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I am reading the paper "Gapless Floquet topology" by Cardoso et al and the following section got me confused.

I understand that now $F$ lives in a space where a 2 dimensional subspace is removed (where $\det ([F,\Gamma]=0)$. But how can this yellow equation give a winding invariant?

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    $\begingroup$ Have you tried reading Appendix A? Apparently the argument appears there. $\endgroup$
    – Ryan Budney
    Commented yesterday
  • $\begingroup$ Yes I have read it. In appendix A they proved that this is an integral of a closed form, hence a topological quantity. But I dont know if the yellow formula is general or only valid after being transformed into some other form. I dont know how to interepret this $\endgroup$
    – wooohooo
    Commented yesterday
  • $\begingroup$ I take it the integrand $[F, \Gamma]$ is the Lie bracket of two matrices, so you are using the vector space structure on matrices, making the integral a matrix that you can apply the trace to. I'm a little foggy on the domain of integration, though. It's unclear if that's compact or not. $\endgroup$
    – Ryan Budney
    Commented yesterday
  • $\begingroup$ It is the commutator $[F,\Gamma]=F\Gamma - \Gamma F$ indeed. I am not sure why this domain is compact or not $\endgroup$
    – wooohooo
    Commented 23 hours ago

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