Abstract
We construct a large class of gauge theories with extended supersymmetry on four-dimensional manifolds with a Killing vector field and isolated fixed points. We extend previous results limited to super Yang-Mills theory to general \( \mathcal{N} \) = 2 gauge theories including hypermultiplets. We present a general framework encompassing equivariant Donaldson-Witten theory and Pestun’s theory on S4 as two particular cases. This is achieved by expressing fields in cohomological variables, whose features are dictated by supersymmetry and require a generalized notion of self-duality for two-forms and of chirality for spinors. Finally, we implement localization techniques to compute the exact partition function of the cohomological theories we built up and write the explicit result for manifolds with diverse topologies.
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G. Festuccia, J. Qiu, J. Winding and M. Zabzine, Twisting with a flip (the art of pestunization), Commun. Math. Phys. 377 (2020) 341 [arXiv:1812.06473] [INSPIRE].
E. Witten, Topological quantum field theory, Commun. Math. Phys. 117 (1988) 353 [INSPIRE].
S.K. Donaldson, Polynomial invariants for smooth manifolds, Topology 29 (1990) 257 [INSPIRE].
J.P. Yamron, Topological actions from twisted supersymmetric theories, Phys. Lett. B 213 (1988) 325 [INSPIRE].
D. Anselmi and P. Fré, Topological twist in four-dimensions, R duality and hyperinstantons, Nucl. Phys. B 404 (1993) 288 [hep-th/9211121] [INSPIRE].
D. Anselmi and P. Fré, Topological σ-models in four-dimensions and triholomorphic maps, Nucl. Phys. B 416 (1994) 255 [hep-th/9306080] [INSPIRE].
D. Anselmi and P. Fré, Gauged hyper-instantons and monopole equations, Phys. Lett. B 347 (1995) 247 [hep-th/9411205] [INSPIRE].
M. Alvarez and J.M.F. Labastida, Breaking of topological symmetry, Phys. Lett. B 315 (1993) 251 [hep-th/9305028] [INSPIRE].
M. Alvarez and J.M.F. Labastida, Topological matter in four-dimensions, Nucl. Phys. B 437 (1995) 356 [hep-th/9404115] [INSPIRE].
J.M.F. Labastida and M. Mariño, A topological Lagrangian for monopoles on four manifolds, Phys. Lett. B 351 (1995) 146 [hep-th/9503105] [INSPIRE].
J.M.F. Labastida and M. Mariño, Non-Abelian monopoles on four manifolds, Nucl. Phys. B 448 (1995) 373 [hep-th/9504010] [INSPIRE].
S. Hyun, J. Park and J.-S. Park, Spin-c topological QCD, Nucl. Phys. B 453 (1995) 199 [hep-th/9503201] [INSPIRE].
A. Losev, N. Nekrasov and S.L. Shatashvili, Issues in topological gauge theory, Nucl. Phys. B 534 (1998) 549 [hep-th/9711108] [INSPIRE].
A. Lossev, N. Nekrasov and S.L. Shatashvili, Testing Seiberg-Witten solution, NATO Sci. Ser. C 520 (1999) 359 [hep-th/9801061] [INSPIRE].
G.W. Moore, N. Nekrasov and S. Shatashvili, Integrating over Higgs branches, Commun. Math. Phys. 209 (2000) 97 [hep-th/9712241] [INSPIRE].
G.W. Moore, N. Nekrasov and S. Shatashvili, D particle bound states and generalized instantons, Commun. Math. Phys. 209 (2000) 77 [hep-th/9803265] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].
N.A. Nekrasov, Localizing gauge theories, in XIVth International Congress on Mathematical Physics, J.C. Zambrini ed., World Scientific, Singapore (2006).
L. Gottsche, H. Nakajima and K. Yoshioka, Instanton counting and Donaldson invariants, J. Diff. Geom. 80 (2008) 343 [math/0606180] [INSPIRE].
L. Gottsche, H. Nakajima and K. Yoshioka, K-theoretic Donaldson invariants via instanton counting, Pure Appl. Math. Quart. 5 (2009) 1029 [math/0611945] [INSPIRE].
