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1. • CommentRowNumber1.
   • CommentAuthorLuigi
   • CommentTimeJun 24th 2019
   • (edited Jun 24th 2019)
   • PermaLink
   Author: Luigi
   Format: MarkdownItexHello, I would like to show a poster at String Math 2019 which summarizes the project in geometry of Double Field Theory I have been working on in last months. I want to disclaim: I have a physical and not mathematical background, so I honestly expect various naiveties (especially in the more abstract notions). But I hope there is at least some reasonable core. ~ Here is the [Link draft poster](https://drive.google.com/file/d/11kFnoMXLzeeS7FLHxw0uBiU5W5h0Fsb1/view) Any feedback (or correction) is really appreciated, thanks in advance.

   Hello, I would like to show a poster at String Math 2019 which summarizes the project in geometry of Double Field Theory I have been working on in last months.

   I want to disclaim: I have a physical and not mathematical background, so I honestly expect various naiveties (especially in the more abstract notions). But I hope there is at least some reasonable core.

   ~ Here is the Link draft poster

   Any feedback (or correction) is really appreciated, thanks in advance.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 24th 2019
   • (edited Jun 24th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexLooks good! I gather you will still add pointer to a reference?! (If I may bluntly ask: Would you mind sharing your style file with me? I will also make a poster contribution at Strings2019, but haven't gotten around to setting myself up...) Regarding the final question on your poster: > Can a super non-abelian Higher Kaluza-Klein Theory on the total space of the (twisted) M2/M5-brane gerbe over the 11d super-spacetime geometrize M-theory and its dualities? As you may know we have some results on exactly this, see _[[schreiber:Higher T-duality of super M-branes|Higher T-duality of super M-branes]]_.

   Looks good! I gather you will still add pointer to a reference?!

   (If I may bluntly ask: Would you mind sharing your style file with me? I will also make a poster contribution at Strings2019, but haven’t gotten around to setting myself up…)

   Regarding the final question on your poster:

   Can a super non-abelian Higher Kaluza-Klein Theory on the total space of the (twisted) M2/M5-brane gerbe over the 11d super-spacetime geometrize M-theory and its dualities?

   As you may know we have some results on exactly this, see Higher T-duality of super M-branes.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeJun 24th 2019
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItex>I will also make a poster contribution at Strings2019 You have good evidence that you've cracked M-theory, and you're there just to give a poster!

   I will also make a poster contribution at Strings2019

   You have good evidence that you’ve cracked M-theory, and you’re there just to give a poster!

4. • CommentRowNumber4.
   • CommentAuthorLuigi
   • CommentTimeJun 24th 2019
   • (edited Jun 24th 2019)
   • PermaLink
   Author: Luigi
   Format: MarkdownItex> I gather you will still add pointer to a reference?! Of course, I am doing it right now EDIT: file updated. > (If I may bluntly ask: Would you mind sharing your style file with me? I will also make a poster contribution at String2019, but haven’t gotten around to setting myself up…) [I used this template here](https://v2.overleaf.com/latex/templates/portrait-beamer-poster-template-jacobs-style/fxfzyznxpghw) EDIT: oh, so it is not StringMath in Uppsala > As you may know we have some results on exactly this, see Higher T-duality of super M-branes (schreiber). I know this paper and I am intrigued by your ideas, as far as I can understand them. But I hope considering this groupoid (locally looking like $T^\ast U_\alpha$) as doubled space makes some sense. I hope that, if this picture works, can be directly extended to Exceptional Field Theory. For instance by dimensional reducing the local data of M2/M5 gerbe on a 6-torus-bundle on a 5-dimensional base you should get 15+6 functions on twofold overlaps of patches. Together with the 6 transition functions of the 6-torus-bundle you should get 27 functions on twofold overlaps which is the dimension expected in ExFT for d=5. If you do the same on a 4-torus bundle on a 7-dimensional base you get 10 functions on twofold overlaps, which is still the dimension expected in ExFT for d=7.

   I gather you will still add pointer to a reference?!

   Of course, I am doing it right now

   EDIT: file updated.

   (If I may bluntly ask: Would you mind sharing your style file with me? I will also make a poster contribution at String2019, but haven’t gotten around to setting myself up…)

   I used this template here

   EDIT: oh, so it is not StringMath in Uppsala

   As you may know we have some results on exactly this, see Higher T-duality of super M-branes (schreiber).

