On the quantum invariants for the spherical Seifert manifolds
In this paper some invariants (Witten-Reshetkhin-Turaev (WRT) quantum invariants) are studied for Seifert manifolds \(S^3/\Gamma\), where \(\Gamma\) is a finite subgroup of SU(2). Quantum invariants are invariants introduced by Witten and Atiyah in the so called topological quantum field theory in the years 1989 and 1990 and further developed by many other mathematicians (Kontsevich, Quinn, Reshetikhin, Turaev, Vassiliev, \dots). (The Chern-Simons field theory and the Wess-Zumino-Witten model can be considered as the physical back-ground of quantum invariants.) A particular attention was put on \(3\)-manifolds, where quantum invariants have been related to so called ribbon Hopf algebras and Hopf diagrams. The author, in this paper, shows that WRT quantum invariants for \(S^3/\Gamma\) can be written in terms of the Eichler integral of modular forms with half-integral weight, and gives an exact asymptotic expansion of the invariants by using the nearly property of the Eichler integral. Furthermore, a relation with polyhedral equations associated to \(\Gamma\) is proved. The paper, after a detailed introduction, splits in five more sections. 2. Preliminaries. 3. Witten-Reshetkhin-Turaev invariant. 4. WRT invariant for the spherical Seifert manifolds. 5. Relationship with the Platonic solids. 6. Conclusions and discussions. Remark. ``Quantum invariants'' were also considered in the geometric theory of PDE's, super PDE's and quantum PDE's by the reviewer of this paper. In particular, these are interpreted as evaluation of ``conservation laws'' on admissible integral Cauchy manifolds. Then, such invariants are related to ``Hopf algebras of PDE's''. Furthermore, more recently, such invariants are related also to subgroups of crystallographic groups. (See works published starting from 1990.)
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