A note on the third way consistent deformation
of Yang-Mills theory

Nihat Sadik Degera,b,c and Henning Samtlebend,e

a Department of Mathematics, Bogazici University,
Bebek, 34342, Istanbul, Turkey

b Feza Gursey Center for Physics and Mathematics,
Bogazici University, Kandilli, 34684, Istanbul, Turkey

c Erwin Schrödinger International Institute for Mathematics and Physics,
University of Vienna, Boltzmanngasse 9, 1090, Vienna, Austria

d ENSL, CNRS, Laboratoire de physique, F-69342 Lyon, France

e Institut Universitaire de France (IUF)

Abstract

> Three-dimensional Yang-Mills theory allows for a deformation quadratic in
> the field strengths which can not be integrated to a local action without
> auxiliary fields. Yet, its covariant divergence consistently vanishes after
> iterating the equation, realizing a spin-1 analogue of ‘minimal massive
> gravity’, which has been dubbed ‘third way consistent’. In this note, we
> show that after dualization of the three-dimensional gauge fields, the model
> possesses a natural action as a Chern-Simons coupled gauged sigma model. In
> this dual formulation, coupling to matter and to gravity becomes
> straightforward. As a direct application, we derive the coupling of the
> model to ${\cal N}=1$ supergravity.

##  1 Introduction

In [1] a three-dimensional gauge theory defined by the following field
equations

$\displaystyle\varepsilon^{\mu\nu\rho}\,{\cal
D}_{\mu}{\tilde{F}}_{\nu}{}^{A}+\mu\,\tilde{F}^{\rho\,A}$ $\displaystyle=$
$\displaystyle-\frac{1}{2m}\,\varepsilon^{\mu\nu\rho}\,f_{BC}{}^{A}{\tilde{F}}_{\mu}{}^{B}\,{\tilde{F}}_{\nu}{}^{C}\;,$
(1.1)

with non-abelian Yang-Mills field strength

$\displaystyle\tilde{F}_{\mu}{}^{A}=\frac{1}{2}\,\varepsilon_{\mu\nu\rho}\,F^{\nu\rho\,A}=\frac{1}{2}\,\varepsilon_{\mu\nu\rho}\left(2\,\partial^{\nu}A^{\rho
A}+f_{BC}{}^{A}\,A^{\nu B}A^{\rho C}\right)\;,$ (1.2)

was considered. Here, the $f_{BC}{}^{A}$ are structure constants of a non-
abelian gauge group $G$, and $m$ and $\mu$ are constants111Equation (1.1) with
some particular choices of these constants appeared earlier in [2, 3].. This
is a deformation of (topologically massive) Yang-Mills theory (TMYM) as the
l.h.s. of (1.1) can be derived from its Lagrangian

$\displaystyle{\cal L}_{\rm TMYM}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\,{\rm
tr}\left[F^{\mu\nu}F_{\mu\nu}\right]+\frac{\mu}{4}\,\varepsilon^{\mu\nu\rho}\,{\rm
tr}\left[F_{\mu\nu}A_{\rho}\right]\;.$ (1.3)

However, in presence of the r.h.s., the equations (1.1) can no longer be
integrated to a gauge-invariant local action without auxiliary fields.
Therefore, consistency of (1.1) is not automatic. Yet, rather surprisingly,
the gauge covariant divergence of (1.1) vanishes on-shell upon iterating the
equation. This novel way of satisfying the consistency requirement is referred
to as ‘third-way consistent’ and reviewed in [4, 5]. The mechanism was first
discovered in the context of three-dimensional, higher curvature gravity
theories [6] and other three-dimensional gravity examples were found later [7,
8]. Recently, third-way consistent $p$-form theories in arbitrary dimensions
have been constructed [9] where a generalization of (1.1) with higher
derivative terms is also given. Moreover, in [10] the ${\cal N}=1$ off-shell
supersymmetric extension of (1.1) has been obtained so that bosonic and
fermionic field equations are mapped to each other under supersymmetry. The
existence of a supersymmetric version of (1.1) calls for a better
understanding of this model. It was observed in [9] that the equations (1.1)
are actually dual to a principal chiral sigma model. In this paper we will
make this correspondence more concrete and use this fact to construct the
coupling of (1.1) to ${\cal N}=1$ supergravity. After decoupling gravity, the
model reduces to the supersymmetric system of [10].

