Abstract
We study the matrix models calculating the sphere partition functions of 3d gauge theories with \( \mathcal{N}=4 \) supersymmetry and a quiver structure of a \( \widehat{D} \) Dynkin diagram (where each node is a unitary gauge group). As in the case of necklace (Â) quivers, we can map the problem to that of free fermion quantum mechanics whose complicated Hamiltonian we find explicitly. Many of these theories are conjectured to be dual under mirror symmetry to certain unitary linear quivers with extra Sp nodes or antisymmetric hypermultiplets. We show that the free fermion formulation of such mirror pairs are related by a linear symplectic transformation.
We then study the large N expansion of the partition function, which as in the case of the  quivers is given to all orders in 1/N by an Airy function. We simplify the algorithm to calculate the numerical coefficients appearing in the Airy function and evaluate them for a wide class of \( \widehat{D} \)-quiver theories.
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References
K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].
J. de Boer, K. Hori, H. Ooguri and Y. Oz, Mirror symmetry in three-dimensional gauge theories, quivers and D-branes, Nucl. Phys. B 493 (1997) 101 [hep-th/9611063] [INSPIRE].
J. de Boer, K. Hori, H. Ooguri, Y. Oz and Z. Yin, Mirror symmetry in three-dimensional theories, SL(2, Z) and D-brane moduli spaces, Nucl. Phys. B 493 (1997) 148 [hep-th/9612131] [INSPIRE].
A. Kapustin, B. Willett and I. Yaakov, Exact results for Wilson loops in superconformal Chern-Simons theories with matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].
A. Kapustin, B. Willett and I. Yaakov, Nonperturbative tests of three-dimensional dualities, JHEP 10 (2010) 013 [arXiv:1003.5694] [INSPIRE].
A. Dey and J. Distler, Three dimensional mirror symmetry and partition function on S 3, JHEP 10 (2013) 086 [arXiv:1301.1731] [INSPIRE].
B. Assel, Hanany-Witten effect and SL(2, Z) dualities in matrix models, JHEP 10 (2014) 117 [arXiv:1406.5194] [INSPIRE].
S. Cremonesi, A. Hanany and A. Zaffaroni, Monopole operators and Hilbert series of Coulomb branches of 3d N = 4 gauge theories, JHEP 01 (2014) 005 [arXiv:1309.2657] [INSPIRE].
S. Cremonesi, A. Hanany, N. Mekareeya and A. Zaffaroni, Coulomb branch Hilbert series and three dimensional Sicilian theories, JHEP 09 (2014) 185 [arXiv:1403.2384] [INSPIRE].
S. Cremonesi, A. Hanany, N. Mekareeya and A. Zaffaroni, T σ ρ (G) theories and their Hilbert series, JHEP 01 (2015) 150 [arXiv:1410.1548] [INSPIRE].
A. Dey, A. Hanany, P. Koroteev and N. Mekareeya, Mirror symmetry in three dimensions via gauged linear quivers, JHEP 06 (2014) 059 [arXiv:1402.0016] [INSPIRE].
M. Porrati and A. Zaffaroni, M theory origin of mirror symmetry in three-dimensional gauge theories, Nucl. Phys. B 490 (1997) 107 [hep-th/9611201] [INSPIRE].
A. Dey, On three-dimensional mirror symmetry, JHEP 04 (2012) 051 [arXiv:1109.0407] [INSPIRE].
N. Drukker and J. Felix, 3d mirror symmetry as a canonical transformation, JHEP 05 (2015) 004 [arXiv:1501.02268] [INSPIRE].
M. Mariño and P. Putrov, ABJM theory as a Fermi gas, J. Stat. Mech. 03 (2012) P03001 [arXiv:1110.4066] [INSPIRE].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M 2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
H. Fuji, S. Hirano and S. Moriyama, Summing up all genus free energy of ABJM matrix model, JHEP 08 (2011) 001 [arXiv:1106.4631] [INSPIRE].
N. Drukker, M. Mariño and P. Putrov, From weak to strong coupling in ABJM theory, Commun. Math. Phys. 306 (2011) 511 [arXiv:1007.3837] [INSPIRE].
N. Drukker, M. Mariño and P. Putrov, Nonperturbative aspects of ABJM theory, JHEP 11 (2011) 141 [arXiv:1103.4844] [INSPIRE].
Y. Hatsuda, S. Moriyama and K. Okuyama, Exact results on the ABJM Fermi gas, JHEP 10 (2012) 020 [arXiv:1207.4283] [INSPIRE].
P. Putrov and M. Yamazaki, Exact ABJM partition function from TBA, Mod. Phys. Lett. A 27 (2012) 1250200 [arXiv:1207.5066] [INSPIRE].
Y. Hatsuda, S. Moriyama and K. Okuyama, Instanton effects in ABJM theory from Fermi gas approach, JHEP 01 (2013) 158 [arXiv:1211.1251] [INSPIRE].
F. Calvo and M. Mariño, Membrane instantons from a semiclassical TBA, JHEP 05 (2013) 006 [arXiv:1212.5118] [INSPIRE].
Y. Hatsuda, S. Moriyama and K. Okuyama, Instanton bound states in ABJM theory, JHEP 05 (2013) 054 [arXiv:1301.5184] [INSPIRE].
Y. Hatsuda, M. Mariño, S. Moriyama and K. Okuyama, Non-perturbative effects and the refined topological string, JHEP 09 (2014) 168 [arXiv:1306.1734] [INSPIRE].
