Quantum gravity and TQFTs with defects Marc Geiller Perimeter - - PowerPoint PPT Presentation

Quantum gravity and TQFTs with defects

Marc Geiller Perimeter Institute Quantum Gravity in Paris March 20th − 23rd 2017

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Introduction

Bottom-up approach to quantum gravity

⋆ Think of quantum gravity as a theory of quantum geometry ⋆ Start with basic building blocks: quanta or atoms of geometry ⋆ Condense the building blocks into spacetime

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Introduction

Bottom-up approach to quantum gravity

⋆ Think of quantum gravity as a theory of quantum geometry ⋆ Start with basic building blocks: quanta or atoms of geometry ⋆ Condense the building blocks into spacetime

1) Choose description of microscopic degrees of freedom

⋆ Gravity as a gauge theory

continuum discrete “position” 1 2 gauge field (connection) A 1 2 holonomies h(A) ∈ G “momentum” 1 2 electric field (metric) E 1 2 fluxes X(E) ∈ Lie(G) 2) Dynamics

⋆ Hamiltonian (canonical LQG) ⋆ Path integral (spin foams) ⋆ Group field theories

3) Existence and characterization of a continuum limit

⋆ GFT renormalization ⋆ Coarse-graining of spin foam models

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Introduction

Encouraging results

⋆ Physics of symmetry-reduced models (black holes, cosmology, . . . ) ⋆ (Tensorial) group field theories are well-defined field theories

(Ben Geloun, Benedetti, Bonzom, Carrozza, Freidel, Gurau, Oriti, Rivasseau, Ryan, . . . )

⋆ Spin foams have non-trivial phase diagrams

(Delcamp, Dittrich, Eckert, Kaminski, Martin-Benito, Mizera, Schnetter, Steinhaus, . . . ) (Delcamp, Dittrich, 2016)

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Introduction

Encouraging results

⋆ Physics of symmetry-reduced models (black holes, cosmology, . . . ) ⋆ (Tensorial) group field theories are well-defined field theories

(Ben Geloun, Benedetti, Bonzom, Carrozza, Freidel, Gurau, Oriti, Rivasseau, Ryan, . . . )

⋆ Spin foams have non-trivial phase diagrams

(Delcamp, Dittrich, Eckert, Kaminski, Martin-Benito, Mizera, Schnetter, Steinhaus, . . . ) (Delcamp, Dittrich, 2016) What could the phases correspond to?

⋆ Change viewpoint and view quantum gravity as a TQFT with defects ⋆ This will give a deeper understanding of vacua, phases, and representations

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Outline

1. Vacua and TQFTs with defects 2. Example 1: vanishing cosmological constant 3. Example 2: positive cosmological constant 4. Conclusion and perspectives

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Vacua and TQFTs with defects

1. Vacua and TQFTs with defects 2. Example 1: vanishing cosmological constant 3. Example 2: positive cosmological constant 4. Conclusion and perspectives

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Vacua and TQFTs with defects

Gravitational degrees of freedom

⋆ Extrinsic and intrinsic spatial geometry encoded in connection and “electric field”

A = Γ + γK E = e det(e)

• A, E
• = 1

Lattice gauge theory variables

⋆ Holonomies of the connection

SU(2) ∋ hℓ = exp

• ℓ

A

• ℓ

⋆ Fluxes of the electric field

su(2) ∋ XS =

• S

h−1Eh S

⋆ These form the holonomy-flux algebra

• hℓ, XS
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Vacua and TQFTs with defects

What is a vacuum?

⋆ Typically, the state of lowest energy ⋆ No generic notion of energy in general relativity (let alone quantum gravity) ⋆ Can be the simplest state to write down ⋆ Other states are obtained from it by excitations (i.e. vacuum is a cyclic state) ⋆ Depends on the type of physics we are interested in (e.g. condensation, . . . )

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Vacua and TQFTs with defects

What is a vacuum?

