Abstract
The Witten index counts the difference in the number of bosonic and fermionic states of a quantum mechanical system. The Schur index, which can be defined for theories with at least \( \mathcal{N}=2 \) supersymmetry in four dimensions is a particular refinement of the index, dependent on one parameter q serving as the fugacity for a particular set of charges which commute with the hamiltonian and some supersymmetry generators. This index has a known expression for all Lagrangian and some non-Lagrangian theories as a finite dimensional integral or a complicated infinite sum. In the case of \( \mathcal{N}=2 \) SYM with gauge group U(N ) we rewrite this as the partition function of a gas of N non interacting and translationally invariant fermions on a circle. This allows us to perform the integrals and write down explicit expressions for fixed N as well as the exact all orders large N expansion.
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References
E. Witten, Constraints on supersymmetry breaking, Nucl. Phys. B 202 (1982) 253 [INSPIRE].
C. Romelsberger, Counting chiral primaries in \( \mathcal{N}=1 \) , 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].
F.A. Dolan and H. Osborn, Applications of the superconformal index for protected operators and q-hypergeometric identities to \( \mathcal{N}=1 \) dual theories, Nucl. Phys. B 818 (2009) 137 [arXiv:0801.4947] [INSPIRE].
L. Rastelli and S.S. Razamat, The superconformal index of theories of class \( \mathcal{S} \), arXiv:1412.7131 [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, The 4d superconformal index from q-deformed 2d Yang-Mills, Phys. Rev. Lett. 106 (2011) 241602 [arXiv:1104.3850] [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 \( \mathcal{N}=2 \) superconformal theories in four dimensions, JHEP 10 (2012) 191 [arXiv:1208.5056] [INSPIRE].
V.P. Spiridonov and G.S. Vartanov, Superconformal indices of \( \mathcal{N}=4 \) SYM field theories, Lett. Math. Phys. 100 (2012) 97 [arXiv:1005.4196] [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.
D.V. Chudnovsky and G.V. Chudnovsky, Hypergeometric and modular function identities, and new rational approximations to and continued fraction expansions of classical constants and functions, Contemp. Math. 143 (1993) 117.
C. Krattenthaler, Advanced determinant calculus: a complement, Linear Algebra Appl. 411 (2005) 68 [math.CO/0503507].
R.P. Feynman, Statistical mechanics: a set of lectures, Frontiers in Physics, W.A. Benjamin (1972).
B. Assel, D. Cassani and D. Martelli, Localization on Hopf surfaces, JHEP 08 (2014) 123 [arXiv:1405.5144] [INSPIRE].
B. Assel et al., The Casimir energy in curved space and its supersymmetric counterpart, JHEP 07 (2015) 043 [arXiv:1503.05537] [INSPIRE].
A. Mikhailov, Giant gravitons from holomorphic surfaces, JHEP 11 (2000) 027 [hep-th/0010206] [INSPIRE].
I. Biswas, D. Gaiotto, S. Lahiri and S. Minwalla, Supersymmetric states of \( \mathcal{N}=4 \) Yang-Mills from giant gravitons, JHEP 12 (2007) 006 [hep-th/0606087] [INSPIRE].
G. Mandal and N.V. Suryanarayana, Counting 1/8-BPS dual-giants, JHEP 03 (2007) 031 [hep-th/0606088] [INSPIRE].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, \( \mathcal{N}=6 \) superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
N. Drukker, M. Mariño and P. Putrov, From weak to strong coupling in ABJM theory, Commun. Math. Phys. 306 (2011) 511 [arXiv:1007.3837] [INSPIRE].
S. Codesido, A. Grassi and M. Mariño, Exact results in \( \mathcal{N}=8 \) Chern-Simons-matter theories and quantum geometry, JHEP 07 (2015) 011 [arXiv:1409.1799] [INSPIRE].
M. Mariño and P. Putrov, ABJM theory as a Fermi gas, J. Stat. Mech. (2012) P03001 [arXiv:1110.4066] [INSPIRE].
A. Dabholkar, N. Drukker and J. Gomes, Localization in supergravity and quantum AdS 4 /CFT 3 holography, JHEP 10 (2014) 090 [arXiv:1406.0505] [INSPIRE].
M.-x. Huang and X.-f. Wang, Topological strings and quantum spectral problems, JHEP 09 (2014) 150 [arXiv:1406.6178] [INSPIRE].
A. Grassi, Y. Hatsuda and M. Mariño, Topological strings from quantum mechanics, arXiv:1410.3382 [INSPIRE].
J.A. Minahan and A.P. Polychronakos, Interacting fermion systems from two-dimensional QCD, Phys. Lett. B 326 (1994) 288 [hep-th/9309044] [INSPIRE].
C. Beem et al., Infinite chiral symmetry in four dimensions, Commun. Math. Phys. 336 (2015) 1359 [arXiv:1312.5344] [INSPIRE].
M. Yamazaki, New integrable models from the gauge/YBE correspondence, J. Statist. Phys. 154 (2014) 895 [arXiv:1307.1128] [INSPIRE].
D. Gang, E. Koh and K. Lee, Line operator index on S 1 × S 3, JHEP 05 (2012) 007 [arXiv:1201.5539] [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].
F.W. Olver, NIST handbook of mathematical functions, Cambridge University Press, Cambridge U.K. (2010).
I.J. Zucker, The summation of series of hyperbolic functions, SIAM J. Math. Anal. 10 (1979) 192.
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Bourdier, J., Drukker, N. & Felix, J. The exact Schur index of \( \mathcal{N}=4 \) SYM. J. High Energ. Phys. 2015, 210 (2015). https://doi.org/10.1007/JHEP11(2015)210
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DOI: https://doi.org/10.1007/JHEP11(2015)210