[Paper Review] The Dijkgraaf-Witten invariants of Seifert 3-manifolds with orientable bases

Haimiao Chen|arXiv (Cornell University)|Jul 1, 2013

Geometric and Algebraic Topology13 references3 citations

TL;DR

This paper derives a closed-form formula for the Dijkgraaf-Witten invariants of Seifert 3-manifolds with orientable bases using the representation theory of the twisted quantum double of a finite group Γ. The key result expresses the invariant in terms of irreducible characters of the twisted quantum double, generalizing prior results for trivial cocycles and providing explicit computational tools via Gauss sums and projective representations.

ABSTRACT

We derive a formula for the Dijkgraaf-Witten invariants of orientable Seifert 3-manifolds with orientable bases.

Motivation & Objective

• To compute the Dijkgraaf-Witten invariants for Seifert 3-manifolds with orientable bases, a class of 3-manifolds with rich topological structure but limited prior computation in the Dijkgraaf-Witten framework.
• To extend previous results on Dijkgraaf-Witten invariants—previously known only for trivial cocycles or specific classes like circle bundles—to the general case with nontrivial U(1)-valued 3-cocycles.
• To establish a formula that expresses the invariant in terms of irreducible characters of the twisted quantum double of a finite group Γ, enabling concrete computation.
• To provide a systematic algebraic framework using group cohomology and functor categories to simplify the computation of these topological invariants.

Proposed method

• The paper uses the simplicial model of the classifying space BΓ and represents the 3-cocycle ω as a function Γ³ → U(1) satisfying the normalized 3-cocycle condition.
• It defines the Dijkgraaf-Witten invariant via a sum over group homomorphisms π₁(M) → Γ, weighted by the pairing of the pullback of ω with the fundamental class [M].
• The computation proceeds by decomposing the Seifert 3-manifold into standard pieces (e.g., Σg;n,1 × S¹ and solid tori), computing the invariant on each piece using representation-theoretic data.
• The invariant on the base space Σg;n,1 × S¹ is computed via the action of the twisted quantum double, using the comultiplication and multiplication maps in the associated Hopf algebra.
• For the Dehn surgery pieces (ST), the paper computes the morphism (f_j)_*Z^ω(ST) using projective representations and character sums involving Gauss sums.
• The final invariant is obtained by composing the maps from the base and the Dehn surgeries, resulting in a trace over the twisted quantum double representations.

Experimental results

Research questions

• RQ1 How can the Dijkgraaf-Witten invariant be computed for Seifert 3-manifolds with orientable bases when the cocycle ω is nontrivial?
• RQ2 What is the algebraic structure underlying the Dijkgraaf-Witten invariant in this class of 3-manifolds, and how does it relate to the twisted quantum double of Γ?
• RQ3 Can the invariant be expressed in terms of irreducible characters of the twisted quantum double, and if so, what is the explicit formula?
• RQ4 What role do projective representations and Gauss sums play in the explicit evaluation of the invariant for specific groups like ℤm?
• RQ5 How do the invariants depend on the Seifert invariants (a_j, b_j) and the group structure, particularly when m is composite?

Key findings

• The Dijkgraaf-Witten invariant for a Seifert 3-manifold M_O(g; (a₁,b₁), ..., (aₙ,bₙ)) with orientable base is given by a sum over irreducible representations ρ of the twisted quantum double, weighted by (D_ρ)^{n+2g-1} and character sums involving η^ω_ρ(a_j,b_j).
• For Γ = ℤm and ω = μ_ℓ, the invariant is expressed using Gauss sums and Legendre symbols, with explicit evaluation via S_p(a) = ∑_{k=0}^{p-1} ζ_p^{ak²} for prime divisors p of m.
• The character sum η^ω_ρ(a_j,b_j) is computed as a sum over solutions to congruences, yielding a formula involving ζ_{m²} and quadratic Gauss sums when m is composite.
• The formula reduces to a finite sum over s ∈ ℤ_m for each h ∈ ℤ_m, with ρ^ℓ_{h,s} given by ρ^ℓ_{h,s}(x) = ζ_{m²}^{ℓh̃x̃ + m(s̃x̃)}.
• The final invariant is a trace over the twisted quantum double, computed as the composition of maps from the base and Dehn surgery pieces, resulting in a finite sum over irreducible representations.
• The computation reveals that the invariant depends on the greatest common divisor d_j = gcd(a_j, m), the modular inverse c_j of a_j/d_j mod m_j, and the structure of the group ℤm via its p-primary components.