Orientifold Planes Research Articles

Orientifolds and twisted boundary conditions

It is argued that the T-dual of a crosscap is a combination of an O+ and an O- orientifold plane. Various theories with crosscaps and D-branes are interpreted as gauge-theories on tori obeying twisted boundary conditions. Their duals live on orientifolds where the various orientifold planes are of different types. We derive how to read off the holonomies from the positions of D-branes in the orientifold background. As an application we reconstruct some results from a paper by Borel, Friedman and Morgan for gauge theories with classical groups, compactified on a 2-- or 3--torus with twisted boundary conditions.

Orientifold planes, type I Wilson lines and non-BPS D-branes

There is a longstanding puzzle concerned with the existence of Op~-planes with p>=6, which are orientifold p-planes of negative charge with stuck Dp-branes. We study the consistency of configurations with various orientifold planes and propose a resolution to this puzzle. It is argued that O6~-planes are possible in massive IIA theory with odd cosmological constant, while O7~-planes and O8~-planes are not allowed. Various relations between orientifold planes and non-BPS D-branes are also addressed.

GEOMETRY OF ORIENTIFOLDS WITH NS–NS B FLUX

We discuss geometry underlying orientifolds with nontrivial NS–NS B flux. If D-branes wrap a torus with B flux the rank of the gauge group is reduced due to noncommuting Wilson lines whose presence is implied by the B flux. In the case of c-branes transverse to a torus with B flux the rank reduction is due to a smaller number of D-branes required by tadpole cancellation conditions in the presence of B flux as some of the orientifold planes now have the opposite orientifold projection. We point out that T duality in the presence of B flux is more subtle than in the case with trivial B flux, and it is precisely consistent with the qualitative difference between the aforementioned two setups. In the case where both types of branes are present, the states in the mixed (e.g. 59) open string sectors come with a nontrivial multiplicity, which we relate to a discrete gauge symmetry due to nonzero B flux, and construct vertex operators for the mixed sector states. Using these results we revisit K3 orientifolds with B flux (where K3 is a T4/ZM orbifold) and point out various subtleties arising in some of these models. For instance, in the Z2 case the conformal field theory orbifold does not appear to be the consistent background for the corresponding orientifolds with B flux. This is related to the fact that nonzero B flux requires the presence of both O5-- as well as O5+-planes at various Z2 orbifold fixed points, which appears to be inconsistent with the presence of the twistedB flux in the conformal field theory orbifold. We also consider four-dimensional [Formula: see text] and [Formula: see text] supersymmetric orientifolds. We construct consistent four-dimensional models with B flux which do not suffer from difficulties encountered in the K3 cases.

The type I D-instanton and its M-theory origin

The tree-level amplitude for the scattering of two gauge particles constrained to move on the two distinct boundaries of eleven-dimensional space-time in the Horava-Witten formulation of M-theory is constructed. At low momenta this reproduces the corresponding tree-level scattering amplitude of the E_8xE_8 heterotic string theory. After compactification to nine dimensions on a large circle with a suitable Wilson line to break the symmetry to SO(16)xSO(16) this amplitude is used to describe the scattering of two massive SO(16) spinor states - one from each factor of the unbroken symmetry group. The amplitude contains a component that is associated with the exchange of a Kaluza-Klein charge between the boundaries, which is interpreted as the exchange of a D-particle between orientifold planes in the Type IA theory. This is related by T-duality to the effect of a non-BPS D-instanton in the Type I theory which is only invariant under those elements of O(16)xSO(16) that are in SO(16)xSO(16).

On orientifolds, discrete torsion, branes and M theory

We find some lifts to M theory of orientifold and orbifold planes including the O1, O3 and O5 planes of Type IIB and their transformations under SL(2,Z). The possible discrete torsion variants (or K theory classes) are explored, and are interpreted as arising from brane intersections with planes. We find new variants of the O0 and of an orbifold line (OF1) and determine their tensions in some cases. A systematic review of orientifolds, M orientifolds, and known M lifts, with some new clarifications is included together with a discussion of the role of T duality.

