Examples of 2d defect TQFTs Trivial defect TQFT Z triv : � � D 2 := C � � D 1 := finite-dimensional C -vector spaces � � D 0 := linear maps � def V 1 Z triv � . . = V 1 ⊗ · · · ⊗ V m . V m � def Z triv � = ( evaluate 0- und 1-strata as string diagrams in Vect) B-twisted sigma models : Calabi-Yau manifolds and holomorphic vector bundles Landau-Ginzburg models : isolated singularities and homological algebra (more soon. . . )
State sum models Input : ∆ -separable symmetric Frobenius C -algebra ( A, µ, ∆) Fukuma/Hosono/Kawai 1992, Lauda/Pfeiffer 2006
State sum models Input : ∆ -separable symmetric Frobenius C -algebra ( A, µ, ∆) (1) Choose oriented triangulation t for every bordism Σ in Bord 2 Fukuma/Hosono/Kawai 1992, Lauda/Pfeiffer 2006
State sum models Input : ∆ -separable symmetric Frobenius C -algebra ( A, µ, ∆) (1) Choose oriented triangulation t for every bordism Σ in Bord 2 (2) Decorate Poincar´ e-dual graph with ( C , A, µ, ∆) : A A A C ∆ C C C C C C µ A A A A C Fukuma/Hosono/Kawai 1992, Lauda/Pfeiffer 2006
State sum models Input : ∆ -separable symmetric Frobenius C -algebra ( A, µ, ∆) (1) Choose oriented triangulation t for every bordism Σ in Bord 2 (2) Decorate Poincar´ e-dual graph with ( C , A, µ, ∆) : A A A C ∆ C C C C C C µ A A A A C (3) Obtain Σ t,A in Bord def 2 ( D triv ) and define Z ss A (Σ) = Z triv (Σ t,A ) Fukuma/Hosono/Kawai 1992, Lauda/Pfeiffer 2006
State sum models Input : ∆ -separable symmetric Frobenius C -algebra ( A, µ, ∆) (1) Choose oriented triangulation t for every bordism Σ in Bord 2 (2) Decorate Poincar´ e-dual graph with ( C , A, µ, ∆) : A A A C ∆ C C C C C C µ A A A A C (3) Obtain Σ t,A in Bord def 2 ( D triv ) and define Z ss A (Σ) = Z triv (Σ t,A ) Theorem. Construction yields TQFT Z ss A : Bord 2 − → Vect . Fukuma/Hosono/Kawai 1992, Lauda/Pfeiffer 2006
State sum models Input : ∆ -separable symmetric Frobenius C -algebra ( A, µ, ∆) (1) Choose oriented triangulation t for every bordism Σ in Bord 2 (2) Decorate Poincar´ e-dual graph with ( C , A, µ, ∆) : A A A C ∆ C C C C C C µ A A A A C (3) Obtain Σ t,A in Bord def 2 ( D triv ) and define Z ss A (Σ) = Z triv (Σ t,A ) Theorem. Construction yields TQFT Z ss A : Bord 2 − → Vect . Proof sketch : Defining properties of ( A, µ, ∆) encode invariance under Pachner moves = ⇒ independent of choice of triangulation: 2-2 1-3 ← → ← → Fukuma/Hosono/Kawai 1992, Lauda/Pfeiffer 2006
State sum models Input : ∆ -separable symmetric Frobenius C -algebra ( A, µ, ∆) (1) Choose oriented triangulation t for every bordism Σ in Bord 2 (2) Decorate Poincar´ e-dual graph with ( C , A, µ, ∆) : A A A C ∆ C C C C C C µ A A A A C (3) Obtain Σ t,A in Bord def 2 ( D triv ) and define Z ss A (Σ) = Z triv (Σ t,A ) Theorem. Construction yields TQFT Z ss A : Bord 2 − → Vect . Proof sketch : Defining properties of ( A, µ, ∆) encode invariance under Pachner moves = ⇒ independent of choice of triangulation: 2-2 1-3 ← → ← → ← → ← → Fukuma/Hosono/Kawai 1992, Lauda/Pfeiffer 2006
Input : ∆ -separable symmetric Frobenius C -algebra ( A, µ, ∆) (1) Choose oriented triangulation t for every bordism Σ in Bord 2 (2) Decorate Poincar´ e-dual graph with ( C , A, µ, ∆) : A A A C ∆ C C C C C C µ A A A A C (3) Obtain Σ t,A in Bord def 2 ( D triv ) and define Z ss A (Σ) = Z triv (Σ t,A ) Theorem. Construction yields TQFT Z ss A : Bord 2 − → Vect .
