Symmetries of supergravity backgrounds and supersymmetric field theory
Sergei M. Kuzenko, Emmanouil S.N. Raptakis
Abstract
A bstract In four spacetime dimensions, all $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 1 supergravity-matter systems can be formulated in the so-called U(1) superspace proposed by Howe in 1981. This paper is devoted to the study of those geometric structures which characterise a background U(1) superspace and are important in the context of supersymmetric field theory in curved space. We introduce (conformal) Killing tensor superfields $$ {\mathrm{\ell}}_{\left({\alpha}_1\dots {\alpha}_m\right)\left({\overset{\cdot }{\alpha}}_1\dots {\overset{\cdot }{\alpha}}_n\right)} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mrow> <mml:mfenced> <mml:mrow> <mml:msub> <mml:mi>α</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>…</mml:mo> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:mrow> </mml:mfenced> <mml:mfenced> <mml:mrow> <mml:msub> <mml:mover> <mml:mi>α</mml:mi> <mml:mo>⋅</mml:mo> </mml:mover> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>…</mml:mo> <mml:msub> <mml:mover> <mml:mi>α</mml:mi> <mml:mo>⋅</mml:mo> </mml:mover> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:msub> </mml:math> , with m and n non-negative integers, m + n > 0, and elaborate on their significance in the following cases: (i) m = n = 1; (ii) m − 1 = n = 0; and (iii) m = n > 1. The (conformal) Killing vector superfields $$ {\mathrm{\ell}}_{\alpha \overset{\cdot }{\alpha }} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mrow> <mml:mi>α</mml:mi> <mml:mover> <mml:mi>α</mml:mi> <mml:mo>⋅</mml:mo> </mml:mover> </mml:mrow> </mml:msub> </mml:math> generate the (conformal) isometries of curved superspace, which are symmetries of every (conformal) supersymmetric field theory. The (conformal) Killing spinor superfields ℓ α generate extended (conformal) supersymmetry transformations. The (conformal) Killing tensor superfields with m = n > 1 prove to generate all higher symmetries of the (massless) massive Wess-Zumino operator.