Topological field theories beyond the semisimplicity Christoph - PowerPoint PPT Presentation

Topological field theories beyond the semisimplicity Christoph Schweigert Mathematics Department Hamburg University based on work with J¨ urgen Fuchs and Gregor Schaumann June 6, 2018

Topological field theory on the way to Cortona

Topological field theory on the way to Cortona

Topological field theory on the way to Cortona Definition: n -dimensional TFT is a symmetric monoidal functor tft : cob n , n − 1 → vect Vector space of inital and final values; transition amplitudes Examples: Chern-Simons theories, Dijkgraaf-Witten theories. Reshetikhin-Turaev: 3d TFT ↔ modular tensor categories

Motivation Some applications of TFT: Holographic constructions of conformal field theories Representation theory ( � many flavours of TFT) Quantum computing (Low-dimensional) quantum gravity

Motivation Some applications of TFT: Holographic constructions of conformal field theories Representation theory ( � many flavours of TFT) Quantum computing (Low-dimensional) quantum gravity Constructions State sum constructions (Turaev-Viro, Kitaev, Dijkgraaf-Witten, Kuperberg) Surgery (Reshetikhin-Turaev, Hennings)

Motivation Some applications of TFT: Holographic constructions of conformal field theories Representation theory ( � many flavours of TFT) Quantum computing (Low-dimensional) quantum gravity Constructions State sum constructions (Turaev-Viro, Kitaev, Dijkgraaf-Witten, Kuperberg) Surgery (Reshetikhin-Turaev, Hennings) This talk Include defects (i.e. admit submanifolds with labels) – Symmetries and dualities implemented by defects – Holographic CFT constructions require boundaries or defects – Representation theory Work with non-semisimple structures: – Logarithmic CFT (e.g. critical percolation) – Representation theory – Conceptual clarity

Introduction: State sum construction Task Input data: Finite tensor category Not necessarily pivotal, not necessarily semisimple, but finiteness conditions.

Introduction: State sum construction Task Input data: Finite tensor category Not necessarily pivotal, not necessarily semisimple, but finiteness conditions. Construct: – Finite C -linear categories, giving labels for generalized Wilson lines – Vector spaces for surfaces with punctures – Representations of mapping class groups

Introduction: State sum construction Task Input data: Finite tensor category Not necessarily pivotal, not necessarily semisimple, but finiteness conditions. Construct: – Finite C -linear categories, giving labels for generalized Wilson lines – Vector spaces for surfaces with punctures – Representations of mapping class groups This is less than a 3d extended TFT: Definition (Extended topological field theory) A 3-2-1 extended oriented topological field theory is a symmetric monoidal 2-functor tft : cob 3 , 2 , 1 → 2- vect . Plan 2. How to “sum over all states”: coends 3. Categories for (generalized) Wilson lines 4. Functors for surfaces

Mathematical motivation for defects: generalized Frobenius Schur indicators Recap V a finite-dimensional irreducible C [ G ]-module. 3 cases: real / pseudoreal / complex with Frobenius-Schur indicator ν FS = +1 / − 1 / 0. Non-deg. invariant bilinear form on V either symmetric or antisymmetric.

Mathematical motivation for defects: generalized Frobenius Schur indicators Recap V a finite-dimensional irreducible C [ G ]-module. 3 cases: real / pseudoreal / complex with Frobenius-Schur indicator ν FS = +1 / − 1 / 0. Non-deg. invariant bilinear form on V either symmetric or antisymmetric. Generalization for pivotal categories: V ∈ C and X ∈ Z ( C ): [Kashina, Sommerh¨ auser, Zhu; Ng, Schauenburg] Generalized Frobenius Schur indicator: ν V , X , ( n , l ) := tr ξ V , X , ( n , l ) . Equivariance under SL (2 , Z ). Congruence subgroup conjecture for Drinfeld doubles of fusion categories FS indicators for big finite groups ( ∼ 2 ∙ 10 18 elements) Goal: Understand equivariance in terms of topological field theory!

2 Coends and sums over all states

Summing over “all” states - towards a mathematical notion Assumption: states are organized in terms of representations of an algebraic structure whose representation category is a finite tensor category C ∼ Fix a set I of representatives of irreducible representations with U ∨ = U i . i Candidate for sum over all states: � U i ⊠ U i i ∈ I Reasonable, whenever all representations are fully reducible.

Summing over “all” states - towards a mathematical notion Assumption: states are organized in terms of representations of an algebraic structure whose representation category is a finite tensor category C ∼ Fix a set I of representatives of irreducible representations with U ∨ = U i . i Candidate for sum over all states: � U i ⊠ U i i ∈ I Reasonable, whenever all representations are fully reducible. Toy example: algebraically closed field K of characteristic 2. – Because of +1 = − 1, only one (one-dimensional) irreducible representation S � 1 � 1 – Indecomposable matrix squares to unit matrix, hence 0 1 indecomposable two-dimensional projective representation P . Should we sum over S ⊠ S , P ⊠ P , S ⊠ P or all of them?

