Integral Bases for Tqft Modules and Unimodular Representations of Mapping Class Groups
نویسندگان
چکیده
We construct integral bases for the SO(3)-TQFT-modules of surfaces in genus one and two at roots of unity of prime order and show that the corresponding mapping class group representations preserve a unimodular Hermitian form over a ring of algebraic integers. For higher genus surfaces the Hermitian form sometimes must be non-unimodular. In one such case, genus 3 and p = 5, we still give an explicit basis.
منابع مشابه
Integral Tqft for a One-holed Torus
We give new explicit formulas for the representations of the mapping class group of a genus one surface with one boundary component which arise from Integral TQFT. Our formulas allow one to compute the h-adic expansion of the TQFT-matrix associated to a mapping class in a straightforward way. Truncating the h-adic expansion gives an approximation of the representation by representations into fi...
متن کاملIntegrality for Tqfts
We discuss two ways that the ring of coeffients for a TQFT can be reduced if one may restrict somewhat the allowed cobordisms. When we apply both methods to a TQFT associated to SO(3) and an odd prime p, we obtain a functor from a somewhat restricted cobordism category to the category of free finitely generated modules over a ring of cyclotomic integers : Z[ζp], if p ≡ −1 (mod 4), and Z[ζ4p], i...
متن کاملIrreducibility of Some Quantum Representations of Mapping Class Groups
The SU(2) TQFT representation of the mapping class group of a closed surface of genus g, at a root of unity of prime order, is shown to be irreducible. Some examples of reducible representations are also given.
متن کاملMonomial Irreducible sln-Modules
In this article, we introduce monomial irreducible representations of the special linear Lie algebra $sln$. We will show that this kind of representations have bases for which the action of the Chevalley generators of the Lie algebra on the basis elements can be given by a simple formula.
متن کاملIntegral Lattices in Tqft
We find explicit bases for naturally defined lattices over a ring of algebraic integers in the SO(3)-TQFT-modules of surfaces at roots of unity of odd prime order. Some applications relating quantum invariants to classical 3-manifold topology are given.
متن کامل