NLSM and Extended Superspace

  • Sergei V. Ketov
Chapter
Part of the Texts and Monographs in Physics book series (TMP)

Abstract

Given a 2d NLSM action of the form (Chap. 2)
$$ {S_{bosonic}}\left[ \phi \right] = \frac{1}{2}\int {{d^2}x} \left\{ {{g_{ij}}\left( \phi \right){\partial ^\mu }{\phi ^i}{\partial _\mu }{\phi ^j} + {\varepsilon ^{\mu \upsilon }}{h_{ij}}\left( \phi \right){\partial _\mu }{\phi ^i}{\partial _\upsilon }{\phi ^j}} \right\},$$
(0.1)
it can be supersymmetrized by using the Noether (trial and error) procedure. The Noether procedure amounts to defining fermionic partners λi of øi by the supersymmetry (susy) transformation law, \(\delta {\phi ^i} = \bar \varepsilon {\lambda ^i}\), where the susy parameter ε and the fields λi(x) may be either chiral or non-chiral, and demanding the action (0.1) to be invariant under susy by modifying it with fermionic-dependent terms. For dimensional reasons, the extra fermionic terms can be either quadratic or quartic in A. The susy algebra requires the susy transformations to be closed to translations on all fields subject to their equations of motion. The outcome of the (non-chiral) Noether (1,1) supersymmetrization of the action (0.1) reads [149, 200]
$${S_{susy}}\left[ {\phi ,\lambda } \right] = {S_{bosonic}} + \frac{1}{2}\int {{d^2}x} \left\{ {i{g_{ij}}\bar \lambda _ + ^i{\gamma ^\mu }{{\left( {{D_\mu }{\lambda _ + }} \right)}^j} + i{g_{ij}}{{\bar \lambda }^i}_ - \gamma {{\left( {{D_\mu }{\lambda _ - }} \right)}^j} + \frac{1}{4}R_{ijkl}^ + \left( {\bar \lambda _ + ^i{\gamma ^\mu }\lambda _ + ^j} \right)\left( {\bar \lambda _ - ^k{\gamma _\mu }\lambda _ - ^l} \right)} \right\}$$
(0.2)
, where we have introduced the notation
$$ {\left( {{D_\mu }{\lambda _ \pm }} \right)^k} = {\partial _\mu }\lambda _ \pm ^k + \Gamma _{ \pm jl}^k\lambda _ \pm ^j{\partial _\mu }{\phi ^l},\quad \lambda _ \pm ^k = \frac{1}{2}\left( {1 \pm {\gamma _3}} \right){\lambda ^k},$$
(0.3)
and
$$\Gamma _{ \pm jk}^i = \Gamma _{jk}^i \pm H_{jk}^i,\quad {H_{ijk}} = \frac{3}{2}{h_{\left[ {ij,k} \right]}},\quad R_{ijkl}^ \pm = {R_{ijkl}}\left( {{\Gamma _ \pm }} \right) $$
(0.4)

Keywords

Manifold Covariance Eter 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Sergei V. Ketov
    • 1
  1. 1.Institut für Theoretische PhysikUniversität HannoverHannoverGermany

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