Topological defects and generalised orbifolds ... - Nils Carqueville

Topological defects and
generalised orbifolds
based on arXiv:1208.1481 with Daniel Murfet, and arXiv:1210.6363 with Ingo Runkel
Nils Carqueville
Hausdorff Institute & Simons Center

Bigger picture
Calabi-Yau
geometry
singularity
theory
homological
mirror
symmetry
matrix factorisations
homological
knot
invariants
Landau-
Ginzburg
models
(topological)
string
theory
tensor
categories
2d TFT
2d CFT

Bigger picture
Calabi-Yau
geometry
singularity
theory
homological
mirror
symmetry
matrix factorisations
homological
knot
invariants
Landau-
Ginzburg
models
(topological)
string
theory
tensor
categories
2d TFT
2d CFT

Bigger picture
Calabi-Yau
geometry
singularity
theory
homological
mirror
symmetry
matrix factorisations
homological
knot
invariants
Landau-
Ginzburg
models
(topological)
string
theory
tensor
categories
2d TFT
2d CFT

Bigger picture
Calabi-Yau
geometry
singularity
theory
homological
mirror
symmetry
matrix factorisations
homological
knot
invariants
Landau-
Ginzburg
models
(topological)
string
theory
tensor
categories
2d TFT
2d CFT

Bigger picture
Calabi-Yau
geometry
singularity
theory
homological
mirror
symmetry
matrix factorisations
homological
knot
invariants
Landau-
Ginzburg
models
(topological)
string
theory
tensor
categories
2d TFT
2d CFT

Bigger picture
Calabi-Yau
geometry
singularity
theory
homological
mirror
symmetry
matrix factorisations
homological
knot
invariants
Landau-
Ginzburg
models
(topological)
string
theory
tensor
categories
2d TFT
2d CFT

Bigger picture
Calabi-Yau
geometry
singularity
theory
homological
mirror
symmetry
matrix factorisations
homological
knot
invariants
Landau-
Ginzburg
models
(topological)
string
theory
tensor
categories
2d TFT
2d CFT

Bigger picture
Calabi-Yau
geometry
singularity
theory
homological
mirror
symmetry
matrix factorisations
homological
knot
invariants
Landau-
Ginzburg
models
(topological)
string
theory
tensor
categories
2d TFT
2d CFT

Bigger picture
Calabi-Yau
geometry
singularity
theory
homological
mirror
symmetry
matrix factorisations
homological
knot
invariants
Landau-
Ginzburg
models
(topological)
string
theory
tensor
categories
2d TFT
2d CFT

Bigger picture
Calabi-Yau
geometry
singularity
theory
homological
mirror
symmetry
matrix factorisations
homological
knot
invariants
Landau-
Ginzburg
models
(topological)
string
theory
tensor
categories
2d TFT
2d CFT

Bigger picture
Calabi-Yau
geometry
singularity
theory
homological
mirror
symmetry
matrix factorisations
homological
knot
invariants
Landau-
Ginzburg
models
(topological)
string
theory
tensor
categories
2d TFT
2d CFT

Summary
2d TFTs with defects are naturally described in terms of bicategories
with extra structure.

Summary
2d TFTs with defects are naturally described in terms of bicategories
with extra structure.
Theorem. The bicategory of Landau-Ginzburg models has adjoints.

Summary
2d TFTs with defects are naturally described in terms of bicategories
with extra structure.
Theorem. The bicategory of Landau-Ginzburg models has adjoints.
(conceptual construction, yet very “computable”)

Summary
2d TFTs with defects are naturally described in terms of bicategories
with extra structure.
Theorem. The bicategory of Landau-Ginzburg models has adjoints.
(conceptual construction, yet very “computable”)
Applications.
understand open/closed TFT universally from within bicategory
compute any correlator as a 2-morphism
new proof of Cardy condition
. . .

Summary
2d TFTs with defects are naturally described in terms of bicategories
with extra structure.
Theorem. The bicategory of Landau-Ginzburg models has adjoints.
(conceptual construction, yet very “computable”)
Applications.
understand open/closed TFT universally from within bicategory
compute any correlator as a 2-morphism
new proof of Cardy condition
. . .
Generalised orbifolds.
conceptual, unified perspective on conventional orbifolds
new equivalences

2d TFTs with defects
Worldsheet, partitioned into domains:
theories
field insertions
T1
X2
φ1
T2
X1
X4
X3
X5
T3
φ6
T4
X7
φ2 φ3 φ4
Petkova/Zuber 2000, Bachas/Gaberdiel 2004, Fuchs/Runkel/Schweigert 2004, Davydov/Kong/Runkel 2010
T5
X8
X6
φ5
defect lines

2d TFTs with defects
Worldsheet, partitioned into domains:
theories
field insertions
T1
X2
φ1
T2
X1
X4
X3
X5
T3
φ6
T4
X7
φ2 φ3 φ4
A TFT assigns a number 〈. . .〉, the correlator, to any worldsheet,
depending only on isotopy class of defect lines:
� �
=
T5
X8
X6
φ5
� �
Petkova/Zuber 2000, Bachas/Gaberdiel 2004, Fuchs/Runkel/Schweigert 2004, Davydov/Kong/Runkel 2010
defect lines