Coends Category theory gives a radical and clear answer: • Do not sum over irreps up to isomorphism. • Sum over all representations modulo all morphisms.

Coends Category theory gives a radical and clear answer: • Do not sum over irreps up to isomorphism. • Sum over all representations modulo all morphisms. Coend: � X ∈C ( Y ∨ ⊗ X ) f ⇒ X ∨ ⊗ X → X ∨ ⊗ X → 0 � � X ∈C X f → Y “Direct sum over all objects, with all morphisms taken into account.” f The components of the two maps are for X → Y ( Y ∨ ⊗ X ) f f ∨ ⊗ id X → X ∨ ⊗ X and ( Y ∨ ⊗ X ) f id Y ∨ ⊗ f → Y ∨ ⊗ Y − − − − − − − − − −

Coends Category theory gives a radical and clear answer: • Do not sum over irreps up to isomorphism. • Sum over all representations modulo all morphisms. Coend: � X ∈C ( Y ∨ ⊗ X ) f ⇒ X ∨ ⊗ X → X ∨ ⊗ X → 0 � � X ∈C X f → Y “Direct sum over all objects, with all morphisms taken into account.” f The components of the two maps are for X → Y ( Y ∨ ⊗ X ) f f ∨ ⊗ id X → X ∨ ⊗ X and ( Y ∨ ⊗ X ) f id Y ∨ ⊗ f → Y ∨ ⊗ Y − − − − − − − − − − Universal property. Coends are generalizations of direct sums.

Some properties of coends Remarks Dual notion: end (reverse all arrows) Examples of ends and coends � v ∈ vect k � v ⊗ v ∗ = k and Nat ( F , G ) = Hom D ( F ( c ) , G ( c )) c ∈ C

Some properties of coends Remarks Dual notion: end (reverse all arrows) Examples of ends and coends � v ∈ vect k � v ⊗ v ∗ = k and Nat ( F , G ) = Hom D ( F ( c ) , G ( c )) c ∈ C Peter-Weyl theorem [FSS, 2017]: as A -bimodules � m ∈ A - mod � m ⊗ k m ∗ = A m ⊗ k m ∗ = A ∗ and m ∈ A - mod

Some properties of coends Remarks Dual notion: end (reverse all arrows) Examples of ends and coends � v ∈ vect k � v ⊗ v ∗ = k and Nat ( F , G ) = Hom D ( F ( c ) , G ( c )) c ∈ C Peter-Weyl theorem [FSS, 2017]: as A -bimodules � m ∈ A - mod � m ⊗ k m ∗ = A m ⊗ k m ∗ = A ∗ and m ∈ A - mod Co-Yoneda lemma: G : D → C linear, then � Y ∈D G ( y ) ⊗ Hom D ( y , u ) ∼ = G ( u )

Some properties of coends Remarks Dual notion: end (reverse all arrows) Examples of ends and coends � v ∈ vect k � v ⊗ v ∗ = k and Nat ( F , G ) = Hom D ( F ( c ) , G ( c )) c ∈ C Peter-Weyl theorem [FSS, 2017]: as A -bimodules � m ∈ A - mod � m ⊗ k m ∗ = A m ⊗ k m ∗ = A ∗ and m ∈ A - mod Co-Yoneda lemma: G : D → C linear, then � Y ∈D G ( y ) ⊗ Hom D ( y , u ) ∼ = G ( u ) Fubini theorem: order of coends can be exchanged.

3 Categories for generalized Wilson lines

Construction of oriented TFTs with defects: labelling 2-cells / surface defects To understand the decoration data, start with a closed oriented 3-manifold + oriented cell decomposition. 3-cells: Turaev-Viro theory � Assign finite tensor category 2-cells: surface defect: Assign an A i - A f -bimodule category

Module categories Monoidal categories are “categorifcations” of rings. Definition (Module categories) Let A be a linear monoidal category. A left A -module category is a linear category M with a bilinear functor 1 ⊗ : A × M → M and natural isomorphisms (mixed associativity, unitality) satisfying obvious pentagon and triangle axioms. We write a . m := a ⊗ m Right module categories and bimodule categories defined analogously. 2 Module functors, module natural transformations defined in obvious way. 3 Example: A = H -mod and A an H -module algebra. Then A -mod is a A -module category. Definition (Finite module categories) Let A be a finite tensor category over k . A left A -module category is finite, if the underlying category is a finite abelian category over k and the action is k -linear in each variable and right exact in the first variable.

Generalized Wilson lines Several surface defects meet in generalized Wilson lines:

Generalized Wilson lines Several surface defects meet in generalized Wilson lines: A decorated 1-manifold S : Decoration: 1-cells: finite categories 0-cells: bimodule categories Question: which C -linear category to assign to S ? (Objects: labels for Wilson lines)