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\begin{center}

{\Large{\bf From $N=1$ to $N=2$ supersymmetries\\
in 2+1 dimensions}}\\

\vspace{1cm}

{\bf Jean Alexandre}\\
Physics Department, King's College \\
WC2R 2LS, London, UK\\
jean.alexandre@kcl.ac.uk

\vspace{3cm}

{\bf Abstract}

\end{center}

\vspace{.2cm}

Starting from $N=1$ scalar and vector supermultiplets in 2+1 dimensions, we
construct superfields which constitute Lagrangians invariant under $N=2$
supersymmetries. We first recover the $N=2$ supersymmetric Abelian-Higgs model
and then the $N=2$ pure super Yang-Mills model. The conditions for this 
elevation are consistent with previous results found by other authors.

\vspace{3cm}



$N=2$ supersymmetry in 2+1 dimensions had been studied
\cite{affleck} in order to
investigate the supersymmetrization of the instantons effects which lead
to a linear confinement \cite{poliakov}.
Since then, systematic studies of $N=2$ supersymmetry in 2+1
dimensions have been done
in \cite{strassler} where exact results were obtained,
such as
superpotentials and topologies of moduli spaces in various cases.

These exact results can be derived
since $N=2$ supersymmetry in 2+1 dimensions can be obtained by
dimensional reduction of $N=1$ supersymmetry in 3+1 dimensions \cite{siegel}
and thus has similar properties, such as non-renormalization theorems.
These theorems are not
present for $N=1$ supersymmetry in 2+1 dimensions
and it is thus interesting to study its elevation to $N=2$.
In this context,
it was shown that the presence of topologically
conserved currents
leads to a centrally extended $N=2$ superymmetry, the
central charge of the
superalebra being the topological charge \cite{hlousek}.
In \cite{edelstein}, the $N=1$ supersymmetric Abelian-Higgs model
was considered and it was shown that the on-shell Lagrangian can be
extended to the $N=2$
Abelian-Higgs model if a relation is imposed between the gauge
coupling
and the Higgs self-coupling. Such a condition is in general expected in a
$N=2$ invariant theory built out of $N=1$ Lagrangians \cite{sohnius}.
Another example of extension 
was given in \cite{ams2} where $N=1$ supersymmetries of composite
operators
was elevated to a $N=2$ Abelian model, up to irrelevant operators.
In this work, the coupling of matter to the gauge field
was obtained with higher order composites, simulating the dynamical
generation of the $N=2$
supersymmetry that would occur after an appropriate functional
integration of a
gauge field coupled to the original $N=1$ supermultiplets.

We propose here another illustration of the supersymmetry extension,
with the superfield construction of $N=2$ Lagrangians in
terms of $N=1$
scalar and vector superfields.
The $N=1$ superspace in 2+1 dimensions contains only one real
two-component Grassmann coordinate $\theta$
and the invariant actions are integrals over superspace which involve $\int
d^2\theta$.
The Lagrangians are constructed out of superfields which mix the
original $N=1$
superfields in such a way that a $N=1$ supersymmetric transformation
on the original superfields
leaves the $N=2$ Lagrangians invariant. 

We first consider the pure $U(1)$ case and
then add matter so as to construct the
Abelian Higgs model, using a Fayet-Iliopoulos term.
We finally make a superfield construction for the
$N=2$ pure Yang-Mills Lagrangian.
As will be seen, the present contruction exhibits naturally the
conditions found in \cite{hlousek} and \cite{edelstein} for the elevation of a $N=1$
to a $N=2$ supersymmetry.


