D-Branes and Quotient Singularities of Calabi-Yau Fourfolds

In these notes we give an introduction to some of the concepts involved in constructing SM-like gauge theories in systems of branes at singularities of CY manifolds. These notes are an expanded version of lectures given by Herman Verlinde at the Cargese 2006 Summer School.

We consider compactifications of the N=1, d=6, E_8 theory on tori to five, four, and three dimensions and learn about some properties of this theory. As a by-product we derive the SL(2,\IZ) duality of the N=2, d=4, SU(2) theory with N_f=4. Using this theory on a D-brane probe we shed new light on the singularities of F-theory compactifications to eight dimensions. As another application we consider compactifications of F-theory, M-theory and the IIA string on (singular) Calab...

We construct intersecting D-brane configurations that encode the gauge groups and field content of dual N=4 supersymmetric gauge theories in three dimensions. The duality which exchanges the Coulomb and Higgs branches and the Fayet-Iliopoulos and mass parameters is derived from the SL(2,Z) symmetry of the type IIB string. Using the D-brane configurations we construct explicitly this mirror map between the dual theories and study the instanton corrections in the D-brane worldv...

We report on a broad new class of N=1 gauge theory dualities which relate the worldvolume gauge theories of D3 branes probing different orientifolds of the same Calabi-Yau singularity. In this paper, we focus on the simplest example of these new dualities, arising from the orbifold singularity C^3/Z_3. We present extensive checks of the duality, including anomaly matching, partial moduli space matching, matching of discrete symmetries, and matching of the superconformal indic...

We extend the string model building rules for the construction of chiral supersymmetric Type I compactifications on smooth Calabi-Yau manifolds. These models contain stacks of D9-branes endowed with general stable U(n) bundles on their world-volume and D5-branes wrapping holomorphic curves on the Calabi-Yau.

We find explicit solutions of Type IIB string theory on R^4/Z_2 corresponding to the classical geometry of fractional D1-branes. From the supergravity solution obtained, we capture perturbative information about the running of the coupling constant and the metric on the moduli space of N=4, D=2 Super Yang Mills.

We argue that D-branes corresponding to rational B boundary states in a Gepner model can be understood as fractional branes in the Landau-Ginzburg orbifold phase of the linear sigma model description. Combining this idea with the generalized McKay correspondence allows us to identify these states with coherent sheaves, and to calculate their K-theory classes in the large volume limit, without needing to invoke mirror symmetry. We check this identification against the mirror s...

We consider D4-branes on toric Calabi-Yau spaces. The quiver gauge theory that describes several D4-branes on the Calabi-Yau has a Higgs branch, that describes configurations of a single large D4-brane with the same charges. We propose that the world volume of such a D4-brane is described by a determinantal variety. We discuss a description of the Higgs branch of the moduli space in terms of a quiver with twice as many nodes and only bifundamental fields, arising from a $D6-\...

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 fun...

These lectures, given at the Chinese Academy of Sciences for the BeiJing/HangZhou International Summer School in Mathematical Physics, are intended to introduce, to the beginning student in string theory and mathematical physics, aspects of the rich and beautiful subject of D-brane gauge theories constructed from local Calabi-Yau spaces. Topics such as orbifolds, toric singularities, del Pezzo surfaces as well as chaotic duality will be covered.