Abstract
We discuss reductions of general \( \mathcal{N}=1 \) four dimensional gauge theories on \( {\mathbb{S}}^2 \). The effective two dimensional theory one obtains depends on the details of the coupling of the theory to background fields, which can be translated to a choice of R-symmetry. We argue that, for special choices of R-symmetry, the resulting two dimensional theory has a natural interpretation as an \( \mathcal{N}=\left(0,2\right) \) gauge theory. As an application of our general observations, we discuss reductions of \( \mathcal{N}=1 \) and \( \mathcal{N}=2 \) dualities and argue that they imply certain two dimensional dualities.
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M. Aganagic, K. Hori, A. Karch and D. Tong, Mirror symmetry in (2 + 1)-dimensions and (1 + 1) - dimensions, JHEP 07 (2001) 022 [hep-th/0105075] [INSPIRE].
O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, 3d dualities from 4d dualities, JHEP 07 (2013) 149 [arXiv:1305.3924] [INSPIRE].
O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, 3d dualities from 4d dualities for orthogonal groups, JHEP 08 (2013) 099 [arXiv:1307.0511] [INSPIRE].
Y. Imamura, Relation between the 4d superconformal index and the S 3 partition function, JHEP 09 (2011) 133 [arXiv:1104.4482] [INSPIRE].
A. Gadde and W. Yan, Reducing the 4d index to the S 3 partition function, JHEP 12 (2012) 003 [arXiv:1104.2592] [INSPIRE].
F.A.H. Dolan, V.P. Spiridonov and G.S. Vartanov, From 4d superconformal indices to 3d partition functions, Phys. Lett. B 704 (2011) 234 [arXiv:1104.1787] [INSPIRE].
F. Benini and A. Zaffaroni, A topologically twisted index for three-dimensional supersymmetric theories, JHEP 07 (2015) 127 [arXiv:1504.03698] [INSPIRE].
M. Honda and Y. Yoshida, Supersymmetric index on T 2 × S 2 and elliptic genus, arXiv:1504.04355 [INSPIRE].
A. Gadde, S. Gukov and P. Putrov, (0, 2) trialities, JHEP 03 (2014) 076 [arXiv:1310.0818] [INSPIRE].
C. Closset and I. Shamir, The N = 1 chiral multiplet on T 2 × S 2 and supersymmetric localization, JHEP 03 (2014) 040 [arXiv:1311.2430] [INSPIRE].
D. Kutasov and J. Lin, (0, 2) dynamics from four dimensions, Phys. Rev. D 89 (2014) 085025 [arXiv:1310.6032] [INSPIRE].
Y. Tachikawa, unpublished.
C. Closset, S. Cremonesi and D.S. Park, The equivariant A-twist and gauged linear σ-models on the two-sphere, JHEP 06 (2015) 076 [arXiv:1504.06308] [INSPIRE].
O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, Two dimensional dualities from three dimensional dualities, to appear.
E. Witten, Some comments on string dynamics, in Future perspectives in string theory. Proceedings, Conference, Strings ′95, Los Angeles U.S.A. March 13-18 1995 [hep-th/9507121] [INSPIRE].
E. Sharpe, Decomposition in diverse dimensions, Phys. Rev. D 90 (2014) 025030 [arXiv:1404.3986] [INSPIRE].
F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of two-dimensional N = 2 gauge theories with rank-one gauge groups, Lett. Math. Phys. 104 (2014) 465 [arXiv:1305.0533] [INSPIRE].
F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of 2d N = 2 gauge theories, Commun. Math. Phys. 333 (2015) 1241 [arXiv:1308.4896] [INSPIRE].
A. Gadde and S. Gukov, 2d index and surface operators, JHEP 03 (2014) 080 [arXiv:1305.0266] [INSPIRE].
N. Seiberg, Electric-magnetic duality in supersymmetric non-Abelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].
S. Gukov and E. Witten, Gauge theory, ramification, and the geometric Langlands program, hep-th/0612073 [INSPIRE].
K.A. Intriligator and P. Pouliot, Exact superpotentials, quantum vacua and duality in supersymmetric SP(N c ) gauge theories, Phys. Lett. B 353 (1995) 471 [hep-th/9505006] [INSPIRE].
P. Putrov, J. Song and W. Yan, (0, 4) dualities, arXiv:1505.07110 [INSPIRE].
A. Gadde, S.S. Razamat and B. Willett, “Lagrangian” for a non-Lagrangian field theory with N = 2 supersymmetry,Phys. Rev. Lett. 115(2015) 171604 [arXiv:1505.05834] [INSPIRE].
K. Hori and D. Tong, Aspects of non-Abelian gauge dynamics in two-dimensional N = (2, 2) theories, JHEP 05 (2007) 079 [hep-th/0609032] [INSPIRE].
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ArXiv ePrint: 1506.08795
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Gadde, A., Razamat, S.S. & Willett, B. On the reduction of 4d \( \mathcal{N}=1 \) theories on \( {\mathbb{S}}^2 \) . J. High Energ. Phys. 2015, 163 (2015). https://doi.org/10.1007/JHEP11(2015)163
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DOI: https://doi.org/10.1007/JHEP11(2015)163