It is well known that the classical two-dimensional topological field theories are in one-to-one correspondence with the commutative Frobenius algebras. An important extension of classical two-dimensional topological field theories is provided by open-closed two-dimensional topological field theories. In this paper we extend open-closed two-dimensional topological field theories to nonorientable surfaces. We call them Klein topological field theories (KTFT).We prove that KTFTs bijectively correspond to (in general noncommutative) algebras with certain additional structures, called structure algebras. The semisimple structure algebras are classified. Starting from an arbitrary finite group, we construct a structure algebra and prove that it is semisimple.We define an analog of Hurwitz numbers for real algebraic curves and prove that they are correlators of a KTFT. The structure algebra of this KTFT is the structure algebra of a symmetric group.

There is now a renewed interest [1]-[4] to a Hurwitz τ -function, counting the isomorphism classes of Belyi pairs, arising in the study of equilateral triangulations and Grothiendicks's dessins d'enfant. It is distinguished by belonging to a particular family of Hurwitz τ -functions, possessing conventional Toda/KP integrability properties. We explain how the variety of recent observations about this function fits into the general theory of matrix model τ -functions. All such quantities possess a number of different descriptions, related in a standard way: these include Toda/KP integrability, several kinds of W -representations (we describe four), two kinds of integral (multi-matrix model) descriptions (of Hermitian and Kontsevich types), Virasoro constraints, character expansion, embedding into generic set of Hurwitz τ -functions and relation to knot theory. When approached in this way, the family of models in the literature has a natural extension, and additional integrability with respect to associated new time-variables. Another member of this extended family is the Itsykson-Zuber integral.

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