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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 20th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am creating a reference-entry [[T-Duality and Differential K-Theory]]. In the course of this I have now first of all created stubs for * [[twisted de Rham cohomology]] * [[twisted differential K-theory]]

   I am creating a reference-entry T-Duality and Differential K-Theory. In the course of this I have now first of all created stubs for

   • twisted de Rham cohomology

   • twisted differential K-theory

2. • CommentRowNumber2.
   • CommentAuthordomenico_fiorenza
   • CommentTimeMay 23rd 2011
   • PermaLink
   Author: domenico_fiorenza
   Format: MarkdownItexEdited and cleaned a bit the first part of [[T-Duality and Differential K-Theory]]

   Edited and cleaned a bit the first part of T-Duality and Differential K-Theory

3. • CommentRowNumber3.
   • CommentAuthordomenico_fiorenza
   • CommentTimeMay 27th 2011
   • PermaLink
   Author: domenico_fiorenza
   Format: MarkdownItexExpanded above Definition 1

   Expanded above Definition 1

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMay 28th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks, good. I'll come back to all this later, need to do something on [[string topology]] first. One discussion piece currently missing in the entry is this: I have indicated how the existence of the degree-3 twist on the fiber product of the two torus bundles follows general abstractly. It remains to describe how this is the pullback of either degree-3 twist on each of the torus bundles separately. Then with that there should be a nice general abstract way to see that the pull-tensor-push operation on twisted differential K-theory is the one given in the article. But I need to think about this a bit more.

   Thanks, good. I’ll come back to all this later, need to do something on string topology first.

   One discussion piece currently missing in the entry is this: I have indicated how the existence of the degree-3 twist on the fiber product of the two torus bundles follows general abstractly. It remains to describe how this is the pullback of either degree-3 twist on each of the torus bundles separately.

   Then with that there should be a nice general abstract way to see that the pull-tensor-push operation on twisted differential K-theory is the one given in the article. But I need to think about this a bit more.

5. • CommentRowNumber5.
   • CommentAuthordomenico_fiorenza
   • CommentTimeMay 28th 2011
   • PermaLink
   Author: domenico_fiorenza
   Format: MarkdownItexI've now completely rewritten the part before Definition 1, and created a stub for [[Beilinson-Deligne cup-product]].

   I’ve now completely rewritten the part before Definition 1, and created a stub for Beilinson-Deligne cup-product.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeMay 28th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks, that's better.

   Thanks, that’s better.

7. • CommentRowNumber7.
   • CommentAuthordomenico_fiorenza
   • CommentTimeMay 28th 2011
   • PermaLink
   Author: domenico_fiorenza
   Format: MarkdownItexI was thinking we could completely treat the untwisted case, first. namely, in the untwisted case, at the chain complex level we want to compute the homotopy fiber of $$ \array{ & &0\\ & & \downarrow\\ \Lambda[2]^\infty_D\otimes\hat{\Lambda}[2]^\infty_D&\to& \mathbb{Z}[4]^\infty_D } $$ and to do this we just have to make a fibrant repalcement of $0\to \mathbb{Z}[4]^\infty_D$, which is easily done: take the shifted mapping cone of the identity of $\mathbb{Z}[4]^\infty_D$. then we are reduced to taking the ordinary pullback of chain complexes $$ \array{ & &Cone(\mathbb{Z}[4]^\infty_D \stackrel{id}{\to} \mathbb{Z}[4]^\infty_D)[-1]\\ & & \downarrow\\ \Lambda[2]^\infty_D\otimes\hat{\Lambda}[2]^\infty_D&\to& \mathbb{Z}[4]^\infty_D } $$ The general untwisted case is more interesting but also more difficult: since now the upper right corner of the diagram is arbitrary, what we have to do is to take a fibrant replacement of the bottom horizontal morphism $\Lambda[2]^\infty_D\otimes\hat{\Lambda}[2]^\infty_D&\to& \mathbb{Z}[4]^\infty_D$.

