Elementary String Research Articles

Nonextremal stringy black hole

We construct a four-dimensional BPS saturated heterotic string solution from the Taub-NUT solution. It is a nonextremal black hole solution since its Euler number is nonzero. We evaluate its black hole entropy semiclassically. We discuss the relation between the black hole entropy and the degeneracy of string states. The entropy of our string solution can be understood as the microscopic entropy which counts the elementary string states without any complications.

Scattering of strings from D-branes

We review a number of perturbative calculations describing the interactions of D-branes with massless elementary string states. The form factors for the scattering of closed strings off D-branes are closely related to the Veneziano amplitude. They show that, in interactions with strings, D-branes acquire many of their physical features: the effective size of D-branes is of order the string scale and expands with the energy of the probe, while the fixed angle scattering amplitudes fall off exponentially. We also calculate the leading process responsible for the absorption of closed strings: the amplitude for a closed string to turn into a pair of open strings attached to the D-brane. The inverse of this process describes the Hawking radiation by an excited D-brane.

Interactions involving D-branes

We investigate some aspects of the spectrum of D-branes and their interactions with closed strings. As argued earlier, a collection of many D-strings behaves at large dilaton values as a single multiply wound string. We use this result and T-duality transformations to show that a similar phenomenon occurs for effective strings produced by wrapping p-branes on a small ( p − 1)-dimensional torus, for suitable coupling. To understand the decay of an excited D-string at large dilaton values, we study the decay of an elementary string at small dilaton values. A long string, multiply wound on a circle, with a small excitation energy is found to predominantly decay into another string with the same winding number, and an unwound closed string (rather than two wound strings). This decay amplitude agrees, under duality, with the decay amplitude computed using the Born-Infeld action for the D-string. We compute the absorption cross section for the D-brane model studied by Callan and Maldacena. The absorption cross section for the dilaton equals that for the scalars obtained by reduction of the graviton, and both agree with the cross section expected from a classical hole with the same charges.

Remark on marginally stable bound states in type II string theory.

The spectrum of elementary string states in type II string theory contains ultrashort multiplets that are marginally stable. $U$-duality transformation converts these states into bound states at the threshold of $p$-branes carrying Ramond-Ramond charges, and wrapped around $p$-cycles of a torus. We propose a test for the existence of these marginally stable bound states. Using the recent results of Polchinski and Witten, we argue that the spectrum of the bound states of $p$-branes is in agreement with the prediction of $U$ duality.

Excitations of D-strings, entropy and duality

We examine the BPS and low energy non-BPS excitations of the D-string in terms of open strings that travel on the D-string. We use this to study the energy thresholds for exciting a long D-string, for arbitrary winding number. We also compute the leading correction to the entropy from non-BPS states for a long D-string, and observe the relation of all these quantities with the corresponding quantities for the elementary string.

Black hole geometry around an elementary BPS string state

We probe the geometry around an elementary BPS (EBPS) state in heterotic string theory compactified on a six-torus by scattering a massless scalar off it and comparing with the corresponding experiment in which the EBPS state is replaced by a classical extremal black hole background satisfying the BPS condition. We find that the low energy limit of the scattering amplitudes precisely agree if one takes the limit m bh ⪢ m p . In the classical experiment, beyond a certain frequency of the incident wave, part of the wave is found to be absorbed by the black hole, whereas in case of the string scattering there is a critical frequency (inelastic threshold) of the probe beyond which the EBPS state gets excited to a higher mass non-BPS elementary state. The classical absorption threshold matches exactly with the inelastic threshold in the limit of maximum degeneracy of the EBPS state of a given mass. In that limit we can therefore identify absorption by the black hole as excitation of the elementary string state to the next vibrational state of the string and consequently also identify the non-BPS string state as a non-extremal black hole.

U duality and intersecting D-branes.

The spectrum of elementary string states in type II string theory compactified on a torus contains short multiplets which are invariant under only one-quarter of the space-time supersymmetry generators. $U$-duality transformation converts these states into bound states of Dirichlet branes which wrap around intersecting cycles of the internal torus. We study a class of these bound states that are dual to the elementary string states at the first excited level, and argue that the degeneracy of these bound states is in agreement with the $U$-duality prediction.