⋆ Typically, the state of lowest energy ⋆ No generic notion of energy in general relativity (let alone quantum gravity) ⋆ Can be the simplest state to write down ⋆ Other states are obtained from it by excitations (i.e. vacuum is a cyclic state) ⋆ Depends on the type of physics we are interested in (e.g. condensation, . . . )

Example: Fock vacuum

⋆ Simplest state, i.e. state with no particles ⋆ Invariant under Poincaré transformations ⋆ Create particle states by exciting with creation operators ⋆ Fock Hilbert space generated from excitations of the cyclic vacuum ⋆ Fock Hilbert space carries a representation of the algebra of observables ⋆ Not a discretization, just a truncation written in a convenient discrete basis

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Vacua and TQFTs with defects

AL representation in LQG (Ashtekar, Isham, Lewandowski, 1992 – 1995)

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Vacua and TQFTs with defects

AL representation in LQG (Ashtekar, Isham, Lewandowski, 1992 – 1995)

⋆ Vacuum: state with no excitations (no graph and no geometry), 0|XS|0AL = 0, ∀ S

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Vacua and TQFTs with defects

AL representation in LQG (Ashtekar, Isham, Lewandowski, 1992 – 1995)

⋆ Vacuum: state with no excitations (no graph and no geometry), 0|XS|0AL = 0, ∀ S ⋆ Spatial geometry vanishes in the vacuum: totally squeezed and degenerate state

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Vacua and TQFTs with defects

AL representation in LQG (Ashtekar, Isham, Lewandowski, 1992 – 1995)

⋆ Vacuum: state with no excitations (no graph and no geometry), 0|XS|0AL = 0, ∀ S ⋆ Spatial geometry vanishes in the vacuum: totally squeezed and degenerate state ⋆ Creation operators: holonomies

ˆ h1ˆ h2ˆ h3 vacuum excited state ∈ HΓ ˆ h1ˆ h2ˆ h3 ⊲ = ∈ HΓ

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Vacua and TQFTs with defects

AL representation in LQG (Ashtekar, Isham, Lewandowski, 1992 – 1995)

⋆ Vacuum: state with no excitations (no graph and no geometry), 0|XS|0AL = 0, ∀ S ⋆ Spatial geometry vanishes in the vacuum: totally squeezed and degenerate state ⋆ Creation operators: holonomies

ˆ h1ˆ h2ˆ h3 vacuum excited state ∈ HΓ ˆ h1ˆ h2ˆ h3 ⊲ = ∈ HΓ

⋆ Discrete basis of excitations labelled by graphs and carrying quanta of geometry

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Vacua and TQFTs with defects

AL representation in LQG (Ashtekar, Isham, Lewandowski, 1992 – 1995)

⋆ Vacuum: state with no excitations (no graph and no geometry), 0|XS|0AL = 0, ∀ S ⋆ Spatial geometry vanishes in the vacuum: totally squeezed and degenerate state ⋆ Creation operators: holonomies

ˆ h1ˆ h2ˆ h3 vacuum excited state ∈ HΓ ˆ h1ˆ h2ˆ h3 ⊲ = ∈ HΓ

⋆ Discrete basis of excitations labelled by graphs and carrying quanta of geometry ⋆ Makes it difficult to construct semi-classical states: need condensates

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Vacua and TQFTs with defects

AL representation in LQG (Ashtekar, Isham, Lewandowski, 1992 – 1995)

⋆ Vacuum: state with no excitations (no graph and no geometry), 0|XS|0AL = 0, ∀ S ⋆ Spatial geometry vanishes in the vacuum: totally squeezed and degenerate state ⋆ Creation operators: holonomies

ˆ h1ˆ h2ˆ h3 vacuum excited state ∈ HΓ ˆ h1ˆ h2ˆ h3 ⊲ = ∈ HΓ

⋆ Discrete basis of excitations labelled by graphs and carrying quanta of geometry ⋆ Makes it difficult to construct semi-classical states: need condensates ⋆ Embedding of Hilbert spaces based on embedding of graphs ⋆ Kinematical continuum limit: H∞ = ∪ΓHΓ/ ∼ = L2

A/G, dµAL

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Vacua and TQFTs with defects

Is LQG discrete or continuous?

