the chernâsimons theory, named after shiing-shen chern and james harris simons, is a 3-dimensional topological quantum field theory of schwarz type, developed by edward witten. it is so named because its action is proportional to the integral of the chernâsimons 3-form. in condensed matter physics, chernâsimons theory describes the topological order in fractional quantum hall effect states. in mathematics, it has been used to calculate knot invariants and three-manifold invariants such as the jones polynomial. particularly, chernâsimons theory is specified by a choice of simple lie group g known as the gauge group of the theory and also a number referred to as the level of the theory, which is a constant that multiplies the action. the action is gauge dependent, however the partition function of the quantum theory is well-defined when the level is an integer and the gauge field strength vanishes on all boundaries of the 3-dimensional spacetime. ==the classical theory== ===mathematical origin=== in the 1940s s. s. chern and a. weil studied the global curvature properties of smooth manifolds m as de rham cohomology (chernâweil theory), which is an important step in the theory of characteristic classes in differential geometry. given a flat g-principal bundle p on m there exists a unique homomorphism, called chernâweil homomorphism, from the algebra of g-adjoint invariant polynomial on g (lie algebra of g) to the cohomology h^*(m,\mathbb{r}). if the invariant polynomial is homogeneous one can write down concretely any k-form of the closed connection Ï as some 2k-form of the associated curvature form Î© of Ï. in 1974 s. s. chern and j. h. simons had concretely constructed a (2k&nbsp;&minus;&nbsp;1)-form df(Ï) such that :dtf(\omega)=f(\omega^k), where t is the chernâweil homomorphism. this form is called chernâsimons form. if df(Ï) is closed one can integrate the above formula :tf(\omega)=\int_c f(\omega^k), where c is a (2k&nbsp;&minus;&nbsp;1)-dimensional cycle on m. this invariant is called chernâsimons invariant. as pointed out in the introduction of the chernâsimons paper, the chernâsimons invariant cs(m) is the boundary term that cannot be determined by any pure combinatorial formulation. it also can be defined as :cs(m)=\int_{s(m)}\tfrac{1}{2}tp_1\in\mathbb{r}/\mathbb{z}, where p_1 is the first pontryagin number and s(m) is the section of the normal orthogonal bundle p.ãmoreover, the chernâsimons term is described as the eta invariant defined by atiyah, patodi a
Quantum_field_theory