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This answer is only a bit of a sketch for (2), because it's been a while since I thought/heard about this. And I interpret 'physical considerations' loosely, since this all comes from string theory, whose status as a physical theory is open to debate (at the very least, nothing below comes from experiments!).

There is, in M-theory, a locally-defined 3-form 'higher connection', whose 'curvature' is a 4-form. This is related to the existence of a bundle 2-gerbe/circle 3-bundle whose characteristic class is this 4-form. Really all that people see of this is the low-energy limit which is 11-dimensional supergravity. Since spacetime $X^{11}$ as conceived in M-theory is 11-dimensional, any classifying map $X^{11} \to K(\mathbb{Z},4)$ for one of these higher structures wouldn't be able to distinguish $K(\mathbb{Z},4)$ from $BE_8$. Thus the bundle 2-gerbe might secretly be an $E_8$-bundle.

Alternatively, in regular string theory one has the Kalb-Ramond field and its $H$-flux, which is 3-form, and is commonly understood to be the curvature 3-form associated to a bundle gerbe/circle 2-bundle. However exactly the same argument as before means that perhaps what is going on is that there is an $\Omega E_8$-bundle instead of a bundle gerbe (this is the point of view espoused to me by Jarah Evslin a few years back). Due to the heterotic string theory $E_8\times E_8$, and various models with compactifications, it is not wholly unreasonable to expect $\Omega E_8$ to turn up at some point.

It should be noted that I don't think that these ideas are widely accepted, though.

If you're really interested in this, look at the work of Evslin (e.g. this which I think kicked off the observation you mention, based on an observation by Hoˇrava which relies on work by Witten, and Dionescu, Moore and Witten on which the paper by Dionescu, Moore and Freed that Jeff cites develops) and also Hisham Sati (e.g. this, joint with Evslin, or this) Sati in particular does a lot of work on anomaly cancellation in M-theory and what one might call 'higher characteristic classes'. Of course, I cannot avoid mentioning Urs Schreiber's work (together with Jim Stasheff and Sati) on higher gauge theory.

Edit: I add that the only source I have for this is various talks at least half a decade ago. There is substantial work by others that I was previously unaware of, including Bergman and Varadarajan's hep-th/0406218I'm sure I have no idea about who did what.

As far as (1) goes, which version of $E_8$ are you after? Assuming it is a compact real Lie group, it is 2-connected and we have $\pi_3(E_8) = \mathbb{Z}$. After that I'm not sure how the homotopy groups are calculated. Morse theory?

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This answer is only a bit of a sketch for (2), because it's been a while since I thought/heard about this. And I interpret 'physical considerations' loosely, since this all comes from string theory, whose status as a physical theory is open to debate (at the very least, nothing below comes from experiments!).

There is, in M-theory, a locally-defined 3-form 'higher connection', whose 'curvature' is a 4-form. This is related to the existence of a bundle 2-gerbe/circle 3-bundle whose characteristic class is this 4-form. Really all that people see of this is the low-energy limit which is 11-dimensional supergravity. Since spacetime $X^{11}$ as conceived in M-theory is 11-dimensional, any classifying map $X^{11} \to K(\mathbb{Z},4)$ for one of these higher structures wouldn't be able to distinguish $K(\mathbb{Z},4)$ from $BE_8$. Thus the bundle 2-gerbe might secretly be an $E_8$-bundle.

Alternatively, in regular string theory one has the Kalb-Ramond field and its $H$-flux, which is 3-form, and is commonly understood to be the curvature 3-form associated to a bundle gerbe/circle 2-bundle. However exactly the same argument as before means that perhaps what is going on is that there is an $\Omega E_8$-bundle instead of a bundle gerbe (this is the point of view espoused to me by Jarah Evslin a few years back). Due to the heterotic string theory $E_8\times E_8$, and various models with compactifications, it is not wholly unreasonable to expect $\Omega E_8$ to turn up at some point.

It should be noted that I don't think that these ideas are widely accepted, though.

If you're really interested in this, look at the work of Evslin (e.g. this which I think kicked off the observation you mention, based on an observation by Hoˇrava which relies on work by Witten, and Dionescu, Moore and Witten on which the paper by Dionescu, Moore and Freed that Jeff cites develops) and also Hisham Sati (e.g. this, joint with Evslin, or this) Sati in particular does a lot of work on anomaly cancellation in M-theory and what one might call 'higher characteristic classes'. Of course, I cannot avoid mentioning Urs Schreiber's work (together with Jim Stasheff and Sati) on higher gauge theory.

Edit: I add that the only source I have for this is various talks at least half a decade ago. There is substantial work by others that I was previously unaware of, (e.g. nLab)including Bergman and Varadarajan's hep-th/0406218.

As far as (1) goes, which version of $E_8$ are you after? Assuming it is a compact real Lie group, it is 2-connected and we have $\pi_3(E_8) = \mathbb{Z}$. After that I'm not sure how the homotopy groups are calculated. Morse theory?

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This answer is only a bit of a sketch for (2), because it's been a while since I thought/heard about this. And I interpret 'physical considerations' loosely, since this all comes from string theory, whose status as a physical theory is open to debate (at the very least, nothing below comes from experiments!).

There is, in M-theory, a locally-defined 3-form 'higher connection', whose 'curvature' is a 4-form. This is related to the existence of a bundle 2-gerbe/circle 3-bundle whose characteristic class is this 4-form. Really all that people see of this is the low-energy limit which is 11-dimensional supergravity. Since spacetime $X^{11}$ as conceived in M-theory is 11-dimensional, any classifying map $X^{11} \to K(\mathbb{Z},4)$ for one of these higher structures wouldn't be able to distinguish $K(\mathbb{Z},4)$ from $BE_8$. Thus the bundle 2-gerbe might secretly be an $E_8$-bundle.

Alternatively, in regular string theory one has the Kalb-Ramond field and its $H$-flux, which is 3-form, and is commonly understood to be the curvature 3-form associated to a bundle gerbe/circle 2-bundle. However exactly the same argument as before means that perhaps what is going on is that there is an $\Omega E_8$-bundle instead of a bundle gerbe (this is the point of view espoused to me by Jarah Evslin a few years back). Due to the heterotic string theory $E_8\times E_8$, and various models with compactifications, it is not wholly unreasonable to expect $\Omega E_8$ to turn up at some point.

It should be noted that I don't think that these ideas are widely accepted, though.

If you're really interested in this, look at the work of Evslin (e.g. this which I think kicked off the observation you mention, based on an observation by Hoˇrava which relies on work by Witten, and Dionescu, Moore and Witten on which the paper by Dionescu, Moore and Freed that Jeff cites develops) and also Hisham Sati (e.g. this, joint with Evslin, or this) Sati in particular does a lot of work on anomaly cancellation in M-theory and what one might call 'higher characteristic classes'. Of course, I cannot avoid mentioning Urs Schreiber's work (together with Jim Stasheff and Sati) on higher gauge theory, (e.g. nLab)

As far as (1) goes, which version of $E_8$ are you after? Assuming it is a compact real Lie group, it is 2-connected and we have $\pi_3(E_8) = \mathbb{Z}$. After that I'm not sure how the homotopy groups are calculated. Morse theory?

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