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Euclidean Quantum Supergravity

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Euclidean Quantum Supergravity Euclidean Quantum Supergravity David Tong Based on work with Carl Turner arXiv:1408.3418 Garyfest, 2015 1 2GM 2GM − distance2 = 1 δt2 + 1 dr2 + r2 δθ2 +sin2 θδφ2 − − rc2 − rc2 x = R sin θ sin φ y = R sin θ cos φ z = R cos θ δx =(R cos θ sin φ)δθ +(R sin θ cos φ)δφ δy =(R cos θ cos φ)δθ (R sin θ sin φ)δφ − δz = (R sin θ)δθ − θ φ Acknowledgement My thanks to Nick Dorey for many useful discussions. I’m supported by the Royal Society. References An Old Idea: Euclidean Quantum Gravity [1] J. Polchinski, “Dirichlet-Branes and Ramond-Ramond Charges,” Phys. Rev. Lett. 75, 4724 (1995) [arXiv:hep-th/9510017]. = g exp d4x √g Z D − R topology 17 1 2GM 2GM − distance2 = 1 δt2 + 1 dr2 + r2 δθ2 +sin2 θδφ2 − − rc2 − rc2 x = R sin θ sin φ y = R sin θ cos φ z = R cos θ δx =(R cos θ sin φ)δθ +(R sin θ cos φ)δφ δy =(R cos θ cos φ)δθ (R sin θ sin φ)δφ − δz = (R sin θ)δθ − θ φ Acknowledgement My thanks to Nick Dorey for many useful discussions. I’m supported by the Royal Society. References An Old Idea: Euclidean Quantum Gravity [1] J. Polchinski, “Dirichlet-Branes and Ramond-Ramond Charges,” Phys. Rev. Lett. 75, 4724 (1995) [arXiv:hep-th/9510017]. = g exp d4x √g Z D − R topology 17 1 2GM 2GM − distance2 = 1 δt2 + 1 dr2 + r2 δθ2 +sin2 θδφ2 − − rc2 − rc2 x = R sin θ sin φ y = R sin θ cos φ z = R cos θ δx =(R cos θ sin φ)δθ +(R sin θ cos φ)δφ δy =(R cos θ cos φ)δθ (R sin θ sin φ)δφ − δz = (R sin θ)δθ − θ A Preview of the Main Result φ 1,d 1 1 Kaluza-Klein Theory: = R − S M × 1 ∂ σ = F νρ µ 2 µνρ Λ M grav pl R Acknowledgement My thanks to Nick Dorey for many useful discussions. I’m supported by the Royal Society.There is a long history of quantum instabilities of these backgrounds • Casimir Forces References • Tunneling to “Nothing” Gross, Perry and Yaffe ‘82 Appelquist and Chodos ‘83 [1] J. Polchinski, “Dirichlet-Branes and Ramond-RamondWitten Charges ‘82 ,” Phys. Rev. Lett. 75, 4724 (1995) [arXiv:hep-th/9510017]. = g exp d4x √g Z D − R topology 17 1. Introduction and Summary 1 2GM 2GM − The purpose of thisdistance paper2 is= to study1 four-dimensionalδt2 + 1 =1supergravitycompact-dr2 + r2 δθ2 +sin2 θδφ2 − − rc2 − rc2 N ified on a spatial circle. We will show that this background is quantum mechanically unstable: the circle dynamically expands and the ground state is Minkowski space with all three spatial dimensions non-compact. x = R sin θ sin φ y = R sin θ cos φ Quantum mechanical instabilities of Kaluza-Kleinz = R cos compactificationsθ have a long his- tory. In the absence of supersymmetry, a Casimir force is generated perturbatively with acompetitionbetweenbosonicfields,whichcausethecircletocontract,andfermionic fields which cause the circle to expandδx =(R [1cos].θ Moresin φ)δθ scary+(R instabilitiessin θ cos φ)δφ lurk at the non- perturbative level, with space teeteringδy =(R oncos θ thecos φ brink)δθ ( ofR sin tunnellingθ sin φ)δφ into a bubble of − nothing [2]. Theδz =Main(R sinResultθ)δθ − The existence of supersymmetry removes both instabilities described above. But another remains. As we show in some detail, a Casimir force is now generated by Kaluza-Klein compactification of N=1 Supergravityθ is unstable. gravitational instantons. This results in a superpotential which schematically takes the form φ πR2 exp iσ (1.1) W ∼ −4GN −1,d 1 1 = R − S M × where R is the radius of the spatial circle and σ is dual to the Kaluza-Klein photon, dσ F . The existence of the superpotential (1.1)wasfirstproposedin[3] on the basis ∼ 1 νρ of fermi zero mode counting.Kaluza-Klein It isdual also photon: closely ∂µσ relatedµνρF to the superpotentials arising ∼ 2 from D6-brane instantons wrapping G2-holonomy manifolds described in [4]. Our goal in this paper is to develop the full quantum supergravityΛ M computation which results in grav pl (1.1). Grimm and Savelli 2011 (See also Harvey and Moore ‘99) 2 1 2 One motivation for1 performing5 theL instanton calculation1 L ∂ inR some detail is11 thatL − L=12 ∂σ = M + M M + N supergravity offLerseff a testing2 3 ground16π R2 inR which(3) − some3 − of6π theR2 oldR ideas− of Euclidean3 24π R2 quantumR4 2π gravity can be explored, but where many of the accompanying difficulties do not arise. It thus providesAcknowledgement an opportunity for precision