Abstract
The five-dimensional \( \mathcal{N}=1 \) supersymmetric gauge theory with Sp(N) gauge group and SO(2N f ) flavor symmetry describes the physics on N D4-branes with N f D8-branes on top of a single O8 orientifold plane in Type I′ theory. This theory is known to be superconformal at the strong coupling limit with the enhanced global symmetry \( {E_{{N_f +1}}} \) for N f ≤ 7. In this work we calculate the superconformal index on S 1 × S 4 for the Sp(1) gauge theory by the localization method and confirm such enhancement of the global symmetry at the superconformal limit for N f ≤ 5 to a few leading orders in the chemical potential. Both perturbative and (anti)instanton contributions are present in this calculation. For N f = 6, 7 cases some issues related the pole structure of the instanton calculation could not be resolved and here we could provide only some suggestive answer for the leading contributions to the index. For the Sp(N) case, similar issues related to the pole structure appear.
Similar content being viewed by others
References
N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett. B 388 (1996) 753 [hep-th/9608111] [INSPIRE].
D.R. Morrison and N. Seiberg, Extremal transitions and five-dimensional supersymmetric field theories, Nucl. Phys. B 483 (1997) 229 [hep-th/9609070] [INSPIRE].
M.R. Douglas, S.H. Katz and C. Vafa, Small instantons, Del Pezzo surfaces and type-I-prime theory, Nucl. Phys. B 497 (1997) 155 [hep-th/9609071] [INSPIRE].
O.J. Ganor, D.R. Morrison and N. Seiberg, Branes, Calabi-Yau spaces and toroidal compactification of the N = 1 six-dimensional E 8 theory, Nucl. Phys. B 487 (1997) 93 [hep-th/9610251] [INSPIRE].
K.A. Intriligator, D.R. Morrison and N. Seiberg, Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces, Nucl. Phys. B 497 (1997) 56 [hep-th/9702198] [INSPIRE].
U.H. Danielsson, G. Ferretti, J. Kalkkinen and P. Stjernberg, Notes on supersymmetric gauge theories in five-dimensions and six-dimensions, Phys. Lett. B 405 (1997) 265 [hep-th/9703098] [INSPIRE].
T. Kugo and K. Ohashi, Off-shell D = 5 supergravity coupled to matter Yang-Mills system, Prog. Theor. Phys. 105 (2001) 323 [hep-ph/0010288] [INSPIRE].
E. Bergshoeff, S. Cucu, M. Derix, T. de Wit, R. Halbersma, et al., Weyl multiplets of N = 2 conformal supergravity in five-dimensions, JHEP 06 (2001) 051 [hep-th/0104113] [INSPIRE].
E. Bergshoeff, S. Cucu, T. De Wit, J. Gheerardyn, R. Halbersma, et al., Superconformal N = 2, D = 5 matter with and without actions, JHEP 10 (2002) 045 [hep-th/0205230] [INSPIRE].
E. Witten, Phase transitions in M-theory and F-theory, Nucl. Phys. B 471 (1996) 195 [hep-th/9603150] [INSPIRE].
O. Aharony, M. Berkooz and N. Seiberg, Light cone description of (2,0) superconformal theories in six-dimensions, Adv. Theor. Math. Phys. 2 (1998) 119 [hep-th/9712117] [INSPIRE].
M.R. Douglas, On D = 5 super Yang-Mills theory and (2,0) theory, JHEP 02 (2011) 011 [arXiv:1012.2880] [INSPIRE].
N. Lambert, C. Papageorgakis and M. Schmidt-Sommerfeld, M5-Branes, D4-branes and Quantum 5D super-Yang-Mills, JHEP 01 (2011) 083 [arXiv:1012.2882] [INSPIRE].
N. Lambert and P. Richmond, (2,0) Supersymmetry and the Light-Cone Description of M5-branes, JHEP 02 (2012) 013 [arXiv:1109.6454] [INSPIRE].
