Abstract
We explicitly find representations for different large N phases of Chern-Simons matter theory on S2 × S1. These representations are characterised by Young diagrams. We show that no-gap and lower-gap phase of Chern-Simons-matter theory correspond to integrable representations of SU(N)k affine Lie algebra, where as upper-cap phase corresponds to integrable representations of SU(k − N)k affine Lie algebra. We use phase space description of [1] to obtain these representations and argue how putting a cap on eigenvalue distribution forces corresponding representations to be integrable. We also prove that the Young diagrams corresponding to lower-gap and upper-cap representations are related to each other by transposition under level-rank duality. Finally we draw phase space droplets for these phases and show how information about eigenvalue and Young diagram descriptions can be captured in topologies of these droplets in a unified way.
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References
S. Dutta and R. Gopakumar, Free fermions and thermal AdS/CFT, JHEP 03 (2008) 011 [arXiv:0711.0133] [INSPIRE].
A. Chattopadhyay, P. Dutta and S. Dutta, Emergent phase space description of unitary matrix model, JHEP 11 (2017) 186 [arXiv:1708.03298] [INSPIRE].
E. Brézin, C. Itzykson, G. Parisi and J.B. Zuber, Planar diagrams, Commun. Math. Phys. 59 (1978) 35 [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].
V.A. Kazakov, M. Staudacher and T. Wynter, Character expansion methods for matrix models of dually weighted graphs, Commun. Math. Phys. 177 (1996) 451 [hep-th/9502132] [INSPIRE].
N. Seiberg, Electric-magnetic duality in supersymmetric nonAbelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].
A. Giveon and D. Kutasov, Seiberg duality in Chern-Simons theory, Nucl. Phys. B 812 (2009) 1 [arXiv:0808.0360] [INSPIRE].
A. Kapustin, B. Willett and I. Yaakov, Tests of Seiberg-like duality in three dimensions, arXiv:1012.4021 [INSPIRE].
S.G. Naculich, H.A. Riggs and H.J. Schnitzer, Group level duality in WZW models and Chern-Simons theory, Phys. Lett. B 246 (1990) 417 [INSPIRE].
E.J. Mlawer, S.G. Naculich, H.A. Riggs and H.J. Schnitzer, Group level duality of WZW fusion coefficients and Chern-Simons link observables, Nucl. Phys. B 352 (1991) 863 [INSPIRE].
S.G. Naculich and H.J. Schnitzer, Duality between SU(N)k and SUk−n WZW models, Nucl. Phys. B 347 (1990) 687 [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.
J. Jurkiewicz and K. Zalewski, Phase structure of U (N → ∞) gauge theory on a two-dimensional lattice for a broad class of variant actions, Nucl. Phys. B 220 (1983) 167.
O. Aharony et al., The Hagedorn-deconfinement phase transition in weakly coupled large N gauge theories, Adv. Theor. Math. Phys. 8 (2004) 603 [hep-th/0310285] [INSPIRE].
P. Basu and S.R. Wadia, R-charged AdS 5 black holes and large N unitary matrix models, Phys. Rev. D 73 (2006) 045022 [hep-th/0506203] [INSPIRE].
L. Álvarez-Gaumé, P. Basu, M. Mariño and S.R. Wadia, Blackhole/string transition for the small Schwarzschild blackhole of AdS 5 × S 5 and critical unitary matrix models, Eur. Phys. J. C 48 (2006) 647 [hep-th/0605041] [INSPIRE].
G. Mandal, Phase structure of unitary matrix models, Mod. Phys. Lett. A 5 (1990) 1147 [INSPIRE].
D. Yamada and L.G. Yaffe, Phase diagram of N = 4 super-Yang-Mills theory with R-symmetry chemical potentials, JHEP 09 (2006) 027 [hep-th/0602074] [INSPIRE].
L. Álvarez-Gaumé, C. Gomez, H. Liu and S. Wadia, Finite temperature effective action, AdS 5 black holes and 1/N expansion, Phys. Rev. D 71 (2005) 124023 [hep-th/0502227] [INSPIRE].
T. Harmark and M. Orselli, Quantum mechanical sectors in thermal N = 4 super Yang-Mills on R × S 3, Nucl. Phys. B 757 (2006) 117 [hep-th/0605234] [INSPIRE].
D. Friedan, Some nonabelian toy models in the large N limit, Commun. Math. Phys. 78 (1981) 353 [INSPIRE].
B. Sundborg, The Hagedorn transition, deconfinement and N = 4 SYM theory, Nucl. Phys. B 573 (2000) 349 [hep-th/9908001] [INSPIRE].
