Lecture 4: State sum models in 2d

John Barrett

School of Mathematical SciencesUniversity of Nottingham

June 4, 2014

State sum model

I M manifold, triangulation ∆.

I A finite set S

I A state is an assignment of an element of S to eachk-simplex (some fixed k < d)

I Weight Zσ ∈ C defined for each simplex σ, using thestates on the simplex

Z (M ,∆) =∑states

∏σ

Zσ

Examples

I Lattice gauge theory

I Topological quantum field theory

I Spin foam models

Example: 2d state sum model

[Fukuma, Hosono, Kawai 1994]

ZT = Cabc ∈ CZE = Bab

Z• = R

Cabc = Ccab Bab = Bba

Dual formalism

Planar model[Lauda, Pfeiffer 2007]

M ⊂ R2

Frobenius property

Replace cyclic symmetry of C and symmetry of B with

CabdBdc = BceCeab

Z is independent of the way the diagram is drawn in theinterior.

Topological QFT

Z depends on M but not on gµν .

Examples

I Chern-Simons QFT d = 3

I Quantum Gravity

I Topological phase in CM

Topological State Sum Model

Z depends on M but not on triangulation.

Example: planar model

C eabC

dec = C d

af Cfbc

=⇒ structure constants of an associative algebra A.

3-1 move

A is a special Frobenius algebra.

Spin models[JWB, Sara Tavares 2013]

MΦ−→ R3 π−→ R2

I Φ is an immersionI π is a coordinate projectionI Planar model with additional data λ : A⊗A → A⊗A.

Axioms for crossing

Spin model: results

I Z is independent of the triangulation

I Z depends on Φ only via the spin structure induced fromR3.

I ψ2 = 1

I If ψ = 1 then Z is independent of the spin structure.

I There exist examples of Cabc ,Bab, λabcd that distinguish

spin structures.

Example: torusA torus has four spin structures.

Torus diagrams