### hep-th/0206158

, 2002

"... Recently it has been shown that D-branes in orientifolds are not always described by equivariant Real K–theory. In this paper we define a previously unstudied twisted version of equivariant Real K–theory which gives the D-brane spectrum for such orientifolds. We find that equivariant Real K–theory c ..."

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Recently it has been shown that D-branes in orientifolds are not always described by equivariant Real K–theory. In this paper we define a previously unstudied twisted version of equivariant Real K–theory which gives the D-brane spectrum for such orientifolds. We find that equivariant Real K–theory can be twisted by elements of a generalised group cohomology. This cohomology classifies all orientifolds just as group cohomology classifies all orbifolds. As an example we consider the Ω × I4 orientifolds. We completely determine the equivariant orthogonal K–theory KOZZ2 (Rp,q) and analyse the twisted versions. Agreement is found between K–theory and Boundary Confromal Field Theory (BCFT) results for both integrally- and torsion-charged D-branes.

### unknown title

, 2000

"... hep-th/0008173 ZN × ZM orientifolds with and without discrete torsion ..."

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### IFT-UAM/CSIC-06-34 LMU-ASC 51/06 Coisotropic D8-branes and Model-building

, 2006

"... Up to now chiral type IIA vacua have been mostly based on intersecting D6-branes wrapping special Lagrangian 3-cycles on a CY3 manifold. We argue that there are additional BPS D-branes which have so far been neglected, and which seem to have interesting model-building features. They are coisotropic ..."

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Up to now chiral type IIA vacua have been mostly based on intersecting D6-branes wrapping special Lagrangian 3-cycles on a CY3 manifold. We argue that there are additional BPS D-branes which have so far been neglected, and which seem to have interesting model-building features. They are coisotropic D8-branes, in the sense of Kapustin and Orlov. The D8-branes wrap 5-dimensional submanifolds of the CY3 which are trivial in homology, but contain a worldvolume flux that induces D6-brane charge on them. This induced D6-brane charge not only renders the D8-brane BPS, but also creates D = 4 chirality when two D8-branes intersect. We discuss in detail the case of a type IIA T6 /(Z2 × Z2) orientifold, where we provide explicit examples of coisotropic D8-branes. We study the chiral spectrum, SUSY conditions, and effective field theory of different systems of D8-branes in this orientifold, and show how the magnetic fluxes generate a superpotential for untwisted Kähler moduli. Finally, using both D6-branes and coisotropic D8-branes we construct new examples of MSSM-like type IIA vacua. 1 On leave from Departamento de Física, Facultad de Ciencias, Universidad Central de Venezuela, A.P.

### IFT-UAM/CSIC-06-34 LMU-ASC 51/06

, 2006

"... Up to now chiral type IIA vacua have been mostly based on intersecting D6-branes wrapping special Lagrangian 3-cycles on a CY3 manifold. We argue that there are additional BPS D-branes which have so far been neglected, and which seem to have interesting model-building features. They are coisotropic ..."

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Up to now chiral type IIA vacua have been mostly based on intersecting D6-branes wrapping special Lagrangian 3-cycles on a CY3 manifold. We argue that there are additional BPS D-branes which have so far been neglected, and which seem to have interesting model-building features. They are coisotropic D8-branes, in the sense of Kapustin and Orlov. The D8-branes wrap 5-dimensional submanifolds of the CY3 which are trivial in homology, but contain a worldvolume flux that induces D6-brane charge on them. This induced D6-brane charge not only renders the D8-brane BPS, but also creates D = 4 chirality when two D8-branes intersect. We discuss in detail the case of a type IIA T6 /(Z2 × Z2) orientifold, where we provide explicit examples of coisotropic D8-branes. We study the chiral spectrum, SUSY conditions, and effective field theory of different systems of D8-branes in this orientifold, and show how the magnetic fluxes generate a superpotential for untwisted Kähler moduli. Finally, using both D6-branes and coisotropic D8-branes we construct new examples of MSSM-like type IIA vacua. 1 On leave from Departamento de Física, Facultad de Ciencias, Universidad Central de Venezuela, A.P.

### hep-th/0509036 String Theory in β Deformed Spacetimes

, 2005

"... Fluxbrane-like backgrounds obtained from flat space by a sequence of T-dualities and shifts of polar coordinates (β deformations) provide an interesting class of exactly solvable string theories. We compute the one-loop partition function for various such deformed spaces and study their spectrum of ..."

