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 : cobn,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 : cob3,2,1 → 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 νFS = +1/ − 1/0. Non-deg. invariant bilinear form on V either symmetric

• r 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 νFS = +1/ − 1/0. Non-deg. invariant bilinear form on V either symmetric

• r antisymmetric.

Generalization for pivotal categories: V ∈ C and X ∈ Z(C): [Kashina, Sommerh¨ auser, Zhu; Ng, Schauenburg] Generalized Frobenius Schur indicator: νV ,X,(n,l) := trξV ,X,(n,l). Equivariance under SL(2, Z). Congruence subgroup conjecture for Drinfeld doubles of fusion categories FS indicators for big finite groups (∼ 2 ∙ 1018 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∨

i

∼ = Ui. Candidate for sum over all states:

• i∈I

Ui ⊠ Ui 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∨

i

∼ = Ui. Candidate for sum over all states:

• i∈I

Ui ⊠ Ui 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 – Indecomposable matrix 1 1 1

• squares to unit matrix, hence

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 f

→Y

(Y ∨ ⊗ X)f ⇒

• X∈C

X ∨ ⊗ X → X∈C X ∨ ⊗ X → 0 “Direct sum over all objects, with all morphisms taken into account.” The components of the two maps are for X

f

→ Y (Y ∨ ⊗ X)f

f ∨⊗idX

− − − − − → X ∨ ⊗ X and (Y ∨ ⊗ X)f

idY ∨ ⊗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 f

→Y

(Y ∨ ⊗ X)f ⇒

• X∈C

X ∨ ⊗ X → X∈C X ∨ ⊗ X → 0 “Direct sum over all objects, with all morphisms taken into account.” The components of the two maps are for X

f

→ Y (Y ∨ ⊗ X)f

f ∨⊗idX

− − − − − → X ∨ ⊗ X and (Y ∨ ⊗ X)f

idY ∨ ⊗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∈vectk v ⊗ v ∗ = k and Nat(F, G) =

• c∈C

HomD(F(c), G(c))

Some properties of coends

Remarks Dual notion: end (reverse all arrows) Examples of ends and coends v∈vectk v ⊗ v ∗ = k and Nat(F, G) =

• c∈C

HomD(F(c), G(c)) Peter-Weyl theorem [FSS, 2017]: as A-bimodules

• m∈A-mod

m ⊗k m∗ = A and m∈A-mod m ⊗k m∗ = A∗

Some properties of coends

Remarks Dual notion: end (reverse all arrows) Examples of ends and coends v∈vectk v ⊗ v ∗ = k and Nat(F, G) =

• c∈C

HomD(F(c), G(c)) Peter-Weyl theorem [FSS, 2017]: as A-bimodules

• m∈A-mod

m ⊗k m∗ = A and m∈A-mod m ⊗k m∗ = A∗ Co-Yoneda lemma: G : D → C linear, then Y ∈D G(y) ⊗ HomD(y, u) ∼ = G(u)

Some properties of coends

Remarks Dual notion: end (reverse all arrows) Examples of ends and coends v∈vectk v ⊗ v ∗ = k and Nat(F, G) =

• c∈C

HomD(F(c), G(c)) Peter-Weyl theorem [FSS, 2017]: as A-bimodules

• m∈A-mod

m ⊗k m∗ = A and m∈A-mod m ⊗k m∗ = A∗ Co-Yoneda lemma: G : D → C linear, then Y ∈D G(y) ⊗ HomD(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 +

• riented cell decomposition.

3-cells: Turaev-Viro theory Assign finite tensor category 2-cells: surface defect: Assign an Ai-Af -bimodule category

Module categories

Monoidal categories are “categorifcations” of rings. Definition (Module categories) Let A be a linear monoidal category.

1

A left A-module category is a linear category M with a bilinear functor ⊗ : A × M → M

and natural isomorphisms (mixed associativity, unitality) satisfying obvious pentagon and triangle axioms. We write a.m := a ⊗ m

2

Right module categories and bimodule categories defined analogously.

3

Module functors, module natural transformations defined in obvious way. 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)

Balnacings and braidings

Definition Let A be a monoidal category, B an A-bimodule and b ∈ B. A balancing for b is a natural family (σa : a.b → b.a)a∈A such that (a ⊗ a′).b → b.(a ⊗ a′) ց ր a.b.a′

Balnacings and braidings

Definition Let A be a monoidal category, B an A-bimodule and b ∈ B. A balancing for b is a natural family (σa : a.b → b.a)a∈A such that (a ⊗ a′).b → b.(a ⊗ a′) ց ր a.b.a′ Facts

1

For AAA get the Drinfeld center.

