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