Built on

Frontiers in physics. W. A. Benjamin, 1972
R. P. Feynman, Statistical mechanics: a set of lectures · 1972
Earlier work this paper cites.
I. J. Zucker, “The summation of series of hyperbolic functions,” SIAM J. Math. Anal. 10 no.~1, (1979) 192–206
1979
Earlier work this paper cites.
E. Witten, “Constraints on supersymmetry breaking,” Nucl. Phys. B202 (1982) 253
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Earlier work this paper cites.
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
1993
Earlier work this paper cites.
J. A. Minahan and A. P. Polychronakos, “Interacting fermion systems from two-dimensional QCD,” Phys. Lett. B326 (1994) 288–294 , hep-th/9309044
1994
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A. Mikhailov, “Giant gravitons from holomorphic surfaces,” JHEP 11 (2000) 027 , hep-th/0010206
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C. Krattenthaler, “Advanced determinant calculus: a complement,” Linear Algebra Appl. 411 (2005) 68–166 , math/0503507
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C. Romelsberger, “Counting chiral primaries in 𝒩 = 1 \mathcal{N}=1 , d = 4 d=4 superconformal field theories,” Nucl. Phys. B747 (2006) 329–353 , hep-th/0510060
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J. Kinney, J. M. Maldacena, S. Minwalla, and S. Raju, “An index for 4 4 dimensional super conformal theories,” Commun. Math. Phys. 275 (2007) 209–254 , hep-th/0510251
2007
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I. Biswas, D. Gaiotto, S. Lahiri, and S. Minwalla, “Supersymmetric states of 𝒩 = 4 {\cal N}=4 Yang-Mills from giant gravitons,” JHEP 12 (2007) 006 , hep-th/0606087
2007
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2015

The exact Schur index of $\mathcal{N}=4$ SYM

Bourdier, Jun, Drukker, Nadav, Felix, Jan

Understand

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 $\mathcal{N}=4$ 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.

Built on

Frontiers in physics. W. A. Benjamin, 1972
R. P. Feynman, Statistical mechanics: a set of lectures · 1972
Earlier work this paper cites.
I. J. Zucker, “The summation of series of hyperbolic functions,” SIAM J. Math. Anal. 10 no.~1, (1979) 192–206
1979
Earlier work this paper cites.
E. Witten, “Constraints on supersymmetry breaking,” Nucl. Phys. B202 (1982) 253
1982
Earlier work this paper cites.
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
1993
Earlier work this paper cites.
J. A. Minahan and A. P. Polychronakos, “Interacting fermion systems from two-dimensional QCD,” Phys. Lett. B326 (1994) 288–294 , hep-th/9309044
1994
Earlier work this paper cites.
A. Mikhailov, “Giant gravitons from holomorphic surfaces,” JHEP 11 (2000) 027 , hep-th/0010206
2000
Earlier work this paper cites.
C. Krattenthaler, “Advanced determinant calculus: a complement,” Linear Algebra Appl. 411 (2005) 68–166 , math/0503507
2005
Earlier work this paper cites.
C. Romelsberger, “Counting chiral primaries in 𝒩 = 1 \mathcal{N}=1 , d = 4 d=4 superconformal field theories,” Nucl. Phys. B747 (2006) 329–353 , hep-th/0510060
2006
Earlier work this paper cites.
J. Kinney, J. M. Maldacena, S. Minwalla, and S. Raju, “An index for 4 4 dimensional super conformal theories,” Commun. Math. Phys. 275 (2007) 209–254 , hep-th/0510251
2007
Earlier work this paper cites.
I. Biswas, D. Gaiotto, S. Lahiri, and S. Minwalla, “Supersymmetric states of 𝒩 = 4 {\cal N}=4 Yang-Mills from giant gravitons,” JHEP 12 (2007) 006 , hep-th/0606087
2007
Earlier work this paper cites.