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Symmetries and defects in 3d TFT

Christoph Schweigert

Hamburg University, Department of Mathematics andCenter for Mathematical Physics

joint with Jürgen Fuchs, Jan Priel and Alessandro Valentino

Plan:

Surface defects in 3d TFTs of Reshetikhin-Turaev type- General theory- Obstructions in Witt groups- Application: bilayer systems

Dijkgraaf-Witten theories:- Defects from relative bundles- Symmetry groups as Brauer-Picard groups

Motivation:

1) TFT construction of RCFT correlators ("holographic description of RCFT")

2) Gapped interfaces of topological phases

3) Symmetries and dualities

TFT construction of RCFT correlators [FRS 2001-05] :

[KS 2010]

conformal surface

right moversleft movers

surface defect

1. Preliminaries

1.1 Open / closed 2d TFT

Frobenius algebra

Objects:

module

Idea: larger category of cobordisms: boundaries

boundary conditionsand

is an

(not necessarily commutative; assume semisimple)

Algebra

is Morita-equivalent to algebra

Boundary conditions form a category:

for all is the center of

1.2. Dijkgraaf-Witten theories as 3d extended TFTs

Defn. [Atiyah]

Example: Dijkgraaf-Witten theories

M closed oriented 3-mfd.

groupoid cardinality

finite group

1.2. Dijkgraaf-Witten theories as 3d extended TFTs

Defn. [Atiyah]

Example: Dijkgraaf-Witten theories

M closed oriented 3-mfd.

closed oriented 2-mfd; given

groupoid cardinality

finite group

function on

1.2. Dijkgraaf-Witten theories as 3d extended TFTs

Extended TFTs

Defn. [Atiyah]

Example: Dijkgraaf-Witten theories Cut surfaces along circles

Vector bundle over space of field configurations

category1-mfd

oriented 1-mfg. For any M closed oriented 3-mfd.

closed oriented 2-mfd; given

groupoid cardinality

finite group

function on

1.3 3d extended TFT of Reshetikhin-Turaev type

symmetric monoidal bifunctor

finitely semi-simple, linear abelian categories

objects 1-morphisms 2-morphisms:mfds. with corner

Wilson lines /ribbon graphs

Evaluation of the functor tft:

a finitely ssi linear category

functor

natural transformation

modular tensor category (MTC)RT

Freed, ...

2 step construction of DW theories

Drinfeld center

Example:

Center of an associative algebra:

equivariant vector bundles on

braided monoidal category

linearize

Drinfeld center2 step construction of DW theories

Drinfeld center

Example:

Center of an associative algebra:

equivariant vector bundles on

braided monoidal category

monoidal category

+ coherence properties

linearize

Drinfeld center2 step construction of DW theories

Drinfeld center

Example:

Center of an associative algebra:

equivariant vector bundles on

braided monoidal category

modular equivalence

monoidal category

braided:

Deligne product

Forgetful functor

braided monoidal category

+ coherence properties

linearize

2.1 The bicategory of boundary conditions

2. Boundary conditions and defects in 3d topological field theories of RT type

Given a MTC , find bicategory of boundary conditions

objects = boundary conditions

2-morphisms = insertions

1-morphisms = boundary Wilson lines

Poincaré dual:

fusion category, but not braided

Category of boundary Wilson lines

linear, finitely semisimple, rigid, monoidal

2.2 The crucial process

Move a bulk Wilson line to the boundary to get a boundary Wilson line

braided not braided

Functor:

2.2 The crucial process

Move a bulk Wilson line to the boundary to get a boundary Wilson line

braided not braided

Functor:

coherentlyis a monoidal functor

"image of F is in the center"

coherently

"image of F is in the center"

coherently

Defn. [B]

Structure of a central functor on the monoidal functor F is a lift

with braided functor

Naturality argument:

is an equivalence of braided categories

Witt group of MTC are equivalence classes

2.3 Definition [DMNO]

Naturality argument:

is an equivalence of braided categories

Witt group of metric abelian groups

Two MTC are Witt equivalent, if there are fusion

categories and a braided equivalence

Witt group of MTC are equivalence classes

2.3 Definition [DMNO]

Naturality argument:

is an equivalence of braided categories

Witt group of metric abelian groups

Two MTC are Witt equivalent, if there are fusion

categories and a braided equivalence

Not every MTC is a center - unlike commutative rings

A boundary condition is a Witt trivialization

Obstruction to boundary conditions:

