Abelian Chern–Simons theory and linking numbers via oscillatory integrals

Abstract

A rigorous mathematical model of Abelian Chern–Simons theory based on the theory of infinite-dimensional oscillatory integrals developed by Albeverio and Ho/egh-Krohn is introduced. A gauge-fixed Chern–Simons path integral is constructed as a Fresnel integral in a certain Hilbert space. Wilson loop variables are defined as Fresnel integrable functions and it is shown in this context that the expectation value of products of Wilson loops with respect to the Chern–Simons path integral is a topological invariant which can be computed in terms of pairwise linking numbers of the loops, as conjectured by Witten. Furthermore, a lattice Chern–Simons action is proposed which converges to the continuum limit.

Geometric discretization scheme applied to the Abelian Chern-Simons theory

“…This can be shown to coincide with the discrete action for the Abelian ChernSimons theory introduced in [6]. This prescription fails however, in the sense that the resulting partition function Z K (λ) is not a topological invariant i.e.…”

“…With this doubling the topological features of Abelian Chern-Simons theory are completely captured by the formulation. An alternate scheme of discretisation proposed previously in [6] which does not involve the doubling of fields fails to capture the topological features of the Abelian Chern-Simons theory as we will show numerically in Section 5. A mathematical treatment of this discretisation scheme has been given in [8,11].…”

“…a decomposition into hyper-tetrahedra rather then hyper-cubes) and a mathematical tool called the Whitney map [4]. These have previously been used to discretise field theories in [5,6] where various convergence results (e.g. convergence of the discrete action to the continuum action) were established.…”

Chern–Simons theory on R3 in axial gauge: a rigorous approach

“…After Witten [20] succeeded in computing the partition function and the Wilson loop observables (WLOs) for various base manifolds M and structure groups G the Chern-Simons gauge theory was studied intensively by many different authors, see, e.g., [1][2][3][4][5][7][8][9]12,14].…”

Theory and Applications of Infinite Dimensional Oscillatory Integrals

“…The application of the infinite dimensional oscillatory integrals to the mathematical definition of the Chern-Simons functional integral described in section 1.3 has been realized in [28] in the case where the gauge group G is abelian. It has been proven in particular that if H 1 (M ) = 0 then I Φ (f ) gives the linking numbers.…”