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I am reading the paper. What is the definition of $\delta^{m|m}$ and $\delta^{m+k}$ in (1.1) and (1.3) on pages 2,3? Are they some kind of delta function?

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  • $\begingroup$ I would guess that part of the delta function is a delta function over Grassmann variables. You might want to trace back the references above the formula (and maybe trace back the references in those papers too) to see if they explain it. Or maybe try arXiv:1308.1697. $\endgroup$ – octonion May 20 '17 at 8:00
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$\delta^{n|m}(z)$ usually denotes the Dirac delta distribution on the superspace $\mathbb{R}^{n|m}$ with $n$ Grassmann-even and $m$ Grassmann-odd dimensions.

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Yes, they're delta-functions. The exponent in $\delta^m$ means that it's an $m$-dimensional delta-function, i.e. the product of $m$ one-dimensional delta-functions of the coordinates collectively indicated as the argument.

The vertical line in $\delta^{m|n}$ indicates a delta-function in a "superspace" that has $m$ bosonic and $n$ Grasmmannian or fermionic variables. Recall that for an anticommuting or Grassmannian variable $\theta$, the delta-function $\delta(\theta)$ is basically the same thing as $\theta$ itself. The integral of this delta function $\int d\theta\,\theta$ equals one due to the axioms of the Berezin integration.

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If you let $m=n=4$ in the Dirac delta function it'll represent that you have four fermionic functions and four bosonic functions and the thing that encompasses the argument or $z$ in this case is actually a matrix if you are referring to that. While in the inside you have some vector in $\mathbb{R}^4$ called $c$ dotted into some $4\times4$ matrix where the points $Z$ are actually vectors, which would give you back a matrix in the argument plus another matrix for the fifth point in case you have five points.

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