Abstract
We study the topologically twisted index of 3d \( \mathcal{N} \) = 2 supersymmetric gauge theories with unitary gauge groups. We implement a Gröbner basis algorithm for computing the Σg × S1 index explicitly and exactly in terms of the associated Bethe ideal, which is defined as the algebraic ideal associated with the Bethe equations of the corresponding 3d A-model. We then revisit recently discovered infrared dualities for unitary SQCD with gauge group U(Nc)k,k+lNc with l ≠ 0, namely the Nii duality that generalises the Giveon-Kutasov duality, the Amariti-Rota duality that generalises the Aharony duality, and their further generalisations in the case of arbitrary numbers of fundamental and antifundamental chiral multiplets. In particular, we determine all the flavour Chern-Simons contact terms needed to make these dualities work. This allows us to check that the twisted indices of dual theories match exactly. We also initiate the study of the Witten index of unitary SQCD with l ≠ 0.
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References
S. Kim, The Complete superconformal index for N = 6 Chern-Simons theory, Nucl. Phys. B 821 (2009) 241 [Erratum ibid. 864 (2012) 884] [arXiv:0903.4172] [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].
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].
A. Kapustin and B. Willett, Generalized Superconformal Index for Three Dimensional Field Theories, arXiv:1106.2484 [INSPIRE].
B. Willett, Localization on three-dimensional manifolds, J. Phys. A 50 (2017) 443006 [arXiv:1608.02958] [INSPIRE].
C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, Supersymmetric Field Theories on Three-Manifolds, JHEP 05 (2013) 017 [arXiv:1212.3388] [INSPIRE].
C. Closset and H. Kim, Three-dimensional \( \mathcal{N} \) = 2 supersymmetric gauge theories and partition functions on Seifert manifolds: A review, Int. J. Mod. Phys. A 34 (2019) 1930011 [arXiv:1908.08875] [INSPIRE].
C. Closset, H. Kim and B. Willett, Supersymmetric partition functions and the three-dimensional A-twist, JHEP 03 (2017) 074 [arXiv:1701.03171] [INSPIRE].
C. Closset, H. Kim and B. Willett, Seifert fibering operators in 3d \( \mathcal{N} \) = 2 theories, JHEP 11 (2018) 004 [arXiv:1807.02328] [INSPIRE].
E. Witten, Dynamical Breaking of Supersymmetry, Nucl. Phys. B 188 (1981) 513 [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Bethe/Gauge correspondence on curved spaces, JHEP 01 (2015) 100 [arXiv:1405.6046] [INSPIRE].
F. Benini and A. Zaffaroni, A topologically twisted index for three-dimensional supersymmetric theories, JHEP 07 (2015) 127 [arXiv:1504.03698] [INSPIRE].
F. Benini and A. Zaffaroni, Supersymmetric partition functions on Riemann surfaces, Proc. Symp. Pure Math. 96 (2017) 13 [arXiv:1605.06120] [INSPIRE].
C. Closset and H. Kim, Comments on twisted indices in 3d supersymmetric gauge theories, JHEP 08 (2016) 059 [arXiv:1605.06531] [INSPIRE].
E. Witten, Supersymmetric index of three-dimensional gauge theory, hep-th/9903005 [https://doi.org/10.1142/9789812793850_0013] [INSPIRE].
K. Intriligator and N. Seiberg, Aspects of 3d N = 2 Chern-Simons-Matter Theories, JHEP 07 (2013) 079 [arXiv:1305.1633] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Supersymmetric vacua and Bethe ansatz, Nucl. Phys. B Proc. Suppl. 192-193 (2009) 91 [arXiv:0901.4744] [INSPIRE].
Y. Jiang and Y. Zhang, Algebraic geometry and Bethe ansatz. Part I. The quotient ring for BAE, JHEP 03 (2018) 087 [arXiv:1710.04693] [INSPIRE].
J. Lykke Jacobsen, Y. Jiang and Y. Zhang, Torus partition function of the six-vertex model from algebraic geometry, JHEP 03 (2019) 152 [arXiv:1812.00447] [INSPIRE].
D.A. Cox, J. Little and D. O’shea, Using algebraic geometry, vol. 185, Springer Science & Business Media (2006) [https://doi.org/10.1007/b138611].
W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, Singular 4-3-0 — A computer algebra system for polynomial computations, http://www.singular.uni-kl.de (2022).
W.R. Inc., Mathematica, Version 13.2, https://www.wolfram.com/mathematica.