⋆ “Both”, since we work on finite graphs, but states live in H∞ ⋆ Field theory with arbitrary finite # of degrees of freedom = TQFT with defects

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Vacua and TQFTs with defects

Is LQG discrete or continuous?

⋆ “Both”, since we work on finite graphs, but states live in H∞ ⋆ Field theory with arbitrary finite # of degrees of freedom = TQFT with defects

Classical level

⋆ Theorem (Freidel, MG, Ziprick, 2013)

(h, X) ∈ spin network phase space ≃ T ∗(space of piecewise-flat connections)

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Vacua and TQFTs with defects

Is LQG discrete or continuous?

⋆ “Both”, since we work on finite graphs, but states live in H∞ ⋆ Field theory with arbitrary finite # of degrees of freedom = TQFT with defects

Classical level

⋆ Theorem (Freidel, MG, Ziprick, 2013)

(h, X) ∈ spin network phase space ≃ T ∗(space of piecewise-flat connections) Quantum theory

⋆ Choice of TQFT ↔ choice of vacuum for the quantum kinematics ⋆ Defects in the TQFT ↔ excitations on top of the vacuum

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Vacua and TQFTs with defects

TQFTs (Atiyah, Kontsevich, Segal, Witten, . . . )

⋆ Field theories with zero degrees of freedom: no local excitations, only global properties ⋆ Their path integral computes topological invariants of manifolds ⋆ Triangulation independence: equivalent continuous / graphical / simplicial definitions ⋆ Conjectured classification of topological order in condensed matter (via categories)

(Etingof, Kitaev, Kong, Laughlin, Levin, Moore, Preskill, Read, Wang, Wen, Wilczek, . . . )

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Vacua and TQFTs with defects

TQFTs (Atiyah, Kontsevich, Segal, Witten, . . . )

⋆ Field theories with zero degrees of freedom: no local excitations, only global properties ⋆ Their path integral computes topological invariants of manifolds ⋆ Triangulation independence: equivalent continuous / graphical / simplicial definitions ⋆ Conjectured classification of topological order in condensed matter (via categories)

(Etingof, Kitaev, Kong, Laughlin, Levin, Moore, Preskill, Read, Wang, Wen, Wilczek, . . . )

⋆ Typical examples (Achucarro, Blau, Horowitz, Townsend, Witten)

• Chern–Simons theory L3 = A ∧ F [A] − 1

3 A ∧ A ∧ A

• d-dimensional BF theory (gravity in d = 3) L3 = B ∧ F [A] + λB ∧ B ∧ B

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Vacua and TQFTs with defects

TQFTs (Atiyah, Kontsevich, Segal, Witten, . . . )

⋆ Field theories with zero degrees of freedom: no local excitations, only global properties ⋆ Their path integral computes topological invariants of manifolds ⋆ Triangulation independence: equivalent continuous / graphical / simplicial definitions ⋆ Conjectured classification of topological order in condensed matter (via categories)

(Etingof, Kitaev, Kong, Laughlin, Levin, Moore, Preskill, Read, Wang, Wen, Wilczek, . . . )

⋆ Typical examples (Achucarro, Blau, Horowitz, Townsend, Witten)

• Chern–Simons theory L3 = A ∧ F [A] − 1

3 A ∧ A ∧ A

• d-dimensional BF theory (gravity in d = 3) L3 = B ∧ F [A] + λB ∧ B ∧ B

⋆ N.B.: non-topological theories

• Yang–Mills L4 = B ∧ F [A] + g2 ⋆ B ∧ B
• gravity L4 = B ∧ F [A] + φB ∧ B

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Vacua and TQFTs with defects

TQFTs (Atiyah, Kontsevich, Segal, Witten, . . . )