   I was thinking we could completely treat the untwisted case, first. namely, in the untwisted case, at the chain complex level we want to compute the homotopy fiber of

   $$\array{ & &0\\ & & \downarrow\\ \Lambda[2]^\infty_D\otimes\hat{\Lambda}[2]^\infty_D&\to& \mathbb{Z}[4]^\infty_D }$$

   and to do this we just have to make a fibrant repalcement of $0\to \mathbb{Z}[4]^\infty_D$, which is easily done: take the shifted mapping cone of the identity of $\mathbb{Z}[4]^\infty_D$. then we are reduced to taking the ordinary pullback of chain complexes

   $$\array{ & &Cone(\mathbb{Z}[4]^\infty_D \stackrel{id}{\to} \mathbb{Z}[4]^\infty_D)[-1]\\ & & \downarrow\\ \Lambda[2]^\infty_D\otimes\hat{\Lambda}[2]^\infty_D&\to& \mathbb{Z}[4]^\infty_D }$$

   The general untwisted case is more interesting but also more difficult: since now the upper right corner of the diagram is arbitrary, what we have to do is to take a fibrant replacement of the bottom horizontal morphism .

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeMay 28th 2011
   • (edited May 28th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes, I agree. Somewhat orthogonal to this: it is also natural, if not even compelling, now to consider genuinely "twisted T-duality", where the underlying topological class of the twisting cocycle is not required to be trivial. We know that this is the generality needed for twisted string- and fivebrane structures, and we see that T-duality pairs are really just a variant of that. But I haven't thought about what the action on twisted differential K-theory would be in that generality. I would like to figure that out. It would also shed light on the ordinary case where the underlying topological class of the topological twist is trivial.

   Yes, I agree. Somewhat orthogonal to this: it is also natural, if not even compelling, now to consider genuinely “twisted T-duality”, where the underlying topological class of the twisting cocycle is not required to be trivial. We know that this is the generality needed for twisted string- and fivebrane structures, and we see that T-duality pairs are really just a variant of that. But I haven’t thought about what the action on twisted differential K-theory would be in that generality. I would like to figure that out. It would also shed light on the ordinary case where the underlying topological class of the topological twist is trivial.

9. • CommentRowNumber9.
   • CommentAuthordomenico_fiorenza
   • CommentTimeMay 29th 2011
   • (edited May 29th 2011)
   • PermaLink
   Author: domenico_fiorenza
   Format: MarkdownItexHi Urs, there's a point which is confusing me: if we consider [[topological T-duality]], then the homotopy fiber $DPair$ of the morphism $\mathbf{B}^2 \Lambda\times \mathbf{B}^2\hat{\Lambda}\to \mathbf{B}^4\mathbb{Z}$ gives rise to the long fiber sequence $$ \dots \mathbf{B}^3\mathbb{Z} \to Dpair\to \mathbf{B}^2\Lambda\times \mathbf{B}^2\hat{\Lambda}\to \mathbf{B}^4\mathbb{Z} $$ this happens because $\mathbf{B}^3 \mathbb{Z}$ is the loop space of $\mathbf{B}^4 \mathbb{Z}$. yet, when we pass to the differential refinements, I'm not sure that the homotopy fiber $DPair {}_{conn}$ of the morphism $\mathbf{B}(\mathbb{R}^n/\Lambda) {}_{conn}\times \mathbf{B}((\mathbb{R}^n)^*/\hat{\Lambda}))_{conn}\to \mathbf{B}^3\mathbf{U}(1)_{conn}$ gives rise to the long fiber sequence $$ \dots \mathbf{B}^2\mathbf{U}(1)_{conn} \to Dpair {}_{conn}\to \mathbf{B}(\mathbb{R}^n/\Lambda)_{conn}\times \mathbf{B}((\mathbb{R}^n)^*/\hat{\Lambda}))_{conn}\to \mathbf{B}^3\mathbf{U}(1)_{conn}. $$ this precisely because $\mathbf{B}^2\mathbf{U}(1) {}_{conn}$ should not be the loop space of $\mathbf{B}^3\mathbf{U}(1) {}_{conn}$

   Hi Urs,

   there’s a point which is confusing me: if we consider topological T-duality, then the homotopy fiber $DPair$ of the morphism $\mathbf{B}^2 \Lambda\times \mathbf{B}^2\hat{\Lambda}\to \mathbf{B}^4\mathbb{Z}$ gives rise to the long fiber sequence

   $$\Lambda[2]^\infty_D\otimes\hat{\Lambda}[2]^\infty_D& \dots \mathbf{B}^3\mathbb{Z} \to Dpair\to \mathbf{B}^2\Lambda\times \mathbf{B}^2\hat{\Lambda}\to \mathbf{B}^4\mathbb{Z}$$

   this happens because $\mathbf{B}^3 \mathbb{Z}$ is the loop space of $\mathbf{B}^4 \mathbb{Z}$. yet, when we pass to the differential refinements, I’m not sure that the homotopy fiber $DPair {}_{conn}$ of the morphism $\mathbf{B}(\mathbb{R}^n/\Lambda) {}_{conn}\times \mathbf{B}((\mathbb{R}^n)^*/\hat{\Lambda}))_{conn}\to \mathbf{B}^3\mathbf{U}(1)_{conn}$ gives rise to the long fiber sequence

