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String Modular Motives of Mirrors of Rigid CalabiYau Varieties
, 2009
"... The modular properties of some higher dimensional varieties of special Fano type are analyzed by computing the Lfunction of their Ω−motives. It is shown that the emerging modular forms are string theoretic in origin, derived from the characters of the underlying rational conformal field theory. Th ..."
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The modular properties of some higher dimensional varieties of special Fano type are analyzed by computing the Lfunction of their Ω−motives. It is shown that the emerging modular forms are string theoretic in origin, derived from the characters of the underlying rational conformal field theory. The definition of the class of Fano varieties of special type is motivated by the goal to find candidates for a geometric realization of the mirrors of rigid CalabiYau varieties. We consider explicitly the cubic sevenfold and the quartic fivefold, and show that their motivic Lfunctions agree with the Lfunctions of their rigid mirror CalabiYau varieties. We also show that the cubic fourfold is string theoretic, with a modular form that is determined by that of an exactly solvable K3 surface.
Emergent Spacetime from Modular Motives
, 2008
"... The program of constructing spacetime geometry from string theoretic modular forms is extended to CalabiYau varieties of dimensions two, three, and four, as well as higher rank motives. Modular forms on the worldsheet can be constructed from the geometry of spacetime by computing the Lfunctions as ..."
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The program of constructing spacetime geometry from string theoretic modular forms is extended to CalabiYau varieties of dimensions two, three, and four, as well as higher rank motives. Modular forms on the worldsheet can be constructed from the geometry of spacetime by computing the Lfunctions associated to omega motives of CalabiYau varieties, generated by their holomorphic n−forms via Galois representations. The modular forms that emerge from the Ω−motive and other motives of the intermediate cohomology are related to characters of the underlying rational conformal field theory. The converse problem of constructing space from string theory proceeds in the class of diagonal theories by determining the motives associated to modular forms in the category of pure motives with complex multiplication. The emerging picture indicates that the Lfunction can be interpreted
The Langlands program and . . .
, 2006
"... A number theoretic approach to string compactification is developed for CalabiYau hypersurfaces in arbitrary dimensions. The motivic strategy involved is illustrated by showing that the Hecke eigenforms derived from Galois group orbits of the holomorphic twoform of a particular type of K3 surfaces ..."
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A number theoretic approach to string compactification is developed for CalabiYau hypersurfaces in arbitrary dimensions. The motivic strategy involved is illustrated by showing that the Hecke eigenforms derived from Galois group orbits of the holomorphic twoform of a particular type of K3 surfaces can be expressed in terms of modular forms constructed from the worldsheet theory. The process of deriving string physics from spacetime geometry can be reversed, allowing the construction of K3 surface geometry from the string characters of the partition function. A general argument for K3 modularity follows from mirror symmetry, in combination with the proof of the ShimuraTaniyama conjecture.