Open String Sector Research Articles

Effective string tension and renormalizability: string theory in a noncommutative space

We show that the one loop amplitudes of open and closed string theory in a constant background two-form tensor field are characterized by an effective string tension larger than the fundamental string tension, and by the appearance of antisymmetric and symmetric noncommutativity parameters. We derive the form of the phase functions normalizing planar and nonplanar tachyon scattering amplitudes in this background, verifying the decoupling of the closed string sector in the regime of infinite momentum transfer. We show that the functional dependence of the phase functions on the antisymmetric star product of external momenta permits interpretation as a finite wavefunction renormalization of vertex operators in the open string sector. Using world-sheet duality we clarify the regimes of finite and zero momentum transfer between boundaries, demonstrating the existence of poles in the nonplanar amplitude when the momentum transfer equals the mass of an on-shell closed string state. Neither noncommutativity parameter has any impact on the renormalizability of open and closed string theory in the Wilsonian sense. We comment on the relationship to noncommutative scalar field theory and the UV-IR correspondence.

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.

Open-Closed String Field Theory in a Background B Field

In this paper, we study open-closed string field theory in the background B-field in the so-called alpha=p^{+} formulation. The string field theory in the infrared gives noncommutative gauge theory in the open string sector. Since this theory includes closed string fields as dynamical variables, we can obtain another string field theory in the same background through the condensation of closed string fields, whose low-energy effective action does not show the noncommutativity of spacetime. Although we have two string field theories in the same background, we show that these theories are equivalent. In fact, we give the unitary transformation from string fields in one of them to string fields in the other.

Discrete torsion in non-geometric orbifolds and their open-string descendants

We discuss some Z N L × Z N R orbifold compactifications of the type IIB superstring to D=4,6 dimensions and their type I descendants. Although the Z N L × Z N R generators act asymmetrically on the chiral string modes, they result into left–right symmetric models that admit sensible unorientable reductions. We carefully work out the phases that appear in the modular transformations of the chiral amplitudes and identify the possibility of introducing discrete torsion. We propose a simplifying ansatz for the construction of the open-string descendants in which the transverse-channel Klein-bottle, annulus and Möbius-strip amplitudes are numerically identical in the proper parametrization of the world-sheet. A simple variant of the ansatz for the Z 2 L × Z 2 R orbifold gives rise to models with supersymmetry breaking in the open-string sector.

Supersymmetric orientifolds in 6D with D-branes at angles

We study a new class of N=1 supersymmetric orientifolds in six space-time dimensions. The world-sheet parity transformation is combined with a permutation of the internal complex coordinates. In contrast to ordinary orientifolds the twisted sectors contribute to the Klein bottle amplitude leading to new tadpoles to be cancelled by twisted open string sectors. They arise from open strings stretched between D7-branes intersecting at non-trivial angles. We study in detail the Z 3 , Z 4 and Z 6 permutational orientifolds obtaining in all cases anomaly free massless spectra.

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.

Non-commutative unification in the brane world

We point out that in (open) string compactifications with non-zero NS–NS B-field we can have large Kaluza–Klein thresholds even in the small volume limit. In this limit the corresponding gauge theory description is in terms of a compactification on a non-commutative space (e.g., a torus or an orbifold thereof). Based on this observation we discuss a brane world scenario of non-commutative unification via Kaluza–Klein thresholds. In this scenario, the unification scale can be lowered down to the TeV-range, yet the corresponding compactification radii are smaller than the string length. We discuss a potential application of this scenario in the context of obtaining mixing between different chiral generations which is not exponentially suppressed – as we point out, such mixing is expected to be exponentially suppressed in certain setups with large volume compactifications. We also point out that T-duality is broken by certain non-perturbative twisted open string sectors which are supposed to give rise to chiral generations, so that in the case of a small volume compactification with a rational B-field we cannot T-dualize to a large volume description. In this sense, the corresponding field theoretic picture of unification via Kaluza–Klein thresholds in this setup is best described in the non-commutative language.

A REVIEW OF SYMMETRY ALGEBRAS OF QUANTUM MATRIX MODELS IN THE LARGE N LIMIT

This is a review article in which we will introduce, in a unifying fashion and with more intermediate steps in some difficult calculations, two infinite-dimensional Lie algebras of quantum matrix models, one for the open string sector and the other for the closed string sector. Physical observables of quantum matrix models in the large N limit can be expressed as elements of these Lie algebras. We will see that both algebras arise as quotient algebras of a larger Lie algebra. We will also discuss some properties of these Lie algebras not published elsewhere yet, and briefly review their relationship with well-known algebras like the Cuntz algebra, the Witt algebra and the Virasoro algebra. We will also review how the Yang–Mills theory, various low energy effective models of string theory, quantum gravity, string-bit models, and the quantum spin chain models can be formulated as quantum matrix models. Studying these algebras thus help us understand the common symmetry of these physical systems.

Non-perturbative K3 orientifolds with NS–NS B-flux

We consider non-perturbative six dimensional N=1 space-time supersymmetric orientifolds of Type IIB on K3 with non-trivial NS–NS B-flux. All of these models are non-perturbative in both orientifold and heterotic pictures. Thus, some states in such compactifications arise in “twisted” open string sectors which lack world-sheet description in terms of D-branes. We also discuss their dual F-theory compactifications on certain Voisin–Borcea orbifolds. In particular, the explicit construction of non-perturbative K3 orientifolds with NS–NS B-flux gives additional evidence for the conjectured extension of Nikulin's classification in the context of Voisin–Borcea orbifolds.