2d TFTs with defects
Defect fusion gives product, unit = “invisible” defect I
�
X Y
�
=
�
X ⊗ Y
� �
I X
�
=
�
X
�

2d TFTs with defects
Defect fusion gives product, unit = “invisible” defect I
�
X Y
�
=
�
X ⊗ Y
� �
I X
operator product of fields, unit = identity field
�
ψ
ϕ
�
=
�
ψϕ
�
=
�
�
=
�
1
ψϕ
X
�
�

2d TFTs with defects
Defect fusion gives product, unit = “invisible” defect I
�
X Y
�
=
�
X ⊗ Y
� �
I X
operator product of fields, unit = identity field
�
ψ
ϕ
�
=
�
ψϕ
Fact. 2d TFTs with defects give bicategory:
objects (domains) = theories
1-morphisms (lines) = defects
2-morphisms (points) = fields
Davydov/Kong/Runkel 2010
�
=
�
�
=
�
1
ψϕ
X
�
�

Diagrammatics in bicategories
X
X
= 1X

Diagrammatics in bicategories
X
X
= 1X
ϕ
Y
X
= ϕ : X −→ Y

Diagrammatics in bicategories
X
X
= 1X
ϕ
Y
X
= ϕ : X −→ Y
ψ
ϕ
Z
X
= ψϕ

Diagrammatics in bicategories
X
X
= 1X
ϕ
Y
X
= ϕ : X −→ Y
ψ
ϕ
Z
X
= ψϕ
X
X
Y
φ ′ φ = φ⊗φ′
Y

Diagrammatics in bicategories
X
X
= 1X
φ
Z
X Y
ϕ
Y
X
= ϕ : X −→ Y
= φ : X ⊗ Y −→ Z
ψ
ϕ
Z
X
= ψϕ
X
X
Y
φ ′ φ = φ⊗φ′
Y

Diagrammatics in bicategories
X
X
= 1X
φ
Z
X Y
ϕ
Y
X
= ϕ : X −→ Y
= φ : X ⊗ Y −→ Z
ψ
ϕ
Z
X
ρX
= ψϕ
X
X I
X
X
Y
φ ′ φ = φ⊗φ′
Y

Diagrammatics in bicategories
X
X
= 1X
φ
Z
X Y
ϕ
Y
X
= ϕ : X −→ Y
= φ : X ⊗ Y −→ Z
ψ
ϕ
Z
X
ρX
= ψϕ
X
X I
I
X
X
Y
φ ′ φ = φ⊗φ′
X
X
Y
λX

Diagrammatics in bicategories
X
X
= 1X
φ
Z
X Y
ϕ
Y
X
= ϕ : X −→ Y
= φ : X ⊗ Y −→ Z
Orientation matters:
ψ
ϕ
T1
Z
X
ρX
= ψϕ
X
X I
X
X
T2
I
X
X
Y
φ ′ φ = φ⊗φ′
X
X
Y
λX

Diagrammatics in bicategories
X
X
= 1X
φ
Z
X Y
ϕ
Y
X
= ϕ : X −→ Y
= φ : X ⊗ Y −→ Z
Orientation matters:
ψ
ϕ
T1
Z
X
ρX
= ψϕ
X
X I
X
X
T2
X †
X †
I
X
X
Y
φ ′ φ = φ⊗φ′
X
X
T1
Y
λX

Orientation and adjoints
I
X † X
= evX : X † ⊗X −→ I
X
I
X †
= coevX : I −→ X ⊗X †

Orientation and adjoints
I
X † X
= evX : X † ⊗X −→ I
Defects are topological:
X
X
=
X
X
X
I
X †
= coevX : I −→ X ⊗X †
X †
X †
=
X †
X †

Orientation and adjoints
I
X † X
= evX : X † ⊗X −→ I
Defects are topological:
1X =
X
X
=
X
X
X
I
X †
= ρ ◦ (1 ⊗ ev) ◦ (coev ⊗1) ◦ λ −1
= coevX : I −→ X ⊗X †
X †
X †
=
X †
X †

Orientation and adjoints
I
X † X
= evX : X † ⊗X −→ I
Defects are topological:
1X =
X
X
=
X
X
X
I
X †
= ρ ◦ (1 ⊗ ev) ◦ (coev ⊗1) ◦ λ −1
= coevX : I −→ X ⊗X †
Definition. A bicategory has adjoints if for each 1-morphism X there is
a 1-morphism X † with 2-morphisms evX, coevX such that the above
Zorro moves hold.
X †
X †
=
X †
X †

Zorro

Landau-Ginzburg models
theories: potentials W ∈ R = C[x1, . . . , xn], dim(R/(∂W )) < ∞

Landau-Ginzburg models
theories: potentials W ∈ R = C[x1, . . . , xn], dim(R/(∂W )) < ∞
defects between W ∈ R and V ∈ S: matrix factorisations of
V − W , i. e. free Z2-graded (R ⊗ S)-modules X = X0 ⊕ X1 with
�
�
0 d1 dX =
X
d0 X 0
Kontsevich, Kapustin/Li 2002, Lazaroiu 2003, Brunner/Roggenkamp 2007
∈ End 1 R⊗S(X) , d 2 X = (V − W ) · 1X