E. Gasparim and C.-C.M. Liu, The Nekrasov conjecture for toric surfaces, Commun. Math. Phys. 293 (2010) 661 [arXiv:0808.0884] [INSPIRE].
M. Bershtein, G. Bonelli, M. Ronzani and A. Tanzini, Exact results for \( \mathcal{N} \) = 2 supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants, JHEP 07 (2016) 023 [arXiv:1509.00267] [INSPIRE].
M. Bershtein, G. Bonelli, M. Ronzani and A. Tanzini, Gauge theories on compact toric surfaces, conformal field theories and equivariant Donaldson invariants, J. Geom. Phys. 118 (2017) 40 [arXiv:1606.07148] [INSPIRE].
A. Bawane, G. Bonelli, M. Ronzani and A. Tanzini, \( \mathcal{N} \) = 2 supersymmetric gauge theories on S2 × S2 and Liouville Gravity, JHEP 07 (2015) 054 [arXiv:1411.2762] [INSPIRE].
M. Sinamuli, On \( \mathcal{N} \) = 2 supersymmetric gauge theories on S2 × S2, JHEP 05 (2016) 062 [arXiv:1411.4918] [INSPIRE].
D. Rodriguez-Gomez and J. Schmude, Partition functions for equivariantly twisted \( \mathcal{N} \) = 2 gauge theories on toric Kähler manifolds, JHEP 05 (2015) 111 [arXiv:1412.4407] [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].
N. Hama and K. Hosomichi, Seiberg-Witten theories on ellipsoids, JHEP 09 (2012) 033 [Addendum ibid. 10 (2012) 051] [arXiv:1206.6359] [INSPIRE].
V. Pestun, Localization for \( \mathcal{N} \) = 2 supersymmetric gauge theories in four dimensions, in New dualities of supersymmetric gauge theories, J. Teschner, Springer, Germany (2016) [arXiv:1412.7134] [INSPIRE].
V. Pestun et al., Localization techniques in quantum field theories, J. Phys. A 50 (2017) 440301 [arXiv:1608.02952] [INSPIRE].
G. Festuccia, J. Qiu, J. Winding and M. Zabzine, Transversally elliptic complex and cohomological field theory, J. Geom. Phys. 156 (2020) 103786 [arXiv:1904.12782] [INSPIRE].
J. Kallen, Cohomological localization of Chern-Simons theory, JHEP 08 (2011) 008 [arXiv:1104.5353] [INSPIRE].
B. Assel, D. Martelli, S. Murthy and D. Yokoyama, Localization of supersymmetric field theories on non-compact hyperbolic three-manifolds, JHEP 03 (2017) 095 [arXiv:1609.08071] [INSPIRE].
A. Pittelli, Supersymmetric localization of refined chiral multiplets on topologically twisted H2 × S1, Phys. Lett. B 801 (2020) 135154 [arXiv:1812.11151] [INSPIRE].
C. Closset and I. Shamir, The \( \mathcal{N} \) = 1 chiral multiplet on T2 × S2 and supersymmetric localization, JHEP 03 (2014) 040 [arXiv:1311.2430] [INSPIRE].
G. Festuccia, J. Qiu, J. Winding and M. Zabzine, \( \mathcal{N} \) = 2 supersymmetric gauge theory on connected sums of S2 × S2, JHEP 03 (2017) 026 [arXiv:1611.04868] [INSPIRE].
A. Bawane, S. Benvenuti, G. Bonelli, N. Muteeb and A. Tanzini, \( \mathcal{N} \) = 2 gauge theories on unoriented/open four-manifolds and their AGT counterparts, JHEP 07 (2019) 040 [arXiv:1710.06283] [INSPIRE].
P. Longhi, F. Nieri and A. Pittelli, Localization of 4d \( \mathcal{N} \) = 1 theories on 𝔻2 × 𝕋2, JHEP 12 (2019) 147 [arXiv:1906.02051] [INSPIRE].
J. Källén and M. Zabzine, Twisted supersymmetric 5D Yang-Mills theory and contact geometry, JHEP 05 (2012) 125 [arXiv:1202.1956] [INSPIRE].
J. Källén, J. Qiu and M. Zabzine, The perturbative partition function of supersymmetric 5D Yang-Mills theory with matter on the five-sphere, JHEP 08 (2012) 157 [arXiv:1206.6008] [INSPIRE].
