Reflection Positivity and Chern-Simons Functional Integrals

TL;DR

This paper constructs a nonperturbative, finite mathematical version of the Chern-Simons functional integral for 3-manifolds with reflection symmetry, leading to a Hilbert space and partition function without renormalization.

Contribution

It introduces a reflection positive functional approach to rigorously define the Chern-Simons integral, yielding a Hilbert space and operators, and constructs a nonperturbative, finite quantum field theory.

Findings

Constructs a Hilbert space from reflection positivity.

Defines a self-adjoint operator from the cubic interaction term.

Obtains a nonperturbative, finite partition function.

Abstract

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.