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Fluxbrane-like backgrounds obtained from flat space by a sequence of T-dualities and shifts of polar coordinates (β deformations) provide an interesting class of exactly solvable string theories. We compute the one-loop partition function for various such deformed spaces and study their spectrum of D-branes. For rational values of the B-field these models are equivalent to ZN × ZN orbifolds with discrete torsion. We also obtain an interesting new class of time-dependent backgrounds which resemble localized closed string

### Boundary Fixed Points, Enhanced Gauge Symmetry and Singular Bundles on K3

, 2000

"... We investigate certain fixed points in the boundary conformal field theory representation of type IIA D-branes on Gepner points of K3. They correspond geometrically to degenerate brane configurations, and physically lead to enhanced gauge symmetries on the world-volume. Non-abelian gauge groups aris ..."

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We investigate certain fixed points in the boundary conformal field theory representation of type IIA D-branes on Gepner points of K3. They correspond geometrically to degenerate brane configurations, and physically lead to enhanced gauge symmetries on the world-volume. Non-abelian gauge groups arise if the stabilizer group of the fixed points is realized projectively, which is similar to D-branes on orbifolds with discrete torsion. Moreover, the fixed point boundary states can be resolved into several irreducible components. These correspond to bound states at threshold and can be viewed as (non-locally free) sub-sheaves of semi-stable sheaves. Thus, the BCFT fixed points appear to carry two-fold geometrical information: on the one hand they probe the boundary of the instanton moduli space on K3, on the other hand they probe discrete torsion in D-geometry.

### ILL-(TH)-00-08 hep-th/0009209 Non-Commutative Calabi-Yau Manifolds

, 2008

"... We discuss aspects of the algebraic geometry of compact non-commutative Calabi-Yau manifolds. In this setting, it is appropriate to consider local holomorphic algebras which can be glued together into a compact Calabi-Yau algebra. We consider two examples: a toroidal orbifold T 6 /Z2 ×Z2, and an orb ..."

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We discuss aspects of the algebraic geometry of compact non-commutative Calabi-Yau manifolds. In this setting, it is appropriate to consider local holomorphic algebras which can be glued together into a compact Calabi-Yau algebra. We consider two examples: a toroidal orbifold T 6 /Z2 ×Z2, and an orbifold of the quintic in CP4, each with discrete torsion. The non-commutative geometry tools are enough to describe various properties of the orbifolds. First, one describes correctly the fractionation of branes at singularities. Secondly, for the first example we show that one can recover explicitly a large slice of the moduli space of complex structures which deform the orbifold. For this example we also show that we get the correct counting of complex structure deformations at the orbifold point by using traces of non-commutative differential forms (cyclic homology).

### Comments on D-branes on Orbifolds and K-theory

, 2007

"... We systematically revisit the description of D-branes on orbifolds and the classification of their charges via K-theory. We include enough details to make the results accessible to both physicists and mathematicians interested in these topics. The minimally charged branes predicted by K-theory in ZN ..."

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We systematically revisit the description of D-branes on orbifolds and the classification of their charges via K-theory. We include enough details to make the results accessible to both physicists and mathematicians interested in these topics. The minimally charged branes predicted by K-theory in ZN orbifolds with N odd are only BPS. We confirm this result using the boundary state formalism for Z3. For ZN × ZN orbifolds with and without discrete torsion, we show that the K-theory classification of charges agrees with the boundary state approach, largely developed by Gaberdiel and Much of the recent progress in string theory revolves around the concept of Dirichlet-branes. These are objects that source Ramond-Ramond bosonic massless fields of type II and type I string theories carrying one unit of charge [1, 2]. More precisely, D-branes are nonperturbative states that enter in the theory as boundaries for the closed string world-sheet,

### BPS branes in discrete torsion orbifolds

, 2005

"... We investigate D-branes in a Z3 × Z3 orbifold with discrete torsion. For this class of orbifolds the only known objects which couple to twisted RR potentials have been non-BPS branes. By using more general gluing conditions we construct here a D-brane which is BPS and couples to RR potentials in the ..."

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We investigate D-branes in a Z3 × Z3 orbifold with discrete torsion. For this class of orbifolds the only known objects which couple to twisted RR potentials have been non-BPS branes. By using more general gluing conditions we construct here a D-brane which is BPS and couples to RR potentials in the twisted and in the untwisted sectors.

### ILL-(TH)-00-08 hep-th/0012050 D-branes on Singularities: New Quivers from Old

, 2008

"... In this paper we present simplifying techniques which allow one to compute the quiver diagrams for various D-branes at (non-Abelian) orbifold singularities with and without discrete torsion. The main idea behind the construction is to take the orbifold of an orbifold. Many interesting discrete group ..."

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In this paper we present simplifying techniques which allow one to compute the quiver diagrams for various D-branes at (non-Abelian) orbifold singularities with and without discrete torsion. The main idea behind the construction is to take the orbifold of an orbifold. Many interesting discrete groups fit into an exact sequence N → G → G/N. As such, the orbifold M/G is easier to compute as (M/N)/(G/N) and we present graphical rules which allow fast computation given the M/N quiver.