An action for a classical string,the equation of motion and group invariant classical solutions

A string action which is essentially a Willmore functional is presented and studied. This action determines the physics of a surface in Euclidean three space which can be used to model classical string configurations. By varying this action an equation of motion for the mean curvature of the surface is obtained which is shown to govern certain classical string configurations. Several classes of classical solutions for this equation are discussed from the symmetry group point of view and an application is presented.      

Antisymmetric string actions

An action is presented for the free bosonic string on external flat space in terms of an antisymmetric second-rank string background tensor which is classically equivalent to the Nambu-Goto action. Both action and field equations are entirely described in terms of 2D world-sheet forms, without any reference to a 2D metric tensor background. The analysis of its canonical formulation shows how the quadratic Virasoro constraints are generated in this case and what their connection with the Bianchi identities are. Since in the orthonormal gauge the reduced action coincides with the standard one, it has the same critical dimension D = 26. The existence of an interaction term of a purely geometric structure stemming in the extrinsic curvature is pointed out. Its action and the new string field equations are then derived. This polynomial antisymmetric string action is uniformly generalized in order to describe d < D-dimensional extended objects in D-dimensional flat space.     

New Lagrangians for Bosonic m-branes with vanishing cosmological constant

A class of conformally invariant σ model actions in 2n dimensions is shown to be classically equivalent to the Nambu-Goto action for an extended object, an m-brane (m+1=2n), embedded in a higher dimensional space-time (d⪖m+1). when m is even, a (2n + 1)-dimensional σ model action is also constructed, which is classically equivalent to the Nambu-Goto action, but in this case there is no conformal invariance. In both cases the cosmological constant can be set to zero.     

Fine structure of strings

We show that the critical behavior of the string tension and most other important qualitative properties of strings are driven by a term in the action containing extrinsic curvatures, which is necessary to add to the Nambu-Goto term. Depending on the renormalization-group structure, strings can be “creased” (with non-zero surface tension at the critical point) or “smooth”. Candidates for both possibilities and their physical relevance are discussed.     

Random surfaces with rigidity at largeD

A model of two-dimensional random surfaces embedded in aD-dimensional space is investigated in the largeD limit. Its action is that of Nambu-Goto, supplemented by an extrinsic curvature term. The surfaces considered have the topology of a torus. For largeD the (renormalized) effective action is constructed using a Pauli-Villars regularization procedure. Our main concern are finite size effects. No nonperturbative modification arises for the Coulomblike term appearing in the effective energy at large distances. The two-point function of the model implies that the fluctuations diverge logarithmically with the size, a behaviour familiar from surface roughening in lattice gauge theories.     

Finite-width corrections to the Nambu action for the Nielsen-Olesen string

The finite-width correction terms to the Nambu action for Nielsen-Olesen strings are calculated. They consist of an extrinsic curvature squared or rigidity term and a new “twist” term. The extrinsic curvature term prevents cusps forming, rounding them off with a curvature radius of the order of the string width.     

Self-duality in the theory of the bosonic p-branes

It is shown that the Nambu-Goto lagrangian for the p-branes embedded in (p+1)-dimensional euclidean spacetime has a topological character and the dynamics is determined completely by the constraints. Self-duality equations are proposed whose solutions satisfy the constraints equations and minimize the action.     

Induced Rigid String Action from Fermions

By adding the Dirac action on the world sheet, an effective action is obtained by integrating over the four-dimensional fermion fields pulled back to the world sheet. This action consists of the Nambu–Goto area term with a right dimensionful constant in front, extrinsic curvature action, and the topological Euler characteristic term. The divergent coefficients in front of these terms are absorbed in the rigid string action without fermions.     

