Analytic Asymptotic Expansions of the Reshetikhin–turaev Invariants of Seifert 3–manifolds for Su(2)

We calculate the large quantum level asymptotic expansion of the RT–invariants associated to SU(2) of all oriented Seifert 3–manifolds X with orientable base or non-orientable base with even genus. Moreover, we identify the Chern–Simons invariants of flat SU(2)– connections on X in the asymptotic formula thereby proving the so-called asymptotic expansion conjecture (AEC) due to J. E. Andersen [An1], [An2] for these manifolds. For the case of Seifert manifolds with base S we actually prove a little weaker result, namely that the asymptotic formula has a form as predicted by the AEC but contains some extra terms which should be zero according to the AEC. We prove that these ‘extra’ terms are indeed zero if the number of exceptional fibers n is less than 4 and conjecture that this is also the case if n ≥ 4. For the case of Seifert fibered rational homology spheres we identify the Casson–Walker invariant in the asymptotic formula. Our calculations demonstrate a general method for calculating the large r asymptotics of a finite sum Σk=1f(k), where f is a meromorphic function depending on the integer parameter r and satisfying certain symmetries. Basically the method, which is due to Rozansky [Ro1], [Ro3], is based on a limiting version of the Poisson summation formula together with an application of the steepest descent method from asymptotic analysis.