Quantum gravity and TQFTs with defects Marc Geiller Perimeter Institute Quantum Gravity in Paris March 20 th − 23 rd 2017 0 / 24
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 / 24
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 1 2 gauge field (connection) A 1 “position” holonomies h ( A ) ∈ G 2 1 2 electric field (metric) E 1 “momentum” fluxes X ( E ) ∈ Lie ( G ) 2 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 1 / 24
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) 2 / 24
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 2 / 24
Outline 1. Vacua and TQFTs with defects 2. Example 1: vanishing cosmological constant 3. Example 2: positive cosmological constant 4. Conclusion and perspectives 3 / 24
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 3 / 24
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) ∋ h ℓ = exp A ℓ ℓ ⋆ Fluxes of the electric field � h − 1 Eh su (2) ∋ X S = S S ⋆ These form the holonomy-flux algebra � � h ℓ , X S 4 / 24
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, . . . ) 5 / 24
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 5 / 24
Vacua and TQFTs with defects AL representation in LQG (Ashtekar, Isham, Lewandowski, 1992 – 1995) 6 / 24
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 | X S | 0 � AL = 0 , ∀ S 6 / 24
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 | X S | 0 � AL = 0 , ∀ S ⋆ Spatial geometry vanishes in the vacuum: totally squeezed and degenerate state 6 / 24
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 | X S | 0 � AL = 0 , ∀ S ⋆ Spatial geometry vanishes in the vacuum: totally squeezed and degenerate state ⋆ Creation operators: holonomies ˆ h 1 ˆ h 2 ˆ h 3 vacuum excited state ∈ H Γ h 1 ˆ ˆ h 2 ˆ h 3 ⊲ = ∈ H Γ 6 / 24
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 | X S | 0 � AL = 0 , ∀ S ⋆ Spatial geometry vanishes in the vacuum: totally squeezed and degenerate state ⋆ Creation operators: holonomies ˆ h 1 ˆ h 2 ˆ h 3 vacuum excited state ∈ H Γ h 1 ˆ ˆ h 2 ˆ h 3 ⊲ = ∈ H Γ ⋆ Discrete basis of excitations labelled by graphs and carrying quanta of geometry 6 / 24
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 | X S | 0 � AL = 0 , ∀ S ⋆ Spatial geometry vanishes in the vacuum: totally squeezed and degenerate state ⋆ Creation operators: holonomies ˆ h 1 ˆ h 2 ˆ h 3 vacuum excited state ∈ H Γ h 1 ˆ ˆ h 2 ˆ h 3 ⊲ = ∈ H Γ ⋆ Discrete basis of excitations labelled by graphs and carrying quanta of geometry ⋆ Makes it difficult to construct semi-classical states: need condensates 6 / 24
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 | X S | 0 � AL = 0 , ∀ S ⋆ Spatial geometry vanishes in the vacuum: totally squeezed and degenerate state ⋆ Creation operators: holonomies ˆ h 1 ˆ h 2 ˆ h 3 vacuum excited state ∈ H Γ h 1 ˆ ˆ h 2 ˆ h 3 ⊲ = ∈ H Γ ⋆ 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 ∞ = ∪ Γ H Γ / ∼ = L 2 � � A / G , d µ AL 6 / 24
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 7 / 24
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 ) 7 / 24
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 7 / 24
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, . . . ) 8 / 24
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 L 3 = A ∧ F [ A ] − 1 3 A ∧ A ∧ A - d -dimensional BF theory (gravity in d = 3 ) L 3 = B ∧ F [ A ] + λB ∧ B ∧ B 8 / 24
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