F. Deformed gauge structure

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

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

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

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

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

$$+2\frac{\delta S^{\mathrm{L}}}{\delta \phi _{\mu \nu }}\frac{d\xi... ...{\delta K^{\nu \rho \lambda }}\frac{ d\xi ^{\mu \nu \rho \lambda }}{d\varphi },$$ (403)

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

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

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

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

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

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

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

where

$$\epsilon ^{\mu \nu } = \lambda \left\{ -\frac{dW_{1}}{d\varphi }\left( \epsilon ^{(1)}\epsilon ^{(2)\mu \nu }-\epsilon ^{(2)}\epsilon ^{(1)\mu \nu }\right) \right. \notag$$ (410)

$$+6\frac{dW_{3}}{d\varphi }\left[ \phi _{\rho \lambda }\left( \eps... ... \lambda }-\epsilon ^{(2)}\xi ^{(1)\mu \nu \rho \lambda }\right) \right. \notag$$ (411)

$$\left. +\tfrac{1}{2}K^{\mu \nu \rho }\left( \epsilon ^{(1)}\xi _{... ...da }-\xi _{\lambda }^{(2)}\xi ^{(1)\mu \nu \rho \lambda }\right) \right] \notag$$ (412)

$$-3\frac{dW_{2}}{d\varphi }\left( \xi _{\rho }^{(1)}\epsilon ^{(2)\mu \nu \rho }-\xi _{\rho }^{(2)}\epsilon ^{(1)\mu \nu \rho }\right) \notag$$ (413)

$$+3\frac{dW_{6}}{d\varphi }\varepsilon _{\rho \alpha \beta \gamma ... ...\epsilon ^{(2)\mu \nu \rho }\xi ^{(1)\alpha \beta \gamma \delta }\right) \notag$$ (414)

$$+6\frac{dW_{4}}{d\varphi }\left[ \varepsilon _{\rho \alpha \beta ... ...ta }-\epsilon ^{(2)}\xi ^{(1)\alpha \beta \gamma \delta }\right) \right. \notag$$ (415)

$$\left. +\tfrac{1}{6}\varepsilon ^{\mu \nu \rho \lambda \sigma }\v... ...pha ^{\prime }\beta ^{\prime }\gamma ^{\prime }\delta ^{\prime }}\right] \notag$$ (416)

$$\left. -\tfrac{1}{2}\varepsilon ^{\mu \nu \rho \lambda \sigma }\f... ...}\right) -2V_{\rho }\xi _{\lambda }^{(1)}\xi _{\sigma }^{(2)}\right] \right\} ,$$ (417)

$$\epsilon ^{\mu \nu \rho } = -8\lambda \left[ W_{3}\left( \xi _{\lambda }^{(1)}\xi ^{(2)\mu \n... ...da }-\xi _{\lambda }^{(2)}\xi ^{(1)\mu \nu \rho \lambda }\right) \right. \notag$$ (418)

$$-\tfrac{1}{12}\varepsilon ^{\mu \nu \rho \lambda \sigma }\left( W... ...pha ^{\prime }\beta ^{\prime }\gamma ^{\prime }\delta ^{\prime }}\right. \notag$$ (419)

$$\left. \left. +W_{5}\xi _{\lambda }^{(1)}\xi _{\sigma }^{(2)}\right) \right] ,$$ (420)

$$\xi _{\mu } = -3\lambda \left[ W_{3}\left( \epsilon ^{(1)}\xi _{\mu }^{(2)}-\epsilon ^{(2)}\xi _{\mu }^{(1)}\right) \right. \notag$$ (421)

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

$$\xi ^{\mu \nu \rho \lambda } = 3\lambda \left[ W_{3}\left( \epsilon ^{(1)}\xi ^{(2)\mu \nu \rho \lambda }-\epsilon ^{(2)}\xi ^{(1)\mu \nu \rho \lambda }\right) \right. \notag$$ (423)

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

In addition, we made the notations

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

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

$$\epsilon ^{\mu \nu }\left( \bar{\Omega}\right) = -3D_{\rho }\bar{\epsilon} ^{\mu \nu \rho }-\lambda \frac{dW_{2}}{... ...xi} -6\phi _{\rho \lambda }\bar{\epsilon}^{\mu \nu \rho \lambda }\right) \notag$$ (426)

$$+3\lambda \frac{dW_{3}}{d\varphi }V_{\rho }\left( K^{\mu \nu \rho... ...10\phi _{\lambda \sigma }\bar{\xi}^{\mu \nu \rho \lambda \sigma }\right) \notag$$ (427)

$$-6\lambda \varepsilon _{\alpha \beta \gamma \delta \varepsilon }\... ...sigma }\frac{dW_{5}}{d\varphi }V_{\rho }\phi _{\lambda \sigma }\bar{\xi} \notag$$ (428)

$$+\lambda \frac{dW_{6}}{d\varphi }\left( \varepsilon _{\alpha \bet... ...lpha \beta \gamma \delta }\bar{\epsilon}^{\alpha \beta \gamma \delta }\right) ,$$ (429)

