Special Geometry and Mirror Symmetry for Open String Backgrounds with N=1 Supersymmetry

We review an approach for computing non-perturbative, exact superpotentials for Type II strings compactified on Calabi-Yau manifolds, with extra fluxes and D-branes on top. The method is based on an open string generalization of mirror symmetry, and takes care of the relevant sphere and disk instanton contributions. We formulate a framework based on relative (co)homology that uniformly treats the flux and brane sectors on a similar footing. However, one important difference is that the brane induced potentials are of much larger functional diversity than the flux induced ones, which have a hidden N=2 structure and depend only on the bulk geometry. This introductory lecture is meant for an audience unfamiliar with mirror symmetry.Comment: latex, 35p, 2 figs, refs added; brief comments about duality to fourfolds added in the concluding sectio

Instanton Geometry and Quantum A-infinity structure on the Elliptic Curve

We first determine and then study the complete set of non-vanishing A-model correlation functions associated with the ``long-diagonal branes'' on the elliptic curve. We verify that they satisfy the relevant A-infinity consistency relations at both classical and quantum levels. In particular we find that the A-infinity relation for the annulus provides a reconstruction of annulus instantons out of disk instantons. We note in passing that the naive application of the Cardy-constraint does not hold for our correlators, confirming expectations. Moreover, we analyze various analytical properties of the correlators, including instanton flops and the mixing of correlators with different numbers of legs under monodromy. The classical and quantum A-infinity relations turn out to be compatible with such homotopy transformations. They lead to a non-invariance of the effective action under modular transformations, unless compensated by suitable contact terms which amount to redefinitions of the tachyon fields.Comment: 24 pages, 6 figures, LaTeX2

On a Boundary CFT Description of Nonperturbative N=2 Yang-Mills Theory

We describe a simple method for determining the strong-coupling BPS spectrum of four dimensional N=2 supersymmetric Yang-Mills theory. The idea is to represent the magnetic monopoles and dyons in terms of D-brane boundary states of a non-compact d=2 N=2 Landau-Ginzburg model. In this way the quantum truncated BPS spectrum at the origin of the moduli space can be directly mapped to the finite number of primary fields of the superconformal minimal models

Emergent Strings, Duality and Weak Coupling Limits for Two-Form Fields

We systematically analyse weak coupling limits for 2-form tensor fields in the presence of gravity. Such limits are significant for testing various versions of the Weak Gravity and Swampland Distance Conjectures, and more broadly, the phenomenon of emergence. The weak coupling limits for 2-forms correspond to certain infinite-distance limits in the moduli space of string compactifications, where asymptotically tensionless, solitonic strings arise. These strings are identified as weakly coupled fundamental strings in a dual frame, which makes the idea of emergence manifest. Concretely we first consider weakly coupled tensor fields in six-dimensional compactifications of F-theory, where the arising tensionless strings play the role of dual weakly coupled heterotic strings. As the main part of this work, we consider certain infinite distance limits of Type IIB strings on K3 surfaces, for which we show that the asymptotically tensionless strings describe dual fundamental Type IIB strings, again on K3 surfaces. By contrast the analogous weak coupling limits of M-theory compactifications are found to correspond to an F-theory limit where an extra dimension emerges rather than tensionless strings. We comment on extensions of our findings to four-dimensional compactifications.Comment: 30 pages, 1 figure; v2: cosmetic changes and minor comments adde