The key to a dual formulation of (1.1) is the on-shell duality between vector
and scalar fields in three dimensions. This is a standard feature for the
dynamics of abelian vector fields satisfying the free Maxwell equations, but
the duality can be extended to non-abelian gauge groups and is central to the
construction of gauged supergravities in three dimensions [11]. In particular,
the topologically massive Yang-Mills Lagrangian (1.3) has a dual description
as a gauged sigma model on the flat target space ${\mathfrak{g}}={\rm
Lie}\,G$, upon introducing an additional set of vector fields $B_{\mu}$
gauging the shift symmetry on the scalar fields [12]. Specifically, the dual
Lagrangian is given by

$\displaystyle\tilde{\cal L}_{\rm\tiny TMYM}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\,{\rm tr}\left[{\cal D}_{\mu}\phi{\cal
D}^{\mu}\phi\right]+\varepsilon^{\mu\nu\rho}\,{\rm
tr}\left[B_{\mu}F_{\nu\rho}\right]+2\,\mu\,\varepsilon^{\mu\nu\rho}\,{\rm
tr}\left[A_{\mu}{}\Big{(}\partial_{\nu}A_{\rho}+\frac{1}{3}\,\left[A_{\nu},A_{\rho}\right]\Big{)}\right]\;,\qquad$
(1.4)

with scalar fields $\phi$ in the adjoint representation of $\mathfrak{g}$, and
their covariant derivatives defined as

$\displaystyle{\cal D}_{\mu}\phi$ $\displaystyle=$
$\displaystyle\partial_{\mu}\phi+\left[A_{\mu}{},\phi\right]+B_{\mu}\;.$ (1.5)

In the dual formulation (1.4), all degrees of freedom are carried by the
scalar fields $\phi$, while the vector fields appear with a Chern-Simons
rather than a Yang-Mills coupling. The full gauge algebra of (1.4) is given by
$\mathfrak{g}\oright\mathfrak{n}$ where $\mathfrak{n}$ denotes a set of
nilpotent generators transforming in the adjoint representation of
$\mathfrak{g}$.

In this note, we point out that a very similar construction can be given for
the dual formulation of (1.1). In this case, the scalar sigma model is built
on the curved target space $G$, with gauging of the full
$\mathfrak{g}\oplus\mathfrak{g}$ algebra of isometries. The Lagrangian is
given by the gauged sigma model coupled to $G\times G$ Chern-Simons vector
fields, see (2.5) below. The rest of this note is organized as follows. In
section 2, we spell out the details of the dual formulation of (1.1). In
section 3 we derive the coupling of (1.1) to matter and to gravity directly in
the dual formulation. In section 4 finally, we use the dual formulation
together with the results of [13] to work out the coupling of (1.1) to ${\cal
N}=1$ supergravity.

##  2 Dual formulation

The dual formulation of (1.1) is a gauged sigma model with Chern-Simons gauge
fields. Explicitly, we let the scalar fields of the model parametrize a ${\rm
G}$-valued matrix $U$ with covariant derivatives given by

$\displaystyle{\cal D}_{\mu}U$ $\displaystyle=$
$\displaystyle\partial_{\mu}U+\beta\,B_{\mu}\,U-\alpha\,UA_{\mu}\;,$ (2.1)

i.e. the vector fields $A_{\mu}$ and $B_{\mu}$ gauge the commuting right and
left group action on $U$ with coupling constants $\alpha$ and $\beta$,
respectively. The left invariant currents

$\displaystyle{\cal J}_{\mu}$ $\displaystyle=$ $\displaystyle U^{-1}{\cal
D}_{\mu}U~{}\in~{}\mathfrak{g}\;,$ (2.2)

satisfy the integrability relations

$\displaystyle{\cal D}_{[\mu}{\cal J}_{\nu]}+\frac{1}{2}\left[{\cal
J}_{\mu},{\cal J}_{\nu}\right]$ $\displaystyle=$
$\displaystyle-\frac{\alpha}{2}\,F_{\mu\nu}+\frac{\beta}{2}\,U^{-1}H_{\mu\nu}U\;,$
(2.3)

with the field strengths given by

$\displaystyle F_{\mu\nu}$ $\displaystyle=$ $\displaystyle
2\,\partial_{[\mu}A_{\nu]}+\alpha\,\left[A_{\mu},A_{\nu}\right]\;,$
$\displaystyle H_{\mu\nu}$ $\displaystyle=$ $\displaystyle
2\,\partial_{[\mu}B_{\nu]}+\beta\,\left[B_{\mu},B_{\nu}\right]\;.$ (2.4)

The model is given by coupling the gauged sigma model with target space ${\rm
G}$ built from (2.2) to a ${\rm G}\times{\rm G}$ Chern-Simons term for the
vector fields

$\displaystyle{\cal L}_{\sigma}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\,{\rm tr}\left[{\cal J}_{\mu}{\cal
J}^{\mu}\right]-g_{1}\,\varepsilon^{\mu\nu\rho}\,{\rm
tr}\left[A_{\mu}{}\Big{(}\partial_{\nu}A_{\rho}+\frac{\alpha}{3}\,\left[A_{\nu},A_{\rho}\right]\Big{)}\right]$
(2.5) $\displaystyle{}-g_{2}\,\varepsilon^{\mu\nu\rho}\,{\rm
tr}\left[B_{\mu}{}\Big{(}\partial_{\nu}B_{\rho}+\frac{\beta}{3}\,\left[B_{\nu},B_{\rho}\right]\Big{)}\right]\;,$

with coupling constants $g_{1}$ and $g_{2}$, respectively. Variation of (2.5)
in particular yields the first-order field equations

