Paper · Feb 7, 2019

Classical and quantum integrable sigma models. Ricci flow, "nice duality" and perturbed rational conformal field theories

Vladimir Fateev

hep-tharXiv:1902.02811
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Abstract

We consider classical and quantum integrable sigma models and their relations with the solutions of renormalization group equations. We say that an integrable sigma model possesses the "nice" duality property if the dual quantum field theory has the weak coupling region. As an example, we consider the deformed sigma model with additional quantum degrees of freedom. We formulate the dual integrable field theory and use perturbed conformal field theory, perturbation theory, -matrix, Bethe Ansatz and renormalization group methods to show that this field theory has the "nice" duality property. We consider also an alternative approach to the analysis of sigma models on the deformed symmetric spaces, based on the perturbed rational conformal field theories.

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Submitted
Feb 7, 2019
Subjects
hep-th

Comments: 37 pages

Ricci flowPhysics

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    Topics Physics · Shared terms “sigma model”, “field theory”, “quantum field”

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    A. V. Litvinov and L. A. Spodyneiko

    Topics Physics · Shared terms “sigma model”, “field theory”, “deformed”

  3. On dual regime in Yang-Baxter deformed sigma models

    Alexey Bychkov and Alexey Litvinov

    Topics Physics · Shared terms “sigma model”, “integrable sigma”, “deformed”

  4. Ricci flow of Integrable Sigma Models, Hitchin Systems and Quantum Seiberg-Witten theory

    G. Bonelli, S. Lacroix, T. Pedroni et al.

    Topics Physics · Shared terms “integrable sigma”, “sigma model”, “conformal field”

  5. Integrable 2d sigma models: quantum corrections to geometry from RG flow

    Ben Hoare, Nat Levine and Arkady A. Tseytlin

    Topics Physics · Shared terms “sigma model”, “integrable”, “coupling”

  6. Conformal invariance from scale invariance in non-linear sigma models

    Yu Nakayama

    Topics Physics · Shared terms “sigma model”, “field theory”, “quantum field”

  7. Flag manifold sigma models: spin chains and integrable theories

    Ian Affleck, Dmitri Bykov and Kyle Wamer

    Topics Physics · Shared terms “sigma model”, “integrable”, “renormalization group”

  8. Scale and Conformal Invariance in 2d Sigma Models, with an Application to N=4 Supersymmetry

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Physics Ricci flow first appeared in physics, as the leading-order renormalization-group flow of the string sigma model (Friedan, 1980). Start with: Nonlinear Models in 2+ε Dimensions (Friedan, 1980)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.