Input : ∆ -separable symmetric Frobenius C -algebra ( A, µ, ∆) (1) Choose oriented triangulation t for every bordism Σ in Bord 2 (2) Decorate Poincar´ e-dual graph with ( C , A, µ, ∆) : A A A C ∆ C C C C C C µ A A A A C (3) Obtain Σ t,A in Bord def 2 ( D triv ) and define Z ss A (Σ) = Z triv (Σ t,A ) Theorem. Construction yields TQFT Z ss A : Bord 2 − → Vect . No need to consider only algebras over C !
Orbifolds Definition. Let Z : Bord def 2 ( D ) − → Vect be defect TQFT. Carqueville/Runkel 2012
Orbifolds Definition. Let Z : Bord def 2 ( D ) − → Vect be defect TQFT. An orbifold datum for Z is A ≡ ( α, A, µ, ∆) : α A A A α α ∆ α α α µ A α α A α A A α ∈ D 2 A ∈ D 1 µ ∈ D 0 ∆ ∈ D 0 such that Pachner moves become identities under Z : � � � � � � � � ! ! Z = Z Z = Z Carqueville/Runkel 2012
Orbifolds Definition. Let Z : Bord def 2 ( D ) − → Vect be defect TQFT. An orbifold datum for Z is A ≡ ( α, A, µ, ∆) : α A A A α α ∆ α α α µ A α α A α A A α ∈ D 2 A ∈ D 1 µ ∈ D 0 ∆ ∈ D 0 such that Pachner moves become identities under Z : � � � � � � � � ! ! Z = Z Z = Z Definition & Theorem. Triangulation + A -decoration + evaluation with Z Carqueville/Runkel 2012
Orbifolds Definition. Let Z : Bord def 2 ( D ) − → Vect be defect TQFT. An orbifold datum for Z is A ≡ ( α, A, µ, ∆) : α A A A α α ∆ α α α µ A α α A α A A α ∈ D 2 A ∈ D 1 µ ∈ D 0 ∆ ∈ D 0 such that Pachner moves become identities under Z : � � � � � � � � ! ! Z = Z Z = Z Definition & Theorem. Triangulation + A -decoration + evaluation with Z = A -orbifold TQFT Z A : Bord 2 − → Vect Carqueville/Runkel 2012
Algebraic characterisation Theorem. 2d defect TQFT Z = ⇒ pivotal 2-category B Z Davydov/Kong/Runkel 2011
Algebraic characterisation Theorem. 2d defect TQFT Z = ⇒ pivotal 2-category B Z Proof sketch : – objects = elements of D 2 = ‘theories’ Davydov/Kong/Runkel 2011
Algebraic characterisation Theorem. 2d defect TQFT Z = ⇒ pivotal 2-category B Z Proof sketch : – objects = elements of D 2 = ‘theories’ – 1-morphisms X : α → β = (lists of) elements of D 1 = ‘defect lines’: . . . α 2 α 1 α β x n x 3 x 2 x 1 Davydov/Kong/Runkel 2011
Algebraic characterisation Theorem. 2d defect TQFT Z = ⇒ pivotal 2-category B Z Proof sketch : – objects = elements of D 2 = ‘theories’ – 1-morphisms X : α → β = (lists of) elements of D 1 = ‘defect lines’: . . . α 2 α 1 α β x n x 3 x 2 x 1 – 2-morphisms = ‘junction fields’: . . . y 2 � � y 1 y m Hom( X, Y ) = Z x 1 x n x 2 . . . Davydov/Kong/Runkel 2011
Algebraic characterisation Theorem. 2d defect TQFT Z = ⇒ pivotal 2-category B Z Proof sketch : – objects = elements of D 2 = ‘theories’ – 1-morphisms X : α → β = (lists of) elements of D 1 = ‘defect lines’: . . . α 2 α 1 α β x n x 3 x 2 x 1 – 2-morphisms = ‘junction fields’: y 1 y 2 y m . . . . . . y 2 � � y 1 y m Hom( X, Y ) = Z ∋ α β x 1 x n x 2 . . . . . . x 1 x 2 x n Davydov/Kong/Runkel 2011
