Down the rabbit hole with theories of class \( \mathcal{S} \)
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Abstract
We review some of the properties of 3d \( \mathcal{N}=4 \) theories obtained by dimensionally reducing theories of class \( \mathcal{S} \). We study 3d partition functions, and certain limits thereof, for such theories, and the properties implied for these by 3d mirror symmetry.
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Supersymmetric gauge theory Field Theories in Lower Dimensions Duality in Gauge Field Theories Download
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References
- [1]D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems and the WKB Approximation, arXiv:0907.3987 [INSPIRE].
- [2]
- [3]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].ADSCrossRefGoogle Scholar
- [4]A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, Gauge Theories and Macdonald Polynomials, Commun. Math. Phys. 319 (2013) 147 [arXiv:1110.3740] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [5]D. Gaiotto, L. Rastelli and S.S. Razamat, Bootstrapping the superconformal index with surface defects, JHEP 01 (2013) 022 [arXiv:1207.3577] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [6]L.F. Alday, M. Bullimore and M. Fluder, On S-duality of the Superconformal Index on Lens Spaces and 2d TQFT, JHEP 05 (2013) 122 [arXiv:1301.7486] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [7]S.S. Razamat and M. Yamazaki, S-duality and the N = 2 Lens Space Index, JHEP 10 (2013) 048 [arXiv:1306.1543] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [8]M. Lemos, W. Peelaers and L. Rastelli, The superconformal index of class S theories of type D, JHEP 05 (2014) 120 [arXiv:1212.1271] [INSPIRE].ADSCrossRefGoogle Scholar
- [9]O. Chacaltana, J. Distler and A. Trimm, Tinkertoys for the Twisted D-Series, arXiv:1309.2299 [INSPIRE].
- [10]O. Chacaltana, J. Distler and A. Trimm, Tinkertoys for the E 6 Theory, arXiv:1403.4604 [INSPIRE].
- [11]A. Gadde, E. Pomoni, L. Rastelli and S.S. Razamat, S-duality and 2d Topological QFT, JHEP 03 (2010) 032 [arXiv:0910.2225] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [12]F. Benini, Y. Tachikawa and D. Xie, Mirrors of 3d Sicilian theories, JHEP 09 (2010) 063 [arXiv:1007.0992] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [13]L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [14]A. Gadde and S. Gukov, 2d Index and Surface operators, JHEP 03 (2014) 080 [arXiv:1305.0266] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [15]A. Gorsky and N. Nekrasov, Hamiltonian systems of Calogero type and two-dimensional Yang-Mills theory, Nucl. Phys. B 414 (1994) 213 [hep-th/9304047] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [16]A. Gorsky and N. Nekrasov, Relativistic Calogero-Moser model as gauged WZW theory, Nucl. Phys. B 436 (1995) 582 [hep-th/9401017] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [17]A. Gorsky and N. Nekrasov, Elliptic Calogero-Moser system from two-dimensional current algebra, hep-th/9401021 [INSPIRE].
- [18]N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, arXiv:0908.4052 [INSPIRE].
- [19]R. Brooks and S.J. Gates Jr., Extended supersymmetry and superBF gauge theories, Nucl. Phys. B 432 (1994) 205 [hep-th/9407147] [INSPIRE].ADSCrossRefMATHGoogle Scholar
- [20]A. Kapustin and M.J. Strassler, On mirror symmetry in three-dimensional Abelian gauge theories, JHEP 04 (1999) 021 [hep-th/9902033] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [21]D. Gaiotto and P. Koroteev, On Three Dimensional Quiver Gauge Theories and Integrability, JHEP 05 (2013) 126 [arXiv:1304.0779] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [22]K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
- [23]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].ADSMathSciNetCrossRefMATHGoogle Scholar
- [24]N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, hep-th/9607163 [INSPIRE].
- [25]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].
- [26]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].ADSMathSciNetCrossRefMATHGoogle Scholar
- [27]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].ADSMathSciNetCrossRefMATHGoogle Scholar
- [28]D.L. Jafferis, The Exact Superconformal R-Symmetry Extremizes Z, JHEP 05 (2012) 159 [arXiv:1012.3210] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [29]F. Benini, T. Nishioka and M. Yamazaki, 4d Index to 3d Index and 2d TQFT, Phys. Rev. D 86 (2012) 065015 [arXiv:1109.0283] [INSPIRE].ADSGoogle Scholar
- [30]T. Dimofte, D. Gaiotto and S. Gukov, 3-Manifolds and 3d Indices, Adv. Theor. Math. Phys. 17 (2013) 975 [arXiv:1112.5179] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
- [31]O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, 3d dualities from 4d dualities, JHEP 07 (2013) 149 [arXiv:1305.3924] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [32]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].ADSMathSciNetCrossRefMATHGoogle Scholar
- [33]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].ADSMathSciNetCrossRefMATHGoogle Scholar
- [34]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].ADSCrossRefGoogle Scholar
- [35]A. Gadde and W. Yan, Reducing the 4d Index to the S 3 Partition Function, JHEP 12 (2012) 003 [arXiv:1104.2592] [INSPIRE].ADSCrossRefGoogle Scholar
- [36]Y. Imamura, Relation between the 4d superconformal index and the S 3 partition function, JHEP 09 (2011) 133 [arXiv:1104.4482] [INSPIRE].ADSCrossRefMATHGoogle Scholar
- [37]S. Pasquetti, Factorisation of N = 2 Theories on the Squashed 3-Sphere, JHEP 04 (2012) 120 [arXiv:1111.6905] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [38]C. Beem, T. Dimofte and S. Pasquetti, Holomorphic Blocks in Three Dimensions, arXiv:1211.1986 [INSPIRE].
