F. Deformed gauge structure

If we denote by $ \Omega _{1}^{\alpha _{1}}$ and $ \Omega _{2}^{\alpha _{1}}$ two independent sets of gauge parameters,

$\displaystyle \Omega _{1}^{\alpha _{1}}$ $\displaystyle \equiv$ $\displaystyle \left( \epsilon ^{(1)\mu \nu },\epsilon
^{(1)},\epsilon ^{(1)\mu ...
...\mu \nu \rho
\lambda },\theta _{\mu \nu }^{(1)},\chi _{\mu \nu }^{(1)}\right) ,$ (399)
$\displaystyle \Omega _{2}^{\alpha _{1}}$ $\displaystyle \equiv$ $\displaystyle \left( \epsilon ^{(2)\mu \nu },\epsilon
^{(2)},\epsilon ^{(2)\mu ...
...\mu \nu \rho
\lambda },\theta _{\mu \nu }^{(2)},\chi _{\mu \nu }^{(2)}\right) ,$ (400)

then the concrete form of the commutators among the deformed gauge transformations of the fields associated with (285) and (286) (and generically written as in (162)) read as

$\displaystyle \left[ \bar{\delta}_{\Omega _{1}},\bar{\delta}_{\Omega _{2}}\right] \varphi =0,$ (401)


$\displaystyle \left[ \bar{\delta}_{\Omega _{1}},\bar{\delta}_{\Omega _{2}}\right] H^{\mu }$ $\displaystyle =$ $\displaystyle \bar{\delta}_{\Omega }H^{\mu }-2\frac{\delta S^{\mathrm{L}}}{\del...
...{L}}
}{\delta B^{\nu \rho }}\frac{d\epsilon ^{\mu \nu \rho }}{d\varphi } \notag$ (402)
    $\displaystyle +2\frac{\delta S^{\mathrm{L}}}{\delta \phi _{\mu \nu }}\frac{d\xi...
...{\delta K^{\nu \rho \lambda }}\frac{
d\xi ^{\mu \nu \rho \lambda }}{d\varphi },$ (403)

$\displaystyle \left[ \bar{\delta}_{\Omega _{1}},\bar{\delta}_{\Omega _{2}}\right] V_{\mu }= \bar{\delta}_{\Omega }V_{\mu },$ (404)

$\displaystyle \left[ \bar{\delta}_{\Omega _{1}},\bar{\delta}_{\Omega _{2}}\righ...
...S^{\mathrm{L}}}{ \delta H^{\rho }}\frac{d\epsilon ^{\mu \nu \rho }}{d\varphi },$ (405)

$\displaystyle \left[ \bar{\delta}_{\Omega _{1}},\bar{\delta}_{\Omega _{2}}\righ...
...\frac{\delta S^{\mathrm{L} }}{\delta H^{[\mu }}\frac{d\xi _{\nu ]}}{d\varphi },$ (406)

$\displaystyle \left[ \bar{\delta}_{\Omega _{1}},\bar{\delta}_{\Omega _{2}}\righ...
...thrm{L }}}{\delta H^{\lambda }}\frac{d\xi ^{\mu \nu \rho \lambda }}{d\varphi },$ (407)

$\displaystyle \left[ \bar{\delta}_{\Omega _{1}},\bar{\delta}_{\Omega _{2}}\right] t_{\mu \nu \vert\alpha }=0.$ (408)

The gauge parameters from the right-hand side of the above formulas are defined through

$\displaystyle \Omega ^{\alpha _{1}}=\left( \epsilon ^{\mu \nu },\epsilon =0,\ep...
...},\xi ^{\mu \nu \rho \lambda },\theta _{\mu \nu }=0,\chi _{\mu \nu }=0\right) ,$ (409)

