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Osp(N|4) supermultiplets as conformal superfields on \partial AdS_4 and the generic form of N=2, D=3 gauge theories

D. Fabbri, P. Fre`, L. Gualtieri, P. Termonia

TL;DR

This work addresses the problem of connecting bulk Osp(2|4) KK multiplets in AdS_4 to boundary conformal superfields in a three-dimensional N=2 CFT framework. Its approach combines an algebraic translation via two five-gradings and a geometric construction of AdS_4 and its boundary using solvable cosets to map bulk states into boundary primaries, including explicit shortening conditions. The main contributions are (i) a detailed transcription of KK Osp(2|4) multiplets into boundary conformal superfields for the hypermultiplet, short vector, gravitino, graviton, and massless multiplets, (ii) a complete rheonomic derivation of a generic N=2, d=3 gauge theory with gauge, chiral multiplets, and superpotential couplings including CS and FI terms, and (iii) analysis of conditions for N=4 and N=8 supersymmetry enhancements in these three-dimensional theories. The results provide a robust framework to compare KK spectra with scaling dimensions of boundary operators for Sasakian AdS_4/CFT_3 setups, enabling systematic studies of M2-brane duals on AdS_4×X^7.

Abstract

In this paper we fill a necessary gap in order to realize the explicit comparison between the Kaluza Klein spectra of supergravity compactified on AdS_4 x X^7 and superconformal field theories living on the world volume of M2-branes. On the algebraic side we consider the superalgebra Osp(N|4) and we study the double interpretation of its irreducible representations either as supermultiplets of particle states in the bulk or as conformal superfields on the boundary. On the lagrangian field theory side we construct, using rheonomy rather than superfield techniques, the generic form of an N=2, d=3 gauge theory. Indeed the superconformal multiplets are supposed to be composite operators in a suitable gauge theory.

Osp(N|4) supermultiplets as conformal superfields on \partial AdS_4 and the generic form of N=2, D=3 gauge theories

TL;DR

This work addresses the problem of connecting bulk Osp(2|4) KK multiplets in AdS_4 to boundary conformal superfields in a three-dimensional N=2 CFT framework. Its approach combines an algebraic translation via two five-gradings and a geometric construction of AdS_4 and its boundary using solvable cosets to map bulk states into boundary primaries, including explicit shortening conditions. The main contributions are (i) a detailed transcription of KK Osp(2|4) multiplets into boundary conformal superfields for the hypermultiplet, short vector, gravitino, graviton, and massless multiplets, (ii) a complete rheonomic derivation of a generic N=2, d=3 gauge theory with gauge, chiral multiplets, and superpotential couplings including CS and FI terms, and (iii) analysis of conditions for N=4 and N=8 supersymmetry enhancements in these three-dimensional theories. The results provide a robust framework to compare KK spectra with scaling dimensions of boundary operators for Sasakian AdS_4/CFT_3 setups, enabling systematic studies of M2-brane duals on AdS_4×X^7.

Abstract

In this paper we fill a necessary gap in order to realize the explicit comparison between the Kaluza Klein spectra of supergravity compactified on AdS_4 x X^7 and superconformal field theories living on the world volume of M2-branes. On the algebraic side we consider the superalgebra Osp(N|4) and we study the double interpretation of its irreducible representations either as supermultiplets of particle states in the bulk or as conformal superfields on the boundary. On the lagrangian field theory side we construct, using rheonomy rather than superfield techniques, the generic form of an N=2, d=3 gauge theory. Indeed the superconformal multiplets are supposed to be composite operators in a suitable gauge theory.

Paper Structure

This paper contains 29 sections, 216 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The conceptual environment and our goals
  3. The specific problems addressed and solved in this paper
  4. Our paper is organized as follows:
  5. The $Osp({\cal N}\vert 4)$ superalgebra: definition, properties and notations
  6. Compact and non compact five gradings of the $Osp({\cal N}|4)$ superalgebra
  7. The lowest weight UIR.s as seen from the compact and non compact five--grading viewpoint
  8. $AdS_4$ and $\partial AdS_4$ as cosets and their Killing vectors
  9. $Osp(2\vert 4)$ superfields in the bulk and on the boundary
  10. Conformal $Osp(2 \vert 4)$ superfields: general discussion
  11. Matching the Kaluza Klein results for $Osp(2 \vert 4)$ supermultiplets with boundary conformal superfields
  12. Chiral superfields are the Hypermultiplets: the basic example
  13. Superfield description of the short vector multiplet
  14. Superfield description of the short gravitino multiplet
  15. Superfield description of the short graviton multiplet
  16. ...and 14 more sections

Figures (6)

  • Figure 1: The metric cone $C\left (X^{D-p-2}\right)$ is transverse to the stack of branes
  • Figure 2: Boundary superfields are obtained as limiting values of superfields in the bulk
  • Figure 3: Schematic representation of the root diagram of $Osp({\cal N}|4)$ in the $SO(2) \times SO(3)$ basis. The grading w.r.t. the energy $E$ is given on the right.
  • Figure 4: Schematic representation of the root diagram of $Osp({\cal N}|4)$ in the $SO(1,1)\times SO(1,2)$ basis. The grading w.r.t. the dilatation $D$ is given on the right.
  • Figure 5: The operator $U=\exp\{i\pi/\sqrt 2(E-D)\}$ rotates the Hilbert space of the physical states. It takes states labeled by the Casimirs ($E,\,s$) of the $SO(2)\times SO(3)\subset Osp({\cal N}|4)$ into states labeled by the Casimirs ($D,\,j$) of $SO(1,1)\times SO(1,2)$.
  • ...and 1 more figures

Theorems & Definitions (1)

  • Definition 4.1