Geometry and duality in supersymmetric σ-models

The Supersymmetric Dual Sigma Model (SDSM) is a local field theory introduced to be nonlocally equivalent to the Supersymmetric Chiral nonlinear σ-Model (SCM), this dual equivalence being proven by explicit canonical transformation in tangent space. This model is here reconstructed in superspace and identified as a chiral-entwined supersymmetrization of the Dual Sigma Model (DSM). This analysis sheds light on the boson fermion symphysis of the dual transition, and on the new geometry of the DSM.     

Nonlinear σ-models and torsion in Kaluza-Klein type theories

It is shown that one can obtain the Lagrangians of nonlinear -models for the scalar curvature of multidimensional spaces of Kaluza-Klein type theories with torsion.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 27–31, June, 1989.     

Heterotic σ-models and conformal supergravity in two dimensions

The (1, 0) and (2, 0) type heterotic σ-models with Wess-Zumino term are coupled to conformal supergravity in two dimensions. There are no new restrictions on the σ-model manifolds in addition to those which arise in the globally supersymmetric cases. In the (1, 0) case possible isometries of the scalar manifold are gauged. A derivation of d = 2 conformal supergravity based on the super Lie algebra OSp(2, N)⊕OSp(2, N) (N = 1, 2) is given.     

‘In’ and ‘out’ vertex operators in a class of UV-finite nonlinear σ-models

In and out scalar vertex operators are constructed perturbatively in a class of recently discovered UV finite nonlinear -models describing the string evolution in gravitational plane wave backgrounds. They exhibit peculiar singularities in the target space related to the focusing phenomena in such backgrounds well known from the classical and quantum gravity theories. The computation is performed up to three loops of the usual perturbation expansion and to all loops of the weak field limit. An argument is given that the vertex operator singularities should persist, even when summing up the all perturbation expansions.     

Reduction of two-dimensional G-invariant σ-models

The Euclidean two-dimensional SO(n)-invariant -model is reduced to the system of equations for the Toda lattice. The similar reduction is studied for the SU(n)-invariant -model.     

Nonlinear σ-models and their gauging in and out of superspace

We analyze and generalize bosonic nonlinear σ-models and their N = 1,2 supersymmetric extensions in (4 spacetime-dimensional) N = 1 superspace. We give a general construction of nonminimal kinetic terms for gauge fields and of N = 1,2 gauging of isometries on Kähler and hyper-Kähler manifolds. In particular, we study the gauging of noncompact groups. We derive the complete component action and supertrace formula. For N = 2 models, the supertrace always vanishes.     

Supersymmetry and geometry in D < 4 nonlinear Σ-models

We discuss the augmentation of D < 4 supersymmetric σ-models by the addition of higher derivative terms, such as the Skyrme and Wess-Zumino actions. These terms must satisfy differential constraints so that they preserve the symmetries of the usual kinetic action. In two dimensions, we explicitly construct the gauged WZ action for group manifolds. Some difficulties with the analogous four dimensional theories are noted.     

Finiteness of Ricci flat supersymmetric non-linear σ-models

Combining the constraints of Kähler differential geometry with the universality of the normal coordinate expansion in the background field method, we study the ultraviolet behavior of 2-dimensional supersymmetric non-linear -models with target space an arbitrary riemannian manifoldM. We show that the constraint ofN=2 supersymmetry requires that all counterterms to the metric beyond one-loop order be cohomologically trivial. It follows that such supersymmetric non-linear -models defined on locally symmetric spaces are super-renormalizable and thatN=4 models are on-shell ultraviolet finite to all orders of perturbation theory.     

A relation between instantons of grassmann σ-models and Toda equations: II

We construct special solutions of Toda equations and show that these are deeply related with irreducible representations of sl(2;). We also show that after taking a continuum limit of Toda equations, these become the well-known solution of the continuum Toda equation.Partially supported by the Grant-in-Aid for Scientific Research, No. 03640085.     

