[Paper Review] Gauge Model in D=3, N=5 Harmonic Superspace

Б. М. Зупник|ArXiv.org|Aug 29, 2007

Geophysics and Sensor Technology1 references3 citations

TL;DR

This paper constructs a non-abelian Chern-Simons-type action in D=3, N=5 harmonic superspace using SO(5)/U(1)×U(1) harmonics, introducing Grassmann-analytic gauge superfields that contain an infinite off-shell multiplet of bosonic and fermionic fields. The model exhibits full N=6 supersymmetry, with the component Lagrangian describing scale-invariant, nontrivial interactions of the gauge field $A_m$ with auxiliary and physical fields, where all auxiliary fields can be eliminated on-shell, yielding a finite-dimensional N=6 multiplet with nontrivial dynamics for $B_m$.

ABSTRACT

We construct the Grassmann-analytic gauge superfields in D=3, N=5 harmonic superspace using the SO(5)/U(1)xU(1) harmonics. These gauge N=5 superfields contain an infinite number of bosonic and fermionic fields arising from decompositions in harmonics and Grassmann coordinates. The bosonic sector of this supermultiplet includes the gauge field A_m, the additional nongauge vector field B_m, the scalar field S, two SO(5)-vector scalar fields and an infinite number of auxiliary fields with SO(5) indices. The nonabelian Chern-Simons-type action in the N=5 analytic harmonic superspace is constructed. This action is also invariant with respect to the sixth supersymmetry realized on the N=5 gauge superfields. The component Lagrangian describes the scale-invariant nontrivial interactions of the gauge Chern-Simons field A_m with B_m, S and other basic and auxiliary fields. All auxiliary fields can be excluded from this Lagrangian.

Motivation & Objective

• To construct a consistent gauge theory in D=3, N=5 harmonic superspace using SO(5)/U(1)×U(1) harmonics.
• To extend the Chern-Simons theory to higher N=5 supersymmetry in three dimensions, where no such formulation existed before.
• To demonstrate that the N=5 gauge superfields admit a sixth supersymmetry, closing the algebra on-shell.
• To derive a component Lagrangian that describes scale-invariant, nontrivial interactions among gauge, vector, scalar, and auxiliary fields.
• To show that all auxiliary fields with more than two SO(5) indices vanish on-shell, reducing the multiplet to a finite-dimensional N=6 structure.

Proposed method

• Utilizes D=3, N=5 harmonic superspace with SO(5)/U(1)×U(1) harmonics to define Grassmann-analytic superfields.
• Introduces three basic gauge superfields in analytic superspace, analogous to D=4,N=3 SYM but adapted to 3D.
• Constructs a Chern-Simons-type (CST) superfield action invariant under N=5 and sixth supersymmetry transformations.
• Derives the component Lagrangian via expansion in harmonics and Grassmann coordinates, identifying all physical and auxiliary fields.
• Applies on-shell constraints to eliminate auxiliary fields with more than two SO(5) indices, reducing the multiplet to N=6.
• Uses dimensional analysis and scale invariance to verify the consistency of the integration measure and action.

Experimental results

Research questions

• RQ1 Can a non-abelian Chern-Simons theory be consistently formulated in D=3, N=5 harmonic superspace?
• RQ2 How do the gauge superfields in this model realize an additional (sixth) supersymmetry beyond N=5?
• RQ3 What is the structure of the off-shell multiplet, and which auxiliary fields can be eliminated on-shell?
• RQ4 Does the component Lagrangian describe scale-invariant, nontrivial interactions among the gauge and auxiliary fields?
• RQ5 Can the infinite off-shell multiplet be reduced to a finite-dimensional on-shell N=6 multiplet?

Key findings

• The Chern-Simons-type action is invariant under both N=5 and a sixth supersymmetry, extending the supersymmetry algebra beyond N=5.
• The component Lagrangian includes a nontrivial interaction between the gauge field $A_m$ and the vector field $B_m$, with $A_m$ in pure gauge and $B_m$ having a nontrivial solution on-shell.
• All auxiliary fields with more than two SO(5) indices vanish on-shell, reducing the multiplet to a finite-dimensional N=6 structure.
• The scalar field $S$ and fermionic field $ u^a_eta$ satisfy free massless equations of motion, while $A_m$ is pure gauge in the abelian case.
• The on-shell auxiliary fields are expressed as total derivatives of basic real fields, such as $F^a = -\frac{1}{2}\partial^m\partial_m v^a$ and $G^a_m = \partial_m S^a$.
• The model realizes scale invariance, with the integration measure and action both scale-invariant, supporting the consistency of the construction.