$\displaystyle{\cal J}_{\mu}$ $\displaystyle=$
$\displaystyle\frac{g_{1}}{\alpha}\,\varepsilon_{\mu\nu\rho}\,F^{\nu\rho}~{}=~{}-\frac{g_{2}}{\beta}\,\varepsilon_{\mu\nu\rho}\,U^{-1}H^{\nu\rho}U\;.$
(2.6)

Plugging these relations back into the integrability relations (2.3) yields

$\displaystyle\varepsilon^{\mu\nu\rho}\,{\cal
D}_{[\mu}{\tilde{F}}_{\nu]}+\frac{g_{1}}{\alpha}\
\varepsilon^{\mu\nu\rho}\,\left[{\tilde{F}}_{\mu},{\tilde{F}}_{\nu}\right]$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\left(\frac{\alpha^{2}}{g_{1}}+\frac{\beta^{2}}{g_{2}}\right){\tilde{F}}^{\rho}\;.$
(2.7)

Setting $\alpha=\beta=1$, these equations precisely reproduce the deformed
Yang-Mills equations (1.1) with the translation of parameters

$\displaystyle m$ $\displaystyle=$
$\displaystyle\frac{1}{2g_{1}}\;,\qquad\mu-m~{}=~{}\frac{1}{2g_{2}}\;.$ (2.8)

Since both left and right group multiplication of $U$ are local symmetries, we
may fix part of the gauge symmetry by setting

$\displaystyle U={\rm I}\;,$ (2.9)

which breaks the gauge group down to the diagonal ${\rm G}_{\rm
diag}\subset{\rm G}\times{\rm G}$. With the translation of constants (2.8),
the action (2.5) then reduces to

$\displaystyle{\cal L}_{0}$ $\displaystyle=$ $\displaystyle-\frac{1}{2}\,{\rm
tr}\left[(B_{\mu}-A_{\mu})(B^{\mu}-A^{\mu})\right]-\frac{1}{2m}\,\varepsilon^{\mu\nu\rho}\,{\rm
tr}\left[A_{\mu}{}\Big{(}\partial_{\nu}A_{\rho}+\frac{1}{3}\,\left[A_{\nu},A_{\rho}\right]\Big{)}\right]$
(2.10) $\displaystyle{}-\frac{1}{2(\mu-m)}\,\varepsilon^{\mu\nu\rho}\,{\rm
tr}\left[B_{\mu}{}\Big{(}\partial_{\nu}B_{\rho}+\frac{1}{3}\,\left[B_{\nu},B_{\rho}\right]\Big{)}\right]\;,$

which (up to a global factor) precisely reproduces the massive Chern-Simons
action found in [1]. We note in passing that while the Lagrangian (2.10)
appears degenerate at the special value $m=\mu$ [1], in this case an action
for the system is straightforwardly obtained from the Lagrangian (2.5) upon
setting $\beta=0=g_{2}$, such that the vector fields $B_{\mu}$ disappear from
the action. Its gauge fixed version then reduces to the massive Chern-Simons
Lagrangian

$\displaystyle{\cal L}_{0,\mu=m}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\,{\rm
tr}\left[A_{\mu}A^{\mu}\right]-\frac{1}{2m}\,\varepsilon^{\mu\nu\rho}\,{\rm
tr}\left[A_{\mu}{}\Big{(}\partial_{\nu}A_{\rho}+\frac{1}{3}\,\left[A_{\nu},A_{\rho}\right]\Big{)}\right]\;.$
(2.11)

Note that the dual formulation (2.5) as well as its gauge fixed version (2.10)
make manifest the duality symmetry of the system under simultaneous exchange
of the vector fields $A_{\mu}$ and $B_{\mu}$ together with the parameters

$\displaystyle g_{1}\leftrightarrow g_{2}\qquad\Longleftrightarrow\qquad
m\leftrightarrow\mu-m\;.$ (2.12)

Since only the vector fields $A_{\mu}$ appear in the original deformed Yang-
Mills equations (1.1), this duality can be employed in order to map solutions
to (1.1) with given values of $m$ and $\mu$ into new solutions to the same
equations with different values of the parameters related by (2.12).

Let us also note, that the equations (1.1) admit a smooth limit to a
deformation of the Maxwell equations (i.e. to abelian field strengths
$F_{\mu\nu}{}^{A}=2\,\partial_{[\mu}A_{\nu]}{}^{A}$) while keeping the
structure constants $f_{BC}{}^{A}$ on the r.h.s. of (1.1), parametrizing the
non-linear deformation of the topologically massive Maxwell equations as was
considered in [9]. This limit is described by rescaling vector fields as

$\displaystyle A_{\mu}{}^{A}$ $\displaystyle\rightarrow$ $\displaystyle
m\,A_{\mu}{}^{A}\;,$ (2.13)

dividing a resulting overall factor $m$ from (1.1), and finally sending
$m\rightarrow 0$ . It is however not clear how this limit can be implemented
directly on the level of the action (2.5).

