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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeAug 25th 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItexFinally I am starting an entry _[[Platonic 2-group]]_. For the moment, all it has is the statement of [Epa-Ganter 16, prop. 4.1](https://ncatlab.org/nlab/show/Platonic+2-group#EpaGanter16), rephrased as the diagram $$ \array{ \mathcal{G}_{uni}[i] &\longrightarrow& \mathbf{B}G_{ADE} &\longrightarrow& \mathbf{B}^3 \mathbb{Z}/{\vert G_{ADE}\vert} \\ \downarrow && \downarrow && \downarrow \\ String(SU(2)) &\longrightarrow& \mathbf{B} SU(2) &\underset{\mathbf{c}_2}{\longrightarrow}& \mathbf{B}^3 U(1) } $$

   Finally I am starting an entry Platonic 2-group.

   For the moment, all it has is the statement of Epa-Ganter 16, prop. 4.1, rephrased as the diagram

   $$\array{ \mathcal{G}_{uni}[i] &\longrightarrow& \mathbf{B}G_{ADE} &\longrightarrow& \mathbf{B}^3 \mathbb{Z}/{\vert G_{ADE}\vert} \\ \downarrow && \downarrow && \downarrow \\ String(SU(2)) &\longrightarrow& \mathbf{B} SU(2) &\underset{\mathbf{c}_2}{\longrightarrow}& \mathbf{B}^3 U(1) }$$
2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 25th 2016
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   Author: David_Corfield
   Format: MarkdownItexI added a comment on what the $i$ in your diagram means. It's reminiscent of my quest to find a basic 2-group for Klein 2-geometry. I was looking for copies of some group sitting on the vertices of a cube, and then wondering about symmetries.

   I added a comment on what the $i$ in your diagram means.

   It’s reminiscent of my quest to find a basic 2-group for Klein 2-geometry. I was looking for copies of some group sitting on the vertices of a cube, and then wondering about symmetries.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeAug 25th 2016
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   Author: Urs
   Format: MarkdownItex> I added a comment on what the ii in your diagram means. Thanks! I also added that to the diagram, and harmonized the notation a little.

   I added a comment on what the ii in your diagram means.

   Thanks! I also added that to the diagram, and harmonized the notation a little.

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeOct 8th 2018
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   Author: David_Corfield
   Format: MarkdownItexIdle thought in passing: can these Platonic 2-groups be lifted to the fivebrane 6-group?

   Idle thought in passing: can these Platonic 2-groups be lifted to the fivebrane 6-group?

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeOct 9th 2018
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   Author: Urs
   Format: MarkdownItexIt sure looks like the pattern wants to continue. But since the Platonic n-groups are, crucially, not just the pullbacks of the Whitehead stages, it is not evident if it will work.

   It sure looks like the pattern wants to continue. But since the Platonic n-groups are, crucially, not just the pullbacks of the Whitehead stages, it is not evident if it will work.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJul 5th 2021
   • (edited Jul 5th 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have recovered the pdf for * [[Narthana Epa]], _Platonic 2-groups_, 2010 ([pdf](https://ncatlab.org/nlab/files/Epa_Platonic2Groups.pdf)) via the WaybackMachine, and have now uploaded it to the $n$Lab server. <a href="https://ncatlab.org/nlab/revision/diff/Platonic+2-group/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/Platonic+2-group/8">v8</a>, <a href="https://ncatlab.org/nlab/show/Platonic+2-group">current</a>

   I have recovered the pdf for

   • Narthana Epa, Platonic 2-groups, 2010 (pdf)

   via the WaybackMachine, and have now uploaded it to the $n$Lab server.