where
$\displaystyle \epsilon ^{\mu \nu }$ $\displaystyle =$ $\displaystyle \lambda \left\{ -\frac{dW_{1}}{d\varphi }\left(
\epsilon ^{(1)}\epsilon ^{(2)\mu \nu }-\epsilon ^{(2)}\epsilon ^{(1)\mu \nu
}\right) \right. \notag$ (410)
    $\displaystyle +6\frac{dW_{3}}{d\varphi }\left[ \phi _{\rho \lambda }\left( \eps...
... \lambda }-\epsilon ^{(2)}\xi ^{(1)\mu \nu \rho
\lambda }\right) \right. \notag$ (411)
    $\displaystyle \left. +\tfrac{1}{2}K^{\mu \nu \rho }\left( \epsilon ^{(1)}\xi _{...
...da }-\xi _{\lambda }^{(2)}\xi
^{(1)\mu \nu \rho \lambda }\right) \right] \notag$ (412)
    $\displaystyle -3\frac{dW_{2}}{d\varphi }\left( \xi _{\rho }^{(1)}\epsilon ^{(2)\mu \nu
\rho }-\xi _{\rho }^{(2)}\epsilon ^{(1)\mu \nu \rho }\right) \notag$ (413)
    $\displaystyle +3\frac{dW_{6}}{d\varphi }\varepsilon _{\rho \alpha \beta \gamma ...
...\epsilon ^{(2)\mu \nu \rho }\xi ^{(1)\alpha \beta \gamma \delta }\right)
\notag$ (414)
    $\displaystyle +6\frac{dW_{4}}{d\varphi }\left[ \varepsilon _{\rho \alpha \beta ...
...ta }-\epsilon ^{(2)}\xi ^{(1)\alpha \beta \gamma \delta }\right) \right.
\notag$ (415)
    $\displaystyle \left. +\tfrac{1}{6}\varepsilon ^{\mu \nu \rho \lambda \sigma }\v...
...pha ^{\prime }\beta ^{\prime }\gamma
^{\prime }\delta ^{\prime }}\right] \notag$ (416)
    $\displaystyle \left. -\tfrac{1}{2}\varepsilon ^{\mu \nu \rho \lambda \sigma }\f...
...}\right) -2V_{\rho }\xi _{\lambda
}^{(1)}\xi _{\sigma }^{(2)}\right] \right\} ,$ (417)


$\displaystyle \epsilon ^{\mu \nu \rho }$ $\displaystyle =$ $\displaystyle -8\lambda \left[ W_{3}\left( \xi _{\lambda
}^{(1)}\xi ^{(2)\mu \n...
...da }-\xi _{\lambda }^{(2)}\xi ^{(1)\mu \nu
\rho \lambda }\right) \right. \notag$ (418)
    $\displaystyle -\tfrac{1}{12}\varepsilon ^{\mu \nu \rho \lambda \sigma }\left(
W...
...pha ^{\prime }\beta ^{\prime
}\gamma ^{\prime }\delta ^{\prime }}\right. \notag$ (419)
    $\displaystyle \left. \left. +W_{5}\xi _{\lambda }^{(1)}\xi _{\sigma }^{(2)}\right)
\right] ,$ (420)


$\displaystyle \xi _{\mu }$ $\displaystyle =$ $\displaystyle -3\lambda \left[ W_{3}\left( \epsilon ^{(1)}\xi _{\mu
}^{(2)}-\epsilon ^{(2)}\xi _{\mu }^{(1)}\right) \right. \notag$ (421)
    $\displaystyle \left. +2W_{4}\varepsilon _{\mu \nu \rho \lambda \sigma }\left( \...
...bda \sigma }-\epsilon ^{(2)}\xi ^{(1)\nu \rho
\lambda \sigma }\right) \right] ,$ (422)


$\displaystyle \xi ^{\mu \nu \rho \lambda }$ $\displaystyle =$ $\displaystyle 3\lambda \left[ W_{3}\left( \epsilon
^{(1)}\xi ^{(2)\mu \nu \rho \lambda }-\epsilon ^{(2)}\xi ^{(1)\mu \nu \rho
\lambda }\right) \right. \notag$ (423)
    $\displaystyle \left. -\tfrac{1}{12}W_{4}\varepsilon ^{\mu \nu \rho \lambda \sig...
...^{(1)}\xi _{\sigma }^{(2)}-\epsilon ^{(2)}\xi _{\sigma
}^{(1)}\right) \right] .$ (424)

In addition, we made the notations

$\displaystyle \theta ^{(i)}=\sigma _{\alpha \beta }\theta ^{(i)\alpha \beta },\qquad i= \overline{1,2}.$ (425)

Related to the first-order reducibility, the transformations (163) are given by

$\displaystyle \epsilon ^{\mu \nu }\left( \bar{\Omega}\right)$ $\displaystyle =$ $\displaystyle -3D_{\rho }\bar{\epsilon}
^{\mu \nu \rho }-\lambda \frac{dW_{2}}{...
...xi}
-6\phi _{\rho \lambda }\bar{\epsilon}^{\mu \nu \rho \lambda }\right) \notag$ (426)
    $\displaystyle +3\lambda \frac{dW_{3}}{d\varphi }V_{\rho }\left( K^{\mu \nu \rho...
...10\phi _{\lambda \sigma }\bar{\xi}^{\mu \nu \rho \lambda \sigma }\right)
\notag$ (427)
    $\displaystyle -6\lambda \varepsilon _{\alpha \beta \gamma \delta \varepsilon }\...
...sigma }\frac{dW_{5}}{d\varphi }V_{\rho }\phi _{\lambda \sigma }\bar{\xi}
\notag$ (428)
    $\displaystyle +\lambda \frac{dW_{6}}{d\varphi }\left( \varepsilon _{\alpha \bet...
...lpha \beta \gamma
\delta }\bar{\epsilon}^{\alpha \beta \gamma \delta }\right) ,$ (429)

