Extended supersymmetry in gauge theory, gravity and integrable models

在本课题中，我们计划考虑以下主要任务：(1)研究具有手性辅助超场的超空间形式体系中的N=1超对称自对偶电动力学，包括协集标量超场，目的是将U(N)对偶推广到Sp（2N）。(2)在具有手性辅助超场的方法中，重新讨论了具有半破N=4超对称的N=2 Born-Infeld作用的构造问题。(3)构建不同维度规范理论的调和超空间表述，包括d=3超共形Chern-Simons规范理论和不同版本的N=4 SYM理论，研究调和超空间中四维N=2超引力的量子性质。(4)建立了新的方法来分析不同容许规下的Yang-Mills理论的量子性质，研究了d=3， N=2超对称非阿贝尔规理论中低能有效作用的双环结构。(5)找到了AdS_5中与物质场耦合的einstein - chen - simons方程的新经典解，并在边界上识别了相应的全息对偶算子。(6)寻找新的可积和不可积的宇宙学和黑洞模型，以适应测量的扩展超引力。(7)研究扩展超对称量子力学，包括超共形量子力学，揭示其与高维超对称场论、AdS/CFT对应和黑洞物理的关系。(8)在超场方法中构造具有变形N=4和N=8超对称性的新经典和量子d=1和d=2模型。(9)在多项式方法中构造具有更高运动积分（能量以外）的d=2可积模型，并利用对称变量研究主方程的新解。（10）在退化轨道上寻找全对称sl(n)-Toda流的不变子变体，构造相应的相变图，并将其推广到基于其他李代数和余集空间的Toda系统。（11）研究了连接R^4上g值Yang-Mills场的调和球猜想和g上黎曼球在环空间中的调和映射。（12）将非交换几何应用于磁Bloch理论和量子霍尔效应，并将狄拉克量子化扩展到非光滑观测。这些任务大多数是早先进行的调查的合乎逻辑的继续。

In the course of this project, we plan to consider the following main tasks:(1) Studying N=1 supersymmetric self-dual electrodynamics in the superspace formalism with chiral auxiliary superfields, including coset scalar superfields into this scheme with the purpose to extend U(N) duality to Sp(2N).(2) Revisiting the problem of constructing the N=2 Born-Infeld action with half-broken N=4 supersymmetry in the approach with chiral auxiliary superfields.(3) Constructing new harmonic superspace formulations of gauge theories in diverse dimensions, including d=3 superconformal Chern-Simons gauge theories and different versions of N=4 SYM theory, and investigating the quantum properties of four-dimensional N=2 supergravity in harmonic superspace.(4) Developing new methods for analyzing the quantum properties of Yang-Mills theories formulated in different admissible gauges and studying the two-loop structure of the low-energy effective action in d=3, N=2 supersymmetric non-Abelian gauge theory.(5) Finding new classical solutions to the Einstein-Chern-Simons equations coupled with matter fields in AdS_5 and identifying the corresponding holographically dual operators on the boundary.(6) Searching for new integrable and non-integrable cosmological and black-hole models which can fit into gauged extended supergavity. (7) Studying quantum mechanics with extended supersymmetry, including superconformal ones, revealing their relation to supersymmetric field theories in higher dimensions, AdS/CFT correspondence and black-hole physics.(8) Constructing new classical and quantum d=1 and d=2 models with deformed N=4 and N=8 supersymmetries in the superfield approach.(9) Constructing d=2 integrable models with a higher integral of motion (besides energy) within the polynomial approach and studying new solutions of the master equations by using symmetric variables. (10) Searching for invariant subvarieties of the full symmetric sl(n)-Toda flow on degenerate orbits, constructing the corresponding phase transition diagrams and generalizing it to Toda systems based on other Lie algebras and coset spaces.(11) Studying the harmonic spheres conjecture linking G-valued Yang-Mills fields on R^4 and harmonic maps of the Riemann sphere into the loop space over G.(12) Applying of non-commutative geometry to the magnetic Bloch theory and the quantum Hall effect as well as extending Dirac quantization to non-smooth observables.Most of these tasks are logical continuations of investigations that were carried out earlier.