Extremal black holes and elementary string states

We study the relationship between entropy of extremal black holes and degeneracy of Bogomol'nyi saturated elementary string states in heterotic string theory compactified on a six dimensional torus.

Supermembranes with fewer supersymmetries

The usual supermembrane solution of D = 11 supergravity interpolates between R 11 and AdS 4 × round R 7, has symmetry P 3 × SO(8) and preserves 1 2 of the spacetime supersymmetries for either orientation of the round S 7. Here we show that more general supermembrane solutions may be obtained by replacing the round S 7 by any seven-dimensional Einstein space M 7. These have symmetry P 3 × G, where G is the isometry group of M 7. For example, G = SO(5) × SO(3) for the squashed S 7. For one orientation of M 7, they preserve N 16 spacetime supersymmetries where 1 ≤ N ≤ 8 is the number of Killing spinors on M 7; for the opposite orientation they preserve no supersymmetries since then M 7 has no Killing spinors. For example N = 1 for the left-squashed S 7 owing to its G 2 Weyl holonomy, whereas N = 0 for the right-squashed S 7. All these solutions saturate the same Bogomol'nyi bound between the mass and charge. Similar replacements of S D- p-2 by Einstein spaces M D- p-2 yield new super p-brane solutions in other spacetime dimensions D ≤ 11. In particular, simultaneous dimensional reduction of the above D = 11 supermembranes on S 1 leads to a new class of D = 10 elementary string solutions which also have fewer supersymmetries.

Dynamics of extreme black holes and massive string states.

In a recent paper, Duff and Rahmfeld argued that certain massive $N_R=1/2$ states of the four-dimensional heterotic string correspond to extreme black hole solutions. We provide further, dynamical, evidence for this identification by comparing the scattering of these elementary string states with that of the corresponding extreme black holes, in the limit of low velocities.

Entropy and supersymmetry of D-dimensional extremal electric black holes versus string states

Following the work of Sen, we consider the correspondence between extremal black holes and string states in the context of the entropy. We obtain and study properties of electrically charged black hole backgrounds of tree level heterotic string theory compactified on a p-dimensional torus, for D = (10 − p) = 4,…,9. We study in particular a one-parameter extremal class of these black holes, the members of which are shown to be supersymmetric. We find that the entropy of such an extremal black hole, when calculated at the stringy stretched horizon, scales in such a way that it can be identified with the entropy of the elementary string state with the corresponding quantum numbers.

EXTREMAL BLACK HOLES AND ELEMENTARY STRING STATES

Some of the extremal black hole solutions in string theory have the same quantum numbers as the Bogomol’nyi saturated elementary string states. We explore the possibility that these black holes can be identified with elementary string excitations. It is shown that stringy effects could correct the Bekenstein-Hawking formula for the black hole entropy in such a way that it correctly reproduces the logarithm of the density of elementary string states. In particular, this entropy has the correct dependence on three independent parameters, the mass and the left-handed charge of the black hole, and the string coupling constant.

Black hole condensation and the unification of string vacua

It is argued that black hole condensation can occur at conifold singularities in the moduli space of type II Calabi-Yau string vacua. The condensate signals a smooth transition to a new Calabi-Yau space with different Euler characteristic and Hodge numbers. In this manner string theory unifies the moduli spaces of many or possibly all Calabi-Yau vacua. Elementary string states and black holes are smoothly interchanged under the transitions, and therefore cannot be invariantly distinguished. Furthermore, the transitions establish the existence of mirror symmetry for many many or possibly all Calabi-Yau manifolds.

