Geodesic
$T\overline T$ deformation of Calogero-Sutherland model via dimensional reduction
Teoretičeskaâ i matematičeskaâ fizika, Tome 217 (2023) no. 2, pp. 358-377 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We derive a general expression for the trace of the energy-momentum tensor $T\overline T$-deformed field theories using a dynamical change of coordinates. Then we perform a dimensional reduction of the bilinear $T\overline T$ operator and obtain a new $T\overline T$-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.
Keywords: Field Theories in Lower Dimensions, Integrability, Quantum Mechanics. 
@article{TMF_2023_217_2_a7,
     author = {D. V. Pavshinkin},
     title = {$T\overline T$ deformation of {Calogero-Sutherland} model via dimensional reduction},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {358--377},
     year = {2023},
     volume = {217},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2023_217_2_a7/}
}
TY  - JOUR
AU  - D. V. Pavshinkin
TI  - $T\overline T$ deformation of Calogero-Sutherland model via dimensional reduction
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2023
SP  - 358
EP  - 377
VL  - 217
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2023_217_2_a7/
LA  - ru
ID  - TMF_2023_217_2_a7
ER  - 
%0 Journal Article
%A D. V. Pavshinkin
%T $T\overline T$ deformation of Calogero-Sutherland model via dimensional reduction
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2023
%P 358-377
%V 217
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2023_217_2_a7/
%G ru
%F TMF_2023_217_2_a7
D. V. Pavshinkin. $T\overline T$ deformation of Calogero-Sutherland model via dimensional reduction. Teoretičeskaâ i matematičeskaâ fizika, Tome 217 (2023) no. 2, pp. 358-377. http://geodesic.mathdoc.fr/item/TMF_2023_217_2_a7/

[1] F. A. Smirnov, A. B. Zamolodchikov, “On space of integrable quantum field theories”, Nucl. Phys. B, 915 (2017), 363–383, arXiv: 1608.05499 | DOI | MR

[2] A. Cavaglià, S. Negro, I. M. Szécsényi, R. Tateo, “$T\overline T$-deformed 2D quantum field theories”, JHEP, 10 (2016), 112, 27 pp., arXiv: 1608.05534 | DOI | MR

[3] Y. Jiang, “A pedagogical review on solvable irrelevant deformations of 2D quantum field theory”, Commun. Theor. Phys., 73:5 (2021), 057201, 39 pp., arXiv: 1904.13376 | DOI | MR

[4] J. Cardy, $T\overline T$ deformations of non-Lorentz invariant field theories, arXiv: 1809.07849

[5] T. Bargheer, N. Beisert, F. Loebbert, “Long-range deformations for integrable spin chains”, J. Phys. A, 42:28 (2009), 285205, 58 pp., arXiv: 0902.0956 | DOI | MR

[6] T. Bargheer, N. Beisert, F. Loebbert, “Boosting nearest-neighbour to long-range integrable spin chains”, J. Stat. Mech., 2008:11 (2008), L11001, 10 pp., arXiv: 0807.5081 | DOI

[7] B. Pozsgay, Y. Jiang, G. Takács, “$T\overline T$-deformation and long range spin chains”, JHEP, 03 (2020), 092, 21 pp., arXiv: 1911.11118 | DOI | MR

[8] E. Marchetto, A. Sfondrini, Z. Yang, “$T\overline T$ deformations and integrable spin chains”, Phys. Rev. Lett., 124:10 (2020), 100601, 6 pp., arXiv: 1911.12315 | DOI | MR

[9] J. Cardy, B. Doyon, “$T\overline T$ deformations and the width of fundamental particles”, JHEP, 04 (2022), 136, 27 pp., arXiv: 2010.15733 | DOI | MR

[10] Y. Jiang, “$\mathrm T\overline{\mathrm T}$-deformed 1d Bose gas”, SciPost Phys., 12:6 (2022), 191, 48 pp., arXiv: 2011.00637 | DOI | MR

[11] B. Chen, J. Hou, J. Tian, “Note on the nonrelativistic $T\overline T$ deformation”, Phys. Rev. D, 104:2 (2021), 025004, 15 pp. | DOI | MR