   I know this paper and I am intrigued by your ideas, as far as I can understand them.

   But I hope considering this groupoid (locally looking like $T^\ast U_\alpha$) as doubled space makes some sense.

   I hope that, if this picture works, can be directly extended to Exceptional Field Theory. For instance by dimensional reducing the local data of M2/M5 gerbe on a 6-torus-bundle on a 5-dimensional base you should get 15+6 functions on twofold overlaps of patches. Together with the 6 transition functions of the 6-torus-bundle you should get 27 functions on twofold overlaps which is the dimension expected in ExFT for d=5. If you do the same on a 4-torus bundle on a 7-dimensional base you get 10 functions on twofold overlaps, which is still the dimension expected in ExFT for d=7.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJun 24th 2019
   • (edited Jun 24th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> [I used this template here](https://v2.overleaf.com/latex/templates/portrait-beamer-poster-template-jacobs-style/fxfzyznxpghw) Ah, I see. Thanks. > EDIT: oh, so it is not StringMath in Uppsala Sure, my bad. > For instance by dimensional reducing the local data of M2/M5 gerbe on a 6-torus-bundle on a 5-dimensional base you should get 15+6 functions on twofold overlaps of patches. Sorry, say again how you are counting here?

   I used this template here

   Ah, I see. Thanks.

   EDIT: oh, so it is not StringMath in Uppsala

   Sure, my bad.

   For instance by dimensional reducing the local data of M2/M5 gerbe on a 6-torus-bundle on a 5-dimensional base you should get 15+6 functions on twofold overlaps of patches.

   Sorry, say again how you are counting here?

6. • CommentRowNumber6.
   • CommentAuthorLuigi
   • CommentTimeJun 25th 2019
   • (edited Jun 25th 2019)
   • PermaLink
   Author: Luigi
   Format: MarkdownItex> Sorry, say again how you are counting here? Yes, sorry. This is a very rough argument. An M2/M5 gerbe should have as local data a collection of 2-forms $\Lambda_{\alpha\beta}^{M2}$ and 5-forms $\Lambda_{\alpha\beta}^{M5}$ on two-fold overlaps $U_\alpha \cap U_\beta$ of patches of $M$ and so on on $n$-fold overlaps. Now, if $M$ is a $T^6$-bundle on some $5$-dimensional base $M_0$, we can KK reduce every form to $M_0$. So $\Lambda_{\alpha\beta}^{M2}$ will give a 2-form $\Lambda_{\alpha\beta}^{M2(2)}$, six 1-forms $\Lambda_{\alpha\beta}^{M2(1) i }$ and 15 scalars $\Lambda_{\alpha\beta}^{M2(0) i j }$. I do the same with $\Lambda_{\alpha\beta}^{M5}$ to get 6 scalars $\Lambda_{\alpha\beta}^{M5(0) i j k l n }$. Hopefully these scalars can be interpreted as the transition functions of some (probably twisted) bundle over $M_0$. Together with the starting $T^6$-bundle it'd be something locally looking like $U_\alpha\times T^6\times T^{15}\times T^6 = U_\alpha\times T^{27}$. This is the dimension expected for the extended manifold of $d=5$ Exceptional Field Theory. One could define an $E_{6(6)}(\mathbb{Z})$-action for U-duality, etc. The U-duals of the starting bundle should be the so-called U-folds. (The others KK reduced forms should give tensor hierarchy). This is basically the line I'd like to follow for ExFT.

   Sorry, say again how you are counting here?

   Yes, sorry. This is a very rough argument.

   An M2/M5 gerbe should have as local data a collection of 2-forms $\Lambda_{\alpha\beta}^{M2}$ and 5-forms $\Lambda_{\alpha\beta}^{M5}$ on two-fold overlaps $U_\alpha \cap U_\beta$ of patches of $M$ and so on on $n$-fold overlaps.

   Now, if $M$ is a $T^6$-bundle on some $5$-dimensional base $M_0$, we can KK reduce every form to $M_0$. So $\Lambda_{\alpha\beta}^{M2}$ will give a 2-form $\Lambda_{\alpha\beta}^{M2(2)}$, six 1-forms $\Lambda_{\alpha\beta}^{M2(1) i }$ and 15 scalars $\Lambda_{\alpha\beta}^{M2(0) i j }$. I do the same with $\Lambda_{\alpha\beta}^{M5}$ to get 6 scalars $\Lambda_{\alpha\beta}^{M5(0) i j k l n }$.

