Abstract
We study 5d \( \mathcal{N} \) = 2 maximally supersymmetric Yang-Mills theory with a gauge group G on S 2 × M 3, where M 3 is a 3-manifold. By explicit localization computation we show that the path-integral of the 5d \( \mathcal{N} \) = 2 theory reduces to that of the 3d \( {G_{\mathbb{C}}} \) Chern-Simons theory on M 3, where \( {G_{\mathbb{C}}} \) is the complexification of G. This gives a direct derivation of the appearance of the Chern-Simons theory from the compactification of the 6d (2, 0) theory, confirms the predictions from the 3d/3d correspondence for G = SU(N), and suggests the generalization of the correspondence to more general gauge groups.
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References
Y. Terashima and M. Yamazaki, SL(2, \( \mathbb{R} \)) Chern-Simons, Liouville and Gauge Theory on Duality Walls, JHEP 08 (2011) 135 [arXiv:1103.5748] [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, Gauge Theories Labelled by Three-Manifolds, arXiv:1108.4389 [INSPIRE].
S. Cecotti, C. Cordova and C. Vafa, Braids, Walls and Mirrors, arXiv:1110.2115 [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, 3-Manifolds and 3d Indices, arXiv:1112.5179 [INSPIRE].
N. Drukker, D. Gaiotto and J. Gomis, The Virtue of Defects in 4D Gauge Theories and 2D CFTs, JHEP 06 (2011) 025 [arXiv:1003.1112] [INSPIRE].
T. Dimofte, S. Gukov and L. Hollands, Vortex Counting and Lagrangian 3-manifolds, Lett. Math. Phys. 98 (2011) 225 [arXiv:1006.0977] [INSPIRE].
K. Hosomichi, S. Lee and J. Park, AGT on the S-duality Wall, JHEP 12 (2010) 079 [arXiv:1009.0340] [INSPIRE].
Y. Terashima and M. Yamazaki, 3d N = 2 Theories from Cluster Algebras, arXiv:1301.5902 [INSPIRE].
Y. Terashima and M. Yamazaki, Emergent 3-manifolds from 4d Superconformal Indices, Phys. Rev. Lett. 109 (2012) 091602 [arXiv:1203.5792] [INSPIRE].
M. Yamazaki, Quivers, YBE and 3-manifolds, JHEP 05 (2012) 147 [arXiv:1203.5784] [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].
D.L. Jafferis, The Exact Superconformal R-Symmetry Extremizes Z, JHEP 05 (2012) 159 [arXiv:1012.3210] [INSPIRE].
N. Hama, K. Hosomichi and S. Lee, Notes on SUSY Gauge Theories on Three-Sphere, JHEP 03 (2011) 127 [arXiv:1012.3512] [INSPIRE].
S. Kim, The complete superconformal index for N = 6 Chern-Simons theory, Nucl. Phys. B 821 (2009) 241 [Erratum ibid. B 864 (2012) 884] [arXiv:0903.4172] [INSPIRE].
Y. Imamura and S. Yokoyama, Index for three dimensional superconformal field theories with general R-charge assignments, JHEP 04 (2011) 007 [arXiv:1101.0557] [INSPIRE].
F. Benini and S. Cremonesi, Partition functions of N = (2,2) gauge theories on S 2 and vortices, arXiv:1206.2356 [INSPIRE].
N. Doroud, J. Gomis, B. Le Floch and S. Lee, Exact Results in D = 2 Supersymmetric Gauge Theories, JHEP 05 (2013) 093 [arXiv:1206.2606] [INSPIRE].
J. Gomis and S. Lee, Exact Kähler Potential from Gauge Theory and Mirror Symmetry, JHEP 04 (2013) 019 [arXiv:1210.6022] [INSPIRE].
E. Witten, Analytic Continuation Of Chern-Simons Theory, arXiv:1001.2933 [INSPIRE].
E. Witten, A New Look At The Path Integral Of Quantum Mechanics, arXiv:1009.6032 [INSPIRE].
Y. Fukuda, T. Kawano and N. Matsumiya, 5D SYM and 2D q-Deformed YM, Nucl. Phys. B 869 (2013) 493 [arXiv:1210.2855] [INSPIRE].
T. Kawano and N. Matsumiya, 5D SYM on 3D Sphere and 2D YM, Phys. Lett. B 716 (2012) 450 [arXiv:1206.5966] [INSPIRE].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, The 4d Superconformal Index from q-deformed 2d Yang-Mills, Phys. Rev. Lett. 106 (2011) 241602 [arXiv:1104.3850] [INSPIRE].
T. Nishioka, Y. Tachikawa and M. Yamazaki, 3d Partition Function as Overlap of Wavefunctions, JHEP 08 (2011) 003 [arXiv:1105.4390] [INSPIRE].
J. Yagi, 3d TQFT from 6d SCFT, JHEP 08 (2013) 017 [arXiv:1305.0291] [INSPIRE].
N. Marcus, The other topological twisting of N = 4 Yang-Mills, Nucl. Phys. B 452 (1995) 331 [hep-th/9506002] [INSPIRE].
M. Blau and G. Thompson, Aspects of N T ≥ 2 topological gauge theories and D-branes, Nucl. Phys. B 492 (1997) 545 [hep-th/9612143] [INSPIRE].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
M.F. Atiyah, Elliptic operators and compact groups, Lecture Notes in Mathematics 401, Springer-Verlag, Berlin Germany (1974).
N. Hama and K. Hosomichi, Seiberg-Witten Theories on Ellipsoids, JHEP 09 (2012) 033 [Addendum ibid. 1210 (2012) 051] [arXiv:1206.6359] [INSPIRE].
E. Witten, Fivebranes and Knots, arXiv:1101.3216 [INSPIRE].
H.–C. Kim, J. Kim and S. Kim, Instantons on the 5-sphere and M5-branes, arXiv:1211.0144 [INSPIRE].
V.V. Fock and A.B. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math-Paris 103 (2006) 1 [math/0311149].
T. Dimofte, M. Gabella and A.B. Goncharov, K-Decompositions and 3d Gauge Theories, arXiv:1301.0192 [INSPIRE].
E. Witten, Quantization of Chern-Simons Gauge Theory With Complex Gauge Group, Commun. Math. Phys. 137 (1991) 29 [INSPIRE].
A. Kapustin and E. Witten, Electric-Magnetic Duality And The Geometric Langlands Program, Commun. Num. Theor. Phys. 1 (2007) 1 [hep-th/0604151] [INSPIRE].
S. Pasquetti, Factorisation of N = 2 Theories on the Squashed 3-Sphere, JHEP 04 (2012) 120 [arXiv:1111.6905] [INSPIRE].
C. Beem, T. Dimofte and S. Pasquetti, Holomorphic Blocks in Three Dimensions, arXiv:1211.1986 [INSPIRE].
N. Doroud and J. Gomis, Gauge Theory Dynamics and Kähler Potential for Calabi-Yau Complex Moduli, arXiv:1309.2305 [INSPIRE].
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ArXiv ePrint: 1305.2429
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Lee, S., Yamazaki, M. 3d Chern-Simons theory from M5-branes. J. High Energ. Phys. 2013, 35 (2013). https://doi.org/10.1007/JHEP12(2013)035
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DOI: https://doi.org/10.1007/JHEP12(2013)035