The integral representation of solutions of KZ equation and a modification by $\mathcal{K}$ operator insertion

A root of unity limit of the $q$-deformed Virasoro algebra is considered. The $\widehat{sl}(2)_k$ current algebra and the integral formulas of the solutions of the KZ equations can be realized by the $q$-deformed boson at the limit and an additional boson. We explicitly construct the integral representation of the four-point blocks with a $\mathcal{K}$-operator insertion.

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