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

$$\epsilon ^{\mu \nu \rho }\left( \bar{\Omega}\right) = 4\partial _{\lambda } \bar{\epsilon}^{\mu \nu \rho \lambda }+2\la... ...bda W_{3}\phi _{\lambda \sigma }\bar{\xi}^{\mu \nu \rho \lambda \sigma } \notag$$ (431)

$$+2\lambda K^{\mu \nu \rho }\left( W_{3}\bar{\xi}-2\varepsilon _{\... ...epsilon }W_{4}\bar{\xi}^{\alpha \beta \gamma \delta \varepsilon }\right) \notag$$ (432)

$$-\tfrac{\lambda }{3}\varepsilon ^{\mu \nu \rho \lambda \sigma }W_{5}\phi _{\lambda \sigma }\bar{\xi},$$ (433)

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

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

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

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

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

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

$$\bar{\delta}_{\Omega \left( \bar{\Omega}\right) }H^{\mu } = \lambda \frac{ \delta S^{\mathrm{L}}}{\delta H^{\nu }}\left\{ 6V_... ... \rho \lambda \sigma }-K^{\mu \nu \rho }\bar{\xi}\right) \right. \right. \notag$$ (439)

$$\left. -2\varepsilon _{\alpha \beta \gamma \delta \varepsilon }\f... ... \frac{d^{2}W_{5}}{d\varphi ^{2}}\phi _{\lambda \sigma }\bar{\xi}\right] \notag$$ (440)

$$+2\frac{d^{2}W_{6}}{d\varphi ^{2}}\left( 3\varepsilon _{\rho \alp... ...}B^{\mu \nu }\bar{\xi} ^{\alpha \beta \gamma \delta \varepsilon }\right) \notag$$ (441)

$$\left. +2\frac{d^{2}W_{2}}{d\varphi ^{2}}\left( 6\phi _{\rho \lam... ... \epsilon}^{\mu \nu \rho \lambda }-B^{\mu \nu }\bar{\xi}\right) \right\} \notag$$ (442)

$$+6\lambda \frac{\delta S^{\mathrm{L}}}{\delta B^{\nu \rho }}\left... ...{\mu \nu \rho \lambda \sigma }-K^{\mu \nu \rho }\bar{\xi}\right) \right. \notag$$ (443)

$$\left. -2\varepsilon _{\alpha \beta \gamma \delta \varepsilon }\f... ...\sigma }\frac{ dW_{5}}{d\varphi }\phi _{\lambda \sigma }\bar{\xi}\right] \notag$$ (444)

$$-2\lambda \frac{\delta S^{\mathrm{L}}}{\delta V_{\mu }}\left( \fr... ...6}}{d\varphi }\bar{\xi}^{\alpha \beta \gamma \delta \varepsilon }\right) \notag$$ (445)

$$+\lambda \frac{\delta S^{\mathrm{L}}}{\delta K^{\nu \rho \lambda ... ...u \rho \lambda \sigma }\frac{dW_{5}}{d\varphi } \bar{\xi}\right) \right. \notag$$ (446)

$$\left. +12\frac{dW_{2}}{d\varphi }\bar{\epsilon}^{\mu \nu \rho \l... ...c{dW_{6}}{d\varphi } \bar{\epsilon}^{\alpha \beta \gamma \delta }\right. \notag$$ (447)

$$\left. +V_{\nu }\left( \frac{dW_{3}}{d\varphi }\bar{\xi}-2\vareps... ...d\varphi }\bar{\xi} ^{\alpha \beta \gamma \delta \varepsilon }\right) \right] ,$$ (448)

