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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 26th 2016
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexSince Urs started [[Freudenthal magic square]], I added [[magic triangle]]. Does the 'magic pyramid' of Duff and colleagues in [A magic pyramid of supergravities](http://arxiv.org/abs/1312.6523) warrant an entry? Hmm, if the triangle extends the square, and the pyramid has the square as a base, shouldn't there be a magic tetrahedron?

   Since Urs started Freudenthal magic square, I added magic triangle. Does the ’magic pyramid’ of Duff and colleagues in A magic pyramid of supergravities warrant an entry?

   Hmm, if the triangle extends the square, and the pyramid has the square as a base, shouldn’t there be a magic tetrahedron?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJan 26th 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItex> Does the 'magic pyramid' of Duff and colleagues in [A magic pyramid of supergravities](http://arxiv.org/abs/1312.6523) warrant an entry? Yes, just yesterday I was talking about this with John Huerta (I am in Lisbon this week) and I thought that we'd need an entry on this. Right now I am busy with something else. But if you feel like making a start, that would be much appreciated.

   Does the ’magic pyramid’ of Duff and colleagues in A magic pyramid of supergravities warrant an entry?

   Yes, just yesterday I was talking about this with John Huerta (I am in Lisbon this week) and I thought that we’d need an entry on this.

   Right now I am busy with something else. But if you feel like making a start, that would be much appreciated.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJan 26th 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItexRegarding what you write in the entry: on the face of it it sounds strange to say that a square has been "extended" to a triangle. Is that the usual was this is phrased? (Just expressing my ignorance here.)

   Regarding what you write in the entry: on the face of it it sounds strange to say that a square has been “extended” to a triangle. Is that the usual was this is phrased? (Just expressing my ignorance here.)

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 26th 2016
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexA $4 \times 4$ square contained in an $8 \times 8$-triangle. Deligne and Gross >The subgroups which occur form a Magic Triangle, which extends Freudenthal's Magic Square of Lie algebras.

   A $4 \times 4$ square contained in an $8 \times 8$-triangle.

   Deligne and Gross

   The subgroups which occur form a Magic Triangle, which extends Freudenthal’s Magic Square of Lie algebras.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJan 26th 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks. Could you say that in the entry?

   Thanks. Could you say that in the entry?

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 26th 2016
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexOK, I added some more bits and pieces.

   OK, I added some more bits and pieces.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeSep 17th 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItexre #1, #2: I have started _[[magic supergravity]]_ with some bare minimum. Mainly a collection of references.

   re #1, #2: I have started magic supergravity with some bare minimum. Mainly a collection of references.

8. • CommentRowNumber8.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 18th 2016
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI really haven't followed all this, but what of #4 Deligne and Gross embedding that base square into a larger triangle? Is this interest now linked to your work with John Huerta?

   I really haven’t followed all this, but what of #4 Deligne and Gross embedding that base square into a larger triangle?

   Is this interest now linked to your work with John Huerta?

