Flat Connections from Irregular Conformal Blocks

In this work we study Liouville conformal blocks with degenerate primaries and one operator in an irregular representation of the Virasoro algebra. Using an algebraic approach, we derive modified BPZ equations satisfied by such blocks and subsequently construct corresponding integral representations based on integration over non-compact Lefschetz cycles. The integral representations are then used to derive novel types of flat connections on the irregular conformal block bundle.