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.     

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.     

New nonlinear σ-model with N=4 extended supersymmetry

A new nonlinear -model with N=4 supersymmetry in d=2 is constructed in terms of a N=2 tensor multiplet in d=4. Four equivalent formulations of the theory in superspace are obtained. The component structure of the model is considered by means of the Legendre transformation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 70–74, September, 1987.     

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.     

The Green-Schwarz superstring σ-model

We study the Green-Schwarz superstring coupled to a ten-dimensional supergravity background. We exhibit the background and quantum invariances of the action expanded in terms of normal coordinates. In the presence of nontrivial fermionic background we go to components and identity the vertex operators for emitting the physical supergravity fields. A comparison with the Neveu-Schwarz-Ramond formalism is then obtained by going to light-cone gauge.     

Geometrical structure and ultraviolet finiteness in the supersymmetric σ-model

A complete geometrical classification of supersymmetric -models is given. Extended supersymmetry requires covariantly constant complex structures, and Kahler and hyperkahler manifolds play a special role. As an application of the classification, it is shown that a particular class of these models is on-shell ultraviolet finite to all orders in perturbation theory.     

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     

σ-model β-functions and ghost-free string effective actions

We explore the relationship between the effective action for a bosonic string propagating in a massless background and the corresponding σ-model, and show that a simple criterion proposed by Mavromatos and Miramontes leads to a manifestly ghost-free form for the O(α'²) action     

Locally supersymmetric σ-model with Wess-Zumino term in two dimensions and critical dimensions for strings

We construct an N = 1 locally supersymmetric σ-model with a Wess-Zumino term coupled to supergravity in two dimensions. If one takes the σ-model manifold to be the product of d-dimensional Minkowski space M^d and a group manifold G, and if the radius of G is quantized in appropriate units of the string tension, then the model describes a Neveu-Schwarz-Ramond (NSR)-type string moving on M_d × G. (Our model generalizes earlier work of refs. [1,2] which do not contain a Wess-Zumino term and that of refs. [5,6] which is not locally supersymmetric.) The zweibein and the gravitino field equations yield constraints which generalize those of the NSR model to the case of a non-abelian group manifold. In particular, the fermionic constraint contains a new term trilinear in the fermionic fields. We quantize the theory in the light-cone gauge and derive the critical dimensions. We compute the mass spectrum of a closed string moving on M_d × G and show that massless fermions do not arise for non-abelian G for the spinning string, in agreement with the result of Friedan and Shenker [22].     

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.     

The background field method and the non-linear σ-model

We prove the renormalizability in the background field method of the two-dimensional bosonic non-linear σ-model with an arbitrary riemannian manifold as target space. Particular attention is paid to the question of non-linear renormalization of the quantum fields and its effect on subdivergences. We give an algorithm that allows one to compute to arbitrary loop order.     

Four-loop β-function for the N = 1 and N = 2 supersymmetric non-linear sigma model in two dimensions

A superspace calculation of the four-loop β-function is presented for two-dimensional N= 1,2 supersymmetric non-linear σ-models. It is concluded that besides the one-loop contribution which vanishes on Ricci-flat manifolds, the β-function receives four-loop contributions which do not vanish in the Ricci-flat case. Implications for the superstring are discussed.     

Finiteness of Ricci flatN=2 supersymmetric σ-models

We study the ultraviolet behavior of two dimensional supersymmetric non-linear -models with target space an arbitrary Kähler manifoldM, so that the models areN=2 supersymmetric. We point out that these models have an additional fermionic axial symmetry if and only if the metric onM is Ricci flat. We show that the preservation of this symmetry in perturbation theory implies that both bare and renormalized metrics onM are Ricci flat. Combining this result with the constraint ofN=2 supersymmetry requiring that all counter-terms to the metric beyond one-loop order be cohomologically trivial, we argue thatN=2 models defined on Ricci flat Kähler manifolds are on-shell ultraviolet finite to all orders of perturbation theory.     

Analyticity of solutions of theO(N) nonlinear σ-model

We consider continuous weak solutions of the Euler-Lagrange equation associated with the Euclideand-dimensionalO(N) nonlinear -model. We show for arbitraryN and arbitraryd that such solutions with locally square integrable gradient are real analytic.     

Continuum limit of the hierarchicalO(N) non-linear σ-model

We construct rigorously the continuum limit of theO(N) non-linear model in two euclidean dimensions with a hierarchical kinetic term. Asymptotic freedom and weak coupling Wilson renormalization group flow are established.     

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.     

Nonlocal field correlators on the lattice in HP1 σ-model

Connected two-point field strength correlators have been measured on the lattice in quaternionic projective σ-model of pure SU(2) Yang-Mills theory. The correlation lengths, extracted from the exponential fit for these correlators, are found to be γ ₁ ^?1 = 1.40(3) GeV and γ^?1 = 1.51(3) GeV in good agreement with other existing calculations. The dependence of bilocal functions on the connector shape was studied.     

String solutions in the S2 nonlinear σ-model with a gauge field

A variant of the S² nonlinear -model of Faddeev with an Abelian gauge field is considered. By using a direct variational method we demonstrate the existence of string-like solutions in this model and we establish the regularity of these solutions.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, No. 5, pp. 79–83, May, 1986.