2

If A has left duals, then σa is an isomorphism.

3

Convenient description: monad on B: ZA : B → B b → a∈A a∨.b.a

Framing

Important point: For a non-semisimple Hopf algebra, S2 = idH. We did not choose a pivotal structure ?∨∨

∼

→⊗ idC (and do not even require its existence)

Framing

Important point: For a non-semisimple Hopf algebra, S2 = idH. We did not choose a pivotal structure ?∨∨

∼

→⊗ idC (and do not even require its existence) Rigidify situation:

• 2-Framing on 2-manifolds, i.e.

nowhere vanishing vector field parallel to defect lines and boundaries

(Σ, δ, χ) =

• Induce winding indices on segments
• f boundary circles

indχ

• = 0 =

+ +

indχ

• ,

− −

indχ

• = −1 ,

+ −

indχ

• = 1 .

− +

Categories for 1-manifolds

S = Decoration: 1-cells: finite tensor categories 0-cells: bimodule categories Framings

Categories for 1-manifolds

S = Decoration: 1-cells: finite tensor categories 0-cells: bimodule categories Framings κ-framed center C(S) := Bǫ1

1 κ1

⊠Bǫ2

2 κ2

⊠Bǫ3

3 κ3

⊠ ∙ ∙ ∙

κn−1

⊠Bǫn

n κn

⊠ with twisted balancings = coherent isomorphisms mi.a ⊠ mi+1 ∼ = mi ⊠ [κi ]a.mi+1 and m1 ⊠ ∙ ∙ ∙ ⊠ mn.a ∼ =

[κn]a.m1 ⊠ ∙ ∙ ∙ ⊠ mn .

Categories for 1-manifolds

S = Decoration: 1-cells: finite tensor categories 0-cells: bimodule categories Framings κ-framed center C(S) := Bǫ1

1 κ1

⊠Bǫ2

2 κ2

⊠Bǫ3

3 κ3

⊠ ∙ ∙ ∙

κn−1

⊠Bǫn

n κn

⊠ with twisted balancings = coherent isomorphisms mi.a ⊠ mi+1 ∼ = mi ⊠ [κi ]a.mi+1 and m1 ⊠ ∙ ∙ ∙ ⊠ mn.a ∼ =

[κn]a.m1 ⊠ ∙ ∙ ∙ ⊠ mn .

Remarks

1

Forgetful functor U : C(S) → Bǫ1

1 ⊠Bǫ2 2 ⊠Bǫ3 3 ⊠ ∙ ∙ ∙ ⊠ Bǫn n

2

The notion contains a category valued trace B

2

⊠

• f a bimodule category and the relative Deligne

product (N

2

⊠M) ∼ = N ⊠A M

Comparison for Dijkgraaf-Witten theories

Dijkgraaf-Witten theory based on a finite group G and 3-cocycle ω ∈ Z 3(G, C×) Facts Correspond to Turaev-Viro theories based on fusion category (G−vect)ω. (Indecomposable) bimodule categories are classified [O] by H ≤ G × G and 2-cochain θ such that dθ = p∗

1 ω ∙ (p∗ 2 ω)−1.

S = tftDW (S) = [G\\G × G//H, vect]τ(ω,θ) = AGdiag-AH,θ-bimod(G-vectω ⊗ G-vectω) which is one realization of the category-valued trace.

4

Functors for surfaces

Functors for surfaces: preblocks

Extended TFT: surface with boundaries left exact k-linear functor. Definition A modular functor is a symmetric monoidal 2-functor T : Bordfr,def,dec

2

− → Lex that is compatible with factorization, actions of mapping class groups and transparency.

Functors for surfaces: preblocks

Extended TFT: surface with boundaries left exact k-linear functor. Definition A modular functor is a symmetric monoidal 2-functor T : Bordfr,def,dec

2

− → Lex that is compatible with factorization, actions of mapping class groups and transparency. Preblocks are left exact functors Tpre(Σ) : T(∂Σ) → vect with Tpre(Σ)(−) := m1,m2,...,mn Hom(m1 ⊠ m1 ⊠ ∙ ∙ ∙ ⊠ mn ⊠ mn, U(−)) . (Note that every edge appears with inital and final point, so that the coend can be taken.)