$\displaystyle \epsilon \left( \bar{\Omega}\right) =2\lambda \left( W_{2}\bar{\x...
...\varepsilon }W_{6}\bar{\xi} ^{\alpha \beta \gamma \delta \varepsilon }\right) ,$ (430)


$\displaystyle \epsilon ^{\mu \nu \rho }\left( \bar{\Omega}\right)$ $\displaystyle =$ $\displaystyle 4\partial _{\lambda }
\bar{\epsilon}^{\mu \nu \rho \lambda }+2\la...
...bda W_{3}\phi _{\lambda \sigma }\bar{\xi}^{\mu \nu \rho \lambda
\sigma } \notag$ (431)
    $\displaystyle +2\lambda K^{\mu \nu \rho }\left( W_{3}\bar{\xi}-2\varepsilon _{\...
...epsilon }W_{4}\bar{\xi}^{\alpha \beta \gamma \delta
\varepsilon }\right) \notag$ (432)
    $\displaystyle -\tfrac{\lambda }{3}\varepsilon ^{\mu \nu \rho \lambda \sigma }W_{5}\phi
_{\lambda \sigma }\bar{\xi},$ (433)

$\displaystyle \xi _{\mu }\left( \bar{\Omega}\right) =D_{\mu }^{(-)}\bar{\xi}+6\...
... _{\mu \nu \rho \lambda \sigma }W_{6}\bar{\epsilon}^{\nu \rho \lambda \sigma },$ (434)

$\displaystyle \xi ^{\mu \nu \rho \lambda }\left( \bar{\Omega}\right) =-5D_{\sig...
...ambda }{4}\varepsilon ^{\mu \nu \rho \lambda \sigma }W_{5}V_{\sigma }\bar{\xi},$ (435)

$\displaystyle \theta _{\mu \nu }\left( \bar{\Omega}\right) =3\partial _{(\mu }\...
...delta \varepsilon }\bar{\xi}^{\alpha \beta \gamma \delta \varepsilon }\right) ,$ (436)

$\displaystyle \chi _{\mu \nu }\left( \bar{\Omega}\right) =\partial _{\lbrack \mu }\bar{ \theta}_{\nu ]},$ (437)

while the first-order reducibility relations (164) read as

$\displaystyle \bar{\delta}_{\Omega \left( \bar{\Omega}\right) }\varphi =0,$ (438)


$\displaystyle \bar{\delta}_{\Omega \left( \bar{\Omega}\right) }H^{\mu }$ $\displaystyle =$ $\displaystyle \lambda \frac{
\delta S^{\mathrm{L}}}{\delta H^{\nu }}\left\{ 6V_...
... \rho
\lambda \sigma }-K^{\mu \nu \rho }\bar{\xi}\right) \right. \right. \notag$ (439)
    $\displaystyle \left. -2\varepsilon _{\alpha \beta \gamma \delta \varepsilon }\f...
...
\frac{d^{2}W_{5}}{d\varphi ^{2}}\phi _{\lambda \sigma }\bar{\xi}\right]
\notag$ (440)
    $\displaystyle +2\frac{d^{2}W_{6}}{d\varphi ^{2}}\left( 3\varepsilon _{\rho \alp...
...}B^{\mu \nu }\bar{\xi}
^{\alpha \beta \gamma \delta \varepsilon }\right) \notag$ (441)
    $\displaystyle \left. +2\frac{d^{2}W_{2}}{d\varphi ^{2}}\left( 6\phi _{\rho \lam...
...
\epsilon}^{\mu \nu \rho \lambda }-B^{\mu \nu }\bar{\xi}\right) \right\}
\notag$ (442)
    $\displaystyle +6\lambda \frac{\delta S^{\mathrm{L}}}{\delta B^{\nu \rho }}\left...
...{\mu \nu \rho \lambda \sigma
}-K^{\mu \nu \rho }\bar{\xi}\right) \right. \notag$ (443)
    $\displaystyle \left. -2\varepsilon _{\alpha \beta \gamma \delta \varepsilon }\f...
...\sigma }\frac{
dW_{5}}{d\varphi }\phi _{\lambda \sigma }\bar{\xi}\right] \notag$ (444)
    $\displaystyle -2\lambda \frac{\delta S^{\mathrm{L}}}{\delta V_{\mu }}\left( \fr...
...6}}{d\varphi }\bar{\xi}^{\alpha \beta \gamma \delta \varepsilon
}\right) \notag$ (445)
    $\displaystyle +\lambda \frac{\delta S^{\mathrm{L}}}{\delta K^{\nu \rho \lambda ...
...u \rho \lambda \sigma }\frac{dW_{5}}{d\varphi }
\bar{\xi}\right) \right. \notag$ (446)
    $\displaystyle \left. +12\frac{dW_{2}}{d\varphi }\bar{\epsilon}^{\mu \nu \rho \l...
...c{dW_{6}}{d\varphi }
\bar{\epsilon}^{\alpha \beta \gamma \delta }\right. \notag$ (447)
    $\displaystyle \left. +V_{\nu }\left( \frac{dW_{3}}{d\varphi }\bar{\xi}-2\vareps...
...d\varphi }\bar{\xi}
^{\alpha \beta \gamma \delta \varepsilon }\right) \right] ,$ (448)