Non-perturbative orientifolds

We consider non-perturbative six and four dimensional N=1 space-time supersymmetric orientifolds. Some states in such compactifications arise in “twisted” open string sectors which lack world-sheet description in terms of D-branes. Using Type I-heterotic duality we are able to obtain the massless spectra for some of such orientifolds. The four dimensional compactification we discus in this context is an example of a chiral N=1 supersymmetric string vacuum which is non-perturbative in both orientifold and heterotic pictures. In particular, it contains both D9- and D5-branes plus non-perturbative “twisted” open string sector states as well.

Type IIB orientifolds with NS-NS antisymmetric tensor backgrounds

We consider six dimensional $\mathcal{N}=1$ space-time supersymmetric type IIB orientifolds with a nonzero untwisted NS-NS sector $B$ field. The $B$ field is quantized due to the requirement that the type IIB spectrum be left-right symmetric. The presence of the $B$ field results in rank reduction of both 99 and 55 open string sector gauge groups. We point out that in some of the models with a nonzero $B$ field there are extra tensor multiplets in the ${\mathbf{Z}}_{2}$ twisted closed string sector, and we explain their origin in a simple example. Also, the 59 open string sector states come with a multiplicity that depends on the $B$ field. These two facts are in accord with anomaly cancellation requirements. We point out relations between various orientifolds with and without the $B$ field, and also discuss the F-theory duals of these models.

Open strings on the Neveu-Schwarz 5-brane

We analyze the propagation of open and unoriented strings on the Neveu-Schwarz 5-brane (N5-brane) along the lines of a similar analysis for the SU(2) WZNW models. We discuss the two classes of open descendants of the diagonal models and a series of Z 2 projected models which exist only for even values of the level k and correspond to branes at D-type orbifold singularities. The resulting configurations of branes and planes are T-dual to those relevant to the study of dualities in super Yang-Mills theories. The association of Chan-Paton factors to D-brane multiplicities is possible in the semi-classical limit k → ∞, but due to strong curvature effects is unclear for finite k. We show that the introduction of a magnetic field implies a twist of the SU(2) current algebra in the open-string sector leading to space-time supersymmetry breaking.

Strings and D-branes at high temperature

The thermodynamics of a gas of strings and D-branes near the Hagedorn transition is described by a coupled set of Boltzmann equations for weakly interacting open and closed long strings. The resulting distributions are dominated by the open string sector, indicating that D-branes grow to fill space at high temperature.

An orientifold of Type-IIB theory on K3

A new orientifold of Type-IIB theory on K3 is constructed that has N=1 supersymmetry in six dimensions. The orientifold symmetry consists ofaa |Z 2 involution of K3 combined with orientation reversal on the world-sheet. The closed-string sector in the resulting theory contains nine tensor multiplets and twelve neutral hypermultiplets in addition to the gravity multiplet, and is anomaly free by itself. The open-string sector contains only five-branes and gives rise to maximal gauge groups SO(16) or U(8) × U(8) at different points in the moduli space. Anomalies are canceled by a generalization of the Green-Schwarz mechanism that involves more than one tensor multiplets.

Planar duality in the discrete series

The models of the discrete series of Belavin, Polyakov and Zamolodchikov may be generalized by allowing for suitable open-string sectors. The resulting open-string disk amplitudes have planar duality and proper factorization in the presence of a global internal Chan-Paton symmetry.

NO GHOST THEOREM AND COHOMOLOGY THEOREM FOR STRINGS IN ARBITRARY STATIC BACKGROUNDS

We consider a string moving in an arbitrary time-independent background given by an arbitrary conformal field theory of appropriate central charge (e.g., c=25 for bosonic string) and one flat time-like dimension. We show that the physical subspace of the Hilbert space is positive semi-definite (no ghost theorem) and that the cohomology of the BRST operator is trivial except for the ghost number one (for open bosonic string) sector (cohomology theorem). Both the proofs are reductio ad absurdum proofs based on the corresponding theorems for the strings moving in flat background. In cases where there is an extra flat space-like dimension (besides the flat time-like one), the transverse subspace with positive-definite norm can be constructed.

Casimir effects in superstring theories

Compactification of the space in superstring theories is studied by calculating the one-loop effective potential provided that the supersymmetry is softly broken by some mechanism. The Casimir energy of closed string modes is shown to compactify the ten-dimensional Minkowski space M 10 into a stable M d ×T 10− d (1⩽ d<10) with a i= α′ R i =1 , where α′ and R i are the Regge slope and the ith radius of the torus T 10− d , respectively. In the SST-I model, although the closed string (gravity) sector decouples from the open string (Yang-Mills) sector in α′ → 0 limit, the former functions as an initiator as well as a stabilizer of the compactification.

Supersymmetrical dual string theory

It is proved that the physical states of the open-string sector of the ten-dimensional string theory form supersymmetry multiplets. The proof is achieved by first constructing a new formulation of the spectrum generating algebra, and then forming the supersymmetry operator (as well as associated operators) and demonstrating that it transforms as a spinor under Lorentz transformations and has the correct anticommutation relations. The results can be interpreted either in terms of a covariant formulation or in terms of a light-cone gauge formulation. In the former case certain formulas are not completely proved, although they are in the latter. When interpreted in four dimensions (by dimensional reduction) the string theory provides an interacting theory of an infinite number of massive representations of N = 4 supersymmetry involving particles of arbitrarily high spin.