S. Jain et al., Phases of large N vector Chern-Simons theories on S 2 × S 1, JHEP 09 (2013) 009 [arXiv:1301.6169] [INSPIRE].
T. Takimi, Duality and higher temperature phases of large N Chern-Simons matter theories on S 2 × S 1, JHEP 07 (2013) 177 [arXiv:1304.3725] [INSPIRE].
S. Codesido, A. Grassi and M. Mariño, Exact results in \( \mathcal{N}=8 \) Chern-Simons-matter theories and quantum geometry, JHEP 07 (2015) 011 [arXiv:1409.1799] [INSPIRE].
M. Mariño and P. Putrov, Interacting fermions and N = 2 Chern-Simons-matter theories, JHEP 11 (2013) 199 [arXiv:1206.6346] [INSPIRE].
S. Jain, S.P. Trivedi, S.R. Wadia and S. Yokoyama, Supersymmetric Chern-Simons theories with vector matter, JHEP 10 (2012) 194 [arXiv:1207.4750] [INSPIRE].
Y. Dandekar, M. Mandlik and S. Minwalla, Poles in the S-matrix of relativistic Chern-Simons matter theories from quantum mechanics, JHEP 04 (2015) 102 [arXiv:1407.1322] [INSPIRE].
S. Jain, S. Minwalla and S. Yokoyama, Chern Simons duality with a fundamental boson and fermion, JHEP 11 (2013) 037 [arXiv:1305.7235] [INSPIRE].
S. Giombi et al., Chern-Simons theory with vector fermion matter, Eur. Phys. J. C 72 (2012) 2112 [arXiv:1110.4386] [INSPIRE].
S. Minwalla, P. Narayan, T. Sharma, V. Umesh and X. Yin, Supersymmetric states in large N Chern-Simons-Matter theories, JHEP 02 (2012) 022 [arXiv:1104.0680] [INSPIRE].
T. Suyama, Eigenvalue distributions in matrix models for Chern-Simons-Matter theories, Nucl. Phys. B 856 (2012) 497 [arXiv:1106.3147] [INSPIRE].
E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
S.G. Naculich and H.J. Schnitzer, Level-rank duality of the U(N) WZW model, Chern-Simons theory and 2D qYM theory, JHEP 06 (2007) 023 [hep-th/0703089] [INSPIRE].
P. Dutta and S. Dutta, Phase space distribution of Riemann zeros, J. Math. Phys. 58 (2017) 053504 [arXiv:1610.07743] [INSPIRE].
O. Aharony et al., The thermal free energy in large N Chern-Simons-Matter theories, JHEP 03 (2013) 121 [arXiv:1211.4843] [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.R. Douglas, Conformal field theory techniques in large N Yang-Mills theory, in the proceedings of the NATO Advanced Research Workshop on New Developments in String Theory, Conformal Models and Topological Field Theory, May 12–21, Cargese, France (1993) hep-th/9311130 [INSPIRE].
M. Lassalle, Explicitation of characters of the symmetric group, Compt. Rend. Math. 341 (2005) 529.
M. Hamermesh, Group theory and its application to physical problems, Dover Publication, U.S.A. (1989).
W. Fulton and J. Harris, Representation theory: a first course, Graduate Texts in Mathematics, Springer, Germany (1999).
P. Dutta and S. Dutta, Phase space distribution for two-gap solution in unitary matrix model, JHEP 04 (2016) 104 [arXiv:1510.03444] [INSPIRE].
S.R. Das and A. Jevicki, String field theory and physical interpretation of D = 1 strings, Mod. Phys. Lett. A 5 (1990) 1639.
A. Jevicki and B. Sakita, Collective field approach to the large-N limit: Euclidean field theories, Nucl. Phys. B 185 (1981) 89.
A. Jevicki and B. Sakita, The quantum collective field method and its application to the planar limit, Nucl. Phys. B 165 (1980) 511.
H. Georgi, Lie algebras in particle physics, Front. Phys. 54 (1999) 1.
P.D. Francesco, P. Mathieu and D. Sénéchal, Conformal field theory, Springer, Germany (1997).
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Chattopadhyay, A., Dutta, P. & Dutta, S. From phase space to integrable representations and level-rank duality. J. High Energ. Phys. 2018, 117 (2018). https://doi.org/10.1007/JHEP05(2018)117
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DOI: https://doi.org/10.1007/JHEP05(2018)117