The conventional constraint arises from the fact that a connection 1-form can be arbitrarily shifted by an adjoint tensor of the gauge group without ceasing to be a connection 1-form. We thus have the freedom to redefine the superconnection as

\begin{align}
\Ag_A &\xrightarrow{\text{redef}} \Ag_A + \Xg_A
\begin{cases}
\Ag_A \to e^X(\Ag_A + D_A)e^{-X}
\\
\Xg_A \to e^X\Xg_Ae^{-X}
\end{cases}
\end{align}

From this it follows that the superfield strength transforms as

\begin{equation}
\Fg_{AB}
\xrightarrow{\text{redef}} \Fg_{AB} + 2\nabla_{[A}\Xg_{B\}} + [\Xg_A,\Xg_B\} - \Tg_{AB}{^C}\Xg_C
\end{equation}

If we now choose

\begin{align}
\Xg_\mu &= \frac{i}{4}\gamma_\mu{^{\alpha\beta}}\Fgo_{\alpha\beta}
\Xg_\alpha &= \Xgt_\alpha = 0
\end{align}

we find that

\begin{equation}
\Fgo_{\alpha\beta}
\xrightarrow{\text{redef}}
\Fgo_{\alpha\beta}
+ \frac{1}{2}\gamma^\mu{_{\alpha\beta}}\gamma_\mu{^{\gamma\delta}}\Fgo_{\gamma\delta}
\overeq{\ref{Lemma symmetric}}
\Fgo_{\alpha\beta} - \Fgo_{(\alpha\beta)}
\Fgo_{[\alpha\beta]}.
\end{equation}

That is, we can generally impose the constraint

\begin{align}
\Fgo_{(\alpha\beta)} &= 0
\Fgo_{\alpha\beta} &= \Fgo_{[\alpha\beta]} =: \varepsilon_{\alpha\beta}\Fgo.
\end{align}

This constraint which can always be imposed is referred to as the conventional constraint [3]. A direct consequence of this is that we can express $\Ag_\mu$ in terms of $\Ag_\alpha$ and $\Agt_\alpha$. Indeed, we see that

\begin{equation}
0 = \Fgo_{(\alpha\beta)} = D_\alpha\Agt_\beta + \Dt_\beta\Ag_\alpha + \{\Ag_\alpha,\Agt_\beta\} - 2i\gamma^\mu{_{\alpha\beta}}\Ag_\mu
\end{equation}

Again making use of equation <ref> we find that

\begin{equation}
\Ag_\mu = \frac{i}{4}\gamma_\mu{^{\alpha\beta}}\Big(D_\alpha\Agt_\beta + \Dt_\beta\Ag_\alpha + \{\Ag_\alpha,\Agt_\beta\}\Big).
\end{equation}

§.§.§ Representation preserving constraint

Now we move on to the second kind of constraint. These constraints are motivated by integrability conditions for so-called gauge-chiral and gauge-antichiral superfields $\Phi$ and $\Phit$ which are constrained by

\begin{align}
\nabla_\alpha\Phi &= 0,
\nablat_\alpha\Phit &= 0.
\end{align}

The representation preserving constraints are now given by

\begin{align}
\Fg_{\alpha\beta} &= 0,
\Fgt_{\alpha\beta} &= 0.
\end{align}

Indeed, if these weren't vanishing and one would take covariant derivatives of the equations above arive at conditions which overconstrain the gauge-chiral and gauge-antichiral multiplets $\Phi$ and $\Phit$ [3].

§.§ Bianchi Identities

We will now go over Bianchi identities as they were described in equation <ref>. These will result in some new information because the constraints we imposed weren't taking account in any way with the superconnection origins of the superfield strengths [21].

* $\nabla_\alpha\nabla_\beta\nablat_\gamma$ and $\nabla_\alpha\nablat_\beta\nablat_\gamma$ Bianchi identities: These respectively yield

\begin{align}
\label{Bianchi sst}
\nabla_{(\alpha}\Fgo\varepsilon_{\beta)\gamma} &= +2i\Fg_{\mu(\alpha}\gamma^\mu{_{\beta)\gamma}}
\nabla_\alpha\Fgo &= +\frac{2i}{3}\gamma^\mu{_\alpha}{^\beta}\Fg_{\mu\beta}
\\
\label{Bianchi stt}
\nablat_{(\alpha}\Fgo\varepsilon_{\beta)\gamma} &= -2i\Fgt_{\mu(\alpha}\gamma^\mu{_{\beta)\gamma}}
\nablat_\alpha\Fgo &= -\frac{2i}{3}\gamma^\mu{_\alpha}{^\beta}\Fgt_{\mu\beta}
\end{align}

Using these equations as well as the Fierz identity <ref> one then finds explicit expressions

\begin{align}
\label{F sv}
\Fg_{\alpha,\beta\gamma}
\Fg_{\alpha\mu}\gamma^\mu{_{\beta\gamma}}
+\frac{3i}{4}\varepsilon_{\alpha\beta}\nabla_\gamma\Fgo + \frac{i}{2}\nabla_{(\alpha}\Fgo\varepsilon_{\beta)\gamma}
\\
\label{Ft sv}
\Fgt_{\alpha,\beta\gamma}
\Fgt_{\alpha\mu}\gamma^\mu{_{\beta\gamma}}
-\frac{3i}{4}\varepsilon_{\alpha\beta}\nablat_\gamma\Fgo - \frac{i}{2}\nablat_{(\alpha}\Fgo\varepsilon_{\beta)\gamma}
\end{align}

* $\nabla_\mu\nabla_\alpha\nabla_\beta$ and $\nabla_\mu\nablat_\alpha\nablat_\beta$ Bianchi identities: These respectively yield

\begin{align}
\nabla_{(\alpha}\Fg_{\beta)\mu} &= 0
&\overset{\ref{Bianchi sst}}{\Longrightarrow}
\nabla^\alpha\nabla_\alpha\Fgo &= 0
\\
\nablat_{(\alpha}\Fgt_{\beta)\mu} &= 0
&\overset{\ref{Bianchi stt}}{\Longrightarrow}
\nablat{^\alpha}\nablat_\alpha\Fgo &= 0
\end{align}

* $\nabla_\mu\nabla_\alpha\nablat_\beta$ Bianchi identity: This identity yields

\begin{equation}
2i\gamma^\nu{_{\alpha\beta}}\Fg_{\mu\nu} + \varepsilon_{\alpha\beta}\nabla_\mu\Fgo
\nablat_\beta\Fg_{\mu\alpha} + \nabla_\alpha\Fgt_{\mu\beta}
\end{equation}

Tracing over $_{\alpha\beta}$ and contracting with the Levi-Civita tensor then yields

\begin{gather}
\label{tussenstap 1}
4i\Fg_{\mu\nu} = \nablat{^\alpha}\gamma_{\nu\alpha}{^\beta}\Fg_{\mu\beta} + \nabla{^\alpha}\gamma_{\nu\alpha}{^\beta}\Fgt_{\mu\beta}
=: \nablat\gamma_\nu\Fg_\mu + \nabla\gamma_\nu\Fgt_\mu
\Big.\\
\Updownarrow\nonumber
\Big.\\
\label{tussenstap 2}
8i{\ast}\Fg^\mu = \epsilon^{\mu\rho\sigma}\Big(\nablat\gamma_\sigma\Fg_\rho + \nabla\gamma_\sigma\Fgt_\rho\Big).
\Big.
\end{gather}

Contracting <ref> with $\gamma^\mu{_{\alpha\beta}}$ and using equation <ref> as well as expressions <ref> and <ref> finally results in

\begin{align}
{\ast}\Fgs_{\alpha\beta} &= \frac{i}{16}[\nablat_{(\alpha},\nabla_{\beta)}]\Fgo
&\overset{\text{\ref{Lemma symmetric}}}{\Leftrightarrow}
{\ast}\Fg^\mu &= -\frac{i}{32}\gamma^{\mu\alpha\beta}[\nablat_\alpha,\nabla_\beta]\Fgo.
\end{align}

On the other hand, contracting $_{\mu\nu}$ in <ref> and using previous Bianchi identities <ref> and <ref> results in

\begin{equation}
\big(\nabla^\alpha\nablat_\alpha - \nablat{^\alpha}\nabla_\alpha\big)\Fgo = 0.
\end{equation}

These sum up the most interesting constraints placed on $\Fgo$ by using the Bianchi identities.

§.§ Solving the Constraints

We now wish to formulate an explicit solution to the constraints placed on the connection. To this end we recall that in the case of gauge theories a flat connection —that is, a vanishing field strength— corresponds to what's called a `pure gauge' connection 1-form

\begin{align}
F &= 0
\exists \theta : A &= e^{-\theta}\extd e^\theta
\end{align}

which can obviously always be gauged away by choosing a gauge transformation with parameter $\theta$ [14]. In the spirit of this fact we choose to write the spinorial superconnection components $\Ag_\alpha$ and $\Agt_\alpha$ as

\begin{align}
\Fg_{\alpha\beta} &= 0
\Ag_\alpha &= e^{-2\Vg}D_\alpha e^{2\Vg}
\\
\Fgt_{\alpha\beta} &= 0
\Agt_\alpha &= e^{-2\Vgtt}\Dt_\alpha e^{2\Vgtt}
\end{align}

which is referred to as a spinorially flat connection. $\Vg$ and $\Vgt$ are $\gLie$-valued superfields which we refer to as the prepotentials [3][21]. One could now wonder whether this means that the superconnection is flat and thus contains no physical degrees of freedom. However it isn't since $\Vg$ and $\Vgt$ aren't generally the same and thus can't be generally gauged away simultaneously.

§.§.§ Pregauge and supergauge transformations

Now we move on to describe how gauge symmetries work in this propotential formalism, and particularly how the prepotentials transform under gauge transformations. Due to the graded Leibniz property of the spinorial covariant derivatives we find that the prepotentials transform under supergauge transformations as

\begin{align}
e^{2\Vg} &\to e^{2\Vg}e^{-X}
e^{2\Vgtt} &\to e^{2\Vgtt}e^{-X}
\end{align}

More concretely using the BCH formulas outlined in the appendix section <ref> we find that

\begin{align}
\label{chiral residual gauge symmetries}
\Vg &\to \frac{1}{2}(2\Vg)\star(-X)
\Vgt &\to \frac{1}{2}(2\Vgt)\star(-X)
\end{align}

as defined in equation <ref> of the appendix.[The five-pointed star $\star$ of the BCH formula is not to be confused with the six-pointed star $\ast$ of the Hodge operator.] Following equation <ref> we find infinitesimal supergauge transformations

\begin{align}
\delta_X\Vg &= \frac{-\ad\Vg}{1-e^{-2\ad\Vg}}X,
\delta_X\Vgt &= \frac{-\ad\Vgt}{1-e^{-2\ad\Vgtt}}X.
\end{align}

Another kind of `gauge' symmetry is one which arises as a redundancy of the way we write down the spinorially flat solution to the constraints we imposed. Namely, we note that the spinorial superconnections $\Ag_\alpha$ and $\Agt_\alpha$ are left invariant under transformations

\begin{align}
e^{2\Vg} &\to e^\Lambdatt e^{2\Vg}
\Vg &\to \frac{1}{2}\big(\Lambdat\star(2\Vg)\big)
\\
e^{2\Vgtt} &\to e^\Lambda e^{2\Vgtt}
\Vg &\to \frac{1}{2}\big(\Lambda\star(2\Vgt)\big)
\end{align}

where $\Lambda$ and $\Lambdat$ are respectively taken to be $\gLie$-valued chiral and antichiral superfields. These transformations are referred to as pregauge transformations. Similarly one finds following the results of equation <ref> that the pregauge transformations are infinitesimally given by

\begin{align}
\delta_\Lambdatt\Vg &= \frac{-\ad\Vg}{1-e^{2\ad\Vg}}\Lambdat,
\delta_\Lambda\Vgt &= \frac{-\ad\Vgt}{1-e^{2\ad\Vgtt}}\Lambda.
\end{align}

In summary, we find that:

§.§.§ Gauge transformations in the prepotential formalism

The prepotentials $\Vg$ and $\Vgt$ contain gauge transformations

\begin{align}
e^{2\Vg} &\to e^\Lambdatt e^{2\Vg} e^{-X}
\Vg &\to \frac{1}{2}\big(\Lambdat\star(2\Vg)\star (-X)\big)
\\
e^{2\Vgtt} &\to e^\Lambda e^{2\Vgt} e^{-X}
\Vgt &\to \frac{1}{2}\big(\Lambda\star(2\Vgt)\star(-X)\big)
\end{align}

infinitesimally given by

\begin{align}
\delta_{X,\Lambdatt}\Vg &= \frac{-\ad\Vg}{1-e^{-2\ad\Vg}}X + \frac{-\ad\Vg}{1-e^{2\ad\Vg}}\Lambdat
\\
\delta_{X,\Lambda}\Vgt &= \frac{-\ad\Vgt}{1-e^{-2\ad\Vgtt}}X + \frac{-\ad\Vgt}{1-e^{2\ad\Vgtt}}\Lambda
\end{align}

where $X$ is a $\gLie$-valued general superfield parameter for the supergauge transformations and $\Lambdat$ and $\Lambda$ are respectively $\gLie$-valued antichiral and chiral superfield parameters of the pregauge transformations.

§.§ Wess-Zumino Gauge

We now move on to make a supergauge choice. This kind of gauge fixing is different from the kind we're used to in gauge theory. Usually when we think of gauge fixing this corresponds to imposing constraints on the field components of the connection 1-form. However, in the case of supergauge fixing the components which will be constrained will be the component fields of the supermultiplet.

§.§.§ The chiral representation

We start off with a gauge consition which is generally preserved by supersymmetry: We note that by supergauge transformations we have enough freedom to set one of the spinorial gauge superfields to zero. Choosing $X = 2\Vgt$ we set

\begin{align}
\Agt_\alpha &= 0
\nablat_\alpha &= \Dt_\alpha,
\Ag_\mu &= -\frac{i}{4}\Dt\gamma_\mu\Ag.
\end{align}

However taking into account the pregauge transformation freedom of $\Vgt$ we see that this doesn't fully use all the freedom of $X$. We're left with the freedom to take $X$ to be chiral. That is, the residual gauge symmetries of $\Vg$ are

\begin{align}
\label{chiral rep gauge transformations}
e^{2\Vg} &\to e^\Lambdatt e^{2\Vg}e^\Lambda
\Vg &\to \frac{1}{2}\big(\Lambdat\star(2\Vg)\star\Lambda\big)
\end{align}

where $\Lambda$ and $\Lambdat$ are taken to be respectively $\gLie$-valued chiral and antichiral superfields. This is the so-called chiral representation.

§.§.§ Wess-Zumino gauge

As it turns out, these residual supergauge transformations are in their turn enough to set to zero the chiral and the antichiral components of $\Vg$. It follows then that we are left with

\begin{equation}
\label{Wess-Zumino gauge}
\Vg = -\theta(i\As+\sigma)\thetat + i\lambda\theta\thetat{^2} + i\lambdat\thetat\theta^2 - \frac{1}{2}D\theta^2\thetat{^2}.
\end{equation}

We hence have the following field content (which are all $\gLie$-valued):

* A real 1-form $A_\mu$ of dimension 1

* A real scalar $\sigma$ of dimension 1

* Two independent complex spinors $\lambda$ and $\lambdat$ of dimensions $\frac{3}{2}$

* A real scalar $D$ of dimension 2

One could now wonder wether taking the bosonic field content to be real is consistent with the supersymmetry transformations. This issue was addressed in [22][23]. There the action is understood as analytically continued to the space of complexified fields. The path integral is then understood as integrating a holomorphic functional over a half-dimensional contour in complexified field space.

§.§ Residual Gauge Symmetries

We have now derived the $\gLie$-valued component fields for the vector multiplet. However, we haven't yet derived if and how their transformational properties relate to those of non-supersymmetric gauge theories.
As it turns out, these are related to the residual gauge symmetries of the Wess-Zumino gauge, which only partially fixes the gauge [3][24]. We note that the choices of $\Lambda$ and $\Lambdat$ in equation <ref> which preserve the Wess-Zumino gauge <ref> are parametrised by a real $\gLie$-valued local parameter $\vartheta$ (not a superfield) and given by

\begin{align}
\Lambda_\res(\vartheta) &= e^{+i\thetatt\dels\theta}(+\vartheta)
= +\vartheta - i\theta(\dels\vartheta)\thetat - \frac{1}{4}\square\vartheta\theta^2\thetat{^2}
\\
\Lambdat_\res(\vartheta) &= e^{-i\thetatt\dels\theta}(-\vartheta) = -\vartheta - i\theta(\dels\vartheta)\thetat + \frac{1}{4}\square\vartheta\theta^2\thetat{^2}
\end{align}

One finds that the prepotential $\Vg$ transforms under these residual gauge transformations as

\begin{equation}
\begin{split}
\delta_\vartheta\Vg
&= \frac{\ad\Vg}{1-e^{-2\ad\Vg}}\Lambda_\res(\vartheta) + \frac{-\ad\Vg}{1-e^{2\ad\Vg}}\Lambdat_\res(\vartheta)
\\
&\overeq{\ref{BCH infinitesimal expansion}} -\frac{1}{2}\big(\Lambdat_\res(\vartheta)+\Lambda_\res(\vartheta)\big) + \frac{1}{2}\big[\Vg, ~ \Lambdat_\res(\vartheta) - \Lambda_\res(\vartheta)\big]
\end{split}
\end{equation}

where higher order terms vanish identically due to the anticommuting nature of Grassmann numbers. Working this out we find that

\begin{equation}
\begin{split}
\delta_\vartheta\Vg
&= -\theta\Big(-i(\dels+\ad\As)\vartheta + [\vartheta,\sigma]\Big)\thetat + i[\vartheta,\lambda]\theta\thetat{^2} + i[\vartheta,\lambdat]\thetat\theta^2 - \frac{1}{2}[\vartheta,D]\theta^2\thetat{^2}
\\
&=: -\theta(i\delta_\vartheta\As + \delta_\vartheta\sigma)\thetat + i\delta_\vartheta\lambda\theta\thetat{^2} + i\delta_\vartheta\lambdat\thetat\theta^2 - \frac{1}{2}\delta_\vartheta D\theta^2\thetat{^2}
\end{split}
\end{equation}

In conclusion, we find that

§.§.§ Residual gauge symmetries of the Wess-Zumino gauge

\begin{align}
\delta_\vartheta A_\mu &= -\Dg_\mu\vartheta
\delta_\vartheta\sigma &= [\vartheta,\sigma]
\delta_\vartheta D &= [\vartheta,D]
\Big.\\
\delta_\vartheta\lambda &= [\vartheta,\lambda]
\delta_\vartheta\lambdat &= [\vartheta,\lambdat]
\Big.
\end{align}

That is, the residual gauge transformations parametrised by $\vartheta$ constitute ordinary gauge transformations, for which $A_\mu$ is a $G$-connection 1-form and $\sigma$, $\lambda$, $\lambdat$ and $D$ lie in the adjoint reperesentation of $G$.