Finally, the limit of (2.5) to the dual formulation of the undeformed Yang-
Mills theory (1.4) requires flattening of the sigma model target space, such
that in particular its isometry group contracts from ${\rm G}\times{\rm G}$ to
${\rm G}\ltimes{\rm N}$ .

##  3 Coupling to matter and gravity

The coupling of the deformed Yang-Mills system (1.1) to gravity and lower-spin
matter has been determined in [1] by a careful analysis of the consistency
conditions of the coupled system. Within the dual formulation (2.5), this
coupling is straightforward. Simply adding a matter Lagrangian ${\cal L}_{\rm
mat}$ to (2.5)

$\displaystyle{\cal L}$ $\displaystyle=$ $\displaystyle{\cal L}_{\sigma}+{\cal
L}_{\rm mat}\;,$ (3.1)

the first-order field equations (2.6) change into

$\displaystyle{\cal J}_{\mu}$ $\displaystyle=$
$\displaystyle\frac{g_{1}}{\alpha}\,\varepsilon_{\mu\nu\rho}\,F^{\nu\rho}+j_{\mu}\;,\qquad
j_{\mu}=\frac{1}{\alpha}\,\frac{\partial{\cal L}_{\rm mat}}{\partial
A^{\mu}}\;.$ (3.2)

For simplicity, we assume that the matter is only charged under the right
factor of the gauge group ${\rm G}\times{\rm G}$, i.e. that ${\cal L}_{\rm
mat}$ does not depend on $B_{\mu}$. After plugging these equations into the
integrability relations (2.3), the equations (2.7) are modified according to

$\displaystyle\varepsilon^{\mu\nu\rho}\,{\cal
D}_{[\mu}{\tilde{F}}_{\nu]}+\frac{g_{1}}{\alpha}\
\varepsilon^{\mu\nu\rho}\,\left[{\tilde{F}}_{\mu},{\tilde{F}}_{\nu}\right]+\frac{1}{2}\left(\frac{\alpha^{2}}{g_{1}}+\frac{\beta^{2}}{g_{2}}\right){\tilde{F}}^{\rho}$
$\displaystyle=$
$\displaystyle-\frac{\alpha}{2g_{1}}\,\varepsilon^{\mu\nu\rho}\,{\cal
D}_{\mu}j_{\nu}-\frac{\alpha}{4g_{1}}\,\varepsilon^{\mu\nu\rho}\left[j_{\mu},j_{\nu}\right]$
(3.3)
$\displaystyle{}-\frac{\alpha\,\beta^{2}}{4\,g_{1}g_{2}}\,j^{\rho}-\varepsilon^{\mu\nu\rho}\,\big{[}{\tilde{F}}_{\mu},j_{\nu}\big{]}\;,$

where the r.h.s. directly reproduces (after setting $\alpha=\beta=1$,
rescaling $j_{\mu}\rightarrow-4\,g_{1}g_{2}\,j_{\mu}$, and translating
parameters by (2.8)) the matter coupling constructed in [1] but with the
opposite sign of the second term on the r.h.s. which agrees with [9].

Similarly, coupling of the system (2.5), (3.1) to three-dimensional gravity is
straightforward. The full energy-momentum tensor $T_{\mu\nu}$ is given by

$\displaystyle T_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{1}{2m^{2}}\,{\rm
tr}\big{[}{\tilde{F}}_{\mu}{\tilde{F}}_{\nu}\big{]}+\frac{1}{4m^{2}}\,g_{\mu\nu}\,{\rm
tr}\big{[}{\tilde{F}}_{\rho}{\tilde{F}}^{\rho}\big{]}-\frac{1}{m}{\rm
tr}\big{[}{\tilde{F}}_{\mu}{j}_{\nu}\big{]}+\frac{1}{2m}\,g_{\mu\nu}\,{\rm
tr}\big{[}{\tilde{F}}_{\rho}{j}^{\rho}\big{]}$ (3.4)
$\displaystyle{}-\frac{1}{2}\,{\rm
tr}\big{[}j_{\mu}j_{\nu}\big{]}+\frac{1}{4}\,g_{\mu\nu}\,{\rm
tr}\big{[}j_{\rho}j^{\rho}\big{]}+\sqrt{|g|}^{-1}\,\frac{\partial{\cal L}_{\rm
mat}}{\partial g^{\mu\nu}}\;,$

with the first six terms derived from $\sqrt{|g|}^{-1}\frac{\partial{\cal
L}_{\sigma}}{\partial g^{\mu\nu}}$ after using (3.2). In absence of ${\cal
L}_{\rm mat}$, i.e. for $j_{\mu}=0$, this precisely reproduces the energy-
momentum tensor given in [1]. The full expression (3.4) directly extends this
result to the presence of additional matter.

