Reviews uses of nonlinear sigma models in various systems.
problem Understanding nonlinear sigma models in different physical contexts.
method General discussion and focus on geometrical interpretations.
result Connection between sigma models and various geometries.
Generalizes momentum sections to higher-dimensional gauged sigma models.
problem Understanding momentum sections in Hamiltonian mechanics and sigma models.
method Introduces a generalization of momentum sections on pre-multisymplectic manifolds.
result Shows a connection between constrained Hamiltonian systems and gauged sigma models.
Generalizes Hamiltonian structures to Dirac structures for new mechanics models.
problem Extending Hamiltonian structures to non-symplectic manifolds.
method Introduces Hamiltonian Lie algebroids and momentum sections over Dirac structures.
result Constructs new sigma models based on these structures.
Study connects supermanifold Laplacian to harmonic superfunctions via sigma model.
problem Understanding harmonic superfunctions on supermanifolds.
method Calculus of variations applied to a supersymmetric sigma model.
result Relates Laplacian to harmonic superfunctions on supermanifolds.
Sigma models linked to Gross-Neveu models via quiver varieties.
problem Understanding the relationship between sigma models and Gross-Neveu models.
method Exploring the mathematical correspondence between sigma models and Gross-Neveu models, including their geometric and trigonometric/elliptic deformations.
result Sigma models are mathematically equivalent to Gross-Neveu models under certain conditions.
This is an introductory review of topological field theories (TFTs) called AKSZ sigma models. The AKSZ construction is a mathematical formulation for the construction and analyses of a large class of TFTs, inspired by the Batalin-Vilkovisky formalism of gauge theories. We begin by considering a simple two-dimensional t…
Constructs zero-curvature representations for sigma-models.
problem Equations of motion for sigma-models with complex homogeneous spaces.
method Constructs zero-curvature representations and shows gauge-equivalence in symmetric cases.
result Zero-curvature representations for sigma-models in non-symmetric cases.
The Poisson--Weil sigma model, worked out by us recently, stems from gauging a Hamiltonian Lie group symmetry of the target space of the Poisson sigma model. Upon gauge fixing of the BV master action, it yields interesting topological field theories such as the 2--dimensional Donaldson-Witten topological gauge theory a…
Summarizes geometric connections between sigma models and Gross-Neveu models.
problem Understanding geometric connections between sigma models and Gross-Neveu models.
method Geometric facts and connections to nilpotent orbits, Springer resolutions, and quiver varieties.
result Sheds light on the general setup of the correspondence.
Relates field theory algebras to sigma model geometric structures.
problem Conformal field theory and sigma model geometric structures.
method Formulate conformal invariance conditions as generalized Maurer-Cartan equations.
result Einstein equations with extra fields in the quasi-classical limit.
Derives energy estimates for supersymmetric models.
problem Critical points of supersymmetric sigma models.
method Gradient and energy estimates.
result Applications of derived estimates.
Grassmannian sigma models extend Gross-Neveu model formulations.
problem Understanding sigma models on Grassmannian targets.
method Chiral Gross-Neveu model formulations for orthogonal and symplectic Grassmannians.
result One-loop β-functions proportional to dual Coxeter numbers. Study shows weak solutions' regularity for a sigma model with coarse gravitino.
problem Analyzing the regularity of solutions to a nonlinear sigma model with limited gravitino regularity.
method Examined solutions in Lp space with p>4 and derived precise regularity results. result Precise regularity results depend on the value of p. Study of classical solutions in a sigma-model on a flag manifold.
problem Explicit complex structure in sigma-model action.
method Analysis of classical solutions for a sphere worldsheet.
result Description of classical solutions for the flag manifold.
Study sigma-models on flag manifolds with curvature zero and relate to chiral models.
problem Understanding sigma-models on flag manifolds with zero curvature.
method Use zero-curvature representation and nilpotent orbits theory.
result Established relation between flag manifold sigma-models and principal chiral models.
Characterizes superforms for a sigma model and describes an integrable system.
problem Characterizing Maurer-Cartan 1-superforms in the CPN−1 sigma model. method Solves the linear spectral problem using solutions to describe an integrable system.
result Describes an integrable system for a su(N)-valued map. New sigma models compute graviton scattering amplitudes from quaternionic geometry.
problem Computing graviton scattering amplitudes from quaternionic geometry.
method Introducing new twistor sigma models that encode finite non-linear perturbations of flat structures.
result Provides a first-principles derivation of Hodges' formula for MHV graviton amplitudes.
