Analytic asymptotic expansions of the Reshetikhin-Turaev invariants of Seifert 3-manifolds for SU(2)

Søren Kold Hansen

Research output: Book/ReportReportResearch

Abstract

We calculate the large quantum level asymptotic expansion of the Reshetikhin-Turaev-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 for these manifolds. For the case of Seifert manifolds with base the 2-sphere 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 is greater than 3. 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 f(1)+...+f(r), where f is a meromorphic function depending on the integer parameter r and satisfying certain symmetries. Basically the method, which is due to L. Rozansky, is based on a limiting version of the Poisson summation formula together with an application of the steepest descent method from asymptotic analysis.
Original languageEnglish
PublisherarXiv.org, e-print service at Cornell University, Ithaca, NY, USA
Number of pages107
Publication statusPublished - 2005
Externally publishedYes

Cite this

Hansen, S. K. (2005). Analytic asymptotic expansions of the Reshetikhin-Turaev invariants of Seifert 3-manifolds for SU(2). arXiv.org, e-print service at Cornell University, Ithaca, NY, USA.
Hansen, Søren Kold. / Analytic asymptotic expansions of the Reshetikhin-Turaev invariants of Seifert 3-manifolds for SU(2). arXiv.org, e-print service at Cornell University, Ithaca, NY, USA, 2005. 107 p.
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abstract = "We calculate the large quantum level asymptotic expansion of the Reshetikhin-Turaev-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 for these manifolds. For the case of Seifert manifolds with base the 2-sphere 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 is greater than 3. 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 f(1)+...+f(r), where f is a meromorphic function depending on the integer parameter r and satisfying certain symmetries. Basically the method, which is due to L. Rozansky, is based on a limiting version of the Poisson summation formula together with an application of the steepest descent method from asymptotic analysis.",
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Hansen, SK 2005, Analytic asymptotic expansions of the Reshetikhin-Turaev invariants of Seifert 3-manifolds for SU(2). arXiv.org, e-print service at Cornell University, Ithaca, NY, USA.

Analytic asymptotic expansions of the Reshetikhin-Turaev invariants of Seifert 3-manifolds for SU(2). / Hansen, Søren Kold.

arXiv.org, e-print service at Cornell University, Ithaca, NY, USA, 2005. 107 p.

Research output: Book/ReportReportResearch

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N2 - We calculate the large quantum level asymptotic expansion of the Reshetikhin-Turaev-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 for these manifolds. For the case of Seifert manifolds with base the 2-sphere 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 is greater than 3. 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 f(1)+...+f(r), where f is a meromorphic function depending on the integer parameter r and satisfying certain symmetries. Basically the method, which is due to L. Rozansky, is based on a limiting version of the Poisson summation formula together with an application of the steepest descent method from asymptotic analysis.

AB - We calculate the large quantum level asymptotic expansion of the Reshetikhin-Turaev-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 for these manifolds. For the case of Seifert manifolds with base the 2-sphere 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 is greater than 3. 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 f(1)+...+f(r), where f is a meromorphic function depending on the integer parameter r and satisfying certain symmetries. Basically the method, which is due to L. Rozansky, is based on a limiting version of the Poisson summation formula together with an application of the steepest descent method from asymptotic analysis.

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Hansen SK. Analytic asymptotic expansions of the Reshetikhin-Turaev invariants of Seifert 3-manifolds for SU(2). arXiv.org, e-print service at Cornell University, Ithaca, NY, USA, 2005. 107 p.