Reps of relative mapping class groups via conformal nets

Abstract: Given a surface with boundary Σ, its relative mapping class group is the quotient of Diff(Σ) by the subgroup of maps which are isotopic to the identity via an isotopy that fixes the boundary pointwise. (If Σ has no boundary, then that's the usual mapping class group; if Σ is a disc, then that's the group Diff(S¹) of diffeomorphisms of S¹.)

Conformal nets are one of the existing axiomatizations of chiral conformal field theory (vertex operator algebras being another one). We will show that, given an arbitrary conformal net and a surface with boundary Σ, we get a continuous projective unitary representation of the relative mapping class group (orientation reversing elements act by anti-unitaries). When the conformal net is rational and Σ is a closed surface (i.e. ∂Σ = ∅), then these representations are finite dimensional and well known. When the conformal net is not rational, then we must require ∂Σ ≠ ∅ for these representations to be defined. We will try to explain what goes wrong when Σ is a closed surface and the conformal net is not rational.

The material presented in this talk is partially based on my paper arXiv:1409.8672 with Arthur Bartels and Chris Douglas.

mathematical physicsalgebraic topologycategory theoryquantum algebra

Audience: researchers in the topic

( video )

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Topological Quantum Field Theory Club (IST, Lisbon)

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