# Lattice topological field theory in two dimensions

@article{Fukuma1994LatticeTF, title={Lattice topological field theory in two dimensions}, author={Masafumi Fukuma and Shinobu Hosono and Hikaru Kawai}, journal={Communications in Mathematical Physics}, year={1994}, volume={161}, pages={157-175} }

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 algebrasR, and the physical Hilbert space is identified with the centerZ(R) of the associative algebraR. Perturbations of TFT's are also considered in this approach, showing that the form of topological…

## 134 Citations

### Lattice topological field theory on nonorientable surfaces

- Mathematics
- 1997

The lattice definition of the two-dimensional topological quantum field theory [Fukuma et al., Commun. Math. Phys. 161, 157 (1994)] is generalized to arbitrary (not necessarily orientable) compact…

### Topological and conformal field theory as Frobenius algebras

- Mathematics
- 2005

Two-dimensional conformal field theory (CFT) can be defined through its correlation functions. These must satisfy certain consistency conditions which arise from the cutting of world sheets along…

### ON LATTICE TOPOLOGICAL FIELD THEORIES WITH FINITE GROUPS

- Mathematics
- 2014

The approach of Lattice Topological Field Theories is used to describe quantities which are independent of manifolds, and the same time Lattice Gauge Theories are important to renormalize continuous…

### Classification and construction of unitary topological field theories in two dimensions

- Mathematics
- 1994

It is proven that unitary two‐dimensional topological field theories are uniquely characterized by n positive real numbers λ1,...,λn, which can be regarded as the eigenvalues of a Hermitian handle…

### Topological lattice field theories from intertwiner dynamics

- Physics
- 2013

We introduce a class of 2D lattice models that describe the dynamics of intertwiners, or, in a condensed matter interpretation, the fusion and splitting of anyons. We identify different families and…

### 2-DIMENSIONAL LATTICE TOPOLOGICAL FIELD THEORIES

- Mathematics
- 2011

Fukuma, Hosono and Kawai (FHK) [8] introduced Lattice Topological Field Theories (2D-LTFTs) to understand certain invariants of 3-manifolds. These “toy models” are not only useful for understanding…

### Construction of two-dimensional topological field theories with non-invertible symmetries

- MathematicsJournal of High Energy Physics
- 2021

Abstract
We construct the defining data of two-dimensional topological field theories (TFTs) enriched by non-invertible symmetries/topological defect lines. Simple formulae for the three-point…

### Homotopy field theory in dimension 2 and group-algebras

- Mathematics
- 1999

We apply the idea of a topological quantum field theory (TQFT) to maps from manifolds into topological spaces. This leads to a notion of a (d+1)-dimensional homotopy quantum field theory (HQFT) which…

### Spin from defects in two-dimensional quantum field theory

- PhysicsJournal of Mathematical Physics
- 2020

We build two-dimensional quantum field theories on spin surfaces starting from theories on oriented surfaces with networks of topological defect lines and junctions. The construction uses a…

## References

SHOWING 1-10 OF 18 REFERENCES

### Lattice Topological Field Theory in Two Dimensions

- Mathematics
- 1992

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…

### Topological quantum field theory

- Physics, Mathematics
- 1988

A twisted version of four dimensional supersymmetric gauge theory is formulated. The model, which refines a nonrelativistic treatment by Atiyah, appears to underlie many recent developments in…

### Discrete and Continuum Approaches to Three-Dimensional Quantum Gravity

- Mathematics, Physics
- 1991

It is shown that, in the three-dimensional lattice gravity defined by Ponzano and Regge, the space of physical states is isomorphic to the space of gauge-invariant functions on the moduli space of…

### On quantum gauge theories in two dimensions

- Mathematics
- 1991

Two dimensional quantum Yang-Mills theory is studied from three points of view: (i) by standard physical methods; (ii) by relating it to the largek limit of three dimensional Chern-Simons theory and…

### N = 2 superconformal models as topological field theories

- Physics
- 1990

We present simple observations on the topological nature of N=2 superconformal field theories. We point out that under a suitable redefinition of the energy-momentum tensor the central charge of N=2…

### Topological models on the lattice and a remark on string theory cloning

- Physics
- 1992

The addition of a topological model to the matter content of a conventional closed-string theory leads to the appearance of many, perturbatively-decoupled spacetime worlds. We illustrate this by…