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Shortening of primary operators in N-extended SCFT_4 and harmonic-superspace analyticity

L. Andrianopoli, S. Ferrara, E. Sokatchev, B. Zupnik

TL;DR

This work classifies all shortening patterns for composite gauge-invariant conformal primaries in SU(2,2/N) theories using extended harmonic superspace. By exploiting G-analyticity and H-analyticity on SU(N)/U(1)^{N-1} cosets, it constructs and catalogs ultrashort, short, and semishort multiplets for N=2,3,4, linking boundary CFT representations to bulk BPS states in AdS$_5$/CFT$_4$. It shows how multitrace operators in N=4 SYM decompose into short UIR blocks, identifying 1/2-, 1/4-, and 1/8-BPS channels corresponding to multiparticle supergravity states, with implications for non-renormalization theorems and protected conformal dimensions. The analysis provides a unified framework for understanding shortening as arising from harmonic-space subspaces and clarifies the spectrum of Kaluza–Klein towers in AdS backgrounds. Overall, the paper advances precise operator realizations of shortened representations and their AdS/CFT significance in extended supersymmetric theories.

Abstract

We present the analysis of all possible shortenings which occur for composite gauge invariant conformal primary superfields in SU(2,2/N) invariant gauge theories. These primaries have top-spin range N/2 \leq J_{max} < N with J_{max} = J_1 + J_2, (J_1,J_2) being the SL(2,C) quantum numbers of the highest spin component of the superfield. In Harmonic superspace, analytic and chiral superfields give J_{max}= N/2 series while intermediate shortenings correspond to fusion of chiral with analytic in N=2, or analytic with different analytic structures in N=3,4. In the AdS/CFT language shortenings of UIR's correspond to all possible BPS conditions on bulk states. An application of this analysis to multitrace operators, corresponding to multiparticle supergravity states, is spelled out.

Shortening of primary operators in N-extended SCFT_4 and harmonic-superspace analyticity

TL;DR

This work classifies all shortening patterns for composite gauge-invariant conformal primaries in SU(2,2/N) theories using extended harmonic superspace. By exploiting G-analyticity and H-analyticity on SU(N)/U(1)^{N-1} cosets, it constructs and catalogs ultrashort, short, and semishort multiplets for N=2,3,4, linking boundary CFT representations to bulk BPS states in AdS/CFT. It shows how multitrace operators in N=4 SYM decompose into short UIR blocks, identifying 1/2-, 1/4-, and 1/8-BPS channels corresponding to multiparticle supergravity states, with implications for non-renormalization theorems and protected conformal dimensions. The analysis provides a unified framework for understanding shortening as arising from harmonic-space subspaces and clarifies the spectrum of Kaluza–Klein towers in AdS backgrounds. Overall, the paper advances precise operator realizations of shortened representations and their AdS/CFT significance in extended supersymmetric theories.

Abstract

We present the analysis of all possible shortenings which occur for composite gauge invariant conformal primary superfields in SU(2,2/N) invariant gauge theories. These primaries have top-spin range N/2 \leq J_{max} < N with J_{max} = J_1 + J_2, (J_1,J_2) being the SL(2,C) quantum numbers of the highest spin component of the superfield. In Harmonic superspace, analytic and chiral superfields give J_{max}= N/2 series while intermediate shortenings correspond to fusion of chiral with analytic in N=2, or analytic with different analytic structures in N=3,4. In the AdS/CFT language shortenings of UIR's correspond to all possible BPS conditions on bulk states. An application of this analysis to multitrace operators, corresponding to multiparticle supergravity states, is spelled out.

Paper Structure

This paper contains 16 sections, 146 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Highest weight UIR's of $SU(2,2/N)$: a review
  3. Extended harmonic superspaces and short superfields
  4. The $N=2$ case
  5. The $N=3$ case
  6. The $N=3$ super Yang--Mills multiplet
  7. Series of short $N=3$ multiplets
  8. The $N=4$ case
  9. The $N=4$ SYM multiplet
  10. Series of short multiplets
  11. Extra shortening of N=4 superfields
  12. Multitrace operators and multiparticle states
  13. Conclusion
  14. Appendix
  15. Harmonic variables
  16. ...and 1 more sections

Figures (3)

  • Figure 1: $[q,p,q]$ representation:
  • Figure 2: $[q+2n-x,p-k+2x,q+2k-x]$ representation, for $k\leq p$; $0\leq k \leq n$, $0\leq x \leq q+2k$:
  • Figure 3: $[p+q+2n-k-x,-p+k+2x,p+q+k-x]$ representation, for $k\geq p$; $0\leq k \leq n$, $0\leq x \leq p+q+k$: