2 d gravity with massive matter Harold Erbin Lptens , École Normale Supérieure (France) SCGSC 2017 Ihp , Paris – 17th February 2017 arXiv: 1612.04097 , 1511.06150 1 / 25
Outline Introduction Classical gravity Quantum gravity Mabuchi spectrum Conclusion 2 / 25
Outline: 1. Introduction Introduction Classical gravity Quantum gravity Mabuchi spectrum Conclusion 3 / 25
Motivations 2 d (quantum) gravity is useful for: ◮ toy model for 4 d quantum gravity ◮ spontaneous dimensional reduction [1605.05694, Carlip] ◮ (non-)critical string theories 4 / 25
Motivations 2 d (quantum) gravity is useful for: ◮ toy model for 4 d quantum gravity ◮ spontaneous dimensional reduction [1605.05694, Carlip] ◮ (non-)critical string theories Real-world requires massive matter 4 / 25
Goals ◮ Study classical gravity coupled to massive matter ◮ Show that (classical) 2 d gravity is not a good toy model ◮ Derive the spectrum of the Mabuchi action (quantum action for the metric) 5 / 25
Outline: 2. Classical gravity Introduction Classical gravity Quantum gravity Mabuchi spectrum Conclusion 6 / 25
Total action Matter ψ + gravity g µν action S [ g , ψ ] = S grav [ g ] + S m [ g , ψ ] Conditions ◮ renormalizability ◮ invariance under diffeomorphisms ◮ no more than first order derivatives ◮ S m [ g , ψ ] obtained from minimal coupling Note: in 2 d g µν has one dynamical component = conformal factor (or Liouville field) φ 7 / 25
Gravity action Gravitational action: two possible terms S grav [ g ] = S EH [ g ] + S µ [ g ] ◮ Einstein–Hilbert � � d 2 σ S EH [ g ] = | g | R = 4 πχ topological invariant (Euler number χ ) → not dynamical, ignore it ◮ Cosmological constant � � d 2 σ S µ [ g ] = µ | g | = µ A [ g ] 8 / 25
Equations of motion ◮ Energy–momentum tensor (with traceless and trace components) T µν = − 4 π δ S δ g µν = T ( m ) µν + 2 πµ g µν � | g | T µν = T µν − T ¯ T = g µν T µν 2 g µν , ◮ Equations of motion δ S δ S δ g µν = 0 , δψ = 0 9 / 25
Equations of motion ◮ Energy–momentum tensor (with traceless and trace components) T µν = − 4 π δ S δ g µν = T ( m ) µν + 2 πµ g µν � | g | T µν = T µν − T ¯ T = g µν T µν 2 g µν , ◮ Equations of motion δ S δ S δ g µν = 0 , δψ = 0 ◮ Metric eom → vanishing of T µν T = T ( m ) + 4 πµ = 0 � T ( m ) T µν = ¯ ¯ = 0 µν → decoupling of traceless component from gravity 9 / 25
Dynamics: conformal matter ◮ Weyl transformation g µν = e 2 ω ( σ ) g ′ µν conformal invariance S m [ η, ψ ] = ⇒ Weyl invariance S m [ g , ψ ] (here ⇐ = also holds) 10 / 25
Dynamics: conformal matter ◮ Weyl transformation g µν = e 2 ω ( σ ) g ′ µν conformal invariance S m [ η, ψ ] = ⇒ Weyl invariance S m [ g , ψ ] (here ⇐ = also holds) ◮ Weyl invariance → traceless T ( m ) µν T ( m ) = 0 = ⇒ µ = 0 from gravity (trace) eom 10 / 25