§.§ Supersymmetries in the WZ Gauge

We now move on to discuss supersymmetry transformations in the Wess-Zumino gauge. These are generated in the same way as for the chiral multiplet, only now we have to include compensating gauge transformations which ensure the Wess-Zumino gauge is upheld. These compensating gauge transfromations can themselves be taken to be residual gauge transformations <ref> of the chiral representation since this condition is supersymmetric. Hence, we find that supersymmetry transformations are covariantly taken to be of the form

\begin{align}
\delta_{\zeta,\zetatt}\Vg &= \zeta Q + \zetat\Qt + \delta_{\Lambda,\Lambdatt},
\Lambda &= \Lambda(\zeta,\zetat,\Vg),
\Lambdat &= \Lambdat(\zeta,\zetat,\Vg)
\end{align}

where the gauge symmetry $\delta_{\Lambda,\Lambdatt}$ is included to ensure the preservation of the Wess-Zumino gauge.

§.§.§ Compensating gauge transformations

We start off by working out the compensating gauge transformations of the Wess-Zumino gauge. To this end we start off by noting that the out of gauge terms are given by

\begin{align}
(\zeta Q)\Vg\big|_{\text{out of gauge}} &= -\zeta(+i\As+\sigma)\thetat + i(\zeta\lambda)\thetat{^2}
\\
(\zetat\Qt)\Vg\big|_{\text{out of gauge}} &= -\zetat(-i\As+\sigma)\theta + i(\zetat\lambdat)\theta^2
\end{align}

From this we can read off that the parameters for the compensating gauge transformations are given by

\begin{align}
\begin{aligned}
\Lambdat &= 2e^{+i\thetatt\dels\theta}\Big[-\zeta(+i\As+\sigma)\thetat + i(\zeta\lambda)\thetat{^2}\Big]
\Big.\\
&= -2\zeta(+i\As+\sigma)\thetat + 2i(\zeta\lambda)\thetat{^2} + i\zeta(+i\As+\sigma)\delsl\theta\thetat{^2}
\Big.
\end{aligned}
\\
\begin{aligned}
\Lambda &= 2e^{-i\thetatt\dels\theta}\Big[-\zetat(-i\As+\sigma)\theta + i(\zetat\lambdat)\theta^2\Big]
\Big.\\
&= -2\zetat(-i\As+\sigma)\theta + 2i(\zetat\lambdat)\theta^2 + i\zetat(-i\As+\sigma)\delsl\thetat\theta^2
\Big.
\end{aligned}
\end{align}

In analogy to the case of the residual gauge symmetries we find that the compensating gauge transformations simplify to

\begin{equation}
\delta_{\Lambda,\Lambdatt}\Vg = -\frac{1}{2}\big(\Lambda+\Lambdat\big) + \frac{1}{2}\big[\Lambda-\Lambdat, ~ \Vg\big]
\end{equation}

again due to the anticommuting nature of Grassmann numbers.

§.§.§ Computing the SUSY transformations

We are now ready to compute the gauge covariant supersymmetry transformations. Inserting the aforementioned results we find (after some tedious computations) that

\begin{equation}
\begin{split}
\delta_{\zeta,\zetatt}\Vg
&= \big(\zeta Q + \zetat\Qt + \delta_{\Lambda,\Lambdatt}\big)\Vg
\\
\big(\zeta Q - \frac{1}{2}\ad\Lambdat\big)\Vg - \frac{1}{2}\Lambdat
\big(\zetat\Qt + \frac{1}{2}\ad\Lambda\big)\Vg - \frac{1}{2}\Lambda
\\
%&= \Big(\zeta\del + \zeta(i\nablas + \ad\sigma)\thetat\Big)\Vg - \frac{1}{2}\Lambdat + \Big(\zetat\delt + \zetat(i\nablas - \ad\sigma)\thetat\Big)\Vg - \frac{1}{2}\Lambda
\begin{aligned}[t]
&+ \theta\bigg[i\big(-\zeta\gamma_\mu\lambdat + \zetat\gamma_\mu\lambda\big) + \big(i\zeta\lambdat + i\zetat\lambda\big)\bigg]\thetat
\\
&+ i\zeta\bigg[iD - \frac{1}{2}\epsilon^{\mu\nu\rho}F_{\mu\nu}\gamma_\rho + \Dgs\sigma\bigg]\theta\thetat{^2}
+ i\zetat\bigg[iD + \frac{1}{2}\epsilon^{\mu\nu\rho}F_{\mu\nu}\gamma_\rho + \Dgs\sigma\bigg]\thetat\theta^2
\\
&- \frac{i}{2}\zeta\big(i\Dgs + \ad\sigma\big)\lambdat - \frac{i}{2}\zetat\big(i\Dgs - \ad\sigma\big)\lambda
\end{aligned}
\\
&=: +\theta\Big(i\gamma^\mu\delta_{\zeta,\zetatt}A_\mu + \delta_{\zeta,\zetatt}\sigma\Big)\thetat + i\delta_{\zeta,\zetatt}\lambda\theta\thetat{^2} + i\delta_{\zeta,\zetatt}\lambdat\thetat\theta^2 - \frac{1}{2}\delta_{\zeta,\zetatt}D\theta^2\thetat{^2}
\end{split}
\end{equation}

From this, we conclude:

§.§.§ Supersymmetry transformations of the WZ gauge vector multiplet

\begin{align}
\label{vector multiplet SUSY A}
\delta_{\zeta,\zetatt} A_\mu &= -\zeta\gamma_\mu\lambdat + \zetat\gamma_\mu\lambda
\bigg.\\
\delta_{\zeta,\zetatt}\sigma &= i\zeta\lambdat + i\zetat\lambda
\bigg.\\
\delta_{\zeta,\zetatt}\lambda &= \zeta\bigg[iD + \frac{1}{2}\epsilon^{\mu\nu\rho}F_{\mu\nu}\gamma_\rho + \Dgs\sigma\bigg]
\bigg.\\
\delta_{\zeta,\zetatt}\lambdat &= \zetat\bigg[iD - \frac{1}{2}\epsilon^{\mu\nu\rho}F_{\mu\nu}\gamma_\rho + \Dgs\sigma\bigg]
\bigg.\\
\label{vector multiplet SUSY D}
\delta_{\zeta,\zetatt} D &= i\zeta\big(i\Dgs + \ad\sigma\big)\lambdat + i\zetat\big(i\Dgs - \ad\sigma\big)\lambda
\bigg.
\end{align}

the gauge covariantised supersymmetry transformations, where henceforth we shall denote

\begin{equation}
\delta_{\zeta,\zetatt} = \delta_\zeta + \deltat_\zetatt
\end{equation}

as this will become convenient notation for localisations.

§.§ Commentary on the Algebra

Let us comment a bit on this algebra. We recall that the supersymmetry algebra is characterised by the anticommutation relation

\begin{align}
\{\delta_\zeta,\deltat_\zetatt\} &= -2i(\zeta\gamma^\mu\zetat)\del_\mu
\text{SUSY}^2 &= \text{transl}
\end{align}

That is, supersymmetry transformations square to translations. However, this doesn't take into account the fact that the Wess-Zumino gauge is not supersymmetric and has to be complemented by gauge transformations to stay in this gauge. Taking this into account the algebra takes on the form

\begin{gather}
\label{algebra WZ gauge vector multipley}
\{\delta_\zeta,\deltat_\zetatt\} = -2iK^\mu\del_\mu + \delta_{\text{gauge}}(-2iK^\mu A_\mu - 2\zeta\zetat\sigma)
\\
\Updownarrow\nonumber
\\
\text{SUSY}^2 = \text{transl} + \text{gauge}
\end{gather}

where we defined the Killing vector $K^\mu = \zeta\gamma^\mu\zetat$. That is, the algebra now becomes accompanied by a field dependent gauge transformation. Indeed, a direct computation yields generalisations of the nilpotency

\begin{equation}
\{\delta_\zeta,\delta_\eta\} = \{\deltat_\zetatt,\deltat_\etatt\} = 0
\end{equation}

as well as non-trivial relations

\begin{align}
\{\delta_\zeta,\deltat_\zetatt\}A_\mu &= -2iK^\nu F_{\nu\mu} + 2\zeta\zetat\Dg_\mu\sigma
\\
\{\delta_\zeta,\deltat_\zetatt\}\sigma &= -2iK^\mu\Dg_\mu\sigma
\\
\{\delta_\zeta,\deltat_\zetatt\}\lambda^\alpha &= -2iK^\mu\Dg_\mu\lambda^\alpha - 2\zeta\zetat[\sigma,\lambda^\alpha]
\\
\{\delta_\zeta,\deltat_\zetatt\}\lambdat{^\alpha} &= -2iK^\mu\Dg_\mu\lambdat{^\alpha} - 2\zeta\zetat[\sigma,\lambdat{^\alpha}]
\\
\{\delta_\zeta,\deltat_\zetatt\}D &= -2iK^\mu\Dg_\mu D - 2\zeta\zetat[\sigma,D]
\end{align}

in agreement with the algebra <ref>.

§.§.§ An aside on odd derivatives

A source of some confusion could the question of whether to take the (Grassmann-odd!) supersymmetry transformation $\delta$ to be a left or a right derivative. As it turns out despite us using left derivatives it are right derivatives which are more fundamental [25]. However, the difference will be at most an overall minus sign if the object acted on isn't the sum of objects with differing Grassmann parities. Hence, for our purposes the distinction won't be relevant. A way to translate between left and right acting derivatives is through

\begin{align}
(\text{even})\deltal &= +\delta(\text{even}),
(\text{odd})\deltal &= -\delta(\text{odd}).
\end{align}

§ SUPERSYMMETRIC ACTIONS AND LOCALISATION

In this section we'll go over two kinds of supersymmetric gauge theories: super-Yang-Mills theory and super-Chern-Simons theory. Particularly, as it turns out, it is in super-Yang-Mills theory that we find the clue to localising supersymmetric gauge theories.

§.§ Super-Yang-Mills Theory

We start off by giving a description of super-Yang-Mills theory. We start off by noting a very interesting and relevant fact to this thesis. The action of super-Yang-Mills theory is $Q$-exact [7][8]! The action of super-Yang-Mills theory is given by

\begin{equation}
\label{SYM action}
\frac{1}{g^2}\int\extd^3x ~ \tr\bigg[-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} - \frac{1}{2}\Dg_\mu\sigma\Dg^\mu\sigma - \frac{1}{2}D^2
- \lambdat\big(i\Dgs - \ad\sigma\big)\lambda\bigg]
\end{equation}

where $g^2$ a constant of mass dimension 1 and where we used $\Vg$ to denote the collective field content of the vector multiplet in WZ gauge. Showing this action is supersymmetric will be done through its $Q$-exactness. Particularly, we note that this action can be written in different ways as

\begin{align}
\label{SYM exact 1}
&= \frac{1}{2|\zeta|^2}\delta_\zeta\int\extd^3x ~ \tr\Big[(\delta_\zeta\lambda)^\dagger\lambda\Big]
\bigg.\\
\label{SYM exact 2}
&= \frac{1}{2|\zeta|^2}\deltat_\zeta\int\extd^3x ~ \tr\Big[(\deltat_\zeta\lambdat)^\dagger\lambdat\Big]
\bigg.\\
\label{SYM exact 3}
&= \frac{1}{4|\zeta|^2}\Qloc_\zeta\int\extd^3x ~ \tr\bigg[(\Qloc_\zeta\lambda)^\dagger\lambda + (\Qloc_\zeta\lambdat)^\dagger\lambdat\bigg]
\bigg.
\end{align}

where we defined

\begin{equation}
\Qloc_\zeta := \delta_\zeta + \deltat_\zetatt
\end{equation}

where now we take the two conjugate SUSY trasformations but with the same non-zero constant bosonic spinor $\zeta$. For conventions regarding the norm $|\bullet|^2$ and Hermitian conjugation as it relates to the index formalism for spinors we refer the reader to section <ref> of the appendix. Let us verify this fact a little more explicitly:

§.§.§ Expressions <ref> and <ref>

We start off by taking a look at the first description, that is, expression <ref>. We find that

\begin{equation}
g^2S_\SYM[\Vg] = \frac{1}{2|\zeta|^2}\int\extd^3x ~ \tr\bigg[|\delta_\zeta\lambda|^2 + \delta_\zeta(\delta_\zeta\lambda)^\dagger\lambda\bigg]
\end{equation}

This gives us respectively bosonic and fermionic sectors

\begin{align}
g^2S_\SYM[\Vg]\big|_\bos &= \frac{1}{2|\zeta|^2}\int\extd^3x ~ \tr|\delta_\zeta\lambda|^2,
\\
g^2S_\SYM[\Vg]\big|_\fer &= \frac{1}{2|\zeta|^2}\int\extd^3x ~ \tr\delta_\zeta(\delta_\zeta\lambda)^\dagger\lambda
\end{align}

Working out the bosonic part in some closer detail will be important since due to it's exactness it will function as a localising term for which the bosonic part will be responsible for the Gaussian damping central to localisations. Noting that

\begin{align}
\delta_\zeta\lambda &= \Big[iD - \frac{1}{2}\epsilon^{\mu\nu\rho}F_{\mu\nu}\gamma_\rho - \Dgs\sigma\Big]\zeta
\\
(\delta_\zeta\lambda)^\dagger &= \zetad\Big[iD + \frac{1}{2}\epsilon^{\mu\nu\rho}F_{\mu\nu}\gamma_\rho + \Dgs\sigma\Big]
\end{align}

we find that

\begin{equation}
\label{computation partial localisation}
\begin{split}
\tr|\delta_\zeta\lambda|^2
%&= \tr\Big[(\delta_\zeta\lambda)^\dagger\delta_\zeta\lambda\Big]
&= \tr\zetad\bigg[-D^2-\Big(\frac{1}{2}\epsilon^{\mu\nu\rho}F_{\mu\nu}\gamma_\rho + \Dgs\sigma\Big)^2\bigg]\zeta
\\
&= |\zeta|^2\tr\bigg[-D^2 - \Big(\frac{1}{2}\epsilon_{\mu\rho\sigma}F^{\rho\sigma} + \Dg_\mu\sigma\Big)\Big(\frac{1}{2}\epsilon^{\mu\rho\sigma}F_{\rho\sigma} + \Dg^\mu\sigma\Big)\bigg]
\\
&= |\zeta|^2\tr\bigg[-D^2 - \frac{1}{2}F_{\mu\nu}F^{\mu\nu} - \Dg_\mu\sigma\Dg^\mu\sigma - \Dg_\mu\big(\epsilon^{\mu\rho\sigma}F_{\rho\sigma}\sigma\big)\bigg]
\end{split}
\end{equation}

where we used the Bianchi identity $\Dg F \equiv 0$ to turn the final term into a total derivative. This yields a bosonic sector

\begin{equation}
= \frac{1}{g^2}\int\extd^3x ~ \tr\bigg[-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} - \frac{1}{2}\Dg_\mu\sigma\Dg^\mu\sigma - \frac{1}{2}D^2\bigg]
\end{equation}

which is indeed in agreement with super-Yang-Mills action <ref>. Doing the fermionic sector is straightforward but tedious and not particularly insightful to any future discussion of the localisation of the Third Way Theory. Hence we will simply give the answer which is

\begin{equation}
\label{SYM action fermionic}
= \frac{1}{g^2}\int\extd^3x ~ \tr\bigg[-\lambdat\big(i\Dgs - \ad\sigma\big)\lambda\bigg]
\end{equation}

also in agreement with the super-Yang-Mills action <ref>. This verifies that the expression <ref> agrees with the super-Yang-Mills action <ref>. The agreement of expresion <ref> follows similarly.