   diff, v8, current

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJul 9th 2021
   • (edited Jul 9th 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI think we finally understand what the main [Epa-Ganter result](https://ncatlab.org/nlab/show/finite+rotation+group#RestrictingTheSecondChernClassFromSU2ToFiniteSubgroup) *means* in light of *[[schreiber:Hypothesis H|Hypothesis H]]*. Namely, recalling ([pp. 38](https://arxiv.org/pdf/1904.10207.pdf#page=38) in [arXiv:1904.10207](https://ncatlab.org/schreiber/show/Twisted+Cohomotopy+implies+M-theory+anomaly+cancellation+on+8-manifolds)) that, by Hypothesis H, the integral shifted C-field flux is witnessed by the universal characteristic class $$ vol_{S^4} + \tfrac{1}{4}p_1 \;\;\in\;\; H^4\big( S^{\mathbb{H}} \!\sslash\! Sp(2);\, \mathbb{Z} \big) $$ on Borel-equivariant 4-Cohomotopy (which, as usual, is named after its rational image, where it is the sum of the fiberwise volume form on the 4-sphere with one quarter of the first Pontrjagin class). Now given an ADE-singularity controlled by a finite subgroup inclusion $$ G \overset{i}{\hookrightarrow} Sp(1) \xhookrightarrow{ q \mapsto \left( \array{ q & 0 \\ 0 & 1} \right) } Sp(2) $$ we are to ask how this class restricts to the $G$-equivariant Cohomotopy. Using Epa-Ganter and [this SS-computation](https://ncatlab.org/nlab/show/Serre%20spectral%20sequence#IntegralCohomologyOfHomotopyQuotientOf4SphereByADEAction), one finds, I think, that the result is $$ i^\ast \big( \tfrac{1}{2}\chi_4 + \tfrac{1}{4}p_1 \big) \;=\; (1, [1]) \;\;\in\;\; \mathbb{Z} \oplus \mathbb{Z}_{\left\vert G \right\vert} \;\simeq\; H^4\big( S^4 \!\sslash\! G;\, \mathbb{Z} \big) \,. $$ This means, in particular, that on a flat $G$-orbifold with vanishing 4-flux form, the residual integral *background 4-flux* is the generator $[1] \in H^4\big( B G;\, \mathbb{Z} \big)$, which classifies the *universal* $B S^1$ 2-group extension of the Platonic group $G$ that Epa-Ganter discuss.

   I think we finally understand what the main Epa-Ganter result means in light of Hypothesis H.

   Namely, recalling (pp. 38 in arXiv:1904.10207) that, by Hypothesis H, the integral shifted C-field flux is witnessed by the universal characteristic class

   $$vol_{S^4} + \tfrac{1}{4}p_1 \;\;\in\;\; H^4\big( S^{\mathbb{H}} \!\sslash\! Sp(2);\, \mathbb{Z} \big)$$

   on Borel-equivariant 4-Cohomotopy (which, as usual, is named after its rational image, where it is the sum of the fiberwise volume form on the 4-sphere with one quarter of the first Pontrjagin class).

   Now given an ADE-singularity controlled by a finite subgroup inclusion

   $$G \overset{i}{\hookrightarrow} Sp(1) \xhookrightarrow{ q \mapsto \left( \array{ q & 0 \\ 0 & 1} \right) } Sp(2)$$

   we are to ask how this class restricts to the $G$-equivariant Cohomotopy. Using Epa-Ganter and this SS-computation, one finds, I think, that the result is

   $$i^\ast \big( \tfrac{1}{2}\chi_4 + \tfrac{1}{4}p_1 \big) \;=\; (1, [1]) \;\;\in\;\; \mathbb{Z} \oplus \mathbb{Z}_{\left\vert G \right\vert} \;\simeq\; H^4\big( S^4 \!\sslash\! G;\, \mathbb{Z} \big) \,.$$

   This means, in particular, that on a flat $G$-orbifold with vanishing 4-flux form, the residual integral background 4-flux is the generator $[1] \in H^4\big( B G;\, \mathbb{Z} \big)$, which classifies the universal $B S^1$ 2-group extension of the Platonic group $G$ that Epa-Ganter discuss.

8. • CommentRowNumber8.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 9th 2021
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   Author: David_Corfield
   Format: MarkdownItexThose Platonic 2-groups were part of a program with ambitious goals, including a Monster 2-group: * (with [[Matthew Ando]]), _Subtle symmetries and the refined Monster_, ([pdf](https://researchers.ms.unimelb.edu.au/~nganter@unimelb/research_program_ganter.pdf)) Is there any sign of further contact with Hypothesis H?

   Those Platonic 2-groups were part of a program with ambitious goals, including a Monster 2-group:

   • (with Matthew Ando), Subtle symmetries and the refined Monster, (pdf)

   Is there any sign of further contact with Hypothesis H?

