aainstitutetext: Instituto Militar de Engenharia (IME),
Praça Gen. Tibúrcio 80, Urca, Rio de Janeiro, RJ, Brazil, CEP 22291-270
bbinstitutetext: Centro Brasileiro de Pesquisas Físicas (CBPF),
Rua Dr. Xavier Sigaud 150, Urca, Rio de Janeiro, RJ, Brazil, CEP 22290-180

# F-term spontaneous breaking of 3D-SUSY: an algebro-geometric treatment

José J. Ramón Marí,111Corresponding author. b Y.M.P. Gomes b J. A. Helayël-
Neto<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We settle a result on generic exactness of SUSY in 3-D, and provide a
mechanism of F-term spontaneous breaking of 3-D SUSY, with a different set of
tools from those used by O’Raifeartaigh in his seminal work on 4-D SUSY. In
our study, we use techniques of projective algebraic geometry so as to deal
successfully with cubic hypersurfaces.

##  1 Introduction

Supersymmetry (SUSY) appears in relativistic field theories (both classical
and quantum) as a space-time symmetry with the remarkable consequence of
placing bosons and fermions in the same (linear) representation space, called
supermultiplet. This is why it is usually referred to as a fermion-boson
symmetry. Besides being an elegant solution to the naturalness and gauge-
hierarchy problems of the Standard Model of Particle Physics, SUSY also
implies that the running coupling constants of the electroweak and strong
interactions can be unified at a high energy scale, $10^{16}$ GeV, providing
then a viable path towards unification. Supersymmetric partners of the
particles of the Standard Model have however not yet been observed at
available accelerator energies (LHC, in Run II, has been exploiting physics up
to 13 TeV centre-of-mass energy). Therefore, to describe some new physics
beyond the Standard Model, SUSY needs to be broken by some particular
mechanism (be it explicit by soft terms, spontaneous or dynamical). Even if
the breaking should occur at very high energies, it can be communicated to the
low-energy sector of the spectrum MSSM . In four space-time dimensions, as is
well-known, SUSY exhibits a complex structure: the fundamental representation
of simple four-dimensional (4D) SUSY is realised in terms of complex
superfields and their associated complex component fields. That is a crucial
point in connection to a particular mechanism of SUSY spontaneous breaking,
namely, F-term SUSY breaking, thoroughly studied by O’Raifeartaigh in his
classical 1975 papers ORafer . In ORafer , the author turns the analysis of
F-term spontaneous simple SUSY breaking into the study of a system of N
quadratic equations in N complex unkowns (these N unkowns being space-time
constant configurations of N complex scalar fields accommodated in N matter
superfields), and shows that spontaneous breaking is possible only if there
are at least $N=3$ superfields that display these scalar fields. The results
of ORafer  gave rise to the so-called O’Raifeartaigh models. (Just to avoid
any possible misunderstanding: N stands for the number of fields present in
the particular model under consideration. Here we shall be dealing only with
simple SUSY, and so N never refers to the number of extended supersymmetries.)

SUSY has been carefully explored in diverse space-time dimensions (more
specifically, from two to eleven). In particular, SUSY in three space-time
dimensions received a remarkable boost in relation to Chern-Simons and
topological field-theoretic models in three dimensions (3D) CSTFT3D . More
recently, renewed interest in 3D SUSY has arisen in connection to topological
materials in lower space dimensions, for instance (1+2)-dimensional
topological superconductors topsemicond , where SUSY appears as an emergent
symmetry of the action which describes the dynamics of the excitations. So,
motivated by the relevance that SUSY is nowadays acquiring as an emergent
symmetry in low-dimensional Condensed Matter systems such as graphene,
topological insulators and topological superconductors, we re-examine the
issue of F-term SUSY spontaneous breaking in three space-time dimensions.

