[Paper Review] Functional integration and abelian link invariants

E. Guadagnini|arXiv (Cornell University)|Jan 26, 2010

Black Holes and Theoretical Physics7 references3 citations

TL;DR

This paper addresses the ill-defined nature of the Chern-Simons path integral in abelian U(1) gauge theory on 3-manifolds by employing Deligne-Beilinson cohomology to regularize the functional integral. It establishes that expectation values of Wilson loops yield well-defined link invariants, and shows that these match the Reshetikhin-Turaev surgery invariants, providing a nonperturbative functional integral derivation of 3-manifold invariants for $S^1 \times S^2$ and other homology spheres.

ABSTRACT

The functional integral computation of the various topological invariants, which are associated with the Chern-Simons field theory, is considered. The standard perturbative setting in quantum field theory is rewieved and new developments in the path-integral approach, based on the Deligne-Beilinson cohomology, are described in the case of the abelian U(1) Chern-Simons field theory formulated in S^1 x S^2.

Motivation & Objective

• To resolve the ill-defined nature of the Chern-Simons path integral in abelian U(1) gauge theory on 3-manifolds.
• To provide a nonperturbative, functional integral-based derivation of topological invariants for 3-manifolds such as $S^1 \times S^2$ and homology spheres.
• To demonstrate that Wilson loop expectation values computed via regularized path integrals match known combinatorial invariants like the Reshetikhin-Turaev surgery invariant.
• To clarify the role of regularization and normalization in making the partition function well defined in topological quantum field theory.

Proposed method

• Utilizes Deligne-Beilinson cohomology to define a consistent regularization of the functional integral measure in abelian Chern-Simons theory.
• Applies a finite-dimensional regularization scheme by truncating the field expansion in an orthonormal basis, replacing infinite-dimensional integrals with limits over finite-dimensional approximations.
• Computes Wilson loop expectation values as ratios of regularized path integrals, ensuring gauge invariance and topological invariance.
• Establishes equivalence between the path-integral result and the Reshetikhin-Turaev surgery invariant via the formula $I_k(M) = (2k)^{-N_{\mathcal{L}}/2} e^{i\pi \sigma(\mathcal{L})/4} \langle W(\mathcal{L}) \rangle|_{S^3}$.
• Uses Kirby calculus and surgery presentations to show that the invariants are independent of the choice of link presentation, relying on invariance under Kirby moves.
• Analyzes the dependence of the partition function on the homology and homotopy type of the 3-manifold, showing that invariants distinguish non-homeomorphic manifolds with identical homology.

Experimental results

Research questions

• RQ1 How can the ill-defined Chern-Simons path integral be regularized to yield meaningful topological invariants in abelian U(1) gauge theory?
• RQ2 What is the precise relationship between the functional integral computation of Wilson loop expectation values and the Reshetikhin-Turaev surgery invariants?
• RQ3 Why does the path integral fail for certain manifolds like $RP^3$ at integer coupling constants, and how can this be resolved?
• RQ4 To what extent do the invariants depend on the homology or homotopy type of the 3-manifold, and can they distinguish non-homeomorphic manifolds with isomorphic homology?

Key findings

• The path-integral computation of Wilson loop expectation values in abelian Chern-Simons theory on $S^1 \times S^2$ yields well-defined, topologically invariant results when regularized via Deligne-Beilinson cohomology.
• The expectation value $\langle W(L)\rangle|_M$ for a framed, colored link $L$ in a 3-manifold $M$ matches the Reshetikhin-Turaev surgery invariant $I_k(M)$, confirming consistency between functional and combinatorial approaches.
• For lens spaces $L(5,1)$ and $L(5,2)$, the invariant $I_2(M)$ takes values $-1$ and $1$ respectively, showing that the invariant distinguishes non-homeomorphic manifolds with isomorphic $H_1(M) = \mathbb{Z}_5$.
• For $L(9,1)$ and $L(9,2)$, $I_3(M)$ yields $i\sqrt{3}$ and $-i\sqrt{3}$, respectively, further demonstrating that the invariant is sensitive to finer topological structure beyond homology.
• The partition function $R_{N_0}(M)$ is well defined only when the normalization $N_0$ is chosen appropriately, and such a choice corresponds to a consistent path-integral regularization.
• The set of invariants for homology spheres $M_0$ coincides with those for $S^3$, implying that $\langle W(L)\rangle|_{M_0} = \langle W(L)\rangle|_{S^3}$ for all links $L$.