W-symmetry and the rigid particle

We prove that W₃ is the gauge symmetry of the scale-invariant rigid particle, whose action is given by the integrated extrinsic curvature of its world line. This is achieved by showing that its equations of motion can be written in terms of the Boussinesq operator. The W₃ generators T and W then appear respectively as functions of the induced world line metric and the extrinsic curvature. We also show how the equations of motion for the standard relativistic particle arise from those of the rigid particle whenever it is consistent to impose the “zero-curvature gauge”, and how to rewrite them in terms of the KdV operator. The relation between particle models and integrable systems is further pursued in the case of the spinning particle, whose equations of motion are closely related to the SKdV operator. We also partially extend our analysis in the supersymmetric domain to the scale-invariant rigid particle by explicitly constructing a supercovariant version of its action.     

Spiky strings on NS5-branes

We study rigidly rotating strings in the near-horizon geometry of a stack of Neveu-Schwarz (NS) 5-branes. We solve the Nambu-Goto action of the fundamental string in the presence of a NS-NS two form (Bμν) and find out limiting cases corresponding to magnon and spike like solutions.     

On the geometrization of matter by exotic smoothness

In this paper we discuss the question how matter may emerge from space. For that purpose we consider the smoothness structure of spacetime as underlying structure for a geometrical model of matter. For a large class of compact 4-manifolds, the elliptic surfaces, one is able to apply the knot surgery of Fintushel and Stern to change the smoothness structure. The influence of this surgery to the Einstein–Hilbert action is discussed. Using the Weierstrass representation, we are able to show that the knotted torus used in knot surgery is represented by a spinor fulfilling the Dirac equation and leading to a Dirac term in the Einstein–Hilbert action. For sufficient complicated links and knots, there are “connecting tubes” (graph manifolds, torus bundles) which introduce an action term of a gauge field. Both terms are genuinely geometrical and characterized by the mean curvature of the components. We also discuss the gauge group of the theory to be U(1) × SU(2) ×?SU(3).     

A background-dependent approach to the theory of gravitation II. Relativistic gravitational equations

On the basis of the results of Paper I and guided by a Machian view of nature, we find new gravitational equations which are background dependent. Such equations describe a purely geometrical theory of gravitation, and their dependence on the background structure is through the total energy-momentum tensor on the past sheet of the light cone of each space-time pointx [θ^μν x, say], i.e., through the integral on the past sheet of the light cone ofx of the parallel transport of the energy-momentum tensor from the space-time point in which it is defined tox along the geodesic connecting the two space-time points. Following Gürsey, we assume that the source of the De Sitter metric is not the cosmological term, but, rather, the energy-momentum tensor of a “uniform distribution of mass scintillations” [T ^μν x, say].T ^μν x, indeed, turns out to be equal to the metric tensor times a constant factor. As a consequence, in any local inhomogeneity A of a space-time whose background structure is determined by the Perfect Cosmological Principle,θ ^μν turns out to be approximately equal to the metric tensor times a constant factor, providedT=g _αβ T ^αβ is sufficiently small and the structure of the past sheet of the light cones of the space-time points belonging to Λ is not too much perturbed by the local gravitational field. As a consequence, in Λ the new equations approximately reduce to Einstein's equations. If one considers a “superuniverse model” in which our universe is considered as a local inhomogeneity in a De Sitter background, then from the above result there follows a fortiori the agreement of the new gravitational equations with the classical tests of gravitation. Furthermore, the dependence on the background structure is such that the new equations (i) incorporate the idea that the frame has to be fixeddirectly in connection with cosmological observations, and (ii) are singular in the absence of matter in the whole space-time. Moreover, (iii) the coupling constant turns out to be dimensionless in natural units (c=1=?), and (iv) a local inertial frame in a De Sitter background is determined by the condition that with respect to it the background structure is homogeneous in space and in time and is Lorentz invariant.     