§.§.§ Expression <ref>

As for the third expression for the super-Yang-Mills action, equation <ref>, there are a couple of interesting remarks to make which will become relevant later on when looking at the Third Way Theory. For this expression we find that

\begin{equation}
g^2S_\SYM[\Vg] = g^2S_\SYM[\Vg]\big|_\bos + g^2S_\SYM[\Vg]\big|_\fer
\end{equation}

with bosonic and fermionic sectors

\begin{align}
\frac{1}{4|\zeta|^2}\int\extd^3x ~ \tr\bigg[|\Qloc_\zeta\lambda|^2 + |\Qloc_\zeta\lambdat|^2\bigg]
\\
\frac{1}{4|\zeta|^2}\int\extd^3x ~ \tr\bigg[\Qloc_\zeta(\Qloc_\zeta\lambda)^\dagger\lambda + \Qloc_\zeta(\Qloc_\zeta\lambdat)^\dagger\lambdat\bigg]
\end{align}

In close analogy to the computations for expression <ref> we find for the bosonic sector that

\begin{align}
\label{complete localisation SYM 1}
\tr|\Qloc_\zeta\lambda|^2
&= |\zeta|^2\tr\bigg[-D^2 - \Big(\frac{1}{2}\epsilon_{\mu\rho\sigma}F^{\rho\sigma} + \Dg_\mu\sigma\Big)\Big(\frac{1}{2}\epsilon^{\mu\rho\sigma}F_{\rho\sigma} + \Dg^\mu\sigma\Big)\bigg]
\\
\label{complete localisation SYM 2}
\tr|\Qloc_\zeta\lambdat|^2
&= |\zeta|^2\tr\bigg[-D^2 - \Big(\frac{1}{2}\epsilon_{\mu\rho\sigma}F^{\rho\sigma} - \Dg_\mu\sigma\Big)\Big(\frac{1}{2}\epsilon^{\mu\rho\sigma}F_{\rho\sigma} - \Dg^\mu\sigma\Big)\bigg]
\end{align}

where the reader is encouraged to note the relative minus signs between the two expressions. Because of these relative minus signs one finds that

\begin{equation}
\label{complete localisation SYM 3}
\frac{1}{2}\tr\Big[|\Qloc_\zeta\lambda|^2 + |\Qloc_\zeta\lambdat|^2\Big]
\tr\bigg[-D^2 - \frac{1}{2}F_{\mu\nu}F^{\mu\nu} - \Dg_\mu\sigma\Dg^\mu\sigma\bigg]
\end{equation}

without invoking the Bianchi identities! Now, one may wonder why this fact could possibly be useful, since it just yields the same result as previous cases modulo a total derivative. The reason that this is interesting is that the localisation of the Third Way Theory will be attempted through a deformation of the supersymmetry transformations. With these deformations it will be the case that we can't use the Bianchi identities as we did for expressions <ref> and <ref>, and hence an analogue of expression <ref> will be our starting point for localising the Third Way Theory.

To continue our discussion, we find that

\begin{equation}
= \frac{1}{g^2}\int\extd^3x ~ \tr\bigg[-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} - \frac{1}{2}\Dg_\mu\sigma\Dg^\mu\sigma - \frac{1}{2}D^2\bigg]
\end{equation}

nicely in agreement with the super-Yang-Mills action <ref> and without having to care about boundary terms. As for the fermionic sector, we start off by noting that

\begin{align}
&= \zetad\Big[iD + \frac{1}{2}\epsilon^{\mu\nu\rho}F_{\mu\nu}\gamma_\rho + \Dgs\sigma\Big]
= \delta_\zetad\lambda
\end{align}

with in analogy following

\begin{align}
(\delta_\zeta\lambda)^\dagger &= \delta_\zetad\lambda,
(\deltat_\zeta\lambdat)^\dagger &= \deltat_\zetad\lambdat,
(\Qloc_\zeta\lambda)^\dagger &= \Qloc_\zetad\lambda,
(\Qloc_\zeta\lambdat)^\dagger &= \Qloc_\zetad\lambdat.
\end{align}

We thus find that

\begin{equation}
\begin{split}
&\Qloc_\zeta(\Qloc_\zeta\lambda)^\dagger\lambda + \Qloc_\zeta(\Qloc_\zeta\lambdat)^\dagger\lambdat
\\
(\delta_\zeta + \deltat_\zeta)\delta_\zetad\lambda\lambda + (\delta_\zeta + \deltat_\zeta)\deltat_\zetad\lambdat\lambdat
\\
\big(\delta_\zeta\delta_\zetad\lambda\big)\lambda + \big(\{\deltat_\zeta,\delta_\zetad\}\lambda\big)\lambda
\big(\deltat_\zeta\deltat_\zetad\lambdat\big)\lambdat + \big(\{\delta_\zeta,\deltat_\zetad\}\lambdat\big)\lambdat
\\
\underbrace{
\Big[\delta_\zeta(\delta_\zeta\lambda)^\dagger\lambda + \deltat_\zeta(\deltat_\zeta\lambdat)^\dagger\lambdat\Big]
}_{\text{compare to \ref{SYM exact 1} and \ref{SYM exact 2}}}
\underbrace{
\Big[\{\deltat_\zeta,\delta_\zetad\}(\lambda^2) + \{\delta_\zeta,\deltat_\zetad\}(\lambdat{^2})\Big]
}_{\text{transl + gauge}}
\end{split}
\end{equation}

Recalling the algebra as well as the results from expressions <ref> and <ref> as well as the fact that $\int\tr(\text{transl + gauge}) = 0$ we find that

\begin{equation}
= \frac{1}{g^2}\int\extd^3x ~ \tr\bigg[-\lambdat\big(i\Dgs - \ad\sigma\big)\lambda\bigg]
\end{equation}

again nicely agreeing with the super-Yang-Mills action <ref>.

§.§.§ Supersymmetry of SYM theory

It now follows almost trivially that the super-Yang-Mills action is invariant under supersymmetry. Indeed, we find that

\begin{align}
g^2\delta_\zeta S_\SYM &= \frac{1}{2|\zeta|^2}\delta^2_\zeta\int\extd^3x ~ \tr\Big[(\delta_\zeta\lambda)^\dagger\lambda\Big] = 0
\\
g^2\deltat_\zeta S_\SYM &= \frac{1}{2|\zeta|^2}\deltat{^2_\zeta}\int\extd^3x ~ \tr\Big[(\deltat_\zeta\lambdat)^\dagger\lambdat\Big] = 0
\end{align}

proving that super-Yang-Mills theory is indeed supersymmetric.

§.§ Super-Chern-Simons Theory

We now move on to describe the superanalogue of Chern-Simons theory [7][11][21][26]. Super-Chern-Simons theory at level $k$ is described through an action

\begin{equation}
\label{SCS action}
= \frac{k}{2\pi}S_\SCS[\Vg]
= \frac{k}{2\pi}S_\CS[A] + \frac{k}{2\pi}\int\extd^3x ~ \tr\Big[\lambda\lambdat - \sigma D\Big].
\end{equation}

This theory has a very interesting property which will become relevant when studying the Third Way Theory: Chern-Simons theory and its superanalogue are physically equivalent! Indeed, the additional field content $\sigma$, $\lambda$, $\lambdat$, $D$ can all be integrated out and the dynamics of the connection 1-form are left unchanged. This will be interesting in the case of the Third Way Theory because if it is physically equivalent to its conjectured superanalogue and it can be localised then we've achieved essentially a localisation of the Third Way Theory as is without having to change its physics. These kind of circumstance is exceedingly rare and being able to achieve this would prove significant theoretical progress.

§.§.§ Supersymmetry of SCS theory

Let us now verify that this theory is supersymmetric. This is a relatively short computation but we will do so explicitly because this is also instructive to understanding progress made towards localising the Third Way Theory.

Now to perform this computation: Varying this action yields

\begin{equation}
\label{SCS computation 1}
\delta S_\SCS[\Vg]
\int\extd^3x ~ \tr\bigg[\frac{1}{2}\epsilon^{\mu\nu\rho}F_{\mu\nu}\delta A_\rho + \delta\lambda\lambdat + \delta\lambdat\lambda - D\delta\sigma - \sigma\delta D\bigg]
\end{equation}

We hence find that the different contributions from supersymmetry transformations are given as follows:
We start off by writing out the most complicated contribution to this transformation which comes from the gaugino fields $\lambda$ and $\lambdat$. Here we find that

\begin{equation}
\label{SCS computation 2}
\begin{multlined}
\int\extd^3x ~ \tr\Big[\delta\lambda\lambdat + \delta\lambdat\lambda\Big]
\int\extd^3x ~ \tr\bigg[\zeta\Big(iD + \frac{1}{2}\epsilon^{\mu\nu\rho}F_{\mu\nu}\gamma_\rho + \Dgs\sigma\Big)\lambdat
\\
+ \zetat\Big(iD - \frac{1}{2}\epsilon^{\mu\nu\rho}F_{\mu\nu}\gamma_\rho + \Dgs\sigma\Big)\lambda\bigg]
\end{multlined}
\end{equation}

This consists of three pairs of terms. We find that the $D$ terms cancel against

\begin{equation}
\label{SCS computation 3}
\int\extd^3x ~ \tr\Big[-D\delta\sigma\Big]
\int\extd^3x ~ \tr\Big[\zeta(-iD)\lambdat + \zetat(-iD)\lambda\Big].
\end{equation}

We furthermore find that the $\Dg\sigma$ terms cancel against

\begin{multline}
\label{SCS computation 4}
\int\extd^3x ~ \tr\Big[-\sigma\delta D\Big]
\int\extd^3x ~ \tr\Big[-i\sigma\zeta\big(i\Dgs + \ad\sigma\big)\lambdat - i\sigma\zetat\big(i\Dgs - \ad\sigma\big)\lambda\Big]
\\
\int\extd^3x ~ \tr\Big[\zeta(-\Dgs\sigma)\lambdat + \zetat(-\Dgs\sigma)\lambda\Big]
\end{multline}

where we used the Leibniz rule as well as the fact that $[\sigma,\sigma] = 0$. Finally we move on to the $F$ terms. Interestingly, these cancel because they are proportional to the Chern-Simons field equations. Indeed, one finds that

\begin{equation}
\label{SCS computation 5}
\int\extd^3x ~ \tr\bigg[\frac{1}{2}\epsilon^{\mu\nu\rho}F_{\mu\nu}\delta A_\rho\bigg]
\int\extd^3x ~ \tr\bigg[-\zeta\Big(\frac{1}{2}\epsilon^{\mu\nu\rho}F_{\mu\nu}\gamma_\rho\Big)\lambdat + \zetad\Big(\frac{1}{2}\epsilon^{\mu\nu\rho}F_{\mu\nu}\gamma_\rho\Big)\lambda\bigg]
\end{equation}

indeed cancelling against the $F$ terms in the variation of the gauginos. We this conclude that

\begin{equation}
\delta_\zeta S_\SCS[\Vg] = \deltat_\zetatt S_\SCS[\Vg] = 0.
\end{equation}

That is, super-Chern-Simons theory is indeed supersymmetric.

§.§ Localisation of SYM and SCS Theory

Let us now move on to describe the way supersymmetric gauge theories are localised. As already mentioned, this will be done through the super-Yang-Mills action. Particularly, through the observation that expectation value of BPS operators in a SYM theory is indepedent on the coupling strength $g^2$ of the theory. As such, we can take the limit $t := g^{-2} \to \infty$ localising the theory. Here we will discuss two differing localisation schemes and discuss their equivalence and their meaning in the context of the Third Way Theory.

§.§.§ $\delta$-localisation scheme

This localisation scheme will be centered around the localising supersymmetry $\delta_\zeta$ (the $\deltat_\zeta$ case follows similarly). To rephrase in the language of section <ref>, we have a fermionic functional given by

\begin{equation}
\Fg_\loc[\Vg] = \frac{1}{2|\zeta|^2}\int\extd^3x ~ \tr\Big[(\delta_\zeta\lambda)^\dagger\lambda\Big].
\end{equation}

The corresponding localising action is then given by

\begin{equation}
tS_\loc[\Vg] = t\delta_\zeta\Fg_\loc[\Vg] = S_\SYM[\Vg]\big|_{t=g^{-2}}.
\end{equation}

The BPS operators then consist of operators which are annihilated by $\delta_\zeta$ and the BPS configurations on which the path integral localises are given by

\begin{align}
F_{\mu\nu} &= 0
\Dg_\mu\sigma &= 0
D &= 0
\end{align}

all in accordance to the discussion of section <ref>. For this localisation argument to make sense we need the fermionic localising functional to be annihilated by $\delta^2_\zeta$. However we recall from our discussion in section <ref> that

\begin{align}
\delta_\zeta^2 &= \frac{1}{2}\{\delta_\zeta,\delta_\zeta\} \equiv 0
\delta^2_\zeta\Fg_\loc[\Vg] &= 0
\end{align}

thus verifying that this is indeed the case.

§.§.§ $\Qloc$-localisation scheme

We now move on to the $\Qloc$-localisation scheme which is based on the localising supersymmetry transformation $\Qloc_\zeta$. In this case the fermionic localising functional is given by

\begin{equation}
\Fg_\loc[\Vg]
\frac{1}{4|\zeta|^2}\int\extd^3x ~ \tr\Big[(\Qloc_\zeta\lambda)^\dagger\lambda + (\Qloc_\zeta\lambdat)^\dagger\lambdat\Big]
\end{equation}

which again yields a localising action

\begin{equation}
tS_\loc[\Vg] = t\Qloc_\zeta\Fg_\loc[\Vg] = S_\SYM[\Vg]\big|_{t=g^{-2}}.
\end{equation}

In this scheme the BPS operators will consist of those annihilated by $\Qloc_\zeta$ and the localisation locus will again consist of BPS configurations

\begin{align}
F_{\mu\nu} &= 0
\Dg_\mu\sigma &= 0
D &= 0.
\end{align}

Most seems to be the same for this localisation argument compared to the previous case. However, now the condition for this localisation argument to make sense is somewhat different. Now, we need the fermionic localising functional to be annihilated by $\Qloc_\zeta^2$. In accordance to our discussion in previous sections we recall that this is indeed the case since

\begin{equation}
\Qloc_\zeta^2 = \frac{1}{2}\{\Qloc_\zeta,\Qloc_\zeta\} = -2iK^\mu\del_\mu + \delta_{\text{gauge}}(-2iK^\mu A_\mu)
\end{equation}

where we recall that we defined the Killing vector $K^\mu = \zeta\gamma^\mu\zeta$ as a spinor bilinear from the Killing spinor $\zeta$. From this it follows that

\begin{equation}
\Qloc_\zeta^2\Fg_\loc[\Vg] = \int\extd^3x ~ \tr\Big[\Qloc_\zeta^2\big\{\dots\big\}\Big] = 0.
\end{equation}

We thus find that the $\Qloc$-localisation scheme also works.

§.§.§ Comparison between the two schemes

Let us now compare these two schemes. We note that

* The $\delta$-localisation scheme uses the Bianchi identity while the $\Qloc$-localisation scheme doesn't. If one considers different supersymmetry algebras (e.g. include a central charge) the Bianchi identity might not be sufficient anymore to seperate the $A_\mu$ and $\sigma$ loci. Furthermore if this isn't the case anymore the resulting locus might not be on-shell anymore.

* The $\delta$-localisation scheme uses the nilpotency of $\delta_\zeta$ while the $\Qloc$-scheme uses the fact that $\Qloc_\zeta^2 = \text{transl} + \text{gauge}$. If one changes to algebra the nilpotency remains generically unaffected while the nontrivial relation becomes augmented, possibly in a way which could obstruct the localisation argument.

We hence find that both schemes in more general contexts have strong and weak points. For more general supersymmetry representations the $\delta$-localisation scheme is more likely to work but might localise to looser, possibly off-shell configurations. On the other hand, the $\Qloc$-localisation scheme usually will localise on-shell but has stronger requirements to work. In more general contexts —particularly the Third Way Theory— these issues will become glaringly clear.

§ PROCA-CHERN-SIMONS THEORY

§.§ Non-Supersymmetric Formulation

Before moving on to discussing how to localise the Third Way Theory, we will discuss a toy model, namely a model which we coin Proca-Chern-Simons theory. As the name suggests, this theory is simply given by combining a Proca term with Chern-Simons theory. That is, we consider a theory given by an action

§.§.§ Proca-Chern-Simons Theory

\begin{equation}
S_\PCS[A] = S_\CS[A] - \int\tr\Big[\frac{m}{2}A\ast A\Big]
\end{equation}

with $A$ a $G$-connection 1-form, where for simplicity we assume Euclidean signature, and where $m$ is a parameter of mass dimension 1. This theory has field equations

\begin{equation}
\label{PCS field equations}
F - m\ast A = 0.
\end{equation}

As is generally the case the case for Proca terms, they break gauge symmetry. The gauge group is broken down to

\begin{equation}
G \to \{1\}.
\end{equation}

Let us now take a closer look at the field equations <ref> of this theory. Taking the covariant exterior derivative of these field equations one finds

\begin{equation}
\label{Lorenz gauge}
0 = \Dg\big(F - m\ast A\big) \equiv -m\extd\ast A
\end{equation}

where we used the Bianchi identity as well as the fact that $[A,\ast A] = 0$. This is very interesting, because the Lorenz gauge $\extd\ast A = 0$ is implied by the field equations themselves. This can be regarded as a consequence of the fact that the Proca mass term breaks gauge invariance.

§.§ Super-Proca-Chern-Simons Theory

Let us now supersymmetrise this theory. The way we supersymmetrise this theory will be by deforming the supersymmetry transformations of super-Chern-Simons theory in such a way that we're forced to include a mass term. That is to say, in strong analogy to the action <ref> of super-Chern-Simons theory we have:

[breakable, enhanced]

§.§.§ Super-Proca-Chern-Simons Action

\begin{multline}
\label{SPCS action}
= S_\SCS[\Vg] - \int\tr\bigg[\frac{m}{2}A\ast A + \frac{m}{2}\sigma\ast\sigma\bigg]
\\
= S_\CS[A] + \int\extd^3x ~ \tr\bigg[\lambda\lambdat - \sigma D - \frac{m}{2}A_\mu A^\mu - \frac{m}{2}\sigma^2\bigg]
\end{multline}

where $\Vg = (A_\mu,\sigma,\lambda,\lambdat,D)$ denotes the field content of the WZ gauge vector multiplet. The field equations are given by

\begin{align}
F - m\ast A &= 0
\lambda &= 0
\lambdat &= 0
D + m\sigma &= 0
\sigma &= 0.
\end{align}

The supersymmetries of this theory are deformed supersymmetry transformations of the vector multiplet $\Vg$, chosen to satisfy three purposes:

* supersymmetrise Proca-Chern-Simons theory,

* yield nilpotent supersymmetry transformations, i.e. $\delta_\zeta^2 = \deltat{^2_\zetatt} = 0$,

* localise Proca-Chern-Simons theory to on-shell field configurations.

