Quantum Dynamics of Supergravity

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1 Quantum Dynamics of Supergravity David Tong Work with Carl Turner Based on arxiv: Crete, September 2014

2 An Old Idea: Euclidean Quantum Gravity Z = Dg exp d 4 x g R topology

3 A Preview of the Main Results Kaluza-Klein Theory: M = R 1,d 1 S 1 R There is a long history of quantum instabilities of these backgrounds Casimir Forces Tunneling to Nothing Appelquist and Chodos 83 Witten 82

4 The Main Result Kaluza-Klein compactification of N=1 Supergravity is unstable. W exp πr2 iσ 4G N Kaluza-Klein dual photon: µ σ 1 2 µνρf νρ Grimm and Savelli 2011

5 And Something Interesting Along the Way Quantum Gravity has a hidden infra-red scale! Λ grav M pl This is the scale at which gravitational instantons contribute

6 The Theory: N=1 Supergravity S = M 2 pl 2 d 4 x g R (4) + ψ µ γ µνρ D ν ψ ρ

7 Compactify on a Circle ds 2 (4) = L2 R 2 ds2 (3) + R2 L 2 dz 2 + A i dx i 2 z [0, 2πL) M = R 1,2 S 1 R Fields R(x i ) and A i (x i ) live here L is fiducial scale

8 Classical Low-Energy Physics S eff = M 2 pl 2 = M 3 2 d 4 x g R (4) d 3 x g (3) R (3) 2 R R R 4 L F ijf ij 4 M 3 =2πLM 2 pl gth. Or, if we work with the dual photon µ σ 1 2 µνρf νρ S eff = d 3 x g (3) M 3 2 R (3) M 3 R R 2 1 M 3 L 2 R 4 σ 2π 2 Goal: Understand quantum corrections to this action.

9 Perturbative Quantum Corrections

10 Perturbative Quantum Corrections One-Loop Effects: UV divergences Running Gauss-Bonnet term The anomaly Finite corrections ~ Casimir forces Other corrections o the low- 1/M 2 pl R2

11 One-Loop Divergences t Hooft and Veltman 74 At one-loop in pure gravity, there are three logarithmic divergences R 2, R µν R µν, R µνρσ R µνρσ These two can be absorbed by a field redefinition of the metric The Riemann 2 term can be massaged into Gauss-Bonnet. χ = 1 8π 2 d 4 x R µνρσ R µνρσ 4R µν R µν + R 2 This is purely topological. It doesn t affect perturbative physics around flat space.

12 The Gauss-Bonnet Term S α = α 8π 2 The coupling runs logarithmically d 4 x R µνρσ R µνρσ 4R µν R µν + R 2 = αχ α(µ) =α 0 α 1 log M 2 UV µ 2 where the beta function is given by α 1 = N2 233N 3/2 52N 1 +7N 1/2 +4N 0 Christensen and Duff 78 Perry 78; Yoneya 78 For us α 1 =41/48.

13 A New RG-Invariant Scale α(µ) =α 0 α 1 log M 2 UV µ 2 As in Yang-Mills, we can replace the log running with an RG invariant scale Λ grav = µ exp α(µ) 2α 1 This scale will be associated with non-trivial spacetime topologies. (We will see an example)

14 Another Divergence: The Anomaly The classical action is invariant under rotations of the phase of the fermion. This U(1) R symmetry does not survive in the quantum theory. µ J µ 5 = 1 21N3/2 N 24 16π 2 1/2 R µνρσ R µνρσ The phase of the fermion can be absorbed by shifting the theta term S θ = θ 16π 2 d 4 x g R µνρσ R µνρσ

15 Topological Terms One-loop effects tell us that we should consider two topological terms S α = α 32π 2 S θ = θ 16π 2 d 4 x g R µνρσ R µνρσ d 4 x g R µνρσ R µνρσ In supergravity, these two coupling constants sit in a chiral multiplet τ grav = α +2iθ

16 Finite Quantum Corrections Casimir Energy: V eff = N B N F 720π L 3 R 6 Appelquist and Chodos 83 Supersymmetry means that N B =N F and this Casimir energy vanishes. But there are other effects.

