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\(T\overline T\) deformation of the Calogero–Sutherland model via dimensional reduction

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Abstract

We derive a general expression for the trace of the energy–momentum tensor \(T \kern0.9pt\overline{\vphantom{T}\kern6.0pt}\kern-6.9pt T \)-deformed field theories using a dynamical change of coordinates. Then we perform a dimensional reduction of the bilinear \(T \kern0.9pt\overline{\vphantom{T}\kern6.0pt}\kern-6.9pt T \) operator and obtain a new \(T \kern0.9pt\overline{\vphantom{T}\kern6.0pt}\kern-6.9pt T \)-like deformation of the quantum mechanics of free nonrelativistic fermions and the interacting Calogero–Sutherland system. The deformation leads to a change in the energy spectrum but does not affect the eigenfunctions. Furthermore, an expression for the deformed classical Lagrangian is obtained. We also study the correspondence between the two-dimensional Yang–Mills theory and the Calogero–Sutherland system in the presence of the deformation.

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Notes

  1. A \(T\overline T\)-perturbed 1D CFT was considered in [14]. The \({T_\lambda}^x_x\) component was identified with the field dual to the dilaton in the bulk.

  2. The additional eigenstates do not appear under the deformation in the case of a system with a finite number of degrees of freedom.

  3. Strictly speaking, a difference may appear because of counterterms related to operator ordering.

  4. The corresponding Young tableau is a row of \(N\gamma\) boxes.

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Acknowledgments

The author is grateful to A. S. Gorsky for inspiring discussions and collaborations on related projects.

Funding

This work was supported by the BASIS Foundation grant No. 20-1-1-23-1 (Sections 1–3) and the Russian Science Foundation grant No. 22-72-10122, https://rscf.ru/en/project/22-72-10122/ (Sections 4–6).

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Authors and Affiliations

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

    D. V. Pavshinkin

  2. Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, Russia

    D. V. Pavshinkin

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Correspondence to D. V. Pavshinkin.

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Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, 2023, Vol. 217, pp. 358–377 https://doi.org/10.4213/tmf10537.

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Pavshinkin, D.V. \(T\overline T\) deformation of the Calogero–Sutherland model via dimensional reduction. Theor Math Phys 217, 1726–1742 (2023). https://doi.org/10.1134/S0040577923110089

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