Abstract
We consider \( \mathcal{N}=2 \) supersymmetric pure gauge theories on toric Kähler manifolds, with particular emphasis on ℂℙ2. By choosing a vector generating a U(1) action inside the torus of the manifold, we construct equivariantly twisted theories. Then, using localization, we compute their supersymmetric partition functions. As expected, these receive contributions from a classical, a one-loop, and an instanton term. It turns out that the one-loop term is trivial and that the instanton contributions are localized at the fixed points of the U(1). In fact the full partition function can be re-written in a factorized form with contributions from each of the fixed points. The full significance of this is yet to be understood.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [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, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, The geometry of supersymmetric partition functions, JHEP 01 (2014) 124 [arXiv:1309.5876] [INSPIRE].
E. Gerchkovitz, J. Gomis and Z. Komargodski, Sphere partition functions and the Zamolodchikov metric, JHEP 11 (2014) 001 [arXiv:1405.7271] [INSPIRE].
J. Gomis and N. Ishtiaque, Kähler potential and ambiguities in 4d N = 2 SCFTs, JHEP 04 (2015) 169 [arXiv:1409.5325] [INSPIRE].
G. Festuccia and N. Seiberg, Rigid supersymmetric theories in curved superspace, JHEP 06 (2011) 114 [arXiv:1105.0689] [INSPIRE].
T.T. Dumitrescu, G. Festuccia and N. Seiberg, Exploring curved superspace, JHEP 08 (2012) 141 [arXiv:1205.1115] [INSPIRE].
E. Witten, Topological quantum field theory, Commun. Math. Phys. 117 (1988) 353 [INSPIRE].
A. Karlhede and M. Roček, Topological quantum field theory and N = 2 conformal supergravity, Phys. Lett. B 212 (1988) 51 [INSPIRE].
C. Klare and A. Zaffaroni, Extended supersymmetry on curved spaces, JHEP 10 (2013) 218 [arXiv:1308.1102] [INSPIRE].
E. Witten, Supersymmetric Yang-Mills theory on a four manifold, J. Math. Phys. 35 (1994) 5101 [hep-th/9403195] [INSPIRE].
A. Johansen, Twisting of N = 1 SUSY gauge theories and heterotic topological theories, Int. J. Mod. Phys. A 10 (1995) 4325 [hep-th/9403017] [INSPIRE].
G.W. Moore and E. Witten, Integration over the u-plane in Donaldson theory, Adv. Theor. Math. Phys. 1 (1997) 298 [hep-th/9709193] [INSPIRE].
L. Gottsche, H. Nakajima and K. Yoshioka, Instanton counting and Donaldson invariants, math/0606180 [INSPIRE].
J. Gomis, T. Okuda and V. Pestun, Exact results for ’t Hooft loops in gauge theories on S 4, JHEP 05 (2012) 141 [arXiv:1105.2568] [INSPIRE].
N. Hama and K. Hosomichi, Seiberg-Witten theories on ellipsoids, JHEP 09 (2012) 033 [arXiv:1206.6359] [INSPIRE].
K. Hosomichi, R.-K. Seong and S. Terashima, Supersymmetric gauge theories on the five-sphere, Nucl. Phys. B 865 (2012) 376 [arXiv:1203.0371] [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 Y p,q Sasaki-Einstein manifolds, Commun. Math. Phys. 333 (2015) 861 [arXiv:1307.3149] [INSPIRE].
J. Schmude, Localisation on Sasaki-Einstein manifolds from holomorphic functions on the cone, JHEP 01 (2015) 119 [arXiv:1401.3266] [INSPIRE].
J. Qiu and M. Zabzine, On twisted N = 2 5D super Yang-Mills theory, arXiv:1409.1058 [INSPIRE].
B. Assel, D. Cassani and D. Martelli, Localization on Hopf surfaces, JHEP 08 (2014) 123 [arXiv:1405.5144] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, hep-th/0306238 [INSPIRE].
A. Bawane, G. Bonelli, M. Ronzani and A. Tanzini, N = 2 supersymmetric gauge theories on S 2 × S 2 and Liouville gravity, arXiv:1411.2762 [INSPIRE].
