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I would like to know the reason why the equation (14) in the paper by Yamada is called the Toda equation. \begin{equation} \left[\frac12\sum_{i=1}^N\left(y_i\frac{\partial}{\partial y_i}-y_{i+1}\frac{\partial}{\partial y_{i+1}} \right)^2+\sum_{i=1}^N u_iy_i\frac{\partial}{\partial y_i}+{\rm const}\sum_{i=1}^N y_i\right]Z=0. \end{equation}

This equation was obtained by taking the decoupling limit of the equation (12) which is the equation conformal blocks in $SL(N)$ WZNW theory satisfy.

  1. Does a certain correlation function in Toda conformal field theory satisfy the equation (14)? If so, there should be a realization of a full surface operator by M2-branes in the context of the AGT relation.

  2. What kind of correlation functions in Toda conformal field theory satisfy the equation (14)?

  3. Is there any good reference which explains the Toda equations?

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