Contrary to the 4D case, (1+2)-D SUSY has a real structure: simple SUSY in
three space-time dimensions is realised in terms of real superfields and their
corresponding real component fields. Renormalisation requirements in 3D allow
the matter superfields to have up to quartic interactions, whereas in 4D, the
coupling amongst matter superfields is at most cubic. This drastically changes
the structure underneath spontaneous SUSY breaking in comparison to its four-
dimensional counterpart: instead of a system of N quadratic equations in N
complex unkowns, the conditions for 3D F-term spontaneous SUSY breaking hinge
on the existence of real solutions of a system of N cubic equations in N real
variables, which renders reference to the seminal results in ORafer  no longer
suitable, and calls for different techniques. We carry out the analysis of the
corresponding system of cubic equations derived from F-term breaking using
projective algebraic geometry. Today, algebraic geometry is no stranger to
theoretical physics, see the work of Candelas et al Candelas  for a recent
example; for a sample of applications to biology, see Casanellas . Here we use
classical tools from this field to solve a physical question.

We wish to emphasize that we do not attempt a phenomenological discussion
here, but stick to the mathematical problem underlying F-term spontaneous
breaking of SUSY, although we keep in mind applications to emergent SUSY in
planar condensed matter phenomena, which shall be addressed elsewhere.

The outline of our paper is as follows: after presenting some background on
algebraic geometry in Section 2, we provide a result on generic exactness in
Section 3 ; next, in Section 4, we introduce an explicit family of potentials
with F-term SUSY breaking, and provide a different example, this time
numerical, of an even potential with F-term breaking in Section 5. Finally, in
Section 6 we derive a small hypersurface of potentials satisfying F-term SUSY
breaking in the 3-superfield case ($33$ dimensions) and its N-superfield
analogue, building upon Section 5, and cast our conclusions and final remarks
in Section 7.

  * Notation:

We denote by $E=E_{3}$ the linear space of quartic scalar potentials
$\Phi=\Phi(x,y,z)\in E$ on $\mathbb{R}^{3}$ ($E_{N}$ will be its
$N$-superfield analogue), and $(F,G,H)$ to be the gradient of $\Phi$ in the
N=3 case (quartic polynomials in N variables). Note that $E$ has dimension
$1+3+6+10+14=34$. We denote by $\mathbb{R}_{d}[x,y,z]$ the space of
polynomials in the variables $x,y,z$ of degree at most $d$, and write as
$\mathbb{R}[x,y,z]_{d}$ the space of homogeneous polynomials in $x,y,z$. Given
a polynomial $F$ in $x,y,z$, its homogenized form is denoted by $\tilde{F}$,
i.e.
$\tilde{F}(X_{0},X_{1},X_{2},X_{3})=X_{0}^{deg(F)}F(X_{1}/X_{0},X_{2}/X_{0},X_{3}/X_{0})$.

##  2 Intersection product in projective spaces

See Sha GH . We work over the complex projective $n$-space
$\mathbb{P}^{n}_{\mathbb{C}}=\mathbb{P}^{n},$ with projective homogeneous
coordinates $X_{0},\ldots,X_{n}.$ The affine coordinates are by default
$x_{i}=\frac{X_{i}}{X_{0}}$, and a form of degree $d$ is a homogeneous
polynomial of degree $d$. We start with a simple situation in the complex
plane: clearly, a complex projective line $\ell\subset\mathbb{P}^{2}$ (say
$X_{1}=0$) and a curve $F=0$ (where $F(X_{0},X_{1},X_{2})$ is a form of degree
$d$ in the $X_{i}$’s) intersect in $d$ complex points, counting
multiplicities, unless $X_{1}$ divides $F$. Likewise, we see that two conics
with no common factors $Q_{1},Q_{2}$ intersect in $4$ points counted with
multiplicities, which we write in intersection product notation as follows:
$\\{Q_{1}=0\\}\bullet\\{Q_{2}=0\\}=\\{Q_{1}=0\\}\bullet\\{Q_{3}=0\\},$ where
$Q_{3}=Q_{2}+\lambda Q_{1}$, where $\lambda$ is a solution of
$det(Q_{2}+xQ_{1})=0$, so $Q_{3}$ decomposes into linear factors, i.e.

