2D Sigma Models Research Articles

Off-shell (4,4) supersymmetric sigma models with torsion as gauge theories in harmonic superspace

We present a manifestly supersymmetric off-shell formulation of a wide class of (4,4) 2D sigma models with torsion and noncommuting left and right complex structures in the harmonic superspace with a double set of SU(2) harmonic variables. The distinguishing features of the relevant superfield action are: (i) a nonabelian and in general nonlinear gauge invariance ensuring a correct number of physical degrees of freedom; (ii) an infinite tower of auxiliary fields. For a particular class of such models we explicitly demonstrate the noncommutativity of complex stuctures on the bosonic target.

Light-cone gauge quantization of 2D sigma models

This work describes the formulation of the manifestly ghost-free (spacetime) light-cone gauge for bosonic string theory with non-trivial spacetime metric, antisymmetric tensor, dilaton and tachyon fields. The action is a general two-dimensional sigma model, corresponding to a closed string theory with a second order action in the Polyakov picture. The spacetime fields must have a symmetry generated by a null, covariantly constant spacetime vector in order for the light-cone gauge to be accessible. Also, the theory must be Weyl invariant. The conditions for Weyl invariance are computed within the light-cone gauge, reproducing the usual beta functions. The calculation of the dilaton beta function and the critical dimension is somewhat novel in this ghost-free theory. Some exactly solvable light-cone theories are discussed.

The general solution of the 2D sigma model stringy black hole and the massless complex sine-Gordon model

The exact general solution for a sigma model having the 2D stringy black hole (SBH) as internal manifold is found in closed form. We also give the exact solution for the massless complex sine-Gordon (MCSG) model. Both, models and their solutions are related by analytic continuation. The solution is expressed in terms of four arbitrary functions of one variable.

2D SIGMA MODEL APPROACH TO 4D INSTANTONS

4D self-dual theories are proposed to generalize 2D conformal field theory. We identify 4D self-dual gravity as well as self-dual Yang-Mills theory with 2D sigma models valued in infinite-dimensional gauge groups. It is shown that these models possess infinite-dimensional symmetries with associated algebras—“CP1 extensions” of respective gauge algebras of 2D sigma models—which generalize the Kac-Moody algebra as well as W∞. We address various issues concerning 2D sigma models, twistors and sheaf cohomology. An attempt to connect 4D self-dual theories with 2D conformal field theory is made through sl (∞) Toda field theory.

Self-dual Yang-Mills (+gravity) as a 2D sigma model

Abstract The 4D self-dual Yang-Mills gauge theory on a self-dual space X is shown to be equivalent to a 2D sigma model valued in infinite dimensional gauge groups. Infinitesimal symmetries of the self-dual equation are obtained in terms of the sheaf cohomology group H 1 ( T , O F (−2)) , of the twistor space T corresponding to X. They are shown to form a symmetry algebra - a “CP1-extension” of the gauge symmetry algebra of a 2D sigma model which generalizes the Kac-Moody algebra as well as Wt8.

Super-Virasoro current algebra of the (4, 0)σ model with the Taub-Nut metric

The author looks at the superconformal current algebra of the torsion-free (4, 0) 2D sigma model with the Taub-Nut metric. After finding a traceless energy-momentum tensor, the central charges of the right and left Virasoro algebras are calculated. CR(g)=6 and CL(g)=4, at both zeroth and first perturbative order. The absence of perturbative corrections to C(g) results from the tracelessness of Tmu nu and the fact that Tmu nu belongs to a supermultiplet containing non-Abelian Kac-Moody currents. Using the four global supersymmetries of the theory, the supermultiplet of chirally conserved currents is found. It is then shown that the operator algebra of these currents is different from the old N=4 algebras of Ademollo et al (1976) and also from the new superalgebras of Schoutens (1988) and Sevrin et al (1988).

On the structure of the renormalisation group beta functions in a class of 2D models

We explain how the results for the beta-functions in a class of 2D field theory models with exponential interactions are related to the Weyl anomaly coefficients (and the action functional) corresponding to a general 2D sigma model with the “tachyon”, “metric” and “dilaton” couplings. We also discuss the Liouville theory and suggest a consistent definition of a scalar theory with an interaction potential depending both on the scalar field and its “dual”.

The partition function of the sigma model of the string on a compact two-space

We develop a manifestly covariant procedure of computation of the partitionfunction Z of a general renormalizable 2D sigma model defined on a compact two-space. Special attention is paid to a proper separation of the path integral into the integrals over the constant and nonconstant parts of the coordinate. The important role of the path integral measure in establishing the covariance and renormalizability properties of Z is stressed. We explicitly check the validity of the representation of the closed string effective action as the partial derivative of Z over the logarithm of the cutoff. We also comment on the status of the “ c-theorem” and suggest a resolution of a problem in the renormalization of the closed string loop corrections corresponding to the disc topology.

Gell-mann-low function of the chiral model on the homogeneous berger space

The asymptotic freedon of two-dimensional nonlinear sigma models (chiral models) is one of the most important properties on which the analogy between 2d sigma models and 4d gauge theories is based. For the two-dimensional nonlinear sigma model on the homogeneous space Sp(2)/SU(2) (Berger manifold) the method of sectional curvatures is used to find the Gell-Mann-Low beta function in the single-loop approximation. The result indicates that the model is an asymptotically free renormalizable nonlinear model.

Wilson's operator expansion: Can it fail?

Within the framework of some simple models we discuss the status of the operator product expansion (OPE) in the presence of nonperturbative effects. We consider, in particular, the 4d Higgs model, 2d sigma model and the Schwinger model. The general formulation of OPE is presented and it is demonstrated that there exists a consistent procedure allowing one to define unambiguously both coefficient functions and matrix elements of composite operators. One of the key elements of the procedure is the introduction of an auxiliary parameter, the normalization point μ. For the simplest T-products discussed in the literature earlier we construct the corresponding OPE explicitly. Then we check its validity by comparing the results for the two-point functions with independent direct calculations of the same correlators. Although the general procedure is standard and does not vary from one theory to another, numerically the relative role of perturbative and nonperturbative contributions in vacuum condensates is different in different theories. The two extremes considered are the λϕ 4 theory with no spontaneous breaking of the symmetry and the O( N) sigma model in the limit N → ∞. In the former case there is only perturbative contribution to ( ϕ 2), while in the latter case the perturbative pieces are suppressed by 1/ N factors. Numerically QCD is much closer to the O( N) sigma model in the large- N limit. Comments on specific features of QCD are presented.