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On cubic hypersurfaces of dimension seven and eight
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abstract
Cubic sevenfolds are examples of Fano manifolds of Calabi-Yau type. We study them in relation with the Cartan cubic, the $E_6$-invariant cubic in $\PP^{26}$. We show that a generic cubic sevenfold $X$ can be described as a linear section of the Cartan cubic, in finitely many ways. To each such "Cartan representation" we associate a rank nine vector bundle on $X$ with very special cohomological properties. In particular it allows to define auto-equivalences of the non-commutative Calabi-Yau threefold associated to $X$ by Kuznetsov. Finally we show that the generic eight dimensional section of the Cartan cubic is rational.
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Special Fano geometry from Feynman integrals
Special Fano varieties, which include Calabi-Yau spaces as the Q=1 case, arise from the Symanzik polynomials of several families of Feynman integrals.
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