Existence and Uniqueness of Irregular Vectors of Integer and Half-Integer Ranks for the Virasoro Algebra

Although irregular vectors for the Virasoro algebra are widely used in modern mathematical physics, a rigorous existence and uniqueness theorem in arbitrary rank has not been available in the literature. In this paper, we develop an algebraic framework, based on Virasoro differential operators on the parameter space, which gives such a theorem for arbitrary integer and half-integer ranks. A key ingredient is the construction of a canonical operator \(L_*\) from the coefficient matrix of the vector-field part of a truncated Virasoro realization. This operator closes the recursive system by isolating the derivative with respect to the highest irregular parameter. Using this mechanism, we prove the existence and uniqueness of formal irregular vectors of arbitrary integer rank. We then construct the truncated Virasoro vector fields required in the half-integer-rank setting and prove the existence and uniqueness of the corresponding half-integer-rank formal irregular vectors. We also prove that, after a scalar gauge normalization, the canonical solutions satisfy the full lower Virasoro deformation equations. These results provide an algebraic foundation for the rigorous construction of irregular conformal blocks built from higher-rank irregular vectors. After passing to eigenvalue coordinates, the vector-field part of the half-integer construction is identified with the differential realizations appearing in the literature, while the zeroth-order terms are explained by scalar gauge freedom.