$\\{Q_{3}=0\\}=\ell+\ell^{\prime}.$

Thus,

$\\{Q_{1}=0\\}\bullet\\{Q_{2}=0\\}=\\{Q_{1}=0\\}\bullet(\ell+\ell^{\prime})=\\{Q_{1}=0\\}\bullet\ell+\\{Q_{1}=0\\}\bullet\ell^{\prime}=2+2=4.$

Bézout’s Theorem in $\mathbb{P}^{2}$ generalises this to the following result:
if $F,G$ are forms of respective degrees $d,e$ in the variables
$X_{0},X_{1},X_{2}$, and have no common factors, their intersection number in
$\mathbb{P}^{2}$, $(F=0)\bullet(G=0)=F\bullet G=de$ (we abuse notation
identifying the locus with its equation).

In the case of $\mathbb{P}^{n},$ one has $D_{1},D_{2},\ldots,D_{n}$
hypersurfaces, where $D_{i}$ is the set of zeros of a form $F_{i}$ in
$X_{0},\ldots,X_{n}$ of degree $d_{i}$. Assume that they properly intersect,
i.e., that their intersection $F_{1}=0,\ldots,F_{n}=0$ is finite. One may
define an intersection $0$-cycle

$D_{1}\bullet\cdots\bullet D_{n}=\sum\mu_{P}P,$

where $\mu_{P}$ is the multiplicity of $P$, and all but a finite number of
$\mu_{P}$ are zero. If ${\mathcal{O}}_{P}$ is the ring of germs of holomorphic
functions on $\mathbb{P}^{n}$ around $P$ and $f_{i}=0$ are local equations
defined by $D_{i}$ (i.e. by $F_{i}$) one has:

$\mu_{P}=\mu_{P}(F_{1},\ldots,F_{n})=\dim_{\mathbb{C}}\frac{{\mathcal{O}}_{P}}{(f_{1},\ldots,f_{n})},$

where ${\mathcal{O}}$ is taken to be the ring of germs of holomorphic
functions at $P$, or the ring of rational functions in $n$ variables that are
regular at $P$.

This generalises to the following. In $\mathbb{P}^{2}$ we may factor every
(homogeneous) polynomial into (homogeneous) irreducible factors, $F=\prod
F_{i}^{e_{i}},$ and there is an intersection product defined on integer linear
combinations of hypersurfaces; thus if $D$ is the divisor associated with $F$,
and $E_{i}$ are the divisors associated with $F_{i}$, one has $E=\sum
e_{k}E_{k}$, and the intersection product is linear on each argument, if each
summand is defined. The intersection number is always an integer, it is
invariant under deformation, depends only on the ideal generated by $(F_{i})$
and if they do not intersect properly (i.e. their intersection does not have
the right dimension) one may tweak the polynomials $F_{i}$ so that their
perturbed cousins $H_{i}$ intersect properly (i.e. the $n$ hypersurfaces
intersect only in a finite set) and the intersection number of the $F_{i}$
equals that of the $H_{i}$. Back to the proper intersection case, the
intersection of $F_{1},\cdots,F_{n-1}$ is a curve, consisting of a finite
number of irreducible components endowed with a multiplicity. If we take a
hypersurface $F_{n}$ which, say, cuts this curve transversally (hence away
from intersections of components), then the result is a $0$-cycle $\sum
n_{i}P_{i}$, where the $P_{i}$ are the points of intersection and the $n_{i}$
are its multiplicities, which sum up to

$\sum n_{i}=d_{1}d_{2}\cdots d_{n}.$

This offers an upper bound in the case of finite intersection of the $F_{i}$.

