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started a bare minimum at Poisson-Lie T-duality, for the moment just so as to have a place to record the two original references
added more references, in particular Ossa-Quevedo 92, which is maybe the first one.
Made “nonabelian T-duality” a redirect
Are you looking at this with regard to Higher T-duality? Any interest in higher non-abelian T-duality for, say, $String(G)$?
Yes and no: The question in the background was whether the “geometric” complement of “topological” 3-spherical T-duality could be Poisson-Lie T-duality for isometry group $SU(2)$. I still don’t know on the math side. But at least the physics contexts do not match: The Poisson-Lie T-duality is based on string sigma-models, while the realization of spherical T-duality that we found replaces strings by 5-branes.
added pointer to Fraser-Manolopoulos-Sfetsos 18
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added the following warning, based on discussion we had at the meeting String and M-theory, Singapre 2018. I am not aware of any written reference that would make this statement, but if anyone has one, it should be added.
It has been proven that these generalized T-dualities are are/induce equivalences of the corresponding string sigma-models at the level of classical field theory. However it seems to be open, and in fact questionable, that away from standard T-duality this yields an equivalence at the level of worldsheet quantum field theories, hence it is open whether Poisson-Lie/non-abelian T-duality is really a duality operation on perturbative string theory vacua.
added pointer to today’s discussion of nonabelian T-folds in
My contact person gave me the following more detailed assessment of the non-duality status of non-abelian T-duality by email. For the moment I have no citations for this, apart from this private correspondence, but since it seems to be very worthwhile information, I have put it into the entry, like so:
While ordinary abelian T-duality is supposedly a full duality in string theory, in particular in that it is an equivalence on the string perturbation series to all orders of the string length/Regge slope $\alpha'$ and the string coupling constant $g_s$, it has apparntly been shown by Martin Roček (citation?) that there are topological obstructions at higher genus for non-abelian T-duality, letting it break down in higher orders of $g_s$; and already a genus-0 (tree level) it apparently breaks down for the open string (i.e. on punctured disks) at some order of $\alpha'$.
If any expert sees this, please feel invited to correct/expand.
Added pointer to today’s
Does this new duality reduce to higher T-duality in the abelian case? At first sight it looks like the dual algebra being shifted $h' = h^\ast[n-2]$ does not allow it
It would be interesting to find a relation, but in the present form I’d think neither can reduce to the other, simply because they formulate different aspects of T-duality:
Here the Poisson-Lie perspective focuses on the local geometric content, while the higher T-duality we described is the higher generalization of topological T-duality, albeit rationalized.
To see how the two higher generalizations relate to each other, it would hence be useful to first have a concrete relation between ordinary Poisson-Lie T-duality and ordinary topological T-duality. Did anyone look into that?
Another vague thought one might have here is that, due to $S^3 \simeq SU(2)$, the higher 3-spherical topological T-duality is somehow related to non-higher but $\mathfrak{su}(2)$-nonabelian T-duality. But beyond the coincidental $S^3 \simeq SU(2)$ there may be no real hint for this, and in fact the physics pictures behind the two sides are disparate (M5-brane physics on one hand, perturbative string sigma models on the other.)
So in summary, I don’t know. And I am not thinking about it. But it seems like something worth exploring.
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discussion of non-abelian T-duality from a comprehensive picture of higher differential geometry, relating Kaluza-Klein compactification on principal ∞-bundles to double field theory, T-folds, type II geometry, exceptional geometry, etc.:
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I was also thinking something about this.
They limited themselves to “Exceptional Drinfel’d doubles” representable by $T D \simeq T G \oplus \wedge^2T^\ast G$ for some subgroup $G$, in analogy with what happens for classical Drinfel’d doubles (i.e. $T D\simeq T G \oplus T^\ast G$).
Now, as far as I understand the Drinfel’d double $D(G)$ of a group $G$ is related to a cocycle $\omega\in H^3(G,\mathbb{Z})$ (in particular, a flat gerbe(?)).
It looks to me that they are trying to define something analogous by using $\omega\in H^4(G,\mathbb{Z})$.
Maybe then a full working definition of an “Exceptional Drinfel’d double” would require a to be in correspondence an element of the cohomotopy $\pi^4(G)$, in the spirit of Exceptional geometry as given by differential cohomotopy theory.
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Falk Hassler, Thomas Rochais, $\alpha'$-corrected Poisson-Lie T-duality (arXiv:2007.07897)
Riccardo Borsato, Linus Wulff, Quantum correction to Poisson-Lie and non-abelian T-duality (arXiv:2007.07902)
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