http://cds.cern.ch/record/994225/files/0610327.pdf

Four-dimensional string compactifications with D-branes, orientifolds and fluxes

We consider families ${\cal F}(\Delta)$ consisting of complex $(n-1)$-dimensional projective algebraic compactifications of $\Delta$-regular affine hypersurfaces $Z_f$ defined by Laurent polynomials $f$ with a fixed $n$-dimensional Newton polyhedron $\Delta$ in $n$-dimensional algebraic torus ${\bf T} =({\bf C}^*)^n$. If the family ${\cal F}(\Delta)$ defined by a Newton polyhedron $\Delta$ consists of $(n-1)$-dimensional Calabi-Yau varieties, then the dual, or polar, polyhedron $\Delta^*$ in the dual space defines another family ${\cal F}(\Delta^*)$ of Calabi-Yau varieties, so that we obtain the remarkable duality between two {\em different families} of Calabi-Yau varieties. It is shown that the properties of this duality coincide with the properties of {\em Mirror Symmetry} discovered by physicists for Calabi-Yau $3$-folds. Our method allows to construct many new examples of Calabi-Yau $3$-folds and new candidats for their mirrors which were previously unknown for physicists. We conjecture that there exists an isomorphism between two conformal field theories corresponding to Calabi-Yau varieties from two families ${\cal F}(\Delta)$ and ${\cal F}(\Delta^*)$.

Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties

We investigate orientifolds of type II string theory on K3 and Calabi-Yau 3-folds with intersecting D-branes wrapping special lagrangian cycles. We determine quite generically the chiral massless spectrum in terms of topological invariants and discuss both orbifold examples and algebraic realizations in detail. Intriguingly, the developed techniques provide an elegant way to figure out the chiral sector of orientifold models without computing any explicit string partition function. As a new example we derive a non-supersymmetric Standard-like Model from an orientifold of type IIA on the quintic Calabi-Yau 3-fold with wrapped D6-branes. In the case of supersymmetric intersecting brane models on Calabi-Yau manifolds we discuss the D-term and F-term potentials, the effective gauge couplings and the Green-Schwarz mechanism. The mirror symmetric formulation of this construction is provided within type IIB theory. We finally include a short discussion about the lift of these models from type IIB on K3 to F-theory and from type IIA on Calabi-Yau 3-folds to M-theory on G2 manifolds.

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Orientifolds of K3 and Calabi-Yau manifolds with intersecting D-branes

We consider the E 8 ? E 8 heterotic string theory compactified on Calabi-Yau manifolds with bundles containing abelian factors in their structure group. Generic low energy consequences such as the generalised Green-Schwarz mechanism for the multiple anomalous abelian gauge groups are studied. We also compute the holomorphic gauge couplings and induced Fayet-Iliopoulos terms up to one-loop order, where the latter are interpreted as stringy one-loop corrections to the Donaldson-Uhlenbeck-Yau condition. Such models generically have frozen combinations of K?hler and dilaton moduli. We study concrete bundles with structure group SU(N) ? U(1) M yielding quasi-realistic gauge groups with chiral matter given by certain bundle cohomology classes. We also provide a number of explicit tadpole free examples of bundles defined by exact sequences of sums of line bundles over complete intersection Calabi-Yau spaces. This includes one example with precisely the Standard Model gauge symmetry.

Loop-corrected compactifications of the heterotic string with line bundles

Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry

(0,2) Mirror symmetry

We generalize the previously established (0,2) triality of exactly solvable models, Landau-Ginzburg theories and Calabi-Yau manifolds to a number of different classes of (0,2) compactifications derived from (2,2) vacua. For the resulting models we show that the known (2,2) mirror constructions induce mirror symmetry in the (0,2) context.

(0,2) Mirror Symmetry

We identify the exactly solvable theory of the conformal fixed point of (0,2) Calabi-Yau sigma-models and their Landau-Ginzburg phases. To this end we consider a number of (0,2) models constructed from a particular (2,2) exactly solvable theory via the method of simple currents. In order to establish the relation between exactly solvable (0,2) vacua of the heterotic string, (0,2) Landau-Ginzburg orbifolds, and (0,2) Calabi-Yau manifolds, we compute the Yukawa couplings in the exactly solvable model and compare the results with the product structure of the chiral ring which we extract from the structure of the massless spectrum of the exact theory. We find complete agreement between the two up to a finite number of renormalizations. For a particularly simple example we furthermore derive the generating ideal of the chiral ring from a (0,2) linear sigma-model which has both a Landau-Ginzburg and a (0,2) Calabi-Yau phase.

The (0,2) Exactly Solvable Structure of Chiral Rings, Landau-Ginzburg Theories and Calabi-Yau Manifolds

Because of the existence of rigid Calabi-Yau manifolds, mirror symmetry cannot be understood as an operation on the space of manifolds with vanishing first Chern class. In this article I continue to investigate a particular type of Kahler manifolds with positive first Chern class which generalize Calabi-Yau manifolds in a natural way and which provide a framework for mirrors of rigid string vacua. This class comprises Fano manifolds of a special type which encode crucial information about ground states of the superstring. It is shown in particular that the massless spectra of (2, 2)-supersymmetric vacua of central charge ĉ=Dcrit can be derived from special Fano varieties of complex dimension Dcrit+2(Q−1), Q>1, and that in certain circumstances it is even possible to embed Calabi-Yau manifolds into such higher dimensional spaces. The constructions described here lead to new insight into the relation between exactly solvable models and their mean field theories on the one hand and their corresponding Calabi-Yau manifolds on the other. Furthermore it is shown that Witten’s formulation of the Landau-Ginzburg/Calabi-Yau relation can be applied to the present framework as well.

Mirror Symmetry and String Vacua from a Special Class of Fano Varieties

We review several aspects of heterotic, type II, F-theory, and M-theory compactifications on Calabi-Yau three-folds and fourfolds. In the context of dualities we focus on the heterotic gauge structure determined by the various types of fibration relevant in the framework of heterotic/type II duality in D=4 as well as 4D F-theory. We also consider transitions between Calabi-Yau manifolds in both three and four dimensions and review some of the consequences for the behavior of the superpotential.

Rolf Schimmrigk

Papers

(0,2) Mirror symmetry

(0,2) Mirror Symmetry

The (0,2) Exactly Solvable Structure of Chiral Rings, Landau-Ginzburg Theories and Calabi-Yau Manifolds

Mirror Symmetry and String Vacua from a Special Class of Fano Varieties

Dualities and phase transitions for Calabi-Yau threefolds and fourfolds