###### Example 2.1

An extreme example of plane cubics having one real intersection point and none
in the affine plane is that of the equations $y^{2}=f(x)$ and $y^{2}=f(x)+1$.
In this case, after homogenisation they intersect only on $(0:0:1)$ with
multiplicity $9$. Thus Bezout’s credits are all spent on one point, which lies
outside the affine part. This example is easily generalised to the case of
three cubic surfaces in projective $3$-space, see for instance Theorem 4.1.

###### Proposition 2.2

Let $F$ be a degree-$d$ form in the variables $Z_{0},\cdots,Z_{n},$ and let
$p\in\mathbb{P}^{n}$ lie on $F=0.$ Being a singular point of $F$ is a
projective property; in other words, if in one affine chart one has the
dehomogenised polynomial $f=0$, and the image $p_{0}$ of $p$ in the affine
chart is a critical point of $f$, then this is so in any coordinate chart.

Proof: One should recall Euler’s formula, which works for homogeneous
functions of degree $d$:

$\sum_{i=0}^{n}Z_{i}\frac{\partial F}{\partial Z_{i}}=dF.$

Now, assume that the $0$-th coordinate of $p$ is nonzero; in the affine chart
$Z_{0}\neq 0$, the partial derivatives of $F(1,x_{1},\ldots,x_{n})$ correspond
to $\partial_{Z_{i}}F(1,x_{1},\ldots,x_{n})$, and by Euler’s formula,
$\partial_{Z_{0}}F(p)=0$ as well. This fact is easily seen to be preserved
under projective linear transformations and by restriction to any other affine
chart. $\blacksquare$

######  Proposition 2.3 (“Liouville’s Theorem")

Let $Z\subset\mathbb{P}^{N}$ be the zero set of $r$ forms
$F_{1},\ldots,F_{r}.$ If $Z$ is not finite, then for any hyperplane $H$ the
intersection $H\cap Z$ is non-empty.

Proof: It suffices to apply (Sha, , Th.I.6.6, p.76). $\blacksquare$

##  3 Generic exactness of $3D$ SUSY

The result holds for $\mathrm{N}\geq 2$ superfields.

######  Theorem 3.1 (Main Result on Exactness)

Given $N\geq 2$, outside a real hypersurface $H$ in $E_{N}$, every potential
$\Phi\in E_{N}$ gives rise to an exact (simple) SUSY in 3D. Equivalently, if
the homogenised components of the gradient of $\Phi$ have no common zeros at
infinity, then $\Phi$ has a real critical point. In particular, 3D SUSY with N
superfields is generically exact.

Proof: By (Sha, , I.6. Exercise 10), $N$ cubic forms (or $N$ forms of
prescribed degrees $m_{1},\ldots,m_{N}$) in $N$ variables have a common
(complex) zero if and only if a certain real polynomial $R$ in their
coefficients vanishes ($R$ has its coefficients in $\mathbb{Q}$, by Galois
theory (Silverman, , Ch.I)). Thus the components of the gradient of the
homogenised $\Phi\in E_{N}$ have a common zero at infinity if and only if
$R(G_{1},\ldots,G_{N})=0$, where
$G_{i}(X_{1},\ldots,X_{N})=\partial_{X_{i}}\tilde{\Phi}(0,X_{1},\ldots,X_{N})$,
and the number of complex critical points of $\Phi$ is finite if $R\neq 0$,
for then the locus of critical points has empty intersection with the
hyperplane $X_{0}=0$, see Proposition 2.3. Outside this real hypersurface
$R=0$ in $E_{N}$, a potential $\Phi\in E_{N}$ satisfies that the real
hypersurfaces $\partial_{X_{i}}\tilde{\Phi}(X_{0},X_{1},\ldots,X_{N})=0,1\leq
i\leq N$ intersect in $3^{N}$ points in $\mathbb{C}^{N}$ (an odd number)
counting multiplicities, as no point of intersection lies in $X_{0}=0$. Now,
imaginary solutions come in complex conjugate pairs (and here
$\mu_{P}=\mu_{\overline{P}}$, the system being defined over $\mathbb{R}$), so
a real common zero of $\partial_{1}\Phi,\ldots,\partial_{N}\Phi$ must exist.
$\blacksquare$

