Two-dimensional state sum models and spin structures

TL;DR

The paper extends two-dimensional state sum models by allowing non-symmetric Frobenius algebra data to yield spin-structure dependent theories. It develops planar diagrammatic and spherical formalisms, derives precise algebraic conditions (special Frobenius structure, crossing axioms), and classifies the resulting models, including explicit spin-sensitive examples. A key contribution is demonstraing how spin parity enters partition functions via central elements η and χ, with explicit constructions where the spin structures on surfaces are topologically distinguished. The work also frames these models categorically, linking them to pivotal/symmetric ribbon categories and outlining paths toward higher-dimensional generalisations with defect structures.

Abstract

The state sum models in two dimensions introduced by Fukuma, Hosono and Kawai are generalised by allowing algebraic data from a non-symmetric Frobenius algebra. Without any further data, this leads to a state sum model on the sphere. When the data is augmented with a crossing map, the partition function is defined for any oriented surface with a spin structure. An algebraic condition that is necessary for the state sum model to be sensitive to spin structure is determined. Some examples of state sum models that distinguish topologically-inequivalent spin structures are calculated.

Figures (9)

• Figure 1:
• Figure 2:
• Figure 3:
• Figure 4: Building $\Sigma_g$. A genus $g$ surface is constructed from a disk with $4g$ boundary edges (internal edges are omitted). Edges are identified following the pattern shown on the right: $a \leftrightarrow c$, $b \leftrightarrow d$. On the left, it can be seen how the glued edges give rise to curves on the surface.
• Figure 5:

Theorems & Definitions (43)

• Theorem 1: \ref{['theo:diagram']}
• Theorem 2: \ref{['theo:main']}
• Definition 2.1
• Theorem 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.4
• Definition 3.1
• Theorem 3.2