Holonomy operation

TV generalizes theories of flat connections → 2-cell implies trivial holonomy

m∈M,n∈N Hom(m ⊠ n ⊠ n ⊠ m, xm ⊠ a.xn ⊠ yn ⊠ ym)

∼ =

− − → Hom(

• m∈M,n∈N m ⊠ n ⊠ n ⊠ m, xm ⊠ a.xn ⊠ yn ⊠ ym)

∼ =

− − → Hom(

• m∈M,n∈N m ⊠ a∨.n ⊠ n ⊠ m, xm ⊠ xn ⊠ yn ⊠ ym)

∼ =

− − → Hom(

• m∈M,n∈N m ⊠ n ⊠ a.n ⊠ m, xm ⊠ xn ⊠ yn ⊠ ym)

∼ =

− − → Hom(

• m∈M,n∈N m ⊠ n ⊠ n ⊠ m, xm ⊠ xn ⊠ a∨.yn ⊠ ym)

∼ =

− − → Hom(

• m∈M,n∈N m ⊠ n ⊠ n ⊠ m, xm ⊠ xn ⊠ yn ⊠ a∨∨.ym)

∼ =

− − → Hom(

• m∈M,n∈N m ⊠ n ⊠ n ⊠ a∨∨∨.m, xm ⊠ xn ⊠ yn ⊠ ym)

∼ =

− − → Hom(

• m∈M,n∈N a∨∨.m ⊠ n ⊠ n ⊠ m, xm ⊠ xn ⊠ yn ⊠ ym)

∼ =

− − → Hom(

• m∈M,n∈N m ⊠ n ⊠ n ⊠ m, a∨∨∨.xm ⊠ xn ⊠ yn ⊠ ym)

AN AM

Holonomy operation

TV generalizes theories of flat connections → 2-cell implies trivial holonomy

m∈M,n∈N Hom(m ⊠ n ⊠ n ⊠ m, xm ⊠ a.xn ⊠ yn ⊠ ym)

∼ =

− − → Hom(

• m∈M,n∈N m ⊠ n ⊠ n ⊠ m, xm ⊠ a.xn ⊠ yn ⊠ ym)

∼ =

− − → Hom(

• m∈M,n∈N m ⊠ a∨.n ⊠ n ⊠ m, xm ⊠ xn ⊠ yn ⊠ ym)

∼ =

− − → Hom(

• m∈M,n∈N m ⊠ n ⊠ a.n ⊠ m, xm ⊠ xn ⊠ yn ⊠ ym)

∼ =

− − → Hom(

• m∈M,n∈N m ⊠ n ⊠ n ⊠ m, xm ⊠ xn ⊠ a∨.yn ⊠ ym)

∼ =

− − → Hom(

• m∈M,n∈N m ⊠ n ⊠ n ⊠ m, xm ⊠ xn ⊠ yn ⊠ a∨∨.ym)

∼ =

− − → Hom(

• m∈M,n∈N m ⊠ n ⊠ n ⊠ a∨∨∨.m, xm ⊠ xn ⊠ yn ⊠ ym)

∼ =

− − → Hom(

• m∈M,n∈N a∨∨.m ⊠ n ⊠ n ⊠ m, xm ⊠ xn ⊠ yn ⊠ ym)

∼ =

− − → Hom(

• m∈M,n∈N m ⊠ n ⊠ n ⊠ m, a∨∨∨.xm ⊠ xn ⊠ yn ⊠ ym)

AN AM

Holonomy of a ∈ A is isomorphism hola,x : m∈M,n∈N Hom(m ⊠ n ⊠ n ⊠ m, xm ⊠ a.xn ⊠ y n ⊠ ym)

∼ =

− − → m∈M,n∈NHom(m ⊠ n ⊠ n ⊠ m, a∨∨∨.xm ⊠ xn ⊠ y n ⊠ ym)

Blocks

For any a ∈ A, consider the composite Tpre(... xm ⊠ xn ⊠ y)

coevl

∗

− − − − − → Tpre(... xm ⊠ (∨a ⊗ a) . xn ⊠ y)

hol

− − − →

∼ =

Tpre(... a∨∨. xm ⊠ a.xn ⊠ y) Dinatural in a, hence holx : Tpre(... xm ⊠ xn ⊠ y) − →

• a∈A

Tpre(... a∨∨. xm ⊠ a . xn ⊠ y) .