Vector field effective action in the open superstring theory

Using a path integral technique we find a closed expression (to all orders in α′) for the abelian, constant field strenght limit of the (tree) effective action for the massless vector field in the open superstring theory. The result is a modification of the Born-Infeld action found in the Bose string theory case. One-loop correction to the effective action is computed and shown to be finite if the gauge group is SO(32). It is demonstrated how the on-shell superstring scattering amplitudes can be calculated in the path-integral approach. We determine the leading (O(α') and O(α′²)) terms in the full non-abelian effective action starting from the known results for the 3-point and 4-point amplitudes. We find that because of the equivalence theorem the coefficients of some of the invariant structures in the effective lagrangian cannot be fixed from the S-matrix. In the path integral approach this ambiguity manifests itself as a 2d renormalization scheme (and Weyl gauge choice) ambiguity. We also discuss the leading terms in the gravitational effective actions in the closed (super) string theories and point out that whether or not the R² terms form the “Gauss-Bonnet” combination depends on choice of a renormalization (massless exchange subtraction) scheme.     

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.     

Monte Carlo meets the fishnet: String formation in a gas of Feynman diagrams

It is likely that the quark confinement mechanism at large N should be understood purely in terms of high-order planar Feynman diagrams; in particular, the center of the gauge group can play no role whatever. With this motivation I consider the diagrammatic expansion of loop integrals 〈PTr exp(g∫ø)〉 in planar wrong-sign ø⁴ theory. It is shown that the sum of all fishnet diagrams contributing to the loop can be expressed as the grand partition function of an unusual gas, whose dynamics can be simulated on a computer. The “molecules” of this gas correspond to vertices of the position-space diagrams, the molecular interactions are determined by the propagators, and the coupling constant plays the role of a chemical potential. The most remarkable feature of this gas is the existence of a critical coupling g_c, where string formation takes place. As g → g_c the fishnet vertices tend to cluster around the minimal surface of the loop, thereby forming a string. This is clearly illustrated with the help of computer graphics. The role of asymptotic freedom in bringing the coupling to the critical point, and the connection to the Polyakov string, are also discussed. In the hamiltonian formulation a new and very straightforward explanation of quark confinement is presented.     

Space-time compactification/decompactification transitions via lightlike branes

We consider Einstein-Maxwell-Kalb-Ramond gravity-matter system in bulk space-time interacting self-consistently with two (widely separated) codimension-one electrically charged lightlike branes. The lightlike brane dynamics is explicitly given by manifestly reparametrization invariant world-volume actions in two equivalent dual to each other formulations (Polyakov-type and Nambu-Goto-type ones) proposed in our previous work. We find an explicit solution of the pertinent Einstein-Maxwell-Kalb-Ramond-lightlike-brane equations of motion describing a “two-throat” wormhole-like space-time consisting of a “left” compactified Bertotti-Robinson universe connected to a “middle” non-compact Reissner-Nordström- de-Sitter space-time region, which in turn is connected to another “right” compactified Bertotti-Robinson universe. Each of the lightlike branes automatically occupies one of the “throats”, so that they dynamically induce a sequence of spontaneous space-time compactification/decompactification transitions.     

The Casimir energy of the rigid string with massive ends

The Casimir energy in the rigid string with massive ends is investigated. This system models the flux tube of finite thickness that connects the massive quarks. The string stiffness and quark mass give additive contributions to the Casimir energy, the second contribution being the same as in the Nambu-Goto string with massive ends. The string stiffness results in an additional positive contribution to the Casimir energy, however it alone does not compensate the negative Casimir energy in the Nambu-Goto string. Only common consideration of the string thickness and the quark mass enables one to get the positive Casimir energy, i.e. to remove the tachyon from the string spectrum.     

Pseudo-supergravity extension of the bosonic string

We construct a “pseudo-supersymmetric” fermionic extension of the effective action of the bosonic string in arbitrary spacetime dimension D. The theory is invariant under pseudo-supersymmetry transformations up to the quadratic fermion order, which is sufficient in order to be able to derive Killing spinor equations in bosonic backgrounds, and hence to define BPS type solutions determined by a system of first-order equations. The pseudo-supersymmetric theory can be extended by coupling it to a Yang-Mills pseudo-supermultiplet. This also allows us to construct “α^′ corrections” involving quadratic curvature terms. An exponential dilaton potential term, associated with the conformal anomaly for a bosonic string outside its critical dimension, can also be pseudo-supersymmetrised.