## Abstract

We study the partition function of a \( T\overline{T} \) -deformed version of Yang-Mills theory on the two-sphere. We show that the Douglas-Kazakov phase transition persists for a range of values of the deformation parameter, and that the critical area is lowered. The transition is of third order and also induced by instantons, whose contributions we characterize.

## References

A.B. Zamolodchikov,

*Expectation value of composite field*\( T\overline{T} \)*in two-dimensional quantum field theory*, hep-th/0401146 [INSPIRE].F.A. Smirnov and A.B. Zamolodchikov,

*On space of integrable quantum field theories*,*Nucl. Phys.***B 915**(2017) 363 [arXiv:1608.05499] [INSPIRE].A. Cavaglià, S. Negro, I.M. Szécsényi and R. Tateo, \( T\overline{T} \)

*-deformed*2*D quantum field theories*,*JHEP***10**(2016) 112 [arXiv:1608.05534] [INSPIRE].S. Dubovsky, V. Gorbenko and M. Mirbabayi,

*Asymptotic fragility, near AdS*_{2}*holography and*\( T\overline{T} \),*JHEP***09**(2017) 136 [arXiv:1706.06604] [INSPIRE].S. Dubovsky, V. Gorbenko and G. Hernández-Chifflet, \( T\overline{T} \)

*partition function from topological gravity*,*JHEP***09**(2018) 158 [arXiv:1805.07386] [INSPIRE].A. Giveon, N. Itzhaki and D. Kutasov, \( T\overline{T} \)

*and LST*,*JHEP***07**(2017) 122 [arXiv:1701.05576] [INSPIRE].A. Giveon, N. Itzhaki and D. Kutasov,

*A solvable irrelevant deformation of AdS*_{3}*/CFT*_{2},*JHEP***12**(2017) 155 [arXiv:1707.05800] [INSPIRE].S. Datta and Y. Jiang, \( T\overline{T} \)

*deformed partition functions*,*JHEP***08**(2018) 106 [arXiv:1806.07426] [INSPIRE].O. Aharony, S. Datta, A. Giveon, Y. Jiang and D. Kutasov,

*Modular invariance and uniqueness of*\( T\overline{T} \)*deformed CFT*, arXiv:1808.02492 [INSPIRE].J. Cardy,

*The*\( T\overline{T} \)*deformation of quantum field theory as random geometry*,*JHEP***10**(2018) 186 [arXiv:1801.06895] [INSPIRE].R. Conti, S. Negro and R. Tateo,

*The*\( T\overline{T} \)*perturbation and its geometric interpretation*, arXiv:1809.09593 [INSPIRE].J. Cardy, \( T\overline{T} \)

*deformations of non-Lorentz invariant field theories*, arXiv:1809.07849 [INSPIRE].R. Conti, L. Iannella, S. Negro and R. Tateo,

*Generalised Born-Infeld models, Lax operators and the*\( T\overline{T} \)*perturbation*,*JHEP***11**(2018) 007 [arXiv:1806.11515] [INSPIRE].S. Cordes, G.W. Moore and S. Ramgoolam,

*Lectures on*2*D Yang-Mills theory, equivariant cohomology and topological field theories*,*Nucl. Phys. Proc. Suppl.***41**(1995) 184 [hep-th/9411210] [INSPIRE].M.R. Douglas and V.A. Kazakov,

*Large N phase transition in continuum QCD in two-dimensions*,*Phys. Lett.***B 319**(1993) 219 [hep-th/9305047] [INSPIRE].D.J. Gross and A. Matytsin,

*Instanton induced large N phase transitions in two-dimensional and four-dimensional QCD*,*Nucl. Phys.***B 429**(1994) 50 [hep-th/9404004] [INSPIRE].A.A. Migdal,

*Recursion equations in gauge theories*,*Sov. Phys. JETP***42**(1975) 413 [*Zh. Eksp. Teor. Fiz.***69**(1975) 810] [INSPIRE].P. Menotti and E. Onofri,