##  4 Coupling to ${\cal N}=1$ supergravity

The ${\cal N}=1$ supersymmetrization of the system (1.1) was constructed in
[10] on the level of the field equations. In terms of the dual formulation
(2.5), we can trace this back to the known structures of a supersymmetric
sigma model with target space ${\rm G}$ [14]. As a new application, we will
employ the dual formulation (2.5) in order to work out the coupling of the
system (1.1) to ${\cal N}=1$ supergravity. The coupling of a general gauged
sigma-model to three-dimensional supergravity has been constructed in [13]. In
particular, any Riemannian target space allows for a coupling to ${\cal N}=1$
supergravity. The fermionic field content in this case is given by $n(={\rm
dim\,G})$ spin-1/2 fermions $\chi^{i}$ together with a gravitino $\psi_{\mu}$.

To translate to the notation from [13], we parametrize the group manifold
${\rm G}$ by coordinates $\phi^{i}$ and define the left and right invariant
vector fields $L_{A}$, $R_{A}$ satisfying the differential relations

$\displaystyle\nabla_{i}L_{A\,j}=-\frac{1}{2}\,{f}_{A}{}^{BC}\,L_{B\,i}L_{C\,j}\;,\qquad\nabla_{i}R_{A\,j}=\frac{1}{2}\,{f}_{A}{}^{BC}\,R_{B\,i}R_{C\,j}\;.$
(4.1)

The metric on the group manifold can be expressed as

$\displaystyle g_{ij}$ $\displaystyle=$ $\displaystyle
L_{A\,i}L^{A}_{j}~{}=~{}R_{A\,i}R^{A}_{j}\;.$ (4.2)

The currents of the gauged sigma model (2.2) are then given by

$\displaystyle{\cal J}_{\mu}{}^{A}$ $\displaystyle=$ $\displaystyle
L^{A}_{i}\,{\cal
D}_{\mu}\phi^{i}~{}=~{}L^{A}_{i}\,\partial_{\mu}\phi^{i}-\alpha\,A^{A}_{\mu}+\beta\,B^{B}_{\mu}\,(L^{A}_{i}R_{B}^{i})\;.$
(4.3)

On the fermionic side, we define the fermions $\chi^{A}=L_{i}^{A}\chi^{i}$
transforming under the right action of ${\rm G}$, with covariant derivatives

$\displaystyle{\cal D}_{\mu}\chi^{A}$ $\displaystyle=$
$\displaystyle\left(\partial_{\mu}+\tfrac{1}{2}\omega_{\mu}{}^{a}\gamma_{a}\right)\chi^{A}+\alpha{f}_{BC}{}^{A}\,A_{\mu}^{B}\chi^{C}\;.$
(4.4)

In terms of these objects, we can obtain the coupling of the gauged sigma
model (2.5) to ${\cal N}=1$ supergravity from the general result of [13]
(after a suitable rescaling of the vector fields) as

$\displaystyle\sqrt{g}^{-1}{\cal L}$ $\displaystyle=$
$\displaystyle-{\textstyle{{\scriptstyle 1}\over{\scriptstyle
2}}}\,\epsilon^{\mu\nu\rho}\left(e_{\mu}{}^{a}\,R_{\nu\rho
a}+\bar{\psi}{}_{\mu}{\cal
D}_{\nu}\psi_{\rho}\right)-4\,C^{2}-g_{1}\,\epsilon^{\mu\nu\rho}\,A_{\mu}{}^{A}\Big{(}\partial_{\nu}A_{\rho}{}^{A}+\tfrac{\alpha}{3}\,f_{BC}{}^{A}A_{\nu}{}^{B}A_{\rho}{}^{C}\Big{)}$
(4.5)
$\displaystyle{}-g_{2}\,\epsilon^{\mu\nu\rho}\,B_{\mu}{}^{A}\Big{(}\partial_{\nu}B_{\rho}{}^{A}+\tfrac{\beta}{3}\,f_{BC}{}^{A}B_{\nu}{}^{B}B_{\rho}{}^{C}\Big{)}-{\textstyle{{\scriptstyle
1}\over{\scriptstyle 2}}}{\cal J}_{\mu}{}^{A}{\cal J}^{\mu
A}-{\textstyle{{\scriptstyle 1}\over{\scriptstyle
2}}}\,\bar{\chi}{}^{A}\gamma^{\mu}{\cal D}_{\mu}\chi^{A}$
$\displaystyle{}+{\textstyle{{\scriptstyle 1}\over{\scriptstyle 4}}}{\cal
J}_{\mu}{}^{A}\,{f}^{A}{}_{BC}\bar{\chi}{}^{B}\gamma^{\mu}\chi^{C}+{\textstyle{{\scriptstyle
1}\over{\scriptstyle 2}}}{\cal
J}_{\nu}{}^{A}\,\bar{\chi}{}^{A}\gamma^{\mu}\gamma^{\nu}\psi_{\mu}+{\textstyle{{\scriptstyle
1}\over{\scriptstyle
2}}}\,C\,\bar{\psi}{}_{\mu}\,\gamma^{\mu\nu}\,\psi_{\nu}\,$
$\displaystyle{}-\tfrac{1}{4}\left(\tfrac{\alpha^{2}}{g_{1}}+\tfrac{\beta^{2}}{g_{2}}-2C\right)\bar{\chi}^{A}\chi^{A}+{\textstyle{{\scriptstyle
1}\over{\scriptstyle
16}}}\,(\bar{\chi}{}^{A}\chi^{A})^{2}-{\textstyle{{\scriptstyle
1}\over{\scriptstyle
8}}}\,(\bar{\chi}{}^{A}\gamma^{\mu}\gamma^{\nu}\psi_{\mu})\,(\bar{\psi}_{\nu}\chi^{A})$
$\displaystyle{}+{\textstyle{{\scriptstyle 1}\over{\scriptstyle
96}}}\,{f}_{AB}{}^{E}{f}_{CDE}\;\bar{\chi}{}^{A}\gamma_{\mu}\chi^{B}\,\bar{\chi}{}^{C}\gamma^{\mu}\chi^{D}\;,$