   $$\dots \mathbf{B}^2\mathbf{U}(1)_{conn} \to Dpair {}_{conn}\to \mathbf{B}(\mathbb{R}^n/\Lambda)_{conn}\times \mathbf{B}((\mathbb{R}^n)^*/\hat{\Lambda}))_{conn}\to \mathbf{B}^3\mathbf{U}(1)_{conn}.$$

   this precisely because $\mathbf{B}^2\mathbf{U}(1) {}_{conn}$ should not be the loop space of $\mathbf{B}^3\mathbf{U}(1) {}_{conn}$

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2011
    • PermaLink
    Author: Urs
    Format: MarkdownItexRight, the loop space of $\mathbf{B}^n U(1)_{conn}$ is instead $\mathbf{B}^{n-1}U(1)_{flat}$.

    Right, the loop space of $\mathbf{B}^n U(1)_{conn}$ is instead $\mathbf{B}^{n-1}U(1)_{flat}$.

11. • CommentRowNumber11.
    • CommentAuthordomenico_fiorenza
    • CommentTimeMay 29th 2011
    • (edited May 29th 2011)
    • PermaLink
    Author: domenico_fiorenza
    Format: MarkdownItex> One discussion piece currently missing in the entry is this: I have indicated how the existence of the degree-3 twist on the fiber product of the two torus bundles follows general abstractly. It remains to describe how this is the pullback of either degree-3 twist on each of the torus bundles separately. the Kahle-Valentino argument is a version of the old pinciple "the cup product of a cocycle with a coboundary is a coboundary". this should be said better, but roughtly things goes as follows: let $c:*\to \mathbf{H}(Y,\mathbf{B}G_{conn})$ be the characteristic map of a principal $G$-bundle $E$ with connection $\theta$ on $Y$, and let $0:*\to \mathbf{H}(Y,\mathbf{B}G_{conn})$ be the characteristic map of the trivial $G$-bundle with teh trivial $G$-connection. Then a trivialization of $(E,\theta)$ is a path between $0$ and $c$ in $\mathbf{H}(Y,\mathbf{B}G_{conn})$. Now consieder a T-duality pair $(P,\theta)$ and $(\hat{P},\hat{\theta})$ over $X$ and denote by $\pi:P\to X$ and $\hat{\pi}:\hat{P}\to X$ the projections. since $(P,\theta)$ and $(\hat{P},\hat{\theta})$ are a duality pair, one of our data is a trivialization of $$ (P,\theta)\cup(\hat{P},\hat{\theta}): *\to \mathbf{H}(X,\mathbf{B}^3\mathbf{U}(1)_{conn}), $$ i.e., a path in $\mathbf{H}(X,\mathbf{B}^3\mathbf{U}(1)_{conn})$ between the charcateristic map of $(P,\theta)\cup(\hat{P},\hat{\theta})$ and the characteristic map of the trivial circle 3-bundle over $X$. pulling this back along $\pi^*$ we get a trivialization of $(\pi^*P,\pi^*\theta)\cup(\pi^*\hat{P},\pi^*\hat{\theta})$, i.e., a path in $\mathbf{H}(P,\mathbf{B}^3\mathbf{U}(1)_{conn})$ between the charcateristic map of $(\pi^*P,\pi^*\theta)\cup(\pi^*\hat{P},\pi^*\hat{\theta})$ and the characteristic map of the trivial circle 3-bundle over $P$. On the other hand, we have another natural trivialization of $(\pi^*P,\pi^*\theta)\cup(\pi^*\hat{P},\pi^*\hat{\theta})$. namely, we have a natural trivialization of $(\pi^*P,\pi^*\theta)$ as a torus bundle over $P$ (this is the "with connection" version of the classical fact that the pullback of a principal bundle to the total space of the bundle is naturally trivialized). therefore we have a path in $\mathbf{H}(P,(\mathbf{B}\mathbb{R}^n/\Lambda)_{conn})$ between the characteristic map of $(\pi^*P,\pi^*\theta)$ and the charcateristic map of the trivial bundle. the cup product of this path with the costant path over the characteristic map of $(\pi^*\hat{P},\pi^*\hat{\theta})$ gives a path in $\mathbf{H}(P,\mathbf{B}^3\mathbf{U}(1)_{conn})$ between the characteristic map of the circle 3-bundle with connection over $P$ whose characteristic map is $(\pi^*P,\pi^*\theta)\cup(\pi^*\hat{P},\pi^*\hat{\theta})$ and the trivial circle 3-bundle with connection. Concatenation of these two paths then gives an element of the loop group of $\mathbf{H}(P,\mathbf{B}^3\mathbf{U}(1)_{conn})$ based at the trivial circle 3-bundle, and so an element of $\mathbf{H}(P,\mathbf{B}^2\mathbf{U}(1)_{flat}))$. As I said above this is very rought and should be refined.

