A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory and the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for work related to topological field theory.

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  • A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory and the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for work related to topological field theory. In condensed matter physics, topological quantum field theories are the low energy effective theories of topologically ordered states, such as fractional quantum Hall states, string-net condensed states, and other strongly correlated quantum liquid states.
  • 拓扑量子场论（Topological Quantum Field Theory; TQFT）是一类量子场论，其共同特征是某些关联函数不依赖于背景时空流形的度量. 20世纪70年代 阿. 施瓦茨就研究过一种拓扑量子场论（阿贝尔的陈-塞蒙斯场论）。80年代末, 在阿蒂亚启发下, 威顿研究了三个拓扑量子场论：一个由超对称杨-米尔斯场论扭变得到，用以将唐纳尔森不变量和弗勒尔瞬子同调解释为量子物理对象；第二个是非阿贝尔的陈-塞蒙斯场论，用以将琼斯多项式及其衍生物解释为量子物理对象；第三个由超对称西格玛模型扭变得到，用以将格罗莫夫的赝全纯曲线和弗勒尔的拉格朗日同调解释为量子物理对象. 1994年威顿应用弦论学家得到的强弱对偶结果将唐纳尔森不变量等价为更易计算的塞伯格-威顿不变量. 进入21世纪, 威顿等人又研究了具有更多超对称的杨-米尔斯场论的扭变, 并将数学中的几何郎兰兹对偶解释为量子场论中的强弱对偶. 威顿等人进一步发现, 西格玛模型, 陈-塞蒙斯场论, 以及超对称杨-米尔斯场论之间有千丝万缕的联系, 它们都可以包含在弦论或者 M-理论中, 在这个大框架之下, 琼斯多项式的范畴化----霍万诺夫同调被解释为量子物理对象。
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  • 拓扑量子场论（Topological Quantum Field Theory; TQFT）是一类量子场论，其共同特征是某些关联函数不依赖于背景时空流形的度量. 20世纪70年代 阿. 施瓦茨就研究过一种拓扑量子场论（阿贝尔的陈-塞蒙斯场论）。80年代末, 在阿蒂亚启发下, 威顿研究了三个拓扑量子场论：一个由超对称杨-米尔斯场论扭变得到，用以将唐纳尔森不变量和弗勒尔瞬子同调解释为量子物理对象；第二个是非阿贝尔的陈-塞蒙斯场论，用以将琼斯多项式及其衍生物解释为量子物理对象；第三个由超对称西格玛模型扭变得到，用以将格罗莫夫的赝全纯曲线和弗勒尔的拉格朗日同调解释为量子物理对象. 1994年威顿应用弦论学家得到的强弱对偶结果将唐纳尔森不变量等价为更易计算的塞伯格-威顿不变量. 进入21世纪, 威顿等人又研究了具有更多超对称的杨-米尔斯场论的扭变, 并将数学中的几何郎兰兹对偶解释为量子场论中的强弱对偶.
  • A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory and the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for work related to topological field theory.
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  • Topological quantum field theory
  • 拓撲量子場論
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