with a cosmological constant $C$. The supersymmetry variations are given by

$\displaystyle\delta e_{\mu}{}^{a}$ $\displaystyle=$
$\displaystyle{\textstyle{{\scriptstyle 1}\over{\scriptstyle
2}}}\,\bar{\epsilon}{}\gamma^{a}\,\psi_{\mu}\;,\qquad\delta\phi^{i}~{}=~{}{\textstyle{{\scriptstyle
1}\over{\scriptstyle 2}}}\,\bar{\epsilon}\,\chi^{i}\;,$ $\displaystyle\delta
A_{\mu}{}^{A}$ $\displaystyle=$
$\displaystyle\tfrac{\alpha}{4g_{1}}\,\bar{\chi}^{A}\gamma_{\mu}\epsilon\;,\qquad\delta
B_{\mu}{}^{A}~{}=~{}-\tfrac{\beta}{4g_{2}}\,(R^{A}_{i}L^{i}_{B})\,\bar{\chi}^{B}\gamma_{\mu}\epsilon\;,$
$\displaystyle\delta\psi_{\mu}$ $\displaystyle=$
$\displaystyle\left(\partial_{\mu}+\tfrac{1}{2}\omega_{\mu}{}^{a}\gamma_{a}\right)\epsilon-{\textstyle{{\scriptstyle
1}\over{\scriptstyle
8}}}\,\bar{\chi}{}^{A}\gamma^{\nu}\chi^{A}\,\gamma_{\mu\nu}\,\epsilon+C\gamma_{\mu}\,\epsilon\;,$
$\displaystyle\delta\chi^{A}$ $\displaystyle=$
$\displaystyle\tfrac{1}{2}\,{{\cal
J}}_{\mu}{}^{A}\,\gamma^{\mu}\epsilon-\tfrac{1}{4}\,(\bar{\psi}{}_{\mu}\chi^{A})\,\gamma^{\mu}\epsilon-\tfrac{1}{4}\,{f}_{BC}{}^{A}\,(\bar{\epsilon}\,\chi^{B})\,\chi^{C}\;,$
(4.6)

and leave the Lagrangian (4.5) invariant. The supersymmetric extension of the
Yang-Mills system (1.1) is then obtained in full analogy to (2.7), and (3.3).
Variation of (4.5) w.r.t. the vector fields, yields the first-order duality
equations

$\displaystyle{\cal J}^{\mu\,A}$ $\displaystyle=$
$\displaystyle\frac{g_{1}}{\alpha}\,\epsilon^{\mu\nu\rho}\,F_{\nu\rho}{}^{A}+\frac{1}{2}\,\bar{\chi}{}^{A}\gamma^{\nu}\gamma^{\mu}\,\psi_{\nu}-\frac{1}{4}\,{f}^{A}{}_{BC}\,\bar{\chi}{}^{B}\gamma^{\mu}\chi^{C}$
(4.7) $\displaystyle=$
$\displaystyle-\frac{g_{2}}{\beta}\,\epsilon^{\mu\nu\rho}\,H_{\nu\rho}{}^{B}\,(R_{B}^{i}L^{A}_{i})+\frac{1}{2}\,\bar{\chi}{}^{A}\gamma^{\nu}\gamma^{\mu}\,\psi_{\nu}+\frac{1}{4}\,{f}^{A}{}_{BC}\,\bar{\chi}{}^{B}\gamma^{\mu}\chi^{C}\;.$