The supersymmetry transformations we arrived at are then given by:

[breakable, enhanced]

§.§.§ Massive Supersymmetry Transformations

\begin{align}
\label{SPCS SUSY transformation A}
\delta A_\mu &= -\zeta\gamma_\mu\lambdat + \zetat\gamma_\mu\lambda
\bigg.\\
\label{SPCS SUSY transformation sigma}
\delta\sigma &= i\zeta\lambdat + i\zetat\lambda
\bigg.\\
\label{SPCS SUSY transformation lambda}
\delta\lambda &= \zeta\bigg[iD + \frac{1}{2}\epsilon^{\mu\nu\rho}F_{\mu\nu}\gamma_\rho + \Dgs\sigma \mathcolor{blue}{- m\As + im\sigma}\bigg]
\bigg.\\
\label{SPCS SUSY transformation lambdab}
\delta\lambdat &= \zetat\bigg[iD - \frac{1}{2}\epsilon^{\mu\nu\rho}F_{\mu\nu}\gamma_\rho + \Dgs\sigma \mathcolor{blue}{+ m\As + im\sigma}\bigg]
\bigg.\\
\label{SPCS SUSY transformation D}
\delta D &= i\zeta\big(i\Dgs + \ad\sigma\big)\lambdat + i\zetat\big(i\Dgs - \ad\sigma\big)\lambda
\bigg.
\end{align}

where the highlighted terms are the `deformations' of the usual supersymmetry transformations <ref>-<ref>. Note that these are proportional to the mass parameter $m$ of Proca-Chern-Simons theory. Again we denote

\begin{equation}
\delta_{\zeta,\zetatt} = \delta_\zeta + \deltat_\zetatt
\end{equation}

from now on.

The supersymmetry of the action follows in complete analogy to the computations <ref>-<ref>. To highlight why this action still is supersymmetric, we note that the new contributions to <ref> are countered by the SUSY transformations of the mass terms in the SPCS action:

\begin{equation}
\begin{split}
\delta S_\SCS[\Vg]\big|_{\textcolor{blue}{\text{new}}}
\int\extd^3x ~ \tr\Big[\delta\lambda\lambdat + \delta\lambdat\lambda\Big]\Big|_{\textcolor{blue}{\text{new}}}
\\
\int\extd^3x ~ \tr\Big[\zeta\big(-m\As+im\sigma\big)\lambdat + \zetat\big(m\As+im\sigma\big)\lambda\Big]
\\
&= -\delta\int\tr\bigg[-\frac{m}{2}A\ast A - \frac{m}{2}\sigma\ast\sigma\bigg].
\end{split}
\end{equation}

As for the nilpotency of the supersymmetry transformations, this follows from the results in section <ref> as well as the observation that

\begin{equation}
\label{tussenstap 3}
\delta(i\As \pm \sigma)_\alpha{^\beta}
\begin{cases}
-2i\zeta_\alpha\lambdat{^\beta} + 2i\zetat{^\beta}\lambda_\alpha & \uparrow
\\
-2i\zeta^\beta\lambdat_\alpha + 2i\zetat_\alpha\lambda^\beta & \downarrow
\end{cases}
\end{equation}

which follows as a direct consequence of the Fierz identity <ref>. The nilpotency of the SUSY transformations on $A_\mu$, $\sigma$ and $D$ follows directly from previous results and for the gauginos $\lambda$ and $\lambdat$ we find

\begin{align}
\delta^2_\zeta\lambda^\alpha\big|_\new
+im\zeta^\beta\delta_\zeta(i\As + \sigma)_\beta{^\alpha}
\overeq{\ref{tussenstap 3}} 2m\zeta^2\lambdat{^\alpha} = 0
\\
\deltat{^2_\zetatt}\lambdat{^\alpha}\big|_\new
-im\zetat{^\beta}\deltat_\zetatt(i\As - \sigma)_\beta{^\alpha}
\overeq{\ref{tussenstap 3}} 2m\zetat{^2}\lambda^\alpha = 0
\end{align}

proving that indeed

\begin{equation}
\label{nilpotency}
\delta^2_\zeta = \deltat{^2_\zetatt} = 0
\end{equation}

the massive SUSY transformations are nilpotent.

§.§ The Algebra of Massive SUSY

Let us now more closely work out the algebra associated to these massive SUSY transformations. A straightforward if not somewhat tedious computation yields

\begin{equation}
\{\delta_\zeta,\delta_\eta\} = \{\deltat_\zetatt,\deltat_\etatt\} = 0.
\end{equation}

As for the remaining anticommutators, we find

\begin{align}
\{\delta_\zeta,\deltat_\zetatt\}A_\mu &= -2iK^\nu (F_{\nu\mu} - m\epsilon_{\nu\mu\rho}A^\rho) + 2\zeta\zetat\Dg_\mu\sigma
\\
\{\delta_\zeta,\deltat_\zetatt\}\sigma &= -2iK^\mu\Dg_\mu\sigma
\\
\{\delta_\zeta,\deltat_\zetatt\}\lambda^\alpha &= -2iK^\mu\Dg_\mu\lambda^\alpha - 2\zeta\zetat[\sigma,\lambda^\alpha] - 2m\zetat{^\alpha}(\zeta\lambda)
\\
\{\delta_\zeta,\deltat_\zetatt\}\lambdat{^\alpha} &= -2iK^\mu\Dg_\mu\lambdat{^\alpha} - 2\zeta\zetat[\sigma,\lambdat{^\alpha}] - 2m\zeta^\alpha(\zetat\lambdat)
\\
\{\delta_\zeta,\deltat_\zetatt\}D &= -2iK^\mu\Dg_\mu (D + im\sigma) - 2\zeta\zetat[\sigma,D] + 2m\zeta\zetat\del_\mu A^\mu
\end{align}

Interestingly, the algebra now takes on the form

\begin{equation}
\text{SUSY}^2 = \text{transl} + \text{gauge} + \text{mass}
\end{equation}

where `mass' refers to the inclusion of terms proportional to the mass $m$. Interestingly, this does not have the interpretation of a central charge. Originally it was attempted to realise the mass term in Proca-Chern-Simons theory as arising through a central charge. However, the relative signs between the new terms in $\delta\lambda$ and $\delta\lambdat$ in equations <ref> and <ref> this can't be the case (compare to e.g. [11]). Furthermore, such relative signs to our knowledge wouldn't allow for the construction of a supersymmetric Proca-Chern-Simons action.

§.§ Localisation of Proca-Chern-Simons Theory

In this section we will study the localisation of Proca-Chern-Simons theory. It should be noted that due to the fact that all additional field content of SPCS being auciliary a localisation of PCS theory and SPCS theory would be equivalent.

§.§.§ $\delta$-Localisation Scheme

Let us start off by studying the $\delta$-localisation scheme. In this scheme the fermionic localising function will be given by

\begin{equation}
\Fg_\loc[\Vg]
\frac{1}{2|\zeta|^2}\int\extd^3x ~ \tr\Big[(\delta_\zeta\lambda)^\dagger\lambda\Big].
\end{equation}

We now work out the corresponding localising action

\begin{equation}
S_\loc[\Vg] = \delta_\zeta\Fg_\loc[\Vg] = \frac{1}{2|\zeta|^2}\int\extd^3x ~ \tr\Big[|\delta_\zeta\lambda|^2 + \delta_\zeta(\delta_\zeta\lambda)^\dagger\lambda\Big].
\end{equation}

Starting off with the bosonic part we find in analogy to computation <ref> that

\begin{equation}
\begin{split}
\frac{1}{2|\zeta|^2}|\delta_\zeta\lambda|^2
\tr\bigg[
\begin{aligned}[t]
&-\frac{1}{2} (D+m\sigma)^2 - \frac{1}{2}\Big(\frac{1}{2}\epsilon_{\mu\rho\sigma}F^{\rho\sigma} - mA_\mu + \Dg_\mu\sigma\Big)
\\
\Big(\frac{1}{2}\epsilon^{\mu\rho\sigma}F_{\rho\sigma} - mA^\mu + \Dg^\mu\sigma\Big)\bigg]
\end{aligned}
\\
\begin{aligned}[t]
&-\frac{1}{2} (D + m\sigma)^2 - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} - \frac{1}{2}\Dg_\mu\sigma\Dg^\mu\sigma
\\
&- \frac{1}{2}\Dg_\mu\Big(\frac{1}{2}\epsilon^{\mu\rho\sigma}F_{\rho\sigma}\sigma-mA^\mu\sigma\Big)
+m\del_\mu A^\mu\sigma\bigg].
\end{aligned}
\end{split}
\end{equation}

It is here that our remarks with regards to the comparison of the $\delta$- and $\Qloc$-localisation schemes become important. Particularly, we find that an obstruction to the seperation of the two field equations in the $\delta$-localisation scheme is proportional to $\del_\mu A^\mu$. Interestingly, if we had a good reason for assuming a Lorenz gauge $\del_\mu A^\mu = 0$ this would seperate the two field equations. We hence arrive at a bosonic localising action

\begin{equation}
\begin{split}
\int\extd^3x ~ \tr\bigg[
\begin{aligned}[t]
&- \frac{1}{4}\Big(F_{\mu\nu} - m\epsilon_{\mu\nu\rho}(A+\Dg\sigma)^\rho\Big)
\Big(F^{\mu\nu} - m\epsilon^{\mu\nu\rho}(A+\Dg\sigma)_\rho\Big)
\\
&-\frac{1}{2} (D+m\sigma)^2\bigg]
\end{aligned}
\\
&= \int\extd^3x ~ \tr\bigg[
\begin{aligned}[t]
&-\frac{1}{2}(D+m\sigma)^2 - \frac{1}{4}\big(F_{\mu\nu} - m\epsilon_{\mu\nu\rho}A^\rho\big)\big(F^{\mu\nu} - m\epsilon_{\mu\nu\rho}A^\rho\big)
\\
&- \frac{1}{2}\Dg_\mu\sigma\Dg^\mu\sigma + m\del_\mu A^\mu\sigma\bigg]
\end{aligned}
\end{split}
\end{equation}

As for the fermionic part of the localising action, we find new contributions

\begin{equation}
\tr\Big[\delta_\zeta(\delta_\zeta\lambda)^\dagger\lambda\Big]\Big|_\new
= \tr\bigg[im\zetad{^\beta}\delta_\zeta(i\As + \sigma)_\beta{^\alpha}\lambda_\alpha\bigg]
\overeq{\ref{tussenstap 3}} \tr\Big[2m|\zeta|^2\lambdat\lambda\Big]
\end{equation}

Comparing this to equation <ref> we hence find find that the fermionic part of the localising functional is given by

\begin{equation}
S_\loc[\Vg]\big|_\fer = \int\extd^3x ~ \tr\bigg[-\lambdat\big(i\Dgs - \ad\sigma - m\big)\lambda\bigg].
\end{equation}

In conclusion, we find that the localising action is given by

[breakable, enhanced]

§.§.§ $\delta$-Localisation of Proca-Chern-Simons Theory

\begin{equation}
\int\extd^3x ~ \tr\bigg[
\begin{aligned}[t]
&-\frac{1}{4}\Big(F_{\mu\nu} - m\epsilon_{\mu\nu\rho}A^\rho + \epsilon_{\mu\nu\rho}\Dg^\rho\sigma)\Big)
\\
&\times\Big(F^{\mu\nu} - m\epsilon^{\mu\nu\rho}A_\rho + \epsilon^{\mu\nu\rho}\Dg_\rho\sigma)\Big)
\\
&-\frac{1}{2}(D + m\sigma)^2 - \lambdat\big(i\Dgs - \ad\sigma - m\big)\lambda\bigg]
\end{aligned}
\end{equation}

which yields a localisation locus

\begin{align}
D+m\sigma &= 0,
F - m\ast A &= -\ast\Dg\sigma.
\end{align}

localising the action on sourced Proca-Chern-Simons $A_\mu$ field equations as well as the $\sigma$ field equations.

We note that for the covariant flatness condition $\Dg\sigma = 0$ to seperate from the $A_\mu$ field equations and hence become its own localisation locus the condition $\del_\mu A^\mu = 0$ has to be satisfied.
A perhaps meaningful way of looking at this is that this would be the case if Noether's second theorem were to hold somehow. However, due to the fact that PCS theory violates gauge symmetry it doesn't have to. That is to say,

\begin{align}
\delta_\theta S_\PCS[A] &~\slashed{\equiv}~ 0,
\Dg_\mu\fder{S_\PCS}{A_\mu} &~\slashed{\equiv}~ 0
\del_\mu A^\mu &~\slashed{\equiv}~ 0.
\end{align}

What we note though is that if the gauge field $A_\mu$ is taken to be on-shell, it does hold:

\begin{align}
\fder{S_\PCS}{A_\mu} &= 0
F_{\mu\nu} - m\epsilon_{\mu\nu\rho}A^\rho &= 0
\del_\mu A^\mu &= 0.
\end{align}

This is an interesting fact, since terms appearing in supersymmetry which vanish on-shell are often symptomatic of the elimination of auxiliary field content. Perhaps this suggests that we need to introduce additional field content for the Proca-Chern-Simons localisation to make sense off-shell.

§.§.§ $\Qloc$-Localisation Scheme

As an attempt to get rid of the terms which appear in the $\delta$-localisation scheme, we will now perform the $\Qloc$-localisation scheme, which isn't equivalent to the former with these supersymmetry transformations.
In analogy to section <ref> we have a fermionic localising functional

\begin{equation}
\Fg_\loc[\Vg] = \frac{1}{4|\zeta|^2}\int\extd^3x ~ \tr\Big[(\Qloc_\zeta\lambda)^\dagger\lambda + (\Qloc_\zeta\lambdat)^\dagger\lambdat\Big]
\end{equation}

In complete analogy to the computations <ref>-<ref> we find

\begin{equation}
\begin{split}
&= \frac{1}{4|\zeta|^2}\int\extd^3x ~ \tr\Big[|\Qloc_\zeta\lambda|^2 + |\Qloc_\zeta\lambdat|^2\Big]
\\
&= \int\extd^3x ~ \tr\bigg[
\begin{aligned}[t]
&-\frac{1}{4}\big(F_{\mu\nu} - m\epsilon_{\mu\nu\rho}A^\rho\big)\big(F^{\mu\nu} - m\epsilon^{\mu\nu\rho}A_\rho\big)
\\
&-\frac{1}{2}\Dg_\mu\sigma\Dg^\mu\sigma - \frac{1}{2}(D + m\sigma)^2\bigg]
\end{aligned}
\end{split}
\end{equation}

which does not involve the term proportional to $\del_\mu A^\mu$. If the total action is then supersymmetric, that is to say, $\Qloc_\zeta S_\loc = 0$, then the localisation argument makes sense. However, before moving on to this let us discuss the fermionic part of this localising action. In this case the fermionic part of the action won't agree with that of tha $\delta$-localisation scheme. We find after some brief computations that we arrive at a fermionic sector

\begin{equation}
\begin{split}
&= \frac{1}{4|\zeta|^2}\int\extd^3x ~ \tr\Big[\Qloc_\zeta(\Qloc_\zeta\lambda)^\dagger\lambda + \Qloc_\zeta(\Qloc_\zeta\lambdat)^\dagger\lambdat\Big]
\\
&= \int\extd^3x ~ \tr\bigg[-\lambdat\big(i\Dgs - \ad\sigma - m\big)\lambda - \frac{m}{2}\big(\lambda^2 + \lambdat{^2}\big)\bigg].
\end{split}
\end{equation}

This action interestingly is augmented by non-standard mass terms. We thus arrive at a total localising action

\begin{equation}
\int\extd^3x ~ \tr\bigg[
\begin{aligned}[t]
&-\frac{1}{4}\big(F_{\mu\nu} - m\epsilon_{\mu\nu\rho}A^\rho\big)\big(F^{\mu\nu} - m\epsilon^{\mu\nu\rho}A_\rho\big)
\\
&-\frac{1}{2}\Dg_\mu\sigma\Dg^\mu\sigma - \frac{1}{2}(D + m\sigma)^2
\\
&-\lambdat\big(i\Dgs - \ad\sigma - m\big)\lambda - \frac{m}{2}\big(\lambda^2 + \lambdat{^2}\big)\bigg].
\end{aligned}
\end{equation}

If this action is supersymmetric, $\Qloc_\zeta S_\loc[\Vg] = 0$, this localises on the localisation locus

\begin{align}
D+m\sigma &= 0,
F_{\mu\nu} - m\epsilon_{\mu\nu\rho}A^\rho &= 0,
\Dg_\mu\sigma &= 0.
\end{align}

consisting of the $\sigma$ field equation, $A_\mu$ field equation and the $\sigma$ field being covariantly constant. However, as it turns out, this action is not supersymmetric under $\Qloc_\zeta$:

\begin{equation}
\Qloc_\zeta S_\loc[\Vg] \neq 0.
\end{equation}

In conclusion, we find that the $\Qloc$-localisation scheme, which aimed at removing the problems which involve using the Bianchi identities in the derivation of the $\delta$-localisation, does not work for localising Proca-Chern-Simons theory, despite its initial appeal. Having set up this toy case we are now finally ready to move on to the supersymmetrisation and localisation of the Third Way Theory.