Landau-Ginzburg models
theories: potentials W ∈ R = C[x1, . . . , xn], dim(R/(∂W )) < ∞
defects between W ∈ R and V ∈ S: matrix factorisations of
V − W , i. e. free Z2-graded (R ⊗ S)-modules X = X0 ⊕ X1 with
�
�
0 d1 dX =
X
d0 X 0
∈ End 1 R⊗S(X) , d 2 X = (V − W ) · 1X
fields between X and Y : (BRST) cohomology of
Hom(X, Y ) ∋ ψ ↦−→ dY ψ − (−1) |ψ| ψdX
Kontsevich, Kapustin/Li 2002, Lazaroiu 2003, Brunner/Roggenkamp 2007

Landau-Ginzburg models
defect fusion: X ⊗ Y , dX⊗Y = dX ⊗ 1 + 1 ⊗ dY
Brunner/Roggenkamp 2007

Landau-Ginzburg models
defect fusion: X ⊗ Y , dX⊗Y = dX ⊗ 1 + 1 ⊗ dY
invisible defect:
IW = (R ⊗ R) ⊕2 , dIW =
�
0
�
x − y
0
Brunner/Roggenkamp 2007, Kapustin/Rozansky 2004
W (x)−W (y)
x−y

Landau-Ginzburg models
defect fusion: X ⊗ Y , dX⊗Y = dX ⊗ 1 + 1 ⊗ dY
invisible defect:
IW = (R ⊗ R) ⊕2 , dIW =
�
0
�
x − y
0
for n = 1, in general:
IW = � � �n
�
(R⊗R)·θi , dIW =
i=1
Brunner/Roggenkamp 2007, Kapustin/Rozansky 2004
W (x)−W (y)
x−y
n�
i=1
�
(xi −yi)·θ ∗ �
i +∂ [i]W ·θi

Landau-Ginzburg models
defect fusion: X ⊗ Y , dX⊗Y = dX ⊗ 1 + 1 ⊗ dY
invisible defect:
IW = (R ⊗ R) ⊕2 , dIW =
�
0
�
x − y
0
for n = 1, in general:
IW = � � �n
�
(R⊗R)·θi , dIW =
i=1
Fact. End(IW ) ∼ = R/(∂W ) = bulk space
Brunner/Roggenkamp 2007, Kapustin/Rozansky 2004
W (x)−W (y)
x−y
n�
i=1
�
(xi −yi)·θ ∗ �
i +∂ [i]W ·θi

Landau-Ginzburg models
I
X
X
defect fusion: X ⊗ Y , dX⊗Y = dX ⊗ 1 + 1 ⊗ dY
invisible defect:
IW = (R ⊗ R) ⊕2 , dIW =
�
0
�
x − y
0
for n = 1, in general:
IW = � � �n
�
(R⊗R)·θi , dIW =
i=1
Fact. End(IW ) ∼ = R/(∂W ) = bulk space
λX
: I ⊗ X
�
Brunner/Roggenkamp 2007, Kapustin/Rozansky 2004
� ��
(R ⊗ R) ⊗ X mult. ��
X
W (x)−W (y)
x−y
n�
i=1
�
(xi −yi)·θ ∗ �
i +∂ [i]W ·θi

Landau-Ginzburg models
I
X
X
defect fusion: X ⊗ Y , dX⊗Y = dX ⊗ 1 + 1 ⊗ dY
invisible defect:
IW = (R ⊗ R) ⊕2 , dIW =
�
0
�
x − y
0
for n = 1, in general:
IW = � � �n
�
(R⊗R)·θi , dIW =
i=1
Fact. End(IW ) ∼ = R/(∂W ) = bulk space
λX
: I ⊗ X
�
Brunner/Roggenkamp 2007, Kapustin/Rozansky 2004
� ��
(R ⊗ R) ⊗ X mult. ��
X ,
W (x)−W (y)
x−y
n�
i=1
�
(xi −yi)·θ ∗ �
i +∂ [i]W ·θi
X
ρX : X ⊗ I ��
X
X I

Landau-Ginzburg models
I
X
defect fusion: X ⊗ Y , dX⊗Y = dX ⊗ 1 + 1 ⊗ dY
invisible defect:
IW = (R ⊗ R) ⊕2 , dIW =
�
0
�
x − y
0
for n = 1, in general:
IW = � � �n
�
(R⊗R)·θi , dIW =
i=1
Fact. End(IW ) ∼ = R/(∂W ) = bulk space
λX
: I ⊗ X
�
X
operator product: matrix multiplication
Brunner/Roggenkamp 2007, Kapustin/Rozansky 2004
� ��
(R ⊗ R) ⊗ X mult. ��
X ,
W (x)−W (y)
x−y
n�
i=1
�
(xi −yi)·θ ∗ �
i +∂ [i]W ·θi
X
ρX : X ⊗ I ��
X
X I

First main result
Theorem. Landau-Ginzburg models give a bicategory, called LG.
Carqueville/Runkel 2009

First main result
Theorem. Landau-Ginzburg models give a bicategory, called LG.
Theorem. LG has adjoints
Carqueville/Runkel 2009, Carqueville/Murfet 2012

First main result
Theorem. Landau-Ginzburg models give a bicategory, called LG.
Theorem. LG has adjoints:
Let W ∈ C[x1, . . . , xn], V ∈ C[z1, . . . , zm], X matrix fact. of V − W :
V W
X
Carqueville/Runkel 2009, Carqueville/Murfet 2012

First main result
Theorem. Landau-Ginzburg models give a bicategory, called LG.
Theorem. LG has adjoints:
Let W ∈ C[x1, . . . , xn], V ∈ C[z1, . . . , zm], X matrix fact. of V − W :
V W
X
W V
X †
Carqueville/Runkel 2009, Carqueville/Murfet 2012

First main result
Theorem. Landau-Ginzburg models give a bicategory, called LG.
Theorem. LG has adjoints:
Let W ∈ C[x1, . . . , xn], V ∈ C[z1, . . . , zm], X matrix fact. of V − W :
V W
X
Carqueville/Runkel 2009, Carqueville/Murfet 2012
X †
W V X † = X ∨ �
0 (d0 [m] dX † =
X ) ∨
−(d1 X )∨ 0
�
[m]

First main result
Theorem. Landau-Ginzburg models give a bicategory, called LG.
Theorem. LG has adjoints:
Let W ∈ C[x1, . . . , xn], V ∈ C[z1, . . . , zm], X matrix fact. of V − W :
V W
X
X
I
X †
X †
W V X † = X ∨ �
0 (d0 [m] dX † =
X ) ∨
−(d1 X )∨ 0
= coevX : 1 ↦−→ ∂ [1]dX . . . ∂ [m]dX ∈ X ⊗ X ∨
�
[m]
Carqueville/Runkel 2009, Carqueville/Murfet 2012 (Note: we suppress various signs)

First main result
Theorem. Landau-Ginzburg models give a bicategory, called LG.
Theorem. LG has adjoints:
Let W ∈ C[x1, . . . , xn], V ∈ C[z1, . . . , zm], X matrix fact. of V − W :
V W
X
X
X †
X †
W V X † = X ∨ �
0 (d0 [m] dX † =
X ) ∨
−(d1 X )∨ 0
= coevX : 1 ↦−→ ∂ [1]dX . . . ∂ [m]dX ∈ X ⊗ X ∨
X
I
† I
X
� �
str (−) ◦ ∂z1
= evX = Res
dX
� �
. . . ∂zmdX dz
+ O(θ)
∂z1V . . . ∂zmV
�
[m]
Carqueville/Runkel 2009, Carqueville/Murfet 2012 (Note: we suppress various signs)

First main result
Theorem. Landau-Ginzburg models give a bicategory, called LG.
Theorem. LG has adjoints:
Let W ∈ C[x1, . . . , xn], V ∈ C[z1, . . . , zm], X matrix fact. of V − W :
V W
X
X
X †
X †
W V X † = X ∨ �
0 (d0 [m] dX † =
X ) ∨
−(d1 X )∨ 0
= coevX : 1 ↦−→ ∂ [1]dX . . . ∂ [m]dX ∈ X ⊗ X ∨
X
I
† I
X
� �
str (−) ◦ ∂z1
= evX = Res
dX
� �
. . . ∂zmdX dz
+ O(θ)
∂z1V . . . ∂zmV
�
[m]
Proof : homological perturbation, associative Atiyah classes
Carqueville/Runkel 2009, Carqueville/Murfet 2012 (Note: we suppress various signs)

Applications
Defect action on bulk fields for defect X between W (x) and V (z):
X
φ
W
V

Applications
Defect action on bulk fields for defect X between W (x) and V (z):
X
φ
W
V
=
ρX
ρ −1
X
X
φ
W
V
= Res
� �
φ str (Πi∂xidX)(Πj∂zj dX) � �
dx
∂x1W . . . ∂xnW

Applications
Defect action on bulk fields for defect X between W (x) and V (z):
Ψ
X
φ
W
V
=
ρX
Ψ
ρ −1
X
X
φ
W
V
= Res
� �
φ str Ψ(Πi∂xidX)(Πj∂zj dX) � �
dx
∂x1W . . . ∂xnW

Applications
Defect action on bulk fields for defect X between W (x) and V (z):
Ψ
X
φ
W
V
Special cases:
=
ρX
Ψ
ρ −1
X
X
φ
W
V
= Res
� �
φ str Ψ(Πi∂xidX)(Πj∂zj dX) � �
dx
∂x1W . . . ∂xnW
φ = 1, Ψ = 1 gives the quantum dimension dim(X)

Applications
Defect action on bulk fields for defect X between W (x) and V (z):
Ψ
X
φ
W
V
Special cases:
=
ρX
Ψ
ρ −1
X
X
φ
W
V
= Res
� �
φ str Ψ(Πi∂xidX)(Πj∂zj dX) � �
dx
∂x1W . . . ∂xnW
φ = 1, Ψ = 1 gives the quantum dimension dim(X)
V = 0 gives Kapustin-Li disk correlator