\vspace{.5cm}

The gamma matrices are given by
$\gamma^0=\sigma^2,\gamma^1=i\sigma^1,\gamma^2=i\sigma^3$,
where $\sigma^1,\sigma^2,\sigma^3$ are the Pauli matrices, such that
$g^{\mu\nu}=diag(1,-1,-1)$
and $[\gamma^\mu,\gamma^\nu]=-2i\epsilon^{\mu\nu\rho}\gamma_\rho$.
We have the following usual properties, valid for any
2-component complex spinors $\eta,\zeta$:

\be
\eta\zeta=\eta^\alpha\zeta_\alpha=\zeta\eta~~~~\mbox{and}~~~~
\eta\gamma^\mu\zeta=-\zeta\gamma^\mu\eta,
\ee

\nin The 2-component real spinor $\theta$, Grassmann coordinate in the superspace,
satisfies the properties

\bea\label{prop}
&&(\theta\eta)(\theta\zeta)=-\hf\theta^2(\eta\zeta)\nonu
&&\theta\gamma^\mu\theta=0\nonu
&&\theta\gamma^\mu\gamma^\nu\theta=-\theta^2g^{\mu\nu}\nonu
&&\theta\gamma^\mu\gamma^\nu\gamma^\rho\theta=
-i\theta^2\epsilon^{\mu\nu\rho}\nonu
&&\theta\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma\theta=
-\theta^2\left(g^{\mu\nu}g^{\rho\sigma}+g^{\mu\rho}g^{\nu\sigma}
-g^{\mu\sigma}g^{\nu\rho}\right),\nonu
&&\int d^2\theta~\theta^2=1
\eea

In 2+1 dimensions, the $N=1$ scalar superfield and the $N=1$ vector
superfield in the Wess-Zumino gauge
are respectively given by

\bea
\Phi&=&\rho+(\theta\xi)+\hf\theta^2 D\nonu
V_\alpha&=&i(\br A\theta)_\alpha+\hf\theta^2\chi_\alpha,
\eea

\nin where all the fields are real.
To form an $N=2$ supermultiplet, we define the complex gaugino
$\lambda=\xi+i\chi$.
The two fermionic degrees of freedom then
balance the two bosonic ones, since $A_\mu$ has one degree of freedom
\cite{binegar}.
The (complex) scalar superfield $G$ containing these degrees of freedom is

\bea
G&=&\Phi+iD^\alpha V_\alpha\\
&=&\rho+(\theta\lambda)+\hf\theta^2 D^2+i\partial_\mu A_\nu
(\theta\gamma^\mu\gamma^\nu\theta)\nonumber,
\eea

\nin where the superderivative is
$D_\alpha=\partial_\alpha+i(\br\partial\theta)_\alpha$ \cite{hitchin}.
Consider now the superfield

\be
D^\beta G=-\lambda^\beta-D\theta^\beta-i\partial_\mu
A_\nu(\gamma^\nu\gamma^\mu\theta)^\beta
+i(\br\partial\rho\theta)^\beta+i(\theta\partial_\mu\lambda)
(\gamma^\mu\theta)^\beta.
\ee

\nin With the properties (\ref{prop}), it is easy to see that

\bea
\int d^2\theta D^\beta G D_\beta \ol G&=&
-\hf F^{\mu\nu}F_{\mu\nu}+i\ol\lambda\br\partial\lambda
+\partial_\mu\rho\partial^\mu\rho+D^2\nonu
&&+\left(\partial^\mu A_\mu\right)^2+\mbox{surface term},
\eea

\nin where the surface term is
$\partial_\mu(\ol\lambda\gamma^\mu\lambda)$ and $F_{\mu\nu}=\partial_\mu A_\nu
-\partial_\nu A_\mu$.
If the gauge condition $\partial^\mu A_\mu=0$ is imposed, we find then
the $N=2$ Abelian gauge
kinetic term. This gauge condition was found in \cite{hlousek} where $A_\mu$ is given 
the role af a conserved current, necessary to elevate the $N=1$ supersymmetry to 
$N=2$, and it reflects the fact that we work in the Wess-Zumino gauge. 
Disregarding the surface term, the expected $N=2$ Lagrangian is then
expressed in terms of the original $N=1$ superfields as follows:

