Counting BPS operators in N=4 SYM

Abstract

The free field partition function for a generic U(N) gauge theory, where the fundamental fields transform in the adjoint representation, is analysed in terms of symmetric polynomial techniques. It is shown by these means how this is related to the cycle polynomial for the symmetric group and how the large N result may be easily recovered. Higher order corrections for finite N are also discussed in terms of symmetric group characters. For finite N, the partition function involving a single bosonic fundamental field is recovered and explicit counting of multi-trace quarter BPS operators in free \N=4 super Yang Mills discussed, including a general result for large N. The partition function for BPS operators in the chiral ring of \N=4 super Yang Mills is analysed in terms of plane partitions. Asymptotic counting of BPS primary operators with differing R-symmetry charges is discussed in both free \N=4 super Yang Mills and in the chiral ring. Also, general and explicit expressions are derived for SU(2) gauge theory partition functions, when the fundamental fields transform in the adjoint, for free field theory.

38 pages, uses harvmac, v.2. references added, typos corrected, discussion of asymptotic counting included for more general chiral ring, v.3. typos corrected, discussion for su(2) simplified, to be published in Nuclear Physics B

References in corpus (26)

• An Index for 4 dimensional Super Conformal Theories
• General supersymmetric AdS5 black holes
• General Non-Extremal Rotating Black Holes in Minimal Five-Dimensional Gauged Supergravity
• The Hagedorn Transition, Deconfinement and N=4 SYM Theory
• Supersymmetric AdS5 black holes
• Counting Gauge Invariants: the Plethystic Program
• Supersymmetric multi-charge AdS_5 black holes
• Gauge Fields and Space-Time
• Fivebranes from gauge theory
• Character Formulae and Partition Functions in Higher Dimensional Conformal Field Theory
• Partition Sums and Entropy Bounds in Weakly Coupled CFT
• N=4 Superconformal Characters and Partition Functions
• Dual Giant Gravitons in Sasaki-Einstein Backgrounds
• Counting BPS Baryonic Operators in CFTs with Sasaki-Einstein duals
• Do supersymmetric anti-de Sitter black rings exist?
• Half-BPS Giants, Free Fermions and Microstates of Superstars
• Systematics of Quarter BPS operators in N=4 SYM
• Partition Functions, the Bekenstein Bound and Temperature Inversion in Anti-de Sitter Space and its Conformal Boundary
• Dual giant gravitons in AdS${}_m$ $\times$ Y${}^n$ (Sasaki-Einstein)
• Counting BPS Operators in the Chiral Ring of N=2 Supersymmetric Gauge Theories or N=2 Braine Surgery
• Two-charge small black hole entropy: String-loops and multi-strings
• Finite N Index and Angular Momentum Bound from Gravity
• The planar spectrum in U(N)-invariant quantum mechanics by Fock space methods: I. The bosonic case
• Generating functions for tensor products
• SU(N) Glueball Masses in 2+1 Dimensions
• Fock space methods and large N

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