2-Dimensional Dijkgraaf-Witten Theory with Defects Beamer.pdfSum over all labelings of triangulation Normalize sum Resulting state-sum invariant is used in de ning Z. Aria Dougherty

2-Dimensional Dijkgraaf-Witten Theory with Defects

Aria Dougherty and Hwajin ParkMentored by David N. Yetter

22 July 2014

Aria Dougherty and Hwajin ParkMentored by David N. Yetter2-Dimensional Dijkgraaf-Witten Theory with Defects22 July 2014 1 / 20

2-Dimensional Topological Quantum Field Theories

A 2-dimensional topological quantum field theory (TQFT) is a map Ztaking

closed 1-manifolds to vector spaces

disjoint unions of 1-manifolds to tensor products of vector spaces

cobordisms to linear maps

A B

C

W

Z(A) = X, Z(B) = Y , Z(C) = U

Z(A tB) = X ⊗ Y

Z(W ) = f , f : X ⊗ Y → U


State Sums Invariants

2D TQFTs may be constructed with state-sum invariants:

Triangulate the oriented surface

Orient edges and label with algebraicor combinatorial data

Assign state values to 2-simplexesbased on labels

Take product of all state values intriangulation

Sum over all labelings of triangulation

Normalize sum

Resulting state-sum invariant is used in defining Z.


Project Motivation

Dijkgraaf-Witten theory: Label edges with elements from a finitegroup and define state values with a group cocycle.

Jurgen Fuchs, et. al.: Adding a defect, i.e. a curve, to a surface inDijkgraaf-Witten theory results in some sort of group action.

The project: We permit an internal structure for the curve on thesurface in Dijkgraaf-Witten theory by introducing a second group. Thenwe construct a state-sum invariant for surfaces with curves.


Project Motivation





Project Motivation





Main Result

Theorem

Let G, H be groups with commuting group actions on a set X. For afield K, let α : G×G→ K×, β : X ×G→ K×, and γ : H ×X → K×

be functions that satisfy, ∀f, g, h ∈ G, w, x ∈ X, η, θ ∈ H,

α(f, g)α−1(f, gh)α(fg, h)α−1(g, h) = 1 (1)

β(x, g)β−1(x, gh)β(x · g, h)α−1(g, h) = 1 (2)

γ(η, x)γ−1(η, x · g)β(η · x, g)β−1(x, g) = 1 (3)

γ(θ, η · w)γ−1(θη, w)γ(η, w)γ−1(θ, η · x)γ(θη, x)γ−1(η, x) = 1. (4)

γ−1(η, x)γ(η, y)γ−1(η−1, η · x)γ(η−1, η · y) = 1 (5)


Main Result

Theorem (Continued)

For any flag-like triangulation T of an oriented surface Σ with curve C,let

Z(Σ, C) := |G|−|T00 ||H|−|T1

0 |∑

λ∈adm(T)

∏σ∈T0

2

αε(σ)(λ(σ0,1), λ(σ1,2))

·∏σ∈T1

2

βε(σ)(λ(σ0,1), λ(σ1,2))

·∏σ∈T2

2

γε(σ)(λ(σ0,1), λ(σ1,2))

,

where ε(σ) = ±1 depending on whether or not the orientation of σagrees with the orientation of Σ. Then Z(Σ, C) is independent of T andtherefore is a topological invariant of Σ ⊇ C.


Flag-like Triangulations

Require flag-like triangulations:

Curve triangulated as edges and vertices

2-simplex intersects curve only at one point or only at one edge


Alexander Moves

If two triangulated surfaces may be obtained from each other byAlexander moves, then they are piecewise-linear homeomorphic.

Thus invariance under Alexander moves shows topological invariance.