[12] P. Ceschin, R. Conti, R. Tateo, “$\mathrm T\overline{\mathrm T}$-deformed nonlinear Schrödinger”, JHEP, 04 (2021), 121, 22 pp., arXiv: 2012.12760 | DOI | MR

[13] R. Conti, S. Negro, R. Tateo, “The $\mathrm T\overline{\mathrm T}$ perturbation and its geometric interpretation”, JHEP, 2 (2019), 085, 28 pp., arXiv: 1809.09593 | DOI | MR

[14] D. J. Gross, J. Kruthoff, A. Rolph, E. Shaghoulian, “$T\overline{T}$ in $\mathrm{AdS}_2$ and quantum mechanics”, Phys. Rev. D, 101:2 (2020), 026011, 20 pp. | DOI | MR

[15] D. J. Gross, J. Kruthoff, A. Rolph, E. Shaghoulian, “Hamiltonian deformations in quantum mechanics, $T\overline T$, and the SYK model”, Phys. Rev. D, 102:4 (2020), 046019, 17 pp. | DOI | MR

[16] L. McGough, M. Mezei, H. Verlinde, “Moving the CFT into the bulk with $T\overline T$ ”, JHEP, 04 (2018), 010, 33 pp., arXiv: 1611.03470 | DOI | MR

[17] F. Calogero, “Ground state of one-dimensional $N$ body system”, J. Math. Phys., 10:12 (1969), 2197–2200 | DOI

[18] B. Sutherland, “Exact results for a quantum many-body problem in one dimension”, Phys. Rev. A, 4:5 (1971), 2019–2021 | DOI

[19] J. Moser, “Three integrable Hamiltonian systems connnected with isospectral deformations”, Adv. Math., 16:2 (1975), 197–220 | DOI | MR

[20] E. Bergshoeff, M. A. Vasiliev, “The Calogero model and the Virasoro symmetry”, Internat. J. Modern Phys. A, 10:24 (1995), 3477–3496, arXiv: hep-th/9411093 | DOI | MR

[21] J. Cardy, “Calogero–Sutherland model and bulk-boundary correlations in conformal field theory”, Phys. Lett. B, 582:1–2 (2004), 121–126, arXiv: hep-th/0310291 | DOI | MR

[22] A. G. Abanov, P. B. Wiegmann, “Quantum hydrodynamics, quantum Benjamin–Ono equation, and Calogero model”, Phys. Rev. Lett., 95:7 (2005), 076402, 4 pp., arXiv: cond-mat/0504041 | DOI

[23] M. Caselle, U. Magnea, “Random matrix theory and symmetric spaces”, Phys. Rept., 394:2–3 (2004), 41–156, arXiv: cond-mat/0304363 | DOI | MR

[24] S. Ouvry, A. P. Polychronakos, “Mapping the Calogero model on the anyon model”, Nucl. Phys. B, 936 (2018), 189–205, arXiv: 1805.09899 | DOI | MR

[25] A. Gorsky, A. Mironov, “Integrable many body systems and gauge theories”, Integrable Hierarchies and Modern Physical Theories, NATO Science Series II: Mathematics, Physics and Chemistry, 18, eds. H. Aratyn, A. S. Sorin, Springer, Dordrecht, 2001, 33–176, arXiv: hep-th/0011197 | MR

[26] J. A. Minahan, A. P. Polychronakos, “Equivalence of two-dimensional QCD and the $c=1$ matrix model”, Phys. Lett. B, 312:1–2 (1993), 155–165, arXiv: hep-th/9303153 | DOI | MR

[27] M. R. Douglas, Conformal field theory techniques for large $N$ group theory, arXiv: hep-th/9303159

[28] A. Gorsky, N. Nekrasov, “Hamiltonian systems of Calogero-type and two-dimensional Yang–Mills theory”, Nucl. Phys. B, 414:1–2 (1994), 213–238, arXiv: hep-th/9304047 | DOI | MR

[29] A. Gorsky, N. Nekrasov, “Relativistic Calogero–Moser model as gauged WZW theory”, Nucl. Phys. B, 436:3 (1995), 582–608, arXiv: hep-th/9401017 | DOI | MR

