Integrability and renormalizability for the fully anisotropic $\mathrm{SU}(2)$ principal chiral field and its deformations

Abstract

For the class of $1+1$ dimensional field theories referred to as the non-linear sigma models, there is known to be a deep connection between classical integrability and one-loop renormalizability. In this work, the phenomenon is reviewed on the example of the so-called fully anisotropic $\mathrm{SU}(2)$ Principal Chiral Field (PCF). Along the way, we discover a new classically integrable four parameter family of sigma models, which is obtained from the fully anisotropic $\mathrm{SU}(2)$ PCF by means of the Poisson-Lie deformation. The theory turns out to be one-loop renormalizable and the system of ODEs describing the flow of the four couplings is derived. Also provided are explicit analytical expressions for the full set of functionally independent first integrals (renormalization group invariants).

21 pages, 4 figures, minor typos fixed, journal version

References in corpus (15)

• On integrability of the Yang-Baxter $\si$-model
• Yang-Baxter $\sigma$-models and dS/AdS T-duality
• On classical q-deformations of integrable sigma-models
• On deformations of AdS_n x S^n supercosets
• Integrability of the bi-Yang-Baxter sigma-model
• Generalised integrable $\lambda$- and $\eta$-deformations and their relation
• The integrable harmonic map problem versus Ricci flow
• Gauged WZW-type theories and the all-loop anisotropic non-Abelian Thirring model
• One loop renormalizability of the Poisson-Lie sigma models
• Yang-Baxter sigma models: Quantum aspects
• On the Yang-Baxter Poisson algebra in non-ultralocal integrable systems
• Yang Baxter and Anisotropic Sigma and Lambda Models, Cyclic RG and Exact S-Matrices
• Ricci flows and expansion in axion-dilaton cosmology
• Geometry of the spectral parameter and renormalisation of integrable $\sigma$-models
• An elliptic integrable deformation of the Principal Chiral Model