Diagramatic construction of non-semisimple TQFT

第一个结果是通过与Blanchet， Habegger， Masbaum和Vogel在半简单情况下的构造类比，在q为奇阶单位根时，与小量子群Uq（sl2）相关的非半简单tqft的串理论公式，建立了非半简单量子不变量和tqft的组合模型。与Christian Blanchet和Jun Murakami一起，我们开发了当q是奇阶单位的根时小量子群Uq（sl2）的表示的图解结构。然后，与Jun Murakami一起，我们得到了当q是奇阶单位的根时与Uq（sl2）相关的非半单量子不变量的一个完全组合的重新表述。更准确地说，当q是奇阶单位的根时，我们定义了Temperley-Lieb范畴的扩展版本。第二个结果是构造非半简单tqft和从模类别映射类组表示。这是Nathan Geer、Bertrand Patureau、Azat Gainutdinov和Ingo Runkel的共同努力。我们共同开发了Lyubashenko量子不变量的重整化版本，并将其扩展到完整的tqft。修正迹的理论使得重新规格化构造成为可能，以便定义完全一元函子，正如我们已经在Geer和Patureau的Hennings不变量中所做的那样。我们还研究了由这种构造产生的映射类群的投影量子表示，并证明它们恢复了Lyubashenko的投影量子表示。

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.