J. Kallen and M. Mariño, Instanton effects and quantum spectral curves, arXiv:1308.6485 [INSPIRE].
J. Kallen, The spectral problem of the ABJ Fermi gas, arXiv:1407.0625 [INSPIRE].
S. Matsumoto and S. Moriyama, ABJ fractional brane from ABJM Wilson loop, JHEP 03 (2014) 079 [arXiv:1310.8051] [INSPIRE].
M. Honda and K. Okuyama, Exact results on ABJ theory and the refined topological string, JHEP 08 (2014) 148 [arXiv:1405.3653] [INSPIRE].
Y. Hatsuda and K. Okuyama, Probing non-perturbative effects in M-theory, JHEP 10 (2014) 158 [arXiv:1407.3786] [INSPIRE].
S. Moriyama and T. Nosaka, Exact instanton expansion of superconformal Chern-Simons theories from topological strings, JHEP 05 (2015) 022 [arXiv:1412.6243] [INSPIRE].
S. Codesido, A. Grassi and M. Mariño, Exact results in N = 8 Chern-Simons-matter theories and quantum geometry, JHEP 07 (2015) 011 [arXiv:1409.1799] [INSPIRE].
A. Grassi, Y. Hatsuda and M. Mariño, Quantization conditions and functional equations in ABJ(M) theories, arXiv:1410.7658 [INSPIRE].
X.-F. Wang, X. Wang and M.-X. Huang, A note on instanton effects in ABJM theory, JHEP 11 (2014) 100 [arXiv:1409.4967] [INSPIRE].
A. Hanany and A. Zaffaroni, Issues on orientifolds: on the brane construction of gauge theories with SO(2N ) global symmetry, JHEP 07 (1999) 009 [hep-th/9903242] [INSPIRE].
D. Gaiotto and E. Witten, S-duality of boundary conditions in N = 4 super Yang-Mills theory, Adv. Theor. Math. Phys. 13 (2009) 721 [arXiv:0807.3720] [INSPIRE].
A. Dey, Mirror symmetry in flavored affine D-type quivers, Proc. Symp. Pure Math. 88 (2014) 259 [arXiv:1401.6462] [INSPIRE].
A. Hanany and E. Witten, Type IIB superstrings, BPS monopoles and three-dimensional gauge dynamics, Nucl. Phys. B 492 (1997) 152 [hep-th/9611230] [INSPIRE].
S. Moriyama and T. Nosaka, Partition functions of superconformal Chern-Simons theories from Fermi gas approach, JHEP 11 (2014) 164 [arXiv:1407.4268] [INSPIRE].
A. Dabholkar, N. Drukker and J. Gomes, Localization in supergravity and quantum AdS 4 /CFT 3 holography, JHEP 10 (2014) 90 [arXiv:1406.0505] [INSPIRE].
M. Mezei and S.S. Pufu, Three-sphere free energy for classical gauge groups, JHEP 02 (2014) 037 [arXiv:1312.0920] [INSPIRE].
S. Moriyama and T. Nosaka, Superconformal Chern-Simons partition functions of affine D-type quiver from Fermi gas, arXiv:1504.07710 [INSPIRE].
G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, Ann. Math. 156 (2002) 835 [math/0008184].
C. Closset, T.T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, Contact terms, unitarity and F-maximization in three-dimensional superconformal theories, JHEP 10 (2012) 053 [arXiv:1205.4142] [INSPIRE].
C. Closset, T.T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, Comments on Chern-Simons contact terms in three dimensions, JHEP 09 (2012) 091 [arXiv:1206.5218] [INSPIRE].
A. Kapustin, D n quivers from branes, JHEP 12 (1998) 015 [hep-th/9806238] [INSPIRE].
B. Feng and A. Hanany, Mirror symmetry by O3 planes, JHEP 11 (2000) 033 [hep-th/0004092] [INSPIRE].
Y. Hatsuda, Spectral zeta function and non-perturbative effects in ABJM Fermi-gas, arXiv:1503.07883 [INSPIRE].
R. Feynman, Statistical mechanics: a set of lectures, U.S.A. (1972).
C. Zachos, D. Fairlie and T. Curtright, Quantum mechanics in phase space: an overview with selected papers, World Scientific Publishing Company, Singapore (2005).
I. Bars, Nonpertubative effects of extreme localization in noncommutative geometry, hep-th/0109132 [INSPIRE].
C.P. Herzog, I.R. Klebanov, S.S. Pufu and T. Tesileanu, Multi-matrix models and tri-Sasaki einstein spaces, Phys. Rev. D 83 (2011) 046001 [arXiv:1011.5487] [INSPIRE].
P.M. Crichigno, C.P. Herzog and D. Jain, Free energy of D n quiver Chern-Simons theories, JHEP 03 (2013) 039 [arXiv:1211.1388] [INSPIRE].
O. Bergman and S. Hirano, Anomalous radius shift in AdS 4 /CFT 3, JHEP 07 (2009) 016 [arXiv:0902.1743] [INSPIRE].
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Assel, B., Drukker, N. & Felix, J. Partition functions of 3d \( \widehat{D} \)-quivers and their mirror duals from 1d free fermions. J. High Energ. Phys. 2015, 71 (2015). https://doi.org/10.1007/JHEP08(2015)071
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DOI: https://doi.org/10.1007/JHEP08(2015)071