Study on smooth solutions of a nonlinear sigma model with gravitino fields.
problem Analyzing smooth solutions of a modified nonlinear sigma model.
method Geometric setup, Euler-Lagrange equations, Rivière's regularity theory, Riesz potential theory.
result Smoothness of weak solutions proven using advanced mathematical theories.
One has believed that low energy effective theories of the Higgs branch of gauged linear sigma models correspond to supersymmetric nonlinear sigma models, which have been already investigated by many works. In this paper we discuss a explicit derivation of supersymmetric nonlinear sigma models from gauged linear sigma …
Solves sigma model on U(3)/U(1)^3, describing geodesics and spectrum.
problem Classical and quantum problems for 1D sigma model with specific target space.
method Mapping to Gaudin model, solving polynomial equations.
result Explicit description of geodesics and spectrum found.
Study of quantum aspects of generalized Gross-Neveu models, focusing on sigma models.
problem Quantum aspects and anomalies in generalized Gross-Neveu models.
method Admissible gauge Aμ=0, study of chiral anomalies, integration over moduli spaces of connections on a Riemann surface. result Integration over moduli spaces of connections on a Riemann surface.
BiHermitian geometry, discovered long ago by Gates, Hull and Roceck, is the most general sigma model target space geometry allowing for (2,2) world sheet supersymmetry. By using the twisting procedure proposed by Kapustin and Li, we work out the type A and B topological sigma models for a general biHermtian target spac…
Reformulated sigma models for complex Grassmannians using Gross-Neveu formalism.
problem Classical aspects of N=(2,2) supersymmetric sigma models with Hermitian symmetric target spaces. method Reformulation using Gross-Neveu formalism, proposing two types of equivalent Lagrangians.
result Proposed two types of equivalent Lagrangians for maximal isotropic Grassmannians, making either supersymmetry or geometry manifest.
We present a unified method of construction of surfaces associated with Grassmannian sigma models, expressed in terms of an orthogonal projector. This description leads to compact formulae for structural equations of two-dimensional surfaces immersed in the su(N) algebra. In the special case of the CP^1 sigma model we …
Generalizes sigma model with Lie algebroid structure and geometric conditions.
problem Consistency of constraints and gauge symmetry in topological sigma models.
method Analysis of geometric conditions and constraints in Hamiltonian and Lagrangian formalisms.
result Identifies universal compatibility condition between Lie algebroid and multi-symplectic structure.
We show that for any given n, there exists a sequence of words a_k in the generators sigma_1, ... sigma_{n-1} of the braid group B_n, representing the identity element of B_n, such that the number of braid relations of the form sigma_i sigma_{i+1} sigma_i = sigma_{i+1} sigma_i sigma_{i+1} needed to pass from a_k to the…
The paper constructs topological sigma models on supermanifolds.
problem Defining topological sigma models on general target supermanifolds.
method Defining BRST operators and physical observables, using supersymmetric localization.
result Obtained integrals over supermoduli spaces for correlation functions.
New sigma models use (4,0) supersymmetry for hyperkähler target spaces.
problem Constructing sigma models with (4,0) off-shell supersymmetry. method Formulated (4,0) supermultiplets, constructed sigma models with hyperkähler target spaces. result Explicit construction of target space geometries using (4,0) supersymmetry. BiKaehler geometry is characterized by a Riemannian metric g_{ab} and two covariantly constant generally non commuting complex structures K_+^a_b, K_-^a_b, with respect to which g_{ab} is Hermitian. It is a particular case of the biHermitian geometry of Gates, Hull and Roceck, the most general sigma model target space …
Introduces a new phase space for 2D supersymmetric sigma models.
problem Developing a new Hamiltonian formulation for 2D supersymmetric sigma models.
method Introduces a phase space with spinorial momenta and derives a covariant Hamiltonian formulation.
result Shows the existence of additional supersymmetries in the new formulation.
We consider the sigma models where the base metric is proportional to the metric of the configuration space. We show that the corresponding sigma model equation admits a Lax pair. We also show that this type of sigma models in two dimensions are intimately related to the minimal surfaces in a flat pseudo Riemannian 3-s…
Ricci flow emerges from renormalizing nonlinear Sigma models.
problem Understanding the renormalization group flow in nonlinear Sigma models.
method Using Euclidean algebraic quantum field theory and Wick ordering.
result The first-order renormalization group flow of nonlinear Sigma models equals the Ricci flow.