    One discussion piece currently missing in the entry is this: I have indicated how the existence of the degree-3 twist on the fiber product of the two torus bundles follows general abstractly. It remains to describe how this is the pullback of either degree-3 twist on each of the torus bundles separately.

    the Kahle-Valentino argument is a version of the old pinciple “the cup product of a cocycle with a coboundary is a coboundary”. this should be said better, but roughtly things goes as follows: let $c:*\to \mathbf{H}(Y,\mathbf{B}G_{conn})$ be the characteristic map of a principal $G$-bundle $E$ with connection $\theta$ on $Y$, and let $0:*\to \mathbf{H}(Y,\mathbf{B}G_{conn})$ be the characteristic map of the trivial $G$-bundle with teh trivial $G$-connection. Then a trivialization of $(E,\theta)$ is a path between $0$ and $c$ in $\mathbf{H}(Y,\mathbf{B}G_{conn})$.

    Now consieder a T-duality pair $(P,\theta)$ and $(\hat{P},\hat{\theta})$ over $X$ and denote by $\pi:P\to X$ and $\hat{\pi}:\hat{P}\to X$ the projections. since $(P,\theta)$ and $(\hat{P},\hat{\theta})$ are a duality pair, one of our data is a trivialization of

    $$(P,\theta)\cup(\hat{P},\hat{\theta}): *\to \mathbf{H}(X,\mathbf{B}^3\mathbf{U}(1)_{conn}),$$

    i.e., a path in $\mathbf{H}(X,\mathbf{B}^3\mathbf{U}(1)_{conn})$ between the charcateristic map of $(P,\theta)\cup(\hat{P},\hat{\theta})$ and the characteristic map of the trivial circle 3-bundle over $X$. pulling this back along $\pi^*$ we get a trivialization of $(\pi^*P,\pi^*\theta)\cup(\pi^*\hat{P},\pi^*\hat{\theta})$, i.e., a path in $\mathbf{H}(P,\mathbf{B}^3\mathbf{U}(1)_{conn})$ between the charcateristic map of $(\pi^*P,\pi^*\theta)\cup(\pi^*\hat{P},\pi^*\hat{\theta})$ and the characteristic map of the trivial circle 3-bundle over $P$.

    On the other hand, we have another natural trivialization of $(\pi^*P,\pi^*\theta)\cup(\pi^*\hat{P},\pi^*\hat{\theta})$. namely, we have a natural trivialization of $(\pi^*P,\pi^*\theta)$ as a torus bundle over $P$ (this is the “with connection” version of the classical fact that the pullback of a principal bundle to the total space of the bundle is naturally trivialized). therefore we have a path in $\mathbf{H}(P,(\mathbf{B}\mathbb{R}^n/\Lambda)_{conn})$ between the characteristic map of $(\pi^*P,\pi^*\theta)$ and the charcateristic map of the trivial bundle. the cup product of this path with the costant path over the characteristic map of $(\pi^*\hat{P},\pi^*\hat{\theta})$ gives a path in $\mathbf{H}(P,\mathbf{B}^3\mathbf{U}(1)_{conn})$ between the characteristic map of the circle 3-bundle with connection over $P$ whose characteristic map is $(\pi^*P,\pi^*\theta)\cup(\pi^*\hat{P},\pi^*\hat{\theta})$ and the trivial circle 3-bundle with connection.

    Concatenation of these two paths then gives an element of the loop group of $\mathbf{H}(P,\mathbf{B}^3\mathbf{U}(1)_{conn})$ based at the trivial circle 3-bundle, and so an element of $\mathbf{H}(P,\mathbf{B}^2\mathbf{U}(1)_{flat}))$.

    As I said above this is very rought and should be refined.

