Calculating Corrections in F-Theory from Refined BPS Invariants and Backreacted Geometries

This thesis presents various corrections to F-theory compactifications which rely on the computation of refined Bogomol'nyi-Prasad-Sommerfield (BPS) invariants and the analysis of backreacted geometries. Detailed information about rigid supersymmetric theories in five dimensions is contained in an index counting refined BPS invariants. These BPS states fall into representations of SU(2) x SU(2), the little group in five dimensions, which has an induced action on the cohomology of the moduli space of stable pairs. In the first part of this thesis, we present the computation of refined BPS state multi-plicities associated to M-theory compactifications on local Calabi-Yau manifolds whose base is given by a del Pezzo or half K3 surface. For geometries with a toric realization we use an algorithm which is based on the Weierstrass normal form of the mirror geometry. In addition we use the refined holomorphic anomaly equation and the gap condition at the conifold locus in the moduli space in order to perform the direct integration and to fix the holomorphic ambiguity. In a second approach, we use the refined Gottsche formula and the refined modular anomaly equation that govern the (refined) genus expansion of the free energy of the half K3 surface. By this procedure, we compute the refined BPS invariants of the half K3 from which the results of the remaining del Pezzo surfaces are obtained by flop transitions and blow-downs. These calculations also make use of the high symmetry of the del Pezzo surfaces whose homology lattice contains the root lattice of exceptional Lie algebras. In cases where both approaches are applicable, we successfully check the compatibility of these two methods. In the second part of this thesis, we apply the results obtained from the calculation of the refined invariants of the del Pezzo respectively the half K3 surfaces to count non-perturbative objects in F-theory. The first application is given by BPS states of the E-String which are counted in the dual F-theory compactification. Using the refined BPS invariants we can count these states and explain their space-time spin content. In addition, we explain that they fall into representations of E8 which can be explicitly determined. The second application is given by a proposal how to count [p,q]-strings within F-theory which is based on the D3 probe-brane picture and the dual Seiberg-Witten description. As a third contribution to F-theory which is independent of the results obtained in the first part, we consider the backreaction of G4-flux onto the geometry of a local model of a Calabi-Yau fourfold geometry. This induces a non-trivial warp-factor and modifies the Kaluza-Klein reduction ansatz. Taking this into account we demonstrate how corrections to the 7-brane gauge coupling function can be computed within F-theory

Discrete Symmetries in Heterotic/F-theory Duality and Mirror Symmetry

We study aspects of Heterotic/F-theory duality for compactifications with Abelian discrete gauge symmetries. We consider F-theory compactifications on genus-one fibered Calabi-Yau manifolds with n-sections, associated with the Tate-Shafarevich group Z_n. Such models are obtained by studying first a specific toric set-up whose associated Heterotic vector bundle has structure group Z_n. By employing a conjectured Heterotic/F-theory mirror symmetry we construct dual geometries of these original toric models, where in the stable degeneration limit we obtain a discrete gauge symmetry of order two and three, for compactifications to six dimensions. We provide explicit constructions of mirror-pairs for symmetric examples with Z_2 and Z_3, in six dimensions. The Heterotic models with symmetric discrete symmetries are related in field theory to a Higgsing of Heterotic models with two symmetric abelian U(1) gauge factors, where due to the Stuckelberg mechanism only a diagonal U(1) factor remains massless, and thus after Higgsing only a diagonal discrete symmetry of order n is present in the Heterotic models and detected via Heterotic/F-theory duality. These constructions also provide further evidence for the conjectured mirror symmetry in Heterotic/F-theory at the level of fibrations with torsional sections and those with multi-sections.Comment: 25 pages, 4 figure

F-Theory Vacua with Z_3 Gauge Symmetry

Discrete gauge groups naturally arise in F-theory compactifications on genus-one fibered Calabi-Yau manifolds. Such geometries appear in families that are parameterized by the Tate-Shafarevich group of the genus-one fibration. While the F-theory compactification on any element of this family gives rise to the same physics, the corresponding M-theory compactifications on these geometries differ and are obtained by a fluxed circle reduction of the former. In this note, we focus on an element of order three in the Tate-Shafarevich group of the general cubic. We discuss how the different M-theory vacua and the associated discrete gauge groups can be obtained by Higgsing of a pair of five-dimensional U(1) symmetries. The Higgs fields arise from vanishing cycles in $I_2$-fibers that appear at certain codimension two loci in the base. We explicitly identify all three curves that give rise to the corresponding Higgs fields. In this analysis the investigation of different resolved phases of the underlying geometry plays a crucial r\^ole.Comment: 13 page

