Supersymmetric Non-linear Σ-models Research Articles

Resonances in a single-lead reflection from a disordered medium: [formula omitted]-model approach

Using the framework of supersymmetric non-linear σ-model we develop a general non-perturbative characterization of universal features of the density ρ(Γ) of the imaginary parts (“width”) for S-matrix poles (“resonances”) describing waves incident and reflected from a disordered medium via M-channel waveguide/lead. Explicit expressions for ρ(Γ) are derived for several instances of systems with broken time-reversal invariance, in particular for quasi-1D and 3D media. In the case of perfectly coupled lead with a few channels (M∼1) the most salient features are tails ρ(Γ)∼Γ−1 for narrow resonances reflecting exponential localization and ρ(Γ)∼Γ−2 for broad resonances reflecting states located in the vicinity of the attached wire. For multimode quasi 1D wires with M≫1, an intermediate asymptotics ρ(Γ)∼Γ−3/2 is shown to emerge reflecting diffusive nature of decay into wide enough contacts.

Estimates for solutions of Dirac equations and an application to a geometric elliptic-parabolic problem

We develop estimates for the solutions and derive existence and uniqueness results of various local boundary value problems for Dirac equations that improve all relevant results known in the literature. With these estimates at hand, we derive a general existence, uniqueness and regularity theorem for solutions of Dirac equations with such boundary conditions. We also apply these estimates to a new nonlinear elliptic-parabolic problem, the Dirac-harmonic heat flow on Riemannian spin manifolds. This problem is motivated by the supersymmetric nonlinear σ-model and combines a harmonic heat flow type equation with a Dirac equation that depends nonlinearly on the flow.

The generalized Kähler geometry of N = (2, 2) WZW-models

N = (2, 2), d = 2 supersymmetric non-linear σ-models provide a physical realization of Hitchin’s and Gualtieri’s generalized Kähler geometry. A large subclass of such models are comprised by WZW-models on even-dimensional reductive group manifolds. In the present paper we analyze the complex structures, type changing, the superfield content and the affine isometries compatible with the extra supersymmetry. The results are illustrated by an exhaustive discussion of the N = (2, 2) WZW-models on S 3 × S 1 and S 3 × S 3 where various aspects of generalized Kähler and Calabi-Yau geometry are verified and clarified. The examples illustrate a slightly weaker definition for an N = (2, 2) superconformal generalized Kähler geometry compared to that for a generalized Calabi-Yau geometry.

Deformed solitons in a 2-dimensional non(anti)commutative Wess–Zumino model

In this paper we examine a deformation of the 1/2 BPS kink solitons of a WZ model with a cubic superpotential in two Euclidean dimensions caused by introducing non(anti)commutativity in the superspace formulation of the theory. The spacetime Lagrangian of the deformed model consists of the ordinary WZ terms plus corrections in the form of a kinetic and a cubic term for the auxiliary field, both in proportion to the determinant of the non(anti)commutativity parameters C α β . We compute the correction to the “orbit” equation, relating the auxiliary and the scalar fields together, and subsequently derive the modifications to the BPS equation and to its solution up to first order in λ ≡ det ( C α β ) . We give a description of the deformed solitons in terms of the 1/2 BPS states of an effective N = 1 supersymmetric non-linear σ-model in two dimensions, using which we derive the first-order correction to the BPS mass formula for these solitons.

AnN= 2 worldsheet approach to D-branes in bihermitian geometries: I. Chiral and twisted chiral fields

We investigate N = (2, 2) supersymmetric nonlinear σ-models in the presence of a boundary. We restrict our attention to the case where the bulk geometry is described by chiral and twisted chiral superfields corresponding to a bihermitian bulk geometry with two commuting complex structures. The D-brane configurations preserving an N = 2 worldsheet supersymmetry are identified. Duality transformations interchanging chiral for twisted chiral fields and vice versa while preserving all supersymmetries are explicitly constructed. We illustrate our results with various explicit examples such as the WZW-model on the Hopf surface S3 × S1. The duality transformations provide e.g new examples of coisotropic A-branes on Kahler manifolds (which are not necessarily hyper-Kahler). Finally, by dualizing a chiral and a twisted chiral field to a semi-chiral multiplet, we initiate the study of D-branes in bihermitian geometries where the cokernel of the commutator of the complex structures is non-empty.