Upon plugging these equations into the integrability relations (2.3) of the
scalar currents, and translating the coupling constants via (2.8) together
with $\alpha=\beta=1$, we finally obtain the desired supersymmetric extension
of (1.1)

$\displaystyle\epsilon^{\mu\nu\rho}\,{\cal
D}_{[\mu}{\tilde{F}}_{\nu]}{}^{A}+\mu\,{\tilde{F}}^{\rho\,A}$ $\displaystyle=$
$\displaystyle-\frac{1}{2m}\
\epsilon^{\mu\nu\rho}\,f_{BC}{}^{A}\,{\tilde{F}}_{\mu}{}^{B}{\tilde{F}}_{\nu}{}^{C}-\epsilon^{\mu\nu\rho}\,f_{BC}{}^{A}\,{\tilde{F}}_{\mu}\,^{B}j^{(+)C}_{\nu}-m\,\epsilon^{\mu\nu\rho}\,{\cal
D}_{\mu}j^{(+)A}_{\nu}$ (4.8) $\displaystyle{}+m(m-\mu)\left(j^{(+)\rho
A}-j^{(-)\rho
A}\right)-\frac{m}{2}\,\epsilon^{\mu\nu\rho}\,f_{BC}{}^{A}\,j^{(+)B}_{\mu}j^{(+)C}_{\nu}\;,$

with matter currents bilinear in the fermions

$\displaystyle{j}_{\mu}^{(\pm)\,A}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\,\bar{\chi}{}^{A}\gamma^{\nu}\gamma_{\mu}\,\psi_{\nu}\mp\frac{1}{4}\,{f}_{BC}{}^{A}\,\bar{\chi}{}^{B}\gamma_{\mu}\chi^{C}\;.$
(4.9)

The appearance of a second current ${j}_{\mu}^{(-)}$ as opposed to the general
form of (3.3) is due to the appearance of the vectors $B_{\mu}{}^{A}$ via the
fermion couplings to ${\cal J}_{\mu}{}^{A}$ in the matter couplings of (4.5).

In order to complete the supersymmetric system, it remains to compute the
remaining equations of motion from (4.5) and replace all the appearing scalar
currents ${\cal J}_{\mu}{}^{A}$ and field strengths $H_{\mu\nu}{}^{A}$ by
means of the duality equations (4.7). For the gravitino and the spin 1/2 field
equations, this yields after some straightforward computation (and notably
some Fierzing) the deformed Rarita-Schwinger equation

$\displaystyle\epsilon^{\mu\nu\rho}{\cal D}_{\nu}\psi_{\rho}$ $\displaystyle=$
$\displaystyle{\textstyle{{\scriptstyle 1}\over{\scriptstyle
2m}}}\gamma^{\nu}\gamma^{\mu}\chi{}^{A}\,\tilde{F}_{\nu}{}^{A}+C\,\gamma^{\mu\nu}\,\psi_{\nu}\,-{\textstyle{{\scriptstyle
1}\over{\scriptstyle
4}}}\,{f}_{BC}{}^{A}\chi{}^{A}\,\,(\bar{\chi}{}^{B}\gamma^{\mu}\chi^{C})-{\textstyle{{\scriptstyle
1}\over{\scriptstyle 4}}}\,(\bar{\chi}{}^{A}\chi^{A})\,\psi^{\mu}\;,\;\;\;$
(4.10)

as well as the deformed Dirac equation

$\displaystyle\gamma^{\mu}{\cal D}_{\mu}\chi^{A}$ $\displaystyle=$
$\displaystyle{\textstyle{{\scriptstyle 1}\over{\scriptstyle
2m}}}\gamma^{\mu}\chi^{B}\tilde{F}_{\mu}{}^{C}\,{f}_{BC}{}^{A}+{\textstyle{{\scriptstyle
1}\over{\scriptstyle
2m}}}\gamma^{\mu}\gamma^{\nu}\psi_{\mu}\,\tilde{F}_{\nu}{}^{A}+\left(C-\mu\right)\chi^{A}+{\textstyle{{\scriptstyle
1}\over{\scriptstyle
4}}}\psi_{\nu}\,{f}_{BC}{}^{A}(\bar{\chi}{}^{B}\gamma^{\nu}\chi^{C})$ (4.11)
$\displaystyle{}-{\textstyle{{\scriptstyle 1}\over{\scriptstyle
4}}}\chi^{A}(\bar{\psi}_{\mu}\psi^{\mu})-{\textstyle{{\scriptstyle
1}\over{\scriptstyle
12}}}\gamma^{\mu}\chi^{B}(\bar{\chi}{}^{D}\gamma_{\mu}\chi^{E})\,{f}_{BC}{}^{A}{f}_{DE}{}^{C}+{\textstyle{{\scriptstyle
1}\over{\scriptstyle 4}}}\,\chi^{A}\,(\bar{\chi}{}_{B}\chi^{B})\;.$