Dynamics: conformal matter ◮ Weyl transformation g µν = e 2 ω ( σ ) g ′ µν conformal invariance S m [ η, ψ ] = ⇒ Weyl invariance S m [ g , ψ ] (here ⇐ = also holds) ◮ Weyl invariance → traceless T ( m ) µν T ( m ) = 0 = ⇒ µ = 0 from gravity (trace) eom Conclusion Conformal matter coupled to µ � = 0 gravity is inconsistent. 10 / 25
Dynamics: non-conformal matter (1) – model ◮ N scalar fields X i S m = − 1 � � d 2 σ � � g µν ∂ µ X i ∂ ν X i + V ( X i ) | g | 4 π ◮ eom = ∂ µ X i ∂ ν X i − 1 T ( m ) ¯ 2 g µν ( g αβ ∂ α X i ∂ β X i ) = 0 µν V ( X ) = 4 πµ − ∆ X i + 1 ∂ V = 0 2 ∂ X i ∆ = g µν ∇ µ ∇ ν curved space Laplacian 11 / 25
Dynamics: non-conformal matter (2) – solution ◮ Conformal gauge (fix diffeomorphisms) g µν = e 2 φ η µν ◮ Traceless eom T 01 ) = ( ∂ 0 X i ± ∂ 1 X i ) 2 = 0 2( ¯ T 00 ± ¯ → sum of squares X i = X 0 ( ∂ 0 ± ∂ 1 ) X i = 0 = ⇒ ∂ µ X i = 0 = ⇒ i = cst 12 / 25
Dynamics: non-conformal matter (2) – solution ◮ Conformal gauge (fix diffeomorphisms) g µν = e 2 φ η µν ◮ Traceless eom T 01 ) = ( ∂ 0 X i ± ∂ 1 X i ) 2 = 0 2( ¯ T 00 ± ¯ → sum of squares X i = X 0 ( ∂ 0 ± ∂ 1 ) X i = 0 = ⇒ ∂ µ X i = 0 = ⇒ i = cst ◮ Trace and matter eom → constraints on X 0 i ∂ V ( X 0 V ( X 0 i ) = 0 , i ) = 4 πµ ∂ X i 12 / 25
Dynamics: non-conformal matter (2) – solution ◮ Conformal gauge (fix diffeomorphisms) g µν = e 2 φ η µν ◮ Traceless eom T 01 ) = ( ∂ 0 X i ± ∂ 1 X i ) 2 = 0 2( ¯ T 00 ± ¯ → sum of squares X i = X 0 ( ∂ 0 ± ∂ 1 ) X i = 0 = ⇒ ∂ µ X i = 0 = ⇒ i = cst ◮ Trace and matter eom → constraints on X 0 i ∂ V ( X 0 V ( X 0 i ) = 0 , i ) = 4 πµ ∂ X i Conclusion Non-conformal matter coupled to gravity is (at best) trivial. 12 / 25
Dynamics: non-conformal matter (3) – example ◮ Free massive scalars m 2 i X 2 � V ( X i ) = i i ◮ Matter eom m 2 i X 0 X 0 i = 0 = ⇒ i = 0 ∀ m i � = 0 13 / 25
Dynamics: non-conformal matter (3) – example ◮ Free massive scalars m 2 i X 2 � V ( X i ) = i i ◮ Matter eom m 2 i X 0 X 0 i = 0 = ⇒ i = 0 ∀ m i � = 0 ◮ Trace eom i ) 2 = 4 πµ m 2 i ( X 0 � = ⇒ µ = 0 i 13 / 25
Dynamics: non-conformal matter (3) – example ◮ Free massive scalars m 2 i X 2 � V ( X i ) = i i ◮ Matter eom m 2 i X 0 X 0 i = 0 = ⇒ i = 0 ∀ m i � = 0 ◮ Trace eom i ) 2 = 4 πµ m 2 i ( X 0 � = ⇒ µ = 0 i Conclusion Massive free scalar fields coupled to gravity are inconsistent for µ � = 0, trivial for µ = 0. 13 / 25
Degrees of freedom: conformal matter ◮ No cosmological constant, µ = 0 ◮ ∃ Weyl invariance → traceless energy–momentum tensor T ( m ) = 0 14 / 25
Degrees of freedom: conformal matter ◮ No cosmological constant, µ = 0 ◮ ∃ Weyl invariance → traceless energy–momentum tensor T ( m ) = 0 ◮ Metric eom T ( m ) = 0 µν ◮ Weyl invariant eom → independent of the conformal factor → 2 constraints on the matter 14 / 25
