\section{Finite Temperature 3pt function}
   
  
   

  
  
  
  
  
  
  
  
  
  
  
  
  
  
 
  
  
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\noindent{{\bf{\it Conclusions.}}}
In this paper we have offered an argument based on the Kerr/CFT correspondence to evaluate the nonlinearities of the Kerr BH ringdown. We have shown that the Kerr/CFT correspondence provides a simple way   to both explain, at the qualitative level, the
quadratic scaling of the second-order  mode amplitude with the product of the amplitudes of the fundamental modes, and also to quantitatively predict   the size of this  effect in striking  agreement with  numerical results found in  recent literature. 

Our findings hold only in the extremal case; our next step will be to study departures from this condition. Numerical results are indeed found close to, but not exactly at,  extremality, i.e.  for remnant dimensionless spins $\simeq 0.7$ \cite{nl2,nl3}. Some corrections might therefore intervene.  For  small deviations from extremality, a different  set of boundary conditions  lead to a second copy of the Virasoro algebra \cite{b1} with indications of a hidden dual CFT even far from extremality~\cite{Castro:2010fd,Hui,K}, where the new excitations correspond to right-movers.  However, the fact that our findings are so close to the numerical fits of Refs. \cite{nl2,nl3}  might indicate that the value of the remnant spin is not so relevant (as it also appears from  numerical results) and that the right-mover sector is  decoupled from the dynamics. We have indeed reasons to believe that the right sector
is associated to the  $s=+2$ degree of freedom whose contribution to the GW strain is subleading at infinity with respect to the $s=-2$ mode \cite{inprepP}. Large nonlinearities have been observed also in the case of head-on collisions, giving rise to non-spinning BHs \cite{nl3}. This fact  as well might be explained  using similar symmetry arguments, but in the context of Schwarzschild BHs~\cite{Hui}.
The CFT approach may also provide new consistency relations among the QNM amplitudes and useful predictions for the nonlinearities involved in the QNM dynamics.
Finally, it would  also be interesting to go beyond the three-point correlator, to see if sizeable effects persist at higher-orders. 

We will investigate these issues in  order to provide generic predictions in terms of the multipole numbers and spins of the remnants in  future work~\cite{inprepP}.


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\noindent{{\bf{\it Acknowledgments.}}}
We thank E. Berti for many useful discussions about the results of Ref. \cite{nl3}. We thank  E. Berti, M. Biagetti, and V. De Luca  
for useful feedback on the draft. The work of D. P. and F. R.  is supported by the Swiss National Science Foundation under grants no. 200021-205016 and PP00P2-206149. A.K. is supported by the PEVE-2020 NTUA programme for basic research with project number 65228100. A. K. thanks the University of Geneva where part of this work was performed. A.R. is funded by the Boninchi Foundation.





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