Ten Flag-like Alexander Moves


Admissible Labeling of TriangulationEdge labels:

g x η

g ∈ G x ∈ X η ∈ H

Edge orientations:

Enumerate vertices intersecting curve

Enumerate remaining vertices

Orient edges from lower vertex order to higher

x

f g

h

�


Admissible Labeling of TriangulationEdge labels:

g x η

g ∈ G x ∈ X η ∈ H

Edge orientations:

Enumerate vertices intersecting curve

Enumerate remaining vertices

Orient edges from lower vertex order to higher

x

f g

h

�


Admissible Labeling of TriangulationAdmissible labeling of each 2-simplex:

f g

h

x g

y

η x

y

fg = h x · g = y η · x = y

f, g, h ∈ G,x, y ∈ X,η ∈ H

0

1

2

η x

η · x 0

1

2

3

η x

η · x

x ·g

gη ·x ·g

∀η, x, g, (η · x) · g = η · (x · g) ⇐⇒ group actions commute



f g

h

x g

y

η x

y

fg = h x · g = y η · x = y

f, g, h ∈ G,x, y ∈ X,η ∈ H

0

1

2

η x

η · x 0

1

2

3

η x

η · x

x ·g

gη ·x ·g




f g

h

x g

y

η x

y

fg = h x · g = y η · x = y

f, g, h ∈ G,x, y ∈ X,η ∈ H

0

1

2

η x

η · x 0

1

2

3

η x

η · x

x ·g

gη ·x ·g



Functions α, β, and γ for each 2-Simplex

We create three functions α, β, γ corresponding to the three flag-like2-simplexes.

Each function maps the labels of the edges oriented in the samedirection to the multiplicative group of a field K.

α : G×G→ K×

β : X ×G→ K×

γ : H ×X → K×

ε(σ) =

{1 if σ is correctly oriented

−1 if σ is incorrectly oriented

σ1

�f g

h

σ2

�x g

y

σ3

�η x

y

αε(σ1)(f, g) βε(σ2)(x, g) γε(σ3)(η, x)


Functions α, β, and γ for each 2-Simplex

We create three functions α, β, γ corresponding to the three flag-like2-simplexes.

Each function maps the labels of the edges oriented in the samedirection to the multiplicative group of a field K.

α : G×G→ K×

β : X ×G→ K×

γ : H ×X → K×ε(σ) =

{1 if σ is correctly oriented

−1 if σ is incorrectly oriented

σ1

�f g

h

σ2

�x g

y

σ3

�η x

y

αε(σ1)(f, g) βε(σ2)(x, g) γε(σ3)(η, x)


Invariance Conditions on α, β, and γ

We derive conditions on α, β, and γ to maintain invariance underflag-like Alexander moves.

Example: Choose orientation of triangulation to be clockwise.

0

1

2

f g

fg

�

0

1

2

3

f g

fg

gh

hfgh

� �

α(f, g) = α(f, gh)α(g, h)α−1(fg, h)



Three conditions arise from the flag-like Alexander moves that subdividea 2-simplex into three 2-simplexes: ∀f, g, h ∈ G, η ∈ H,x ∈ X,

α(f, g)α−1(f, gh)α(fg, h)α−1(g, h) = 1 (1)

β(x, g)β−1(x, gh)β(x · g, h)α−1(g, h) = 1 (2)

γ(η, x)γ−1(η, x · g)β(η · x, g)β−1(x, g) = 1 (3)

It turns out that most of the other flag-like Alexander moves do notintroduce any additional conditions on α, β, and γ.



One flag-like Alexander move introduces a fourth condition:

2

1

0

3

x

y

η · y

η · x

η

�

2

1

0

3

-1x

y

η · y

η · x

θ

θηθη · y

θη · x

�

�

γ−1(η, x)γ(η, y) = γ−1(θη, x)γ(θη, y)γ−1(θ, η · y)γ(θ, η · x)

Then ∀η, θ ∈ H,w, x ∈ X,

γ(θ, η · w)γ−1(θη, w)γ(η, w)γ−1(θ, η · x)γ(θη, x)γ−1(η, x) = 1. (4)


Condition from Edge OrientationsWe require that the state-sum be independent of the orientations ofedges labeled with group elements.