Nonhermitian supersymmetric partition functions: The case of one bosonic flavor

We discuss the supersymmetric formulation of the nonhermitian β = 2 random matrix partition function with one bosonic flavor. This partition function is regularized by adding one conjugate boson and fermion each. A supersymmetric nonlinear σ-model for the resulting Goldstone degrees of freedom is obtained using symmetry arguments only. For a Gaussian probability distribution the same results are derived using superbosonization and the complex orthogonal polynomial method. The symmetry arguments apply to any model with the same symmetries and a mass gap, and demonstrate the universality of the nonlinear σ-model.

Gauged supersymmetric σ-models and soft breaking terms

Supersymmetric nonlinear σ-models in four dimensions with D-term potentials can sometimes have a singular metric. As the kinetic terms of scalar fields and their chiral fermionic partners are determined by this metric, it follows that their kinetic energy vanishes in the vacuum. In previous work we have shown for a simple model that this degeneracy of the sigma-model metric can be lifted by soft supersymmetry breaking terms. In this Letter I introduce soft breaking terms in more realistic models based on SO ( 10 ) and E 6 and compute their resulting mass spectra.

Supersymmetric non-linear σ-models with boundaries revisited

We study two-dimensional supersymmetric non-linear sigma-models with boundaries. We derive the most general family of boundary conditions in the non-supersymmetric case. Next we show that no further conditions arise when passing to the N=1 model. We present a manifest N=1 off-shell formulation. The analysis is greatly simplified compared to previous studies and there is no need to introduce non-local superspaces nor to go (partially) on-shell. Whether or not torsion is present does not modify the discussion. Subsequently, we determine under which conditions a second supersymmetry exists. As for the case without boundaries, two covariantly constant complex structures are needed. However, because of the presence of the boundary, one gets expressed in terms of the other one and the remainder of the geometric data. Finally we recast some of our results in N=2 superspace and discuss applications.

Singular supersymmetric σ-models

Supersymmetric non-linear σ-models are described by a field dependent Kähler metric determining the kinetic terms. In general it is not guaranteed that this metric is always invertible. Our aim is to investigate the symmetry structure of supersymmetric models in four-dimensional space–time in which metric singularities occur. For this purpose we study a simple anomaly-free extension of the supersymmetric CP 1 model from a classical point of view. We show that the metric singularities can be regularized by the addition of a soft supersymmetry-breaking mass parameter.

N = 1 supersymmetric extension of the QCD effective action

We present a new 4D, N = 1 supersymmetric nonlinear σ-model using complex linear and chiral superfields that generalizes the effective action of Gasser and Leutwyler for the massless limit of QCD.

Antibrackets, supersymmetric σ-model and localization

We consider supersymmetrization of Hamiltonian dynamics via antibrackets for systems whose Hamiltonian generates an isometry of the phase space. We find that the models are closely related to the supersymmetric non-linear σ-model. We interpret the corresponding path integrals in terms of super loop space equivariant cohomology. It turns out that they can be evaluated exactly using localizations techniques.

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.

T-duality and non-local supersymmetries

We study the non-localization of extended worldsheet supersymmetry under T-duality, when the associated complex structure depends on the coordinate with respect to which duality is performed. First, the canonical transformation which implements T-duality is generalized to the supersymmetric non-linear σ-models. Then, we obtain the non-local object which replaces the complex structure in the dual theory and write down the condition it should satisfy so that the dual action is invariant under the non-local supersymmetry. For the target space, this implies that the supersymmetry transformation parameter is a non-local spinor. The analogue of the Killing equation for this non-local spinor is obtained. It is argued that in the target space the supersymmetry is no longer realized in the standard way. The string theoretic origin of this phenomenon is briefly discussed.