⋆ Field theories with zero degrees of freedom: no local excitations, only global properties ⋆ Their path integral computes topological invariants of manifolds ⋆ Triangulation independence: equivalent continuous / graphical / simplicial definitions ⋆ Conjectured classification of topological order in condensed matter (via categories)

(Etingof, Kitaev, Kong, Laughlin, Levin, Moore, Preskill, Read, Wang, Wen, Wilczek, . . . )

⋆ Typical examples (Achucarro, Blau, Horowitz, Townsend, Witten)

• Chern–Simons theory L3 = A ∧ F [A] − 1

3 A ∧ A ∧ A

• d-dimensional BF theory (gravity in d = 3) L3 = B ∧ F [A] + λB ∧ B ∧ B

⋆ N.B.: non-topological theories

• Yang–Mills L4 = B ∧ F [A] + g2 ⋆ B ∧ B
• gravity L4 = B ∧ F [A] + φB ∧ B

Including defects of dim < d

⋆ Support local excitations (quasi-particles) ⋆ Simplest example: boundary degrees of freedom (Brown, Carlip, Henneaux, WZNW, . . . ) ⋆ (d − 1)-dimensional defects can separate phases ⋆ Finding the nature of the degrees of freedom supported by the defects is non-trivial:

classification of extended TQFTs (Fuchs, Lurie, Schweigert, Schommer-Pries, . . . )

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Vacua and TQFTs with defects

Which TQFT and which defects for quantum gravity? (Dittrich, MG)

⋆ AL representation: vacuum is E ≡ B|Σ = 0 and defects carry geometry

(∼ strong coupling limit (confining phase) of lattice gauge theory)

⋆ BF representation: vacuum is F [A] = 0 and defects carry curvature

(∼ zero coupling limit (deconfining phase) of lattice gauge theory)

⋆ TV representation: vacuum is F [A] = λ (B ∧ B) and defects carry curvature ⋆ Depends on dimension, signature, gauge group, sign(λ), . . . ⋆ Need a Hilbert space supporting arbitrarily many excitations (defects) and the observable

algebra generating them

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Vacua and TQFTs with defects

Which TQFT and which defects for quantum gravity? (Dittrich, MG)

⋆ AL representation: vacuum is E ≡ B|Σ = 0 and defects carry geometry

(∼ strong coupling limit (confining phase) of lattice gauge theory)

⋆ BF representation: vacuum is F [A] = 0 and defects carry curvature

(∼ zero coupling limit (deconfining phase) of lattice gauge theory)

⋆ TV representation: vacuum is F [A] = λ (B ∧ B) and defects carry curvature ⋆ Depends on dimension, signature, gauge group, sign(λ), . . . ⋆ Need a Hilbert space supporting arbitrarily many excitations (defects) and the observable

algebra generating them A priori obstruction (Fleischhack, Lewandowski, Okołów, Sahlmann, Thiemann, 2005)

⋆ Uniqueness of AL representation under technical assumptions, including

• Diffeomorphism invariance
• Weak continuity of exponentiated fluxes (i.e. the fluxes exist as operators)

⋆ Very different from QFT, where there exist inequivalent representations ⋆ Need to violate some assumptions in order to have a new representation

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Example 1: vanishing cosmological constant

1. Vacua and TQFTs with defects 2. Example 1: vanishing cosmological constant 3. Example 2: positive cosmological constant 4. Conclusion and perspectives

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Example 1: vanishing cosmological constant

Vacuum

⋆ Triangulation ∆ of d-dimensional spatial manifold Σ (here for d = 2) ⋆ Curvature encoded in cycle holonomies around (d − 2)-dimensional simplices (defects) ⋆ Moduli space of flat connections

A0 =

• A ∈ A
• F (A) = 0 on Σ\∆(d−2)
• /G = Hom
• π1
• Σ\∆(d−2)
• , G
• /G

⋆ Vacuum state ψ{gc} =

• c

δ(gc, 1)

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Example 1: vanishing cosmological constant