Degrees of freedom: conformal matter ◮ No cosmological constant, µ = 0 ◮ ∃ Weyl invariance → traceless energy–momentum tensor T ( m ) = 0 ◮ Metric eom T ( m ) = 0 µν ◮ Weyl invariant eom → independent of the conformal factor → 2 constraints on the matter Conclusion Gravity reduces the dofs of conformal matter from N to N − 2. 14 / 25
Degrees of freedom: non-conformal matter ◮ Action linear in g µν S m = 1 L = − 1 � � � � d 2 σ g µν L µν ( ψ ) + V ( ψ ) | g | L , 2 π 2 ◮ Metric eom � = 0 , T µν = L µν − 1 ¯ � g αβ L αβ T = − V + 4 πµ = 0 2 g µν ◮ Weyl invariant eom → independent of the conformal factor → 3 constraints on the matter 15 / 25
Degrees of freedom: non-conformal matter ◮ Action linear in g µν S m = 1 L = − 1 � � � � d 2 σ g µν L µν ( ψ ) + V ( ψ ) | g | L , 2 π 2 ◮ Metric eom � = 0 , T µν = L µν − 1 ¯ � g αβ L αβ T = − V + 4 πµ = 0 2 g µν ◮ Weyl invariant eom → independent of the conformal factor → 3 constraints on the matter ◮ Abolishing gauge invariance (Weyl) removes dofs Conclusion Gravity reduces the dofs of generic non-conformal matter from N to N − 3, instead of N − 1. 15 / 25
Outline: 3. Quantum gravity Introduction Classical gravity Quantum gravity Mabuchi spectrum Conclusion 16 / 25
Functional integration ◮ Partition functions � d g g µν e − S µ [ g ] Z m | g ] Z = � d g ψ e − S m [ g ,ψ ] Z m [ g ] = ◮ Quantum effects → dynamics for the conformal factor ◮ For computations: fix diffeomorphisms 17 / 25
Conformal gauge ◮ Conformal gauge g = e 2 φ g 0 φ Liouville mode, g 0 (fixed) background metric ◮ Partition function S grav = − ln Z m [ e 2 φ g 0 ] Z [ φ ] = e − S grav [ g 0 ,φ ] Z m [ g 0 ] , Z m [ g 0 ] (ignore ghosts from gauge fixing) 18 / 25
Conformal gauge ◮ Conformal gauge g = e 2 φ g 0 φ Liouville mode, g 0 (fixed) background metric ◮ Partition function S grav = − ln Z m [ e 2 φ g 0 ] Z [ φ ] = e − S grav [ g 0 ,φ ] Z m [ g 0 ] , Z m [ g 0 ] (ignore ghosts from gauge fixing) ◮ Typically [1112.1352, Ferrari-Klevtsov-Zelditch] S grav = S µ + c 6 S L + β 2 S M + · · · S µ cosmological constant, S L Liouville action, S M Mabuchi action 18 / 25
Outline: 4. Mabuchi spectrum Introduction Classical gravity Quantum gravity Mabuchi spectrum Conclusion 19 / 25
Mabuchi action ◮ Kähler potential (work at fixed area) e 2 φ = A � 1 + A 0 � 2 πχ ∆ 0 K A 0 ◮ Mabuchi action (Euclidean) [Mabuchi ’86] S M = 1 d 2 σ √ g 0 � 4 πχ K + 4 πχ � � � � − g µν φ e 2 φ 0 ∂ µ K ∂ ν K + − R 0 4 π A 0 A 20 / 25
Mabuchi action ◮ Kähler potential (work at fixed area) e 2 φ = A � 1 + A 0 � 2 πχ ∆ 0 K A 0 ◮ Mabuchi action (Euclidean) [Mabuchi ’86] S M = 1 d 2 σ √ g 0 � 4 πχ K + 4 πχ � � � � − g µν φ e 2 φ 0 ∂ µ K ∂ ν K + − R 0 4 π A 0 A ◮ eom (same as Liouville) R = 4 πχ A ◮ Note: ill-defined on the torus/cylinder ( χ = 0) 20 / 25
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