§ LOCALISATION OF THE THIRD WAY THEORY

§.§ Motivation

Having treated Proca-Chern-Simons theory as a toy model we are now ready to move on to the localisation of the Third Way Theory. Let us motivate again why this localisation —if successful— is so interesting:

As we saw in previous chapters, the Third Way Theory originated as a new kind of consistent sourcing of the Yang-Mills field equations, with a source which is conserved under the remarkable condition that the gauge fields are taken to be on-shell. Due to this rather peculiar fact it followed that no gauge invariant action could yield the Third Way field equations. The way to solve this issue was then to introduce auxiliary gauge fields which transform diagonally with the original gauge fields and which can't be integrated out of the action, but for which the combined field equations yield the Third Way field equation. This action, rather than taking on the form of some augmented TMYM action, took on the form of two Chern-Simons actions connected with each other through a peculiar gauge invariant mass term.

This then leads us to our results in previous sections on localisation: One of the remarkable facts of the supersymmetrisation of Chern-Simons theory is that the field content added to supersymmetrise the theory is all auxiliary. This means that Chern-Simons theory and its supersymmetrisation are physically equivalent. This then in turn means that results derived from localising super-Chern-Simons theory yield results in Chern-Simons theory. Due to the Chern-Simons-like action of the Third Way Theory, this then leads to the hypothesis that this theory can be supersymmetrised, and that its supersymmetrisation is such that in analogy to Chern-Simons theory it is physically equivalent to the original theory. The results from a conjectured localisation of the super-Third Way Theory would then lead to exact results in the Third Way Theory. That is to say, if we can localise the Third Way Theory what we have localised is a deformation of topologically massive Yang-Mills theory with a highly non-trivial source term and which by its own formulation isn't inherently supersymmetric.

§.§ Arvanitakis $\delta$-Localisation Scheme

We start off by briefly describing a localisation scheme originally suggested by Arvanitakis. This scheme does not closely resemble the localisation scheme developed for Proca-Chern-Simons Theory which was developed throughout this thesis. For example, rather than forcing the mass turn upon us through deformed supersymmetry transformations it seeks to eliminate it through parity properties of the transformations. It further also assumes that $\mu = 0 \Leftrightarrow m = \mb \Leftrightarrow k = \kb$. Instead of trying to localise on the $A$ and $\Ab$ field equations this scheme tries to localise on the diagonal and antidiagonal field equations. That is,

\begin{align}
\begin{cases}
F - m\ast(\Ab-A) = 0
\\
\Fb - m\ast(\Ab-A) = 0
\end{cases}
\begin{cases}
F - \Fb = 0
\\
F + \Fb = 2m\ast(\Ab-A)
\end{cases}
\end{align}

§.§.§ Supersymmetric Action and Supersymmetries

In this scheme we have a supersymmetric action which depends on field content $(A,\Ab,\sigma,\lambda,\lambdat,D)$. That is, the field content of a vector multiplet $\Vg$ alongside the `auxiliary' gauge field $\Ab$. This is a little strange because one would typically expect that supersymmetric theories have matching fermionic and bosonic degrees of freedom. However, this isn't a problem since in terms of field content this isn't a hard rule, as we can always just add auxiliary fields to a theory indefinitely. The corresponding action is now given by

\begin{equation}
= \frac{k}{2\pi}S_\CS[\Ab] - \frac{k}{2\pi}S_\SCS[\Vg] - \frac{k}{2\pi}S_\mass[\Ab-A]
\end{equation}

with the Third Way mass term given by

\begin{equation}
S_\mass[\Ab-A] = \int\tr\bigg[\frac{1}{2\ell}(\Ab-A)\ast(\Ab-A)\bigg].
\end{equation}

As is the case for Chern-Simons theory the additional field content $(\sigma,\lambda,\lambdat,D)$ is all auxiliary. The proposed supersymmetries of this action satisfy the property $\delta A = \delta\Ab$, thus automatically annihilating the mass term. These are given by

\begin{align}
\delta A_\mu &= \delta\Ab_\mu = -\zeta\gamma_\mu\lambdat + \zetat\gamma_\mu\lambda
\bigg.\\
\delta\sigma &= i\zeta\lambdat + i\zetat\lambda
\bigg.\\
\delta\lambda &= \zeta\bigg[iD + \frac{1}{2}\epsilon^{\mu\nu\rho}(\Fb-F)_{\mu\nu}\gamma_\rho + [\Abs-\As,\sigma]\bigg]
\bigg.\\
\delta\lambdat &= \zetat\bigg[iD - \frac{1}{2}\epsilon^{\mu\nu\rho}(\Fb-F)_{\mu\nu}\gamma_\rho + [\Abs-\As,\sigma]\bigg]
\bigg.\\
\delta D &= -\zeta[\Abs-\As,\lambdat] - \zetat[\Abs-\As,\lambda]
\bigg.
\end{align}

Originally these only included the $\zeta$ components, since this scheme only cared about a $\delta$-localisation (for good reasons).
It is worth noting that these transformations don't resemble standard supersymmetry transformations as much anymore, due to the fact that they don't contain derivatives anymore except for the field strengths.

Let us verify whether the action is eliminated by these supersymmetry transformations. First and foremost we note that

\begin{align}
\delta A &= \delta\Ab
\delta S_\mass[\Ab-A] &= 0.
\end{align}

We further also note that in close analogy to computations <ref>-<ref> one finds

\begin{equation}
\begin{split}
\delta\Big\{S_\CS[\Ab] - S_\CS[A]\Big\}
\int\extd^3x ~ \tr\bigg[
\begin{aligned}[t]
&- \frac{1}{2}\epsilon^{\mu\nu\rho}(\Fb-F)_{\mu\nu}\zeta\gamma_\rho\lambdat
\\
&+ \frac{1}{2}\epsilon^{\mu\nu\rho}(\Fb-F)_{\mu\nu}\zetat\gamma_\rho\lambda\bigg]
\end{aligned}
\\
&= - \delta\int\extd^3x ~ \tr\Big[\lambda\lambdat - \sigma D\Big]
\end{split}
\end{equation}

We hence conclude that this action is supersymmetric.

§.§.§ Localisation

The localisation scheme for this procedure is the $\delta$-localisation scheme. The reason for this is that due to the unconventional form of the supersymmetries the $\Qloc$-localisation scheme is hopeless in yielding any results. The fermionic localising functional is again given by

\begin{equation}
\Fg_\loc[\Vg,\Ab] = \frac{1}{2|\zeta|^2}\int\extd^3x ~ \tr\Big[(\delta_\zeta\lambda)^\dagger\lambda\Big].
\end{equation}

Working out the $\delta$-localisation scheme we find in analogy to computation <ref> that

\begin{multline}
\int\extd^3x ~ \tr\bigg[
-\frac{1}{2}D^2 - \frac{1}{4}\Big(\Fb_{\mu\nu} - F_{\mu\nu} + \epsilon_{\mu\nu\rho}[(\Ab-A)^\rho,\sigma]\Big)
\\
\times
\Big(\Fb{^{\mu\nu}} - F^{\mu\nu} + \epsilon^{\mu\nu\rho}[(\Ab-A)_\rho,\sigma]\Big)\bigg]
\end{multline}

It is clear now that we don't have any Bianchi identities at our disposal to make meaningful simplifications. As such we arrive at a localisation locus

\begin{align}
\Fb_{\mu\nu} - F_{\mu\nu} + \epsilon_{\mu\nu\rho}[(\Ab-A)^\rho,\sigma] &= 0,
D &= 0.
\end{align}

This localises the action to the off-shell sourced diagonal field equations and the $\sigma$ field equation.

§.§ The Super-Third Way Theory

We now formulate the super-Third Way Theory. The construction of this theory follows in close analogy to super-Proca-Chern-Simons theory. Again we deform the supersymmetry transformations in such a way that a mass term is forced upon us. In strong analogy to the super-Proca-Chern-Simons action we find

[breakable, enhanced]

§.§.§ The Super-Third Way Action

\begin{equation}
S\SuperThirdWay[\Vg,\Vgb] =
\frac{k}{2\pi}S_\SCS[\Vgb] - \frac{\kb}{2\pi}S_\SCS[\Vg]
- \frac{k\kb}{4\pi^2}S_\mass[\Vgb-\Vg]
\end{equation}

the supersymmetrisation of the Third Way Action. Its field content consists of two vector multiplets $\Vg = (A,\sigma,\lambda,\lambdat,D)$ and $\Vgb = (\Ab,\sigmab,\lambdab,\lambdabt,\Db)$. Its super-Chern-Simons component actions are given by <ref> and the mass term is in this case given by

\begin{equation}
S_\mass[\Vgb-\Vg] = \int\tr\bigg[\frac{1}{2\ell}(\Ab-A)\ast(\Ab-A)+ \frac{1}{2\ell}(\sigmab-\sigma)\ast(\sigmab-\sigma)\bigg]
\end{equation}

in close analogy to the super-Proca-Chern-Simons case <ref>. Its field equations are given by

\begin{align}
F + \frac{k}{2\pi\ell}\ast(\Ab-A) &= 0
\Fb + \frac{\kb}{2\pi\ell}\ast(\Ab-A) &= 0
\bigg.\\
D + \frac{k}{2\pi\ell}(\sigmab - \sigma) &= 0
\Db + \frac{\kb}{2\pi\ell}(\sigmab - \sigma) &= 0
\bigg.\\
\sigma = \lambda = \lambdat &= 0
\sigmab = \lambda = \lambdabt &= 0
\bigg.
\end{align}

Its supersymmetries are also derived in close analogy to super-Proca-Chern-Simons theory. We recall that the criteria which were used to derive these supersymmetry transformations were given by

* Supersymmetrising the Third Way Theory.

* Yielding nilpotent supersymmetry transformations.

* Ideally localising the action on on-shell field configuration.

These conditions yield the following supersymmetry transformations:

§.§.§ Third Way Supersymmetry Transformations

We denote the Third Way supersymmetry transformation by

\begin{equation}
\delta_{\zeta,\zetatt,\zetab,\zetabtt}
\delta_\zeta + \deltat_\zetatt + \deltab_\zetab + \deltabt_\zetabtt
\end{equation}

this time parametrised by four independent constant 2-spinors $\zeta$, $\zetat$, $\zetab$, $\zetabt$. The supersymmetry transformations read

\begin{align}
\delta A_\mu &= -\zeta\gamma_\mu\lambdat + \zetat\gamma_\mu\lambda
\bigg.\\
\delta\Ab_\mu &= -\zetab\gamma_\mu\lambdabt + \zetabt\gamma_\mu\lambdab
\bigg.\\
\delta\sigma &= i\zeta\lambdat + i\zetat\lambda
\bigg.\\
\delta\sigmab &= i\zetab\lambdabt + i\zetabt\lambdab
\bigg.\\
\delta\lambda &= \zeta\bigg[iD + \frac{1}{2}\epsilon^{\mu\nu\rho}F_{\mu\nu}\gamma_\rho + \Dgs\sigma \mathcolor{blue}{- \frac{k}{2\pi\ell}(\Abs-\As) + \frac{ik}{2\pi\ell}(\sigmab-\sigma)}\bigg]
\bigg.\\
\delta\lambdab &= \zetab\bigg[i\Db + \frac{1}{2}\epsilon^{\mu\nu\rho}\Fb_{\mu\nu}\gamma_\rho + \Dgbs\sigmab \mathcolor{blue}{- \frac{\kb}{2\pi\ell}(\Abs-\As) + \frac{i\kb}{2\pi\ell}(\sigmab-\sigma)}\bigg]
\bigg.\\
\delta\lambdat &= \zetat\bigg[iD - \frac{1}{2}\epsilon^{\mu\nu\rho}F_{\mu\nu}\gamma_\rho + \Dgs\sigma \mathcolor{blue}{+ \frac{k}{2\pi\ell}(\Abs-\As) + \frac{ik}{2\pi\ell}(\sigmab-\sigma)}\bigg]
\bigg.\\
\delta\lambdabt &= \zetabt\bigg[i\Db - \frac{1}{2}\epsilon^{\mu\nu\rho}\Fb_{\mu\nu}\gamma_\rho + \Dgbs\sigmab \mathcolor{blue}{+ \frac{\kb}{2\pi\ell}(\Abs-\As) + \frac{i\kb}{2\pi\ell}(\sigmab-\sigma)}\bigg]
\bigg.\\
\delta D &= i\zeta\big(i\Dgs + \ad\sigma\big)\lambdat + i\zetat\big(i\Dgs - \ad\sigma\big)\lambda
\bigg.\\
\delta\Db &= i\zetab\big(i\Dgbs + \ad\sigmab\big)\lambdabt + i\zetabt\big(i\Dgbs - \ad\sigmab\big)\lambdab
\bigg.
\end{align}

where the highlighted terms are the new terms which diverge from the standard supersymmetry transformations of the vector multiplet. Note that these terms indeed vanish as $\ell\to\infty$.

Let us now verify that these indeed leave the super-Third Way action invariant. We find that the new contributions are given by

\begin{align}
\frac{k}{2\pi}\delta S_\SCS[\Vgb]\Big|_\new &= \frac{k\kb}{4\pi^2\ell}\int\extd^3x ~ \tr\bigg[
\begin{aligned}[t]
&\zetab\Big(-(\Abs-\As) + i(\sigmab-\sigma)\Big)\lambdabt
\\
+~&\zetabt\Big(+(\Abs-\As) + i(\sigmab-\sigma)\Big)\lambdab\bigg]
\end{aligned}
\\
-\frac{\kb}{2\pi}\delta S_\SCS[\Vg]\Big|_\new &= \frac{k\kb}{4\pi^2\ell}\int\extd^3x ~ \tr\bigg[
\begin{aligned}[t]
&\zeta\Big(+(\Abs-\As) - i(\sigmab-\sigma)\Big)\lambdat
\\
+~&\zetat\Big(-(\Abs-\As) - i(\sigmab-\sigma)\Big)\lambda\bigg]
\end{aligned}
\end{align}

These cancel precisely against the supersymmetry transformations of the mass term:

\begin{equation}
-\frac{k\kb}{4\pi^2}\delta S_\mass[\Vgb-\Vg]
= -\frac{k}{2\pi}\delta S_\SCS[\Vgb]\Big|_\new + \frac{\kb}{2\pi}\delta S_\SCS[\Vg]\Big|_\new
\end{equation}

hence proving that the super-Third Way Theory is supersymmetric

\begin{equation}
\delta_{\zeta,\zetatt,\zetab,\zetabtt}S\SuperThirdWay[\Vg,\Vgb] = 0.
\end{equation}

As for the nilpotency of these transformations, these follow from precisely the same steps as equations <ref>-<ref>, thus resulting in

\begin{equation}
\delta^2_\zeta = \deltat{^2_\zetatt} = \deltab{^2_\zetab} = \deltab{^2_\zetabtt} = 0.
\end{equation}

That is, the Third Way supersymmetries are nilpotent.

§.§ The Third Way Supersymmetry Algebra

§.§.§ Four Killing Spinors

Let us now take a closer look at the supersymmetry algebra of the super-Third Way Theory. This algebra will take on a significantly more complicated form, since now we have split it up into four different supersymmetries. We split these up into three different cathegories:

* $\delta^2$, $\deltab{^2}$, $\deltat{^2}$, $\deltabt{^2}$: These all vanish generalising the nilpotency of the transformations as

\begin{equation}
\{\delta_\zeta,\delta_\eta\} = \{\deltat_\zetatt,\deltat_\etatt\} = \{\deltab_\zetab,\deltab_\etab\} = \{\deltabt_\zetabtt,\deltabt_\etabtt\} = 0
\end{equation}

* $\delta\deltab$, $\deltat\deltab$, $\delta\deltabt$, $\deltat\deltabt$: These interestingly —or perhaps alarmingly— show some non-vanishing results:

\begin{align}
\label{unusual 1}
\{\delta_\zeta,\deltab_\zetab\}\lambda^\alpha &= \frac{k}{\pi\ell}(\zeta\zetab)\lambdabt{^\alpha}
\\
\label{unusual 2}
\{\delta_\zeta,\deltab_\zetab\}\lambdab{^\alpha} &= \frac{\kb}{\pi\ell}(\zeta\zetab)\lambdat{^\alpha}
\{\delta_\zeta,\deltab_\zetab\}\big|_\other &= 0
\\
\{\deltat_\zetatt,\deltab_\zetab\}\lambdat{^\alpha} &= -\frac{k}{\pi\ell}\zetab{^\alpha}(\zetat\lambdabt)
\\
\{\deltat_\zetatt,\deltab_\zetab\}\lambdab{^\alpha} &= \frac{\kb}{\pi\ell}\zetat{^\alpha}(\zetab\lambda)
\{\deltat_\zetatt,\deltab_\zetab\}\big|_\other &= 0
\\
\{\delta_\zeta,\deltabt_\zetabtt\}\lambda^\alpha &= -\frac{k}{\pi\ell}\zetabt{^\alpha}(\zeta\lambdab)
\\
\{\delta_\zeta,\deltabt_\zetabtt\}\lambdabt{^\alpha} &= \frac{\kb}{\pi\ell}\zeta^\alpha(\zetabt\lambdat)
\{\delta_\zeta,\deltabt_\zetabtt\}\big|_\other &= 0
\\
\label{unusual 3}
\{\deltat_\zetatt,\deltabt_\zetabtt\}\lambdat{^\alpha} &= -\frac{k}{\pi\ell}(\zetat\zetabt)\lambdab{^\alpha}
\\
\label{unusual 4}
\{\deltat_\zetatt,\deltabt_\zetabtt\}\lambdabt{^\alpha} &= - \frac{\kb}{\pi\ell}(\zetat\zetabt)\lambda^\alpha
\{\deltat_\zetatt,\deltabt_\zetabtt\}\big|_\other &= 0
\end{align}