Guideline for Trustworthy Artificial Intelligence -- AI Assessment Catalog

Artificial Intelligence (AI) has made impressive progress in recent years and represents a key technology that has a crucial impact on the economy and society. However, it is clear that AI and business models based on it can only reach their full potential if AI applications are developed according to high quality standards and are effectively protected against new AI risks. For instance, AI bears the risk of unfair treatment of individuals when processing personal data e.g., to support credit lending or staff recruitment decisions. The emergence of these new risks is closely linked to the fact that the behavior of AI applications, particularly those based on Machine Learning (ML), is essentially learned from large volumes of data and is not predetermined by fixed programmed rules. Thus, the issue of the trustworthiness of AI applications is crucial and is the subject of numerous major publications by stakeholders in politics, business and society. In addition, there is mutual agreement that the requirements for trustworthy AI, which are often described in an abstract way, must now be made clear and tangible. One challenge to overcome here relates to the fact that the specific quality criteria for an AI application depend heavily on the application context and possible measures to fulfill them in turn depend heavily on the AI technology used. Lastly, practical assessment procedures are needed to evaluate whether specific AI applications have been developed according to adequate quality standards. This AI assessment catalog addresses exactly this point and is intended for two target groups: Firstly, it provides developers with a guideline for systematically making their AI applications trustworthy. Secondly, it guides assessors and auditors on how to examine AI applications for trustworthiness in a structured way

Fluxes and Warping for Gauge Couplings in F-theory

We compute flux-dependent corrections in the four-dimensional F-theory effective action using the M-theory dual description. In M-theory the 7-brane fluxes are encoded by four-form flux and modify the background geometry and Kaluza-Klein reduction ansatz. In particular, the flux sources a warp factor which also depends on the torus directions of the compactification fourfold. This dependence is crucial in the derivation of the four-dimensional action, although the torus fiber is auxiliary in F-theory. In M-theory the 7-branes are described by an infinite array of Taub-NUT spaces. We use the explicit metric on this geometry to derive the locally corrected warp factor and M-theory three-from as closed expressions. We focus on contributions to the 7-brane gauge coupling function from this M-theory back-reaction and show that terms quadratic in the internal seven-brane flux are induced. The real part of the gauge coupling function is modified by the M-theory warp factor while the imaginary part is corrected due to a modified M-theory three-form potential. The obtained contributions match the known weak string coupling result, but also yield additional terms suppressed at weak coupling. This shows that the completion of the M-theory reduction opens the way to compute various corrections in a genuine F-theory setting away from the weak string coupling limit.Comment: 46 page

Origin of Abelian gauge symmetries in heterotic/F-theory duality

We study aspects of heterotic/F-theory duality for compactifications with Abelian gauge symmetries. We consider F-theory on general Calabi-Yau manifolds with a rank one Mordell-Weil group of rational sections. By rigorously performing the stable degeneration limit in a class of toric models, we derive both the Calabi-Yau geometry as well as the spectral cover describing the vector bundle in the heterotic dual theory. We carefully investigate the spectral cover employing the group law on the elliptic curve in the heterotic theory. We find in explicit examples that there are three different classes of heterotic duals that have U(1) factors in their low energy effective theories: split spectral covers describing bundles with S(U(m) x U(1)) structure group, spectral covers containing torsional sections that seem to give rise to bundles with SU(m) X Z(k) structure group and bundles with purely non-Abelian structure groups having a centralizer in E-8 containing a U(1) factor. In the former two cases, it is required that the elliptic fibration on the heterotic side has a non-trivial Mordell-Weil group. While the number of geometrically massless U(1)\u27s is determined entirely by geometry on the F-theory side, on the heterotic side the correct number of U(1)\u27s is found by taking into account a Stiickelberg mechanism in the lower-dimensional effective theory. In geometry, this corresponds to the condition that sections in the two half K3 surfaces that arise in the stable degeneration limit of F-theory can be glued together globally

Type II string theory on Calabi-Yau manifolds with torsion and non-Abelian discrete gauge symmetries

Abstract We provide the first explicit example of Type IIB string theory compactification on a globally defined Calabi-Yau threefold with torsion which results in a four-dimensional effective theory with a non-Abelian discrete gauge symmetry. Our example is based on a particular Calabi-Yau manifold, the quotient of a product of three elliptic curves by a fixed point free action of ℤ 2 × ℤ 2 $${\mathbb{Z}}_2\times {\mathbb{Z}}_2$$ . Its cohomology contains torsion classes in various degrees. The main technical novelty is in determining the multiplicative structure of the (torsion part of) the cohomology ring, and in particular showing that the cup product of second cohomology torsion elements goes non-trivially to the fourth cohomology. This specifies a non-Abelian, Heisenberg-type discrete symmetry group of the cfour-dimensional theory