The Witten-Reshetikhin-Turaev invariants of finite order mapping tori I (Q2848373)

From the path integral point of view, Witten-Reshetikhin-Turaev invariants will have an asymptotic expansion. Former formulations have the common drawback that asymptotics were expressed in terms of sums of terms which cannot be uniquely recovered from the sums over flat \(G\)-connections. Then this has become a conjecture which can be precisely stated as follows; it has the property that each term in the expansion can be uniquely recovered from the Reshetikhin-Turaev quantum invariants at the level \(k\) for the semisimple, simply connected Lie group \(G\):NEWLINENEWLINE \textbf{Conjecture 1.1} (Asymptotic Expansion Conjecture (AEC)). There exist constants (depending on the manifold \(X\)) \(d_{j,r}\in \mathbb Q\) and \(b_{j,r} \in \mathbb C\) for \(r=1,\dots, u_j , j=0, 1,\dots,n\), and \(a^l_{j,r}\in \mathbb C\) for \(j=0,1,\dots,n, l=1,2,\dots\) such that the asymptotic expansion of \(\mathcal Z^{(k)}_G(X)\) in the limit \(r\to \infty\) is given byNEWLINENEWLINE\[NEWLINE\mathcal Z^{(k)}_G(X)\sim \sum_{j=0}^n e^{2\pi ikq_j} \sum_{r=1}^{u_j} k^{d_{j,r}}b_{j,r} \left(1+\sum_{l=1}^\infty a^l_{j,r} k^{-l}\right) ,NEWLINE\]NEWLINENEWLINEwhere \(q_0=0,q_1,\dots ,q_n\) are the finitely many different values of the Chern-Simons functional on the space of flat \(G\)-connections on \(X\).NEWLINENEWLINE For finite order mapping tori, the author proves that the quantum invariants can be expressed as a sum over the components of the moduli space of flat connections on the mapping torus via geometric gauge theory. Moreover, a deep relation with the fixed point set of the moduli space is also discussed. In addition, the author obtains an explicit formula in terms of the Seifert invariants of the mapping torus for the contributions from each of the smooth components of the moduli space.