[30] J. A. Minahan, A. P. Polychronakos, “Interacting fermion systems from two-dimensional QCD”, Phys. Lett. B, 326:3–4 (1994), 288–294, arXiv: hep-th/9309044 | DOI

[31] R. Conti, L. Iannella, S. Negro, R. Tateo, “Generalised Born–Infeld models, Lax operators and the $T\overline{T}$ perturbation”, JHEP, 11 (2018), 007, 22 pp., arXiv: 1806.11515 | DOI | MR

[32] L. Santilli, M. Tierz, “Large $N$ phase transition in $T\overline T$-deformed 2d Yang–Mills theory on the sphere”, JHEP, 01 (2019), 054, 24 pp., arXiv: 1810.05404 | DOI | MR

[33] T. D. Brennan, C. Ferko, S. Sethi, “A non-abelian analogue of DBI from $T\overline T$ ”, SciPost Phys., 8:4 (2020), 052, 18 pp., arXiv: 1912.12389 | DOI

[34] A. Ireland, V. Shyam, “$T\overline T$ deformed YM$_2$ on general backgrounds from an integral transformation”, JHEP, 07 (2020), 058, 17 pp., arXiv: 1912.04686 | DOI | MR

[35] H. Babaei-Aghbolagh, K. B. Velni, D. M. Yekta, H. Mohammadzadeh, “$ T\overline T $-like flows in non-linear electrodynamic theories and S-duality”, JHEP, 04 (2021), 187, 24 pp., arXiv: 2012.13636 | DOI | MR

[36] L. Santilli, R. J. Szabo, M. Tierz, “$T\overline T$-deformation of $q$-Yang–Mills theory”, JHEP, 11 (2020), 086, 45 pp., arXiv: 2009.00657 | DOI | MR

[37] A. Gorsky, D. Pavshinkin, A. Tyutyakina, “$T\overline T$-deformed 2D Yang–Mills at large $N$: collective field theory and phase transitions”, JHEP, 03 (2021), 142, 21 pp., arXiv: 2012.09467 | DOI | MR

[38] G. Arutyunov, “Lectures on integrable systems”, PoS (Regio2020), 394 (2021), 001, 76 pp. | DOI

[39] A. P. Polychronakos, “Physics and mathematics of Calogero particles”, J. Phys. A: Math. Gen., 39:41 (2006), 12793–12845 | DOI | MR

[40] J. Kruthoff, J. Parrikar, On the flow of states under $T\overline T$, arXiv: 2006.03054

[41] S. Cordes, G. Moore, S. Ramgoolam, “Lectures on 2D Yang–Mills theory, equivariant cohomology and topological field theories”, Nucl. Phys. B Proc. Suppl., 41:1–3 (1995), 184–244, arXiv: hep-th/9411210 | DOI | MR

[42] S. de Haro, M. Tierz, “Brownian motion, Chern–Simons theory, and 2d Yang–Mills”, Phys. Lett. B, 601:3–4 (2004), 201–208, arXiv: hep-th/0406093 | DOI | MR

[43] P. J. Forrester, S. N. Majumdar, G. Schehr, “Non-intersecting Brownian walkers and Yang–Mills theory on the sphere”, Nucl. Phys. B, 844:3 (2011), 500–526 ; Erratum, 857:3 (2012), 424–427, arXiv: 1009.2362 | DOI | MR | DOI | MR

[44] G. Schehr, S. N. Majumdar, A. Comtet, P. J. Forrester, “Reunion probability of $N$ vicious walkers: Typical and large fluctuations for large $N$”, J. Stat. Phys., 150:3 (2012), 491–530 | DOI

[45] A. Gorsky, A. Milekhin, S. Nechaev, “Two faces of Douglas–Kazakov transition: from Yang–Mills theory to random walks and beyond”, Nucl. Phys. B, 950 (2020), 114849, 23 pp., arXiv: 1604.06381 | DOI | MR

[46] D. J. Gross, W. Taylor IV, “Two-dimensional QCD is a string theory”, Nucl. Phys. B, 400:1–3 (1993), 181–208, arXiv: hep-th/9301068 | DOI | MR