Algebraic characterisation Theorem. 2d defect TQFT Z = ⇒ pivotal 2-category B Z Davydov/Kong/Runkel 2011
Algebraic characterisation Theorem. 2d defect TQFT Z = ⇒ pivotal 2-category B Z ր objects = bulk theories, 1-morphisms = defect lines, 2-morphisms = junction fields Davydov/Kong/Runkel 2011
Algebraic characterisation Theorem. 2d defect TQFT Z = ⇒ pivotal 2-category B Z Examples. – vector spaces : BVect ∗ , finite-dimensional C -vector spaces, linear maps Davydov/Kong/Runkel 2011, Carqueville 2016
Algebraic characterisation Theorem. 2d defect TQFT Z = ⇒ pivotal 2-category B Z Examples. – vector spaces : BVect ∗ , finite-dimensional C -vector spaces, linear maps – state sum models ∆ -separable symmetric Frobenius C -algebras, bimodules, intertwiners Davydov/Kong/Runkel 2011, Carqueville 2016
Algebraic characterisation Theorem. 2d defect TQFT Z = ⇒ pivotal 2-category B Z Examples. – vector spaces : BVect ∗ , finite-dimensional C -vector spaces, linear maps – state sum models ∆ -separable symmetric Frobenius C -algebras, bimodules, intertwiners – B-twisted sigma models Calabi-Yau varieties, Fourier-Mukai kernels, RHom – A-twisted sigma models symplectic manifolds, Lagrangian correspondences, Floer homology – Landau-Ginzburg models isolated singularities, matrix factorisations – differential graded categories smooth and proper dg categories, dg bimodules, intertwiners – categorified quantum groups weights, functors E i , F j . . . , string diagrams. . . Davydov/Kong/Runkel 2011, Carqueville 2016
Algebraic characterisation of orbifolds Theorem. 2d defect TQFT Z = ⇒ pivotal 2-category B Z Davydov/Kong/Runkel 2011
Algebraic characterisation of orbifolds Theorem. 2d defect TQFT Z = ⇒ pivotal 2-category B Z Lemma. � ∼ � � � orbifold data for Z ∆ -separable symmetric Frobenius algebras in B Z = Davydov/Kong/Runkel 2011
Algebraic characterisation of orbifolds Theorem. 2d defect TQFT Z = ⇒ pivotal 2-category B Z Lemma. � ∼ � � � orbifold data for Z ∆ -separable symmetric Frobenius algebras in B Z = = = = = Davydov/Kong/Runkel 2011
Algebraic characterisation of orbifolds Theorem. 2d defect TQFT Z = ⇒ pivotal 2-category B Z Lemma. � ∼ � � � orbifold data for Z ∆ -separable symmetric Frobenius algebras in B Z = = = = = ⇐ ⇒ � � � � � � � � Z = Z Z = Z Davydov/Kong/Runkel 2011
Algebraic characterisation of orbifolds Theorem. 2d defect TQFT Z = ⇒ pivotal 2-category B Z Lemma. � ∼ � � � orbifold data for Z ∆ -separable symmetric Frobenius algebras in B Z = Davydov/Kong/Runkel 2011
Algebraic characterisation of orbifolds Theorem. 2d defect TQFT Z = ⇒ pivotal 2-category B Z Lemma. � ∼ � � � orbifold data for Z ∆ -separable symmetric Frobenius algebras in B Z = Examples. – ∆ -separable symmetric Frobenius algebras in BVect Davydov/Kong/Runkel 2011