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeJul 10th 2021
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   Author: Urs
   Format: MarkdownItexI don't currently know what relates the finite subgroups of $SU(2)$ to the Monster, other than that both carry interesting 3-cocycles (when regarded with $S^1$ coefficients -- the "Moonshine cocycle" in the case of the Monster) whose transgression to 2-cocycles on their inertia groupoids is of interest. For the finite subgroups of $SU(2)$ these 3-cocycles are naturally understood, via #2, as background C-fields in M-theory on ADE-orbifolds, which conceptually explains the relevance of their transgression to the inertia: That's the corresponding $H_3$-field! For the Monster the key observation seems to be that the transgression of the Moonshine cocycle to [[Huan's inertia orbifold]] of the Monster has structural properties similar to those appearing in Norton's Moonschine Conjecture. That's Observation 2.1.3 in Nora's [arXiv:1301.2754](https://arxiv.org/abs/1301.2754), in view of the more recent re-interpretation of her "rotation condition" by way of Huan's inertia referenced [here](https://ncatlab.org/nlab/show/Tate+K-theory#IdeaEquivariantVersion). The review in [Dove 19](https://arxiv.org/abs/1912.02374) provides a good cleaned-up account with comprehensive referencing. So that's interesting, but also mysterious (to me). The Monster should be symmetries of a heterotic string background, but not spacetime orbifold singularities as for the ADE-subgroups above. Are we to think of the Moonshine cocycle as an aspect of the C-field in heterotic M-theory? I don't see that but I really don't know.

   I don’t currently know what relates the finite subgroups of $SU(2)$ to the Monster, other than that both carry interesting 3-cocycles (when regarded with $S^1$ coefficients – the “Moonshine cocycle” in the case of the Monster) whose transgression to 2-cocycles on their inertia groupoids is of interest.

   For the finite subgroups of $SU(2)$ these 3-cocycles are naturally understood, via #2, as background C-fields in M-theory on ADE-orbifolds, which conceptually explains the relevance of their transgression to the inertia: That’s the corresponding $H_3$-field!

   For the Monster the key observation seems to be that the transgression of the Moonshine cocycle to Huan’s inertia orbifold of the Monster has structural properties similar to those appearing in Norton’s Moonschine Conjecture. That’s Observation 2.1.3 in Nora’s arXiv:1301.2754, in view of the more recent re-interpretation of her “rotation condition” by way of Huan’s inertia referenced here. The review in Dove 19 provides a good cleaned-up account with comprehensive referencing.

   So that’s interesting, but also mysterious (to me). The Monster should be symmetries of a heterotic string background, but not spacetime orbifold singularities as for the ADE-subgroups above. Are we to think of the Moonshine cocycle as an aspect of the C-field in heterotic M-theory? I don’t see that but I really don’t know.

10. • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 10th 2021
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    Author: David_Corfield
    Format: MarkdownItexThanks! On another point, funny to see as part of the Klein 2-geometry [series](https://golem.ph.utexas.edu/category/2007/05/klein_2geometry_viiis.html) I was looking for something for a categorified platonic group to act on.

    Thanks!

    On another point, funny to see as part of the Klein 2-geometry series I was looking for something for a categorified platonic group to act on.

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJul 10th 2021
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    Author: Urs
    Format: MarkdownItexJust to recall that, generally, the $String(G)$ 2-group (of which the "Platonic 2-groups" are the restriction to finite subgroups of $SU(2)$) is the [[Heisenberg 2-group]] of the WZW-model on $G$, acting by 2-symplectomorphisms covering the left action of $G$ on itself. This is on [p. 42](https://arxiv.org/pdf/1304.0236.pdf#page=42) of our [arXiv:1304.0236](https://arxiv.org/abs/1304.0236), which Nora Ganter is referring to in the note you mention in #8. Another interesting question here is: The transgression of the corresponding group 3-cocycle on $G$ to a 2-cocycle on the inertia groupoid classifies a degree 3-twist for orbifold K-theory, hence an equivariant bundle of Fredholm operators. Via the transchromatic characted idea this may be understood as the degree-4 twist of Tate-elliptic cohomology, before transgression. But it would be interesting to have the degree 4-twist upstairs classify a twist for elliptic cohomology directly, as some bundle of vertex operators or the like.

    Just to recall that, generally, the $String(G)$ 2-group (of which the “Platonic 2-groups” are the restriction to finite subgroups of $SU(2)$) is the Heisenberg 2-group of the WZW-model on $G$, acting by 2-symplectomorphisms covering the left action of $G$ on itself. This is on p. 42 of our arXiv:1304.0236, which Nora Ganter is referring to in the note you mention in #8.

    Another interesting question here is: The transgression of the corresponding group 3-cocycle on $G$ to a 2-cocycle on the inertia groupoid classifies a degree 3-twist for orbifold K-theory, hence an equivariant bundle of Fredholm operators. Via the transchromatic characted idea this may be understood as the degree-4 twist of Tate-elliptic cohomology, before transgression. But it would be interesting to have the degree 4-twist upstairs classify a twist for elliptic cohomology directly, as some bundle of vertex operators or the like.