Blocks

For any a ∈ A, consider the composite Tpre(... xm ⊠ xn ⊠ y)

coevl

∗

− − − − − → Tpre(... xm ⊠ (∨a ⊗ a) . xn ⊠ y)

hol

− − − →

∼ =

Tpre(... a∨∨. xm ⊠ a.xn ⊠ y) Dinatural in a, hence holx : Tpre(... xm ⊠ xn ⊠ y) − →

• a∈A

Tpre(... a∨∨. xm ⊠ a . xn ⊠ y) . Definition Let Tpre

{v} be the pre-block functor associated with a labeled 2-framed defect

• surface. The block functor T{v} associated with the surface is the equalizer

T{v}(... xm ⊠ xn ⊠ y) − − → Tpre(... xm ⊠ xn ⊠ y)

holx

− − − − − − − − − − ⇒

(μx )∗

• x
• a∈A Tpre(... a∨∨. xm ⊠ a . xn ⊠ y) .

Results

Theorem (Fuchs, Schaumann, CS) This defines a modular functor. In particular, it obeys factorization

−κ

AM A

κ

AM A AM A

Iκ

AM A

Isomorphism z T(Σ)(z ⊠ z) ∼ = T(Σ′)

• f left exact functors.

Results

Theorem (Fuchs, Schaumann, CS) This defines a modular functor. In particular, it obeys factorization

−κ

AM A

κ

AM A AM A

Iκ

AM A

Isomorphism z T(Σ)(z ⊠ z) ∼ = T(Σ′)

• f left exact functors.

Fusion of defect lines

Results

Theorem (Fuchs, Schaumann, CS) This defines a modular functor. In particular, it obeys factorization

−κ

AM A

κ

AM A AM A

Iκ

AM A

Isomorphism z T(Σ)(z ⊠ z) ∼ = T(Σ′)

• f left exact functors.

Fusion of defect lines Result for discs using Eilenberg-Watts calculus: M ⊠ N ∼ = Lex(M, N) for x ∈ T(S1) = M

1

⊠ N ∼ = LexA(M, N) and y ∈ T(S2) = N

−1

⊠ M ∼ = T(S1)opp Preblocks Tpre(x ⊠ y) = Nat(φy, φx) Blocks T(x ⊠ y) = NatA(φy, φx) Generalizes Turaev-Viro

Application to the Generalized Frobenius-Schur indicator

Generalized Frobenius Schur indicator: νV ,X,(n,l) := trξV ,X,(n,l). Equivariance under SL(2, Z). To get objects both in A and in Z(A), consider a disc: Spherical fusion category A for surface Boundary labelled by A as a module category over itself n boundary insertion labeled by V ∈ A Bulk insertion X ∈ Z(A) Hom(V ⊗n, X) Solid torus with Wilson line νV ,X,(n,l)

Outlook: Towards a derived modular functor

Lyubashenko (∼ 1995): C modular, not necessarly semisimple Modular functor: to surface Σg,n associate left exact functor (Copp)⊠n → vect with v1 ⊠ . . . ⊠ vn → HomC(v1 ⊗ . . . ⊗ vn, L⊗g) Actions of the mapping class group Mapg,n on these functors.

Outlook: Towards a derived modular functor

Lyubashenko (∼ 1995): C modular, not necessarly semisimple Modular functor: to surface Σg,n associate left exact functor (Copp)⊠n → vect with v1 ⊠ . . . ⊠ vn → HomC(v1 ⊗ . . . ⊗ vn, L⊗g) Actions of the mapping class group Mapg,n on these functors. Fact: Hom is only left exact and can be derived. Derive HomC to obtain Extn

C

Theorem (Lentner, Mierach, CS, Sommerh¨ auser, 2018)

1

The mapping class group Mapg,n naturally acts on Extn

C(v1 ⊗ . . . vn, L⊗g).

2

In particular, the modular group SL(2, Z) acts on the Hochschild complex

• f a factorizable ribbon Hopf algebra.

Question: Physical role of derived conformal blocks?

Concluding comments

Extension to (classes of) 3-manifolds with corners.

(Cf. monoidal structure on transmission functor)

Existence of a surgery approach, generalizing Reshetikhin-Turaev? Applications to representation theory Applications to logarithmic conformal field theory.

Concluding comments

Extension to (classes of) 3-manifolds with corners.

(Cf. monoidal structure on transmission functor)

Existence of a surgery approach, generalizing Reshetikhin-Turaev? Applications to representation theory Applications to logarithmic conformal field theory. Ceterum censeo

Concluding comments

Extension to (classes of) 3-manifolds with corners.

(Cf. monoidal structure on transmission functor)

Existence of a surgery approach, generalizing Reshetikhin-Turaev? Applications to representation theory Applications to logarithmic conformal field theory. Ceterum censeo librum Fredenhaginis esse scribendum