D-branes on some one- and two-parameter Calabi-Yau hypersurfaces

D-branes on one-parameter Calabi-Yau spaces and two-parameter K3-fibered Calabi-Yau manifolds are analyzed from both the Gepner model point of view and the geometric perspective. We compute part of the spectrum of the boundary states and comment on the appearance of the D0-brane as well as on nonsupersymmetric large volume configurations becoming supersymmetric at the Gepner point.Comment: LaTeX2e, 23 pages. v2: typos corrected, references added, JHEP styl

D-branes on Calabi-Yau Spaces

In this thesis the properties of D-branes on Calabi–Yau spaces are investigated. Compactifications of type II string theories on these spaces to which D-branes are added lead to N = 1 supersymmetric gauge theories on the world-volume of these D-branes. Both the Calabi–Yau spaces and the D-branes have in general a moduli space. We examine the dependence of the gauge theory on the choice of the moduli, in particular those of the K¨ahler structure of the Calabi–Yau manifold. For this purpose we choose two points in this moduli space which are distinguished by the fact that there exists an explicit description of the spectrum of the D-branes. One of these points corresponds to a manifold in the large volume limit on which the D-branes are described by classical geometry of vector bundles. At the other points the size of the manifold is smaller than its quantum fluctuations such that the classical geometry looses its meaning and has to be replaced by a conformal field theory. The Witten index in the open string sector is independent of the variation of these moduli and serves, together with mirror symmetry, as a tool to compare the two descriptions. We give an extensive and general presentation of these two descriptions for the class of Fermat hypersurfaces in weighted projective spaces. We explicitly carry out the comparison in many representative examples. Among them are manifolds admitting elliptic and K3-fibrations and manifolds whose moduli space can be embedded into the moduli space of another manifold. One main focus is on D4-branes, in particular on the dimension of their moduli space. Using the methods developed we are able to further confirm with our results the modified geometric hypothesis by Douglas. It essentially states that the properties of these D-branes or of these gauge theories can be determined partly by classical geometry, partly by mirror symmetry. A peculiarity of these gauge theories is the appearance of lines of marginal stability at which BPS states can decay. We show the existence of such lines in the framework of this class of Calabi–Yau spaces in two di®erent ways and discuss the connection to the formation of bound states. Of particular interest is the D0-brane whose appearance in this framework is explained

On the quantum K-theory of the quintic

Quantum K-theory of a smooth projective variety at genus zero is a collection of integers that can be assembled into a generating series $J(Q,q,t)$ that satisfies a system of linear differential equations with respect to $t$ and $q$-difference equations with respect to $Q$. With some mild assumptions on the variety, it is known that the full theory can be reconstructed from its small $J$-function $J(Q,q,0)$ which, in the case of Fano manifolds, is a vector-valued $q$-hypergeometric function. On the other hand, for the quintic 3-fold we formulate an explicit conjecture for the small $J$-function and its small linear $q$-difference equation expressed linearly in terms of the Gopakumar-Vafa invariants. Unlike the case of quantum knot invariants, and the case of Fano manifolds, the coefficients of the small linear $q$-difference equations are not Laurent polynomials, but rather analytic functions in two variables determined linearly by the Gopakumar-Vafa invariants of the quintic. Our conjecture for the small $J$-function agrees with a proposal of Jockers-Mayr.Comment: 22 page

Towards Open String Mirror Symmetry for One-Parameter Calabi-Yau Hypersurfaces

This work is concerned with branes and differential equations for one-parameter Calabi-Yau hypersurfaces in weighted projective spaces. For a certain class of B-branes we derive the inhomogeneous Picard--Fuchs equations satisfied by the brane superpotential. In this way we arrive at a prediction for the real BPS invariants for holomorphic maps of worldsheets with low Euler characteristics, ending on the mirror A-branes.Comment: 68+1 pages, 4 figures, v2: references added, typos correcte

Matrix Factorizations, Massey Products and F-Terms for Two-Parameter Calabi-Yau Hypersurfaces

We discuss B-type tensor product branes in mirrors of two-parameter Calabi-Yau hypersurfaces, using the language of matrix factorizations. We determine the open string moduli of the branes at the Gepner point. By turning on both bulk and boundary moduli we then deform the brane away from the Gepner point. Using the deformation theory of matrix factorizations we compute Massey products. These contain the information about higher order deformations and obstructions. The obstructions are encoded in the F-term equations, which we obtain from the Massey product algorithm. We show that the F-terms can be integrated to an effective superpotential. Our results provide an ingredient for open/closed mirror symmetry for these hypersurfaces.Comment: 62 page