Lattice Topological Field Theory in Two Dimensions

TL;DR

The paper provides a lattice formulation of two‑dimensional topological field theories (LTFTs) and solves them exactly, establishing a one‑to‑one correspondence between LTFTs and associative algebras $R$ with the physical Hilbert space identified as the center $Z(R)$. Observables live in $Z(R)$ and correlators are computed by projecting onto the center, with genus dependence encoded by a handle operator; in particular, genus‑$g$ correlators reproduce continuum TFT results for suitable algebras. It further shows that every TFT can be obtained from a standard topological field theory (STFT) via perturbations determined by locality, thereby mapping the moduli space of TFTs to commutative centers and one‑point data. The paper provides explicit examples, including $R=oldsymbol{C}[G]$ and twisted $N=2$ minimal topological matter, and discusses how perturbations generate the full landscape of TFTs, laying groundwork for incorporating gravity and connections to matrix models like Kontsevich models in future work.

Abstract

The lattice definition of a two-dimensional topological field theory (TFT) is given generically, and the exact solution is obtained explicitly. In particular, the set of all lattice topological field theories is shown to be in one-to-one correspondence with the set of all associative algebras $R$, and the physical Hilbert space is identified with the center $Z(R)$ of the associative algebra $R$. Perturbations of TFT's are also considered in this approach, showing that the form of topological perturbations is automatically determined, and that all TFT's are obtained from one TFT by such perturbations. Several examples are presented, including twisted $N=2$ minimal topological matter and the case where $R$ is a group ring.

Theorems & Definitions (6)

• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8