Instantons in [formula omitted]Sp( N) superconformal gauge theories and the AdS/CFT correspondence

We study, using ADHM construction, instanton effects in an N=2 superconformal Sp( N) gauge theory, arising as effective field theory on a system of N D3-branes near an orientifold 7-plane and eight D7-branes in type I′ string theory. We work out the measure for the collective coordinates of multi-instantons in the gauge theory and compare with the measure for the collective coordinates of (−1)-branes in the presence of 3- and 7-branes in type I′ theory. We analyse the large- N limit of the measure and find that it admits two classes of saddle points: In the first class the space of collective coordinates has the geometry of AdS 5× S 3 which on the string theory side has the interpretation of the D-instantons being stuck on the 7-branes and therefore the resulting moduli space being AdS 5× S 3, In the second class the geometry is AdS 5× S 5/ Z 2 and on the string theory side it means that the D-instantons are free to move in the ten-dimensional bulk. We discuss in detail a correlator of four O(8) flavour currents on the Yang–Mills side, which receives contributions from the first type of saddle points only, and show that it matches with the correlator obtained from F 4 coupling on the string theory side, which receives contribution from D-instantons, in perfect accord with the AdS/CFT correspondence. In particular we observe that the sectors with an odd number of instantons give contributions to an O(8)-odd invariant coupling, thereby breaking O(8) down to SO(8) in type I′ string theory. We finally discuss correlators related to R 4, which receive contributions from both saddle points.

Comments on non-supersymmetric orientifolds at strong coupling

We consider several properties of a set of anti-D$p$-branes in the presence of orientifold $p$-planes in type II theory. This system breaks all the supersymmetries of the theory, but is free of tachyons. In particular, we center on the case of a single anti-D$p$-brane stuck at a negatively charged orientifold $p$-plane, and study its strong coupling behaviour for $p=2,3,4$. Interestingly enough, as the coupling increases the system undergoes a phase transition where an additional antibrane is created. We conclude with some remarks on the limit of large number of antibranes on top of orientifold planes.

Brane descent relations in K-theory

The various descent and duality relations among BPS and non-BPS D-branes are classified using topological K-theory. It is shown how the descent procedures for producing type-II D-branes from brane–antibrane bound states by tachyon condensation and (−1) F L projections arise as natural homomorphisms of K-groups generating the brane charges. The transformations are generalized to type-I theories and type-II orientifolds, from which the complete set of vacuum manifolds and field contents for tachyon condensation is deduced. A new set of internal descent relations is found which describes branes over orientifold planes as topological defects in the worldvolumes of brane–antibrane pairs on top of planes of higher dimension. The periodicity properties of these relations are shown to be a consequence of the fact that all fundamental bound state constructions and hence the complete spectrum of brane charges are associated with the topological solitons which classify the four Hopf fibrations.

Geometry of Orientifolds with NS-NS B-flux

We discuss geometry underlying orientifolds with non-trivial NS-NS B-flux. If D-branes wrap a torus with B-flux the rank of the gauge group is reduced due to non-commuting Wilson lines whose presence is implied by the B-flux. In the case of D-branes transverse to a torus with B-flux the rank reduction is due to a smaller number of D-branes required by tadpole cancellation conditions in the presence of B-flux as some of the orientifold planes now have the opposite orientifold projection. We point out that T-duality in the presence of B-flux is more subtle than in the case with trivial B-flux, and it is precisely consistent with the qualitative difference between the aforementioned two setups. In the case where both types of branes are present, the states in the mixed (e.g., 59) open string sectors come with a non-trivial multiplicity, which we relate to a discrete gauge symmetry due to non-zero B-flux, and construct vertex operators for the the mixed sector states. Using these results we revisit K3 orientifolds with B-flux (where K3 is a T^4/Z_M orbifold) and point out various subtleties arising in some of these models. For instance, in the Z_2 case the conformal field theory orbifold does not appear to be the consistent background for the corresponding orientifolds with B-flux. This is related to the fact that non-zero B-flux requires the presence of both O5^- as well as O5^+ planes at various Z_2 orbifold fixed points, which appears to be inconsistent with the presence of the twisted B-flux in the conformal field theory orbifold. We also consider four dimensional N=2 and N=1 supersymmetric orientifolds. We construct consistent four dimensional models with B-flux which do not suffer from difficulties encountered in the K3 cases.