J. Qiu and M. Zabzine, 5D Super Yang-Mills on Yp,q Sasaki-Einstein manifolds, Commun. Math. Phys. 333 (2015) 861 [arXiv:1307.3149] [INSPIRE].
J. Qiu and M. Zabzine, Factorization of 5D super Yang-Mills theory on Yp,q spaces, Phys. Rev. D 89 (2014) 065040 [arXiv:1312.3475] [INSPIRE].
J. Qiu and M. Zabzine, On twisted N = 2 5D super Yang-Mills theory, Lett. Math. Phys. 106 (2016) 1 [arXiv:1409.1058] [INSPIRE].
J.A. Minahan and M. Zabzine, Gauge theories with 16 supersymmetries on spheres, JHEP 03 (2015) 155 [arXiv:1502.07154] [INSPIRE].
K. Polydorou, A. Rocén and M. Zabzine, 7D supersymmetric Yang-Mills on curved manifolds, JHEP 12 (2017) 152 [arXiv:1710.09653] [INSPIRE].
N. Iakovidis, J. Qiu, A. Rocén and M. Zabzine, 7D supersymmetric Yang-Mills on hypertoric 3-Sasakian manifolds, JHEP 06 (2020) 026 [arXiv:2003.12461] [INSPIRE].
E. Friedman and S. Ruijsenaars, Shintani–Barnes ζ and γ functions, Adv. Math. 187 (2004) 362.
C. Vafa and E. Witten, A strong coupling test of S duality, Nucl. Phys. B 431 (1994) 3 [hep-th/9408074] [INSPIRE].
G. Festuccia and N. Seiberg, Rigid supersymmetric theories in curved superspace, JHEP 06 (2011) 114 [arXiv:1105.0689] [INSPIRE].
M.F. Sohnius, The multiplet of currents for N = 2 extended supersymmetry, Phys. Lett. B 81 (1979) 8 [INSPIRE].
B. de Wit, J.W. van Holten and A. Van Proeyen, Transformation rules of N = 2 supergravity multiplets, Nucl. Phys. B 167 (1980) 186 [INSPIRE].
B. de Wit, J.W. van Holten and A. Van Proeyen, Structure of N = 2 supergravity, Nucl. Phys. B 184 (1981) 77 [Erratum ibid. 222 (1983) 516] [INSPIRE].
B. de Wit, P.G. Lauwers and A. Van Proeyen, Lagrangians of N = 2 supergravity-matter systems, Nucl. Phys. B 255 (1985) 569 [INSPIRE].
D.Z. Freedman and A. Van Proeyen, Supergravity, Cambridge University Press, Cambridge U.K. (2012).
C. Klare and A. Zaffaroni, Extended supersymmetry on curved spaces, JHEP 10 (2013) 218 [arXiv:1308.1102] [INSPIRE].
D. Butter, G. Inverso and I. Lodato, Rigid 4D \( \mathcal{N} \) = 2 supersymmetric backgrounds and actions, JHEP 09 (2015) 088 [arXiv:1505.03500] [INSPIRE].
J. Labastida and M. Mariño, Topological quantum field theory and four manifolds, Springer, Germany (2005).
J. Qiu, L. Tizzano, J. Winding and M. Zabzine, Gluing Nekrasov partition functions, Commun. Math. Phys. 337 (2015) 785 [arXiv:1403.2945] [INSPIRE].
G. Felder and A. Varchenko, The elliptic gamma function and sl(3, z) ⋉ z3, Adv. Math. 156 (2000) 44.
J. Wess and J. Bagger, Supersymmetry and supergravity, Princeton University Press, Princeton U.S.A. (1992).
Y. Kosmann, Dérivées de Lie des spineurs, Ann. Mat. Pura Appl. 91 (1971) 317.
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Festuccia, G., Gorantis, A., Pittelli, A. et al. Cohomological localization of \( \mathcal{N} \) = 2 gauge theories with matter. J. High Energ. Phys. 2020, 133 (2020). https://doi.org/10.1007/JHEP09(2020)133
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DOI: https://doi.org/10.1007/JHEP09(2020)133