Sigma models with local couplings: a new integrability -- RG flow connection

Abstract

We consider several classes of $\sigma$-models (on groups and symmetric spaces, $\eta$-models, $\lambda$-models) with local couplings that may depend on the 2d coordinates, e.g. on time $\tau$. We observe that (i) starting with a classically integrable 2d $\sigma$-model, (ii) formally promoting its couplings $h_{\alpha }$ to functions $h_{\alpha }(\tau )$ of 2d time, and (iii) demanding that the resulting time-dependent model also admits a Lax connection implies that $h_{\alpha }(\tau )$ must solve the 1-loop RG equations of the original theory with $\tau$ interpreted as RG time. This provides a novel example of an 'integrability - RG flow' connection. The existence of a Lax connection suggests that these time-dependent $\sigma$-models may themselves be understood as integrable. We investigate this question by studying the possibility of constructing non-local and local conserved charges. Such $\sigma$-models with $D$-dimensional target space and time-dependent couplings subject to the RG flow naturally appear in string theory upon fixing the light-cone gauge in a $(D+2)$-dimensional conformal $\sigma$-model with a metric admitting a covariantly constant null Killing vector and a dilaton linear in the null coordinate.

arXiv comment31 pages. v2: comments on earlier related work; v3: references added

References in corpus (39)

• Yang-Baxter $\sigma$-models and dS/AdS T-duality
• On classical q-deformations of integrable sigma-models
• Integrable interpolations: From exact CFTs to non-Abelian T-duals
• Spinning strings in AdS_5 x S^5: new integrable system relations
• Integrable Deformations of Strings on Symmetric Spaces
• Gauge Theory and Integrability, II
• Solvable model of strings in a time-dependent plane-wave background
• Gauge Theory and Integrability, I
• String vacuum backgrounds with covariantly constant null Killing vector and 2d quantum gravity
• The all-loop non-Abelian Thirring model and its RG flow
• Analytic Non-integrability in String Theory
• Local conserved charges in principal chiral models
• Solvable models of strings in homogeneous plane wave backgrounds
• Beta Function of k Deformed AdS_5 x S^5 String Theory
• Non-local charges, Zm gradings and coset space actions
• Strings on pp-waves and massive two dimensional field theories
• The integrable harmonic map problem versus Ricci flow
• Gauged WZW-type theories and the all-loop anisotropic non-Abelian Thirring model
• A class of exact pp-wave string models with interacting light-cone gauge actions
• On (non)integrability of classical strings in p-brane backgrounds
• Finite sigma models and exact string solutions with Minkowski signature metric
• A unifying 2d action for integrable $\sigma$-models from 4d Chern-Simons theory
• On string cosmology and the RG flow in 2d field theory
• Conformal Sigma-Models on Supercoset Targets
• Integrable 2d sigma models: quantum corrections to geometry from RG flow
• Assembling integrable sigma-models as affine Gaudin models
• Gauge Theory And Integrability, III
• One loop renormalizability of the Poisson-Lie sigma models
• Commuting charges and symmetric spaces
• A class of finite two - dimensional sigma models and string vacua
• Integrable sigma models and 2-loop RG flow
• Local charges in involution and hierarchies in integrable sigma-models
• The classical r-matrix method for nonlinear sigma-model
• Quantum flag manifold $\sigma$-models and Hermitian Ricci flow
• On integrable models from pp-wave string backgrounds
• Yang-Baxter sigma models and Lax pairs arising from $\kappa$-Poincaré $r$-matrices
• Integrability in time-dependent systems with one degree of freedom
• Lax pairs for deformed Minkowski spacetimes
• Chaotic solitons in driven sine-Gordon model

Cited by in corpus (13)

• Yang-Baxter deformations of the Principal Chiral Model plus Wess-Zumino term
• Flag manifold sigma models: spin chains and integrable theories
• RG flows of integrable $\sigma$-models and the twist function
• Sigma models as Gross-Neveu models
• Semiclassical spectrum of a Jordanian deformation of $Ad{S}_5\times S^5$
• O($D$,$D$)-covariant two-loop $\beta$-functions and Poisson-Lie T-duality
• Integrability vs. RG flow in $G\times G$ and $G\times G/H$ sigma models
• On loop corrections to integrable $2D$ sigma model backgrounds
• Demystifying integrable QFTs in AdS: No-go theorems for higher-spin charges
• Integrability in Gravity from Chern-Simons Theory
• Integrability of the $\lambda$-deformation of the PCM with spectators
• Integrable Deformations of the Breitenlohner-Maison Model from 4d Chern-Simons Theory
• Geometric Approaches to Quantum Fields and Strings at Strong Couplings