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Weissenbacher, Matthias (2015): On geometric corrections to effective actions of string theory. Dissertation, LMU München: Fakultät für Physik



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.