STRONG–WEAK COUPLING DUALITY IN FOUR-DIMENSIONAL STRING THEORY

We present several pieces of evidence for strong–weak coupling duality symmetry in the heterotic string theory, compactified on a six-dimensional torus. They include symmetry of the (1) low energy effective action, (2) allowed spectrum of electric and magnetic charges in the theory, (3) allowed mass spectrum of particles saturating the Bogomol'nyi bound, and (4) Yukawa couplings between massless neutral particles and massive charged particles saturating the Bogomol'nyi bound. This duality transformation exchanges the electrically charged elementary string excitations with the magnetically charged soliton states in the theory. It is shown that the existence of a strong–weak coupling duality symmetry in four-dimensional string theory makes definite predictions about the existence of new stable monopole and dyon states in the theory with specific degeneracies, including certain supersymmetric bound states of monopoles and dyons. The relationship between strong–weak coupling duality transformation in string theory and target space duality transformation in the five-brane theory is also discussed.

Electric-magnetic duality in string theory

The electric-magnetic duality transformation in four-dimensional heterotic string theory discussed by Shapere, Trivedi and Wilczek is shown to be an exact symmetry of the equations of motion of low-energy effective field theory even after including the scalar and the vector fields, arising due to compactification, in the effective field theory. Using this duality transformation we construct rotating black hole solutions in the effective field theory carrying both electric and magnetic charges. The spectrum of extremal magnetically charged black holes turns out to be similar to that of electrically charged elementary string excitations. We also discuss the possibility that the duality symmetry is an exact symmetry of the full string theory under which electrically charged elementary string excitations get exchanged with magnetically charged soliton-like solutions. This proposal might be made concrete following the suggestion of Dabholkar et al. that fundamental strings may be regarded as soliton-like classical solutions in the effective field theory.

MAGNETIC MONOPOLES, BOGOMOL’NYI BOUND AND SL(2, ℤ) INVARIANCE IN STRING THEORY

We show that in heterotic string theory compactified on a six-dimensional torus, the lower bound (Bogomol’nyi bound) on the dyon mass is invariant under the SL (2, ℤ) transformation that interchanges strong and weak coupling limits of the theory. Elementary string excitations are also shown to satisfy this lower bound. Finally, we identify specific monopole solutions that are related via the strong-weak coupling duality transformation to some of the elementary particles saturating the Bogomol’nyi bound, and these monopoles are shown to have the same mass and degeneracy of states as the corresponding elementary particles.

A duality between strings and 5-branes

Not only does the heterotic string admit a heterotic 5-brane as a soliton but the heterotic 5-brane admits as a soliton a heterotic string. This provides further evidence that the two theories may be dual descriptions of the same physics, with the strong coupling regime of one theory described by the weak coupling regime of the other. To illustrate this, the authors show how the energy-momentum tensor of the quadratic Yang-Mills action associated with the string reduces to that of an elementary 5-brane, and how the energy-momentum tensor of the quartic Yang-Mills action associated with the 5-brane reduces to that of an elementary string.

Syzygies among elementary string interactions in 2+1 dimensions

The elementary string interactions are the Reidemeister moves, birth/death, and fusion/fission. Such interactions have as their trace generically mapped surfaces, and these combine to form knotted surfaces in 4-space. The syzygies among these interactions are moves to such surfaces analogous to the Reidemeister moves for knots. ‘Movie’ parametrizations of these syzygies are given and interpreted in dimension 2+2. A Morse theoretic argument shows there are 15 such movie moves.

Praise and Corrective Feedback in the Remediation of Incorrect Left-Hand Positions of Elementary String Players

Subjects of the study were five elementary string class students who performed consistently with incorrect left-hand positions. Three teaching procedures were used in an attempt to modify the incorrect hand positions: corrective (negative) verbal feedback and physical prompts in a schedule of twice per minute for two minute training sessions, positive feedback on the same schedule, and increasing the training session from two to ten minutes and the praise schedule from two to four times per minute. Data were collected for two minutes during each training session in all conditions and, to assess generalization effects, during a later period of class each day when the left-hand position was not taught. Results indicated improvement for some subjects during the first and second interventions, although the improvements were not substantial nor enduring. The third intervention resulted in greater response improvement for all subjects and more generalization. Although students varied in response to different teaching styles, praise was always as effective as corrective feedback in changing the left-hand position and more durable in its effectiveness.

Pedagogy – Early Training: Elementary String Commission Report

Pedagogy – Early Training: Elementary String Commission Report