Nonlinear σ-models as S-matrix generating functionals of strings

A formalism is presented for generating string theories from two-dimensional σ-models. Proceeding from an approach wherein the σ-models is understood as a S-matrix generating functional of string amplitudes, both the ϵb-function-zero slope limit correspondence and a systematic α′ expansion for the effective action are developed. Explicit perturbative calculations are provided, applicable to both open and closed strings which can be extended to include loop effect.     

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.     

The geometry of the quantum correction for topological σ-models

The ring (Frobenius algebra) of local observables for topological -models on ¹ with values in the grassmannianG(s, n) is known to be the same as the quotient of the homology ring of the target space by the (inhomogeneous) ideal generated by the so-called quantum correction. While the need for a quantum correction comes from algebraic motivations in field theory, the aim of this paper is to understand its geometric meaning. The simple examples of ^{1 n} models tell us that the quantum correction comes by restriction on the boundary of the moduli spaces which allows to compute intersections on moduli spaces of lower degrees. We will check this point of view for the case of ¹ G(s,n) models, yielding a proof of the algebraic result from physics in terms of the geometry of the -model itself.Work partially supported by National Project 40% Probabilistic and geometrical methods in Mathematical Physics and by CNR-Gruppo Nazionale di Fisica Matematica.     

Critical behaviour of SU(n) quantum chains and topological non-linear σ-models

The critical behaviour of SU(n) quantum “spin” chains, Wess-Zumino-Witten σ-models and grassmanian σ-models at topological angle θ = π (of possible relevance to the quantum Hall effect) is reexamined. It is argued that an additional Z_n symmetry is generally necessary to stabilize the massless phase. This symmetry is not present for the σ-models for n > 2 and is only present for certain representations of “spin” chains.     

One-loop renormalization in 4-dimensional σ-models taking into account generalized Skyrme and Wess-Zumino-Witten terms

One-loop divergences of the quantum effective action are computed for (supersymmetric) nonlinear -models with higher derivatives in 4-dimensional space-time taking into account the standard -model kinetic terms. The renormalizability of the theories under consideration is proved in the one-loop approximation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 23–29, January, 1990.     

Renormalizability and infrared finiteness of non-linear σ-models: A regularization-independent analysis for compact coset spaces

The non-linear models in two space-time dimensions corresponding to compact homogeneous coset spacesG/H are studied with particular attention to three problems: first, independence of coordinate choice and regularization, second, the physical content of the theory, and finally the regularity of the physics in the infrared limit. Concerning in particular the physical content of the theory, we construct a set of local observables whose correlation functions depend on a finite number of parameters identified among those defining the metric tensor of the coset space. For these models, we give a general proof of renormalizability based on the introduction of a nilpotent BRS operator which describes the non-linear isometries and a classical action which contains a mass term for all quantized fields. The mass term belongs to a finite dimensional representation of the groupG, which allows us to prove the conjecture that the correlation functions of local observables, i.e., the local operators invariant underG, are regular in the infrared limit.     

Dephasing effects in nonlinear δ-models: A phenomenological approach

We introduce a technique of simulating the destruction of mesoscopic conductance fluctuations phenomenologically within the nonlinear -formalism. A similar approach has already been applied within diagrammatic models. Dephasing is generated by adding a second random potential to the Hamilton operator which differs from the potential describing the metallic disorder only in its statistical properties. For that reason, the approach cannot serve to describe physical effects relying on dynamic aspects of dephasing interactions (e.g. the destruction of weak localization effects). The technique is introduced within the framework of a scattering-theoretical model of mesoscopic systems.     

Exact critical exponents for quantum spin chains,non-linear σ-models at θ=π and the quantum hall effect

We use non-abelian bosonization to predict critical exponents for quantum chains of arbitrary spin and arbitrary symmetry SU(n). Passing to the large representation limit gives non-linear σ-models on the manifolds U(2n)/U(n)×U(n) at topological angle θ=π. For n=1 this is the familiar “O(3)” model; taking the replica limit n→0 given critical exponents for the localization transition in the quantum Hall effect. Given certain assumptions, these exponents should be exact.