Paper · Jun 29, 2021

Sigma models as Gross-Neveu models

Dmitri Bykov

hep-tharXiv:2106.15598
View PDF View HTML arXiv page

As on arXiv

Abstract

We review the correspondence between integrable sigma models with complex homogeneous target spaces and chiral bosonic (and possibly mixed bosonic/fermionic) Gross-Neveu models. Mathematically, the latter are models with quiver variety phase spaces, which reduce to the more conventional sigma models in special cases. We discuss the geometry of the models, as well as their trigonometric and elliptic deformations, Ricci flow and the inclusion of fermions.

View PDF View HTML arXiv page
Submitted
Jun 29, 2021
Subjects
hep-th, math-ph, math.DG
Journal
Theor Math Phys 208, 993-1003 (2021)
DOI
10.1134/S0040577921080018

Comments: 17 pages, 1 figure

Ricci flowHomogeneous spacesPhysics

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper On dual description of the deformed O(N) sigma model (A. V. Litvinov and L. A. Spodyneiko, 2018)4Litvinov 2018 Classical and quantum integrable sigma models. Ricci flow, "nice duality" and perturbed rational conformal field theories (Vladimir Fateev, 2019)6Fateev 2019 Integrable 2d sigma models: quantum corrections to geometry from RG flow (Ben Hoare, Nat Levine and Arkady A. Tseytlin, 2019)8Hoare 2019 Quantum flag manifold σ-models and Hermitian Ricci flow (Dmitri Bykov, 2020)2Bykov 2020 Flag manifold sigma models: spin chains and integrable theories (Ian Affleck, Dmitri Bykov and Kyle Wamer, 2021)1Affleck 2021 Strings in Bimetric Spacetimes (Ziqi Yan, 2021)7Yan 2021 Integrability and RG flow in 2d sigma models (Nat Levine, 2021)3Levine 2021 On dual regime in Yang-Baxter deformed O(2N) sigma models (Alexey Bychkov and Alexey Litvinov, 2025)5Bychkov 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Flag manifold sigma models: spin chains and integrable theories

    Ian Affleck, Dmitri Bykov and Kyle Wamer

    Topics Physics · Same author Dmitri Bykov · Shared terms “sigma model”, “Neveu models”, “Gross Neveu”

  2. Quantum flag manifold -models and Hermitian Ricci flow

    Dmitri Bykov

    Topics Ricci flow · Same author Dmitri Bykov · Shared terms “Gross Neveu”, “integrable”

  3. Integrability and RG flow in 2d sigma models

    Nat Levine

    Topics Physics, Homogeneous spaces · Shared terms “sigma model”, “target space”, “integrable”

  4. On dual description of the deformed sigma model

    A. V. Litvinov and L. A. Spodyneiko

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

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

    Alexey Bychkov and Alexey Litvinov

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

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

    Vladimir Fateev

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

  7. Strings in Bimetric Spacetimes

    Ziqi Yan

    Topics Physics · Shared terms “sigma model”, “target space”

  8. 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”

Read around it

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) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992) 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. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  2. 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.

  3. 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.