12. • CommentRowNumber12.
    • CommentAuthordomenico_fiorenza
    • CommentTimeMay 29th 2011
    • PermaLink
    Author: domenico_fiorenza
    Format: MarkdownItex> Right, the loop space of $\mathbf{B}^n U(1)_{conn}$ is instead $\mathbf{B}^{n-1}U(1)_{flat}$. Cool! I just edited accordingly post 11 above.

    Right, the loop space of $\mathbf{B}^n U(1)_{conn}$ is instead $\mathbf{B}^{n-1}U(1)_{flat}$.

    Cool! I just edited accordingly post 11 above.

13. • CommentRowNumber13.
    • CommentAuthordomenico_fiorenza
    • CommentTimeMay 30th 2011
    • PermaLink
    Author: domenico_fiorenza
    Format: MarkdownItexsince I wasn't able to convince myself that the construction I sketch in post 11 is compleletly canonical, I went trough the proof of Kahle-Valentino's lemma 2.2, and I'm not sure about the details of it: in the notations of the paper it is claimes that $\hat{\pi}^*\delta_{\mathcal{P}}\cdot \pi^*\hat{\delta}_{\hat{\mathcal{P}}}:\hat{\pi}^*\tau\to \pi^*\hat{\tau}$. in the proof it is said that pull-backs are omitted for larity of notation; yet, with the explicit pullbacks written out, the term $\sigma-\sigma$ in their formula would be $\pi^*\sigma-\hat{\pi}^*\sigma$, and I don't see how this should canonically cancel. I'm asking Alessandro Valentino about this.

    since I wasn’t able to convince myself that the construction I sketch in post 11 is compleletly canonical, I went trough the proof of Kahle-Valentino’s lemma 2.2, and I’m not sure about the details of it: in the notations of the paper it is claimes that $\hat{\pi}^*\delta_{\mathcal{P}}\cdot \pi^*\hat{\delta}_{\hat{\mathcal{P}}}:\hat{\pi}^*\tau\to \pi^*\hat{\tau}$. in the proof it is said that pull-backs are omitted for larity of notation; yet, with the explicit pullbacks written out, the term $\sigma-\sigma$ in their formula would be $\pi^*\sigma-\hat{\pi}^*\sigma$, and I don’t see how this should canonically cancel. I’m asking Alessandro Valentino about this.

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2011
    • PermaLink
    Author: Urs
    Format: MarkdownItexThanks for looking into this. Maybe I should clarify: what I meant that I am after is a general abstrac construciton of the twisting cocycle on the separate torus bundles: as I discuss in the entry, the twisting cocycle on the fiber product of the two torus bundles is simply the canonical map out of a certain canonical homotopy pullback. I was looking for a similar general abstract construction of the cocycles on the separate torus bundles. On the other hand, I also still need to have a closer look at the construciton: are the twists on the separate torus bundles strictly needed for the construction? Or isn't it only the one on their fiber product that is relevant? I don't have more time to look into this right this moment, though.

    Thanks for looking into this.

    Maybe I should clarify: what I meant that I am after is a general abstrac construciton of the twisting cocycle on the separate torus bundles: as I discuss in the entry, the twisting cocycle on the fiber product of the two torus bundles is simply the canonical map out of a certain canonical homotopy pullback. I was looking for a similar general abstract construction of the cocycles on the separate torus bundles.

    On the other hand, I also still need to have a closer look at the construciton: are the twists on the separate torus bundles strictly needed for the construction? Or isn’t it only the one on their fiber product that is relevant?

    I don’t have more time to look into this right this moment, though.

15. • CommentRowNumber15.
    • CommentAuthordomenico_fiorenza
    • CommentTimeMay 30th 2011
    • PermaLink
    Author: domenico_fiorenza
    Format: MarkdownItex> Maybe I should clarify [...] I was looking for a similar general abstract construction of the cocycles on the separate torus bundles yes, this was clear. what I tried to do in post 11 was to carry out the construction of the twisting cocycles on the separate torus bundles at the highest possible canonical level. and the fact that this construction was by far more obscure and involved than the one of the twisting cocycle on the fiber product was sugegsting me that there was something odd going on there. so I decided to look at the role played by the twists on the separate torus bundles, to find out that with these twists involved I'm not abel to see how a piece of the construction should be canonical (this is post 13). so I'm convinced only the twisting cocycle on the fiber product will be relevant when things will be understood correctly. I have to think more on this.