Monopoles in string theory

A realization of E_{n+1} monopoles in string theory is given. The NS five brane stuck to an Orientifold eight plane is identified as the 't Hooft Polyakov monopole. Correspondingly, the moduli space of many such NS branes is identified with the moduli space of SU(2) monopoles. These monopoles transform in the spinor representation of an SO(2n) gauge group when n D8 branes are stacked upon the orientifold plane. This leads to a realization of E_{n+1} monopole moduli spaces. Charge conservation leads to a dynamical effect which does not allow the NS branes to leave the orientifold plane. This suggests that the monopole moduli space is smooth for n<8. Odd n>8 obeys a similar condition. Using a chain of dualities, we also connect our system to an Heterotic background with Kaluza-Klein monopoles.

A remark on brane stabilization in brane world

In this note we discuss dynamical mechanisms for brane stabilization in the brane world context. In particular, we consider supersymmetry preserving brane stabilization, and also brane stabilization accompanied by supersymmetry breaking. These mechanisms are realized in some four dimensional N=1 supersymmetric orientifold models. For illustrative purposes we consider two explicit orientifold models previously constructed by the author [Nucl. Phys. B 535 (1998) 311]. In both of these models branes are stabilized at a finite distance from the orientifold planes. In the first model brane stabilization occurs via supersymmetry preserving non-perturbative gauge dynamics. In the second model supersymmetry is dynamically broken, and brane stabilization is due to an interplay between non-perturbatively generated superpotential and tree-level Kähler potential.

Non-supersymmetric gauge theories from D-branes in type 0 string theory

We construct non-supersymmetric four dimensional gauge theories arising as effective theories of D-branes placed on various singularities in Type 0B string theory. We mostly focus on models which are conformal in the large N limit and present both examples appearing on self-dual D3-branes on orbifold singularities and examples including orientifold planes. Moreover, we derive type 0 Hanany-Witten setups with NS 5-branes intersected by D-branes and the corresponding rules for determining the massless spectra. Finally, we discuss possible duality symmetries (Seiberg-duality) for non-supersymmetric gauge theories within this framework.

Gravitational couplings, orientifolds and M-planes

We examine string-theory orientifold planes of various types including the Sp and SO-odd planes, and deduce the gravitational Chern-Simons couplings on their world-volumes. Consistency checks are carried out in different spacetime dimensions using various dualities, including those relating string theory with F-theory and M-theory. It is shown that when an orientifold 3-plane crosses a 5-brane, the jump in the charge is accompanied by a corresponding change in the gravitational couplings.

String junctions on backgrounds with a positively charged orientifold plane

By means of the heterotic/type IIB duality, we study properties of junctions on backgrounds with a positively charged orientifold seven-plane and D-branes, which are expected to give seven dimensional Sp(r) gauge theories. We give a modified BPS condition for the junctions and show that it reproduces the adjoint representation of the Sp(r) group.

M-theory realization of a an N = 1 supersymmetric chiral gauge theory in four dimensions

We study the M5-brane configuration of the chiral gauge theory whose type IIA brane configuration with an orientifold 6-plane (06) is studied by various authors. Much of thepaper is devoted to studying the related M-theory picture of SO/ Sp gauge theory with fundamental flavors realized in a type IIA setup with the 06-plane. The Higgs branch of N = 2 SO/ Sp gauge theory is studied and the curve corresponding to the rotated brane configuration is presented. In the chiral gauge theory, the middle NS′5-brane on top of the 06-plane is realized as a detached rational curve in M-theory. Depending on the location of the rational curve in the x 7 direction, the same curve plus the rational curve can be interpreted as describing the Coulomb branch of SU(2 N c ) chiral gauge theory, SO(2 N c )/ Sp( N c ) gauge theory with N f /( N f+4 ) fundamental hypermultiplets. Various consistency checks for this proposal are made. By introducing two more rational curves corresponding to NS′5-branes one can produce a non-trivial fixed point which mediates chiral-non-chiral transition.