Algebraic characterisation of orbifolds Theorem. 2d defect TQFT Z = ⇒ pivotal 2-category B Z Lemma. � ∼ � � � orbifold data for Z ∆ -separable symmetric Frobenius algebras in B Z = Examples. – ∆ -separable symmetric Frobenius algebras in BVect = ∆ -separable symmetric Frobenius C -algebras � Davydov/Kong/Runkel 2011
Algebraic characterisation of orbifolds Theorem. 2d defect TQFT Z = ⇒ pivotal 2-category B Z Lemma. � ∼ � � � orbifold data for Z ∆ -separable symmetric Frobenius algebras in B Z = Examples. – ∆ -separable symmetric Frobenius algebras in BVect = ∆ -separable symmetric Frobenius C -algebras � ⇒ Z ss A = ( Z triv ) A (“ State sum models are orbifolds of the trivial TQFT. ”) = Davydov/Kong/Runkel 2011
Algebraic characterisation of orbifolds Theorem. 2d defect TQFT Z = ⇒ pivotal 2-category B Z Lemma. � ∼ � � � orbifold data for Z ∆ -separable symmetric Frobenius algebras in B Z = Examples. – ∆ -separable symmetric Frobenius algebras in BVect = ∆ -separable symmetric Frobenius C -algebras � ⇒ Z ss A = ( Z triv ) A (“ State sum models are orbifolds of the trivial TQFT. ”) = G – A G G -action in B Z is 2-functor ρ : B G − → B Z . Davydov/Kong/Runkel 2011
Algebraic characterisation of orbifolds Theorem. 2d defect TQFT Z = ⇒ pivotal 2-category B Z Lemma. � ∼ � � � orbifold data for Z ∆ -separable symmetric Frobenius algebras in B Z = Examples. – ∆ -separable symmetric Frobenius algebras in BVect = ∆ -separable symmetric Frobenius C -algebras � ⇒ Z ss A = ( Z triv ) A (“ State sum models are orbifolds of the trivial TQFT. ”) = G – A G G -action in B Z is 2-functor ρ : B G − → B Z . Lemma. A G := � g ∈ G ρ ( g ) is ∆ -separable Frobenius algebra in B Z . Davydov/Kong/Runkel 2011, Fr¨ ohlich/Fuchs/Runkel/Schweigert 2009, Brunner/Carqueville/Plencner 2014
Algebraic characterisation of orbifolds Theorem. 2d defect TQFT Z = ⇒ pivotal 2-category B Z Lemma. � ∼ � � � orbifold data for Z ∆ -separable symmetric Frobenius algebras in B Z = Examples. – ∆ -separable symmetric Frobenius algebras in BVect = ∆ -separable symmetric Frobenius C -algebras � ⇒ Z ss A = ( Z triv ) A (“ State sum models are orbifolds of the trivial TQFT. ”) = G – A G G -action in B Z is 2-functor ρ : B G − → B Z . Lemma. A G := � g ∈ G ρ ( g ) is ∆ -separable Frobenius algebra in B Z . Z G = Z A G Lemma. G -orbifolds are orbifolds: � Davydov/Kong/Runkel 2011, Fr¨ ohlich/Fuchs/Runkel/Schweigert 2009, Brunner/Carqueville/Plencner 2014
