quantum field theory - Witten Index and letter partition function

I haven't seen any reference which explains these things and I am not sure of all the steps of the argument or the equations. I am trying to reproduce here a sequence of arguments that I have mostly picked up from discussions and I would like to know of references for the background details and explanations.

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  It seems that in general the Witten Index (${\cal I}$) can be written as $ Tr (-1)^F \prod_i x_{i}^{C_i}$ where $x_is$ are like the fugacities of the conserved quantity $C_is$ and $C_is$ form a complete set of simultaneously measurable operators.

  
  

  I would like to know the motivation for the above and especially about calling the above an ``index" even if it depends on the fugacities which doesn't seem to be specified by the Lagrangian of the theory or by the underlying manifold? Even with this dependence does it reproduce some intrinsic property of either the underlying manifold or the theory?

  
  


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  Say $\phi$ and $\psi$ are the bosonic and fermionic component fields of a superfield and ${\cal D}$ be the superderivative. Now when the above tracing is done over operators of the kind $Tr({\cal D}^n\phi)$ and $Tr({\cal D}^n\psi)$ it is probably called a "single trace letter partition function (STLP)" and when it also includes things like products of the above kind of stuff it is called a ``multi-trace letter partition function (MTLP)"

  
  
  

I would like to know about the exact/general definitions/references for the above terminologies and the motivations behind them.

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  Probably for adjoint fields, if $f(x)$ is a STLP then apparently in some limit ("large N"?) one has, MTLP = $\prod _{n=1} ^{\infty} \frac{1}{1-f(x^n)}$

  
  


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  One also defines something called the ``full STLP (FSTLP)" where one probably includes terms also of the kind $Tr({\cal D}^n \phi ^m {\cal D}^p \psi ^q)$ and then the MTLP can be gotten from that as MTLP = $exp [ \sum _{m=1} ^ {\infty} \frac{1}{m} FSTLP]$

  
  



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  The upshot all this is probably to show that MTLP is the same as Witten Index (in some limit?).

  
  

I don't understand most of the above argument and I would be happy to know of explanations and expository references which explain the above concepts (hopefully beginner friendly!).

(I see these most often come up in the context of superconformal theories and hence references along that might be helpful especially about the representations of the superconformal group.)