Vacuum

⋆ Triangulation ∆ of d-dimensional spatial manifold Σ (here for d = 2) ⋆ Curvature encoded in cycle holonomies around (d − 2)-dimensional simplices (defects) ⋆ Moduli space of flat connections

A0 =

• A ∈ A
• F (A) = 0 on Σ\∆(d−2)
• /G = Hom
• π1
• Σ\∆(d−2)
• , G
• /G

⋆ Vacuum state ψ{gc} =

• c

δ(gc, 1) Creation / excitation operator

⋆ Exponentiated fluxes (Wilson surface operators)

• eαXi

⊲ ψ{gc} = ψ(g1, . . . , giα, . . . , g|c|) exp

• gi{Xi, ·}
• ⊲ vacuum

excited state ∈ H∆ exp

• α{Xi, ·}
• ⊲

= ∈ H∆ exp

• gi{Xi, ·}
• ⊲ g = 1

g = α = 1 ∈ H∆

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Example 1: vanishing cosmological constant

Vacuum

⋆ Triangulation ∆ of d-dimensional spatial manifold Σ (here for d = 2) ⋆ Curvature encoded in cycle holonomies around (d − 2)-dimensional simplices (defects) ⋆ Moduli space of flat connections

A0 =

• A ∈ A
• F (A) = 0 on Σ\∆(d−2)
• /G = Hom
• π1
• Σ\∆(d−2)
• , G
• /G

⋆ Vacuum state ψ{gc} =

• c

δ(gc, 1) Creation / excitation operator

⋆ Exponentiated fluxes (Wilson surface operators)

• eαXi

⊲ ψ{gc} = ψ(g1, . . . , giα, . . . , g|c|) exp

• gi{Xi, ·}
• ⊲ vacuum

excited state ∈ H∆ exp

• α{Xi, ·}
• ⊲

= ∈ H∆ exp

• gi{Xi, ·}
• ⊲ g = 1

g = α = 1 ∈ H∆ Non-separable Hilbert space

⋆ Discrete Hilbert space topology to accommodate flat vacuum and inductive limit ⋆ Fluxes don’t exist as operators (polymer QM, loop cosmology, Bohr compactification) ⋆ Different spectra for e.g. translation or area operators

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Example 1: vanishing cosmological constant

20 40 60 80 100 20 40 60 80 100 120

⋆ Eigenvalue (y axis) of squared area operator as a function of spin j (x axis)

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Example 1: vanishing cosmological constant

AL BF TQFT state E = 0 F (A) = 0 vacuum state |∅ = nothing |∅ =

cycles δ(gc, 1)

holonomies exponentiated fluxes excitations hℓ ⊲ |∅ = hℓ Rα

i ⊲ |∅ = . . . δ(giα, 1) . . .

dual graphs codimension-1 simplices defects dual graphs codimension-2 simplices refinement j = 0 flatness measure Haar discrete H∞ ∪ΓHΓ/ ∼ ∪∆H∆/ ∼ generalization background Eo constant curvature (λ = 0)

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Example 2: positive cosmological constant

1. Vacua and TQFTs with defects 2. Example 1: vanishing cosmological constant 3. Example 2: positive cosmological constant 4. Conclusion and perspectives

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Example 2: positive cosmological constant

Difficulty

⋆ Three-dimensional quantum gravity with λ = 0 involves quantum groups (Hopf algebras) ⋆ For Euclidean signature and λ > 0 the TQFT is well-known (and involves SU(2)k)

(Reshetikhin, Turaev, Viro, Virelizier, Witten)

⋆ But the quantum group has no group representation!

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Example 2: positive cosmological constant

Difficulty

⋆ Three-dimensional quantum gravity with λ = 0 involves quantum groups (Hopf algebras) ⋆ For Euclidean signature and λ > 0 the TQFT is well-known (and involves SU(2)k)

(Reshetikhin, Turaev, Viro, Virelizier, Witten)

⋆ But the quantum group has no group representation!