For H, this requirement is

2

1

0

3

x

y

η · y

η · x

η

�

2

0

1

3

x

y

η · y

η · x

η−1

�

γ−1(η, x)γ(η, y) = γ(η−1, η · x)γ−1(η−1, η · y)

Then the fifth condition is ∀x, y ∈ X, η ∈ H,

γ−1(η, x)γ(η, y)γ−1(η−1, η · x)γ(η−1, η · y) = 1 (5)


Condition from Edge OrientationsWe require that the state-sum be independent of the orientations ofedges labeled with group elements.For H, this requirement is

2

1

0

3

x

y

η · y

η · x

η

�

2

0

1

3

x

y

η · y

η · x

η−1

�

γ−1(η, x)γ(η, y) = γ(η−1, η · x)γ−1(η−1, η · y)

Then the fifth condition is ∀x, y ∈ X, η ∈ H,

γ−1(η, x)γ(η, y)γ−1(η−1, η · x)γ(η−1, η · y) = 1 (5)


Normalizing the State-Sum

0

1

2

f g

fg0

1

2

3

f g

fg

gh

hfgh

With LHS labels fixed, the RHS may be admissibly labeled for anychoice of h ∈ G.

We normalize the state-sum by dividing by |G| or |H| as many times asthere are vertices of corresponding type.


Normalizing the State-Sum

0

1

2

f g

fg0

1

2

3

f g

fg

gh

hfgh

With LHS labels fixed, the RHS may be admissibly labeled for anychoice of h ∈ G.

We normalize the state-sum by dividing by |G| or |H| as many times asthere are vertices of corresponding type.


Main Theorem Restated

Let T be a flag-like triangulation and Tkn the set of n-simplexes with kvertices intersecting the curve. Let σ ∈ T be any 2-simplex andλ ∈ adm(T) be any admissible labeling of T.

If G and H have commuting group actions on X and α, β, and γ satisfythe five conditions, then for a surface with curve Σ ⊇ C, the quantity

|G|−|T00 ||H|−|T1

0 |∑

λ∈adm(T)

∏σ∈T0

2

αε(σ)(λ(σ0,1), λ(σ1,2))

·∏σ∈T1

2

βε(σ)(λ(σ0,1), λ(σ1,2))

·∏σ∈T2

2

γε(σ)(λ(σ0,1), λ(σ1,2))

is a topological invariant of Σ ⊇ C.


Main Theorem Restated

Let T be a flag-like triangulation and Tkn the set of n-simplexes with kvertices intersecting the curve. Let σ ∈ T be any 2-simplex andλ ∈ adm(T) be any admissible labeling of T.

If G and H have commuting group actions on X and α, β, and γ satisfythe five conditions, then for a surface with curve Σ ⊇ C, the quantity

|G|−|T00 ||H|−|T1

0 |∑

λ∈adm(T)

∏σ∈T0

2

αε(σ)(λ(σ0,1), λ(σ1,2))

·∏σ∈T1

2

βε(σ)(λ(σ0,1), λ(σ1,2))

·∏σ∈T2

2

γε(σ)(λ(σ0,1), λ(σ1,2))

is a topological invariant of Σ ⊇ C.


Examples

Let α ≡ β ≡ γ ≡ 1. Then

|G|−|T00 ||H|−|T1

0 |∑

λ∈adm(T)

1

is an invariant.

Let X be a group, X be the set of all elements of X, and G and Hbe subgroups of X that act on X by group multiplication. Then forany 2-cocycle α of X, define

α = α|G×Gβ = α|X×Gγ = α|H×X .

Let α ≡ γ ≡ 1. Let φ : G→ U(1) ⊆ C be a unitary character on G.Then define ∀g ∈ G, x ∈ X, β(x, g) = φ(g).


Examples


|G|−|T00 ||H|−|T1

0 |∑

λ∈adm(T)

1

is an invariant.





Examples


|G|−|T00 ||H|−|T1

0 |∑

λ∈adm(T)

1

is an invariant.





Acknowledgements

This research was carried out at the Summer Undergraduate Mathematics

Research (SUMaR) math REU at Kansas State University funded by NSF

under DMS award #1262877.

This research was partially carried out at Summer Undergraduate Research

Opportunity Program (SUROP) funded by the Graduate School at Kansas

State University.

We give special thanks to our mentor David Yetter for his guidancethroughout this project.


Documents

2-Dimensional Dijkgraaf-Witten Theory with Defects Beamer.pdfSum over all labelings of triangulation Normalize sum Resulting state-sum invariant is used in de ning Z. Aria Dougherty