M. Sinamuli, On N = 2 supersymmetric gauge theories on S 2 × S 2, arXiv:1411.4918 [INSPIRE].
A. Van Proeyen, N = 2 supergravity in d = 4, 5, 6 and its matter couplings, extended version of lectures given during the semester Supergravity, superstrings and M-theory, http://itf.fys.kuleuven.be/~toine/LectParis.pdf, Institut Henri Poincaré, Paris France November 2000.
D.Z. Freedman and A. Van Proeyen, Supergravity, Cambridge Univ. Pr., Cambridge U.K. (2012) [INSPIRE].
P. van Nieuwenhuizen and A. Waldron, On Euclidean spinors and Wick rotations, Phys. Lett. B 389 (1996) 29 [hep-th/9608174] [INSPIRE].
T. Delzant, Hamiltoniens périodiques et images convexes de l’application moment (in French), Bull. Soc. Math. France 116 (1988) 315.
V. Guillemin, Kähler structures on toric varieties, J. Diff. Geom. 40 (1994) 285.
M. Abreu, Kähler geometry of toric manifolds in symplectic coordinates, in Symplectic and contact topology: interactions and perspectives 35, (2003), pg. 1 [math/0004122].
D. Martelli, J. Sparks and S.-T. Yau, Sasaki-Einstein manifolds and volume minimisation, Commun. Math. Phys. 280 (2008) 611 [hep-th/0603021] [INSPIRE].
D. Huybrechts, Complex geometry: an introduction, Universitext, U.S. Government Printing Office, U.S.A. (2005).
J. Källén, Cohomological localization of Chern-Simons theory, JHEP 08 (2011) 008 [arXiv:1104.5353] [INSPIRE].
J. Källén and M. Zabzine, Twisted supersymmetric 5D Yang-Mills theory and contact geometry, JHEP 05 (2012) 125 [arXiv:1202.1956] [INSPIRE].
M. Blau and G. Thompson, Topological gauge theories of antisymmetric tensor fields, Annals Phys. 205 (1991) 130 [INSPIRE].
L. Baulieu and M. Schaden, Gauge group TQFT and improved perturbative Yang-Mills theory, Int. J. Mod. Phys. A 13 (1998) 985 [hep-th/9601039] [INSPIRE].
R. Goodman and N.R. Wallach, Symmetry, representations and, invariants, Graduate Texts in Mathematics 65, Springer, Germany (2009).
H.-C. Kim, J. Kim and S. Kim, Instantons on the 5-sphere and M5-branes, arXiv:1211.0144 [INSPIRE].
S. Shadchin, On certain aspects of string theory/gauge theory correspondence, hep-th/0502180 [INSPIRE].
S. Pasquetti, Factorisation of N = 2 theories on the squashed 3-sphere, JHEP 04 (2012) 120 [arXiv:1111.6905] [INSPIRE].
F. Nieri, S. Pasquetti and F. Passerini, 3d and 5d gauge theory partition functions as q-deformed CFT correlators, Lett. Math. Phys. 105 (2015) 109 [arXiv:1303.2626] [INSPIRE].
F. Nieri, S. Pasquetti, F. Passerini and A. Torrielli, 5D partition functions, q-Virasoro systems and integrable spin-chains, JHEP 12 (2014) 040 [arXiv:1312.1294] [INSPIRE].
J. Qiu and M. Zabzine, Factorization of 5D super Yang-Mills theory on Y p,q spaces, Phys. Rev. D 89 (2014) 065040 [arXiv:1312.3475] [INSPIRE].
J. Qiu, L. Tizzano, J. Winding and M. Zabzine, Gluing Nekrasov partition functions, Commun. Math. Phys. 337 (2015) 785 [arXiv:1403.2945] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1412.4407
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Rodriguez-Gomez, D., Schmude, J. Partition functions for equivariantly twisted \( \mathcal{N}=2 \) gauge theories on toric Kähler manifolds. J. High Energ. Phys. 2015, 111 (2015). https://doi.org/10.1007/JHEP05(2015)111
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2015)111