Jeffrey-Weitsman-Witten Invariant for 3-Manifolds

• The Jeffrey-Weitsman-Witten invariant is a quantum topological invariant defined via geometric quantization on the moduli space of flat SU(2) connections in 3-manifolds.
• It exploits the stratification of Lagrangian subspaces and incorporates half-density structures and torsion factors to handle both reducible and irreducible connections.
• Constructed using the BKS pairing and analytic integration over moduli space strata, it links classical Chern–Simons theory with modern quantum invariants such as the WRT invariants.

The Jeffrey-Weitsman-Witten invariant is a quantum topological invariant of 3-manifolds rooted in the geometric quantization of the moduli space of flat connections. It provides an analytic formulation for 3-manifold quantum invariants via integration over stratified Lagrangian subspaces associated with Heegaard splittings, blending geometric quantization, symplectic geometry, and topological quantum field theory. This invariant refines Witten’s formal Chern–Simons path integral by incorporating half-density structures, exact sequence–derived torsion factors, and a precise stratification of the underlying moduli space, allowing for a nuanced treatment of reducible and irreducible flat connections.

1. Geometric Quantization of the Moduli Space

The invariant is constructed using geometric quantization on the moduli space of flat connections, primarily for SU(2) bundles over a surface $\Sigma$. The foundational symplectic structure is the Atiyah–Bott symplectic form: $$(\alpha, \beta) \mapsto \int_{\Sigma} \operatorname{tr}(\alpha \wedge \beta)$$ After symplectic reduction at curvature zero, the moduli space becomes $\operatorname{Hom}(\pi_1(\Sigma), SU(2)) / SU(2)$, which carries a natural stratified symplectic structure: the irreducible locus is smooth and symplectic, while loci of connections with nontrivial stabilizer (reducibles) form lower-dimensional singular strata.

A real polarization arises from a trinion (three-holed sphere) decomposition of $\Sigma$ by considering trace functions associated with curves $C_i$, $$f_i(\rho) = \operatorname{tr}(\operatorname{hol}_\rho([C_i]))$$. The fibers of the resulting map are generically Lagrangian subvarieties.

The wavefunctions in geometric quantization are sections of the Chern–Simons line bundle tensored with a half-density bundle on the moduli space, with the prequantum data including the line bundle and a stratified half-density induced from a bi-invariant Riemannian metric on $SU(2)$ (Chitan, 22 Sep 2025).

2. Stratification of the Lagrangian Leaf

A defining feature of the invariant is its use of the stratified structure of the Lagrangian leaf $L_H = \pi^{-1}(2,2,\dots,2)$, corresponding to flat connections with trivial holonomy around set curves. This leaf is canonically isomorphic to $SU(2)^g / SU(2)$ (where $g$ is the genus), which is not a manifold but rather a smooth complex stratified space:

• $L_H^0$: The 0-dimensional stratum with only the trivial connection, stabilizer $=SU(2)$.
• $L_H^1$: Reducible nontrivial connections, modeled locally by the maximal torus $T^g$, dimension $g$.
• $L_H^3$: Dense open smooth stratum of irreducible connections, stabilizer $=Z(SU(2))$, dimension $3g-3$.

The stratification enables the use of volumes and covariant constant half-densities adapted to the specific tangent structures of each stratum, notably reducing the structure group for integration over the reducible locus. An explicit local model for $L_H^1$ is given by $(T^g)^* \times (G/T)\,/\,W$ (Chitan, 22 Sep 2025).

3. Construction of the Invariant via the BKS Pairing

Given a Heegaard splitting $N=H_1 \cup_\Sigma H_2$, the invariant is realized as a Blattner–Kostant–Sternberg (BKS) pairing between half-densities supported on the Lagrangian subspaces $L_{H_1}$ and $L_{H_2}$ associated with the two handlebodies. The intersection $\mathcal{M}(N) = L_{H_1} \cap L_{H_2}$ is itself stratified, and the pairing integrates over each stratum: $$Z(N,k) = \sum_{i=0,1,3} \int_{\mathcal{M}^i(N)} e^{ik\,CS(x)}\,\tau(N,x) \otimes v_x^{-2}$$ where $CS(x)$ is the classical Chern–Simons invariant, $\tau(N,x)$ is a torsion factor constructed from the cohomological exact sequences on the connection, and $v_x^{-2}$ encodes the contraction of half-densities.

On irreducible strata, $\tau(N,x)$ matches the standard Reidemeister torsion; on reducible strata, a careful splitting of tangent bundles (using, for example, $T_x L_H^1 \cong \operatorname{Lie}(T)^g$) ensures the correct normalization and orientation (Chitan, 22 Sep 2025). Canonically constructed (covariantly constant) half-densities play a critical role, particularly inferred from the Levi–Civita connection on $SU(2)$.

4. Analytic and Arithmetic Structure

The invariant extends the heuristic Chern–Simons path integral by recasting the oscillatory integral as a sum over contributions from connected components of the moduli space, paralleling stationary phase approximation: $$Z(N,k) \sim \sum_x k^{d_x} e^{ik\,CS(x)} \sqrt{\tau(N,x)}\,e^{\frac{\pi i}{4}\rho_x}$$ where $d_x$ is one half the expected real dimension of the critical set, $\rho_x$ is related to the Atiyah–Patodi–Singer $\rho$-invariant or spectral flow term, and $\sqrt{\tau(N,x)}$ arises from the square root of the Reidemeister torsion (Andersen et al., 2011, Andersen et al., 2012, Charles, 2016).

The inclusion of discrete, lower-dimensional strata yields additional terms in the expansion, governed by singular stationary phase analysis for discrete oscillatory sums. The contraction of half-densities and torsion factors continues to provide consistency with both semiclassical and quantum group–derived invariants.

5. Relationship with Witten–Reshetikhin–Turaev and Other Quantum Invariants

While the Witten–Reshetikhin–Turaev (WRT) invariants are constructed via combinatorial, quantum group, or skein-theoretic approaches (and formal quantization of the Jones polynomial), the Jeffrey–Weitsman–Witten invariant realizes the same objects analytically through geometric quantization and symplectic methods. Notably, on integral homology spheres and for cases admitting clean intersection, the invariants agree; the analytic formulation refines and in some cases extends the domain of these quantum invariants to include new types of contributions from reducibles (Beliakova et al., 2010, Andersen, 2012, Habiro et al., 2015, Chitan, 22 Sep 2025).

Splitting the SU(2) WRT invariant into SO(3) and abelian factors aligns directly with the structure found in geometric quantization: the abelian factors correspond to contributions from torus bundles and abelian gauge theory, as isolated in the stratification (Beliakova et al., 2010).

6. Applications, Extensions, and Structural Impact

The stratified, half-density construction enables rigorous evaluation of quantum invariants for a broad class of 3-manifolds—specifically, those whose moduli spaces of flat connections intersect cleanly in a stratified sense. The invariants reproduce known results for lens spaces, $S^1\times S^2$, tori, and Seifert manifolds, capturing both irreducible and reducible data (Chitan, 22 Sep 2025, Charles, 2016). Moreover, this analytic framework admits potential extension to manifolds with more complicated moduli space structures, and to quantum invariants valued in cyclotomic completions (Habiro et al., 2015).

The explicit identification of the torsion and spectral invariants in the integrand clarifies the semiclassical expansion (i.e., the large level $k$ regime) and connects to categorification and quantum modularity phenomena (Chae, 2022). The stratification ensures that no contribution—topological or analytic—is omitted due to singularities or reducibles in the underlying geometric data, yielding a comprehensive and unified approach to quantum invariants in low-dimensional topology.