标题： 反射正性和Chern-Simons泛函积分 显示英文标题

标题： Reflection Positivity and Chern-Simons Functional Integrals

摘要： 我们证明，对于配备反射的流形，可以构造出Witten关于形式Chern-Simons泛函积分的数学版本，该构造基于一个与Chern-Simons拉格朗日量二次项相关的反射正泛函，在连接在底层3-流形上的巴拿赫空间${\bf A}$上的函数代数上。 该构造产生了一个与被反射保持的曲面相关的希尔伯特空间。 Chern-Simons拉格朗日量中的三次玻色相互作用项的一个版本在该希尔伯特空间上给出了一个自伴算子，并通过指数化，得到一个酉的一参数子群。 该一参数子群的真空期望值与一个与鬼场及其相互作用相关的附加项结合，适当的弱极限给出了量子场论的划分函数。 该构造是非微扰的。 该理论是有限的，不需要重整化，这从微扰理论中可以预期。 自然地会问，所得的划分函数是否与Witten和Reshetikhin-Turaev的流形不变量有关，或者是否需要更复杂的构造。 显示英文摘要

摘要： We show that a mathematical version of the formal Chern-Simons functional integral of Witten for manifolds equipped with a reflection may be constructed in terms of a reflection positive functional, associated to the quadratic term in the Chern-Simons Lagrangian, on an algebra of functions on a Banach space ${\bf A}$ of connections on the underlying 3-manifold. This construction yields a Hilbert space associated to a surface preserved by the reflection. A version of the cubic Bosonic interaction term in the Chern-Simons Lagrangian gives a self-adjoint operator on this Hilbert space, and by exponentiation, a unitary one parameter subgroup of operators. The vacuum expectation value of this one parameter subgroup is combined with an additional term associated to the ghost fields and their interaction, and an appropriate weak limit gives a partition function for the quantum field theory. This construction is nonperturbative. The theory is finite and does not require renormalization, as may be expected from perturbation theory. It is natural to ask whether the resulting partition function is related to the manifold invariants of Witten and Reshetikhin-Turaev, or whether a more elaborate construction may be needed.