M. Kauers and V. Levandovskyy, Singular.m, https://www3.risc.jku.at/research/combinat/software/Singular.
J. Eckhard, H. Kim, S. Schafer-Nameki and B. Willett, Higher-Form Symmetries, Bethe Vacua, and the 3d-3d Correspondence, JHEP 01 (2020) 101 [arXiv:1910.14086] [INSPIRE].
J. Gu, Y. Jiang and M. Sperling, Rational Q-systems, Higgsing and Mirror Symmetry, SciPost Phys. 14 (2023) 034 [arXiv:2208.10047] [INSPIRE].
K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].
O. Aharony, IR duality in d = 3 N = 2 supersymmetric USp(2Nc) and U(Nc) gauge theories, Phys. Lett. B 404 (1997) 71 [hep-th/9703215] [INSPIRE].
A. Giveon and D. Kutasov, Seiberg Duality in Chern-Simons Theory, Nucl. Phys. B 812 (2009) 1 [arXiv:0808.0360] [INSPIRE].
F. Benini, C. Closset and S. Cremonesi, Comments on 3d Seiberg-like dualities, JHEP 10 (2011) 075 [arXiv:1108.5373] [INSPIRE].
V. Niarchos, Seiberg Duality in Chern-Simons Theories with Fundamental and Adjoint Matter, JHEP 11 (2008) 001 [arXiv:0808.2771] [INSPIRE].
S. Cremonesi, Type IIB construction of flavoured ABJ(M) and fractional M2 branes, JHEP 01 (2011) 076 [arXiv:1007.4562] [INSPIRE].
A. Kapustin, Seiberg-like duality in three dimensions for orthogonal gauge groups, arXiv:1104.0466 [INSPIRE].
A. Kapustin, H. Kim and J. Park, Dualities for 3d Theories with Tensor Matter, JHEP 12 (2011) 087 [arXiv:1110.2547] [INSPIRE].
O. Aharony and I. Shamir, On O(Nc) d = 3 N = 2 supersymmetric QCD Theories, JHEP 12 (2011) 043 [arXiv:1109.5081] [INSPIRE].
J. Park and K.-J. Park, Seiberg-like Dualities for 3d N = 2 Theories with SU(N) gauge group, JHEP 10 (2013) 198 [arXiv:1305.6280] [INSPIRE].
O. Aharony and D. Fleischer, IR Dualities in General 3d Supersymmetric SU(N) QCD Theories, JHEP 02 (2015) 162 [arXiv:1411.5475] [INSPIRE].
F. Benini, S. Benvenuti and S. Pasquetti, SUSY monopole potentials in 2 + 1 dimensions, JHEP 08 (2017) 086 [arXiv:1703.08460] [INSPIRE].
A. Amariti, D. Orlando and S. Reffert, Monopole Quivers and new 3D N=2 dualities, Nucl. Phys. B 924 (2017) 153 [arXiv:1705.09297] [INSPIRE].
T. Dimofte, D. Gaiotto and N.M. Paquette, Dual boundary conditions in 3d SCFT’s, JHEP 05 (2018) 060 [arXiv:1712.07654] [INSPIRE].
C. Hwang, H. Kim and J. Park, On 3d Seiberg-Like Dualities with Two Adjoints, Fortsch. Phys. 66 (2018) 1800064 [arXiv:1807.06198] [INSPIRE].
M. Fazzi, A. Lanir, S.S. Razamat and O. Sela, Chiral 3d SU(3) SQCD and \( \mathcal{N} \) = 2 mirror duality, JHEP 11 (2018) 025 [arXiv:1808.04173] [INSPIRE].
S. Benvenuti, A tale of exceptional 3d dualities, JHEP 03 (2019) 125 [arXiv:1809.03925] [INSPIRE].
K. Nii, Duality and Confinement in 3d \( \mathcal{N} \) = 2 “chiral” SU(N) gauge theories, Nucl. Phys. B 939 (2019) 507 [arXiv:1809.10757] [INSPIRE].
K. Nii, 3d “chiral” Kutasov-Schwimmer duality, Nucl. Phys. B 952 (2020) 114920 [arXiv:1901.08642] [INSPIRE].
S. Giacomelli, Dualities for adjoint SQCD in three dimensions and emergent symmetries, JHEP 03 (2019) 144 [arXiv:1901.09947] [INSPIRE].
K. Nii, “Chiral” and “non-chiral” 3d Seiberg duality, JHEP 04 (2020) 098 [arXiv:1907.03340] [INSPIRE].
K. Nii, Generalized Giveon-Kutasov duality, JHEP 08 (2021) 130 [arXiv:2005.04858] [INSPIRE].
N. Kubo and K. Nii, 3d \( \mathcal{N} \) = 3 generalized Giveon-Kutasov duality, JHEP 04 (2022) 158 [arXiv:2111.13366] [INSPIRE].