##  4 Explicit family with F-term breaking

###### Theorem 4.1

Let $a,D\neq 0$ be real parameters, and let $A(u,v)$ be a quartic homogeneous
polynomial. Define the potential $\Phi=\Phi_{1}+\Phi_{4}$, where

$\Phi_{4}(x,y,z)=A(x+ay,z)$

and $\Phi_{1}(x,y,z)=Dx.$ The gradient $\nabla\Phi=(F,G,H)$ is nonvanishing in
$\mathbb{R}^{3}.$ The above forms a $6$-parameter family of such potentials.

Proof: Consider the equations $\partial_{x}\Phi=\partial_{y}\Phi=0,$ which in
their explicit form are:

$\partial_{u}A=0,\quad D+a\partial_{u}A=0,$

which is clearly impossible for $D\neq 0$. This produces a $6$-parameter
example, counting $D$ and the coefficients of $A$. The Theorem is thus
settled. $\blacksquare$

  * Remark:

The family obtained in Theorem 4.1 is indeed not very big. Non-existence of
affine real solutions relies on the fact that every complex common zero
$P_{i}$ of the homogenised versions of $F,G,H$ lies at infinity, and the
(projective) intersection cycle equals $3\sum P_{i}$ ($\sum P_{i}$ being the
intersection cycle of the plane curves
$\tilde{F}(0,X_{1},X_{2},X_{3}),\tilde{H}(0,X_{1},X_{2},X_{3})$) if the
intersection be proper. Indeed, the monomial $Z_{0}^{3}$ is a linear
combination of $\tilde{F},\tilde{G}$. The rest follows from Section 2.

##  5 The even potential case

Let $\Phi$ be even. This means that the cubic polynomials $F,G,H$ are of the
form

$F=F_{1}+F_{3},G=G_{1}+G_{3},H=H_{1}+H_{3}.$

We wish to study the cases where no solution other than the origin exists. The
assumption of having an ordinary double point below is at this stage
unnecessary, but will later be very fruitful.

###### Theorem 5.1

Assume that there is an even quartic potential $V\in E$, which has an ordinary
double point at the origin (i.e. such that the Hessian at $0$ is
nondegenerate), and no other critical points in $\mathbb{P}^{3}_{\mathbb{R}}$.
There exists an even quartic potential on $\mathbb{R}^{3}$, $\Phi^{\prime}$,
with no critical points in $\mathbb{R}^{3}$.

Proof: In order to study the system $F=0,G=0,H=0$ outside the origin,
substitute $\mathbf{x}=\lambda v$, where $v\in S^{2},\lambda>0.$ The resulting
system of equations is as follows:

$\left(\begin{array}[]{ccc}F_{1}(v)&F_{3}(v)\\\ G_{1}(v)&G_{3}(v)\\\
H_{1}(v)&H_{3}(v)\end{array}\right)\left(\begin{array}[]{ccc}1\\\
\lambda^{2}\end{array}\right)=\left(\begin{array}[]{cc}0\\\ 0\\\
0\end{array}\right).$ (1)

Since $\lambda$ is real, this tells us that not only the rank of the matrix is
less than $2$, but also that the signs of $F_{1}(v),F_{3}(v)$ must be
opposite, just as those of $G_{1}(v),G_{3}(v)$ and of $H_{1}(v),H_{3}(v)$. The
condition on the Hessian of $V$ means that $F_{1},G_{1},H_{1}$ form a basis of
the dual space $(\mathbb{R}^{3})^{\vee}$ (i.e. transversal cut of
$F=0,G=0,H=0$ at the origin).