\bea\label{lgauge}
{\cal L}_{gauge}&=&\hf\int d^2\theta D^\beta G D_\beta \ol G\nonu
&=&-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\frac{i}{2}\ol\lambda\br\partial\lambda
+\hf \partial_\mu\rho\partial^\mu\rho+\hf D^2.
\eea



\vspace{.5cm}


Matter is included with a complex $N=1$ scalar superfield $Q$:

\be
Q=\phi+(\theta\psi)+\hf\theta^2 F.
\ee

\nin So as to avoid the generation
of parity violating terms in the quantum corrections,
we can introduce an even number of superfields \cite{strassler}, but we do not consider
this problem here. 
We remind that a $N=1$ scalar superfield in 2+1 dimensions cannot be
chiral:
since $\theta$ is real, the chirality
condition $D^\alpha Q=0$ would constraint the space-time dependence of
the component fields
$\phi,\psi,F$ \cite{hitchin}.

The derivatives of the fields are obtained with the highest component
of
$D^\alpha Q D_\alpha \ol Q$ which reads

\be
\left.D^\alpha Q D_\alpha \ol Q\right|_{\theta^2}=
\theta^2\left(\partial_\mu\phi\partial^\mu\phi+i\ol\psi\br\partial
\psi+F\ol F+
\mbox{surface term}\right),
\ee

\nin where the surface term is
$\partial_\mu(\ol\psi\partial^\mu\psi)$.
The coupling to the gauge multiplet
is obtained with the highest components of the following superfields:

\bea
\left.D^\alpha QV_\alpha\ol Q\right|_{\theta^2}&=&-\hf\theta^2
\left(\ol\phi(\psi\chi)-\ol\phi A_\mu\partial^\mu\phi+i\ol\psi\br
A\psi\right)\nonu
\left.Q\Phi\ol Q\right|_{\theta^2}&=&\hf\theta^2
\left(\phi\ol\phi D-\phi\ol\psi\xi-\ol\phi\psi\xi+\rho\phi\ol
F+\rho\ol\phi F-
\rho\ol\psi\psi\right)\nonu
\left.QV^\alpha V_\alpha\ol Q\right|_{\theta^2}&=&-\theta^2\phi\ol\phi
A^\mu A_\mu,
\eea

\nin such that the matter Lagrangian is

\bea\label{lmat}
{\cal L}_{matter}
&=&\hf\int d^2\theta\left\{(D^\alpha-igV^\alpha)Q(D_\alpha+igV_\alpha)\ol Q
+2gQ\Phi\ol Q\right\}\nonu
&=&\frac{i}{2}\ol\psi\br D\psi+\hf D_\mu\phi
D^\mu\ol\phi-\frac{g}{2}(\phi\ol\psi\ol\lambda+
\ol\phi\psi\lambda)-\frac{g}{2}\rho\ol\psi\psi\nonu
&&+\frac{g}{2}\phi\ol\phi D+\hf F\ol F+\frac{g}{2}\rho\phi\ol F+\frac{g}{2}\rho\ol\phi F,
\eea

\nin where $g$ is a dimensionfull gauge coupling and
$D_\mu=\partial_\mu+ig A_\mu$. The Lagrangian 
(\ref{lmat}) was found in \cite{siegel} as
a consequence of the dimensional reduction of a $N=1$ theory in 3+1 dimensions.

\vspace{.5cm}

We can recover the
scalar interactions if we write the Lagrangians (\ref{lgauge}) and
(\ref{lmat}) on-shell.
We write for this the equations of motion of the auxiliary fields $D$
and $F$:

\bea
\ol F+g\rho\ol\phi&=&0\nonu
D+\frac{g}{2}\phi\ol\phi&=&0,
\eea

\nin such that the terms depending on the auxiliary fields
lead to the following potential

