# Isomorphisms between the combinatorial and geometric construction of the fundamental TQFT's

Groundbreaking research result

2016.05.11 | Christine Dilling

QGM Center leader, Professor Jørgen Ellegaard Andersen and Professor Kenji Ueno from Kyoto University have provided an explicit isomorphism from the modular functor underlying the skein-theoretic model for the Witten–Reshetikhin–Turaev TQFT due to C. Blanchet, N. Haebegger, G. Masbaum and P. Vogel to the vacua modular functor coming from the conformal field theory. This thus provides a geometric construction of the TQFT first proposed by Witten and constructed first by Reshetikhin–Turaev from the quantum group Uq(sl (N)). The proof is presented in a series of four papers, of which the final one is published in the top international journal “Inventiones Mathematicae”. This isomorphisms offers the possibility of port constructions from one construction of this TQFT to other, which surely will lead to interesting new results on both sides. As an example of such an application, one can mention that, that Andersen and Ueno's isomorphisms allows one to conclude that the Hitchin connection preserves a mapping class group invariant Hermitian structure. Something which is yet to be constructed purely by geometric means.

Construction of the Witten–Reshetikhin–Turaev TQFT from conformal field theory (link to paper online)

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