2.4 Other boundary conditions

Fusion of boundary Wilson lines

is a module category over

2.4 Other boundary conditions

Module categories

Bicategory

Fusion of boundary Wilson lines

is a module category over

For a fixed monoidal category

Module categories overModule functorsModule natural transformations

Bifunctor

(+ mixed associativity constraints + coherence conditions)

a monoidal category, a category

Fusion of boundary Wilson lines

is a module category over

Naturalityright module category

right module category

Fusion of boundary Wilson lines

is a module category over

Theorem [S]

module categorybraided

Naturalityright module category

right module category

2.5 Summary

Find one boundary condition

Wilson lines

fusion

Warning:

Bicategory of boundary conditions is

Example: ("Kitaev's toric code")

modules, but only 2 modules

Thus modules are - modules,

but not all -modules are boundary conditions

2.6 Surface defects

Two functors

combine

Central functor

2.6 Surface defects

Two functors

combine

Obstruction in Witt group for existence of defects

Central functor

Thus itself describes a surface defect: the transparent defect

Recall: modular

2.7 A remark on the TFT construction of RCFT correlators

Folding

2.7 A remark on the TFT construction of RCFT correlators

Folding

modular

2.8 Application: Quantum codes from twist defects

Problems:

Representation of braid group gives quantum gates

surface toric codequantum code

low genus of small codes

simple systems no universal gates

2.8 Application: Quantum codes from twist defects

(cf. permutation orbifolds)

Idea: Bilayer systems and twist defects create branch cuts

Problems:

Representation of braid group gives quantum gates

surface toric codequantum code

low genus of small codes

simple systems no universal gates

3.1. Recap

3. Symmetries of Dijkgraaf-Witten theories

Topological Lagragian from CS 2-gerbe on

Transgress to loop groupoid

twistedlinearization

3.2 Include defects3.1. Recap

3. Symmetries of Dijkgraaf-Witten theories

Topological Lagragian from CS 2-gerbe on

Transgress to loop groupoid

twistedlinearization

Idea: relative bundles

Given relative mfd

and groups

3.2 Categories from 1-manifolds

Example: Interval

3.2 Categories from 1-manifolds

Example: Interval

Data:

bulk Lagrangian

bdry Lagrangian

3.2 Categories from 1-manifolds

Example: Interval

Data:

bulk Lagrangian

bdry Lagrangian

Transgress to 2-cocycle on

(for twisted linearization)

Check:

Module category over

cf. [O]

3.3 Symmetries from defects

General insight:

Idea:

(2d: [FFRS '04])

Symmetries invertible topological defects

"contourdeformation"

Important:

Natural action of symmetries on otherfield theoretic data (boundary conditions, defects, fields)

3.3 Symmetries from defects

General insight:

Idea:

(2d: [FFRS '04])

Symmetries invertible topological defects

"contourdeformation"

Important:

Natural action of symmetries on otherfield theoretic data (boundary conditions, defects, fields)

For 3d TFT:

Brauer-Picard group

Symmetries for with

are invertible bimodule categories

Bicategory ("categorical 2-group")

3.3 Symmetries from defects

General insight:

Idea:

(2d: [FFRS '04])

Symmetries invertible topological defects

"contourdeformation"

Important:

Natural action of symmetries on otherfield theoretic data (boundary conditions, defects, fields)

For 3d TFT:

Brauer-Picard group

"Symmetries can be detected from actionon bulk Wilson lines"

Symmetries for with

are invertible bimodule categories

Bicategory ("categorical 2-group")

Braided equivalence, if D invertible

Transmission functor

(explicitly computable for DW theories)

3.4 Symmetries for abelian DW

Braided equivalence:

Subgroup:

quadratic form

with

abelianSpecial case:

Obvious symmetries:

1) Symmetries of

3.4 Symmetries for abelian DW

Braided equivalence:

Braided equivalence:

Subgroup:

Subgroup:

(transgression)

quadratic form

with

abelianSpecial case:

Obvious symmetries:

2) Automorphisms of CS 2-gerbe

1) Symmetries of

1-gerbe on "B-field"

Braided equivalence:

Subgroup:

3) Partial e-m dualities:

Example: A cyclic, fix

Braided equivalence:

Subgroup:

3) Partial e-m dualities:

Example: A cyclic, fix

Theorem [FPSV]

These symmetries form a set of generatorsfor

4. Conclusions

Surface defects in 3d TFTs of Reshetikhin-Turaev type

- Obstructions in Witt groups of MTC

- TFT approach to RCFT

- Quantum codes

Dijkgraaf-Witten theories:

- Defects from relative bundles

- Symmetry groups are Brauer-Picard groups