Ten-dimensional Superstring Research Articles

Stable vacua for tachyonic strings

In addition to the familiar string theories with spacetime supersymmetry, there exist a number of non-supersymmetric ten-dimensional superstrings. Nearly all of these theories have closed string tachyons, indicating a misidentification of the vacuum around which they are to be quantized. In this work, we identify (meta)stable vacua for all known tachyonic superstrings. These vacua are all lower-dimensional, with most of them being familiar two-dimensional string theories. However, there are also intriguing examples of solutions in dimensions 6, 8, and 9, with respective gauge groups $E_7 \times E_7$, $SU(16)$, and $E_8$. These special vacua have positive vacuum energy and no moduli.

Non-Abelian vortex in four dimensions as a critical superstring

We discuss recent progress in describing a certain non-Abelian vortex string as a critical superstring on a conifold and clarify some subtle points. This particular solitonic vortex is supported in four-dimensional N=2 supersymmetric QCD with the U(2) gauge group, N_f=4 quark flavors and the Fayet-Iliopoulos term. Under certain conditions the non-Abelian vortex can become infinitely thin and can be interpreted as a critical ten-dimensional superstring. In addition to four translational moduli the non-Abelian vortex under consideration carries six orientational and size moduli. The vortex moduli dynamics are described by a two-dimensional sigma model with the target space {R}^4\times Y_6 where Y_6 is a non-compact Calabi-Yau conifold. The closed string states which emerge in four dimensions (4D) are identified with hadrons of 4D bulk N=2 QCD. It turns out that most of the states arising from the ten-dimensional graviton spectrum are non-dynamical in 4D. A single dynamical massless hypermultiplet associated with the deformation of the complex structure of the conifold is found. It is interpreted as a monopole-monopole baryon of the 4D theory (at strong coupling).

Superstring limit of Yang–Mills theories

It was pointed out by Shifman and Yung that the critical superstring on $X^{10}={\mathbb R}^4\times Y^6$, where $Y^6$ is the resolved conifold, appears as an effective theory for a U(2) Yang-Mills-Higgs system with four fundamental Higgs scalars defined on $\Sigma_2\times{\mathbb R}^2$, where $\Sigma_2$ is a two-dimensional Lorentzian manifold. Their Yang-Mills model supports semilocal vortices on ${\mathbb R}^2\subset\Sigma_2\times{\mathbb R}^2$ with a moduli space $X^{10}$. When the moduli of slowly moving thin vortices depend on the coordinates of $\Sigma_2$, the vortex strings can be identified with critical fundamental strings. We show that similar results can be obtained for the low-energy limit of pure Yang-Mills theory on $\Sigma_2\times T^2_p$, where $T^2_p$ is a two-dimensional torus with a puncture $p$. The solitonic vortices of Shifman and Yung then get replaced by flat connections. Various ten-dimensional superstring target spaces can be obtained as moduli spaces of flat connections on $T^2_p$, depending on the choice of the gauge group. The full Green-Schwarz sigma model requires extending the gauge group to a supergroup and augmenting the action with a topological term.

The Quantum Consistency of the Ten-Dimensional Heterotic String Effective Action

The finiteness of superstring theory at each order in perturbation theory is considered with respect to the ten-dimensional effective action. The quantum consistency of the ten-dimensional superstring effective action is confirmed with an analysis of the perturbative expansion of the quartic sector. It is found to be compatible with the finiteness of reduced four-dimensional theory. Furthermore, implications for the validity of superstring perturbation theory at lower energies is considered.

Fermionic superstring loop amplitudes in the pure spinor formalism

The pure spinor formulation of the ten-dimensional superstring leads to manifestly supersymmetric loop amplitudes, expressed as integrals in pure spinor superspace. This paper explores different methods to evaluate these integrals and then uses them to calculate the kinematic factors of the one-loop and two-loop massless four-point amplitudes involving two and four Ramond states.

THREE GENERATIONS OF SUSY STANDARD MODEL OF NAMBU–GOTO STRING

A four-dimensional superstring have been constructed starting from a 26-dimensional bosonic string. Fermions are introduced by noting the Mandelstam's proof of equivalence of two fermions to one boson in 1+1 dimensions. The action of the superstring is invariant under SO (6) ⊗ SO (5) in the worldsheet. It has four bosonic coordinates and 44 Majorana fermions of SO (3, 1). Here the superstring action is shown to be invariant under conformal and superconformal transformations with the usual conformal and superconformal ghosts of the ten-dimensional superstrings. The massless spectrum obtained by quantizing the action, contain vector mesons which are generators of the SO (6) ⊗ SO (5). Using Wilson loops, this product group is proven to descend to Z3 ⊗ SU (3) ⊗ SU (2) ⊗ U (1) without breaking supersymmetry. Thus there are just three generations of quarks and leptons.