Algebraic characterisation of orbifolds Theorem. 2d defect TQFT Z = ⇒ pivotal 2-category B Z Lemma. � ∼ � � � orbifold data for Z ∆ -separable symmetric Frobenius algebras in B Z = Examples. – ∆ -separable symmetric Frobenius algebras in BVect = ∆ -separable symmetric Frobenius C -algebras � ⇒ Z ss A = ( Z triv ) A (“ State sum models are orbifolds of the trivial TQFT. ”) = G – A G G -action in B Z is 2-functor ρ : B G − → B Z . Lemma. A G := � g ∈ G ρ ( g ) is ∆ -separable Frobenius algebra in B Z . Z G = Z A G Lemma. G -orbifolds are orbifolds: � Orbifolds unify gauging of symmetry groups and state sum models. Davydov/Kong/Runkel 2011, Fr¨ ohlich/Fuchs/Runkel/Schweigert 2009, Brunner/Carqueville/Plencner 2014
Orbifold equivalence: main idea Let X : α − → β be line defect such that � = 0 in correlators. α X β Carqueville/Runkel 2012
Orbifold equivalence: main idea Let X : α − → β be line defect such that � = 0 in correlators. α X β Then with A := X † ◦ X : α − → α we have: Carqueville/Runkel 2012
Orbifold equivalence: main idea Let X : α − → β be line defect such that � = 0 in correlators. α X β Then with A := X † ◦ X : α − → α we have: � � β Carqueville/Runkel 2012
Orbifold equivalence: main idea Let X : α − → β be line defect such that � = 0 in correlators. α X β Then with A := X † ◦ X : α − → α we have: � � � � α X ∼ β β Carqueville/Runkel 2012
Orbifold equivalence: main idea Let X : α − → β be line defect such that � = 0 in correlators. α X β Then with A := X † ◦ X : α − → α we have: � � � � α X ∼ β β � � = Carqueville/Runkel 2012
Orbifold equivalence: main idea Let X : α − → β be line defect such that � = 0 in correlators. α X β Then with A := X † ◦ X : α − → α we have: � � � � α X ∼ β β � � � � = = A α Carqueville/Runkel 2012
Orbifold equivalence: main idea Let X : α − → β be line defect such that � = 0 in correlators. α X β Then with A := X † ◦ X : α − → α we have: � � � � α X ∼ β β � � � � = = A α Theorem. ( orbifold equivalence α ∼ β ) � ∼ � � � theory β A -orbifold of theory α = Carqueville/Runkel 2012
Orbifolds of Landau-Ginzburg models Theorem. There is a pivotal 2-category LG with: – objects = potentials W ∈ C [ x 1 , . . . , x n ] Eisenbud 1980, Carqueville/Murfet 2012
Orbifolds of Landau-Ginzburg models Theorem. There is a pivotal 2-category LG with: – objects = potentials W ∈ C [ x 1 , . . . , x n ] Examples. W A n − 1 = x n 1 + x 2 W D n +1 = x n 1 + x 1 x 2 W E 7 = x 3 1 + x 1 x 3 2 , 2 , 2 Eisenbud 1980, Carqueville/Murfet 2012
Orbifolds of Landau-Ginzburg models Theorem. There is a pivotal 2-category LG with: – objects = potentials W ∈ C [ x 1 , . . . , x n ] Eisenbud 1980, Carqueville/Murfet 2012
Orbifolds of Landau-Ginzburg models Theorem. There is a pivotal 2-category LG with: – objects = potentials W ∈ C [ x 1 , . . . , x n ] – LG ( W, V ) = homotopy category of matrix factorisations D of V − W Eisenbud 1980, Carqueville/Murfet 2012