17 Finite Quantum Corrections One loop corrections to the kinetic terms give L eff = 1 2 M π L R 2 R (3) M 3 1 6π M π L R R 2 R L R L 2 R 4 σ 2π 2

18 Finite Quantum Corrections One loop corrections to the kinetic terms give L eff = 1 2 M π L R 2 R (3) M 3 1 6π M π L R R 2 R L R L 2 R 4 σ 2π 2 Something important: these two numbers are different!

19 The Complex Structure The two fields R and σ must combine in a complex number Classically: L eff = = 2 R + 1 L 2 R 4 R M3 2 1 (S + S ) 2 S S σ 2π 2 S =2π 2 M 2 plr 2 + iσ

20 The Complex Structure The two fields R and σ must combine in a complex number At one-loop L eff = 1 1 6π S S S S We want to write this in the form L 2 L R M 3 R 2 R 24π 1 L L 2 M 3 R 2 R 4 σ 2π 2 L eff = K(S, S ) S S S =2π 2 M 2 plr log(m 2 plr 2 )+iσ

21 Non-Perturbative Quantum Corrections

22 Gravitational Instantons Look for other saddle points of the action We want these to contribute to the (super)potential. They must obey R µνρσ = ± R µνρσ

23 Taub-NUT Instantons The appropriate metrics are given by the multi-taub-nut solutions Gibbons and Hawking 78 ds 2 = U(x)dx dx + U(x) 1 (dz + A dx) 2 with U(x) =1+ L 2 k a=1 1 x X a and A = ± U From the low-energy 3d perspective, these look like Dirac monopoles. This is the gravitational verson of Polyakov s famous calculation. Polyakov 77 Gross 84 Hartnoll and Ramirez 13

24 The Boundary of the Space The boundary of Taub-NUT is not the same as the boundary of flat space. (R 3 S 1 )=S 2 S 1 but (TN k )=S 3 /Z k Should we include such geometries in the path integral?

25 The Boundary of the Space The boundary of Taub-NUT is not the same as the boundary of flat space. (R 3 S 1 )=S 2 S 1 but (TN k )=S 3 /Z k Should we include such geometries in the path integral? Yes!

26 The Boundary of the Space The boundary of Taub-NUT is not the same as the boundary of flat space. (R 3 S 1 )=S 2 S 1 but (TN k )=S 3 /Z k Should we include such geometries in the path integral? Yes! c.f. Atiyah-Hitchin with boundary a circle fibre over RP 2 for which the answer is probably no!

27 Zero Modes of Taub-NUT ds 2 = U(x)dx dx + U(x) 1 (dz + A dx) 2 U(x) =1+ L 2 k a=1 1 x X a and A = ± U 3k bosonic zero modes 2k fermionic zero modes Only k=1 solution contributes to the superpotential

28 Doing the Computation Action, Zero Modes, Jacobians, Determinants, Propagators.

29 The Determinants dets = det(fermions) det(bosons) Supersymmetry dets = 1? Hawking and Pope 78

30 The Determinants In a self-dual background, you can write the determinants as dets = det /D /D det /D /D A somewhat detailed calculation gives 1/4 spin 3/2 det /D /D det /D /D 1/2 spin 1/2 dets = A µ 2 41/48 1 R 2 7/48 An ugly number UV cut-off scale We ve seen these fractions before!

31 The calculation gives The Superpotential W = C µ 2 M 2 pl 41/48 1 M 2 pl R2 7/48 e STN iσ e τgrav C = 4e 24ζ ( 1) 1 7/48 2(4π) 3/2 S TN =2π 2 M 2 plr 2 Topological terms All the pieces now fit together W = C Λ 2 grav M 2 pl 41/48 e S with S =2π 2 M 2 plr log(m 2 plr 2 )+iσ

32 The Potential Kaluza-Klein compactification of N=1 supergravity is unstable V M 3 3 (RΛ grav ) 41/24 exp 4π 2 M 2 plr 2 N The ground state has R R

33 Open Questions Λ grav What is this good for? What is it in our Universe?

34 Thank you for your attention