What this reveals is that, if $\tilde{F},\tilde{G},\tilde{H}$ are the
respective (cubic) homogenised forms corresponding to $F,G,H$, then the
intersection cycle

$\tilde{F}\bullet\tilde{G}\bullet\tilde{H}=O+Z,$

where $Z$ consists of pairs of complex conjugate points, with multiplicities,
and $O$ is the origin.

Note that, should we homogenise our quartic potential, the resulting quartic
form $\tilde{V}$ in the variables $Z_{0},Z_{1},Z_{2},Z_{3}$ has respective
partial derivatives $\tilde{K},\tilde{F},\tilde{G},\tilde{H}$. Euler’s formula
yields

$Z_{0}\tilde{K}+Z_{1}\tilde{F}+Z_{2}\tilde{G}+Z_{3}\tilde{H}=4\tilde{V}.$

Being a singular point of a hypersurface is a projective matter, by
Proposition 2.2, and therefore shows in any affine chart we choose. Likewise,
the fact of $\tilde{V}$ having precisely one critical point in
$\mathbb{P}^{3}$, which is an ordinary double point (i.e. the Hessian of
$\tilde{V}$ has rank $2$ at that point) is invariant by real projective linear
transformations and, since the origin becomes the point $O=(1:0:0:0)$ in
$\mathbb{P}^{3}$, placing infinity at, say, the plane $Z_{2}=0$ will furnish a
dehomogenised quartic potential $\Phi^{\prime}$ with no real critical points,
for the only possible candidate is now stashed at infinity. Thus, the new set
of real cubic equations, $F^{\prime}=G^{\prime}=H^{\prime}=0,$ has no real
solutions in affine real space, as desired. $\blacksquare$

###  5.1 Final Ansatz and numerical example

Let $\Phi$ be an even potential on $\mathbb{R}^{3}$ with $\Phi(0)=0$, and
assume that $F_{1},G_{1},H_{1}$ are linearly independent linear forms. Assume
further that, say, $H_{1}=z,H_{3}=zQ(x,y,z)$, where $Q$ is a positive definite
quadratic form (that is an open condition on the $6$ parameters that form
$Q$). The condition $z+\lambda^{2}zQ(x,y,z)=0$ for
$(x,y,z)\neq(0,0,0),\lambda\neq 0$ forces $z=0$, and it remains to find $\Phi$
with these constraints so that the remaining equations restricted to $z=0$
have no solution other than the origin. Denote by $Q_{0}(x,y)=Q(x,y,0)$ the
$z$-free part of $Q$, and write $Q(x,y,z)=Q_{0}(x,y)+zL(x,y)+fz^{2},$ where
$L$ is a linear form and $f>0.$ Thus
$\Phi_{4}=\frac{z^{2}}{2}Q_{0}(x,y)+\frac{z^{3}}{3}L(x,y)+\frac{f}{4}z^{4}+A(x,y),$
where $A(x,y)=\Phi_{4}(x,y,0)$ is an arbitrary quartic form. Now, by
construction, a common zero of $F,G,H$ over $\mathbb{R}$ must have $z=0,$ and
clearly $F_{3}(x,y,0)=\partial_{x}A(x,y),G_{3}(x,y,0)=\partial_{y}A(x,y).$

Take now $\Phi_{2}=\frac{1}{2}(z^{2}+ax^{2}+2bxy+dy^{2})$ nondegenerate. Then
(1) becomes, after imposing the necessary $z=0$:

$\left(\begin{array}[]{ccc}ax+by&\partial_{x}A(x,y)\\\
bx+dy&\partial_{y}A(x,y)\\\
0&0\end{array}\right)\left(\begin{array}[]{ccc}1\\\
\lambda^{2}\end{array}\right)=\left(\begin{array}[]{cc}0\\\ 0\\\
0\end{array}\right).$ (2)

Here, $ad-b^{2}\neq 0$, which is tantamount to saying that $F_{1},G_{1}$ and
$H_{1}=z$ are linearly independent. If the determinant