Orbifolds of Landau-Ginzburg models Theorem. There is a pivotal 2-category LG with: – objects = potentials W ∈ C [ x 1 , . . . , x n ] – LG ( W, V ) = homotopy category of matrix factorisations D of V − W � 0 0 x y � � 0 u n − i y 2 − x for x 3 + xy 3 0 0 � for u n , Examples. D = D = x 2 u i 0 0 0 xy xy 2 − x 2 0 0 Eisenbud 1980, Carqueville/Murfet 2012
Orbifolds of Landau-Ginzburg models Theorem. There is a pivotal 2-category LG with: – objects = potentials W ∈ C [ x 1 , . . . , x n ] – LG ( W, V ) = homotopy category of matrix factorisations D of V − W Eisenbud 1980, Carqueville/Murfet 2012
Orbifolds of Landau-Ginzburg models Theorem. There is a pivotal 2-category LG with: – objects = potentials W ∈ C [ x 1 , . . . , x n ] – LG ( W, V ) = homotopy category of matrix factorisations D of V − W � � � � �� � � str i ∂ x i D j ∂ z j D d x – = Res for D : W − → V W ∂ x 1 W . . . ∂ x n W D V Eisenbud 1980, Carqueville/Murfet 2012
Orbifolds of Landau-Ginzburg models Theorem. There is a pivotal 2-category LG with: – objects = potentials W ∈ C [ x 1 , . . . , x n ] – LG ( W, V ) = homotopy category of matrix factorisations D of V − W � � � � �� � � str i ∂ x i D j ∂ z j D d x – = Res for D : W − → V W ∂ x 1 W . . . ∂ x n W D V Theorem. ( Orbifold equivalences in LG ) x k + xy 2 u 2 k + v 2 � � ∼ D k +1 ∼ A 2 k − 1 x 3 + y 4 u 12 + v 2 � � ∼ E 6 ∼ A 11 x 3 + xy 3 u 18 + v 2 � � ∼ E 7 ∼ A 17 x 3 + y 5 u 30 + v 2 � � ∼ E 8 ∼ A 29 Carqueville/Murfet 2012, Carqueville/Runkel 2012, Carqueville/Ros Camacho/Runkel 2013
Orbifolds of Landau-Ginzburg models Theorem. There is a pivotal 2-category LG with: – objects = potentials W ∈ C [ x 1 , . . . , x n ] – LG ( W, V ) = homotopy category of matrix factorisations D of V − W � � � � �� � � str i ∂ x i D j ∂ z j D d x – = Res for D : W − → V W ∂ x 1 W . . . ∂ x n W D V Theorem. ( Orbifold equivalences in LG ) x k + xy 2 u 2 k + v 2 � � ∼ D k +1 ∼ A 2 k − 1 x 3 + y 4 u 12 + v 2 � � ∼ E 6 ∼ A 11 x 3 + xy 3 u 18 + v 2 � � ∼ E 7 ∼ A 17 x 3 + y 5 u 30 + v 2 � � ∼ E 8 ∼ A 29 x 5 y + y 3 u 3 v + v 5 � � ∼ E 13 ∼ Z 11 x 6 + xy 3 + z 2 vw 3 + v 3 + u 2 w � � ∼ Z 13 ∼ Q 11 Carqueville/Murfet 2012, Carqueville/Runkel 2012, Carqueville/Ros Camacho/Runkel 2013, Recknagel/Weinreb 2017
Orbifold equivalence: application (simple) (complicated) Theorem. A 11 ∼ E 6 etc. Carqueville/Runkel 2012, Carqueville/Ros Camacho/Runkel 2013, Recknagel/Weinreb 2017
Orbifold equivalence: application (simple) (complicated) Theorem. A 11 ∼ E 6 etc. � � u 12 + v 2 = 0 � x 3 + y 4 = 0 � A 11 : E 6 : � = Carqueville/Runkel 2012, Carqueville/Ros Camacho/Runkel 2013, Recknagel/Weinreb 2017
Orbifold equivalence: application (simple) (complicated) Theorem. A 11 ∼ E 6 etc. � � u 12 + v 2 = 0 � x 3 + y 4 = 0 � A 11 : E 6 : ≇ Carqueville/Runkel 2012, Carqueville/Ros Camacho/Runkel 2013, Recknagel/Weinreb 2017
