Complete Set of Cut-and-Join Operators in Hurwitz-Kontsevich Theory

Understand

We define cut-and-join operator in Hurwitz theory for merging of two branching points of arbitrary type.

• These operators have two alternative descriptions:(i) they have the GL characters as eigenfunctions and the symmetric-group characters as eigenvalues; (ii) they can be represented as differential operators of the $W$-type (in particular, acting on the time-variables in the Hurwitz-Kontsevich tau-function).
• The operators have the simplest form if expressed in terms of the matrix Miwa-variables.
• They form an important commutative associative algebra, a Universal Hurwitz Algebra, generalizing all group algebra centers of particular symmetric groups which are used in description of the Universal Hurwitz numbers of particular orders.