$\displaystyle \bar{\delta}_{\Omega \left( \bar{\Omega}\right) }V_{\mu }=2\lambd...
...dW_{6}}{ d\varphi }\bar{\xi}^{\alpha \beta \gamma \delta \varepsilon }\right) ,$ (449)


$\displaystyle \bar{\delta}_{\Omega \left( \bar{\Omega}\right) }B^{\mu \nu }$ $\displaystyle =$ $\displaystyle 6\lambda
\frac{\delta S^{\mathrm{L}}}{\delta H^{\rho }}\left[ -\f...
... }\phi _{\lambda
\sigma }\bar{\xi}^{\mu \nu \rho \lambda \sigma }\right. \notag$ (450)
    $\displaystyle \left. -K^{\mu \nu \rho }\left( \frac{dW_{3}}{d\varphi }\bar{\xi}...
... \sigma }\frac{dW_{5}}{d\varphi }\phi
_{\lambda \sigma }\bar{\xi}\right] \notag$ (451)
    $\displaystyle +\lambda \frac{\delta S^{\mathrm{L}}}{\delta K^{\rho \lambda \sig...
...sigma }+\varepsilon ^{\mu \nu
\rho \lambda \sigma }W_{5}\bar{\xi}\right) \notag$ (452)
    $\displaystyle +6\lambda \frac{\delta S^{\mathrm{L}}}{\delta \phi _{\mu \nu }}\l...
...\varepsilon }W_{4}\bar{
\xi}^{\alpha \beta \gamma \delta \varepsilon }\right) ,$ (453)


$\displaystyle \bar{\delta}_{\Omega \left( \bar{\Omega}\right) }\phi _{\mu \nu }$ $\displaystyle =$ $\displaystyle -3\lambda \frac{\delta S^{\mathrm{L}}}{\delta H^{[\mu }}V_{\nu ]}...
...4}}{d\varphi }\bar{\xi}^{\alpha \beta \gamma \delta
\varepsilon }\right) \notag$ (454)
    $\displaystyle -6\lambda \frac{\delta S^{\mathrm{L}}}{\delta B^{\mu \nu }}\left(...
...psilon }W_{4}\bar{
\xi}^{\alpha \beta \gamma \delta \varepsilon }\right) \notag$ (455)
    $\displaystyle -3\lambda \frac{dW_{6}}{d\varphi }\frac{\delta S^{\mathrm{L}}}{\d...
...\nu ]\alpha \beta \gamma \delta }\bar{\epsilon}
^{\alpha \beta \gamma \delta },$ (456)


$\displaystyle \bar{\delta}_{\Omega \left( \bar{\Omega}\right) }K^{\mu \nu \rho }$ $\displaystyle =$ $\displaystyle \lambda \frac{\delta S^{\mathrm{L}}}{\delta H^{\lambda }}\left[
-...
...u \rho \lambda \sigma }\frac{dW_{5}}{d\varphi }
\bar{\xi}\right) \right. \notag$ (457)
    $\displaystyle \left. +12\frac{dW_{2}}{d\varphi }\bar{\epsilon}^{\mu \nu \rho \l...
...bda \sigma }+\varepsilon ^{\mu \nu
\rho \lambda \sigma }W_{5}\bar{\xi}\right) ,$ (458)

