Diagramatic construction of non-semisimple TQFT

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The first result is a construction of a combinatorial model for non-semisimple quantum invariants and TQFTs by developing a skein theoretic formulation of non-semisimple TQFTs associated with the small quantum group Uq(sl2) when q is a root of unity of odd order, by analogy with the construction of Blanchet, Habegger, Masbaum, and Vogel in the semisimple case. With Christian Blanchet and Jun Murakami, we developed a diagrammatic construction of representations of the small quantum group Uq(sl2) when q is a root of unity of odd order. Then, with Jun Murakami, we obtained a fully combinatorial reformulation of the non-semisimple quantum invariants associated with Uq(sl2) when q is a root of unity of odd order. More precisely, we defined an extended version of the Temperley-Lieb category when q is a root of unity of odd order.The second result is a construction of non-semisimple TQFTs and mapping class group representations from modular categories. This is a joint effort with Nathan Geer, Bertrand Patureau, Azat Gainutdinov, and Ingo Runkel. Together, we developed a renormalized version of the quantum invariants of Lyubashenko, which we extended to full TQFTs. The theory of modified traces then makes it possible to renormalize the construction in order to define fully monoidal functors, as we have already done in the case of Hennings invariants with Geer and Patureau. We also study the projective quantum representations of mapping class groups produced by this construction, and show that they recover Lyubashenko’s ones.