   Hopefully these scalars can be interpreted as the transition functions of some (probably twisted) bundle over $M_0$. Together with the starting $T^6$-bundle it’d be something locally looking like $U_\alpha\times T^6\times T^{15}\times T^6 = U_\alpha\times T^{27}$. This is the dimension expected for the extended manifold of $d=5$ Exceptional Field Theory. One could define an $E_{6(6)}(\mathbb{Z})$-action for U-duality, etc. The U-duals of the starting bundle should be the so-called U-folds.

   (The others KK reduced forms should give tensor hierarchy).

   This is basically the line I’d like to follow for ExFT.

7. • CommentRowNumber7.
   • CommentAuthorDavid_Corfield
   • CommentTimeJun 25th 2019
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexIn case you're wondering, Luigi (and maybe if you start editing pages on the nLab), one difference with the Itex used here is the need to have spaces, so e.g., those superscripts are italicized via $\Lambda_{\alpha\beta}^{M5(0)i j k l n}$ (\Lambda_{\alpha\beta}^{M5(0)i j k l n}).

   In case you’re wondering, Luigi (and maybe if you start editing pages on the nLab), one difference with the Itex used here is the need to have spaces, so e.g., those superscripts are italicized via $\Lambda_{\alpha\beta}^{M5(0)i j k l n}$ (\Lambda_{\alpha\beta}^{M5(0)i j k l n}).

8. • CommentRowNumber8.
   • CommentAuthorLuigi
   • CommentTimeJun 25th 2019
   • PermaLink
   Author: Luigi
   Format: MarkdownItexThanks!

   Thanks!

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeJun 25th 2019
   • (edited Jun 25th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks. So it seems in this counting you are disregarding some of the KK-modes. For instance the 2-form descendant of the 2-form, and all the higher form descendants of the 5-form. I am used to counting as [here](https://ncatlab.org/nlab/show/E7#SmallestFundamentalRepresentation). But that gives $E_7$-reps, while I gather you are after the $E_6$-reps. Allright. Anyway, I see now what you are saying. Thanks. Yes, I certainly agree that exceptional geometry is all about including the local freedom in the higher gauge fields into the spacetime coordinates, absolutely. I think at the bottom of it is really the fact that $D=11$, $\mathcal{N} = 1$ exceptional super-spacetimes $\mathbb{R}^{10,1\vert \mathbf{32}}_{\mathrm{exc},s}$ (as in [FSS18, 4.6](https://ncatlab.org/schreiber/show/Higher+T-duality+of+super+M-branes), but really being the "hidden supergroup" all the way back in [DF81, Sec. 6](https://ncatlab.org/nlab/files/GeometricSupergravity.pdf)) is the supermanifold version of the classifying space for M5-sigma model fields (as in [FSS19b Prop. 4.31](https://ncatlab.org/schreiber/show/Twisted+Cohomotopy+implies+M-theory+anomaly+cancellation), [FSS19c Prop. 4.4](https://ncatlab.org/schreiber/show/Twisted+Cohomotopy+implies+M5+WZ+term+level+quantization)), which itself is the pullback of the quaternionic Hopf fibration along the map classifying the C-field in Cohomotopy. As illustrated in the graphics [here](https://twitter.com/SchreiberUrs/status/1143467112012406784).

   Thanks. So it seems in this counting you are disregarding some of the KK-modes. For instance the 2-form descendant of the 2-form, and all the higher form descendants of the 5-form. I am used to counting as here. But that gives $E_7$-reps, while I gather you are after the $E_6$-reps. Allright.

   Anyway, I see now what you are saying. Thanks. Yes, I certainly agree that exceptional geometry is all about including the local freedom in the higher gauge fields into the spacetime coordinates, absolutely.

   I think at the bottom of it is really the fact that $D=11$, $\mathcal{N} = 1$ exceptional super-spacetimes $\mathbb{R}^{10,1\vert \mathbf{32}}_{\mathrm{exc},s}$ (as in FSS18, 4.6, but really being the “hidden supergroup” all the way back in DF81, Sec. 6) is the supermanifold version of the classifying space for M5-sigma model fields (as in FSS19b Prop. 4.31, FSS19c Prop. 4.4), which itself is the pullback of the quaternionic Hopf fibration along the map classifying the C-field in Cohomotopy. As illustrated in the graphics here.