Off-shell closed string amplitudes: Towards a computation of the tachyon potential

We derive an explicit formula for the evaluation of the classical closed string action for any off-shell string field, and for the calculation of arbitrary off-shell amplitudes. The formulae require a parametrization, in terms of some moduli space coordinates, of the family of local coordinates needed to insert the off-shell states on Riemann surfaces. We discuss in detail the evaluation of the tachyon potential as a power series in the tachyon field. The expansion coefficients in this series are shown to be geometrical invariants of Strebel quadratic differentials whose variational properties imply that closed string polyhedra, among all possible choices of string vertices, yield a tachyon potential which is as small as possible order by order in the string coupling constant. Our discussion emphasizes the geometrical meaning of off-shell amplitudes.     

QCD/string holographic mapping and high-energy scattering amplitudes

We find a one-to-one mapping between low-energy string dilaton states in AdS bulk and high-energy glueball states on the corresponding boundary. This holographic mapping leads to a relation between bulk and boundary scattering amplitudes. From this relation and the dilaton action we find the appropriate momentum scaling for high-energy QCD amplitudes at fixed angles.     

Field theory of the interacting string: The closed string

We recast dual models in the language of a quantum field theory of functional fields, restricting ourselves for simplicity to the closed string model. We derive the dynamics for both scalar and spinor functional fields from a unique parametrization invariant action. Passages to the Hamiltonian formalism and second quantization are explicitly worked out for the closed mesonic string in the front form. The results are equivalent to those obtained earlier by GGRT, i.e. no ghost fields at the price of an anomalous number of space-time dimensions and a tachyon. Finally, we use the parametrization invariance requirement on the action to derive couplings of closed strings to local fields by extending a global U(1) invariance built in the free action into a local one in the functional sense. We construct the self-couplings of closed string fields by requiring these to be consistent with the gauge invariance of the external fields. This procedure uniquely leads (for the open string) to the string splitting picture of Nambu and Mandelstam. The closed string is found to cross itself and then split up into two others. Both open and closed strings have quartic interactions corresponding to strand (for the open string) and lobe (for the closed string) exchanges. The case of the interacting fermionic model is not treated here.     

Explicit parametrizations of degenerate surfaces from the KP equation and closed bosonic string amplitudes

The p-loop amplitude of closed bosonic string theory involves the integration over the moduli space. We seek an explicit parametrization of Riemann matrices in terms of 3p - 3 complex variables by solving the Kadomcev-Petviasvili (KP) equation. We find explicit solutions of this problem (Schottky problem) for certain types of degenerate surfaces. For these classes of surfaces, we obtain closed bosonic string amplitudes from the Belavin-Knizhnik theorem using our parametrizations. We show in what precise way they are related to the correlation functions on the Riemann surfaces.     

High-energy zero-norm states and symmetries of string theory

High-energy limit of zero-norm states in the old covariant first quantized spectrum of the 26D open bosonic string, together with the assumption of a smooth behavior of string theory in this limit, are used to derive infinitely many linear relations among the leading high-energy, fixed-angle behavior of four-point functions of different string states. As a result, ratios among all high-energy scattering amplitudes of four arbitrary string states can be calculated algebraically and the leading order amplitudes can be expressed in terms of that of four tachyons as conjectured by Gross in 1988. A dual calculation can also be performed and equivalent results are obtained by taking the high-energy limit of Virasoro constraints. Finally, we compute all high-energy scattering amplitudes of three tachyons and one massive state at the leading order by saddle-point approximation to verify our results.     

Extended holomorphic anomaly and loop amplitudes in open topological string

Open topological string amplitudes on compact Calabi–Yau threefolds are shown to satisfy an extension of the holomorphic anomaly equation of Bershadsky, Cecotti, Ooguri and Vafa. The total topological charge of the D-brane configuration must vanish in order to satisfy tadpole cancellation. The boundary state of such D-branes is holomorphically captured by a Hodge theoretic normal function. Its Griffiths' infinitesimal invariant is the analogue of the closed string Yukawa coupling and plays the role of the terminator in a Feynman diagram expansion for the topological string with D-branes. The holomorphic anomaly equation is solved and the holomorphic ambiguity is fixed for some representative worldsheets of low genus and with few boundaries on the real quintic.     

Quadratic differentials and closed string vertices

Existence theorems for quadratic differentials on Riemann surfaces are used to explore the possible structure of local vertices in closed string field theory. The cell decomposition of the moduli space of the three-holed sphere is studied in some detail, and the problem of obtaining a single covering of moduli space is investigated.     

