On geometric corrections to effective actions of string theory

In this thesis we study geometric corrections to the low-energy
effective actions of string theory. More concretely, we compute
higher-derivative corrections to the couplings of
three-dimensional, N = 2 supergravity theories and interpret the
induced α′-corrections in N = 1, minimal supergravity theories in
four dimensions, in the framework of F-theory. These allow for
chiral spectra and are therefore phenomenological relevant. We
analyzed higher-derivative corrections to M-theory, accessible
through its low-energy effective theory, given by
eleven-dimensional supergravity. The next to leading order terms to
eleven-dimensional supergravity carry eight-derivatives, and are
suppressed by lM6 compared to the classical terms, with lM being
the eleven-dimensional Planck-Length - the only scale in eleven
dimensions. These corrections are lifted from IIA supergravity
corrections, which are derived from string scattering amplitudes.
The common theme of this thesis is to compactify the bosonic sector
of the eleven-dimensional supergravity action, including all known
eight-derivative corrections, on a supersymmetric background to
find a 3d, N = 2 theory, which then can be lifted to a 4d, N = 1
theory. This goal is approached in several steps. In the classical
reduction of eleven-dimensional supergravity the metric background
is a direct product of the external space, consisting of two space
and one time dimension and the internal eight spacelike-dimensional
Calabi-Yau manifold. However, when considering higher-derivative
corrections the background has to be altered by introducing a
dependence of the external space on the warp- factor, which is a
function of the internal space. We find an explicit warped
background solution to the eleven-dimensional E.O.M.’s including
non-vanishing flux. To check the background for its supersymmetry
features one would need to consider the eleven-dimensional
gravitino variations at this order in lM . However, these are not
known, which leads us to propose higher-order lM -corrected
gravitino variations consistent with our background solution. As a
next step we dimensionally reduce the bosonic sector of the
eleven-dimensional supergravity action including all
eight-derivative terms on this warped background and analyze the
resulting three- dimensional theory. In this context the interplay
of the warp-factor and the higher-derivative terms is of crucial
importance. To identify the N = 2 properties of the resulting
three-dimensional theory obtained by dimensional reduction, we
compare it to the canonical from of three-dimensional N = 2
supergravtiy. We conclude that the reduced action is compatible
with N = 2 supersymmetry and give a proposal for the K ̈ahler
potential and the complex coordinates, which receive lM6
corrections. Besides a warp-factor contribution, the K ̈ahler
potential receives a correction proportional to the third Chern- 2
form of the zeroth order internal background, being the Calabi-Yau
fourfold. The complex coordinates are defined as divisor-integrals
and are corrected by a warp-factor dependent term as well as one
related to the non-harmonic part of the fourth Chern-form, of the
zeroth order Calabi-Yau manifold. Thus the couplings of the
resulting theory receive besides the warp-factor, in particular
geometric corrections of order lM6 . In the first part of this
thesis we study a simplified setup, only considering a subset of
the relevant eight-derivative corrections in eleven dimensions.
Furthermore, we do compactify on the classical background,
consisting of the internal Calabi-Yau fourfold without warping and
fluxes, to gain a three-dimensional theory. However, we use the
M/F-theory duality to uplift the yielded corrections, which results
in corrections to the couplings of the four-dimensional theory. In
the weak coupling limit we find that these are sourced by the
self-intersection curves of D7-branes.