T. Okazaki and D.J. Smith, Web of Seiberg-like dualities for 3D N = 2 quivers, Phys. Rev. D 105 (2022) 086023 [arXiv:2112.07347] [INSPIRE].
S. Benvenuti, I. Garozzo and G. Lo Monaco, Monopoles and dualities in 3d \( \mathcal{N} \) = 2 quivers, JHEP 10 (2021) 191 [arXiv:2012.08556] [INSPIRE].
C. Hwang, S. Kim and J. Park, Monopole deformations of 3d Seiberg-like dualities with adjoint matters, JHEP 11 (2022) 111 [arXiv:2202.09000] [INSPIRE].
A. Amariti and M. Fazzi, Dualities for three-dimensional \( \mathcal{N} \) = 2 SU(Nc) chiral adjoint SQCD, JHEP 11 (2020) 030 [arXiv:2007.01323] [INSPIRE].
A. Amariti and S. Rota, Webs of 3d \( \mathcal{N} \) = 2 dualities with D-type superpotentials, JHEP 01 (2023) 124 [arXiv:2204.06961] [INSPIRE].
A. Amariti and D. Morgante, Chiral dualities for SQCD3 with D-type superpotential, JHEP 02 (2023) 032 [arXiv:2209.12673] [INSPIRE].
O. Aharony, Baryons, monopoles and dualities in Chern-Simons-matter theories, JHEP 02 (2016) 093 [arXiv:1512.00161] [INSPIRE].
N. Seiberg, T. Senthil, C. Wang and E. Witten, A Duality Web in 2+1 Dimensions and Condensed Matter Physics, Annals Phys. 374 (2016) 395 [arXiv:1606.01989] [INSPIRE].
A. Karch and D. Tong, Particle-Vortex Duality from 3d Bosonization, Phys. Rev. X 6 (2016) 031043 [arXiv:1606.01893] [INSPIRE].
G. Gur-Ari and R. Yacoby, Three Dimensional Bosonization From Supersymmetry, JHEP 11 (2015) 013 [arXiv:1507.04378] [INSPIRE].
S. Kachru, M. Mulligan, G. Torroba and H. Wang, Bosonization and Mirror Symmetry, Phys. Rev. D 94 (2016) 085009 [arXiv:1608.05077] [INSPIRE].
A. Karch, B. Robinson and D. Tong, More Abelian Dualities in 2 + 1 Dimensions, JHEP 01 (2017) 017 [arXiv:1609.04012] [INSPIRE].
O. Aharony, F. Benini, P.-S. Hsin and N. Seiberg, Chern-Simons-matter dualities with SO and USp gauge groups, JHEP 02 (2017) 072 [arXiv:1611.07874] [INSPIRE].
F. Benini, P.-S. Hsin and N. Seiberg, Comments on global symmetries, anomalies, and duality in (2 + 1)d, JHEP 04 (2017) 135 [arXiv:1702.07035] [INSPIRE].
A. Amariti and S. Rota, 3d N = 2 dualities for SU(Nc) × U(1) Chern-Simons gauge theories, Nucl. Phys. B 976 (2022) 115710 [arXiv:2106.13762] [INSPIRE].
A. Kapustin and M.J. Strassler, On mirror symmetry in three-dimensional Abelian gauge theories, JHEP 04 (1999) 021 [hep-th/9902033] [INSPIRE].
E. Witten, SL(2, ℤ) action on three-dimensional conformal field theories with Abelian symmetry, in the proceedings of the From Fields to Strings: Circumnavigating Theoretical Physics: A Conference in Tribute to Ian Kogan, (2003), p. 1173–1200 [hep-th/0307041] [INSPIRE].
P.-S. Hsin and N. Seiberg, Level/rank Duality and Chern-Simons-Matter Theories, JHEP 09 (2016) 095 [arXiv:1607.07457] [INSPIRE].
C. Closset et al., Contact Terms, Unitarity, and F-Maximization in Three-Dimensional Superconformal Theories, JHEP 10 (2012) 053 [arXiv:1205.4142] [INSPIRE].
C. Closset et al., Comments on Chern-Simons Contact Terms in Three Dimensions, JHEP 09 (2012) 091 [arXiv:1206.5218] [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].
A.N. Redlich, Parity Violation and Gauge Noninvariance of the Effective Gauge Field Action in Three-Dimensions, Phys. Rev. D 29 (1984) 2366 [INSPIRE].