\bea
\left({\cal L}_{gauge}+{\cal L}_{matter}\right)_{pot}&=&
\hf D^2+\frac{g}{2}\phi\ol\phi D+\hf F\ol F+\frac{g}{2}\rho\ol\phi F+
\frac{g}{2}\rho\phi\ol F\nonu
&=&-\frac{g^2}{2}\rho^2\phi\ol\phi-\frac{g^2}{8}(\phi\ol\phi)^2.
\eea

The Abelian Higgs model is obtained by adding a
Fayet-Iliopoulos term which in
the present context is

\be
{\cal L}_{F.I.}=-\frac{g}{2}\phi_0^2\int d^2\theta(G+\ol G)
=-g\phi_0^2\int d^2\theta \Phi=-\frac{g}{2}\phi_0^2 D,
\ee

\nin where $\phi_0$ is a real parameter.
The addition of this term to the Lagrangian leads to the following
equation of motion
for the auxiliary field $D$:

\be
D+\frac{g}{2}\phi\ol\phi-\frac{g}{2}\phi_0^2=0,
\ee

\nin such that we obtain the expected gauge-symmetry breaking potential

\be\label{lah}
\left({\cal L}_{gauge}+{\cal L}_{matter}+{\cal L}_{F.I.}\right)_{pot}
=-\frac{g^2}{2}\rho^2\phi\ol\phi-\frac{g^2}{8}\left(\phi\ol\phi-\phi_0^2\right)^2.
\ee

\nin Note that the Higgs self-coupling is $g^2/8$, what was found in \cite{edelstein}
as a consistency condition for the elevation of the $N=1$ on-shell Lagrangian
to $N=2$.
The result (\ref{lah}) shows that the moduli space contains a
Higgs branch only,
where the vacuum expectation values of the scalar fields satisfy

\be
<\phi\ol\phi>=\phi_0^2 ~~~~\mbox{and}~~~~ <\rho>=0.
\ee

\vspace{.5cm}

The extension to a non-Abelian gauge group necessitates the
introduction of quadratic
superfields to generate the interactions. We will consider $SU(N)$
dynamics, with
structure constants $f^{abc}$ and coupling constant $g$.
A non-Abelian supermultiplet contains gauginos and scalars in the
adjoint representation, so that
the starting point is the set of scalar and vector $N=1$ superfields

\bea
\Phi^a&=&\rho^a+(\theta\xi^a)+\hf\theta^2D^a\nonu
V_\alpha^a&=&i(\br A^a\theta)_\alpha+\hf\theta^2\chi^a_\alpha,
\eea

\nin where $a=1,...,N^2-1$ is the gauge indice.
We then introduce the complex superfields

\be
G^a=\Phi^a+iD^\alpha V^a_\alpha,
\ee

\nin and, as in the Abelian case, the derivatives of the component fields are obtained with
the term $D^\beta G^aD_\beta\ol G^a$, provided that the gauge condition
$\partial^\mu A_\mu^a=0$ holds.
To generate the interactions of the superpartners, we will add to
$D^\beta G^a$ linear combinations of the following two superfields

\be
G^bV^{c\beta},~~~~~~~~D^\beta(V^{b\alpha} V^c_\alpha),
\ee

\nin and the remaining terms for the covariant derivatives
are obtained with the products

\bea\label{quad1}
\left.f^{abc}D^\beta \ol G^a G^b V^c_\beta\right|_{\theta^2}&=&
\theta^2f^{abc}\left(-\rho^b\xi^a\chi^c+\hf(i\ol\lambda^b\br
A^c\lambda^a+\mbox{c.c}.)
-2\rho^b\partial^\mu\rho^a A_\mu^c\right)\nonu
\left.f^{abc}f^{ade}G^bV^{c\beta}\ol G^d V^e_\beta\right|_{\theta^2}&=&
f^{abc}f^{ade}\theta^2\rho^b\rho^d A_\mu^c A^{e\mu}.
\eea