* $\delta\deltat$, $\deltab\deltabt$: These are the Third Way analogues of the non-trivial relations:

\begin{align}
\{\delta_\zeta,\deltat_\zetatt\}A_\mu &= -2iK^\nu\Big(F_{\nu\mu} - \frac{k}{2\pi\ell}\epsilon_{\nu\mu\rho}(\Ab-A)^\rho\Big) + 2\zeta\zetat\Dg_\mu\sigma
\\
\{\delta_\zeta,\deltat_\zetatt\}\sigma &= -2iK^\mu\Dg_\mu\sigma
\\
\{\delta_\zeta,\deltat_\zetatt\}\lambda^\alpha &= -2iK^\mu\Dg_\mu\lambda^\alpha - 2\zeta\zetat[\sigma,\lambda^\alpha] + \frac{k}{\pi\ell}\zetat{^\alpha}(\zeta\lambda)
\\
\{\delta_\zeta,\deltat_\zetatt\}\lambdat{^\alpha} &= -2iK^\mu\Dg_\mu\lambdat{^\alpha} - 2\zeta\zetat[\sigma,\lambdat{^\alpha}] + \frac{k}{\pi\ell}\zeta^\alpha(\zetat\lambdat)
\\
\{\delta_\zeta,\deltat_\zetatt\}D &=
\begin{aligned}[t]
&-2iK^\mu\Dg_\mu\Big(D + \frac{k}{2\pi\ell}(\sigmab-\sigma)\Big) - 2\zeta\zetat[\sigma,D]
\\
&+ \frac{k}{\pi\ell}\Dg_\mu(\Ab-A)^\mu\zeta\zetat
\end{aligned}
\\
\{\delta_\zeta,\deltat_\zetatt\}\Vgb &= 0
\end{align}

and similarly

\begin{align}
\{\deltab_\zetab,\deltabt_\zetabtt\}\Ab_\mu &= -2i\Kb{^\nu}\Big(\Fb_{\nu\mu} - \frac{\kb}{2\pi\ell}\epsilon_{\nu\mu\rho}(\Ab-A)^\rho\Big) + 2\zetab\zetabt\Dgb_\mu\sigmab
\\
\{\deltab_\zetab,\deltabt_\zetabtt\}\sigmab &= -2i\Kb{^\mu}\Dgb_\mu\sigmab
\\
\{\deltab_\zetab,\deltabt_\zetabtt\}\lambdab{^\alpha} &= -2i\Kb{^\mu}\Dgb_\mu\lambdab{^\alpha} - 2\zetab\zetabt[\sigmab,\lambdab{^\alpha}] - \frac{\kb}{\pi\ell}\zetabt{^\alpha}(\zetab\lambdab)
\\
\{\deltab_\zetab,\deltabt_\zetabtt\}\lambdabt{^\alpha} &= -2i\Kb{^\mu}\Dgb_\mu\lambdabt{^\alpha} - 2\zetab\zetabt[\sigmab,\lambdabt{^\alpha}] - \frac{\kb}{\pi\ell}\zetab{^\alpha}(\zetabt\lambdabt)
\\
\{\deltab_\zetab,\deltabt_\zetabtt\}\Db &=
\begin{aligned}[t]
&-2i\Kb{^\mu}\Dgb_\mu\Big(\Db + \frac{\kb}{2\pi\ell}(\sigmab-\sigma)\Big) - 2\zetab\zetabt[\sigmab,\Db]
\\
&+ \frac{\kb}{\pi\ell}\Dgb_\mu(\Ab-A)^\mu\zetab\zetabt
\end{aligned}
\\
\{\deltab_\zetab,\deltabt_\zetabtt\}\Vg &= 0
\end{align}

where by $\Vg$ and $\Vgb$ we denoted the collective field content of the respective vector multiplets and where we defined the Killing vectors

\begin{align}
K^\mu &= \zeta\gamma^\mu\zetat,
\Kb{^\mu} &= \zetab\gamma^\mu\zetabt.
\end{align}

§.§.§ Two Killing Spinors

We note that due to the relations <ref>-<ref>, and in particular equations <ref>, <ref>, <ref> and <ref> the algebra takes on quite an unusual form. However, we note that these are all proportional to either $\zeta\zetab$ or $\zetat\zetabt$. Hence this suggests that to make these vanish we could take $\zeta = \zetab$ and $\zetat = \zetabt$ and redefine

\begin{align}
\label{redefinition}
\delta_\zeta,\deltab_\zetab &\to \delta_\zeta := \delta_{\zeta,0,\zeta,0}
\deltat_\zetatt,\deltabt_\zetabtt &\to \deltat_\zetatt := \delta_{0,\zetatt,0,\zetatt}.
\end{align}

The algebra then takes on the more familiar form

\begin{align}
\{\delta_\zeta,\delta_\eta\} &= 0
\{\deltat_\zetatt,\deltat_\etatt\} &= 0
\end{align}

with non-trivial anticommutators

\begin{align}
\{\delta_\zeta,\deltat_\zetatt\}A_\mu &= -2iK^\nu\Big(F_{\nu\mu} - \frac{k}{2\pi\ell}\epsilon_{\nu\mu\rho}(\Ab-A)^\rho\Big) + 2\zeta\zetat\Dg_\mu\sigma
\bigg.\\
\{\delta_\zeta,\deltat_\zetatt\}\Ab_\mu &= -2iK^\nu\Big(\Fb_{\nu\mu} - \frac{\kb}{2\pi\ell}\epsilon_{\nu\mu\rho}(\Ab-A)^\rho\Big) + 2\zeta\zetat\Dgb_\mu\sigmab
\bigg.\\
\{\delta_\zeta,\deltat_\zetatt\}\sigma &= -2iK^\mu\Dg_\mu\sigma
\bigg.\\
\{\delta_\zeta,\deltat_\zetatt\}\sigmab &= -2iK^\mu\Dgb_\mu\sigmab
\bigg.\\
\{\delta_\zeta,\deltat_\zetatt\}\lambda^\alpha &= -2iK^\mu\Dg_\mu\lambda^\alpha - 2\zeta\zetat[\sigma,\lambda^\alpha] - \frac{k}{\pi\ell}\zetat{^\alpha}\zeta(\lambdab - \lambda)
\bigg.\\
\{\delta_\zeta,\deltat_\zetatt\}\lambdab{^\alpha} &= -2iK^\mu\Dgb_\mu\lambdab{^\alpha} - 2\zeta\zetat[\sigmab,\lambdab{^\alpha}] - \frac{\kb}{\pi\ell}\zetat{^\alpha}\zeta(\lambdab - \lambda)
\bigg.\\
\{\delta_\zeta,\deltat_\zetatt\}\lambdat{^\alpha} &= -2iK^\mu\Dg_\mu\lambdat{^\alpha} - 2\zeta\zetat[\sigma,\lambdat{^\alpha}] - \frac{k}{\pi\ell}\zeta^\alpha\zetat(\lambdabt - \lambdat)
\bigg.\\
\{\delta_\zeta,\deltat_\zetatt\}\lambdabt{^\alpha} &= -2iK^\mu\Dgb_\mu\lambdabt{^\alpha} - 2\zeta\zetat[\sigmab,\lambdabt{^\alpha}] - \frac{\kb}{\pi\ell}\zeta^\alpha\zetat(\lambdabt - \lambdat)
\bigg.\\
\{\delta_\zeta,\deltat_\zetatt\}D &=
\begin{aligned}[t]
&-2iK^\mu\Dg_\mu\Big(D + \frac{k}{2\pi\ell}(\sigmab-\sigma)\Big) - 2\zeta\zetat[\sigma,D]
\\
&+ \frac{k}{\pi\ell}\Dg_\mu(\Ab-A)^\mu\zeta\zetat
\end{aligned}
\bigg.\\
\{\delta_\zeta,\deltat_\zetatt\}\Db &=
\begin{aligned}[t]
&-2iK^\mu\Dgb_\mu\Big(\Db + \frac{\kb}{2\pi\ell}(\sigmab-\sigma)\Big) - 2\zeta\zetat[\sigmab,\Db]
\\
&+ \frac{\kb}{\pi\ell}\Dgb_\mu(\Ab-A)^\mu\zeta\zetat
\end{aligned}
\bigg.
\end{align}

where we now only defined a single Killing vector

\begin{equation}
K^\mu = \zeta\gamma^\mu\zetat.
\end{equation}

In analogy to the Proca-Chern-Simons case this transformation takes on the form

\begin{equation}
\text{SUSY}^2 = \text{transl} + \text{gauge} + \text{mass}.
\end{equation}

However, an aspect in which it is different from Proca-Chern-Simons theory is that the field dependent gauge transformation are different between the two multiplets $\Vg$ and $\Vgb$. Aside from the issues we incountered in Proca-Chern-Simons theory for the $\Qloc$-localisation argument this also will hinder it.

§.§ Localising the Super-Third Way Theory

Finally, we move on to perform the localisation procedure of the super-Third Way Theory. This will follow in close analogy to the Proca-Chern-Simons case.

§.§.§ $\Qloc$-Localisation Scheme

Let us now start with the $\Qloc$-localisation scheme instead of the $\delta$-localisation scheme, because this is the less promising one for similar reasons as the Proca-Chern-Simons case. In this case the fermionic localising functional is given by

\begin{equation}
\Fg_\loc[\Vg,\Vgb] = \frac{1}{4|\zeta|^2}\int\extd^3x ~ \tr\bigg[(\Qloc_\zeta\lambda)^\dagger\lambda + (\Qloc_\zeta\lambdat)^\dagger\lambdat + (\Qloc_\zeta\lambdab)^\dagger\lambdab + (\Qloc_\zeta\lambdabt)^\dagger\lambdabt\bigg]
\end{equation}

where we defined a localising supersymmetry

\begin{equation}
\Qloc_\zeta = \delta_\zeta + \deltat_\zeta = \delta_{\zeta,\zeta,\zeta,\zeta}
\end{equation}

again following the definition <ref>. However, for reasons similar to the Proca-Chern-Simons case it sadly doesn't work, since

\begin{equation}
\Qloc_\zeta S_\loc[\Vg,\Vgb] \neq 0.
\end{equation}

The explicit expression isn't given because as one might suspect it's quite a mess, and it doesn't seem to vanish on-shell, suggesting that adding auxiliary field content won't obviously help. However, this doesn't stop one from trying and multiple ways of adjusting the supersymmetries were tried out, though all being unsuccessful. Most notably a way to come up with the `right' supersymmetries was starting from superspace, and breaking the group of supergauge transformations. This would lead to supersymmetries somewhat resembling the Third Way supersymmetries but which didn't seem fit for constructing a supersymmetric action.

§.§.§ $\delta$-localisation scheme

Now, we treat the more promising $\delta$-localisation scheme. Here the fermionic localising functional is given by

\begin{equation}
\Fg_\loc[\Vg,\Vgb] = \frac{1}{2|\zeta|^2}\int\extd^3x ~ \tr\bigg[(\delta_\zeta\lambda)^\dagger\lambda + (\delta_\zeta\lambdab)^\dagger\lambdab\bigg]
\end{equation}

where we defined the supersymmetry transformation $\delta_\zeta = \delta_{\zeta,0,\zeta,0}$ as in equation <ref>. The systematics of the computations are identitcal to Proca-Chern-Simons theory but the equations are over twice as long. Hence, we will only present the reader with the results, these are given by:

§.§.§ $\delta$-Localisation of the Super-Third Way Theory

\begin{equation}
\begin{split}
&S_\loc[\Vg,\Vgb] = \delta_\zeta\Fg_\loc[\Vg,\Vgb]
\\
\int\extd^3x ~ \tr\bigg[
\begin{aligned}[t]
&- \frac{1}{4}\Big(F_{\mu\nu} - \frac{k}{2\pi\ell}\epsilon_{\mu\nu\rho}(\Ab-A)^\rho + \epsilon_{\mu\nu\rho}\Dg^\rho\sigma\Big)
\\
&\times\Big(F^{\mu\nu} - \frac{k}{2\pi\ell}\epsilon^{\mu\nu\rho}(\Ab-A)_\rho + \epsilon^{\mu\nu\rho}\Dg_\rho\sigma\Big)
\\
&- \frac{1}{4}\Big(\Fb_{\mu\nu} - \frac{\kb}{2\pi\ell}\epsilon_{\mu\nu\rho}(\Ab-A)^\rho + \epsilon_{\mu\nu\rho}\Dgb{^\rho}\sigmab\Big)
\\
&\times\Big(\Fb{^{\mu\nu}} - \frac{\kb}{2\pi\ell}\epsilon^{\mu\nu\rho}(\Ab-A)_\rho + \epsilon^{\mu\nu\rho}\Dgb_\rho\sigmab\Big)
\\
&-\frac{1}{2}\Big(D + \frac{k}{2\pi\ell}(\sigmab-\sigma)\Big)^2
-\frac{1}{2}\Big(\Db + \frac{\kb}{2\pi\ell}(\sigmab-\sigma)\Big)^2
\\
&- \lambdat\Big(i\Dgs - \ad\sigma + \frac{k}{2\pi\ell}\Big)\lambda + \frac{k}{2\pi\ell}\lambdabt\lambda
\\
&- \lambdabt\big(i\Dgbs - \ad\sigmab - \frac{\kb}{2\pi\ell}\big)\lambdab - \frac{\kb}{2\pi\ell}\lambdat\lambdab\bigg]
\end{aligned}
\end{split}
\end{equation}

which yields a localisation locus

\begin{align}
F - \frac{k}{2\pi\ell}\ast(\Ab-A) &= -\ast\Dg\sigma
D + \frac{k}{2\pi\ell}(\sigmab-\sigma) &= 0
\\
\Fb - \frac{\kb}{2\pi\ell}\ast(\Ab-A) &= -\ast\Dgb\sigmab
\Db + \frac{\kb}{2\pi\ell}(\sigmab-\sigma) &= 0
\end{align}

where the Third Way field equations become sourced by the scalar fields.

In analogy to the Proca-Chern-Simons case, we note that for the covariant flatness $\Dg\sigma = \Dgb\sigmab = 0$ to seperate from the $A$, $\Ab$ field equations the condition $\Dg_\mu(\Ab-A)^\mu \equiv \Dgb_\mu(\Ab-A)^\mu = 0$ would have to be satisfied. The deeper reason why this isn't the case can again be traced back to the violation of Noether's second theorem.
Due to the fact that the action isn't invariant under a full $G\times G$ but rather the diagonal subgroup $G$, Noether's second theorem doesn't have to hold anymore:

\begin{equation}
\delta_{\theta,\thetab}S\ThirdWay ~\slashed{\equiv}~ 0
\end{equation}

where we now have

\begin{align}
\Dg_\mu\fder{S\ThirdWay}{A_\mu} &~\slashed{\equiv}~ 0
\Dgb_\mu\fder{S\ThirdWay}{\Ab_\mu} &~\slashed{\equiv}~ 0
\Dg_\mu(\Ab-A)^\mu \equiv \Dgb_\mu(\Ab-A)^\mu &~\slashed{\equiv}~ 0.
\end{align}

However, if the fields are taken to be on-shell, this requirement is satisfied:

\begin{align}
\fder{S\ThirdWay}{A_\mu} &= 0
\Dg_\mu(\Ab-A)^\mu &= 0
\fder{S\ThirdWay}{\Ab_\mu} &= 0
\end{align}

Again it is worth noting that the appearance of terms which vanish on-shell could be an indicator for auxiliary fields which have been integrated away. Perhaps there exists some addition of auciliary fields, which could after integrating them away impose the condition $\Dg_\mu(\Ab-A)^\mu \equiv \Dgb_\mu(\Ab-A)^\mu = 0$. Attempts have been made to do this consistently, and while the possibility of this isn't excluded, no such addition of additional auxiliary fields is currently known.

CHAPTER: CONCLUSION

Let us now conclude this thesis. The main goal of this thesis was to localise the Third Way Theory. However, a lot of interesting original computations also emerged from our treatment of the Witten index in chapter 2.

For the Witten index, a number of original computations have been performed. These derived already established results which —to the best of our knowledge— haven't been applied elsewhere in the literature. For starters, the $\beta$-independence of the Witten index is typically only derived using the Hilbert space formalism. Instead in this thesis we used the path integral formalism to derive this result. We arrived at this result by using the superspace formalism to note that the action is $Q$-exact. Similarly, the `kinetic' and `potential' rescalings were shown to be symmetries of the Witten index, again making use of superspace methods. These methods had the advantage of showing that a term is manifestly $Q$-exact, rather than forcing us to engage in educated guess work. Finally, the computations were overall more thorough than those in the source material (particularly [13]).

As for the Third Way Theory, let us remind the reader of the significance of localising this theory again: On-shell, the Third Way Theory is a continuous deformation of Yang-Mills theory. However, the action which describes this theory consists of two independent Chern-Simons theories over the same gauge group and a mass term which breaks the two copies of the gauge group to its diagonal subgroup. Combined with the observation that $\NSUSY = 0$ Chern-Simons theory can be localised due to its equivalence with $\NSUSY = 2$ Chern-Simons theory, one is lead to hypothesise that the $\NSUSY = 0$ and $\NSUSY = 2$ Third Way theories are equivalent. As a corollary to this, it would then follow that localising the $\NSUSY = 2$ Third Way Theory is equivalent to localising the $\NSUSY = 0$ Third Way Theory, thus achieving a method of doing non-perturbative computations in a highly non-trivial deformation of $d=3$ Yang-Mills theory.