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

$$\bar{\delta}_{\Omega \left( \bar{\Omega}\right) }B^{\mu \nu } = 6\lambda \frac{\delta S^{\mathrm{L}}}{\delta H^{\rho }}\left[ -\f... ... }\phi _{\lambda \sigma }\bar{\xi}^{\mu \nu \rho \lambda \sigma }\right. \notag$$ (450)

$$\left. -K^{\mu \nu \rho }\left( \frac{dW_{3}}{d\varphi }\bar{\xi}... ... \sigma }\frac{dW_{5}}{d\varphi }\phi _{\lambda \sigma }\bar{\xi}\right] \notag$$ (451)

$$+\lambda \frac{\delta S^{\mathrm{L}}}{\delta K^{\rho \lambda \sig... ...sigma }+\varepsilon ^{\mu \nu \rho \lambda \sigma }W_{5}\bar{\xi}\right) \notag$$ (452)

$$+6\lambda \frac{\delta S^{\mathrm{L}}}{\delta \phi _{\mu \nu }}\l... ...\varepsilon }W_{4}\bar{ \xi}^{\alpha \beta \gamma \delta \varepsilon }\right) ,$$ (453)

$$\bar{\delta}_{\Omega \left( \bar{\Omega}\right) }\phi _{\mu \nu } = -3\lambda \frac{\delta S^{\mathrm{L}}}{\delta H^{[\mu }}V_{\nu ]}... ...4}}{d\varphi }\bar{\xi}^{\alpha \beta \gamma \delta \varepsilon }\right) \notag$$ (454)

$$-6\lambda \frac{\delta S^{\mathrm{L}}}{\delta B^{\mu \nu }}\left(... ...psilon }W_{4}\bar{ \xi}^{\alpha \beta \gamma \delta \varepsilon }\right) \notag$$ (455)

$$-3\lambda \frac{dW_{6}}{d\varphi }\frac{\delta S^{\mathrm{L}}}{\d... ...\nu ]\alpha \beta \gamma \delta }\bar{\epsilon} ^{\alpha \beta \gamma \delta },$$ (456)

$$\bar{\delta}_{\Omega \left( \bar{\Omega}\right) }K^{\mu \nu \rho } = \lambda \frac{\delta S^{\mathrm{L}}}{\delta H^{\lambda }}\left[ -... ...u \rho \lambda \sigma }\frac{dW_{5}}{d\varphi } \bar{\xi}\right) \right. \notag$$ (457)

$$\left. +12\frac{dW_{2}}{d\varphi }\bar{\epsilon}^{\mu \nu \rho \l... ...bda \sigma }+\varepsilon ^{\mu \nu \rho \lambda \sigma }W_{5}\bar{\xi}\right) ,$$ (458)

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

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

$$\bar{\epsilon}^{\mu \nu \rho }\left( \check{\Omega}\right) =4D_{\... ...u \nu \rho }\check{\epsilon}^{\alpha \beta \gamma \delta \varepsilon }\right) ,$$ (460)

$$\bar{\epsilon}^{\mu \nu \rho \lambda }\left( \check{\Omega}\right... ...a \varepsilon }W_{6}\check{\epsilon}^{\alpha \beta \gamma \delta \varepsilon },$$ (461)

$$\bar{\xi}^{\mu \nu \rho \lambda \sigma }\left( \check{\Omega}\rig... ...ho \lambda \sigma },\qquad \bar{ \theta}_{\mu }\left( \check{\Omega}\right) =0,$$ (462)

such that the second-order reducibility relations (166) become

$$\epsilon ^{\mu \nu }\left( \bar{\Omega}\left( \check{\Omega}\right) \right) = 3\lambda \frac{\delta S^{\mathrm{L}}}{\delta H^{\rho }}\left( 4\f... ...{\lambda \sigma }\check{\epsilon} ^{\mu \nu \rho \lambda \sigma }\right. \notag$$ (463)

$$\left. +\varepsilon _{\alpha \beta \gamma \delta \varepsilon }\fr... ...m{L}}}{\delta B^{\rho \lambda }}\check{\epsilon}^{\mu \nu \rho \lambda } \notag$$ (464)

$$+30\lambda \frac{dW_{2}}{d\varphi }\frac{\delta S^{\mathrm{L}}}{\... ...a \phi _{\mu \nu }}\check{ \epsilon}^{\alpha \beta \gamma \delta \varepsilon },$$ (465)

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

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

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

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

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

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

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

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

$$\bar{\epsilon}^{\mu \nu \rho }\left( \check{\Omega}\left( \hat{\O... ...\frac{dW_{1}}{d\varphi }\right) \hat{ \epsilon}^{\mu \nu \rho \lambda \sigma },$$ (472)

$$\bar{\epsilon}^{\mu \nu \rho \lambda }\left( \check{\Omega}\left(... ...sigma }} \frac{dW_{1}}{d\varphi }\hat{\epsilon}^{\mu \nu \rho \lambda \sigma },$$ (473)

$$\bar{\xi}\left( \check{\Omega}\left( \hat{\Omega}\right) \right) ... ...\bar{\theta}_{\mu }\left( \check{\Omega} \left( \hat{\Omega}\right) \right) =0.$$ (474)