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.)


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