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where ˆ F is a D × D matrix. At P , in the same system of coordinates, we can prove the following general fact: 

where g 0 is the standard metric of the sphere. This shows that the quantum metric and the standard metric of the sphere are related by a conformal transformation, which involves the trace of the curvature tensor. n − 1 

We start the proof of Eq. (40). The 2 many body wavefunctions Ψ$_{i}$ of the zero modes depend parametri- cally on the position w of the mobile anyon and take the following general form: 23 

where the crucial fact is that the vectors ˜ Ψ$_{i}$( w ) are all analytic of w . A$_{i}$ ( w ) are normalization factors that are not analytic of w . We have: 

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Due to the presence of 1 − P$_{w}$ , we can ignore the action of the derivatives on the normalization constants A$_{i}$ ( w ), which then can be pulled out of the scalar product as below: 

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Next we use the fact that ˜ Ψ$_{i}$ are analytic functions of w , i.e. they are functions of the single variable w rather than of two independent variables w$_{1}$ and w$_{2}$ , in which case: 

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The scalar product is real, which allows us to easily com- pute the imaginary part, resulting in: 

If one repeats exactly the same arguments for the quan- tum metric tensor given in Eq. 38, the conclusion will be that: 

Since w$_{1}$ and w$_{2}$ coincide with the geodesic coordinates of the sphere at P , i.e. the coordinates in which the metric tensor becomes the identity matrix when evaluated at P : g $^{0}$( P )$_{µν}$ = δ$_{µν}$ , the proof of Eq. 40 is completed. 

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