How to count flat directions after supersymmetry breaking

Question

In short my question is how do I show that a superpotential of the form,
$$\begin{equation}  W = \sum _{i = 1 } ^{ N _Y } Y _i f _i  ( X _1 , ... , X _{ N _X } )  \end{equation}$$  
(generic O'Raifertaigh models) lead to $N _Y - N _X$ flat directions after spontaneous SUSY breaking? I describe the context and the details below.

In Weinberg Vol III (pg. 84) he introduces two fields, 

$$\begin{equation} Y = \left( Y _1, ... , Y _{N _Y } \right) , \quad X = \left( X _1 , ... , X _{ N _X } \right) \end{equation}$$  

where $Y$ has an $R$ charge of $2$ and $X$ has an $R$ charge of $0$. Then superpotential has to take the form,
$$\begin{equation}  W = \sum _i Y _i f _i  ( X _1 , ... , X _{ N _X } )  \end{equation}$$  
which gives the SUSY conserving conditions,
$$\begin{align}   f _i  ( X ) & = 0 \mbox{ for $ i = 1 , ... , N _Y $}\\   \sum _i Y _i \frac{ \partial f ( X ) }{ \partial X _n } & = 0 \mbox{ for $n = 1 , ... , N _X $} \end{align}$$  
The first set of equations is made up of $N _X$ unknowns and $N _Y$ equations, thus if $N _Y > N _X$ it can't be generically solved. He goes on to define 
$$\begin{equation}   v _{n,i} \equiv \frac{ \partial f _i }{ \partial x _n } \bigg|_{ x = x _0 } \end{equation}$$  
which gives the potential,
$$\begin{equation}  V ( x, y ) = \sum _i \left| f _i \right| ^2 + \sum _n \left| \sum _i y _i v _{n,i}  \right| ^2  \end{equation}$$  

Up to this point I have no problems. However, then he goes on to say that the second term vanishes if $y$ is orthogonal to $v _{ n}$. Since ``$v _n$ cannot span the space of $y _i$s and there will be at least $N _Y - N _X$ flat directions''. Where does this conclusion come from? How can I see these ``flat directions''?  

Answer 1

$V(x,y)$ is a function on the space of fields $x$ and $y$. This space of fields is the product of the space of $x$, of dimension $N_X$ by the space of $y$, of dimension $N_Y$. A minimum point of $V$ is of the form $(x_0,0)$ (with $x_0$ minimizing the first term in $V$ and $y=0$ giving the vanishing of the second term). We have spontaneous symmetry breaking at this point if $V(x_0,0)$ is non-zero. We want to show that there exists at least $N_Y-N_X$ flat directions i.e. that there exists at least $N_Y-N_X$ directions in the space of fields around $(x_0,0)$ where $V$ remains at the value $V(x_0,0$ (the graph of the function $V$ is flat in restriction to these directions). We will find these directions in the space of $y$ : we fix $x$ at the value $x_0$ and we consider $V(x_0,y)$ as a function on the space of $y$, which is of dimension $N_Y$. In the formula defining $V$, the first term does not depend on $y$ so we can forget about it. The second vanishes at $y=0$. So we are looking at directions in the space of y such that the second term still vanishes. This means we are looking for $y$ such that the scalar products $y.v_n$ are 0 for every n, i.e to the space inside the space of $y$ orthogonal to the space generated by the $v_n$. As the space of y is of dimension $N_Y$ and the space generated by the $v_n$ is of dimension at most $N_X$, the orthogonal of the space generated by the $v_n$ is at least of dimension $N_Y-N_X$.