Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Higher-Spin Currents and Flows in Auxiliary Field Sigma Models

Daniele Bielli, Christian Ferko, Michele Galli, Gabriele Tartaglino-Mazzucchelli

TL;DR

This work develops a unified framework to study local higher-spin conserved currents and integrability-preserving Smirnov-Zamolodchikov flows in auxiliary-field (AF) sigma models. It proves the existence of spin-n currents for the AF free boson, extends the construction to even spins in a wide class of AF models, and demonstrates how SZ flows can be solved perturbatively by reducing to free-boson-type equations. The paper also treats spin-3 SZ flows in SU(3)-based AF models, showing the necessity of additional Lorentz-invariant invariants and providing explicit series solutions and deformed Lagrangians up to O(λ^2) that contain non-chiral trace structures. Overall, it unifies many AF constructions, extends the set of admissible interaction functions, and opens avenues for quantum and holographic explorations of higher-spin SZ deformations.

Abstract

We study local, higher-spin conserved currents in integrable $2d$ sigma models that have been deformed via coupling to auxiliary fields. These currents generate integrability-preserving flows introduced by Smirnov and Zamolodchikov. For auxiliary field (AF) deformations of a free boson, we prove that local spin-$n$ currents exist for all $n$ and give recursion relations that characterize Smirnov-Zamolodchikov (SZ) flows driven by these currents. We then show how to construct spin-$2n$ currents in a unified class of auxiliary field sigma models with common structure -- including AF theories based on the principal chiral model (PCM), its non-Abelian T-dual, (bi-)Yang-Baxter deformations of the PCM, and symmetric space models -- for interaction functions of one variable, and describe SZ flows driven by any function of the stress tensor in these cases. Finally, we give perturbative solutions for spin-$3$ SZ flows in any member of our unified class of AF models with underlying $\mathfrak{su}(3)$ algebra. Part of our analysis shows that the class of AF deformations can be extended by allowing the interaction function to depend on a larger set of variables than has previously been considered.

Higher-Spin Currents and Flows in Auxiliary Field Sigma Models

TL;DR

This work develops a unified framework to study local higher-spin conserved currents and integrability-preserving Smirnov-Zamolodchikov flows in auxiliary-field (AF) sigma models. It proves the existence of spin-n currents for the AF free boson, extends the construction to even spins in a wide class of AF models, and demonstrates how SZ flows can be solved perturbatively by reducing to free-boson-type equations. The paper also treats spin-3 SZ flows in SU(3)-based AF models, showing the necessity of additional Lorentz-invariant invariants and providing explicit series solutions and deformed Lagrangians up to O(λ^2) that contain non-chiral trace structures. Overall, it unifies many AF constructions, extends the set of admissible interaction functions, and opens avenues for quantum and holographic explorations of higher-spin SZ deformations.

Abstract

We study local, higher-spin conserved currents in integrable sigma models that have been deformed via coupling to auxiliary fields. These currents generate integrability-preserving flows introduced by Smirnov and Zamolodchikov. For auxiliary field (AF) deformations of a free boson, we prove that local spin- currents exist for all and give recursion relations that characterize Smirnov-Zamolodchikov (SZ) flows driven by these currents. We then show how to construct spin- currents in a unified class of auxiliary field sigma models with common structure -- including AF theories based on the principal chiral model (PCM), its non-Abelian T-dual, (bi-)Yang-Baxter deformations of the PCM, and symmetric space models -- for interaction functions of one variable, and describe SZ flows driven by any function of the stress tensor in these cases. Finally, we give perturbative solutions for spin- SZ flows in any member of our unified class of AF models with underlying algebra. Part of our analysis shows that the class of AF deformations can be extended by allowing the interaction function to depend on a larger set of variables than has previously been considered.

Paper Structure

This paper contains 17 sections, 1 theorem, 156 equations.

Table of Contents

  1. Introduction
  2. Review of auxiliary field (AF) deformations
  3. An instructive toy model -- the AF free boson
  4. Higher-spin currents
  5. Smirnov-Zamolodchikov flows
  6. Spin-$2k$ flows of AF sigma models as free boson flows
  7. Unification of auxiliary field models
  8. Even higher-spin currents
  9. Even Smirnov-Zamolodchikov flows
  10. Spin-$3$ flows of AF sigma models based on $\mathfrak{su}(3)$
  11. Need for new variables
  12. Series solution for the spin-$3$ flow
  13. Deformed AF Lagrangian at $O(\lambda_{3}^2)$
  14. Conclusion
  15. ODEs for the AF free boson
  16. ...and 2 more sections

Key Result

Theorem 3.1

Consider an auxiliary field sigma model which describes a deformation of a single free boson and is characterized by an interaction function $E = E ( \nu )$. Suppose that we make an ansatz for higher-spin conserved currents in the model of the form where the functions $F$ and $G$ depend only on dimensionless combinations involving the interaction function and finitely many of its derivatives. Her

Theorems & Definitions (2)

  • Theorem 3.1
  • proof