The Grassmannian Gr(2,8) and an E7 surprise

Question

Are there any mathematical explanations for the following surprising facts?

$$\int_{Gr(2,8)} c_{\text{top}}(TX(-2)) = 6556 = \frac{1}{2} \deg(E_7/P(\alpha_7)) + 1,$$ and $$\int_{Gr(2,6)} c_{\text{top}}(TX(-2)) = 143 = \frac{1}{2} \deg(\mathbb{S}_{12}).$$ where $\mathbb{S}_{12}$ is the spinor variety of type $D_6.$ The homogeneous varieties $E_7/P(\alpha_7)$ and $\mathbb{S}_{12}$ appear in Landsberg-Manivel's spaces subexceptional series.

Related surprises appear in supersymmetric gauge theories studied by Dimofte and Gaiotto.

Added (Dec 2018):

$$\int_{Gr(2,5)} c_{\text{top}}(TX(-2)) = 22 = \frac{1}{2} \deg(Gr(3,6)) + 1.$$

This post imported from StackExchange MathOverflow at 2019-03-15 09:55 (UTC), posted by SE-user Richard Eager

Comments

could it be related to the law of small numbers?

This post imported from StackExchange MathOverflow at 2019-03-15 09:55 (UTC), posted by SE-user bump

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6556 is a little too large to make that seem likely.

This post imported from StackExchange MathOverflow at 2019-03-15 09:55 (UTC), posted by SE-user Ben McKay

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The subexceptional series is $\mathrm{LG}(3,6)$, $\mathrm{Gr}(3,6)$, $\mathbb{S}_{12}$ and $\mathrm{E}_7/P_7$. I believe that, related to this question, is the fundamental relation: $$\frac{1}{10} \left( \frac{1}{2} \deg \mathrm{E}_7/P_7 - 2 + \frac{1}{2} \deg \mathbb{S}_{12} - 2 - \frac{1}{2} \deg \mathrm{Gr}(3,6) - 2 - \frac{1}{2} \deg \mathrm{LG}(3,6) - 2 \right) = 666$$

This post imported from StackExchange MathOverflow at 2019-03-15 09:55 (UTC), posted by SE-user Libli

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I don't know notation $E_7/P(\alpha_7)$ - which of the symmetric spaces it is using Cartan notation - see here: en.wikipedia.org/wiki/Symmetric_space ? In $EIII$ we have naturally embedded grassmanian $G_{2,8}$. In $EVI$ we have naturally embedded grassmanian $G_{2,10}$. They can be considered as 1-dimensional projective spaces over $\mathbb C\otimes \mathbb O$ and $\mathbb H\otimes \mathbb O$

This post imported from StackExchange MathOverflow at 2019-03-15 09:55 (UTC), posted by SE-user Marek Mitros

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Sorry I made mistake above. Please replace $G_{2,10}$ with $G_{4,8}$ in my previoius comment. These are oriented grassmanians. So for example $G_{1,8}$ is sphere $S^8$. Normally $+$ sign is added for oriented grassmanian. Grassmanian $G_{2,6}^+$ is quotient $SO_8/U_4$. Namely it is set of complex structures on $R^8$.

This post imported from StackExchange MathOverflow at 2019-03-15 09:55 (UTC), posted by SE-user Marek Mitros