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The Tannakian formalism allows to attach to any subvariety of an abelianvariety an algebraic group in a natural way. The arising groups are closelyrelated to moduli questions such as the Schottky problem, but their geometricinterpretation is still mysterious. We show that for the theta divisor on theintermediate Jacobian of a cubic threefold, the Tannaka group is an exceptionalgroup of type E_6. This is the first known exceptional case, and it suggests aconnection with the monodromy of the Gauss map.

Cubic threefolds, Fano surfaces and the monodromy of the Gauss map