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

Abstract

We formulate the Asymptotic Expansion Conjecture for the Witten–Reshetikhin–Turaev quantum invariants of closed oriented 3-manifolds. For of 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 & 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 for manifolds and we verify the Growth Rate Conjecture for these finite order mapping tori.