Abstract
We study the superconformal index of the \( \mathcal{N} = 4 \) super-Yang-Milles theory on S 3 × S 1 with the half BPS superconformal surface operator (defect) inserted at the great circle of S 3. The half BPS superconformal surface operators preserve the same supersymmetry as well as the symmetry of the chemical potential used in the definition of the superconformal index, so the structure and the parameterization of the superconformal index remain unaffected by the presence of the surface operator. On the surface defect, a two-dimensional (4, 4) superconformal field theory resides, and the four-dimensional super-conformal index may be regarded as a superconformal index of the two-dimensional (4, 4) superconformal field theory coupled with the four-dimensional bulk system. We construct the matrix model that computes the superconformal index with the surface operator when it couples with the bulk \( \mathcal{N} = 4 \) super-Yang-Milles theory through the defect hypermultiplets on it.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Romelsberger, Counting chiral primaries in N = 1, D = 4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [SPIRES].
J. Kinney, J.M. Maldacena, S. Minwalla and S. Raju, An index for 4 dimensional super conformal theories, Commun. Math. Phys. 275 (2007) 209 [hep-th/0510251] [SPIRES].
Y. Nakayama, Index for orbifold quiver gauge theories, Phys. Lett. B 636 (2006) 132 [hep-th/0512280] [SPIRES].
Y. Nakayama, Index for supergravity on AdS 5 × T (1,1) and conifold gauge theory, Nucl. Phys. B 755 (2006) 295 [hep-th/0602284] [SPIRES].
C. Romelsberger, Calculating the Superconformal Index and Seiberg Duality, arXiv:0707.3702 [SPIRES].
F.A. Dolan and H. Osborn, Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N = 1 Dual Theories, Nucl. Phys. B 818 (2009) 137 [arXiv:0801.4947] [SPIRES].
V.P. Spiridonov and G.S. Vartanov, Superconformal indices for \( \mathcal{N} = 1 \) theories with multiple duals, Nucl. Phys. B 824 (2010) 192 [arXiv:0811.1909] [SPIRES].
A. Gadde, E. Pomoni, L. Rastelli and S.S. Razamat, S-duality and 2 d Topological QFT, JHEP 03 (2010) 032 [arXiv:0910.2225] [SPIRES].
V.P. Spiridonov and G.S. Vartanov, Elliptic hypergeometry of supersymmetric dualities, Commun. Math. Phys. 304 (2011) 797 [arXiv:0910.5944] [SPIRES].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, The Superconformal Index of the E 6 SCFT, JHEP 08 (2010) 107 [arXiv:1003.4244] [SPIRES].
V.P. Spiridonov and G.S. Vartanov, Supersymmetric dualities beyond the conformal window, Phys. Rev. Lett. 105 (2010) 061603 [arXiv:1003.6109] [SPIRES].
V.P. Spiridonov and G.S. Vartanov, Superconformal indices of \( \mathcal{N} = 4 \) SYM field theories, arXiv:1005.4196 [SPIRES].
G.S. Vartanov, On the ISS model of dynamical SUSY breaking, Phys. Lett. B 696 (2011) 288 [arXiv:1009.2153] [SPIRES].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, On the Superconformal Index of N = 1 IR Fixed Points: A Holographic Check, JHEP 03 (2011) 041 [arXiv:1011.5278] [SPIRES].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, The 4 d Superconformal Index from q-deformed 2 d Yang-Mills, Phys. Rev. Lett. 106 (2011) 241602 [arXiv:1104.3850] [SPIRES].
N.R. Constable, J. Erdmenger, Z. Guralnik and I. Kirsch, Intersecting D3-branes and holography, Phys. Rev. D 68 (2003) 106007 [hep-th/0211222] [SPIRES].
S. Gukov and E. Witten, Gauge theory, ramification and the geometric Langlands program, hep-th/0612073 [SPIRES].
J. Gomis and S. Matsuura, Bubbling Surface Operators And S-duality, JHEP 06 (2007) 025 [arXiv:0704.1657] [SPIRES].
E.I. Buchbinder, J. Gomis and F. Passerini, Holographic Gauge Theories in Background Fields and Surface Operators, JHEP 12 (2007) 101 [arXiv:0710.5170] [SPIRES].
N. Drukker, J. Gomis and S. Matsuura, Probing N = 4 SYM With Surface Operators, JHEP 10 (2008) 048 [arXiv:0805.4199] [SPIRES].
S. Gukov and E. Witten, Rigid Surface Operators, arXiv:0804.1561 [SPIRES].
N.N. Shapovalov, On a bilinear form on the universal enveloping algebra of a complex semisimple Lie algebra, Funct. Anal. Appl. 6 (1972) 307.
S. Nawata, Localization of N = 4 Superconformal Field Theory on S 1 × S 3 and Index, arXiv:1104.4470 [SPIRES].
M. Bianchi, F.A. Dolan, P.J. Heslop and H. Osborn, N = 4 superconformal characters and partition functions, Nucl. Phys. B 767 (2007) 163 [hep-th/0609179] [SPIRES].
Y. Nakayama, Finite N Index and Angular Momentum Bound from Gravity, Gen. Rel. Grav. 39 (2007) 1625 [hep-th/0701208] [SPIRES].
F.A.H. Dolan, V.P. Spiridonov and G.S. Vartanov, From 4 d superconformal indices to 3 d partition functions, arXiv:1104.1787 [SPIRES].
A. Gadde and W. Yan, Reducing the 4d Index to the S 3 Partition Function, arXiv:1104.2592 [SPIRES].
Y. Imamura, Relation between the 4 d superconformal index and the S 3 partition function, arXiv:1104.4482 [SPIRES].
T. Nishioka, Y. Tachikawa and M. Yamazaki, 3 d Partition Function as Overlap of Wavefunctions, JHEP 08 (2011) 003 [arXiv:1105.4390] [SPIRES].
V.P. Spiridonov, On the elliptic beta function, Russ. Math. Surv. 56 (2001) 185.
V.P. Spirindonov, Theta hypergeometric integrals, Algebra i Analiz (St. Petersburg Math. J.) 15 (2003) 161 [math/0303205].
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] [SPIRES].
A. Kapustin, B. Willett and I. Yaakov, Nonperturbative Tests of Three-Dimensional Dualities, JHEP 10 (2010) 013 [arXiv:1003.5694] [SPIRES].
L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde, Loop and surface operators in N = 2 gauge theory and Liouville modular geometry, JHEP 01 (2010) 113 [arXiv:0909.0945] [SPIRES].
D. Gaiotto, Surface Operators in N = 2 4 d Gauge Theories, arXiv:0911.1316 [SPIRES].
E. Koh and S. Yamaguchi, Surface operators in the Klebanov-Witten theory, JHEP 06 (2009) 070 [arXiv:0904.1460] [SPIRES].
Y. Imamura, private communication.
A. Kapustin and E. Witten, Electric-magnetic duality and the geometric Langlands program, hep-th/0604151 [SPIRES].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1105.4883
Rights and permissions
About this article
Cite this article
Nakayama, Y. 4D and 2D superconformal index with surface operator. J. High Energ. Phys. 2011, 84 (2011). https://doi.org/10.1007/JHEP08(2011)084
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2011)084