On a modification of the boundary-state formalism in off-shell string theory

We examine the application of boundary states in computing amplitudes in off-shell open string theory. We find a straightforward generalization of a boundary state which produces the correct matrix elements with on-shell closed string states.     

An interplay between static potential and Reggeon trajectory for QCD string

I consider two cases where QCD string is described by an effective theory of long strings: the static potential and meson scattering amplitudes in the Regge regime. I show how they can be solved in the mean-field approximation, justified by the large number of space–time dimensions, and argue that it turns out to be exact. I compare contributions from QCD string and perturbative QCD and discuss experimental consequences for the scattering amplitudes.     

Scattering amplitudes of QCD string in the worldline formalism

I review the derivation of large-N QCD meson scattering amplitudes in the Regge regime, where the effective theory of long strings applies in d = 4. A special attention is payed to the reparametrization path integral which plays a crucial role in the consistency of off-shell amplitudes. I show how the linear Reggeon trajectory is obtained for QCD string in the mean-field approximation, which turns out to be exact for the Nambu-Goto string, and discuss the interrelation with perturbative QCD.     

From conformal quantum mechanics on bordered Riemann surfaces to covariant string field theory

The Faddeev-Popov determinant of the reparametrization ghosts of the closed bosonic string on bordered Riemann surfaces is investigated. It is found that the standard ghost action has to be supplemented by additional boundary terms. These additional terms lead to a nontrivial ghost transition amplitude. It is shown that boundary variations of ghost transition amplitudes are generated by the ghost Virasoro operators. The relevance of the boundary variation operator for string field theory is illustrated. A new interpretation of the BRST operator in terms of boundary variations is obtained and the BRST invariance of the transition amplitudes is demonstrated. A generalization of the Siegel gauge of second quantized closed bosonic string field theory is presented.     

Dilaton vertices of the closed superstring and the heterotic string

BRST - and conformally invariant dilaton vertices of the closed superstring and the heterotic string are constructed, whose counter terms are consistent with the dilaton terms introduced in the non-linear sigma model approach of string theory.     

Lorentz invariance of lightcone string field theory: (I). Bosonic closed string

We construct the Lorentz generators for bosonic closed lightcone string field theory with cubic interactions, and prove that the Lorentz algebra closes if the spacetime dimension is 26.     

Four-photon scattering amplitude in bosonic open string field theory

The covariant string field theory, which is formulated as a natural generalization of the light-cone gauge approach, is used to calculate three- and four-photon scattering amplitudes explicitly at tree level. The results are found to be consistent with the first quantized covariant approach.     

Gravity beyond quantum theory: Electron as a closed heterotic string

The observable gravitational and electromagnetic parameters of an electron determine that its background should be the Kerr-Newman (KN) solution of the rotating black hole without horizons. This metric has a topological defect—the Kerr singular ring which, as we show, is a closed heterotic string of the Compton radius a = ?/(2m). We show that the Dirac equation emerges as a consequence of the underlying KN gravity and string theory. Regularization of the KN solution leads to a model of gravitating soliton of the oscillon type, in which the closed heterotic string is positioned on the edge rim of a disklike vacuum bubble. It is suggested that the string-like core of the electron should be experimentally observable by the novel methods of the “Deeply Virtual (nonforward) Compton scattering”.     

Evaluation of the one loop string path integral

We evaluate Polyakov's path integral for the sum over all closed surfaces with the topology of a torus, in the critical dimensiond = 26. The result is applied to the partition function and cosmological constant of the free bosonic string, and to tachyon scattering amplitudes.     

Relativistic-electron scattering on an atomic string of a crystal at ultrasmall angles of particle incidence on a string

The classical and quantum scattering of fast electrons on an atomic string of a crystal is considered at angles of particle incidence on the string that are much smaller than the critical angle of axial channeling. The investigation was performed within the simplest approximation of the continuous atomic-string potential in the form of a cutoff Coulomb potential. For this case, the azimuthal scattering of particles at an angle exceeding 180° in the plane orthogonal to the string axis is shown to be possible for all impact-parameter values. It is demonstrated that, in particle scattering on a string, an effect can occur that is similar to the Ramsauer-Townsend effect, which consists in a considerable reduction of the total cross section for slow-electron scattering on atoms.     

The cosmological constant of modular closed string field theory. I

We formulate closed string field theory as a quantum theory of modular geometry. We determine the full interacting quantum hamiltonian to all loop orders in perturbation theory. The free action has a new highly non-linear symmetry acting on the string field, and the kinetic operator. Perturbatively we demonstrate that the new theory gives the correct expression for the cosmological constant that is ultraviolet finite to one-loop order.