Applications
Defect action on bulk fields for defect X between W (x) and V (z):
Ψ
X
φ
W
V
Special cases:
=
ρX
Ψ
ρ −1
X
X
φ
W
V
= Res
� �
φ str Ψ(Πi∂xidX)(Πj∂zj dX) � �
dx
∂x1W . . . ∂xnW
φ = 1, Ψ = 1 gives the quantum dimension dim(X)
V = 0 gives Kapustin-Li disk correlator
W = 0 gives boundary-bulk map
β X (Ψ) = str � Ψ ∂z1dX �
. . . ∂zmdX

Applications
Defect action on bulk fields for defect X between W (x) and V (z):
Ψ
X
φ
W
V
Special cases:
=
ρX
Ψ
ρ −1
X
X
φ
W
V
= Res
� �
φ str Ψ(Πi∂xidX)(Πj∂zj dX) � �
dx
∂x1W . . . ∂xnW
φ = 1, Ψ = 1 gives the quantum dimension dim(X)
V = 0 gives Kapustin-Li disk correlator
W = 0 gives boundary-bulk map
β X (Ψ) = str � Ψ ∂z1dX �
. . . ∂zmdX
ch(X) := β X (1) is the Chern character

Open/closed topological field theory
2d open/closed TFT via “generators and relations”
Moore 2000, Lazaroiu 2000, Moore/Segal 2006

Open/closed topological field theory
2d open/closed TFT via “generators and relations”:
commutative Frobenius algebra C,
Calabi-Yau category O,
bulk-boundary maps and boundary-bulk maps for all X ∈ O:
Moore 2000, Lazaroiu 2000, Moore/Segal 2006
βX : C −→ EndO(X) , β X : EndO(X) −→ C

Open/closed topological field theory
2d open/closed TFT via “generators and relations”:
commutative Frobenius algebra C,
Calabi-Yau category O,
bulk-boundary maps and boundary-bulk maps for all X ∈ O:
βX : C −→ EndO(X) , β X : EndO(X) −→ C
The βX are morphisms of unital algebras that map into the centre of
EndO(X).
� βX(φ), Ψ �
X = � φ, β X (Ψ) �
Cardy condition:
str(ΨmΦ) = � β X (Φ), β Y (Ψ) �
for all Φ : X −→ X, Ψ : Y −→ Y where ΨmΦ(α) = Ψ ◦ α ◦ Φ
Moore 2000, Lazaroiu 2000, Moore/Segal 2006

Open/closed TFT from Landau-Ginzburg models
βX(φ) =
W
φ
X
β X (Ψ) =
Ψ
W
X
IW
IW

Open/closed TFT from Landau-Ginzburg models
βX(φ) =
� �
Ψ1, Ψ2 X =
W
W
φ
X
X
Ψ1
Ψ2
β X (Ψ) =
Ψ
W
X
IW
IW
� �
φ1, φ2 W = φ1(x)
W (x) − W (y)
φ2(x)
IW

Open/closed TFT from Landau-Ginzburg models
βX(φ) =
� �
Ψ1, Ψ2 X =
W
W
φ
X
X
Ψ1
Ψ2
β X (Ψ) =
Ψ
W
X
IW
IW
� �
φ1, φ2 W = φ1(x)
W (x) − W (y)
φ2(x)
Theorem. Every W ∈ LG gives a 2d open/closed TFT with
C = R/(∂W ) and O = LG(0, W ).
Vafa 1990, Kapustin/Li 2003, Lazaroiu/Herbst 2004, Kapustin/Rozansky 2004, Murfet 2009, Polishchuk/Vaintrob 2010
IW

Applications
Theorem. The Cardy condition holds in LG
Polishchuk/Vaintrob 2010, Carqueville/Murfet 2012

Applications
Theorem. The Cardy condition holds in LG: for matrix factorisations
X, Y of W and maps Φ : X −→ X, Ψ : Y −→ Y we have
str � � �
� βX (Φ) βY (Ψ) dx
ΨmΦ = Res
∂1W . . . ∂nW
Polishchuk/Vaintrob 2010, Carqueville/Murfet 2012

Applications
Theorem. The Cardy condition holds in LG: for matrix factorisations
X, Y of W and maps Φ : X −→ X, Ψ : Y −→ Y we have
str � � �
� βX (Φ) βY (Ψ) dx
ΨmΦ = Res
∂1W . . . ∂nW
Proof:
Polishchuk/Vaintrob 2010, Carqueville/Murfet 2012
Φ
Y
X Ψ
W

Applications
Theorem. The Cardy condition holds in LG: for matrix factorisations
X, Y of W and maps Φ : X −→ X, Ψ : Y −→ Y we have
str � � �
� βX (Φ) βY (Ψ) dx
ΨmΦ = Res
∂1W . . . ∂nW
Proof:
Polishchuk/Vaintrob 2010, Carqueville/Murfet 2012
Φ
evY
�evX
coevX
coevY �
W
Ψ

Applications
Theorem. The Cardy condition holds in LG: for matrix factorisations
X, Y of W and maps Φ : X −→ X, Ψ : Y −→ Y we have
str � � �
� βX (Φ) βY (Ψ) dx
ΨmΦ = Res
∂1W . . . ∂nW
Proof:
Polishchuk/Vaintrob 2010, Carqueville/Murfet 2012
Φ
evY
�evX
coevX
coevY �
W
Ψ
evY
= W
βX (Φ)
coevY �
Ψ

