Symmetries and defects in 3d TFT - uni-hamburg.de€¦ · Symmetries and defects in 3d TFT...
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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
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TFT construction of RCFT correlators [FRS 2001-05] :
[KS 2010]
conformal surface
right moversleft movers
surface defect
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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)
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Algebra
is Morita-equivalent to algebra
Boundary conditions form a category:
for all is the center of
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1.2. Dijkgraaf-Witten theories as 3d extended TFTs
Defn. [Atiyah]
Example: Dijkgraaf-Witten theories
M closed oriented 3-mfd.
groupoid cardinality
finite group
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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
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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
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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
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Evaluation of the functor tft:
a finitely ssi linear category
functor
natural transformation
modular tensor category (MTC)RT
Freed, ...
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2 step construction of DW theories
Drinfeld center
Example:
Center of an associative algebra:
equivariant vector bundles on
braided monoidal category
linearize
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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
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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
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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
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2.2 The crucial process
Move a bulk Wilson line to the boundary to get a boundary Wilson line
braided not braided
Functor:
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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
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"image of F is in the center"
coherently
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"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
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Naturality argument:
is an equivalence of braided categories
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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
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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:
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2.4 Other boundary conditions
Fusion of boundary Wilson lines
is a module category over
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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
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Fusion of boundary Wilson lines
is a module category over
Naturalityright module category
right module category
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Fusion of boundary Wilson lines
is a module category over
Theorem [S]
module categorybraided
Naturalityright module category
right module category
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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
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2.6 Surface defects
Two functors
combine
Central functor
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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
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2.7 A remark on the TFT construction of RCFT correlators
Folding
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2.7 A remark on the TFT construction of RCFT correlators
Folding
modular
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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
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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
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3.1. Recap
3. Symmetries of Dijkgraaf-Witten theories
Topological Lagragian from CS 2-gerbe on
Transgress to loop groupoid
twistedlinearization
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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
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3.2 Categories from 1-manifolds
Example: Interval
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3.2 Categories from 1-manifolds
Example: Interval
Data:
bulk Lagrangian
bdry Lagrangian
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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]
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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)
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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")
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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)
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3.4 Symmetries for abelian DW
Braided equivalence:
Subgroup:
quadratic form
with
abelianSpecial case:
Obvious symmetries:
1) Symmetries of
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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"
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Braided equivalence:
Subgroup:
3) Partial e-m dualities:
Example: A cyclic, fix
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Braided equivalence:
Subgroup:
3) Partial e-m dualities:
Example: A cyclic, fix
Theorem [FPSV]
These symmetries form a set of generatorsfor
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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