Flux compactifications and Feynman Integrals - Calabi-Yau geometries in physics

Abstract

Calabi-Yau manifolds define fascinating geometric structures that find several ap- plications in theoretical physics. Naturally, this special class of manifolds appears in the context of compactifications of superstring, M- and F-theory. This thesis begins with an examination of the existence of supersymmetric flux vacua in type IIB string and M-theory compactifications on a Calabi-Yau manifold. The vacuum conditions for non-trivial background fluxes on the compactification space are conveniently formulated in terms of N = 1 supergravity. For type IIB string compactifications on a Calabi-Yau threefold we provide an equivalent description for the N = 1 flux vacuum constraints in terms of Minkowski vacua of an associated gauged N = 2 supergravity theory. These describe vacua in the landscape of UV consistent effective field theories with partially spontaneously broken supersymme- try. The existence of supersymmetric flux vacua is related to an arithmetic property of the underlying Calabi-Yau manifold which is called modularity. Generalizing ex- isting methods for Calabi-Yau threefolds, we derive an algorithm, which provides a systematic search for modular points on the corresponding complex structure moduli space of certain types of Calabi-Yau fourfolds with one complex structure modulus. Compactifying M-theory on such modular Calabi-Yau fourfolds may lead to non- trivial supersymmetric flux vacua. We demonstrate the application of this method for several examples. Most interestingly, we identify a modular Calabi-Yau fourfold within the family of Hulek-Verrill fourfolds and verify this observation by several independent consistency checks. Furthermore, Calabi-Yau geometries appear prominently in the framework of multi- loop Feynman integrals. It is well-known that many multi-loop Feynman integrals can be realized in terms of period integrals of certain algebraic varieties such as Calabi-Yau manifolds and hyperelliptic curves. Using the construction of intermedi- ate Jacobians, we derive a correspondence between families of Calabi-Yau threefolds and suitable families of genus-g curves that realize the same family of Feynman in- tegrals. As an explicit example, we discuss this Calabi-Yau-to-curve correspondence for the four-loop equal mass banana integral which is realized by a one-parameter family of Hulek-Verrill Calabi-Yau threefolds.