Applications
Theorem. The Cardy condition holds in LG: for matrix factorisations
X, Y of W and maps Φ : X −→ X, Ψ : Y −→ Y we have
str � � �
� βX (Φ) βY (Ψ) dx
ΨmΦ = Res
∂1W . . . ∂nW
Proof:
1 ⊗ Φ
ev X † ⊗Y
coev � X † ⊗Y
1 ⊗ Ψ
Polishchuk/Vaintrob 2010, Carqueville/Murfet 2012
evY
�evX Ψ
=
Φ
W
coevX
coevY �
evY
= W
βX (Φ)
coevY �
Ψ

Generalised orbifolds
Theorem. Let X ∈ LG(W, V ) have invertible quantum dimensions.
Carqueville/Runkel 2012

Generalised orbifolds
Theorem. Let X ∈ LG(W, V ) have invertible quantum dimensions.
A = X † ⊗ X is a special symmetric Frobenius algebra in LG(W, W ).
Everything about theory V can be recovered from A
Carqueville/Runkel 2012

Generalised orbifolds
Theorem. Let X ∈ LG(W, V ) have invertible quantum dimensions.
A = X † ⊗ X is a special symmetric Frobenius algebra in LG(W, W ).
Everything about theory V can be recovered from A:
◮ LG(0, V ) ∼ = mod(A) (boundary sector)
◮ LG(V, V ) ∼ = bimod(A) (defect sector)
Carqueville/Runkel 2012

Generalised orbifolds
Theorem. Let X ∈ LG(W, V ) have invertible quantum dimensions.
A = X † ⊗ X is a special symmetric Frobenius algebra in LG(W, W ).
Everything about theory V can be recovered from A:
◮ LG(0, V ) ∼ = mod(A) (boundary sector)
◮ LG(V, V ) ∼ = bimod(A) (defect sector)
Idea. Introducing X-bubbles in V -correlator is scaling by dim(X).
Blowing up all X-bubbles produces W -correlator with A-defect network.
�
Carqueville/Runkel 2012
V
�

Generalised orbifolds
Theorem. Let X ∈ LG(W, V ) have invertible quantum dimensions.
A = X † ⊗ X is a special symmetric Frobenius algebra in LG(W, W ).
Everything about theory V can be recovered from A:
◮ LG(0, V ) ∼ = mod(A) (boundary sector)
◮ LG(V, V ) ∼ = bimod(A) (defect sector)
Idea. Introducing X-bubbles in V -correlator is scaling by dim(X).
Blowing up all X-bubbles produces W -correlator with A-defect network.
�
Carqueville/Runkel 2012
V
�
∼
�
X
W
V
�

Generalised orbifolds
Theorem. Let X ∈ LG(W, V ) have invertible quantum dimensions.
A = X † ⊗ X is a special symmetric Frobenius algebra in LG(W, W ).
Everything about theory V can be recovered from A:
◮ LG(0, V ) ∼ = mod(A) (boundary sector)
◮ LG(V, V ) ∼ = bimod(A) (defect sector)
Idea. Introducing X-bubbles in V -correlator is scaling by dim(X).
Blowing up all X-bubbles produces W -correlator with A-defect network.
=
�
V
�
� �
Carqueville/Runkel 2012
∼
�
X
W
V
�

Generalised orbifolds
Theorem. Let X ∈ LG(W, V ) have invertible quantum dimensions.
A = X † ⊗ X is a special symmetric Frobenius algebra in LG(W, W ).
Everything about theory V can be recovered from A:
◮ LG(0, V ) ∼ = mod(A) (boundary sector)
◮ LG(V, V ) ∼ = bimod(A) (defect sector)
Idea. Introducing X-bubbles in V -correlator is scaling by dim(X).
Blowing up all X-bubbles produces W -correlator with A-defect network.
=
�
V
�
� �
Carqueville/Runkel 2012
∼
=
�
�
X
W
V
A
W
�
�

Generalised orbifolds
Let B be pivotal bicategory with adjoints.
Carqueville/Runkel 2012, Fröhlich/Fuchs/Runkel/Schweigert 2009, (Fjelstad/Fröhlich/)Fuchs/Runkel/Schweigert 200x

Generalised orbifolds
Let B be pivotal bicategory with adjoints. Orbifold completion Borb:
objects: pairs (W, A) with W ∈ B and A ∈ EndB(W ) special
symmetric Frobenius algebra:
: A⊗A −→ A : I −→ A
Carqueville/Runkel 2012, Fröhlich/Fuchs/Runkel/Schweigert 2009, (Fjelstad/Fröhlich/)Fuchs/Runkel/Schweigert 200x

Generalised orbifolds
Let B be pivotal bicategory with adjoints. Orbifold completion Borb:
objects: pairs (W, A) with W ∈ B and A ∈ EndB(W ) special
symmetric Frobenius algebra:
: A⊗A −→ A : I −→ A = = =
Carqueville/Runkel 2012, Fröhlich/Fuchs/Runkel/Schweigert 2009, (Fjelstad/Fröhlich/)Fuchs/Runkel/Schweigert 200x

