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Sigma models as Gross–Neveu models

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Abstract

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

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Notes

  1. The fact that the \(B\) field is topological is characteristic of symmetric space models. In general, it is a nonclosed 2-form.

  2. Integrability-preserving deformations of sigma models have a long history, cf. [22]–[26].

  3. The \(r\)-matrix used in that paper is related to (3.3) by a similarity transformation. As a consequence of this, an additional coordinate transformation was needed in [2] in order to satisfy the RG equation, which is not necessary in the present formulation.

  4. The relation between integrability and renormalizability has been discussed throughout the years, cf. [30]–[37].

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Acknowledgments

I thank A. A. Slavnov for support and A. K. Pogrebkov for the invitation to give a talk at the “Polivanov-90” conference. Some of this material was also presented at the conferences “GLSMs-2020” (Virginia Tech, USA) and “RAQIS-2020” (Annecy, France), and I am grateful to the respective organizers E. Sharpe and E. Ragoucy for the invitations.

Funding

This work was supported by the Russian Science Foundation grant RSCF-20-72-10144.

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  1. Steklov Mathematical Institute of Russian Academy of Science, Moscow, Russia

    D. V. Bykov

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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2021, Vol. 208, pp. 165-179 https://doi.org/10.4213/tmf10103.

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Bykov, D.V. Sigma models as Gross–Neveu models. Theor Math Phys 208, 993–1003 (2021). https://doi.org/10.1134/S0040577921080018

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