Why auxiliary fields matter: the strange case of the 4D, N = 1 supersymmetric QCD effective action

Within a four dimensional manifestly N = 1 supersymmetric action, we show that Wess-Zumino-Novikov-Witten (WZNW) terms can be embedded in an extraordinarily simple manner into a purely chiral superaction. In order to achieve this result it is necessary to assign spin-0 and spin- 1 2 degrees of freedom both to chiral superfields and as well to non-minimal scalar multiplets. We propose a new formulation for the effective low-energy action of 4D, N = 1 supersymmetric QCD that is consistent with holomorphy to all orders in the pion superfield. After reduction to a 2D, N = 2 theory we find a new class of manifestly supersymmetric non-linear σ-models with torsion.

Two length scales near the Anderson transition and the critical diffusive behavior

We present results of an analytical and numerical investigation of the Anderson metal-insulator transition based on a supersymmetric non-linear σ-model in effective medium approximation. In the critical metallic regime two characteristic length scales exist which diverge according to a power law but have different critical exponents. We put forward a physical picture that enables us to express the diffusion coefficient in terms of these two lengths and to explain its exponential behavior obtained previously.

The finite two-loop effective action in the N = 1 supersymmetric sigma-model

The finite two-loop effective action of the N = 1 supersymmetric non-linear σ-model is investigated. Two distict regulating schemes, operator regularization and dimensional regularization, are employed to compute this quantity. The superfield formalism is used in our analysis in conjunction with an infrared (IR) regulating mass; we point out that operator regularization manifestly preserves all manipulations involving the D-algebra. The β-function is derived from the application of the renormalization group equation to the finite effective action. We find that the β-function is both method-dependent and sensitive to the IR mass-limit and that the renormalization group equation is not consistent. It is possible to perform a finite renormalization on the one-loop Green's functions to make the effective action method-independent at the price of introducing a one-parameter scheme dependence. Neither the inconsistency of the renormalization group equation nor the IR mass-limit problem are affected by this operation. By introducing a generalized mass-potential with off-diagonal terms, the latter problem may be treated: we show that the vanishing of the β-function to two-loop order then becomes insensitive to the IR mass-limit and agrees with previous results in each method.

Four-loop divergences of the two-dimensional (1,1) supersymmetric non-linear sigma model with a Wess-Zumino-Witten term

The four-loop divergences of the (1,1) supersymmetric two-dimensional non-linear σ-model with a Wess-Zumino-Witten term are analyzed. All the four-loop 1/ϵ-divergences in the general case (and an overall coefficient at the total four-loop contribution to the β-function) are shown to be reducible to only structures proportional to ξ(3). We explicitly calculate non-derivative contributions to the four-loop β-function from logarithmically divergent graphs. As a by-product, we obtain the complete four-loop β-function for the supersymmetric Wess-Zumino-Witten model. We use the partial results for the general four-loop β-function to shed some light on the structure of the ( α ′) 3 -corrections to the superstring effective-action with antisymmetric-tensor field coupling. An inconsistency of the supersymmetrical dimensional regularisation via dimensional reduction in the presence of torsion is discovered at four loops, unless the string interpretation for the σ-model is adopted.

GAUGING σ-MODEL ISOMETRIES IN (2, 0)-SUPERSPACE

Considering (2, 0)-supersymmetric non-linear σ-models defined over Kählerian coset manifolds, we discuss the gauging of the isotropy and isometry groups in (2, 0)-superspace and present the action coupling these σ-models to the (2, 0)-Yang-Mills supermultiplets.

Spontaneous parity violation in a supersymmetric nonlinear σ-model in 2+1 dimensions

A supersymmetric nonlinear σ-model in 2+1 dimensions is constructed. The renormalization of the coupling constant makes the model finite to leading order in the 1 N expansion. Mass generation and supersymmetric bound states occur as consequences of spontaneous violation of parity.

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

One-loop divergences of the quantum effective action are computed for (supersymmetric) nonlinear σ-models with higher derivatives in 4-dimensional space-time taking into account the standard σ-model kinetic terms. The renormalizability of the theories under consideration is proved in the one-loop approximation.