*The action of*SU(*N*)*lattice gauge theory in terms of the heat kernel on the group manifold*,*Nucl. Phys.***B 190**(1981) 288 [INSPIRE].B.E. Rusakov,

*Loop averages and partition functions in*U(*N*)*gauge theory on two-dimensional manifolds*,*Mod. Phys. Lett.***A 5**(1990) 693 [INSPIRE].M.R. Douglas,

*Conformal field theory techniques in large N Yang-Mills theory*, in*NATO Advanced Research Workshop on New Developments in String Theory, Conformal Models and Topological Field Theory*, Cargese, France, 12-21 May 1993 [hep-th/9311130] [INSPIRE].J.A. Minahan and A.P. Polychronakos,

*Classical solutions for two-dimensional QCD on the sphere*,*Nucl. Phys.***B 422**(1994) 172 [hep-th/9309119] [INSPIRE].D.J. Gross and E. Witten,

*Possible third order phase transition in the large N lattice gauge theory*,*Phys. Rev.***D 21**(1980) 446 [INSPIRE].S.R. Wadia,

*N*= ∞*phase transition in a class of exactly soluble model lattice gauge theories*,*Phys. Lett.***B 93**(1980) 403 [INSPIRE].S.R. Wadia,

*A study of*U(*N*)*lattice gauge theory in*2*-dimensions*, arXiv:1212.2906 [INSPIRE].M. Caselle, A. D’Adda, L. Magnea and S. Panzeri,

*Two-dimensional QCD on the sphere and on the cylinder*, in*Proceedings, Summer School in High-energy physics and cosmology*, Trieste, Italy, 15 June-31 July 1992, pg. 0245 [hep-th/9309107] [INSPIRE].D.J. Gross,

*Two-dimensional QCD as a string theory*,*Nucl. Phys.***B 400**(1993) 161 [hep-th/9212149] [INSPIRE].D.J. Gross and W. Taylor,

*Two-dimensional QCD is a string theory*,*Nucl. Phys.***B 400**(1993) 181 [hep-th/9301068] [INSPIRE].W. Donnelly and G. Wong,

*Entanglement branes in a two-dimensional string theory*,*JHEP***09**(2017) 097 [arXiv:1610.01719] [INSPIRE].A.C. Pipkin,

*A course on integral equations*,*Texts Appl. Math.***9**, Springer, New York, U.S.A. (1991).N.I. Muskhelishvili,

*Singular integral equations*, Noordhoff, The Netherlands (1977).D. Jafferis and J. Marsano,

*A DK phase transition in q-deformed Yang-Mills on S*^{2}*and topological strings*, hep-th/0509004 [INSPIRE].R.J. Szabo and M. Tierz,

*q-deformations of two-dimensional Yang-Mills theory: classification, categorification and refinement*,*Nucl. Phys.***B 876**(2013) 234 [arXiv:1305.1580] [INSPIRE].E. Witten,

*On quantum gauge theories in two-dimensions*,*Commun. Math. Phys.***141**(1991) 153 [INSPIRE].E. Witten,

*Two-dimensional gauge theories revisited*,*J. Geom. Phys.***9**(1992) 303 [hep-th/9204083] [INSPIRE].M. Blau and G. Thompson,

*Derivation of the Verlinde formula from Chern-Simons theory and the G/G model*,*Nucl. Phys.***B 408**(1993) 345 [hep-th/9305010] [INSPIRE].M. Blau and G. Thompson,

*Lectures on*2*D gauge theories: topological aspects and path integral techniques*, in*Proceedings, Summer School in High-energy physics and cosmology*, Trieste, Italy, 14 June-30 July 1993, pg. 0175 [hep-th/9310144] [INSPIRE].N. Caporaso, M. Cirafici, L. Griguolo, S. Pasquetti, D. Seminara and R.J. Szabo,

*Topological strings and large N phase transitions. I. Nonchiral expansion of q-deformed Yang-Mills theory*,*JHEP***01**(2006) 035 [hep-th/0509041] [INSPIRE].S. Giombi and V. Pestun,