$(ax+by)\partial_{y}A(x,y)-(bx+dy)\partial_{x}A(x,y)$ (3)

is a (quartic) polynomial with no real linear factors, then no zero of (2)
will exist in $\mathbb{P}^{1}_{\mathbb{R}}$. In that case, the homogenisations
of $F,G,H$ have no common zeros in $\mathbb{P}^{3}_{\mathbb{R}}$ either.
Indeed, $H_{3}(x,y,z)$ has no real zeros outside $z=0$, but since
$F_{3}(x,y,0)=\partial_{x}A(x,y),G_{3}(x,y,0)=\partial_{y}A(x,y)$, these
cannot have common zeros in $\mathbb{P}^{1}_{\mathbb{R}}$ if the quartic form
in (3) has no real zeros in $\mathbb{P}^{1}_{\mathbb{R}}$. In fact, no zeros
of $(F,G,H)$ exist in $\mathbb{R}^{3}$ outside the origin, for if there was
one $(x_{0},y_{0},0)$, it should be a zero of (3).

Thus, if (3) has no nontrivial real zeros, the homogenised potential
$\tilde{V}$ has precisely one critical point in $\mathbb{P}^{3}_{\mathbb{R}}$,
which is the origin in $\mathbb{R}^{3}$, and is an ordinary double point (the
Hessian of $V$ is nonsingular), since this condition forces too $ad-b^{2}\neq
0$. That is how we show that lying in the three surfaces
$\tilde{F}=0,\tilde{G}=0,\tilde{H}=0$ there is only one real point, of
multiplicity one (i.e. a transversal intersection), which is the origin in the
affine part $\mathbb{R}^{3}$.

Take, for instance, the polynomial

$A(x,y)=0.673447x^{4}+0.299177x^{3}y+0.269692x^{2}y^{2}+0.818559xy^{3}+0.44846y^{4},$
(4)

and take any positive definite quadratic form $Q(x,y,z)$ on $\mathbb{R}^{3}$
(this sweeps an open set of an $\mathbb{R}^{6}$). Take now
$F_{1}=0.834982x+0.547667y,G_{1}=0.547667x+0.13926y$ (this determines
$\Phi_{2}$ in our case). Then (3) has the following form:

$1.36194x^{4}-0.340464x^{3}y-0.643023x^{2}y^{2}+0.768934xy^{3}+0.72017y^{4},$

and has no nontrivial real zeros. We are now ready to apply Theorem 5.1, and
thus establish the following result.

###### Theorem 5.2

There is an even quartic potential $\Phi=\Phi_{2}+\Phi_{4}$ on
$\mathbb{R}^{3}$ with no critical points. $\blacksquare$

##  6 Sharp result on SUSY breaking

The numerical example given in Section 5 brings about a $33-$parameter family
out of the $34$ parameters defining the potential $\Phi$ (in the case of 3
superfields).

###### Theorem 6.1

Consider the potential $\Phi^{0}$ obtained in our computer calculations (see
proof of Theorem 5.1). Consider the following four equations in
$p\in\mathbb{R}^{3}$ and in the coefficients of $\Phi$:

$\partial_{x}\Phi(p)=0,\,\partial_{y}\Phi(p)=0,\,\partial_{z}\Phi(p)=0,\,p_{2}=0.$

The four equations are satisfied by our initial, even potential $\Phi^{0}$
from Section 5, and this yields a family of $34+3-4=33$ parameters around the
potential $\Phi^{0},$ satisfying the four equations. Performing the
‘projective trick’ of swapping infinity planes (see Theorem 5.1) on them does
provide a family of quartic potentials $V$ with no critical points. In other
words, the $33$-parameter family of scalar potentials $V$ obtained out of the
$\Phi$ by homogenising, then dehomogenising on the second variable, provides
abundant examples of SUSY breaking at the $F$-term in 3D SUSY, with $N=3$
superfields of simple SUSY.

