The Witten-Reshetikhin-Turaev invariants of finite order mapping tori I

TL;DR

This paper formulates and proves the Asymptotic Expansion Conjecture for Witten-Reshetikhin-Turaev invariants of finite order mapping tori, expressing invariants as sums over moduli space components with explicit formulas.

Contribution

It provides a geometric gauge theory approach to analyze quantum invariants of finite order mapping tori and explicitly describes their asymptotic behavior.

Findings

Quantum invariants expressed as sums over moduli space components

Polynomial dependence on level k with phase factors related to Chern-Simons invariants

Explicit formulas using Seifert invariants for smooth components

Abstract

We formulate the Asymptotic Expansion Conjecture for the Witten-Reshetikhin-Turaev quantum invariants of closed oriented three manifolds. For finite order mapping tori, we study these quantum invariants via the geometric gauge theory approach to the corresponding quantum representations and prove, using a version of the Lefschetz-Riemann-Roch Theorem due to Baum, Fulton, MacPherson and Quart, that the quantum invariants can be expressed as a sum over the components of the moduli space of flat connections on the mapping torus. Moreover, we show that the term corresponding to a component is a polynomial in the level $k$, weighted by a complex phase, which is $k$ times the Chern-Simons invariant corresponding to the component. We express the coefficients of these polynomials in terms of cohomological pairings on the fixed point set of the moduli space of flat connections on the surface. We explicitly describe the fixed point set in terms of moduli spaces of the quotient orbifold Riemann surface and for the smooth components we express the aforementioned coefficients in terms of the known generators of the cohomology ring. We provide an explicit formula in terms of the Seifert invariants of the mapping torus for the contributions from each of the smooth components. We further establish that the Asymptotic Expansion Conjecture and the Growth Rate Conjecture for these finite order mapping tori.