Euclidean quantum gravity. Indeed, as we will see, we willMy thanks be able to to Nick compute Dorey for the many numerical useful discussions. prefactor in I’m (1.1 supported). In doing by the these Royal calculations, weSociety. met a number of issues that were (at least to us) surprising and we think worth highlighting. The Scale of Gravitational Instantons 17 The natural energy scale associated to any quantum gravity effect is usually thought to lie far in the ultra-violet, whether Planck scale, string scale or something else. How- ever, in situations where gravitational instantons play a role, this is not the only scale 2 2. Classical Aspects The Theory: N=1 Supergravity We work with =1supergravityind =3+1dimensions.Throughoutthepaper, N we focus on the minimal theory containing only a graviton and gravitino. The bulk four-dimensional action is given by M 2 S = pl d4x √ g + ψ¯ γµνρ ψ (2.1) 2 − R(4) µ Dν ρ 2 We use the notation of the (reduced) Planck mass Mpl =1/8πGN instead of the Newton constant G . Here is the 4d Ricci scalar, with the subscript to distinguish it from N R(4) its 3d counterpart that we will introduce shortly. There is also the standard Gibbons- Hawking boundary term which we have not written explicitly. The action is to be thought of as a functional of the Majorana gravitino ψµ and the a vierbein eµ where µ, ν =0, 1, 2, 3 are spacetime indices and a, b =0, 1, 2, 3aretangent space indices. Here we follow the standard notation of suppressing the spinor indices on the gravitino, whose covariant derivative is given by 1 ψ = ∂ ψ + ωˆ γabψ Dν ρ ν ρ 4 abν ρ In this formalism, the spin connection ωabµ that appears in the covariant derivative differs from the purely geometric spin connection by the addition of a gravitino torsion term:ω ˆabµ = ωabµ(e)+Habµ with 1 H = eνeρ ψ¯ γ ψ ψ¯ γ ψ ψ¯ γ ψ abµ −4 a b µ ρ ν − ν µ ρ − ρ ν µ The action is, of course, invariant under diffeomorphisms and local supersymmetry transformations. The latter act as δea = 1 γ¯ aψ u and δψ = . µ 2 m µ Dµ The classical theory also enjoys a U(1)R symmetry which acts by axial rotations on ψ. As we describe in more detail in Sections 3 and 4,thisU(1)R symmetry is anomalous in the quantum theory. (Although, as we will see, it mixes with a U(1)J bosonic symmetry that will be described shortly and a combination of the two survives.) 2.1 Reduction on a Circle Our interest in this paper is in the dynamics of =1supergravitywhencompactified N on a manifold = R1,2 S1. We denote the physical radius of the circle as R. M ∼ × We choose the spin structure such that the fermions are periodic around the compact direction and supersymmetry is preserved. 5 1 2GM 2GM − distance2 = 1 δt2 + 1 dr2 + r2 δθ2 +sin2 θδφ2 − − rc2 − rc2 x = R sin θ sin φ y = R sin θ cos φ z = R cos θ δx =(R cos θ sin φ)δθ +(R sin θ cos φ)δφ δy =(R cos θ cos φ)δθ (R sin θ sin φ)δφ At distances larger than the compactification scale R, the dynamics is effectively At distances− larger thanCompactify the compactification on a scaleCircleR, the dynamicsthree dimensional. is effectively The metric degrees of freedom are parameterised by the familiar δz = (threeR sin dimensional.θ)δθ The metric degrees of freedom are parameterisedKaluza-Klein by the ansatz, familiar − 2 2 Kaluza-Klein ansatz, L R 2 ds2 = ds2 + dz2 + A dxi (2.2) (4) R2 (3) L2 i 2 2 2 L 2 R 2 i 2 (3) ds = ds + dz + Aidx where z [0, 2πL) is the(2.2) periodic coordinate. Here R, A and the 3d metric g are (4) R2 (3) L2 ∈ i ij θ dynamical degrees of freedom, while L is a fixed, fiducial scale. It is natural to pick coordinates such that(3)R(x) L asymptotically and we will eventually do so but, for where z [0, 2πL) is the periodic coordinate. Here R, A and the 3d metric g are→ ∈ i now, we leave L arbitrary.ij dynamical degreesφ of freedom, while L is a fixed, fiducial scale. It is natural to pick coordinates such that R(x) L asymptotically and we will eventuallyEvaluated do on so this but, background, for the Einstein-Hilbert action becomes → M 2 now, we leave L arbitrary. pl 4 Seff = d x √ g (4) 1,2 1 2 − R = S R 2 4 EvaluatedM on× this background, the Einstein-Hilbert actionR becomes M3 ∂R 1 R = d3x g 2 F F ij 2 − (3) R(3) − R − 4 L4 ij 2 Mpl 4 Acknowledgement Seff = d x √ g (4) with M =2πLM 2 the 3d Planck scale and F = ∂ A ∂ A the graviphoton field 2 − R 3 pl ij i j − j i strength.
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