An Index for 4 Dimensional Super Conformal Theories

By Justin B. Kinney, J. Maldacena, Shiraz Minwalla, S. Raju. 2005

1,024 citations recorded by the source corpus.

We present a trace formula for an index over the spectrum of four dimensional superconformal field theories on S3 × 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 AdS5 × S5 and find perfect agreement at large N and small charges. Our index does not reproduce the entropy of supersymmetric black holes in AdS5, 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 AdS5 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.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/81c57ba974668185eddfaf29e1d04d14a7939047

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Astronomy and Astrophysics

In Nuclear and High Energy Physics

Citations in the corpus

Gauge Theories and Macdonald Polynomials (2011, background): “cannot capture the non-perturbative states of the bulk theory [12, 48, 49, 23].”
Superconformal indices of three-dimensional theories related by mirror symmetry (2011, methodology): “Formulas (3) and (4) resemble to some extent the procedure of calculating SCIs in 4d supersymmetric field theories [1, 4], but they are much more involved.”
Sasaki–Einstein Manifolds and Volume Minimisation (2006, background): “In [47,48] some indices counting BPS operators of superconformal field theories have been introduced and studied.”
Localization on Hopf surfaces (2014, background): “In four dimensions the supersymmetric index scales like N0 at large N [12, 25], implying that the N2 scaling of the logarithm of the partition function must arise as an extra contribution.”
Black hole microstates in AdS4 from supersymmetric localization (2015, background): “Various attempts have been made to derive the entropy of a class of rotating black holes in AdS5 in terms of states of the dual N = 4 super-Yang-Mills (SYM) theory [2, 3] but none was completely successful.”
Comments on twisted indices in 3d supersymmetric gauge theories (2016, background): “Introduction Supersymmetric indices [1] are simple yet powerful tools for studying supersymmetric field theories [2, 3, 4, 5, 6, 7].”
Supersymmetry on curved spaces and holography (2012, background): “The partition function defined on S × S computes what is known as the supersymmetric index [22, 23] of the theory.”
General instanton counting and 5d SCFT (2014, methodology): “They studied the Nekrasov’s partition function, and also the superconformal index [29, 30] which one can calculate using the former.”
Elliptic Hypergeometry of Supersymmetric Dualities (2009, methodology): “Second, we have deliberately skipped consideration of the superconformal indices for extended N > 1 supersymmetric field theories [6, 49].”
M5-branes from gauge theories on the 5-sphere (2012, methodology): “The index which counts BPS states saturating this bound is studied in [21, 22].”
Black Holes in 4D N=4 Super-Yang-Mills Field Theory (2018, methodology): “Those states can be counted (with sign) by computing the superconformal index [30,36], i.”
Deformations of superconformal theories (2016, methodology): “All marginal N = 2 preserving deformations must be exactly marginal, because the chiral LB1[0; 0] (0 ; 4) 2 multiplet containing the marginal deformation is absolutely protected: there is no recombination rule that allows it to pair up with other multiplets into a long multiplet (see for instance [1”
The superconformal index of the E6 SCFT (2010, methodology): “(5)Our normalization convention for the R-symmetry charges is as in [15] and differs from [6]: Rhere = Rthere/2, rhere = rthere/2.”
N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals (2008, background): “We can find the protected chiral operators by noticing that these operators are given by considering a certain quantum mechanics on the moduli space, as in [26-28].”
Towards lattice simulation of the gauge theory duals to black holes and hot strings (2007, background): “Even protected quantities that can give information about the strongly coupled theories through weak coupling calculations have not yet revealed information about black holes [9] and if they do in the future, they will remain indirect probes of quantum black hole physics.”
The N=2$$ \mathcal{N}=2 $$ superconformal bootstrap (2014, neutral): “We also list two common notations used to designate the different representations — one from [76] (DO) and the other from [77] (KMMR).”