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While working on the Atiyah problem on configurations of points, I came across formulas involving products of traces of products of singular hermitian positive semidefinite $2$ by $2$ matrices.

To illustrate the type of formulas I am talking about, here is one of them:

$$\DeclareMathOperator{\tr}{tr} D = \frac{1}{4 \tr(n_{01}) \tr(n_{02}) \tr(n_{12})} \left( \tr(n_{01}n_{02}) \tr(n_{12}) + \tr(n_{01}n_{21}) \tr(n_{02}) + \tr(n_{02}n_{12}) \tr(n_{01}) + \tr(n_{01}n_{02}n_{12}) - \tr(n_{01}n_{20}n_{12}) + \tr(n_{01}n_{20}n_{21}) + \tr(n_{01}n_{21}n_{02}) + \tr(n_{01}) \tr(n_{02}) \tr(n_{12}) \right).$$

In the previous formula, each $n_{ij}$ is a singular positive semidefinite hermitian $2$ by $2$ matrix satisfying the following constraints: $$\tr(n_{ij}) = \tr(n_{ji})$$ $$n_{ij} + n_{ji} = \tr(n_{ij}) I$$ $$n_{ij} + n_{jk} + n_{ki} = \frac{1}{2}(\tr(n_{ij}) + \tr(n_{jk}) + \tr(n_{ki}))I,$$ where $I$ is the $2$ by $2$ identity matrix, and $i$, $j$ and $k$ are different indices in $\{0,1,2\}$.

I am interested in showing that the lower bound for a formula such as the above is $1$. This is the Atiyah-Sutcliffe conjecture $2$. Kindly note that I do know how to show that for the formula above, and it is not what I am asking for (of course, this is a formula for $n = 3$ and I do not know how to show the lower bound for $n > 4$, which is still open).

I would like to know whether such expressions such as the numerator of the formula above, have been studied before in the literature, whether in the Math or Physics literature. Indeed, I am interested in finding a physical interpretation of the Atiyah-Sutcliffe determinant $D$. I am also interested in knowing whether some tools have been developed to prove lower bounds for expressions such as the above, or perhaps to simplify them.

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    $\begingroup$ Not sure if it helps but a singular positive semidefinite hermitian 2 by 2 matrix can also be written as the outer product of a 2x1 vector with itself, i.e. there exists $u_{ij}\in \mathbb{R}^{2\times 1}$ s.t. $n_{ij}=\bar{u}_{ij} u^T_{ij}$. Then you have for example $tr(n_{ij})=\|u_{ij}\|_2^2$, $tr(n_{ij} n_{kl})=(u^T_{ij} \bar{u}_{kl})^2$ and $tr(n_{ij} n_{kl}n_{mo} )=(u^T_{ij} \bar{u}_{kl})(u^T_{kl} \bar{u}_{mo})(u^T_{mo} \bar{u}_{ij})$ $\endgroup$
    – user35593
    Commented Apr 9, 2021 at 9:35
  • $\begingroup$ @user35593, yes I know. I kind of went in the opposite direction. I started from these Weyl spinors, if I may call them that, and got to the $2$ by $2$ matrices. $\endgroup$
    – Malkoun
    Commented Apr 9, 2021 at 13:58

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