String theory effective actions and σ-model β-functions

String theory effective actions compatible with two recent computations of the two-loop β-functions in σ-models (including torsion) are constructed. The equivalence of these actions is explicitly demonstrated by means of field redefinitions.     

On the relationship between string low-energy effective actions and O(α′3) σ-model β-functions

We examine in detail the connection between low-energy string effective actions and the σ-model β-functions up to O(α^′3) in the torsion-free case. In particular we show that the relationship between, on the one hand, the equations of motion derived from the O(α^′3) string effective action, and, on the other hand, the O(α^′3) σ-model β-functions necessarily involves derivative operators acting on the β-functions.     

Three-loop β-functions for the superstring and heterotic string

The β-function for the (1,1) superstring is calculated explicitly up to three loops, using two different choices of the fermion quantum field. In both cases the bosonic contribution is exactly cancelled by the fermionic contribution. The calculation is repeated for the heterotic string and the non-vanishing β-function is shown to be compatible with an O(α′²) effective action in which there are no terms cubic in the Riemann tensor- or gauge-field strength. A renormalization group equation for the gauge field is constructed and its β-function is shown to vanish at the two-loop level.     

An exact identity for σ-model renormalisation β-functions

An alternative derivation of the generalised integrability condition for conformal invariance of Curi and Paffuti is given. In the torsion-free case the perturbation series can be summed up showing that the equation relates at most third derivatives of the β- functions.     

The two-dimensionalO(N) nonlinear σ-model: Renormalisation and effective actions

We establish the existence of the Wilson Renormalised trajectory of theO(N) sigma model in perturbation theory in the effective charge. This yields a proof of perturbative renormalisability, and is also relevant in the small-field analysis of the rigorous renormalisation group construction of the continuum theory.Laboratoire associé au CNRS UA 280On leave from the Tata Institute of Fundamental Research, Bombay, 400 005, IndiaResearch funded in part by NSF grant DMS 8601978     

Superstring loop corrections to σ-model ß-functions

We present an algorithm for string one-loop corrections to σ-model ß-functions in closed superstring theories, by identifying the massless poles in string one-loop amplitudes with the ultraviolet divergences in the corresponding two-dimensional σ-model. The string one-loop corrected ß-functions for the type-II superstring are given.     

Nonlinear order-parameter relaxation in the nonlinearσ-model

An equation of motion for the order parameter of a relaxational nonlinear-model is derived by renormalization-group considerations ind=2+ dimensions. The result is then used to describe the nonlinear relaxation of the order parameter after switching off an initial symmetry-breaking field below the critical temperature.     

Heterotic — type I superstring duality and low-energy effective actions

We compare order R⁴ terms in the 10-dimensional effective actions of SO(32) heterotic and type I superstrings from the point of view of duality between the two theories. Some of these terms do not receive higher-loop corrections being related by supersymmetry to ‘anomaly-cancelling’ terms which depend on the antisymmetric 2-tensor. At the same time, the consistency of duality relation implies that the ‘tree-level’ R⁴ super-invariant (the one which has ζ(3)-coefficient in the sphere part of the action) should appear also at higher orders of loop expansion, i.e. should be multiplied by a non-trivial function of the dilaton.     

Nonlinear σ-model and nucleon structure

The nonlinear σ-model with the Wess-Zumino action describes the nucleon as a soliton and incorporates the non-abelian chiral anomalies. Several studies have shown that the model works well except for the nucleon mass, which comes out consistently too large. We investigate this question beginning with the more general framework of the linear σ-model, which has besides a pseudoscalar meson sector, a fermion or quark sector, a scalar field and an interaction between the fermions via the scalar field. Using a path integral formulation, we express the fermion measure of the model as the product of a Jacobian and an invariant measure. Identifying this Jacobian as exp[iΓ _wz] , we find that the model breaks up into two parts, when in the pseudoscalar meson sector the scalar field is replaced by its vacuum value. The pseudoscalar part of the model becomes the nonlinear σ-model with the Wess-Zumino actionΓ _wz. The other part involves chiral fermions, the scalar field and their interaction. We continue this part back to the Minkowski space to determine its ground state and energy levels. We find that for a scalar field that vanishes at smallr, but rises sharply to its vacuum value at someR, the ground state energy of the interacting quark-scalar-field system can be lower than the ground state energy of the non-interacting quark system. This means the interaction between quarks and the scalar field can lead to a condensed ground state or vacuum and can reduce the overall energy of the system (a phase transition as in superconductivity). It is, therfore, not surprising that the nonlinear σ-model predicts too large a nucleon mass, since it implicitly assumes a normal non-interacting vacuum in the quark sector. Quarks are now quasiparticles that appear as excitations of the condensed vacuum. The nucleon structure that emerges from this investigation agrees fully with the phenomenological nucleon structure found from analysis of high energy elasticpp and \(\bar p\) p scattering at CERN ISR and SPS Collider.     

Necessary and sufficient conditions for all order vanishing β-functions in supersymmetric Yang-Mills theories

If the matter self-couplings in SYM theories are certain uniquely determined power series of the coupling then it is necessary and sufficient, for the vanishing of all β-functions to all orders, that the gauge β-function and the anomalous dimensions of the matter fields vanish at the one-loop order.     

Symmetry-relations for the classical two-dimensional nonlinear O(n-σ)-model

We prove the identity of two Bäcklund transformations discussed in the literature and the corresponding conservation laws for the nonlinear O(n-) model. Furthermore, we exhibit more clearly the relation between the dual transformation and the Noether symmetry.     

Almost periodic solutions to SO(2,1)σ-model field theory

We construct almost periodic conoidal wave solutions toSO(2,1)-model field theory with the help of a periodic inverse problem suggested by Dote. The solutions can be explicitly constructed in terms of Riemann functions.     

Parity operator and quantization of δ-functions

In the Weyl quantization scheme, the -function at the origin of phase space corresponds to the parity operator. The quantization of a functionf() on phase space is the operator f(/2)W()dM, whereM is the parity andW() the Weyl operator.     

Renormalization of Möbius infinities and partition function representation for the string theory effective action

We show that subtraction of the Möbius group volume in the open string massless amplitudes can be realized as a renormalization of linear 2D ultraviolet divergences in the generating functional (“partition function”). This implies that the vector field effective action can be represented as a renormalized partition function (i.e. as a path integral of the “Wilson factor”). We check this by computing several leading terms in the non-abelian effective action.     

ß-functions and the exact renormalization group

We relate ß-functions to the flow of relevant couplings in the exact renormalization group. The specific case of a cutoff γφ⁴ theory in four dimensions is discussed in detail. The underlying idea of convergence of the flow of effective lagrangians is developed to identify the ß-functions. A perturbative calculation of the ß-functions using the exact flow equations is sketched. The analysis may be extended to any system with a cutoff.     

Lie algebraic approach to τ-functions and its equations

A general construction of equations satisfied by the components of -functions is given by considering the tensor products of modules. As an example, the homogeneous basic realization of is given, leading to NLS equations.     

Keldysh technique and non-linear σ-model: basic principles and applications

The purpose of this review is to provide a comprehensive pedagogical introduction into Keldysh technique for interacting out-of-equilibrium fermionic and bosonic systems. The emphasis is placed on a functional integral representation of the underlying microscopic models. A large part of the review is devoted to derivation and applications of the non-linear σ-model for disordered metals and superconductors. We discuss topics such as transport properties, mesoscopic effects, counting statistics, interaction corrections, kinetic equations, etc. The section devoted to disordered superconductors includes the Usadel equation, fluctuation corrections, time-dependent Ginzburg–Landau theory, proximity and Josephson effects, etc.