Generalised orbifolds
Let B be pivotal bicategory with adjoints. Orbifold completion Borb:
objects: pairs (W, A) with W ∈ B and A ∈ EndB(W ) special
symmetric Frobenius algebra:
: A⊗A −→ A : I −→ A = = =
= = = =
Carqueville/Runkel 2012, Fröhlich/Fuchs/Runkel/Schweigert 2009, (Fjelstad/Fröhlich/)Fuchs/Runkel/Schweigert 200x

Generalised orbifolds
Let B be pivotal bicategory with adjoints. Orbifold completion Borb:
objects: pairs (W, A) with W ∈ B and A ∈ EndB(W ) special
symmetric Frobenius algebra:
: A⊗A −→ A : I −→ A = = =
= = = =
1-morphisms: X ∈ B(W, V ) that are bimodules
X
X
Carqueville/Runkel 2012, Fröhlich/Fuchs/Runkel/Schweigert 2009, (Fjelstad/Fröhlich/)Fuchs/Runkel/Schweigert 200x

Generalised orbifolds
Let B be pivotal bicategory with adjoints. Orbifold completion Borb:
objects: pairs (W, A) with W ∈ B and A ∈ EndB(W ) special
symmetric Frobenius algebra:
: A⊗A −→ A : I −→ A = = =
= = = =
1-morphisms: X ∈ B(W, V ) that are bimodules
X
X
= =
Carqueville/Runkel 2012, Fröhlich/Fuchs/Runkel/Schweigert 2009, (Fjelstad/Fröhlich/)Fuchs/Runkel/Schweigert 200x

Generalised orbifolds
Let B be pivotal bicategory with adjoints. Orbifold completion Borb:
objects: pairs (W, A) with W ∈ B and A ∈ EndB(W ) special
symmetric Frobenius algebra:
: A⊗A −→ A : I −→ A = = =
= = = =
1-morphisms: X ∈ B(W, V ) that are bimodules
X
X
= = =
Carqueville/Runkel 2012, Fröhlich/Fuchs/Runkel/Schweigert 2009, (Fjelstad/Fröhlich/)Fuchs/Runkel/Schweigert 200x

Generalised orbifolds
Let B be pivotal bicategory with adjoints. Orbifold completion Borb:
objects: pairs (W, A) with W ∈ B and A ∈ EndB(W ) special
symmetric Frobenius algebra:
: A⊗A −→ A : I −→ A = = =
= = = =
1-morphisms: X ∈ B(W, V ) that are bimodules
X
= = =
X
horizontal composition: tensor product over algebra
Carqueville/Runkel 2012, Fröhlich/Fuchs/Runkel/Schweigert 2009, (Fjelstad/Fröhlich/)Fuchs/Runkel/Schweigert 200x

Generalised orbifolds
Let B be pivotal bicategory with adjoints. Orbifold completion Borb:
objects: pairs (W, A) with W ∈ B and A ∈ EndB(W ) special
symmetric Frobenius algebra:
: A⊗A −→ A : I −→ A = = =
= = = =
1-morphisms: X ∈ B(W, V ) that are bimodules
X
= = =
X
horizontal composition: tensor product over algebra
2-morphisms: φ ∈ Hom(X, Y ) that are bimodule maps
Carqueville/Runkel 2012, Fröhlich/Fuchs/Runkel/Schweigert 2009, (Fjelstad/Fröhlich/)Fuchs/Runkel/Schweigert 200x

Generalised orbifolds
Theorem.
A = X † ⊗ X is a symmetric Frobenius algebra for any 1-morphism X in B.
Carqueville/Runkel 2012

Generalised orbifolds
Theorem.
A = X † ⊗ X is a symmetric Frobenius algebra for any 1-morphism X in B.
If dim(X) is invertible (easy to check for B = LG) then:
A = X † ⊗ X is a special symmetric Frobenius algebra.
X ⊗A X † ∼ = I.
X and X † are mutually inverse in Borb.
Carqueville/Runkel 2012

Generalised orbifolds
Theorem.
A = X † ⊗ X is a symmetric Frobenius algebra for any 1-morphism X in B.
If dim(X) is invertible (easy to check for B = LG) then:
A = X † ⊗ X is a special symmetric Frobenius algebra.
X ⊗A X † ∼ = I.
X and X † are mutually inverse in Borb.
“If B has nondegenerate pairings, so does Borb.”
Carqueville/Runkel 2012

Generalised orbifolds
Theorem.
A = X † ⊗ X is a symmetric Frobenius algebra for any 1-morphism X in B.
If dim(X) is invertible (easy to check for B = LG) then:
A = X † ⊗ X is a special symmetric Frobenius algebra.
X ⊗A X † ∼ = I.
X and X † are mutually inverse in Borb.
“If B has nondegenerate pairings, so does Borb.”
Theorem. Every (W, A) ∈ LGorb gives rise to 2d open/closed TFT:
� �
φ1, φ2 (W,A) =
�
Carqueville/Runkel 2012
φ1
φ2
�
β orb
X (φ) =
φ
X
X
β X orb (Ψ) =
Ψ
X