H.-C. Kim, S. Kim, E. Koh, K. Lee and S. Lee, On instantons as Kaluza-Klein modes of M5-branes, JHEP 12 (2011) 031 [arXiv:1110.2175] [INSPIRE].
W. Nahm, Supersymmetries and their Representations, Nucl. Phys. B 135 (1978) 149 [INSPIRE].
L. Romans, The F(4) Gauged Supergravity In Six-Dimensions, Nucl. Phys. B 269 (1986) 691 [INSPIRE].
S. Minwalla, Restrictions imposed by superconformal invariance on quantum field theories, Adv. Theor. Math. Phys. 2 (1998) 781 [hep-th/9712074] [INSPIRE].
J. Bhattacharya, S. Bhattacharyya, S. Minwalla and S. Raju, Indices for Superconformal Field Theories in 3,5 and 6 Dimensions, JHEP 02 (2008) 064 [arXiv:0801.1435] [INSPIRE].
O.J. Ganor and A. Hanany, Small E 8 instantons and tensionless noncritical strings, Nucl. Phys. B 474 (1996) 122 [hep-th/9602120] [INSPIRE].
P. Hořava and E. Witten, Heterotic and type-I string dynamics from eleven-dimensions, Nucl. Phys. B 460 (1996) 506 [hep-th/9510209] [INSPIRE].
P. Hořava and E. Witten, Eleven-dimensional supergravity on a manifold with boundary, Nucl. Phys. B 475 (1996) 94 [hep-th/9603142] [INSPIRE].
J. Polchinski and E. Witten, Evidence for heterotic - type-I string duality, Nucl. Phys. B 460 (1996) 525 [hep-th/9510169] [INSPIRE].
J. Polchinski, S. Chaudhuri and C.V. Johnson, Notes on D-branes, hep-th/9602052 [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].
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].
M.F. Atiyah, Elliptic operators and compact groups, in Lecture Notes in Mathematics 401 Springer-Verlag (1974).
O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas and M. Van Raamsdonk, The Hagedorn - deconfinement phase transition in weakly coupled large-N gauge theories, Adv. Theor. Math. Phys. 8 (2004) 603 [hep-th/0310285] [INSPIRE].
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].
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].
H. Nakajima and K. Yoshioka, Instanton counting on blowup. 1., math/0306198 [INSPIRE].
N. Nekrasov and S. Shadchin, ABCD of instantons, Commun. Math. Phys. 252 (2004) 359 [hep-th/0404225] [INSPIRE].
S. Shadchin, On certain aspects of string theory/gauge theory correspondence, hep-th/0502180 [INSPIRE].
S. Kim, K.-M. Lee and S. Lee, Dyonic Instantons in 5-dim Yang-Mills Chern-Simons Theories, JHEP 08 (2008) 064 [arXiv:0804.1207] [INSPIRE].
B. Collie and D. Tong, Instantons, Fermions and Chern-Simons Terms, JHEP 07 (2008) 015 [arXiv:0804.1772] [INSPIRE].
Y. Tachikawa, Five-dimensional Chern-Simons terms and Nekrasov’s instanton counting, JHEP 02 (2004) 050 [hep-th/0401184] [INSPIRE].
U. Bruzzo, F. Fucito, J.F. Morales and A. Tanzini, Multiinstanton calculus and equivariant cohomology, JHEP 05 (2003) 054 [hep-th/0211108] [INSPIRE].
M. Mariño and N. Wyllard, A Note on instanton counting for N = 2 gauge theories with classical gauge groups, JHEP 05 (2004) 021 [hep-th/0404125] [INSPIRE].
F. Fucito, J.F. Morales and R. Poghossian, Instantons on quivers and orientifolds, JHEP 10 (2004) 037 [hep-th/0408090] [INSPIRE].
