Integrable Field Theories in Higher Dimensions and Supersymmetry

在过去的四年中，我们按照最初的计划对几个主题进行了研究：可积模型在各方向上的扩展。场论的新模型，新类别的超标准模型。高维规范理论关于这些学科的几个问题，我们已经取得了一些结果。(a)低维模型可解性的Guttmann-Enting猜想的检验(b)边界为XYZ自旋1/2链的晶格模型的连续统极限和sin - gordon理论(c) d = 2晶格上包含手性费米子和SUSY的可积模型(d) d = 3II中扩展（N = 2,4） SUSY NLSM的紫外散度性质。(a) i) Wess-Zumino-Witten模型的非交换扩展；ii) d = 3时非交换CP(n)模型中孤子的低能动力学；(b)不同维场论之间的对偶性；(c)考虑高维规范理论的3-3-1规范模型(a)作为希格斯场起源的高维规范理论；(b)高维超对称性规范理论：瞬子方程的模空间和超对称环的几何；五维超对称性规范理论：镜像对称和前势场的瞬子膨胀。(a)特殊完整群流形的几何：具有G_2和Spin(7)完整群的流形的度规，以及高维的引力瞬子

During the last four years we have carried out research on a few subjects as we palanned originally :I. Extension of integrable models in various directionsII. New models of field theories, new class of beyond-the-standard modelsIII. Higher-dimensional gauge theoriesIV. M theory and geometryWe have obtained results on a few problems concerning these subjects.I. (a) Test of the Guttmann-Enting conjecture on solvability of low- dimensional models(b) Continuum limit of lattice models with a boundary : XYZ spin 1/2 chain and the sine-Gordon theory(c) Integrable models containing chiral fermions and SUSY on the lattice in d = 2(d) Ultraviolet divergence properties of the extended (N = 2, 4) SUSY NLSM in d = 3II. (a) i) Non-commutative extension of the Wess-Zumino-Witten model, ii) Low -energy dynamics of solitons in the non-commutative CP(n) model in d = 3(b) Duality between field theories in diverse dimensions(c) The 3-3-1 gauge model in view of higher-dimensional gauge theoriesIII. (a) Higher-dimensional gauge theories as the origin of the Higgs field(b) Higher-dimensional SUSY gauge theories : Moduli space of the instanton equation and geometry of supsersymmetric cyclesc Five-dimensional SUSY gauge theories : Mirror symmetry and the instanton expansion of the prepotentialIV. (a) Geometry of manifolds with exceptional holonomy group : the metric of the manifolds with G_2 and Spin(7) holonomy groups, and gravitational instantons in higher dimensions