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The Dijkgraaf-Witten invariants of Seifert 3-manifolds with orientable bases

Haimiao Chen

Abstract

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

The Dijkgraaf-Witten invariants of Seifert 3-manifolds with orientable bases

Abstract

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

Paper Structure

This paper contains 10 sections, 3 theorems, 76 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Exposition of Dijkgraaf-Witten theory
  3. Preparation
  4. The DW invariant of a closed surface
  5. The DW invariant of a 3-manifold
  6. The TQFT axioms
  7. Computing the DW invariants of Seifert 3-manifolds
  8. The Hermitian vector space $Z^{\omega}(\Sigma_{1})$
  9. Cycle-cocycle calculus
  10. Formula for Seifert 3-manifolds with orientable bases

Key Result

Theorem 2.4

To each closed surface $N$ is assigned a Hermitian space $Z^{\omega}(N)$, and to each 3-manifold $M$ is assigned a vector $Z^{\omega}(M)\in Z^{\omega}(\partial M)$. They satisfy the following: (a) (Functorality) Each diffeomorphism $f:N\rightarrow N'$ induces an isometry For each diffeomorphism $F:M\rightarrow M'$, one has $(\partial F)_{\ast}(Z^{\omega}(M))=Z^{\omega}(M').$ (b) (Orientation) The

Figures (3)

  • Figure 1: Computing the cycle $f^{-1}_{\#}(\xi_{0})-\xi_{0}$
  • Figure 2: $\Sigma_{1;1,1}=-P\cup P$
  • Figure 3: Computing $q^{-1}_{\#}(\xi_{0})-\xi_{0}$

Theorems & Definitions (12)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Lemma 3.4
  • proof
  • Remark 3.5
  • Theorem 3.6
  • ...and 2 more