Generalised orbifolds
Theorem.
A = X † ⊗ X is a symmetric Frobenius algebra for any 1-morphism X in B.
If dim(X) is invertible (easy to check for B = LG) then:
A = X † ⊗ X is a special symmetric Frobenius algebra.
X ⊗A X † ∼ = I.
X and X † are mutually inverse in Borb.
“If B has nondegenerate pairings, so does Borb.”
Theorem. Every (W, A) ∈ LGorb gives rise to 2d open/closed TFT:
� �
φ1, φ2 (W,A) =
�
φ1
φ2
�
β orb
X (φ) = φ
X
X
β X orb (Ψ) =
Ψ
X
Corollary. Cardy condition holds for G-equivariant matrix factorisations.
Carqueville/Runkel 2012

Generalised orbifolds
Examples.
“ordinary” orbifolds: for discrete symmetry group G of W we have
hmf(W ) G = LG(0, W ) G � � �
∼ = mod
Carqueville/Runkel 2012, see also Ashok/Dell’Aquila/Diaconescu 2004, Brunner/Roggenkamp 2007, Brunner/Carqueville/Plencner
g∈G
Ig

Generalised orbifolds
Examples.
“ordinary” orbifolds: for discrete symmetry group G of W we have
hmf(W ) G = LG(0, W ) G � �
∼ = mod
�
(& Cardy condition)
Carqueville/Runkel 2012
g∈G
Ig

Generalised orbifolds
Examples.
“ordinary” orbifolds: for discrete symmetry group G of W we have
hmf(W ) G = LG(0, W ) G � �
∼ = mod
�
(& Cardy condition)
Knörrer periodicity
Carqueville/Runkel 2012
g∈G
Ig

Generalised orbifolds
Examples.
“ordinary” orbifolds: for discrete symmetry group G of W we have
hmf(W ) G = LG(0, W ) G � �
∼ = mod
�
(& Cardy condition)
g∈G
Knörrer periodicity
Z2-orbifold between A- and D-type minimal models
Carqueville/Runkel 2012
Ig

Generalised orbifolds
Examples.
“ordinary” orbifolds: for discrete symmetry group G of W we have
hmf(W ) G = LG(0, W ) G � �
∼ = mod
�
(& Cardy condition)
g∈G
Knörrer periodicity
Z2-orbifold between A- and D-type minimal models:
�
x 0
X =
d−u2d x−u2 − y2 x − u2 � �
�
0 z + uy
⊗
0
z − uy 0
is defect between WA = u 2d and WD = x d − xy 2 + z 2 , has invertible
quantum dimensions ⇒ hmf(WD) ∼ = mod(X † ⊗ X)
Carqueville/Runkel 2012
Ig

Generalised orbifolds
Examples.
“ordinary” orbifolds: for discrete symmetry group G of W we have
hmf(W ) G = LG(0, W ) G � �
∼ = mod
�
(& Cardy condition)
g∈G
Knörrer periodicity
Z2-orbifold between A- and D-type minimal models:
�
x 0
X =
d−u2d x−u2 − y2 x − u2 � �
�
0 z + uy
⊗
0
z − uy 0
is defect between WA = u 2d and WD = x d − xy 2 + z 2 , has invertible
quantum dimensions ⇒ hmf(WD) ∼ = mod(X † ⊗ X)
similar equivalences expected e. g. between A- and E-type
Carqueville/Runkel 2012
Ig

Generalised orbifolds
Examples.
“ordinary” orbifolds: for discrete symmetry group G of W we have
hmf(W ) G = LG(0, W ) G � �
∼ = mod
�
(& Cardy condition)
g∈G
Knörrer periodicity
Z2-orbifold between A- and D-type minimal models:
�
x 0
X =
d−u2d x−u2 − y2 x − u2 � �
�
0 z + uy
⊗
0
z − uy 0
is defect between WA = u 2d and WD = x d − xy 2 + z 2 , has invertible
quantum dimensions ⇒ hmf(WD) ∼ = mod(X † ⊗ X)
similar equivalences expected e. g. between A- and E-type
Task. Classify all defects with invertible quantum dimensions (and find
new equivalences this way)!
Carqueville/Runkel 2012
Ig

Conclusions
“2d TFT with defects = bicategory + x”

Conclusions
“2d TFT with defects = bicategory + x”
Theorem. The bicategory of Landau-Ginzburg models has adjoints.
(conceptual construction, yet very “computable”)

Conclusions
“2d TFT with defects = bicategory + x”
Theorem. The bicategory of Landau-Ginzburg models has adjoints.
(conceptual construction, yet very “computable”)
Description naturally incorporates known structure:
compute any correlator from defect picture
encode all open/closed TFT data
easy proof of Cardy condition
defect action on bulk fields, quantum dimensions

Conclusions
“2d TFT with defects = bicategory + x”
Theorem. The bicategory of Landau-Ginzburg models has adjoints.
(conceptual construction, yet very “computable”)
Description naturally incorporates known structure:
compute any correlator from defect picture
encode all open/closed TFT data
easy proof of Cardy condition
defect action on bulk fields, quantum dimensions
Generalised orbifolds
natural description of conventional orbifolds
give rise to new equivalences