Progress to this end has been made in various ways: Significant progress has been made into understanding Proca-Chern-Simons theory, the toy model which leads to the Third Way localisation. A new kind of supersymmetry transformation has been introduced to supersymmetrise this theory, which we coined `massive supersymmetry'. Using this supersymmetry, Proca-Chern-Simons theory was then superymmetrised, such that its superanalogue is physically equivalent to the original theory. After this, the theory was localised on its field equations, sourced by auxiliary fields. Then, a proposed localisation scheme by Arvanitakis was reviewed which localised the Third Way Theory on its diagonal field equation, sourced by auxiliary fields. Finally, the results of Proca-Chern-Simons theory were generalised to the Third Way Theory, leading to a supersymmetrisation of the Third Way Theory which is physically equivalent to the original theory. Using this supersymmetrisation, the Third Way Theory was then localised on both its field equations, sourced by auxiliary fields.

In conclusion, significant progress has been made towards localising the Third Way Theory. We found a localising action which localises the theory on sourced field equations. One assuption which has been made throughout the thesis though is that the path integral measure is supersymmetric. This is at this moment the only barrier between definitively being able to say that the theory is localised.

CHAPTER: APPENDIX

§ DIFFERENTIAL GEOMETRY

§.§ Riemannian Geometry

We denote the flat space metric as

\begin{equation}
\eta_{ab} = \diag(-1,\dots,-1,+1,\dots,+1).
\end{equation}

The curved space metric is given by

\begin{equation}
g_{\mu\nu} = e_\mu{^a}e_\nu{^b}\eta_{ab}
\end{equation}

where $e_\mu{^a}$ the vielbeins with inverses $e^\mu{_a}$ which we take such that

\begin{align}
\eta &:= \det (\eta_{ab}) = (-)^{\text{\# time directions}}
\\
e &:= \det (e_\mu{^a}) > 0
\\
g &:= \det (g_{\mu\nu}) = \eta e^2
\end{align}

We denote the Christoffel symbol by $\Gamma_{\mu\sigma}{^\rho}$ and the spin connection by $\omega_\mu{^{ab}}$ [14][27].

§.§ Lie algebras and Connections

§.§.§ Lie algebras

We consider a simple compact Lie group $G$ with Lie algebra $\gLie$. The Lie algebra $\gLie$ is taken to be generated by $T_I$ with structure constants $f_{IJ}{^K}$ and Killing form $\kappa_{IJ}$ defined through

\begin{align}
[T_I,T_J] &= f_{IJ}{^K}T_K
\kappa_{IJ} &= \tr\big[T_IT_J\big]
\end{align}

in our conventions $f_{IJ}{^K}$ are real and $\kappa_{IJ}$ non-degenerate and negative semi-definite. Let $\omega_p$ and $\eta_q$ be $\gLie$-valued forms. Their commutator is then defined as

\begin{align}
[\omega_p,\eta_q] &:= \omega_p\eta_q - (-)^{pq}\eta_q\omega_p
[\omega_p,\eta_q]^I &:= f_{JK}{^I}\omega_p^J\eta_q^K.
\end{align}

The $\mathbb{Z}_2$-graded Jacobi identity here takes on the form

\begin{equation}
\label{Z2 graded Jacobi identity}
[\alpha_r,[\omega_p,\eta_q]] = [[\alpha_r,\omega_p],\eta_q] + (-)^{rp}[\omega_p,[\alpha_r,\eta_q]].
\end{equation}

One should think of this as analogous to the super-Jacobi identities for Lie superalgebras [9][14].

§.§.§ Connections

Connections of this group are $\gLie$-valued 1-forms $A$ which transform as

\begin{align}
A &\to e^\theta(A+\extd)e^{-\theta}
\delta_\theta A &= -\extd \theta - [A,\theta] =: -\Dg\theta
\end{align}

and field strengths are denoted as

\begin{align}
F &:= \extd A + A^2
F &\to e^\theta Fe^{-\theta}
\delta_\theta F &= [\theta,F]
\end{align}

where $\theta = \theta^IT_I$ a $\gLie$-valued local gauge parameters. Field strengths satisfy the Bianchi identity

\begin{align}
\Dg F &\equiv 0
\Dg_{[\mu}F_{\rho\sigma]} &\equiv 0.
\end{align}

where we defined $\Dg = \extd + \ad A$ [14].

§.§ Levi-Civita Symbol and Hodge Dualities

§.§.§ Levi-Civita symbol and tensor

The Levi-Civita symbol is characterised by

\begin{align}
\varepsilon_{[a_1\dots a_d]} &= \varepsilon_{a_1\dots a_d}
\varepsilon_{0\dots(d-1)} &= 1
\\
\varepsilon^{[a_1\dots a_d]} &= \varepsilon^{a_1\dots a_d}
\varepsilon^{0\dots(d-1)} &= 1
\end{align}

Similarly, we define the Levi-Civita tensor for flat indices as

\begin{align}
\epsilon_{[a_1\dots a_d]} &= \epsilon_{a_1\dots a_d}
\epsilon_{0\dots(d-1)} &= 1
\\
\epsilon^{[a_1\dots a_d]} &= \epsilon^{a_1\dots a_d}
\epsilon^{0\dots(d-1)} &= \eta% := (-)^{\text{\# time directions}}
\end{align}

and the Levi-Civita tensor for curved indices is obtained as

\begin{equation}
\epsilon_{\mu_1\dots\mu_d} := e_{\mu_1}{^{a_1}}\dots e_{\mu_d}{^{a_d}}\varepsilon_{a_1\dots a_d} = e\varepsilon_{\mu_1\dots\mu_d}.
\end{equation}

Contractions between Levi-Civita symbols are computed as

\begin{equation}
\varepsilon^{a_1\dots a_pc_{p+1}\dots c_d}\varepsilon_{b_1\dots b_pc_{p+1}\dots c_d} = p!(d-p)!\delta^{a_1\dots a_p}_{b_1\dots b_p}
\end{equation}

where the generalised Kronecker delta symbol is defined as

\begin{equation}
\delta^{a_1\dots a_p}_{b_1\dots b_p} = \delta^{[a_1}{_{b_1}}\dots\delta^{a_d]}{_{b_d}}.
\end{equation}

For the Levi-Civita tensor this gives

\begin{align}
\epsilon^{a_1\dots a_pc_{p+1}\dots c_d}\epsilon_{b_1\dots b_pc_{p+1}\dots c_d} &= p!(d-p)!\cdot\eta\cdot\delta^{a_1\dots a_p}_{b_1\dots b_p}
\\
\epsilon^{\mu_1\dots\mu_p\rho_{p+1}\dots\mu_d}\epsilon_{\nu_1\dots\nu_p\rho_{p+1}\dots\mu_d}
\end{align}

with additional factors $\eta$ [14][27].

§.§.§ Hodge dualities

We define the Hodge star operator $\ast:\Omega^p(M)\to\Omega^{d-p}(M)$ as

\begin{equation}
\ast\omega_{\mu_{p+1}\dots\mu_d} := \frac{1}{p!}\epsilon_{\mu_1\dots\mu_d}\omega^{\mu_1\dots\mu_p}
\end{equation}

The Hodge star satisfies

\begin{align}
\ast_{d-p}\ast_p &= \eta (-)^{p(d-p)}
\ast_p^{-1} &= \eta(-)^{p(d-p)}\ast_{d-p}
\end{align}

giving us an expression for inverse Hodge operators [14][27].

§.§.§ Inner products between forms

Making use of the Hodge star we can construct inner products

\begin{align}
(\bullet|\bullet) &: \Omega^p(M)^{\otimes 2}\to\mathbb{R}
(\bullet|\bullet) &: (\Omega^p(M)\otimes\gLie)^{\otimes 2}\to\mathbb{R}
\end{align}

between ($\gLie$-valued) forms as

\begin{equation}
(\omega|\eta) := \int\tr\big[\omega\ast\eta\big] = \int\extd^dx ~ \frac{e}{p!}\kappa_{IJ}\omega^I_{\mu_1\dots\mu_p}\eta^{J\mu_1\dots\mu_p}.
\end{equation}

The non-$\gLie$-valued case is simply obtained by not taking the trace as it isn't defined. This product is symmetric and non-degenerate. On Riemannian manifolds it's also negative semi-definite in the $\gLie$-valued case and positive semi-definite in the non-$\gLie$-valued case [14].

§.§.§ Adjoint exterior derivatives

Using the aforementioned inner product we can now define adjoints of exterior derivatives. Let $\omega_p$ and $\eta_{p+1}$ be respectively $p$- and $(p+1)$-forms. We define the adjoint $\extdd_{p+1} : \Omega^{p+1}(M)\to\Omega^p(M)$ of the exterior derivative [14][27] through the relation

\begin{gather}
(\extd_p\omega_p|\eta_{p+1}) = (\omega_p|\extdd_{p+1}\eta_{p+1})
\\
\Downarrow\nonumber
\\
\extdd = -\eta(-)^{p(d-p)}\ast\extd\ast = -(-)^{d+p}\ast\extd\invast = (-)^p\invast\extd\ast = -\extd\inn
\end{gather}

where we defined the contraction between forms as

\begin{gather}
\eta_p\inn\omega_{p+q} = \omega_{p+q}\innr\eta_p := (-)^{r(d-r+s)}\invast(\eta_s\ast\omega_{r+s})\Big.
\\
\mathcal{m}\Big.\nonumber
\\
(\eta_p\inn\omega_{p+q})_{\mu_1\dots\mu_p} = (\omega_{p+q}\innr\eta_p)_{\mu_1\dots\mu_p} := \frac{1}{q!}\eta^{\nu_1\dots\nu_q}\omega_{\nu_1\dots\nu_q\mu_1\dots\mu_p}
\end{gather}

Similarly it follows that

\begin{align}
\Dgd &= -\Dg\inn
(\ad A)^\dagger &= -\ad A\inn
\end{align}

§.§ Matrix Exponentiation and BCH Formula

Let us mention some standard formulas related to matrix exponentiation. Let $X,Y$ be arbitrary matrices. One could verify that

\begin{align}
e^XYe^{-X} &= e^{\ad X}Y = Y + [X,Y] + \frac{1}{2}[X,[X,Y]] + \dots
\\
e^X\delta e^{-X} &= \frac{1 - e^{-\ad X}}{\ad X}\delta X = \delta X - \frac{1}{2}[X,\delta X] + \frac{1}{6}[X,[X,\delta X]] - \dots
\end{align}

where $\delta$ is an arbitrary derivative with respect to matrix multiplication [28].

§.§.§ Baker-Campbell-Hausdorff (BCH) formula

For matrices which don't commute the identity $e^xe^y = e^{x+y}$ doesn't generally hold anymore. Instead, it is replaced by the Baker-Campbell-Hausdorff (BCH) formula [29]

\begin{equation}
e^Xe^Y = e^{X\star Y}
\end{equation}

where $X\star Y$ is formally expressed as

\begin{equation}
\label{BCH formula}
X\star Y = \ln(e^Xe^Y) = Y + \int_0^1\extd t ~ g\big(e^{t\ad X}e^{\ad Y}\big)X = -(-Y)\star(-X)
\end{equation}

with $g(z)$ given by

\begin{align}
g(z) &= \frac{\ln z}{1-z} = \sum_{n=0}^\infty\frac{(1-z)^n}{n+1},
|1-z| &< 1.
\end{align}

Some of the leading order terms are given by

\begin{equation}
X\star Y
X+Y + \frac{1}{2}[X,Y] + \frac{1}{12}[[X,Y],Y] + \frac{1}{12}[X,[X,Y]] + \Og(X^pY^q)_{p+q\geq 4}
\end{equation}

Another interesting way to expand $X\star Y$ is as

\begin{equation}
\label{BCH formula infinitesimal}
\begin{split}
X\star Y
&= X + \frac{\ad X}{1 - e^{-\ad X}}Y + \Og(Y^2)
\\
&= Y - \frac{\ad Y}{1 - e^{+\ad Y}}X + \Og(X^2)
\end{split}
\end{equation}

to leading orders in respectively $Y$ and $X$ [9]. Interestingly, it is the case that

\begin{equation}
\label{BCH infinitesimal expansion}
\frac{z}{e^z-1} = \sum_{n=0}^\infty \frac{B_n}{n!}z^n = 1 - \frac{1}{2}z + \frac{1}{12}z^2 + \Og(z^4)
\end{equation}

where $B_n$ the Bernoulli numbers [30] recursively defined as

\begin{align}
B_0 &= 1,
B_{n+1} &= \sum_{k=0}^{n+1}\binom{n+1}{k}B_k
B_n &= \frac{-1}{n+1}\sum_{k=0}^{n-1}\binom{n+1}{k}B_k.
\end{align}

Some of the leading values are given by

\begin{align}
B_0 &= 1,
B_1 &= -\frac{1}{2},
B_2 &= \frac{1}{6},
B_3 &= 0,
B_4 &= -\frac{1}{30},
\end{align}

§ $D = 0 + 3$, $\NSUSY = 2$ SUPERSYMMETRY

§.§ Spinors and Gamma Matrices

The gamma matrices are $2\times 2$ matrices given by

\begin{equation}
\gamma^\mu = (\gamma^\mu{_\alpha}{^\beta}) = (\sigma^3,~-\sigma^1,~-\sigma^2)
\end{equation}

with $\mu = 1,2,3$ and $\alpha,\beta = 1,2$, where lower spinorial indices are column indices and upper spinorial indices are row indices, and $\sigma^\mu$ are the Pauli matrices [11][26]. These satisfy an algebra

\begin{equation}
\label{Clifford algebra}
\gamma^\mu\gamma^\nu = \eta^{\mu\nu}\idop + i\epsilon^{\mu\nu\rho}\gamma_\rho
\end{equation}

where we normalised the $x$-space Levi-Civita symbol as $\epsilon_{012} = 1$ and the metric is simply given by $\eta_{\mu\nu} = \delta_{\mu\nu}$.
Indices are raised and lowered using the charge conjugation matrix which in this case is just the Levi-Civita symbol $\varepsilon_{\alpha\beta}$ in spinor space, where we follow NW-SE conventions:

\begin{align}
\psi^\alpha &= \varepsilon^{\alpha\beta}\psi_\beta,
\psi_\alpha &= \psi^\beta\varepsilon_{\beta\alpha}.
\end{align}

where we normalised $\varepsilon_{12} = \varepsilon^{12} = 1$. Let us name a few interesting properties of the tensors $\varepsilon_{\alpha\beta}$ and $\gamma^\mu{_\alpha}{^\beta}$:

\begin{align}
\varepsilon_\alpha{^\beta} &= -\varepsilon^\beta{_\alpha} = \delta_\alpha{^\beta}
\gamma^\mu{_\alpha}{^\beta} &= \gamma^{\mu\beta}{_\alpha}
\end{align}

Similarly, we have for example

\begin{align}
\psi\chi &:= \psi^\alpha\chi_\alpha = (-)^{|\psi||\chi|+1}\chi\psi
\\
\psi\gamma^\mu\chi &:= \psi^\alpha\gamma^\mu{_\alpha}{^\beta}\chi_\beta = (-)^{|\psi||\chi|}\chi\gamma^\mu\psi
\end{align}

where $|\bullet|$ denotes the Grassmann parity of an object. In case there is no(t much) room for confusion, contracted indices will usually be dropped, with always an assumption of NW-SE contraction. Some more advanced identities of the gamma matrices include the Fierz identity

\begin{align}
\label{Fierz identity}
M &= \frac{1}{2}\idop \tr M + \frac{1}{2}\gamma^\mu \tr(M\gamma_\mu)
M_\alpha{^\beta} &= \frac{1}{2}M_\gamma{^\gamma}\varepsilon_\alpha{^\beta} + \frac{1}{2}M_\gamma{^\delta}\gamma_{\mu\delta}{^\gamma}\gamma^\mu{_\alpha}{^\beta}
\end{align}

as well as the following few lemmas [11][26]:

§.§ Some identities

\begin{equation}
\label{Lemma symmetric}
\gamma^\mu{_{\alpha\beta}}\gamma_\mu{^{\gamma\delta}} = -2\varepsilon_{(\alpha}{^\gamma}\varepsilon_{\beta)}{^\delta}
\end{equation}

Proof: We start off by noting that

\begin{equation}
\label{tussenstap}
\begin{aligned}
\gamma^\mu\gamma_\mu &\overeq{\ref{Clifford algebra}} \delta^\mu{_\mu}\idop = 3\idop,
\\
\gamma^\nu\gamma^\mu\gamma_\nu
&\overeq{\ref{Clifford algebra}}
\gamma^\mu + i\epsilon^{\nu\mu\rho}\gamma_\rho\gamma_\nu
\overeq{\ref{Clifford algebra}}
\gamma^\mu - \epsilon^{\nu\mu\rho}\epsilon_{\rho\nu\sigma}\gamma^\sigma = -\gamma^\mu.
\end{aligned}
\end{equation}

