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Integral bases for TQFT modules and unimodular representations of mapping class groups. (English) Zbl 1055.57026
The authors study a form of the \(SO(3)\)-TQFT described in [C. Blanchet, N. Habegger, G. Masbaum and P.Vogel, Topology 34, 883–927 (1995; Zbl 0887.57009)]. Specifically, they give two different explicit bases for the module \(S_p(\Sigma)\) generated by connected vacuum states over the ring of algebraic integers in a cyclotomic number field. A vacuum state is \(Z_p(M)(1)\in V(\Sigma)\) where \(M\) is a 3-manifold with boundary \(\Sigma\), viewed as a morphism from \(\emptyset\) to \(\Sigma\); \(Z_p(M)\) is the corresponding homomorphism between \(V(\emptyset)\) and \(V(\Sigma)\), and \(V\) is a TQFT functor. The authors also show that there is a non-degenerate unimodular Hermitian form on this \(S_p(\Sigma)\) when \(\Sigma\) is a surface of genus 1 or 2, and that in some higher genus cases the form must be non-unimodular.

57M99 General low-dimensional topology
57R56 Topological quantum field theories (aspects of differential topology)
HYP; Mathematica; TQFT
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