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeSep 18th 2016
   • (edited Sep 18th 2016)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> but what of #4 Deligne and Gross embedding that base square into a larger triangle? I still don't know, I have no good grasp of these triangles. > Is this interest now linked to your work with John Huerta? It's part of my general quest of understanding if there is a theory of gravity that sort of canonically emanates from the superpoint. With John I have been talking about some of these things, and he worked out answers to some questions that I had. So, remember, the beginning of the story is the observation in _[[schreiber:Modern Physics formalized in Modal Homotopy Type Theory]]_ in the section _[Space-time-matter](https://ncatlab.org/schreiber/show/Modern+Physics+formalized+in+Modal+Homotopy+Type+Theory#spacetimematter)_, that once the progression on modalities has completed its cycle, we are left with three special objects: the real line $\mathbb{R} = \mathbb{R}^{1\vert 0}$, the odd line $\mathbb{R}^{0\vert 1}$ and the infinitesimal disks $\mathbb{D}$. Namely these "represent" the three modalities on the vertical axis of the progression, in that these are given by universally contracting these objects to points: $$ \array{ Rh & \simeq & L_{\mathbb{R}^{0 \vert 1}} \\ \vee \\ \Im & \simeq & L_{\mathbb{D}} L_{\mathbb{R}^{0 \vert 1}} \\ \vee \\ \int & \simeq & L_{\mathbb{R}} L_{\mathbb{D}} L_{\mathbb{R}^{0 \vert 1}} } $$ Now the idea is that we apply the magnifying class of homotopy theory to look what structures we find inside these objects, by computing something like their Whitehead tower. So we consider all 2-cocycles, then pass to the extension these classify, then all 3-cocycles, and so forth. Something like this. On $\mathbb{R}^{0\vert 1}$ we find a single 2-cocycle up to scale. The extension that it classifies is the super-translation group $\mathbb{R}^{1\vert 1}$. On $\mathbb{R}^{0\vert 2 } = \mathbb{R}^{0\vert 1} \times \mathbb{R}^{0\vert 1}$ we find a a 3-dimensional space of 2-cocycles. The extension that these classify is the super-Minowski spacetime group $\mathbb{R}^{3\vert 2}$. On $\mathbb{R}^{3\vert 4} = \mathbb{R}^{3\vert 2} \underset{\mathbb{R}^3}{\times} \mathbb{R}^{3\vert 2}$ we find another 2-cocycle, up to scale. The extension that this classifies is the super-Minkowski spacetime group $\mathbb{R}^{4\vert 4}$ which underlies 4d supergravity. In this story one may replace the real numbers with any real normed division algebra $\mathbb{A}$. IF one then restricts attention to 2-cocycles on $\mathbb{A}^{0\vert 2}$ that are _sesquilinear_, then the same story yields the super-Minkowski spacetimes in dimensions 3,4,6,10 and 4,5,7,11, respectively. This is because one sees that this story of 2-cocycles classifying extensions of superpoints gives just another perspective on the familiar relation between [[supersymmetry and division algebras]]. Based on this, I tried to understand what first principle might make us want to consider those superpoints $\mathbb{A}^{0 \vert 2} = \mathbb{R}^{0 \vert 2 dim(\mathbb{A})}$ with sesquilinear 2-cocycles on them. While it is an impressive re-packaging that just saying "sesquilinear over a real normed division algebra" makes all these super-spacetime groups appear, I presently fail to see how it is more than a neat re-packaging. But then, it's not all that bad to have spacetime dimension 3+1 singled out, and common lore has it that all the higher super-spacetimes, together with their "hidden degrees of freedom" (U-duality, gauge enhancement), is reflected in those "magic" extensions of 3+1 dimensional supergravity anyway, that's more or less what the magic pyramid parameterizes. So now I am thinking that I should exercise trust in where the formalism is leading us, and concentrate more on how the story continues from $\mathbb{R}^{4\vert 4}$ on. The magic pyramid knows at least something about this.

   but what of #4 Deligne and Gross embedding that base square into a larger triangle?

   I still don’t know, I have no good grasp of these triangles.

   Is this interest now linked to your work with John Huerta?

   It’s part of my general quest of understanding if there is a theory of gravity that sort of canonically emanates from the superpoint. With John I have been talking about some of these things, and he worked out answers to some questions that I had.

   So, remember, the beginning of the story is the observation in Modern Physics formalized in Modal Homotopy Type Theory (schreiber) in the section Space-time-matter, that once the progression on modalities has completed its cycle, we are left with three special objects: the real line $\mathbb{R} = \mathbb{R}^{1\vert 0}$, the odd line $\mathbb{R}^{0\vert 1}$ and the infinitesimal disks $\mathbb{D}$. Namely these “represent” the three modalities on the vertical axis of the progression, in that these are given by universally contracting these objects to points:

   $$\array{ Rh & \simeq & L_{\mathbb{R}^{0 \vert 1}} \\ \vee \\ \Im & \simeq & L_{\mathbb{D}} L_{\mathbb{R}^{0 \vert 1}} \\ \vee \\ \int & \simeq & L_{\mathbb{R}} L_{\mathbb{D}} L_{\mathbb{R}^{0 \vert 1}} }$$

   Now the idea is that we apply the magnifying class of homotopy theory to look what structures we find inside these objects, by computing something like their Whitehead tower.