T. Okuda and V. Pestun, On the instantons and the hypermultiplet mass of N = 2* super Yang-Mills on S 4, JHEP 03 (2012) 017 [arXiv:1004.1222] [INSPIRE].
M.R. Douglas, Gauge fields and D-branes, J. Geom. Phys. 28 (1998) 255 [hep-th/9604198] [INSPIRE].
O. Aharony, M. Berkooz, S. Kachru and E. Silverstein, Matrix description of (1,0) theories in six-dimensions, Phys. Lett. B 420 (1998) 55 [hep-th/9709118] [INSPIRE].
O. Bergman, M.R. Gaberdiel and G. Lifschytz, String creation and heterotic type-I’ duality, Nucl. Phys. B 524 (1998) 524 [hep-th/9711098] [INSPIRE].
J. Kallen and M. Zabzine, Twisted supersymmetric 5D Yang-Mills theory and contact geometry, JHEP 05 (2012) 125 [arXiv:1202.1956] [INSPIRE].
J. Kallen, 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].
H.-C. Kim and S. Kim, M5-branes from gauge theories on the 5-sphere, arXiv:1206.6339 [INSPIRE].
N. Hama and K. Hosomichi, Seiberg-Witten Theories on Ellipsoids, JHEP 09 (2012) 033 [arXiv:1206.6359] [INSPIRE].
N. Hama, K. Hosomichi and S. Lee, SUSY Gauge Theories on Squashed Three-Spheres, JHEP 05 (2011) 014 [arXiv:1102.4716] [INSPIRE].
Y. Imamura and D. Yokoyama, N=2 supersymmetric theories on squashed three-sphere, Phys. Rev. D 85 (2012) 025015 [arXiv:1109.4734] [INSPIRE].
F. Dolan, V. Spiridonov and G. Vartanov, From 4d superconformal indices to 3d partition functions, Phys. Lett. B 704 (2011) 234 [arXiv:1104.1787] [INSPIRE].
Y. Imamura, Relation between the 4d superconformal index and the S 3 partition function, JHEP 09 (2011) 133 [arXiv:1104.4482] [INSPIRE].
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].
A. Gadde and W. Yan, Reducing the 4d Index to the S 3 Partition Function, arXiv:1104.2592 [INSPIRE].
O. DeWolfe, A. Hanany, A. Iqbal and E. Katz, Five-branes, seven-branes and five-dimensional E(n) field theories, JHEP 03 (1999) 006 [hep-th/9902179] [INSPIRE].
S. Ferrara, A. Kehagias, H. Partouche and A. Zaffaroni, AdS 6 interpretation of 5 − D superconformal field theories, Phys. Lett. B 431 (1998) 57 [hep-th/9804006] [INSPIRE].
A. Brandhuber and Y. Oz, The D-4 - D-8 brane system and five-dimensional fixed points, Phys. Lett. B 460 (1999) 307 [hep-th/9905148] [INSPIRE].
O. Bergman and D. Rodriguez-Gomez, 5d quivers and their AdS 6 duals, JHEP 07 (2012) 171 [arXiv:1206.3503] [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].
M. Atiyah, N.J. Hitchin, V. Drinfeld and Y. Manin, Construction of Instantons, Phys. Lett. A 65 (1978) 185 [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].
J. Choi, S. Lee and J. Song, Superconformal Indices for Orbifold Chern-Simons Theories, JHEP 03 (2009) 099 [arXiv:0811.2855] [INSPIRE].
R. Feger and T.W. Kephart, LieART - A Mathematica Application for Lie Algebras and Representation Theory, arXiv:1206.6379 [INSPIRE].
LieART project home page: http://lieart.hepforge.org.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1206.6781
Rights and permissions
About this article
Cite this article
Kim, HC., Kim, SS. & Lee, K. 5-dim superconformal index with enhanced E n global symmetry. J. High Energ. Phys. 2012, 142 (2012). https://doi.org/10.1007/JHEP10(2012)142
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2012)142