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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.

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