Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Mixed-symmetry tensor conserved currents and AdS/CFT correspondence

K. B. Alkalaev

TL;DR

This work constructs and analyzes conserved mixed-symmetry tensor currents built from two massless spinor fields in Minkowski space, focusing on hooks described by Young diagrams with one row and one column. Using a bi-local generating function with auxiliary variables, the author identifies two families of currents: Flato–Frønsdal (FF) currents with critical dimension $\Delta_1=s+d-2$ and their on-shell improvements with $\Delta_2=s+d-1$, together with trace structures that correspond to Yukawa-like couplings; these currents realize the Flato–Frønsdal spectrum predicted by $o(d,2)$ representations of two spinor singletons. The paper derives conformally invariant conditions, two-point functions, and explicit examples (including a traceless $\{2,2\}$ current) and demonstrates how traces reduce to on-shell improvements, thereby supporting the $AdS_{d+1}/CFT_d$ correspondence for mixed-symmetry fields. Overall, it provides a concrete, field-theoretic realization of boundary currents dual to bulk mixed-symmetry AdS fields and outlines implications for higher-spin algebras and holographic dualities. The results open paths for computing correlation functions, exploring global symmetry algebras, and constructing explicit bulk/boundary realizations of hook-type fields in various dimensions.

Abstract

We present the full list of conserved currents built of two massless spinor fields in Minkowski space and their derivatives multiplied by Clifford algebra elements. The currents have particular mixed-symmetry type described by Young diagrams with one row and one column of arbitrary lengths and heights. Along with Yukawa-like totally antisymmetric currents the complete set of constructed currents exactly matches the spectrum of AdS mixed-symmetry fields arising in the generalized Flato-Fronsdal theorem for two spinor singletons. As a by-product, we formulate and study general properties of primary fields and conserved currents of mixed-symmetry type.

Mixed-symmetry tensor conserved currents and AdS/CFT correspondence

TL;DR

This work constructs and analyzes conserved mixed-symmetry tensor currents built from two massless spinor fields in Minkowski space, focusing on hooks described by Young diagrams with one row and one column. Using a bi-local generating function with auxiliary variables, the author identifies two families of currents: Flato–Frønsdal (FF) currents with critical dimension and their on-shell improvements with , together with trace structures that correspond to Yukawa-like couplings; these currents realize the Flato–Frønsdal spectrum predicted by representations of two spinor singletons. The paper derives conformally invariant conditions, two-point functions, and explicit examples (including a traceless current) and demonstrates how traces reduce to on-shell improvements, thereby supporting the correspondence for mixed-symmetry fields. Overall, it provides a concrete, field-theoretic realization of boundary currents dual to bulk mixed-symmetry AdS fields and outlines implications for higher-spin algebras and holographic dualities. The results open paths for computing correlation functions, exploring global symmetry algebras, and constructing explicit bulk/boundary realizations of hook-type fields in various dimensions.

Abstract

We present the full list of conserved currents built of two massless spinor fields in Minkowski space and their derivatives multiplied by Clifford algebra elements. The currents have particular mixed-symmetry type described by Young diagrams with one row and one column of arbitrary lengths and heights. Along with Yukawa-like totally antisymmetric currents the complete set of constructed currents exactly matches the spectrum of AdS mixed-symmetry fields arising in the generalized Flato-Fronsdal theorem for two spinor singletons. As a by-product, we formulate and study general properties of primary fields and conserved currents of mixed-symmetry type.

Paper Structure

This paper contains 12 sections, 6 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Review of the generalized Flato-Frø nsdal theorem
  3. Conformal primary fields of mixed-symmetry type
  4. Primary fields and conformally invariant conditions
  5. Two-point correlation functions
  6. Mixed-symmetry type conserved currents
  7. General formulation
  8. Generating function in auxiliary variables
  9. Flato-Frø nsdal conserved currents
  10. Traceless conserved currents: $s=2$, $p=2$ example
  11. Summary and discussion
  12. Acknowledgements.

Key Result

Lemma 3.1

Let a given $\{s,p\}$-type current satisfy both conservation conditions curcon and curcon2. The current represents the trivial current cohomology class iff it has the form $J_n(x|\xi_1) = \partial^m I_{nm}(x|\xi_1)$ and $J_n(x|\xi_2) = \partial^m K_{nm}(x|\xi_2)$, with where $P_{k|n}(x|\xi_1)$ and $P_{k|n}(x|\xi_2)$ are defined by relations Ps, while $P_n(x|\xi_3)$P3 satisfies

Theorems & Definitions (6)

  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Proposition 3.5
  • Proposition 3.6