*Correlators of local operators and*1*/*8*BPS Wilson loops on S*^{2}*from*2*d YM and matrix models*,*JHEP***10**(2010) 033 [arXiv:0906.1572] [INSPIRE].S. Giombi and V. Pestun,

*The*1*/*2*BPS ’t Hooft loops in N*= 4*SYM as instantons in*2*d Yang-Mills*,*J. Phys.***A 46**(2013) 095402 [arXiv:0909.4272] [INSPIRE].A. Bassetto and L. Griguolo,

*Two-dimensional QCD, instanton contributions and the perturbative Wu-Mandelstam-Leibbrandt prescription*,*Phys. Lett.***B 443**(1998) 325 [hep-th/9806037] [INSPIRE].N. Drukker and D.J. Gross,

*An exact prediction of N*= 4*SUSYM theory for string theory*,*J. Math. Phys.***42**(2001) 2896 [hep-th/0010274] [INSPIRE].B. Durhuus and P. Olesen,

*The spectral density for two-dimensional continuum QCD*,*Nucl. Phys.***B 184**(1981) 461 [INSPIRE].J.-P. Blaizot and M.A. Nowak,

*Large N*_{c}*confinement and turbulence*,*Phys. Rev. Lett.***101**(2008) 102001 [arXiv:0801.1859] [INSPIRE].H. Neuberger,

*Complex Burgers’ equation in*2*D*SU(*N*)*YM*,*Phys. Lett.***B 670**(2008) 235 [arXiv:0809.1238] [INSPIRE].M.J. Crescimanno and W. Taylor,

*Large N phases of chiral QCD in two-dimensions*,*Nucl. Phys.***B 437**(1995) 3 [hep-th/9408115] [INSPIRE].M.R. Douglas, K. Li and M. Staudacher,

*Generalized two-dimensional QCD*,*Nucl. Phys.***B 420**(1994) 118 [hep-th/9401062] [INSPIRE].O. Ganor, J. Sonnenschein and S. Yankielowicz,

*The string theory approach to generalized*2*D Yang-Mills theory*,*Nucl. Phys.***B 434**(1995) 139 [hep-th/9407114] [INSPIRE].A. Gadde, L. Rastelli, S.S. Razamat and W. Yan,

*The*4*d superconformal index from q-deformed*2*d Yang-Mills*,*Phys. Rev. Lett.***106**(2011) 241602 [arXiv:1104.3850] [INSPIRE].M. Aganagic, H. Ooguri, N. Saulina and C. Vafa,

*Black holes, q-deformed*2*d Yang-Mills and non-perturbative topological strings*,*Nucl. Phys.***B 715**(2005) 304 [hep-th/0411280] [INSPIRE].C. Beasley and E. Witten,

*Non-Abelian localization for Chern-Simons theory*,*J. Diff. Geom.***70**(2005) 183 [hep-th/0503126] [INSPIRE].J. Kallen,

*Cohomological localization of Chern-Simons theory*,*JHEP***08**(2011) 008 [arXiv:1104.5353] [INSPIRE].X. Arsiwalla, R. Boels, M. Mariño and A. Sinkovics,

*Phase transitions in q-deformed*2*D Yang-Mills theory and topological strings*,*Phys. Rev.***D 73**(2006) 026005 [hep-th/0509002] [INSPIRE].

###
**Open Access**

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

ArXiv ePrint: 1810.05404

## Rights and permissions

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.

The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.

## About this article

### Cite this article

Santilli, L., Tierz, M. Large *N* phase transition in \( T\overline{T} \) -deformed 2*d* Yang-Mills theory on the sphere.
*J. High Energ. Phys.* **2019**, 54 (2019). https://doi.org/10.1007/JHEP01(2019)054

Received:

Accepted:

Published:

DOI: https://doi.org/10.1007/JHEP01(2019)054

### Keywords

- Field Theories in Lower Dimensions
- Matrix Models
- Nonperturbative Effects