A Hermitian TQFT from a non-semisimple category of quantum $${\mathfrak {sl}(2)}$$-modules

Abstract

We study the density of the Burau representation from the perspective of a nonsemisimple TQFT at a fourth root of unity. This gives a TQFT construction of Squier's Hermitian form on the Burau representation with possibly mixed signature. We prove that the image of the braid group in the space of possibly indefinite unitary representations is dense. We also argue for the potential applications of non-semisimple TQFTs toward topological quantum computation.

Non‐semisimple Levin–Wen models and Hermitian TQFTs from quantum (super)groups

“…We now assume that $$H$$‐mod has a fixed choice of half twist. By [23, Proposition 4.12], a half twist exists if $${\mathbb {k}}$$ is an algebraically closed field of characteristic 0. For objects $$W_1,W_2$$ of $$H$$‐mod, define the isomorphism $${\mathsf {X}}: W_1\otimes W_2 \rightarrow W_2\otimes W_1$$ by $$$$\begin{equation} {\mathsf {X}}_{W_1,W_2}={{\left({\sqrt \theta _{W_2\otimes W_1}}\right)}}^{-1}c_{W_1,W_2}\left(\sqrt \theta _{W_1}\otimes \sqrt \theta _{W_2}\right).$$…”

“…Proof This proof is reproduced from [23], but we now include all the relevant signs arising from super considerations. Let $$u\in H$$ and write $$\Delta (u)=u_1\otimes u_2$$.…”

“…Recall that a sesquilinear module $$W$$ is Hermitian if its sesquilinear form has Hermitian symmetry: if for any $$v_1,v_2\in W$$, $$(v_2|v_1)=\overline{(v_1|v_2)}$$. The next lemma is similar to [23, Proposition 4.5]. Lemma Any simple module $$V$$ in $$\mathcal {D}$$ with real highest weight has a sesquilinear structure.…”

“…The unital axiom follows from the fact that $$$$\begin{equation*} {{\big({W\stackrel{\sim }{\rightarrow }W\otimes \mathbb {I}\stackrel{{\mathsf {X}}}{\rightarrow }\mathbb {I}\otimes W\stackrel{\sim }{\rightarrow }W}\big)}} =\mathrm{Id}_W={{\big({W\stackrel{\sim }{\rightarrow }\mathbb {I}\otimes W\stackrel{{\mathsf {X}}}{\rightarrow }W\otimes \mathbb {I}\stackrel{\sim }{\rightarrow }W}\big)}}, \end{equation*}$$$$which is a direct consequence of the definition of $${\mathsf {X}}$$. The associativity axiom follows from the equality $$$$\begin{equation*} {\mathsf {X}}_{V^{\prime }\otimes V,V^{\prime \prime }}({\mathsf {X}}_{V,V^{\prime }}\otimes \mathrm{Id}_{V^{\prime \prime }}) ={\mathsf {X}}_{V,V^{\prime \prime }\otimes V^{\prime }}(\mathrm{Id}_{V}\otimes {\mathsf {X}}_{V^{\prime },V^{\prime \prime }}): V\otimes V^{\prime }\otimes V^{\prime \prime }\rightarrow V^{\prime \prime }\otimes V^{\prime }\otimes V, \end{equation*}$$$$which is proved in [23, Lemma 4.13].$$\Box$$…”

“…We now show that $$\operatorname{\mathsf {t}}$$ is real. This follows as in [23, Lemma 4.19] from the unicity of the trace and the fact that $$f\mapsto \operatorname{\mathsf {t}}^\dagger (f):=\overline{\operatorname{\mathsf {t}}(f^\dagger)}$$ is first also a trace, thus proportional to $$\operatorname{\mathsf {t}}$$, and second $$\operatorname{\mathsf {t}}^\dagger (\mathrm{Id}_{V_0})=\overline{\operatorname{\mathsf {d}}(V_0)}=\operatorname{\mathsf {d}}(V_0)=\operatorname{\mathsf {t}}(\mathrm{Id}_{V_0})\ne 0$$. From this, we get $$\operatorname{\mathsf {t}}=\operatorname{\mathsf {t}}^\dagger$$.…”