Orbifold equivalence: application (simple) (complicated) Theorem. A 11 ∼ E 6 etc. � � u 12 + v 2 = 0 � x 3 + y 4 = 0 � A 11 : E 6 : ∼ Carqueville/Runkel 2012, Carqueville/Ros Camacho/Runkel 2013, Recknagel/Weinreb 2017
Aside: Non-semisimple fully extended TQFTs
Aside: Non-semisimple fully extended TQFTs Theorem. For every potential W , the associated Landau-Ginzburg model Bord 2 , 1 − → Vect can be lifted to a fully extended TQFT − → LG Bord 2 , 1 , 0 pt + �− → W S 1 �− → C [ x 1 , . . . , x n ] / ( ∂ x 1 W, . . . , ∂ x n W ) Carqueville/Montiel Montoya 2018
Aside: Non-semisimple fully extended TQFTs Theorem. For every potential W , the associated Landau-Ginzburg model Bord 2 , 1 − → Vect can be lifted to a fully extended TQFT − → LG Bord 2 , 1 , 0 pt + �− → W S 1 �− → C [ x 1 , . . . , x n ] / ( ∂ x 1 W, . . . , ∂ x n W ) Remarks. – Jacobi algebra C [ x 1 , . . . , x n ] / ( ∂ x 1 W, . . . , ∂ x n W ) is non-semisimple . Carqueville/Montiel Montoya 2018
Aside: Non-semisimple fully extended TQFTs Theorem. For every potential W , the associated Landau-Ginzburg model Bord 2 , 1 − → Vect can be lifted to a fully extended TQFT − → LG Bord 2 , 1 , 0 pt + �− → W S 1 �− → C [ x 1 , . . . , x n ] / ( ∂ x 1 W, . . . , ∂ x n W ) Remarks. – Jacobi algebra C [ x 1 , . . . , x n ] / ( ∂ x 1 W, . . . , ∂ x n W ) is non-semisimple . – Need SO(2) -homotopy fixed points for fully extended oriented TQFTs. Carqueville/Montiel Montoya 2018
Aside: Non-semisimple fully extended TQFTs Theorem. For every potential W , the associated Landau-Ginzburg model Bord 2 , 1 − → Vect can be lifted to a fully extended TQFT − → LG Bord 2 , 1 , 0 pt + �− → W S 1 �− → C [ x 1 , . . . , x n ] / ( ∂ x 1 W, . . . , ∂ x n W ) Remarks. – Jacobi algebra C [ x 1 , . . . , x n ] / ( ∂ x 1 W, . . . , ∂ x n W ) is non-semisimple . – Need SO(2) -homotopy fixed points for fully extended oriented TQFTs. For Q -graded LG models, get constraint on central charge c ( W ) = 3 � i (1 − | x i | ) . Carqueville/Montiel Montoya 2018
Summary so far A 11 E 6 ∼
Summary so far A 11 E 6 S 11 W 13 ∼ ∼
Summary so far A 11 E 6 S 11 W 13 ∼ ∼ Z G = Z A G Z ss A = ( Z triv ) A
Summary so far A 11 E 6 S 11 W 13 ∼ ∼ Z G = Z A G Z ss A = ( Z triv ) A 2d orbifolds – encode triangulation invariance in algebraic structure – involve representation theory of algebras in 2-categories – unify gauging of symmetry groups and state sum models – uncover new dualities
The orbifold construction can be generalised to n -dimensional defect TQFTs Z : Bord def n ( D ) − → Vect in any dimension n � 1 . Carqueville/Meusburger/Schaumann 2016, Carqueville/Runkel/Schaumann 2017–2018
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