Modular properties of algebraic structures arising from conformal field theory

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Currently I am looking at whether it is possible to find a modular transformation formula for conserved charges in different 2d CFTs. I started with the free fermion theory where the the KdV charges simplify to be bilinear in the fermion modes. This allowed me to calculate the expectation values for any power of the KdV charges and then find the modular transform of these expectation values. I will now start to look at other theories, starting with some of the minimal models to see if the modular transform of the KdV charges can be determined in these theories. Initially I will see if I can compute the expectation value of the KdV charges and if this is possible then I will try to find the modular transform of these expectation values. The final aim is to find a general transformation formula for these expectation values and see if there are any general features that hold across different theories.