By analogous arguments, one obtains a codimension-1 family with F-term
supersymmetry breaking with $N=2$ superfields of simple SUSY.

Proof: The four equations imposed on pairs $(\Phi,p)\in
E\times\mathbb{R}^{3}=E^{(1)}$ provide a $33-$parameter family in $E^{(1)}$,
by the Implicit Function Theorem. Should we drop the fourth equation
$p_{2}=0,$ we would have a diffeomorphism around the point
$(\Phi^{0},(0,0,0))$ with an open neighbourhood ${\mathcal{U}}$ of $\Phi^{0}$
in $E$, where every $\Phi\in{\mathcal{U}}$ has a unique critical point on the
plane $y=0$ in $\mathbb{R}^{3}$, and no critical points at infinity (this
property is preserved by deforming the equations for the gradient within the
realm of real polynomials). Thus, by adding the condition $p_{2}=0$, we have
found a hypersurface ${\mathcal{H}}$ in the small open subset ${\mathcal{U}}$,
which makes for the $33$ parameters, where every member of ${\mathcal{H}}$ has
exactly one (real) critical point, which will lie on the coordinate plane
$y=0$. Now we homogenise $\Phi$ and dehomogenise it again (analytic
diffeomorphism of $E_{N}$), so the plane $y=0$ will turn into the plane at
infinity, and this produces the desired family of $V$’s with no critical
points (see Proposition 2.2). This settles the case $N=3$.

For the 2-superfield case, one may use the starting example in Section 5:

$\Phi^{0}(x,y)=\frac{1}{2}[0.834982x+2\times 0.547667xy+0.13926y^{2}]+A(x,y),$

with $A(x,y)$ as in (4), which will produce the expected family, after
homogenising and dehomogenising again (a ball around the transformed potential
$\Phi^{1}$). $\blacksquare$

Just a little more work gives us an N-superfield analogue.

###### Theorem 6.2

Given $N\geq 2$, there is a quartic potential in $N$ variables $\Phi^{1}$ with
no critical points in $\mathbb{R}^{N}$, which gives rise to a small ball of
dimension $\dim E_{N}-1$ around $\Phi^{1}$ in $E_{N}$ satisfying this
condition (spontaneous F-term breaking of 3D SUSY with maximal dimension).

Proof: It suffices to take the example in $3$ variables, $\Phi^{0}(x,y,z)$,
and to define
$\Phi^{1}(x_{1},\ldots,x_{N})=\Phi^{0}(x_{1},x_{2},x_{3})+\sum_{i=4}^{N}x_{i}^{2}+x_{i}^{4}$
($x_{i}=X_{i}/X_{0}$). The very same argument that settled Theorem 6.1 works
here: Implicit Function Theorem and swapping coordinates, hence infinity
hyperplanes, to move the unique critical point to the infinity hyperplane.
$\blacksquare$

##  7 Conclusion

We have found a family of scalar potentials with maximal number of free
parameters, featuring F-term spontaneous breaking of 3D SUSY in the case of
$N\geq 2$ superfields. However, only small balls around explicit even
potentials have been obtained, without showing the global shape of the F-term
breaking locus. An explicit, simpler albeit ‘smaller’ family in the case of 3
superfields has been provided in Theorem 4.1. In the latter case, the real
zeros lie at infinity, but are multiple, which prevents from perturbing to
bigger families. In our case, the locus of F-term spontaneous breaking within
the parameter space of space-time configurations of $N\geq 2$ superfields is
contained in a real (algebraic) hypersurface $H$, and contains at least small
ball in $H$, whereas in 4D SUSY ORafer  its corresponding F-term breaking
locus (only for $N\geq 3$) is contained within a complex hypersurface (real
codimension two), as our proof of Theorem 3.1 and the reference therein show.
This phenomenon is explained by the fact that simple SUSY in 4D corresponds to
double SUSY in 3D.

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