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Euclidean E-models

Ctirad Klimcik

Abstract

We study a class of E-models, referred to as Euclidean E-models, in which the operator E acting on the Drinfeld double squares to minus the identity rather than to the identity. This modification leads to significant structural differences from the standard E-model framework. Most notably, the sigma-models naturally associated with these E-models possess a Euclidean world-sheet. Although, for some Drinfeld doubles, every Lorentzian E-model admits a natural Euclidean counterpart, the duality, integrability, and renormalization properties of Euclidean E-models are independent of the Lorentzian case and must be studied separately. We illustrate the general constructions using the example of a Euclidean bi-Yang-Baxter deformation.

Euclidean E-models

Abstract

We study a class of E-models, referred to as Euclidean E-models, in which the operator E acting on the Drinfeld double squares to minus the identity rather than to the identity. This modification leads to significant structural differences from the standard E-model framework. Most notably, the sigma-models naturally associated with these E-models possess a Euclidean world-sheet. Although, for some Drinfeld doubles, every Lorentzian E-model admits a natural Euclidean counterpart, the duality, integrability, and renormalization properties of Euclidean E-models are independent of the Lorentzian case and must be studied separately. We illustrate the general constructions using the example of a Euclidean bi-Yang-Baxter deformation.
Paper Structure (14 sections, 4 theorems, 174 equations)

This paper contains 14 sections, 4 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Lorentzian and Euclidean $\mathcal{E}$-models
  3. First order formalism
  4. Lorentzian and Euclidean Poisson-Lie T-duality
  5. Perfect Poisson-Lie T-duality
  6. $\mathcal{E}$-Wick rotation
  7. Integrability of Euclidean $\mathcal{E}$-models
  8. Renormalization of Euclidean $\mathcal{E}$-models
  9. Euclidean bi-Yang-Baxter deformation
  10. Lu-Weinstein Drinfeld double
  11. Bi-Yang-Baxter deformations
  12. Integrability
  13. Conclusions and outlook
  14. Appendix

Key Result

Theorem A.1

Consider a Drinfeld double $\mathcal{D}$ and a linear operator $\mathcal{E}:\mathcal{D}\to \mathcal{D}$ such that for all $x,y\in \mathcal{D}$ it holds $(\mathcal{E} x,y)_\mathcal{D}=(x,\mathcal{E} y)_\mathcal{D}$ and such that $\mathcal{E}^2=-\mathbbm{1}$. It follows that the signature of the symme

Theorems & Definitions (10)

  • Remark 2.1
  • Remark 6.1
  • Theorem A.1
  • proof
  • Theorem A.2
  • proof
  • Theorem A.3
  • proof
  • Theorem A.4
  • proof