Auxiliary tensor fields for Sp(2, ℝ) self-duality

Evgeny A Ivanov,Olaf Lechtenfeld,Boris M Zupnik

doi:10.1007/jhep03(2015)123

Evgeny A Ivanov, Olaf Lechtenfeld + Show 1 more

Open Access

https://doi.org/10.1007/jhep03(2015)123

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Abstract

The coset Sp(2, ℝ)/U(1) is parametrized by two real scalar fields. We generalize the formalism of auxiliary tensor (bispinor) fields in U(1) self-dual nonlinear models of abelian gauge fields to the case of Sp(2, ℝ) self-duality. In this new formulation, Sp(2, ℝ) duality of the nonlinear scalar-gauge equations of motion is equivalent to an Sp(2, ℝ) invariance of the auxiliary interaction. We derive this result in two different ways, aiming at its further application to supersymmetric theories. We also consider an extension to interactions with higher derivatives.

Highlights

Solving the algebraic equations for Vαβ as Vαβ = Vαβ(F ), we regain the self-dual nonlinear Lagrangian L(F ) in the standard representation
We generalize the formalism of auxiliary tensor fields to Sp(2, R) duality invariant theories
In the previous section we started from the renowned GR action (1.1), (2.12) and picked up the nonlinearly transforming bispinor auxiliary fields Vαβ so as to construct the natural generalization of the extended formulation of the U(1) self-dual electrodynamics to the case of the Sp(2, R) self-dual systems with the coset scalar fields

Summary

Legendre transformation for the nonlinear auxiliary fields

The Legendre transformation for the auxiliary field formulation of the U(1) self-dual electrodynamics was discussed in [11, 12]. The transformation parameter ρ = cS2 in the NR representation (3.1) depends on the scalar field, so in the case under consideration we need to properly Sp(2, R) covariantize the space-time derivatives of the auxiliary fields involved This can be done following the standard routine of the nonlinear realizations [21, 22]. The Sp(2, R)-covariant local equations of motion for the auxiliary fields in this case contain the Euler-Lagrange derivative Solving this equation (e.g., by recursions), we obtain the Sp(2, R) self-dual Lagrangian in the initial (F, S) representation. In the previous section we started from the renowned GR action (1.1), (2.12) and picked up the nonlinearly transforming bispinor auxiliary fields Vαβ so as to construct the natural generalization of the extended formulation of the U(1) self-dual electrodynamics to the case of the Sp(2, R) self-dual systems with the coset scalar fields.

Auxiliary fields

More on the interplay between the linear and nonlinear formulations

Conclusions