We need to construct 1) A way of writing down and manipulating states (since there is no group picture) 2) A Hilbert space which contains vacuum and excited states 3) A creation / excitation operator

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Example 2: positive cosmological constant

1) Writing and manipulating states: graphical calculus

⋆ SU(2)k at root of unity is a modular braided fusion category C

• Category: irreps j ∈ {0, 1/2, . . . , k/2} with k =
• G

√ λ −1 and v 2

j := (−1)2jdj

• Fusion: there are fusion coefficients such that i ⊗ j =
• k

Nk

ijk

• Braided: there is an R-matrix such that

i j k

= Rij

k

j k i

• Modular: det(S) = 0 with Sij = 1

D

i j

and D2 :=

k/2

• j=0

v 4

j

⋆ Topological invariance encoded in F -symbol i m k l j

=

• n

F ijm

kln

i n k l j

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Example 2: positive cosmological constant

10 20 30 40 2 4 6 8 10 12

⋆ Quantum dimension (y axis) as a function of (2j + 1) ∈ 1, k + 1 (x axis) for k = 40

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Example 2: positive cosmological constant

2) Graph Hilbert space

⋆ Consider a p-punctured 2d manifold Σp ⋆ Define Hp as the span of trivalent graphs modulo the local equivalence relations (TQFT) j

=

j i m k l j

=

• n

F ijm

kln

i n k l j k i l j

= vivj vk δklNk

ij

k l

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Example 2: positive cosmological constant

Examples

⋆ 2-sphere with 0 punctures: dim H0 = 1

= (some number) ×

⋆ 2-sphere with 1 puncture: dim H1 = 1

= (some number) ×

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Example 2: positive cosmological constant

Examples

⋆ 2-sphere with 2 punctures ≃ cylinder ⋆ States correspond to allowed spin labelings of ⋆ Punctures can carry curvature (non-contractible cycles) and torsion (open links) ⋆ Vacuum (physical states)

• No curvature: F = 0
• No torsion: dAB = 0

⋆ Introduce Q basis of H2 given by

Qij

rs :=

r s j i

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Example 2: positive cosmological constant

Vacuum state

⋆ The vacuum is selected by two projections imposing (Levin, Wen)

• Gauge-invariance: Pv ⊲ vertex(i, j, k) = Nk

ij vertex(i, j, k)

• Flatness: Pp ⊲

= 1 D2

• j

v 2

j

j := 1

D

⋆ These dotted (vacuum) lines have the property that j j

=

j j

i.e. make the punctures invisible, i.e. remove the curvature that they carry

⋆ In the SU(2) case, D2Pp becomes

• j

djχj(g) = δ(g)

⋆ If Γ is dual to a triangulation, one gets the Turaev–Viro invariant (Kirillov Jr.)

• Γ, {j}
• faces

Pp

• Γ, {j′}
• = TV
• Σ × [0, 1], {j, j′}
• ⋆ Embedding map: add a puncture in the vacuum state, i.e.

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Example 2: positive cosmological constant

Quasi-particle excitations: the tube algebra

⋆ States Q on the cylinder can be thought of as operators ⋆ Stacking two cylinders gives back a cylinder ⋆ In terms of Q’s this multiplication defines the tube algebra (Ocneanu, Müger)

Qij

rsQkl su =

u l k j s i r

= · · · =

• mn
• F F F
• Qmn

ru

⋆ The Q states therefore define both a vector space and an algebra ⋆ The quasi-particle excitations are irreps. of (or modules over) this algebra (Lan, Wen) ⋆ More precisely, we look for states ξ with a stability property Q ⊲ ξ = ξ

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Example 2: positive cosmological constant

Drinfeld center

⋆ Instead of the Q basis, consider the basis states (Ocneanu, König, Kuperberg, Reichardt)

Oij

rs :=

r i j s

:=

ξ

r ⋆ These satisfy the projection condition Oij

rsOi′j′ su ∝ δii′δjj′Oij ru, and are stable in the sense

ξ

r l

=

• pq

Ωij

rp,ql

r q p l l

ξ

⋆ The tensors

Ω : ξ ⊗ l → l ⊗ ξ

ξ

r l

Ωij

rp,ql =

• mn

vmvn vrv 2

l

Ril

mRlj n F nmr ijl

F qlp

ijm F rlq jmn

are called half-braidings and label the objects of the Drinfeld center category Z(C)

