A What is this unitary invariant?

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The discussion centers on the concept of a unitary invariant related to the eigenvalues of the Dirac operator, specifically referred to as spec_,N (M). Participants seek clarification on whether this invariant is a set of real numbers and its significance as an invariant. The construction of a unitary matrix similar to the CKM matrix is noted, along with the implications of mapping the unit sphere to the projective plane. Additionally, questions arise regarding the parameters in a 3x3 CKM matrix, particularly the Euler angles and CP-violating phase in various parametrizations. The conversation highlights the complexity of understanding unitary invariants in the context of quantum mechanics and particle physics.
Heidi
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Hi Pfs
I am reading this article:
https://arxiv.org/abs/0810.2091
It is know that hearing the possible frequencies emitted by a drum are not enough to know its shape.
Here the frequencies are the eigenvalues of the Dirac operator.
the missing information is the unitary invariant of the title.
The autor call it spec_,N (M)
I have several questions:
Is it a set of real numbers or something else ? (the word unitary..)
why is it invariant?
I see that he builds a nxn unitary matrix (like a ckm matrix)
Lines are equivalent when multiplied by a complex with module = 1. This means that each line is equivalent to its opposite. So the unit sphere is mapped to the projective plane Pn. I do not undertand what is done with the comums....
thanks.
 
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It seems that all is here.
 
We have on page 24 a CKM matrix:
1 1 1
1 z z*
1 z* z
where z = exp(2i pi /3)
In a 3*3 ckm matrix we have 4 free real parameters: 3 Euler's angles and a cp violating phase.
What are they here?
thanks.
 

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