合微小位移的“耦合流算符”指数化。它导致了超对称微扰理论的另一种表述，补充了标准超空间或分量方法。这个主题几乎被遗忘了，直到2020年，赫尔曼·尼古拉和他的合作者重新审视了超级杨-米尔斯理论的地图，现在也是在更高的维度上，发现它只存在于临界维度三、四、六和十，并将其微扰结构推到(目前)四级。从那时起，出现了一系列新的结果：映射的普适公式，它的非唯一性和模糊性，它的线性判据，它的规范相关性，高阶微扰行为和非微扰存在性。此外，我们还看到了在N=4个超杨-米尔斯相关器、矩阵模型和最大超膜中的应用。这些发现为对一侧Nicolai映象的更深层次的结构分析开辟了新的视角，并为超对称性、超规范理论和超引力中的关联函数的实际应用开辟了新的视角。这个项目由四个部分组成。首先，我们计划扩展超对称标量理论的Nicolai-map处理。这里的具体主题是映射的重整化，西格玛模型的扩展，超对称破缺的研究，以及尼古拉映射的非相对论版本。第二部分是关于超杨-米尔斯理论。在这里，我们将把映射推广到调和超空间，推广到超对称保持规范，阐明线性映射在光锥规范中的失效，并对N=4个超级Yang-Mills映射进行优化。第三，我们将尝试建立超引力的Nicolai图，第四，我们打算重温超级矩阵模型和超级薄膜，以便通过小张力极限下的Nicolai图克服其量子相关器的技术障碍。预计这个项目的成功完成将使尼古拉映射成为超对称量子场理论中一个可行的工具和一种有竞争力的计算技术，甚至为量子化超重力或超薄膜等以前难以处理的情况提供一种新的方法。

In 1980 Hermann Nicolai discovered that globally supersymmetric theories can be characterized without fermionic degrees of freedom: The nonlocal purely bosonic theory obtained after integrating out all fermions (and potential ghosts) at any coupling g admits a nonlinear field transformation to the free theory (at g=0), the so-called Nicolai map. Alternatively, this map can be defined by equating the interacting correlation function of any field functional with the free correlation function of the inversely transformed field functional. In the early 1980s a perturbative construction scheme for the Nicolai map was developed in one, two and four spacetime dimensions, for Wess-Zumino models as well as for super Yang-Mills theory, by exponentiating a "coupling-flow operator" which induces an infinitesimal shift in the coupling. It led to an alternative formulation of supersymmetric perturbation theory, complementary to the standard super-space or component approach. The subject was all but forgotten until 2020, when Hermann Nicolai and collaborators revisited the map for super Yang-Mills theory, now also in higher dimensions, found that it exists only in the critical dimensions three, four, six and ten, and pushed its perturbative construction to (currently) fourth order. Since then, there has been a flurry of new results: a universal formula for the map, its non-uniqueness and ambiguities, criteria for its linearity, its gauge dependence, the large-order perturbation behavior and non-perturbative existence. In addition, we have seen applications to N=4 super Yang-Mills correlators, matrix models and the maximal supermembrane. These findings open a new perspective for a deeper structural analysis of the Nicolai map on one side and for practical applications to correlation functions in supersymmetry, supergauge theory and perhaps supergravity. This project has four parts. First, we plan to expand the Nicolai-map treatment of supersymmetric scalar theories. Specific topics here are the renormalization of the map, an extension to sigma models, the investigation of supersymmetry breaking, and nonrelativistic versions of the Nicolai map. The second part concerns super Yang-Mills theory. Here, we shall generalize the map to harmonic super-space, to supersymmetry-preserving gauge fixings, clarify the failure of linear maps in the light-cone gauge, and optimize the map for N=4 super Yang-Mills. Third, we will try to formulate a Nicolai map for supergravity, and fourth, we intend to revisit the super matrix model and the super-membrane, in order to overcome the technical obstacles with its quantum correlators via the Nicolai map in the small-tension limit. It is expected that successful completion of this project will establish the Nicolai map as a viable tool and a competitive calculational technique in supersymmetric quantum field theories and even offer a new way to quantize previously intractable cases like supergravity or the super-membrane.