Finally, the Einstein field equations are of the standard form with the
energy-momentum tensor given by (3.4) and again all scalar currents eliminated
by (4.7). By construction, the system of (4.8), (4.10), (4.11), and the
Einstein equations yields a supersymmetric system, with the equations
transforming into each other under the transformations (4.6) of the remaining
fields

$\displaystyle\delta e_{\mu}{}^{a}$ $\displaystyle=$
$\displaystyle\tfrac{1}{2}\,\bar{\epsilon}{}\gamma^{a}\,\psi_{\mu}\;,$
$\displaystyle\delta A_{\mu}{}^{A}$ $\displaystyle=$
$\displaystyle\tfrac{m}{2}\bar{\chi}^{A}\gamma_{\mu}\epsilon\;,$
$\displaystyle\delta\psi_{\mu}$ $\displaystyle=$
$\displaystyle\left(\partial_{\mu}+\tfrac{1}{2}\omega_{\mu}{}^{a}\gamma_{a}\right)\epsilon-\tfrac{1}{8}\,\bar{\chi}{}^{A}\gamma^{\nu}\chi^{A}\,\gamma_{\mu\nu}\,\epsilon+C\gamma_{\mu}\,\epsilon\;,$
$\displaystyle\delta\chi^{A}$ $\displaystyle=$
$\displaystyle\tfrac{1}{2m}\,{{\tilde{F}}}_{\mu}{}^{A}\,\gamma^{\mu}\epsilon+\tfrac{1}{4}\,(\bar{\chi}^{A}\gamma^{\mu\nu}\psi_{\mu})\,\gamma_{\nu}\epsilon\;.$
(4.12)

A direct construction of this system starting from the supersymmetrization of
(4.8) would have represented a formidable technical challenge with an
uncertain outcome. In the dual picture, the entire system is straightforwardly
obtained by variation of the supersymmetric action (4.5).

As a consistency check, we may study the rigid limit of the model. In order to
decouple gravity, we first set $\psi_{\mu}=C=0$, and then scale fields and
structure constants with a constant $k$ as $\chi^{A}\rightarrow
k\chi^{A},A_{\mu}{}^{A}\rightarrow kA_{\mu}{}^{A},f_{BC}{}^{A}\rightarrow
f_{BC}{}^{A}/k$. After cancelling the overall $k$ and taking the limit
$k\rightarrow 0$ together with a few redefinitions,222Explicitly,
$m\rightarrow m/4,\,\mu\rightarrow
2\mu,\,{j}_{\mu}^{(+)\,A}\rightarrow-2j_{\mu}^{\,A}/m$. one arrives precisely
at the field equations of ${\cal N}=1$ off-shell supersymmetric massive Yang-
Mills theory constructed in [10]. This limit can also be taken at the action
(4.5) by setting $g_{\mu\nu}=\eta_{\mu\nu}$ and making additional scalings
$\phi^{i}\rightarrow k\phi^{i}$, $B_{\mu}{}^{A}\rightarrow kB_{\mu}{}^{A}$.

Let us finally note that the field equations (4.8), (4.10), (4.11), as well as
the supersymmetry transformations (4.12) are compatible with the limit
$m\rightarrow 0$ after rescaling the vector fields as (2.13). This is the
limit to abelian field strengths
$F_{\mu\nu}{}^{A}=2\,\partial_{[\mu}A_{\nu]}{}^{A}$ while keeping all the
remaining structure constants $f_{BC}{}^{A}$ in the various couplings [9].

##  5 Conclusions

In this note, we have shown that the dual formulation (2.5) of the deformed
Yang-Mills equations (1.1) provides a natural explanation for many features of
the model and in particular allows for a straightforward coupling of the model
to ${\cal N}=1$ supergravity. The dual formulation also shows that the
construction cannot be extended to higher supersymmetry ${\cal N}>1$, since
for a sigma model higher supersymmetry requires further constraints on the
scalar target space [15] which are not satisfied by the real compact Lie
groups $G$ that appear here. In particular, an extension to ${\cal N}=2$ would
require $G$ to be a Kähler manifold, thus the gauge group to be complex [16].

Among further applications, it would be interesting to work out the analogous
dual models to the higher dimensional third way consistent $p$-form theories
of [9]. Higher derivative extensions of (1.1) were constructed in [9] and
finding supersymmetric versions of these would also be desirable.

Another interesting extension of the present construction is its analogue in
the context of third way consistent gravitational theories, such as minimal
massive gravity[6], and extensions thereof [7, 8]. We will come back to this
in [17].

## Acknowledgements

We would like to thank Marc Geiller and Jan Rosseel for discussions and Craig
van Coevering for pointing out reference [16] to us. NSD is grateful to ENS de
Lyon, Laboratoire de Physique for hospitality where this work was initiated
and Research Fellowship Program of Embassy of France in Turkey for financial
support during this visit. NSD would also like to thank ESI, Vienna for
hospitality and financial support where this work was completed.

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