Emergent Spacetime from Modular Motives

Abstract

The program of constructing spacetime geometry from string theoretic modular forms is extended to Calabi-Yau 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 L-functions associated to omega motives of Calabi-Yau 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 L-function can be interpreted as a map from the geometric category of motives to the category of conformal field theories on the worldsheet.