Applying the Fierz identity to respectively $_\alpha{^\gamma}$ and $_\beta{^\gamma}$ we now find that

\begin{equation}
\begin{split}
\gamma^\mu{_{\alpha\beta}}\gamma_\mu{^{\gamma\delta}}
&\overeq{\ref{Fierz identity}} -\frac{1}{2}\gamma^\mu\gamma_{\mu(\beta}{^\delta}\varepsilon_{\alpha)}{^\gamma} + \frac{1}{2}\gamma^\mu\gamma_\nu\gamma_{\mu(\beta}{^\delta}\gamma^\nu{_{\alpha)}}{^\gamma}
\\
&\overeq{\ref{tussenstap}} -\frac{3}{2}\varepsilon_{(\alpha}{^\gamma}\varepsilon_{\beta)}{^\delta} - \frac{1}{2}\gamma_{\mu(\alpha}{^\gamma}\gamma^\mu{_{\beta)}}{^\delta}
\\
&\overeq{\ref{Fierz identity}} -\frac{3}{2}\varepsilon_{(\alpha}{^\gamma}\varepsilon_{\beta)}{^\delta} - \frac{1}{4}\gamma^\mu\gamma^\nu\gamma_{\mu\gamma\delta}\gamma_\nu{^{\alpha\beta}}
\\
&\overeq{\ref{tussenstap}} -\frac{3}{2}\varepsilon_{(\alpha}{^\gamma}\varepsilon_{\beta)}{^\delta} + \frac{1}{4}\gamma^\mu{_{\alpha\beta}}\gamma_\mu{^{\gamma\delta}}
\end{split}
\end{equation}

which is clearly equivalent to the desired identity.

\begin{equation}
\label{Lemma antisymmetric}
\epsilon^{\mu\rho\sigma}\gamma_{\rho\alpha}{^\beta}\gamma_{\sigma\gamma}{^\delta} = i\gamma^\mu{_\alpha}{^\delta}\varepsilon_\gamma{^\beta} - i\varepsilon_\alpha{^\delta}\gamma^\mu{_\gamma}{^\beta}
\end{equation}

Proof: Application of the Fierz identity on $^\beta{_\gamma}$ yields

\begin{equation}
\epsilon^{\mu\rho\sigma}\gamma_{\rho\alpha}{^\beta}\gamma_{\sigma\gamma}{^\delta}
\overeq{\ref{Fierz identity}} \frac{1}{2}\epsilon^{\mu\rho\sigma}\Big(\gamma_\rho\gamma_{\sigma\alpha}{^\delta}\varepsilon_\gamma{^\beta} + \gamma_\rho\gamma_\nu\gamma_{\sigma\alpha}{^\delta}\gamma^\nu{_\gamma}{^\beta}\Big).
\end{equation}

For the first term we find

\begin{equation}
\frac{1}{2}\epsilon^{\mu\rho\sigma}\gamma_\rho\gamma_{\sigma\alpha}{^\delta}
\overset{\text{\ref{Clifford algebra}}}{=} \frac{i}{2}\epsilon^{\mu\rho\sigma}\epsilon_{\rho\sigma\nu}\gamma^\nu{_\alpha}{^\delta} = i\gamma^\mu{_\alpha}{^\delta}.
\end{equation}

As for the second term we find

\begin{equation}
\begin{split}
\frac{1}{2}\epsilon^{\mu\rho\sigma}\gamma_\rho\gamma_\nu\gamma_{\sigma\alpha}{^\delta}
&\overeq{\ref{Clifford algebra}}
\frac{1}{2}\epsilon^{\mu\rho\sigma}\big(\eta_{\rho\nu} + i\epsilon_{\rho\nu\tau}\gamma^\tau\big)\gamma_{\sigma\alpha}{^\delta}
\\
&\overeq{\ref{Clifford algebra}}
\frac{1}{2}\epsilon^\mu{_{\nu\sigma}}\gamma^\sigma{_\alpha}{^\delta} - i\delta^{\mu\sigma}_{\nu\tau}\big(\delta^\tau{_\sigma}\varepsilon_\alpha{^\delta} + i\epsilon^\tau{_{\sigma\kappa}}\gamma^\kappa{_\alpha}{^\delta}\big)
\\
&= \frac{1}{2}\epsilon^\mu{_{\nu\sigma}}\gamma^\sigma{_\alpha}{^\delta} - i\delta^\mu{_\nu}\varepsilon_\alpha{^\beta} - \frac{1}{2}\epsilon^\mu{_{\nu\sigma}}\gamma^\sigma{_\alpha}{^\delta}
\\
&= -i\delta^\mu{_\nu}\varepsilon_\alpha{^\beta}
\end{split}
\end{equation}

Combining these results we find

\begin{equation}
\epsilon^{\mu\rho\sigma}\gamma_{\rho\alpha}{^\beta}\gamma_{\sigma\gamma}{^\delta}
= \frac{1}{2}\epsilon^{\mu\rho\sigma}\Big(\gamma_\rho\gamma_{\sigma\alpha}{^\delta}\varepsilon_\gamma{^\beta} + \gamma_\rho\gamma_\nu\gamma_{\sigma\alpha}{^\delta}\gamma^\nu{_\gamma}{^\beta}\Big)
= i\gamma^\mu{_\alpha}{^\delta}\varepsilon_\gamma{^\beta} - i\varepsilon_\alpha{^\delta}\gamma^\mu{_\gamma}{^\beta}
\end{equation}

thus proving our lemma.

Finally, an interesting identity which could prove useful is

Lemma: Let $\Og_{\alpha\beta\gamma}$ be any spinor space tensor. Then

\begin{equation}
\Og^\beta{_{\alpha\beta}}
= \Og_\alpha{^\beta}{_\beta} + \Og^\beta{_{\beta\alpha}}.
\end{equation}

Proof: Since antisymmetric 3-tensors in spinor space always vanish, we find that

\begin{equation}
0 \equiv 3\Og_{[\alpha\beta\gamma]} = \Og_{\alpha[\beta\gamma]} + \Og_{[\gamma|\alpha|\beta]} + \Og_{[\beta\gamma]\alpha}.
\end{equation}

Contracting $\beta$ and $\gamma$ then yields the desired identity.

§.§ Hermitian Conjugation

Let us now make a few comments on Hermitian conjugation. Throughout this thesis we will repeatedly make use of norms of vectors, which arise from Hilbert space inner products, as opposed to the contractions of indices through the charge conjugation matrix $\varepsilon_{\alpha\beta}$. The former is more easy to understand using column matrices and the other in index notation. In this section of the appendix we wish to shed some light on this potentially confusing tension between these two notations.

To be specific, we note that

\begin{align}
\zeta\eta &:= \zeta^\alpha\eta_\alpha = \zeta_\alpha\eta_\beta\varepsilon^{\beta\alpha}
(\zeta|\eta) &:= (\zeta_\alpha)^\ast\eta_\alpha
|\zeta|^2 &:= (\zeta_\alpha)^\ast\zeta_\alpha
\end{align}

These are covariant expressions, however, the way the indices are contracted may be confusing to readers. Now looking at the column vector notation we find that

\begin{align}
(\zeta|\eta) &:= \zeta^\dagger\eta
|\zeta|^2 &:= \zeta^\dagger\zeta
\end{align}

interpreting now $\zeta$ as a column matrix and $\zeta^\dagger$ as a row matrix. As it might be regarded as rather confusing that there are two ways to look at these inner products we harmonise the two by introducing the spinor notation

\begin{equation}
\zeta^{\dagger\alpha} := (\zeta_\alpha)^\ast
\end{equation}

in terms of representations this also makes sense, since if one is in the defining representation of SU(2) the other will be in the conjugate representation, which makes sense index-wise. We can now harmoniously write

\begin{align}
(\zeta|\eta) &= \zeta^\dagger\eta = \zeta^{\dagger\alpha}\eta_\alpha
|\zeta|^2 &= \zeta^\dagger\zeta = \zeta^{\dagger\alpha}\zeta_\alpha
\end{align}

Let us now move on to the gamma matrices. We note that

\begin{align}
(\gamma^\mu)^\dagger &= \gamma^\mu
(\gamma^\mu{_\alpha}{^\beta})^\ast &= \gamma^\mu{_\beta}{^\alpha}
\end{align}

As an example we find that

\begin{equation}
(\gamma^\mu\zeta)^{\dagger\alpha} = (\gamma^\mu{_\alpha}{^\beta}\zeta_\beta)^\ast = (\zeta_\beta)^\ast(\gamma^\mu{_\alpha}{^\beta})^\ast = \zeta^{\dagger\beta}\gamma^\mu{_\beta}{^\alpha} = \zeta^\dagger\gamma^{\mu\alpha}
\end{equation}

which is nicely in agreement with the results one would expect from using matrices instead of index notation [11].

[1]
A. S. Arvanitakis, A. Sevrin, and P. K. Townsend, “Yang-mills as massive
chern-simons theory: a third way to three-dimensional gauge theories,” Physical Review Letters, vol. 114, no. 18, p. 181603, 2015.

[2]
S. Cremonesi, “An introduction to localisation and supersymmetry in curved
space.,” Proceedings of Science (PoS)., p. 02, 2013.

[3]
M. F. Sohnius, “Introducing supersymmetry,” Physics reports, vol. 128,
no. 2-3, pp. 39–204, 1985.

[4]
M. Peskin, An introduction to quantum field theory.
CRC press, 2018.

[5]
J. Wess and J. Bagger, Supersymmetry and supergravity, vol. 25.
Princeton university press, 1992.

[6]
R. Blumenhagen, D. Lüst, and S. Theisen, Basic concepts of string
theory, vol. 17.
Springer, 2013.

[7]
B. Willett, “Localization on three-dimensional manifolds,” Journal of
Physics A: Mathematical and Theoretical, vol. 50, no. 44, p. 443006, 2017.

[8]
A. Kapustin, B. Willett, and I. Yaakov, “Exact results for wilson loops in
superconformal chern-simons theories with matter,” Journal of High
Energy Physics, vol. 2010, no. 3, pp. 1–29, 2010.

[9]
J. Fuchs and C. Schweigert, Symmetries, Lie algebras and representations:
A graduate course for physicists.
Cambridge University Press, 2003.

[10]
D. Z. Freedman and A. Van Proeyen, Supergravity.
Cambridge university press, 2012.

[11]
C. Closset, T. T. Dumitrescu, G. Festuccia, and Z. Komargodski,
“Supersymmetric field theories on three-manifolds,” Journal of High
Energy Physics, vol. 2013, no. 5, pp. 1–45, 2013.

[12]
E. Witten, “Mirror manifolds and topological field theory,” 1991.

[13]
D. Tong, “Supersymmetric quantum mechanics.” Online source; available at

[14]
M. Nakahara, Geometry, topology and physics.
CRC press, 2018.

[15]
E. Bergshoeff, O. Hohm, W. Merbis, A. J. Routh, and P. K. Townsend, “Minimal
massive 3d gravity,” Classical and Quantum Gravity, vol. 31,
p. 145008, jul 2014.

[16]
A. S. Arvanitakis, A. J. Routh, and P. K. Townsend, “Matter coupling in 3d
`minimal massive gravity',” Classical and Quantum Gravity, vol. 31,
p. 235012, nov 2014.

[17]
A. S. Arvanitakis and P. K. Townsend, “Minimal massive 3d gravity unitarity
redux,” Classical and Quantum Gravity, vol. 32, p. 085003, mar 2015.

[18]
S. Deser, R. Jackiw, and S. Templeton, “Topologically massive gauge
theories,” Annals of Physics, vol. 281, no. 1-2, pp. 409–449, 2000.

[19]
S. Mukhi and C. Papageorgakis, “M2 to d2,” Journal of High Energy
Physics, vol. 2008, pp. 085–085, may 2008.

[20]
S. Mukhi, “Unravelling the novel higgs mechanism in (2+ 1) d chern-simons
theories,” Journal of High Energy Physics, vol. 2011, no. 12,
pp. 1–26, 2011.

[21]
H. Nishino and S. J. Gates Jr, “Chern–simons theories with supersymmetries in
three dimensions,” International Journal of Modern Physics A, vol. 8,
no. 19, pp. 3371–3421, 1993.

[22]
V. Pestun, “Localization of gauge theory on a four-sphere and supersymmetric
wilson loops,” Communications in Mathematical Physics, vol. 313,
pp. 71–129, may 2012.

[23]
J. Källén, “Cohomological localization of chern-simons theory,” Journal of High Energy Physics, vol. 2011, aug 2011.

[24]
D. Tong, “Supersymmetric field theory.” Online source; available at

[25]
S. Ferrara, M. A. Lledó, and O. Maciá, “Supersymmetry in
noncommutative superspaces,” Journal of High Energy Physics,
vol. 2003, no. 09, p. 068, 2003.

[26]
F. R. Ruiz and P. Van Nieuwenhuizen, “Lectures on supersymmetry and
supergravity in (2+ 1)-dimensions and regularization of supersymmetric gauge
theories,” Recent developments in gravitation and mathematical physics.
Proceedings of the 2nd Mexican School, Tlaxcala Mexico, pp. 1–7, 1996.

[27]
T. Ortín, Gravity and strings.
Cambridge University Press, 2004.

[28]
C. Pope, “Kaluza-klein theory.” Online source; available at
<http://people.tamu.edu/ c-pope/ihplec.pdf>.

[29]
R. Gilmore, “Baker-campbell-hausdorff formulas,” Journal of Mathematical
Physics, vol. 15, no. 12, pp. 2090–2092, 1974.

[30]
L. Carlitz, “Bernoulli numbers,” Fib. Quart, vol. 6, no. 1, p. 968,

[1]
A. S. Arvanitakis, A. Sevrin, and P. K. Townsend, “Yang-mills as massive
chern-simons theory: a third way to three-dimensional gauge theories,” Physical Review Letters, vol. 114, no. 18, p. 181603, 2015.

[2]
S. Cremonesi, “An introduction to localisation and supersymmetry in curved
space.,” Proceedings of Science (PoS)., p. 02, 2013.

[3]
M. F. Sohnius, “Introducing supersymmetry,” Physics reports, vol. 128,
no. 2-3, pp. 39–204, 1985.

[4]
M. Peskin, An introduction to quantum field theory.
CRC press, 2018.

[5]
J. Wess and J. Bagger, Supersymmetry and supergravity, vol. 25.
Princeton university press, 1992.

[6]
R. Blumenhagen, D. Lüst, and S. Theisen, Basic concepts of string
theory, vol. 17.
Springer, 2013.

[7]
B. Willett, “Localization on three-dimensional manifolds,” Journal of
Physics A: Mathematical and Theoretical, vol. 50, no. 44, p. 443006, 2017.

[8]
A. Kapustin, B. Willett, and I. Yaakov, “Exact results for wilson loops in
superconformal chern-simons theories with matter,” Journal of High
Energy Physics, vol. 2010, no. 3, pp. 1–29, 2010.

[9]
J. Fuchs and C. Schweigert, Symmetries, Lie algebras and representations:
A graduate course for physicists.
Cambridge University Press, 2003.

[10]
D. Z. Freedman and A. Van Proeyen, Supergravity.
Cambridge university press, 2012.

[11]
C. Closset, T. T. Dumitrescu, G. Festuccia, and Z. Komargodski,
“Supersymmetric field theories on three-manifolds,” Journal of High
Energy Physics, vol. 2013, no. 5, pp. 1–45, 2013.

[12]
E. Witten, “Mirror manifolds and topological field theory,” 1991.

[13]
D. Tong, “Supersymmetric quantum mechanics.” Online source; available at

[14]
M. Nakahara, Geometry, topology and physics.
CRC press, 2018.

[15]
E. Bergshoeff, O. Hohm, W. Merbis, A. J. Routh, and P. K. Townsend, “Minimal
massive 3d gravity,” Classical and Quantum Gravity, vol. 31,
p. 145008, jul 2014.

[16]
A. S. Arvanitakis, A. J. Routh, and P. K. Townsend, “Matter coupling in 3d
`minimal massive gravity',” Classical and Quantum Gravity, vol. 31,
p. 235012, nov 2014.

[17]
A. S. Arvanitakis and P. K. Townsend, “Minimal massive 3d gravity unitarity
redux,” Classical and Quantum Gravity, vol. 32, p. 085003, mar 2015.

[18]
S. Deser, R. Jackiw, and S. Templeton, “Topologically massive gauge
theories,” Annals of Physics, vol. 281, no. 1-2, pp. 409–449, 2000.

[19]
S. Mukhi and C. Papageorgakis, “M2 to d2,” Journal of High Energy
Physics, vol. 2008, pp. 085–085, may 2008.

[20]
S. Mukhi, “Unravelling the novel higgs mechanism in (2+ 1) d chern-simons
theories,” Journal of High Energy Physics, vol. 2011, no. 12,
pp. 1–26, 2011.

[21]
H. Nishino and S. J. Gates Jr, “Chern–simons theories with supersymmetries in
three dimensions,” International Journal of Modern Physics A, vol. 8,
no. 19, pp. 3371–3421, 1993.

[22]
V. Pestun, “Localization of gauge theory on a four-sphere and supersymmetric
wilson loops,” Communications in Mathematical Physics, vol. 313,
pp. 71–129, may 2012.

[23]
J. Källén, “Cohomological localization of chern-simons theory,” Journal of High Energy Physics, vol. 2011, aug 2011.

[24]
D. Tong, “Supersymmetric field theory.” Online source; available at

[25]
S. Ferrara, M. A. Lledó, and O. Maciá, “Supersymmetry in
noncommutative superspaces,” Journal of High Energy Physics,
vol. 2003, no. 09, p. 068, 2003.

[26]
F. R. Ruiz and P. Van Nieuwenhuizen, “Lectures on supersymmetry and
supergravity in (2+ 1)-dimensions and regularization of supersymmetric gauge
theories,” Recent developments in gravitation and mathematical physics.
Proceedings of the 2nd Mexican School, Tlaxcala Mexico, pp. 1–7, 1996.

[27]
T. Ortín, Gravity and strings.
Cambridge University Press, 2004.

[28]
C. Pope, “Kaluza-klein theory.” Online source; available at
<http://people.tamu.edu/ c-pope/ihplec.pdf>.

[29]
R. Gilmore, “Baker-campbell-hausdorff formulas,” Journal of Mathematical
Physics, vol. 15, no. 12, pp. 2090–2092, 1974.

[30]
L. Carlitz, “Bernoulli numbers,” Fib. Quart, vol. 6, no. 1, p. 968,