A.J. Niemi and G.W. Semenoff, Axial Anomaly Induced Fermion Fractionization and Effective Gauge Theory Actions in Odd Dimensional Space-Times, Phys. Rev. Lett. 51 (1983) 2077 [INSPIRE].
L. Alvarez-Gaume, S. Della Pietra and G.W. Moore, Anomalies and Odd Dimensions, Annals Phys. 163 (1985) 288 [INSPIRE].
Y. Jiang, R. Wen and Y. Zhang, Exact Quench Dynamics from Algebraic Geometry, arXiv:2109.10568 [INSPIRE].
N. Seiberg, Electric-magnetic duality in supersymmetric nonAbelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].
O. Aharony et al., Aspects of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 499 (1997) 67 [hep-th/9703110] [INSPIRE].
C. Closset and O. Khlaif, On the Witten index of 3d \( \mathcal{N} \) = 2 unitary SQCD with general CS levels, arXiv:2305.00534 [INSPIRE].
C. Córdova, P.-S. Hsin and N. Seiberg, Time-Reversal Symmetry, Anomalies, and Dualities in (2 + 1)d, SciPost Phys. 5 (2018) 006 [arXiv:1712.08639] [INSPIRE].
L. Bhardwaj, M. Bullimore, A.E.V. Ferrari and S. Schafer-Nameki, Anomalies of Generalized Symmetries from Solitonic Defects, arXiv:2205.15330 [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].
F. Benini, C. Closset and S. Cremonesi, Quantum moduli space of Chern-Simons quivers, wrapped D6-branes and AdS4/CFT3, JHEP 09 (2011) 005 [arXiv:1105.2299] [INSPIRE].
C. Closset, H. Kim and B. Willett, \( \mathcal{N} \) = 1 supersymmetric indices and the four-dimensional A-model, JHEP 08 (2017) 090 [arXiv:1707.05774] [INSPIRE].
K. Ohta, Supersymmetric index and s rule for type IIB branes, JHEP 10 (1999) 006 [hep-th/9908120] [INSPIRE].
M. Bullimore and A. Ferrari, Twisted Hilbert Spaces of 3d Supersymmetric Gauge Theories, JHEP 08 (2018) 018 [arXiv:1802.10120] [INSPIRE].
M. Bullimore, A.E.V. Ferrari, H. Kim and G. Xu, The twisted index and topological saddles, JHEP 05 (2022) 116 [arXiv:2007.11603] [INSPIRE].
N. Dorey and D. Tong, Mirror symmetry and toric geometry in three-dimensional gauge theories, JHEP 05 (2000) 018 [hep-th/9911094] [INSPIRE].
A. Kapustin and B. Willett, Wilson loops in supersymmetric Chern-Simons-matter theories and duality, arXiv:1302.2164 [INSPIRE].
H. Jockers, P. Mayr, U. Ninad and A. Tabler, Wilson loop algebras and quantum K-theory for Grassmannians, JHEP 10 (2020) 036 [arXiv:1911.13286] [INSPIRE].
K. Ueda and Y. Yoshida, 3d \( \mathcal{N} \) = 2 Chern-Simons-matter theory, Bethe ansatz, and quantum K-theory of Grassmannians, JHEP 08 (2020) 157 [arXiv:1912.03792] [INSPIRE].
W. Gu, L. Mihalcea, E. Sharpe and H. Zou, Quantum K theory of symplectic Grassmannians, J. Geom. Phys. 177 (2022) 104548 [arXiv:2008.04909] [INSPIRE].
H. Jockers, P. Mayr, U. Ninad and A. Tabler, BPS indices, modularity and perturbations in quantum K-theory, JHEP 02 (2022) 044 [arXiv:2106.07670] [INSPIRE].
W. Gu, L.C. Mihalcea, E. Sharpe and H. Zou, Quantum K theory of Grassmannians, Wilson line operators, and Schur bundles, arXiv:2208.01091 [INSPIRE].
Acknowledgments
We thank Mathew Bullimore, Stefano Cremonesi, Heeyeon Kim, Horia Magureanu, Sakura Schafer-Nameki, and Eric Sharpe for discussions and correspondence. CC is a Royal Society University Research Fellow and a Birmingham Fellow, and his work is supported by the University Research Fellowship Renewal 2022 ‘Singularities, supersymmetry and SQFT invariants’. The work of OK is supported by the School of Mathematics at the University of Birmingham.
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Closset, C., Khlaif, O. Twisted indices, Bethe ideals and 3d \( \mathcal{N} \) = 2 infrared dualities. J. High Energ. Phys. 2023, 148 (2023). https://doi.org/10.1007/JHEP05(2023)148
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DOI: https://doi.org/10.1007/JHEP05(2023)148