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
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