# $\delta^{(2)}$ convention

In this note: https://arxiv.org/abs/hep-th/0410165 at page 12 there is a delta-function constraint written as: \begin{align} \delta \left( ^ { U } M \right) = \prod _ { i < j } \delta ^ { ( 2 ) } \left( U M _ { i j } \right). \end{align} Here $$^ { U } M =UMU^\dagger$$ where $$M$$ is Hermitian and $$U$$'s are unitary $$N\times N$$ matrices.

If we have a func $$f$$ in its $$n$$'th derivative we write $$f^{(n)}$$.

Question: Does $$\delta ^ { ( 2 ) }$$ mean second derivative of the dirac delta function?

I) In general, the notation $$\delta^{(n)}$$ denotes either:
1. the $$n$$'th derivative of the Dirac delta distribution. (For low number of derivatives, one can alternatively use primes: $$\delta^{\prime}$$, $$\delta^{\prime\prime}$$, $$\delta^{\prime\prime\prime}$$, and so forth.)
2. the $$n$$-dimensional Dirac delta distribution. (We recommend to write the latter as $$\delta^n$$ to avoid confusion with derivatives).
II) In the present case, $$\delta^{(2)}$$ means the 2-dimensional Dirac delta distribution (where the complex plane $$\mathbb{C}\cong \mathbb{R}^2$$ is identified with the 2-dimensional real plane.)