$\displaystyle \bar{\delta}_{\Omega \left( \bar{\Omega}\right) }t_{\mu \nu \vert\alpha }=0.$ (459)

Regarding the second-order reducibility, the transformations (165) take the concrete form

$\displaystyle \bar{\epsilon}^{\mu \nu \rho }\left( \check{\Omega}\right) =4D_{\...
...u \nu \rho }\check{\epsilon}^{\alpha \beta \gamma \delta \varepsilon }\right) ,$ (460)
$\displaystyle \bar{\epsilon}^{\mu \nu \rho \lambda }\left( \check{\Omega}\right...
...a \varepsilon }W_{6}\check{\epsilon}^{\alpha \beta \gamma \delta \varepsilon },$ (461)
$\displaystyle \bar{\xi}^{\mu \nu \rho \lambda \sigma }\left( \check{\Omega}\rig...
...ho \lambda \sigma },\qquad \bar{ \theta}_{\mu }\left( \check{\Omega}\right) =0,$ (462)

such that the second-order reducibility relations (166) become
$\displaystyle \epsilon ^{\mu \nu }\left( \bar{\Omega}\left( \check{\Omega}\right) \right)$ $\displaystyle =$ $\displaystyle 3\lambda \frac{\delta S^{\mathrm{L}}}{\delta H^{\rho }}\left( 4\f...
...{\lambda \sigma }\check{\epsilon}
^{\mu \nu \rho \lambda \sigma }\right. \notag$ (463)
    $\displaystyle \left. +\varepsilon _{\alpha \beta \gamma \delta \varepsilon }\fr...
...m{L}}}{\delta B^{\rho \lambda }}\check{\epsilon}^{\mu \nu
\rho \lambda } \notag$ (464)
    $\displaystyle +30\lambda \frac{dW_{2}}{d\varphi }\frac{\delta S^{\mathrm{L}}}{\...
...a \phi _{\mu \nu }}\check{
\epsilon}^{\alpha \beta \gamma \delta \varepsilon },$ (465)

$\displaystyle \epsilon \left( \bar{\Omega}\left( \check{\Omega}\right) \right) =0,$ (466)

$\displaystyle \epsilon ^{\mu \nu \rho }\left( \bar{\Omega}\left( \check{\Omega}...
... S^{\mathrm{L}}}{ \delta H^{\lambda }}\check{\epsilon}^{\mu \nu \rho \lambda },$ (467)

$\displaystyle \xi _{\mu }\left( \bar{\Omega}\left( \check{\Omega}\right) \right...
...}}{\delta H^{\mu }}\check{\epsilon} ^{\alpha \beta \gamma \delta \varepsilon },$ (468)

$\displaystyle \xi ^{\mu \nu \rho \lambda }\left( \bar{\Omega}\left( \check{\Ome...
...athrm{L}}}{ \delta H^{\sigma }}\check{\epsilon}^{\mu \nu \rho \lambda \sigma },$ (469)

$\displaystyle \theta _{\mu \nu }\left( \bar{\Omega}\left( \check{\Omega}\right)...
...uad \chi _{\mu \nu }\left( \bar{\Omega}\left( \check{\Omega}\right) \right) =0.$ (470)

Finally, we investigate the third-order reducibility, for which the transformations (167) can be written as

$\displaystyle \check{\epsilon}^{\mu \nu \rho \lambda }\left( \hat{\Omega}\right...
...at{\Omega}\right) =2\lambda W_{1}\hat{\epsilon}^{\mu \nu \rho \lambda \sigma },$ (471)

while that the third-order reducibility relations (168) are listed below

$\displaystyle \bar{\epsilon}^{\mu \nu \rho }\left( \check{\Omega}\left( \hat{\O...
...\frac{dW_{1}}{d\varphi }\right) \hat{ \epsilon}^{\mu \nu \rho \lambda \sigma },$ (472)
$\displaystyle \bar{\epsilon}^{\mu \nu \rho \lambda }\left( \check{\Omega}\left(...
...sigma }} \frac{dW_{1}}{d\varphi }\hat{\epsilon}^{\mu \nu \rho \lambda \sigma },$ (473)
$\displaystyle \bar{\xi}\left( \check{\Omega}\left( \hat{\Omega}\right) \right) ...
...\bar{\theta}_{\mu }\left( \check{\Omega} \left( \hat{\Omega}\right) \right) =0.$ (474)

Ashkbiz Danehkar
2018-03-26