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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
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