- [39]I.G. Macdonald, Symmetric functions and hall polynomials, Oxford University Press, (1995).Google Scholar
- [40]S. Cremonesi, A. Hanany and A. Zaffaroni, Monopole operators and Hilbert series of Coulomb branches of 3d \( \mathcal{N}=4 \) gauge theories, JHEP 01 (2014) 005 [arXiv:1309.2657] [INSPIRE].ADSCrossRefGoogle Scholar
- [41]S. Cremonesi, A. Hanany, N. Mekareeya and A. Zaffaroni, Coulomb branch Hilbert series and Hall-Littlewood polynomials, arXiv:1403.0585 [INSPIRE].
- [42]S. Cremonesi, A. Hanany, N. Mekareeya and A. Zaffaroni, Coulomb branch Hilbert series and Three Dimensional Sicilian Theories, arXiv:1403.2384 [INSPIRE].
- [43]C. Krattenthaler, V.P. Spiridonov and G.S. Vartanov, Superconformal indices of three-dimensional theories related by mirror symmetry, JHEP 06 (2011) 008 [arXiv:1103.4075] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [44]F.J. van de Bult, Hyperbolic Hypergeometric Functions, Ph.D. Thesis, University of Amsterdam, Amsterdam, Netherlands (2007), http://www.its.caltech.edu/~vdbult/Thesis.pdf.
- [45]I. Yaakov, Redeeming Bad Theories, JHEP 11 (2013) 189 [arXiv:1303.2769] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [46]O. Aharony, IR duality in D = 3 N = 2 supersymmetric USp(2N(c)) and U(N(c)) gauge theories, Phys. Lett. B 404 (1997) 71 [hep-th/9703215] [INSPIRE].ADSCrossRefGoogle Scholar
- [47]B. Willett and I. Yaakov, N = 2 Dualities and Z Extremization in Three Dimensions, arXiv:1104.0487 [INSPIRE].
- [48]K. Hosomichi, S. Lee and J. Park, AGT on the S-duality Wall, JHEP 12 (2010) 079 [arXiv:1009.0340] [INSPIRE].ADSCrossRefMATHGoogle Scholar
- [49]D. Gang, E. Koh, S. Lee and J. Park, Superconformal Index and 3d-3d Correspondence for Mapping Cylinder/Torus, JHEP 01 (2014) 063 [arXiv:1305.0937] [INSPIRE].ADSCrossRefGoogle Scholar
- [50]O. Aharony, N. Seiberg and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP 08 (2013) 115 [arXiv:1305.0318] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [51]D. Gang, E. Koh and K. Lee, Superconformal Index with Duality Domain Wall, JHEP 10 (2012) 187 [arXiv:1205.0069] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
- [52]S. Benvenuti and S. Pasquetti, 3D-partition functions on the sphere: exact evaluation and mirror symmetry, JHEP 05 (2012) 099 [arXiv:1105.2551] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [53]T. Nishioka, Y. Tachikawa and M. Yamazaki, 3d Partition Function as Overlap of Wavefunctions, JHEP 08 (2011) 003 [arXiv:1105.4390] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [54]V.P. Spiridonov and G.S. Vartanov, Vanishing superconformal indices and the chiral symmetry breaking, JHEP 06 (2014) 062 [arXiv:1402.2312] [INSPIRE].ADSCrossRefGoogle Scholar
- [55]A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and Duality, JHEP 04 (2014) 001 [arXiv:1401.0740] [INSPIRE].ADSCrossRefGoogle Scholar
- [56]M. Bullimore, M. Fluder, L. Hollands and P. Richmond, The superconformal index and an elliptic algebra of surface defects, arXiv:1401.3379 [INSPIRE].
- [57]V. Fock, A. Gorsky, N. Nekrasov and V. Rubtsov, Duality in integrable systems and gauge theories, JHEP 07 (2000) 028 [hep-th/9906235] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [58]A. Gorsky and A. Mironov, Integrable many body systems and gauge theories, hep-th/0011197 [INSPIRE].
- [59]S.S. Razamat, On the N = 2 superconformal index and eigenfunctions of the elliptic RS model, Lett. Math. Phys. 104 (2014) 673 [arXiv:1309.0278] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [60]A. Hanany and N. Mekareeya, Tri-vertices and SU(2)’s, JHEP 02 (2011) 069 [arXiv:1012.2119] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
- [61]M. Rahman and A. Verma, A q-integral representation of Rogers’ q-ultraspherical polynomials and some applications, Constr. Approx. 2 (1986) 1.MathSciNetCrossRefMATHGoogle Scholar
- [62]H. Awata, S. Odake and J. Shiraishi, Integral representations of the Macdonald symmetric functions, Commun. Math. Phys. 179 (1996) 647 [q-alg/9506006] [INSPIRE].ADSCrossRefMATHGoogle Scholar
- [63]A. Okounkov, (Shifted) Macdonald Polynomials: q-Integral Representation and Combinatorial Formula, Compos. Math. 112 (1998) 147.MathSciNetCrossRefMATHGoogle Scholar
- [64]L.F. Alday, D. Martelli, P. Richmond and J. Sparks, Localization on Three-Manifolds, JHEP 10 (2013) 095 [arXiv:1307.6848] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
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