(see section \[sec:poles\] for further discussion on this point), or through the connection with $SL(2,{\mathbb C})$ representation theory that this EFT enjoys, which is the subject to which we now turn.

The result for general $N$ from $SL(2,{\mathbb C})$ {#sec:su2}
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It turns out that our one-dimensional theory with scalars has an interesting connection with $SL(2,{\mathbb C})$ representation theory[^9] that allows us to obtain a closed form of the Hilbert series relatively simply for an arbitrary number of flavors. As discussed in the last section, when we only include relations from equations of motion, all operators are generated by the set $\{\ph_i, \pd \ph_i\}$. Pairing $\ph_i$ and $\pd \ph_i$ together into a doublet, the derivative acts as a lowering operator since $\pd^2\ph_i = 0$. This is the origin of an underlying $SL(2,{\mathbb C})$ structure which can be seen acting on the complex weights we use to construct the Hilbert series.

The $SL(2,{\mathbb C})$ structure can be made manifest in the Hilbert series by a simple change of variables on the weights: rescale $u_i \to \a \bau_i$ and $t \to 1/\a^2$. For example, the single flavor Hilbert series with just equations of motion, eq. , becomes

$$\begin{aligned}
\frac{1}{(1-u)(1-tu)} &\to \frac{1}{(1-\bau\a)(1-\bau\a^{-1})} \\