    Maybe I should clarify […] I was looking for a similar general abstract construction of the cocycles on the separate torus bundles

    yes, this was clear. what I tried to do in post 11 was to carry out the construction of the twisting cocycles on the separate torus bundles at the highest possible canonical level. and the fact that this construction was by far more obscure and involved than the one of the twisting cocycle on the fiber product was sugegsting me that there was something odd going on there. so I decided to look at the role played by the twists on the separate torus bundles, to find out that with these twists involved I’m not abel to see how a piece of the construction should be canonical (this is post 13).

    so I’m convinced only the twisting cocycle on the fiber product will be relevant when things will be understood correctly. I have to think more on this.

16. • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2011
    • PermaLink
    Author: Urs
    Format: MarkdownItex> so I'm convinced only the twisting cocycle on the fiber product will be relevant when things will be understood correctly. I have to think more on this. Okay, great. That sounds good. Did you hear back from Alessandro?

    so I’m convinced only the twisting cocycle on the fiber product will be relevant when things will be understood correctly. I have to think more on this.

    Okay, great. That sounds good. Did you hear back from Alessandro?

17. • CommentRowNumber17.
    • CommentAuthordomenico_fiorenza
    • CommentTimeMay 30th 2011
    • PermaLink
    Author: domenico_fiorenza
    Format: MarkdownItexnot yet. I'll report here as I'll hear from him

    not yet. I’ll report here as I’ll hear from him

18. • CommentRowNumber18.
    • CommentAuthordomenico_fiorenza
    • CommentTimeMay 31st 2011
    • PermaLink
    Author: domenico_fiorenza
    Format: MarkdownItexno reply from Alessandro, yet, but I looked back at what I wrote and it was absolute nonsense (I also wrote this to Alessandro, so in the end there will be no reply at all :) ). namely, I was forgetting a further pullback and the meaningless formula $\pi^*\sigma-\hat{\pi}^*\sigma$ I was writing in post 13 should have been $\hat{\pi}^*\pi^*\sigma-\pi^*\hat{\pi}^*\sigma$, which cancels by the very definition of fibered product. but then this makes everything clear, avoiding the twisting cocycles on the separate torus bundles which were confusing us: on $P\times_X\hat{P}$ we have two 3-circle bundles with _two_ isomorphisms between them! indeed, the two 3-circle bundles I'm referring to are the two pullbacks via $\pi^*\hat{\pi}^*$ and $\hat{\pi}^*\pi^*$ of the 3-circle bundle over $X$. they are isomorphic by the isomorphism induced by the very definition of pullback. but they are also isomorphic via the composition of the trivializations induced by the trivialization $\sigma$ of the circle 3-bundle over $X$. all this gives a natural loop (based at the trivial circle 3-bundle) in the space of circle 3-bundles over $P\times_X\hat{P}$, i.e., a natural circle 2-bundle over $P\times_X\hat{P}$. and this is the canonical circle 2-bundle one can construct from the fiber sequence $DPairs(X)\to \mathbf{B}(\mathbb{R}^n/\Lambda)\times \mathbf{B}((\mathbb{R}^n)^*/\hat{\Lambda})\to \mathbf{B}^3\mathbf{U}(1)$.

    no reply from Alessandro, yet, but I looked back at what I wrote and it was absolute nonsense (I also wrote this to Alessandro, so in the end there will be no reply at all :) ). namely, I was forgetting a further pullback and the meaningless formula $\pi^*\sigma-\hat{\pi}^*\sigma$ I was writing in post 13 should have been $\hat{\pi}^*\pi^*\sigma-\pi^*\hat{\pi}^*\sigma$, which cancels by the very definition of fibered product.

    but then this makes everything clear, avoiding the twisting cocycles on the separate torus bundles which were confusing us: on $P\times_X\hat{P}$ we have two 3-circle bundles with two isomorphisms between them! indeed, the two 3-circle bundles I’m referring to are the two pullbacks via $\pi^*\hat{\pi}^*$ and $\hat{\pi}^*\pi^*$ of the 3-circle bundle over $X$. they are isomorphic by the isomorphism induced by the very definition of pullback. but they are also isomorphic via the composition of the trivializations induced by the trivialization $\sigma$ of the circle 3-bundle over $X$.

    all this gives a natural loop (based at the trivial circle 3-bundle) in the space of circle 3-bundles over $P\times_X\hat{P}$, i.e., a natural circle 2-bundle over $P\times_X\hat{P}$. and this is the canonical circle 2-bundle one can construct from the fiber sequence $DPairs(X)\to \mathbf{B}(\mathbb{R}^n/\Lambda)\times \mathbf{B}((\mathbb{R}^n)^*/\hat{\Lambda})\to \mathbf{B}^3\mathbf{U}(1)$.