⋆ Because here C is modular, Z(C) = C ⊠ Cop and dim Z(C) = (dim C)2 (Müger)

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Example 2: positive cosmological constant

3) Excitations

⋆ The excitation operators are given by the following oriented ribbons

ξ ¯ ξ

:=

• k

vk vivj

r i j s2 i j s1 ⋆ By definition, the half-braiding tensors have to satisfy so-called naturality conditions

⇔ fixed point conditions for the Q algebra ⇔ sliding property

ξ ¯ ξ

=

ξ ¯ ξ

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Example 2: positive cosmological constant

Action of the ribbons

⋆ Given by the fusion of excitations, i.e. cutting or gluing of 3-punctured 2-sphere basis ⋆ For example, acting on the cylinder vacuum state

gives a basis state O

⋆ Ribbons can be

• Glued

ξ ¯ ξ ξ ¯ ξ

=

¯ ξ ξ

• Closed

ξ

= v 2

i v 2 j =

• k∈ξ

v 2

k

• Linked

ξ1 ξ2

=: sξ1ξ2 = s(ij)(ab) = siasjb

⋆ Open and closed ribbon operators represent resp. exponentiated fluxes and holonomies ⋆ Description of (2 + 1) Euclidean λ > 0 gravity with massive spinning point particles ⋆ Generalizes results for SU(2) and λ = 0 (Bais, Meusburger, Muller, Noui, Perez, Schroers)

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Conclusion and perspectives

1. Vacua and TQFTs with defects 2. Example 1: vanishing cosmological constant 3. Example 2: positive cosmological constant 4. Conclusion and perspectives

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Conclusion and perspectives

Why different vacua and representations?

⋆ Needed in order to describe phase transitions and condensation in QFT ⋆ Physical states of quantum gravity will not be in the kinematical Hilbert space . . . ⋆ . . . but might be easier to reach starting from certain vacua ⋆ Path integral dynamics is a projector onto physical states ⋆ Generalized lattice path integral = spin foam, which is built upon BF theory

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Conclusion and perspectives

Why different vacua and representations?

⋆ Needed in order to describe phase transitions and condensation in QFT ⋆ Physical states of quantum gravity will not be in the kinematical Hilbert space . . . ⋆ . . . but might be easier to reach starting from certain vacua ⋆ Path integral dynamics is a projector onto physical states ⋆ Generalized lattice path integral = spin foam, which is built upon BF theory

What to expect

⋆ Spin foams have non-trivial phase diagrams ⋆ Each new phase is a TQFT and gives a new vacuum ⋆ Defects of the TQFT are excitations on top of the vacuum ⋆ Encodes new realizations of quantum geometry and new physics

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Conclusion and perspectives

New framework and results

⋆ New vacua, Hilbert spaces, and representations ⋆ New take on the dynamics and extraction of physics ⋆ Establishes link with other areas:

extended TQFTs, anyonic statistics, topological order, topological quantum computation

⋆ Right mathematical framework to describe fixed points of coarse-graining flow

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Conclusion and perspectives

New framework and results

⋆ New vacua, Hilbert spaces, and representations ⋆ New take on the dynamics and extraction of physics ⋆ Establishes link with other areas:

extended TQFTs, anyonic statistics, topological order, topological quantum computation

⋆ Right mathematical framework to describe fixed points of coarse-graining flow

Generalizations and applications

⋆ Other (quantum) groups, signs of λ, dimension, and signature ⋆ Anyon condensation (Bais, Slingerland): flow of λ and dynamical change of vacuum ⋆ Domain walls (Kitaev, Kong): between e.g. Z(Cq) and Z(Cq′), or vacuum1 and vacuum2 ⋆ Revisit loop quantum cosmology and black holes

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Thanks!

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