On the Worldsheet Derivation of Large N Dualities for the Superstring

Large N topological string dualities have led to a class of proposed open/closed dualities for superstrings. In the topological string context, the worldsheet derivation of these dualities has already been given. In this paper we take the first step in deriving the full ten-dimensional superstring dualities by showing how the dualities arise on the superstring worldsheet at the level of F terms. As part of this derivation, we show for F-term computations that the hybrid formalism for the superstring is equivalent to a $\hat c=5$ topological string in ten-dimensional spacetime. Using the $\hat c=5$ description, we then show that the D brane boundary state for the ten-dimensional open superstring naturally emerges on the worldsheet of the closed superstring dual.

Ten-dimensional supergravity constraints from the pure spinor formalism for the superstring

It has recently been shown that the ten-dimensional superstring can be quantized using the BRST operator Q=∮ λ α d α , where λ α is a pure spinor satisfying λγ m λ=0 and d α is the fermionic supersymmetric derivative. In this paper, the pure spinor version of superstring theory is formulated in a curved supergravity background and it is shown that nilpotency and holomorphicity of the pure spinor BRST operator imply the on-shell superspace constraints of the supergravity background. This is shown to lowest order in α′ for the heterotic and Type II superstrings, thus providing a compact pure spinor version of the ten-dimensional superspace constraints for N=1 Type IIA and Type IIB supergravities. Since quantization is straightforward using the pure spinor version of the superstring, it is expected that these methods can also be used to compute higher-order α′ corrections to the ten-dimensional superspace constraints.

The masses of elementary particles

The calculation for the masses of elementary particles is based on the ten dimensional superstring. A Kaluza--Klein substructure is proposed for the six extra spatial dimensions in the ten-dimensional superstring. The periodic table of elementary particles can be constructed from the Kaluza--Klein substructure. All leptons, quarks, and gauge bosons can be placed in the periodic table. The masses of elementary particles can be generated using only four known constants: the number of the extra spatial dimensions in the ten-dimensional superstring, the mass of the electron, the mass of Z0, and αe. The calculated masses derived from the periodic table are in good agreement with the observed values. The calculated mass (176.6 GeV) of the top quark is in excellent agreement with the observed mass (176 GeV). The periodic table also provides an explanation for why the cross section for jet transverse energy, >200 GeV, is greater than the perturbative QCD prediction. The periodic table based on the ten-dimensional superstring provides the most comprehensive explanation and calculation for the properties and the masses of all elementary particles.

Ten to eleven: It is not too late

Superunification underwent a major paradigm shift in 1984 when eleven-dimensional supergravity was knocked off its pedastal by ten-dimensional superstrings. This last year has witnessed a new shift of equal proportions: perturbative ten-dimensional superstrings have in their turn been superseded by a new non-perturbative theory called M-theory, which describes supermembranes and superfivebranes, which subsumes all five consistent string theories and whose low energy limit is, ironically, eleven-dimensional supergravity. In particular, six-dimensional string/string duality follows from membrane/fivebrane duality by compactifying M-theory on S 1/Z 2×K3 (heterotic/heterotic duality) or S 1×K3 (Type IIA/heterotic duality) or S 1/Z 2×T 4 (heterotic/Type IIA duality) or S 1×T 4 (Type IIA/Type IIA duality).

M THEORY (THE THEORY FORMERLY KNOWN AS STRINGS)

Superunification underwent a major paradigm shift in 1984 when eleven-dimensional supergravity was knocked off its pedestal by ten-dimensional superstrings. This last year has witnessed a new shift of equal proportions: perturbative ten-dimensional superstrings have in their turn been superseded by a new nonperturbative theory called M theory, which describes supermembranes and superfivebranes, which subsumes all five consistent string theories and whose low energy limit is, ironically, eleven-dimensional supergravity. In particular, six-dimensional string/string duality follows from membrane/fivebrane duality by compactifying M theory on S1/Z2×K3 (heterotic/heterotic duality) or S1×K3 (Type IIA/heterotic duality) or S1/Z2×T4 (heterotic/Type IIA duality) or S1×T4 (Type IIA/Type IIA duality).

Maximally supersymmetric string theories in D

The existence of maximally supersymmetric solutions to heterotic string theory that are not toroidal compactifications of the ten-dimensional superstring is established. We construct an exact fermionic realization of an N=1 supersymmetric string theory in D=8 with non-simply-laced gauge group Sp(20). Toroidal compactification to six and four dimensions gives maximally extended supersymmetric theories with reduced rank (4,12) and (6,14) respectively.

