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created an entry self-dual object in order to record how $\dagger$-structures are induced from self-duality structures (“transposition”). Copied the respective paragraph also into dagger-compact category.
Have added more references,
in particular (here) from the functorial field theory literature, concerning the case of un-oriented 1d field theories.
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I feel there is an elementary gap in the field’s (everyone’s) understanding of this issue in actual quantum theory:
Namely, starting with an oriented 1d (T)QFT with coefficients in complex vector spaces and classified by some $\mathscr{H} \in \mathbb{C}Vect$, interpreted as the (fin dim Hilbert) space of quantum states on the positively oriented point, it actually makes $\;$no$\;$ physical sense to ask if this lifts to an unoriented 1d (T)QFT:
The issue is that such a lift is equivalently a plain self-duality $\mathscr{H} \overset{\sim}{\to} \mathscr{H}^\ast$ in $\mathbb{C}Vect$, which is a complex bi-linear form on a complex vector space.
But such a structure has no meaning in actual quantum physics. The two options that would make physical sense are:
either the self-duality is complex anti-linear, making a sesqui-linear form on a complex vector space,
or we have an actual bilinear form but on a real vector space.
I believe the solution is that $\mathbb{C}Vect$ is not actually the correct coefficient category for physically meaningful 1d (T)QFTs.
The correct coefficients is more like the category of “Real complex modules”, as described there. Here (fin dim) complex Hermitian spaces do embed as self-dual objects (fully, if these are regarded as equipped with “internal orthogonal complex structure”), and self-dual real vector spaces arise as their $\mathbb{Z}/2$-fixed loci.
One solution to this issue goes as follows:
Physical TQFTs are not just symmetric monoidal functors $Bord \to Coefficients$ but dagger functors. The dagger functors $Bord^{O(1)} \to \mathbb{C}Mod$ are classified by a “dagger-self-dual” structure on a findim Hermitian space, and that is equivalently a real structure (as now discussed there).
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