   So we consider all 2-cocycles, then pass to the extension these classify, then all 3-cocycles, and so forth. Something like this.

   On $\mathbb{R}^{0\vert 1}$ we find a single 2-cocycle up to scale. The extension that it classifies is the super-translation group $\mathbb{R}^{1\vert 1}$.

   On $\mathbb{R}^{0\vert 2 } = \mathbb{R}^{0\vert 1} \times \mathbb{R}^{0\vert 1}$ we find a a 3-dimensional space of 2-cocycles. The extension that these classify is the super-Minowski spacetime group $\mathbb{R}^{3\vert 2}$.

   On $\mathbb{R}^{3\vert 4} = \mathbb{R}^{3\vert 2} \underset{\mathbb{R}^3}{\times} \mathbb{R}^{3\vert 2}$ we find another 2-cocycle, up to scale. The extension that this classifies is the super-Minkowski spacetime group $\mathbb{R}^{4\vert 4}$ which underlies 4d supergravity.

   In this story one may replace the real numbers with any real normed division algebra $\mathbb{A}$. IF one then restricts attention to 2-cocycles on $\mathbb{A}^{0\vert 2}$ that are sesquilinear, then the same story yields the super-Minkowski spacetimes in dimensions 3,4,6,10 and 4,5,7,11, respectively. This is because one sees that this story of 2-cocycles classifying extensions of superpoints gives just another perspective on the familiar relation between supersymmetry and division algebras.

   Based on this, I tried to understand what first principle might make us want to consider those superpoints $\mathbb{A}^{0 \vert 2} = \mathbb{R}^{0 \vert 2 dim(\mathbb{A})}$ with sesquilinear 2-cocycles on them. While it is an impressive re-packaging that just saying “sesquilinear over a real normed division algebra” makes all these super-spacetime groups appear, I presently fail to see how it is more than a neat re-packaging.

   But then, it’s not all that bad to have spacetime dimension 3+1 singled out, and common lore has it that all the higher super-spacetimes, together with their “hidden degrees of freedom” (U-duality, gauge enhancement), is reflected in those “magic” extensions of 3+1 dimensional supergravity anyway, that’s more or less what the magic pyramid parameterizes.

   So now I am thinking that I should exercise trust in where the formalism is leading us, and concentrate more on how the story continues from $\mathbb{R}^{4\vert 4}$ on. The magic pyramid knows at least something about this.

10. • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 18th 2016
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexYou always [knew](https://nforum.ncatlab.org/discussion/4973/synthetic-quantum-field-theory/?Focus=40071#Comment_40071) those three kinds of dimension were the basic ones, so it's pleasant to see them appear lined up down that central axis. I wonder if we could ever see why the respective localizations have: two adjoints to the right; one adjoint either side; two adjoints to the left.

    You always knew those three kinds of dimension were the basic ones, so it’s pleasant to see them appear lined up down that central axis. I wonder if we could ever see why the respective localizations have: two adjoints to the right; one adjoint either side; two adjoints to the left.

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 20th 2016
    • (edited Sep 20th 2016)
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have separated _[[magic pyramid]]_ from _[[magic supergravity]]_ (both entries remain stubs for the moment.). Because, even though both are about magic squares in super-gravity and generally closely related, the "magic supergravity" theories are not in the "magic pyramid". (As highlighted on p. 4 of [ABDHN 13](https://ncatlab.org/nlab/show/magic+pyramid#ABDHN13)).

    I have separated magic pyramid from magic supergravity (both entries remain stubs for the moment.). Because, even though both are about magic squares in super-gravity and generally closely related, the “magic supergravity” theories are not in the “magic pyramid”. (As highlighted on p. 4 of ABDHN 13).