Communications in Mathematical Physics

, Volume 275, Issue 1, pp 209–254 | Cite as

An Index for 4 Dimensional Super Conformal Theories

  • Justin Kinney
  • Juan Maldacena
  • Shiraz Minwalla
  • Suvrat RajuEmail author


We present a trace formula for an index over the spectrum of four dimensional superconformal field theories on S 3 ×  time. Our index receives contributions from states invariant under at least one supercharge and captures all information – that may be obtained purely from group theory – about protected short representations in 4 dimensional superconformal field theories. In the case of the \({\mathcal N}=4\) theory our index is a function of four continuous variables. We compute it at weak coupling using gauge theory and at strong coupling by summing over the spectrum of free massless particles in AdS 5 × S 5 and find perfect agreement at large N and small charges. Our index does not reproduce the entropy of supersymmetric black holes in AdS 5, but this is not a contradiction, as it differs qualitatively from the partition function over supersymmetric states of the \({\mathcal N}=4\) theory. We note that entropy for some small supersymmetric AdS 5 black holes may be reproduced via a D-brane counting involving giant gravitons. For big black holes we find a qualitative (but not exact) agreement with the naive counting of BPS states in the free Yang Mills theory. In this paper we also evaluate and study the partition function over the chiral ring in the \({\mathcal N}=4\) Yang Mills theory.


Black Hole Partition Function Giant Graviton Superconformal Algebra Chiral Ring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Justin Kinney
    • 1
  • Juan Maldacena
    • 2
  • Shiraz Minwalla
    • 3
    • 4
  • Suvrat Raju
    • 4
    Email author
  1. 1.Department of PhysicsPrinceton UniversityPrincetonUSA
  2. 2.Institute for Advanced StudyPrincetonUSA
  3. 3.Tata Institute of Fundamental ResearchMumbaiIndia
  4. 4.Jefferson Physical LaboratoryHarvard UniversityCambridgeUSA

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