\nin The term (\ref{quad1}) also generates the Yukawa interactions
since

\be
2\rho^b\xi^a\chi^c=\rho^b(i\ol\lambda^a\lambda^c+\mbox{c.c.}).
\ee

\nin The non-Abelian gauge kinetic term is obtained with the products

\bea
\left.f^{abc}D^\beta\ol G^a
D_\beta(V^{b\alpha}V^c_\alpha)\right|_{\theta^2}&=&
2f^{abc}(\partial_\mu A_\nu^a) A_\rho^b
A_\sigma^c(\theta\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma\theta)\nonu
\left.f^{abc}f^{ade}D^\beta(V^{b\alpha}
V^c_\alpha)D_\beta(V^{b\alpha}V^c_\alpha)\right|_{\theta^2}&=&
f^{abc}f^{ade}A_\mu^b A_\nu^c A_\rho^d A_\sigma^e
(\theta\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma\theta),
\eea

\nin since we have, using the properties (\ref{prop}) and
$f^{abc}+f^{acb}=0$,

\bea\label{eq25}
&&f^{abc}\int d^2\theta(\partial_\mu A_\nu^a) A_\rho^b A_\sigma^c
(\theta\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma\theta)\nonu
&=&f^{abc}(\partial^\mu A_\mu^a) A_\nu^b A^{c\nu}+
f^{abc}A_\mu^b A_\nu^c(\partial^\mu A^{a\nu}-\partial^\nu A^{a\mu}),
\eea

\nin and

\bea
&&f^{abc}f^{ade}\int d^2\theta A_\mu^b A_\nu^c A_\rho^d A_\sigma^e
(\theta\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma\theta)\nonu
&=&2f^{abc}f^{ade}A_\mu^b A^{d\mu}A_\nu^c A^{e\nu}.
\eea

\nin With the gauge condition $\partial^\mu A_\mu^a=0$, 
the first term in the right-hand side of Eq.(\ref{eq25}) vanishes and only the
expected term remains.
Gathering these results, we find that the extension to an
$N=2$ pure
super-Yang-Mills Lagrangian is given by

\bea
{\cal L}_{Y.M.}&=&\hf\int d^2\theta\left|D^\beta
G^a+gf^{abc}\left(G^bV^{c\beta}
+\frac{i}{2}D^\beta(V^{b\alpha}V_\alpha^c)\right)\right|^2\nonu
&=&-\frac{1}{4}F^{a\mu\nu}F_{\mu\nu}^a+\frac{i}{2}\ol\lambda^a\br D\lambda^a
+\hf D^\mu\rho^a D_\mu\rho^a\nonu
&&-\frac{g}{2}f^{abc}\left(i\rho^b\ol\lambda^a\lambda^c+\mbox{c.c.}\right)+\hf D^aD^a,
\eea

\nin where $F^a_{\mu\nu}=\partial_\mu A_\nu^a-\partial_\nu A_\mu^a+gf^{abc}A_\mu^b A_\nu^c$
and $D_\mu(...)^a=\partial_\mu(...)^a+gf^{abc}A_\mu^b(...)^c$.

\vspace{1cm}

To conclude, let us stress the central point of these results. Whereas in \cite{edelstein},
the starting point was a given dynamics (Abelian Higgs model) described by a $N=1$
on-shell Lagrangian which was then elevated to $N=2$, we do not start here
with any specific dynamics but instead build directly $N=2$
off-shell Lagrangians with $N=1$ superfields.
This allows us to generate different dynamics and we generalize the elevation 
to a $N=2$ non-Abelian theory. Clearly, one could consider with the same method
other $N=2$ dynamics.

Finally, this work might be used in the context of effective models
for high-temperature (planar) superconductivity \cite{mavromatos},
where the initial $N=1$ supermultiplets are built out of composites of spinons and holons in the 
spin-charge separation framework.

\vspace{1cm}

\nin {\bf Acknowledgments} This work is supported by the Leverhulme Trust (U.K.) and I
would like to thank Sarben Sarkar and Nick Mavromatos for introducing me to this
subject.



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