ON THE HEISENBERG INDETERMINACY PRINCIPLE AND THE FORMATION OF STRUCTURE IN THE UNIVERSE

The Heisenberg indeterminacy principle ΔpaΔqa ~ ħ, relating canonically conjugate variables pa and qa, is quantified for the classical action obtained by the reduction of the ten-dimensional heterotic superstring theory to four dimensions, in the mini-superspace (Friedmann space-time) [Formula: see text]. There are two coordinates, α and [Formula: see text], representing position and velocity, respectively, the canonical momenta being [Formula: see text] and [Formula: see text]. In both cases, the result can be expressed as an indeterminacy in the time, (Δt/t)2. The fluctuations connecting position and velocity decrease with time and are always undetectably small, Δt/t ≲ 10−44. But the fluctuations involving velocity and acceleration increase with time, and are evaluated at the time te of equipartition of radiation and matter in the universe. Translated first into a metric fluctuation [Formula: see text], this is equivalent to a Gaussian, scale-invariant spectrum of density fluctuations of magnitude [Formula: see text], where the dimensionless constant B depends only on the compactification scheme. For a Calabi–Yau internal space, the estimate B ≈ 3 implies that ζ ≈ 2 × 10−4, which is sufficient for the creation of galaxies and in approximate agreement with observations of the anisotropy of the cosmic microwave background radiation by COBE and at Tenerife.

Finiteness and unitarity of Lorentz-covariant Green-Schwarz superstring amplitudes

In two recent papers, a new method was developed for calculating ten-dimensional superstring amplitudes with an arbitrary number of loops and external massless bosons or fermions, and for expressing the amplitudes in manifestly Lorentz-invariant form. In this paper, all amplitudes with external massless bosons are proven to be finite by explicitly checking for divergences when the Riemann surface degenerates. These amplitudes are then proven to be unitary by choosing light-cone moduli for the surface and comparing with the light-cone Green-Schwarz formalism.

DOES HETEROTIC STRING GENERATE CHERN-SIMONS ACTION IN THREE DIMENSIONS?

We study the possibility that the ten-dimensional (D=10) heterotic superstring generates a class of Chern-Simons theories in three dimensions, via compactifications on the internal seven-dimensional manifolds. We give an explicit example of such compactifications on (Calabi-Yau)6×S1×(Mink.)3, using the dual formulation of the heterotic string. The string tension 1/α′ as well as the Yang-Mills coupling constant g2 in D=10 is double-quantized in terms of two integers, through a condensate of the antisymmetric field strength NM1…M7, which is dual to the ordinary field strength GMNP. The resultant Chern-Simons theory has the gauge group E6⊗E8, containing also the usual kinetic terms of the gauge fields, which may be interpreted as regulator terms. This result suggests a close relationship between the D=10 heterotic string and D=3 Chern-Simons theory.

AN ESTIMATION OF THE ONE-LOOP MASS SHIFTS IN OPEN SUPERSTRING THEORY

The quantitative studies of the one-loop mass shifts for the first excited states in ten-dimensional open superstring are given. Removing the apparent singularities in the on-shell integral expressions by an off-shell analytic continuation, we obtain the finite mass shifts with imaginary parts. We numerically evaluate the real part to find the smallness of the magnitude of one-loop effect.

The four-gravitino tree amplitude in the superstring

Using superconformal field theoretic methods, the tree amplitude for four massive gravitinos in the open ten-dimensional superstring is calculated. The amplitude is given in a manifestly Lorentz-covariant form.

Gravitational equations of motion at the string scale

We consider the magnitude of string corrections to Einstein's equations to first order in the inverse string tension α′ and their importance at the “limiting” temperature implied by the exponential mass spectrum. In particular we consider both four- and ten-dimensional superstring theories with and without the inclusion of dilaton effects and the role of compactification. Cosmological applications such as inflation are also discussed. We find that a period of inflation necessarily requires a non-trivial role for the dilaton field.

Superstring effects on D=4, N=1 supergravity

We give a lagrangian and supersymmetry transformation rules for the four-dimensional N =1 supergravity sector of superstring theories with their O ( α ′) corrections, obtained by the dimensional reduction á la Witten of the effective action of the ten-dimensional heterotic superstring. We also give general forms of O ( α ′ ″) corrections to supersymmetry transformation rules which arise through an axial vector superfield. Since our system is based on the ten-dimensional superstring without any auxiliary fields, our four-dimensional N =1 supergravity is free of auxiliary fields. Our point-field theory lagrangian is supposed to describe the mass-less fields in the untwisted sector of the ten-dimensional heterotic superstring propagating on orbifolds.

Anomalies in super p-branes

The Green-Schwarz action for the superstring may be generalised to the case of extended objects of p spatial dimensions. Twelve such super p-branes exist classically, including the superstring in ten, six, four and three spacetime dimensions. It is known for the superstring that only the ten-dimensional theory is consistent at the quantum level. In the light-cone quantisation that the authors use, consistency of the theory is tested by the closure of the spacetime Lorentz and supersymmetry algebras. For the higher-dimensional super p-branes, the corresponding analysis is much more complicated, owing to the intrinsic non-linearity of the theories. However, certain consistency checks are possible, based on the fact that the degeneracies of the states of the theories should be compatible with the appropriate little groups for massless and massive modes. The authors apply such counting tests to the twelve super p-branes and conclude that all, except for the ten-dimensional superstring and the eleven-dimensional supermembrane, are quantum mechanically inconsistent. The eleven-dimensional case passes all the tests that they could perform. Further analysis is required in order to settle whether this remaining case is consistent.