Homology TQFT’s via Hopf Algebras
نویسنده
چکیده
In [5] Frohman and Nicas define a topological quantum field theory via the intersection homology of U(1)-representation varieties J(X) = Hom(π1(X), U(1)). We show that this TQFT is equivalent to the combinatorially constructed Hennings-TQFT based on the quasitriangular Hopf algebra N = Z/2 ⋉ ∧ ∗ R. The natural SL(2,R)-action on N is identified with the SL(2,R)action for the Lefschetz decomposition of H∗(J(Σ)) implied by the Kähler structure on J(Σ) for a surface, Σ. We compare peculiarities of both theories, such as the Z/2-projectivity and vanishing phenomena due to non-semisimplicty. This equivalence induces a graded Hopf algebra structure on H∗(J(Σ)), which is isomorphic to the canonical one but at the same time compatible with the Hard-Lefschetz decomposition. We discuss generalizations to higher rank gauge theories and a relation between the semisimple and non-semisimple TQFT’s associated to quantum sl2. 1
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