Non-commutative extensions of two-dimensional topological field theories and Hurwitz numbers for real algebraic curves

TL;DR

The paper extends 2D topological field theories to nonorientable surfaces by introducing Klein topological field theories (KTFTs) and proves a bijective correspondence between KTFTs and structure algebras, including a complete classification of semisimple cases. It provides explicit constructions of semisimple structure algebras from any finite group and demonstrates that Hurwitz numbers for real algebraic curves arise as KTFT correlators, with the symmetric group governing the underlying structure algebra. The framework uses stratified surfaces and cut systems to translate topological data into algebraic axioms, enabling systematic calculation of invariants and connecting open-closed TFTs (Lazaroiu-Moore) to KTFTs. The results unify real Hurwitz theory with KTFTs and yield computable expressions via structure constants of relevant groups, highlighting deep links between topology, algebra, and enumerative geometry.

Abstract

It is well-known that classical two-dimensional topological field theories are in one-to-one correspondence with 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 algebras with certain additional structures, called structure algebras. 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 the symmetric group.

Figures (6)

• Figure 1: The construction of a contracted cut surface. Fig.1a. Cut system $\Gamma=\{\gamma_1,\gamma_2,\gamma_3\}$ of surface $\Omega$. Fig.1b. Cut surface $(\Omega_*,\Gamma_*,\tau)$ obtained from cut system $\Gamma$. Here $\Gamma_*=\{\gamma_{1*}',\gamma_{1*}",\gamma_{2*}', \gamma_{2*}", \gamma_{3*}',\gamma_{3*}"\}$. Fig.1c. Contracted cut surface $\Omega_\#=\Omega/\Gamma$. Special points obtained as contractions of the connected components of $\Gamma_*$ are marked by $*$.
• Figure 2: Examples of cuts of classes 1-9. Each cut is marked by $\gamma$. Fig.2a. A separating contour. Fig.2b. A cut of a handle. Fig.2c. A cut of a neck of Klein bottle. Fig.2d. A Möbius cut. Fig.2e. A cut between two holes. Fig.2f. A separating segment. Fig.2g. A cut of a handle through a hole. Fig.2h. A cut of a neck of a Klein bottle through a hole. Fig.2i. A cut across Möbius band
• Figure 3: Non-equivalent complete cut systems on simple stratified surfaces. Different complete cut systems are drawn by diffent line types. Cuts in a cut system are denoted by $\gamma$ with the same number of accents and are distinguished by lower indeces. Surfaces are numbered according to lemma \ref{['3.9']}. Fig.3b presents Klein bottle cutted by cut $\gamma$ forming one cut system.
• Figure 4: Admissible set of local orientations of (4a) oriented surface; (4b) non-oriented surface.
• Figure 5: Stratified covering $\pi$ over segment $I$.

Theorems & Definitions (113)

• Definition
• Definition
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• Lemma 2.5