We introduce a new topological sigma model, whose fields are bundle maps from the tangent bundle of a 2-dimensional world-sheet to a Dirac subbundle of an exact Courant algebroid over a target manifold. It generalizes simultaneously the (twisted) Poisson sigma model as well as the G/G-WZW model. The equations of motion…
Deformed sigma-models with complex structures have zero-curvature representations.
problem Deformation of sigma-models with complex structures.
method B-field proportional to Kähler form, zero-curvature representation.
result Equations of motion admit zero-curvature representations.
In this contribution we review some of the interplay between sigma models in theoretical physics and novel geometrical structures such as Lie (n-)algebroids. The first part of the article contains the mathematical background, the definition of various algebroids as well as of Dirac structures, a joint generalization of…
We study N=2 nonlinear two dimensional sigma models with boundaries and their massive generalizations (the Landau-Ginzburg models). These models are defined over either Kahler or bihermitian target space manifolds. We determine the most general local N=2 superconformal boundary conditions (D-branes) for these sigma mod…
We present a sigma model field theoretic realization of Hitchin's generalized complex geometry, which recently has been shown to be relevant in compactifications of superstring theory with fluxes. Hitchin sigma model is closely related to the well known Poisson sigma model, of which it has the same field content. The c…
Defines Killing sections for Lie algebroids, generalizing Riemannian geometry concepts.
problem Extending Killing vector field concepts to Lie algebroids.
method Defines Killing sections and examines their properties in the context of Lie algebroids.
result Generalizes Killing vector field concepts to Lie algebroids and applies to sigma models.
Modified SPSNN reduces Pi nodes using adaptive multinomial choice.
problem Reduce the number of Pi nodes in SPSNNs.
method Adaptive approach to find better multinomial for a given problem.
result MSPSNN behaves better than traditional SPSNN with P_s.
The G/G WZW model results from the WZW-model by a standard procedure of gauging. G/G WZW models are members of Dirac sigma models, which also contain twisted Poisson sigma models as other examples. We show how the general class of Dirac sigma models can be obtained from a gauging procedure adapted to Lie algebroids in …
Investigates geometric aspects of double field theory and its membrane sigma-model formulation.
problem Capturing geometric and non-geometric flux backgrounds in DFT.
method Determines a splitting and projection of the Courant algebroid, constructs a membrane sigma-model, and analyzes gauge invariance.
result Unified description of geometric and non-geometric flux backgrounds in DFT.
We discuss additional supersymmetries for N = (2, 2) supersymmetric non-linear sigma models described by left and right semichiral superfields.
Study of energy quantization in a nonlinear sigma model with critical gravitinos.
problem Analyzing properties of a nonlinear sigma model with gravitinos in string theory.
method Analytical and geometric properties, Pohozaev type identity, weakly convergent sequence of fields.
result Established energy identities and obtained a holomorphic quadratic differential.
Let B_n be the Artin braid group on n strings with standard generators sigma_1, ..., sigma_{n-1}, and let SB_n be the singular braid monoid with generators sigma_1^{+-1}, ..., sigma_{n-1}^{+-1}, tau_1, ..., tau_{n-1}. The desingularization map is the multiplicative homomorphism eta: SB_n --> Z[B_n] defined by eta(sigma…
The nonlinear sigma model with gravitino exhibits symmetries and conservation laws.
problem Exploring symmetries and conservation laws in a nonlinear sigma model with gravitino.
method Geometric analysis of rescaled conformal transformations, super Weyl transformations, and diffeomorphisms.
result The model possesses degenerate super symmetry leading to geometric interpretations of energy-momentum tensor and supercurrent.
We give a physical derivation of generalized Kahler geometry. Starting from a supersymmetric nonlinear sigma model, we rederive and explain the results of Gualtieri regarding the equivalence between generalized Kahler geometry and the bi-hermitean geometry of Gates-Hull-Rocek. When cast in the language of supersymmetri…
Vanishing result for supersymmetric model in higher dimensions.
problem Critical points of supersymmetric model on manifolds.
method Proof using Riemannian manifolds, Sobolev inequality, and energy conditions.
result Vanishing of critical points for certain conditions.
Maps strong Kähler with torsion to Generalised Complex Geometry.
problem Representing strong Kähler with torsion in Generalised Complex Geometry.
method Analogy to Gualtieri map for (2,2) models. result Representation of strong Kähler with torsion in Generalised Complex Geometry.