Abstract
Recently the Schur index of \( \mathcal{N}=4 \) SYM was evaluated in closed form to all orders including exponential corrections in the large N expansion and for fixed finite N. This was achieved by identifying the matrix model which calculates the index with the partition function of a system of free fermions on a circle. The index can be enriched by the inclusion of loop operators and the case of Wilson loops is particularly easy, as it amounts to inserting extra characters into the matrix model. The Fermi-gas approach is applied here to this problem, the formalism is explored and explicit results at large N are found for the fundamental as well as a few other symmetric and antisymmetric representations.
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] [INSPIRE].
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] [INSPIRE].
J. Bourdier, N. Drukker and J. Felix, The exact Schur index of \( \mathcal{N}=4 \) SYM, arXiv:1507.08659 [INSPIRE].
J. Bourdier, N. Drukker and J. Felix, The \( \mathcal{N}=2 \) Schur index from free fermions, arXiv:1510.07041 [INSPIRE].
D. Gang, E. Koh and K. Lee, Line operator index on S 1 × S 3, JHEP 05 (2012) 007 [arXiv:1201.5539] [INSPIRE].
Y. Ito, T. Okuda and M. Taki, Line operators on S 1 × R 3 and quantization of the Hitchin moduli space, JHEP 04 (2012) 010 [arXiv:1111.4221] [INSPIRE].
N. Mekareeya and D. Rodriguez-Gomez, 5D gauge theories on orbifolds and 4D ‘t Hooft line indices, JHEP 11 (2013) 157 [arXiv:1309.1213] [INSPIRE].
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] [INSPIRE].
A. Gadde, E. Pomoni, L. Rastelli and S.S. Razamat, S-duality and 2D topological QFT, JHEP 03 (2010) 032 [arXiv:0910.2225] [INSPIRE].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, Gauge theories and Macdonald polynomials, Commun. Math. Phys. 319 (2013) 147 [arXiv:1110.3740] [INSPIRE].
S.S. Razamat, On a modular property of N = 2 superconformal theories in four dimensions, JHEP 10 (2012) 191 [arXiv:1208.5056] [INSPIRE].
G. Frobenius and L. Stickelberger, Über die Addition und Multiplication der elliptischen Functionen, J. Reine Angew. Math. 88 (1879) 146.
G. Frobenius, Über die elliptischen Functionen zweiter Art, J. Reine Angew. Math. 93 (1882) 53.
C. Krattenthaler, Advanced determinant calculus: a complement, Linear Alg. Appl. 411 (2005) 68 [math/0503507].
M. Mariño and P. Putrov, ABJM theory as a Fermi gas, J. Stat. Mech. 1203 (2012) P03001 [arXiv:1110.4066] [INSPIRE].
A. Klemm, M. Mariño, M. Schiereck and M. Soroush, Aharony-Bergman-Jafferis-Maldacena Wilson loops in the Fermi gas approach, Z. Naturforsch. A 68 (2013) 178 [arXiv:1207.0611] [INSPIRE].
Y. Hatsuda, M. Honda, S. Moriyama and K. Okuyama, ABJM Wilson loops in arbitrary representations, JHEP 10 (2013) 168 [arXiv:1306.4297] [INSPIRE].
S. Hirano, K. Nii and M. Shigemori, ABJ Wilson loops and Seiberg duality, PTEP 2014 (2014) 113B04 [arXiv:1406.4141] [INSPIRE].
H. Ouyang, J.-B. Wu and J.-j. Zhang, Exact results for Wilson loops in orbifold ABJM theory, arXiv:1507.00442 [INSPIRE].
I.J. Zucker, The summation of series of hyperbolic functions, SIAM J. Math. Anal. 10 (1979) 192.
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: 1510.02480
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
Drukker, N. The \( \mathcal{N}=4 \) Schur index with Polyakov loops. J. High Energ. Phys. 2015, 1–15 (2015). https://doi.org/10.1007/JHEP12(2015)012
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/JHEP12(2015)012