Integrable theories and loop spaces: fundamentals, applications and new developments

Abstract

We review our proposal to generalize the standard two-dimensional flatness construction of Lax-Zakharov-Shabat to relativistic field theories in d+1 dimensions. The fundamentals from the theory of connections on loop spaces are presented and clarified. These ideas are exposed using mathematical tools familiar to physicists. We exhibit recent and new results that relate the locality of the loop space curvature to the diffeomorphism invariance of the loop space holonomy. These result are used to show that the holonomy is abelian if the holonomy is diffeomorphism invariant. These results justify in part and set the limitations of the local implementations of the approach which has been worked out in the last decade. We highlight very interesting applications like the construction and the solution of an integrable four dimensional field theory with Hopf solitons, and new integrability conditions which generalize BPS equations to systems such as Skyrme theories. Applications of these ideas leading to new constructions are implemented in theories that admit volume preserving diffeomorphisms of the target space as symmetries. Applications to physically relevant systems like Yang Mills theories are summarized. We also discuss other possibilities that have not yet been explored.

64 pages, 8 figures

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• Some aspects of self-duality and generalised BPS theories
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• The gauged BPS baby Skyrme model
• Exact vortex solutions in a CP^N Skyrme-Faddeev type model
• Magnetothermodynamics of BPS baby skyrmions
• Compact boson stars in K field theories
• Vortices in the extended Skyrme-Faddeev model
• Gauge and Integrable Theories in Loop Spaces
• The integral equations of Yang-Mills and its gauge invariant conserved charges
• Static Hopfions in the extended Skyrme-Faddeev model
• Self-dual solitons in a generalized Chern-Simons baby Skyrme model
• Neutron stars within the Skyrme model
• The vector BPS Skyrme model
• Lifshitz field theories with SDiff symmetries
• Some properties of (3+1) dimensional vortex solutions in the extended CP^N Skyrme-Faddeev model
• Compact shell solitons in K field theories
• A mild source for the Wu-Yang magnetic monopole
• The vector BPS baby Skyrme model
• Self-dual Hopfions
• Integrability structures in string theory
• A Generalised Self-Duality for the Yang-Mills-Higgs System
• Waves in the Skyrme--Faddeev model and integrable reductions
• Characteristic analysis for integrable soliton models on two-dimensional target spaces
• BPS states for scalar field theories based on ${\mathfrak{g}}_2$ and $\mathfrak{su}(4)$ algebras
• A direct test of the integral Yang-Mills equations through SU(2) monopoles
• Chiral Dirac-Born-Infeld solitons with SDiff symmetry
• Quantum Yang-Mills Charges in Strongly Coupled 2D Lattice QCD with Three Flavors
• Some vortex solutions in the extended Skyrme-Faddeev model
• A zero-curvature representation of electromagnetism and the conservation of electric charge