State sum construction of two-dimensional topological quantum field theories on spin surfaces

TL;DR

This work builds a comprehensive state-sum framework for two-dimensional topological field theories on spin surfaces by developing a combinatorial model of spin triangulations and a parallel algebraic structure. The geometric part characterises spin structures via marked triangulations and admissible edge signs, establishing a bijection with spin structures and proving invariance under Pachner moves. The algebraic part shows that a Δ-separable Frobenius algebra with an involutive Nakayama automorphism yields a triangulation-independent invariant T_A(Σ); the extra condition N * id = 0 enforces admissibility to zero when edge signs are non-admissible, thereby linking local data to global spin structure. The construction naturally yields NS and R state spaces as centers of A, and provides explicit cylinder, pair-of-pants, and torus computations, along with examples illustrating the dependence on spin structure and connections to existing spin TFT formalisms. These results offer a robust, locally defined approach to spin TFTs with potential links to defects, higher-dimensional analogues, and fully extended TFT frameworks.

Abstract

We provide a combinatorial model for spin surfaces. Given a triangulation of an oriented surface, a spin structure is encoded by assigning to each triangle a preferred edge, and to each edge an orientation and a sign, subject to certain admissibility conditions. The behaviour of this data under Pachner moves is then used to define a state sum topological field theory on spin surfaces. The algebraic data is a Delta-separable Frobenius algebra whose Nakayama automorphism is an involution. We find that a simple extra condition on the algebra guarantees that the amplitude is zero unless the combinatorial data satisfies the admissibility condition required for the reconstruction of the spin structure.

Figures (22)

• Figure 3.1: Orientation convention for a triangle in a combinatorial surface. The circular arrow gives the order on the vertices defining the orientation. The arrows on the edges give the edge orientation induced by the orientation of the triangle. When comparing orientation of simplices to orientations of surfaces, our convention is that the above orientation matches that of the paper plane, thought of as $\mathbb{R}^2$ with its standard orientation.
• Figure 3.2: An edge $e$ with orientation from $d_0^1(e)$ to $d_1^1(e)$.
• Figure 3.3: Standard triangle $\underline\Delta$; the small numbers $0$, $1$, $2$ indicate the numbering of the edges. The first edge has also been marked with a fat green line.
• Figure 3.4: Marking of boundary edges: The marking has to be such that the orientation induced on the edge as in Figure \ref{['fig:label_convention']} agrees with the orientation induced by the parametrisation maps, see Definition \ref{['def:comb-spin-surf']}. Equivalently, the orientation of boundary edges induced by the marking is opposite to that induced by the adjacent triangle via Figure \ref{['fig:triangle-orientation']}.
• Figure 3.5: An inner edge with left and right adjacent faces. In the configuration above $k_L=2$ and $k_R=1$.

Theorems & Definitions (105)

• Definition 2.1
• Proposition 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.5
• Remark 2.6
• Lemma 2.7
• proof
• Lemma 2.8
• proof