Modular Fluxes, Elliptic Genera, and Weak Gravity Conjectures in Four Dimensions

We analyse the Weak Gravity Conjecture for chiral four-dimensional F-theory compactifications with N=1 supersymmetry. Extending our previous work on nearly tensionless heterotic strings in six dimensions, we show that under certain assumptions a tower of asymptotically massless states arises in the limit of vanishing coupling of a U(1) gauge symmetry coupled to gravity. This tower contains super-extremal states whose charge-to-mass ratios are larger than those of certain extremal dilatonic Reissner-Nordstrom black holes, precisely as required by the Weak Gravity Conjecture. Unlike in six dimensions, the tower of super-extremal states does not always populate a charge sub-lattice. The main tool for our analysis is the elliptic genus of the emergent heterotic string in the chiral N=1 supersymmetric effective theories. This also governs situations where the heterotic string is non-perturbative. We show how it can be computed in terms of BPS invariants on elliptic four-folds, by making use of various dualities and mirror symmetry. Compared to six dimensions, the geometry of the relevant elliptically fibered four-folds is substantially richer than that of the three-folds, and we classify the possibilities for obtaining critical, nearly tensionless heterotic strings. We find that the (quasi-)modular properties of the elliptic genus crucially depend on the choice of flux background. Our general results are illustrated in a detailed example.Comment: 72 pages, 2 figure

A Stringy Test of the Scalar Weak Gravity Conjecture

We prove a version of the Weak Gravity Conjecture for 6d F-theory or heterotic string compactifications with 8 supercharges. This sharpens our previous analysis by including massless scalar fields. The latter are known to modify the Weak Gravity Conjecture bound in two a priori independent ways: First, the extremality condition of a charged black hole is modified, and second, the test particles required to satisfy the Weak Gravity Conjecture are subject to additional Yukawa type interactions. We argue on general grounds that at weak coupling, the two types of effects are equivalent for a tower of asymptotically massless charged test particles predicted by the Swampland Distance Conjecture. We then specialise to F-theory compactified on elliptic Calabi-Yau three-folds and prove that the precise numerical bound on the charge-to-mass ratio is satisfied at weak coupling. This amounts to an intriguing coincidence of two a priori different notions of extremality, namely one based on the balance of gauge, gravitational and scalar forces for extremal (non-BPS) black holes, and the other encoded in the modular properties of certain Jacobi forms. In the presence of multiple abelian gauge group factors, the elliptic genus counting these states is a lattice quasi-Jacobi form of higher rank, and we exemplify this in a model with two abelian gauge group factors.Comment: 31 pages, 2 figure

D-brane effective action and tachyon condensation in topological minimal models

We study D-brane moduli spaces and tachyon condensation in B-type topological minimal models and their massive deformations. We show that any B-type brane is isomorphic with a direct sum of `minimal' branes, and that its moduli space is stratified according to the type of such decompositions. Using the Landau-Ginzburg formulation, we propose a closed formula for the effective deformation potential, defined as the generating function of tree-level open string amplitudes in the presence of D-branes. This provides a direct link to the categorical description, and can be formulated in terms of holomorphic matrix models. We also check that the critical locus of this potential reproduces the D-branes' moduli space as expected from general considerations. Using these tools, we perform a detailed analysis of a few examples, for which we obtain a complete algebro-geometric description of moduli spaces and strata.Comment: 36 page

Matrixfaktorisierungen

Im Folgenden soll ein kurzer Abriss des Themas Matrixfaktorisierungen gegeben werden. Wir werden darlegen, warum dieses recht simple Konzept zu erstaunlich tiefen mathematischen Gedankengängen führt und auch in der modernen theoretischen Physik wichtige Anwendungen hat

Superpotentials, A-infinity Relations and WDVV Equations for Open Topological Strings

We give a systematic derivation of the consistency conditions which constrain open-closed disk amplitudes of topological strings. They include the A-infinity relations (which generalize associativity of the boundary product of topological field theory), as well as certain homotopy versions of bulk-boundary crossing symmetry and Cardy constraint. We discuss integrability of amplitudes with respect to bulk and boundary deformations, and write down the analogs of WDVV equations for the space-time superpotential. We also study the structure of these equations from a string field theory point of view. As an application, we determine the effective superpotential for certain families of D-branes in B-twisted topological minimal models, as a function of both closed and open string moduli. This provides an exact description of tachyon condensation in such models, which allows one to determine the truncation of the open string spectrum in a simple manner.Comment: 53 page