Solutions to the reflection equation and integrable systems for N = 2 SQCD with classical groups

Integrable systems underlying the Seiberg-Witten solutions for the N = 2 SQCD with gauge groups SO( n) and Sp( n) are proposed. They are described by the inhomogeneous XXX spin chain with specific boundary conditions given by reflection matrices. We attribute reflection matrices to orientifold planes in the brane construction and briefly discuss its possible deformations.

Gravitational couplings of D-branes and O-planes

An explicit calculation is performed to check all the tangent bundle gravitational couplings of Dirichlet branes and Orientifold planes by scattering q gravitons with a p + 1 form Ramond-Ramond potential in the world-volume of a D( p+2 q)-brane. The structure of the D-brane Wess-Zumino term in the world-volume action is confirmed, while a different O-plane Wess-Zumino action is obtained.

Large N gauge theories from orientifolds with NS-NS B-flux

We consider D-branes and orientifold planes embedded in non-compact (orbifolded) space-time. We point out that even in the non-compact cases we can turn on non-zero (quantized) NS-NS antisymmetric B-field. In particular, we study the effect of the B- field on four-dimensional large N gauge theories from orientifolds. Thus, in theories with both D3- and D7-branes, the effect of the B-field is non-trivial: the number of D7-branes (of each species) is reduced from 8 (which is the required number if the B-field is trivial) to 4. This results in a different orientifold string theory, and, subsequently, the corresponding large N gauge theory is also different. We explicitly construct large N gauge theories from orientifolds with non-zero B- field backgrounds with N = 2, 1, 0 supersymmetries. These theories, just as their counterparts without the B-field, have the property that in the large N limit computation of any M-point correlation function reduces to the corresponding computation in the parent N = 4 supersymmetric theory with a unitary gauge group.

Brane inflation

We present a novel inflationary scenario in theories with low scale (TeV) quantum gravity, in which the standard model particles are localized on the branes whereas gravity propagates in the bulk of large extra dimensions. This inflationary scenario is natural in the brane world picture. In the lowest energy state, a number of branes sit on top of each other (or at an orientifold plane), so the vacuum energy cancels out. In the cosmological setting, some of the branes “start out” relatively displaced in the extra dimensions and the resulting vacuum energy triggers the exponential growth of the 3 non-compact dimensions. The number of e-foldings can be very large due to the very weak brane-brane interaction at large distances. In the effective four-dimensional field theory, the brane motion is described by a slowly rolling scalar field with an extremely flat plateau potential. When branes approach each other to a critical distance, the potential becomes steep and inflation ends rapidly. Then the branes “collide” and oscillate about the equilibrium point, releasing energy mostly into radiation on the branes. In this scenario, it is even possible for the electroweak Higgs field to be the inflaton.

M-theory five-brane wrapped on curves for exceptional groups

We study the M-theory five-brane wrapped around the Seiberg-Witten curves for pure classical and exceptional groups given by an integrable system. Generically, the D4-branes arise as cuts that collapse to points after compactifying the eleventh dimension and going to the semiclassical limit, producing brane configurations of NS5- and D4-branes with N = 2 gauge theories on the world volume of the four-branes. We study the symmetries of the different curves to see how orientifold planes are related to the involutions needed to obtain the distinguished Prym variety of the curve. This approach explains some subtleties encountered for the Sp(2 n) and SO(2 n + 1). Using this approach we investigate the curves for exceptional groups, especially G 2 and E 6, and show that unlike for classical groups taking the semiclassical ten-dimensional limit does not reduce the cuts to D4-branes. For G 2 we find a genus-2 quotient curve that contains the Prym and has the right properties to describe the G2 field theory, but the involutions are far more complicated than the ones for classical groups. To realize them in M-theory instead of an orientifold plane we would need another object, a kind of curved orientifold surface.