October 11, 2021
We study regularization scheme dependence of $\beta$-function for sigma models with two-dimensional target space. Working within four-loop approximation, we conjecture the scheme in which the $\beta$-function retains only two tensor structures up to certain terms containing $\zeta_3$. Using this scheme, we provide explicit solutions to RG flow equation corresponding to Yang-Baxter- and $\lambda$-deformed $SU(2)/U(1)$ sigma models, for which these terms disappear.
Similar papers 1
November 23, 2023
We study regularization scheme dependence of K\"ahler ($N=2$) supersymmetric sigma models. At the one-loop order the metric $\beta$ function is the same as in non-supersymmetric case and coincides with the Ricci tensor. First correction in MS scheme is known to appear in the fourth loop. We show that for certain integrable K\"ahler backgrounds, such as complete $T-$dual of $\eta$-deformed $\mathbb{CP}(n)$ sigma models, there is a scheme in which the fourth loop contribution v...
July 10, 2023
In this paper we consider perturbation theory in generic two-dimensional sigma models in the so-called first-order formalism, using the coordinate regularization approach. Our goal is to analyze the first-order formalism in application to $\beta$ functions and compare its results with the standard geometric calculations. Already in the second loop, we observe deviations from the geometric results that cannot be explained by the regularization/renormalization scheme choices. M...
December 4, 2023
We consider the renormalization group flow equation for the two-dimensional sigma models with the K\"ahler target space. The first-order formulation allows us to treat perturbations in these models as current-current deformations. We demonstrate, however, that the conventional first-order formalism misses certain anomalies in the measure, and should be amended. We reconcile beta functions obtained within the conformal perturbation theory for the current-current deformations w...
July 10, 2019
Classically integrable $\sigma$-models are known to be solutions of the 1-loop RG equations, or "Ricci flow", with only a few couplings running. In some of the simplest examples of integrable deformations we find that in order to preserve this property at 2 (and higher) loops the classical $\sigma$-model should be corrected by quantum counterterms. The pattern is similar to that of effective $\sigma$-models associated to gauged WZW theories. We consider in detail the examples...
October 1, 2019
Following arXiv:1907.04737, we continue our investigation of the relation between the renormalizability (with finitely many couplings) and integrability in 2d $\sigma$-models. We focus on the "$\lambda$-model," an integrable model associated to a group or symmetric space and containing as special limits a (gauged) WZW model and an "interpolating model" for non-abelian duality. The parameters are the WZ level $k$ and the coupling $\lambda$, and the fields are $g$, valued in a ...
May 14, 1997
Starting from a consistency requirement between T-duality symmetry and renormalization group flows, the two-loop metric beta function is found for a d=2 bosonic sigma model on a generic, torsionless background. The result is obtained without Feynman diagram calculations, and represents further evidence that duality symmetry severely constrains renormalization flows.
December 7, 2021
Motivated by the search for solvable string theories, we consider the problem of classifying the integrable bosonic 2d $\sigma$-models. We include non-conformal $\sigma$-models, which have historically been a good arena for discovering integrable models that were later generalized to Weyl-invariant ones. General $\sigma$-models feature a quantum RG flow, given by a 'generalized Ricci flow' of the target-space geometry. This thesis is based on the conjecture that integrable ...
February 22, 2023
Nonlinear sigma models on de Sitter background possess the same kind of derivative interactions as gravity, and show the same sorts of large spacetime logarithms in correlation functions and solutions to the effective field equations. It was recently demonstrated that these logarithms can be resummed by combining a variant of Starobinsky's stochastic formalism with a variant of the renormalization group. This work considers one of these models and completes two pieces of anal...
August 20, 1997
The requirement that duality and renormalization group transformations commute as motions in the space of a theory has recently been explored to extract information about the renormalization flows in different statistical and field theoretical systems. After a review of what has been accomplished in the context of 2d sigma models, new results are presented which set up the stage for a fully generic calculation at two-loop order, with particular emphasis on the question of sch...
September 20, 2021
We study non-linear sigma models on target manifolds with constant (positive or negative) curvature using the functional renormalization group and the background field method. We pay particular attention to the splitting Ward identities associated to the invariance under reparametrization of the background field. Implementing these Ward identities imposes to use the curvature as a formal expansion parameter, which allows us to close the flow equation of the (scale-dependent) ...