Quantum Dynamics of Supergravity

Home , Dual photon, Graviphoton, Graviton, Quantum gravity, Supergravity, Superpotential

David Tong

Work with Carl Turner Based on arXiv:1408.3418

Crete, September 2014 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 −

Acknowledgement

My thanks to Nick Dorey for many useful discussions. I’m supported by the Royal Society. An Old Idea: Euclidean Quantum Gravity References

[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 Results 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” 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 history. 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

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 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 −

φ And Something Interesting Along the Way… = R1,2 S1 M ×

1 Quantum∂ Gravityσ = has a hiddenF νρ infra-red scale! µ 2 µνρ

Λ M grav pl

AcknowledgementThis is the scale at which gravitational instantons contribute

My thanks to Nick Dorey for many useful discussions. I’m supported by the Royal Society.

References

[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 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 âbµ = 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. 2 4 My thanks to Nick Dorey for many useful discussions.M3 3 I’m supported by the∂Ri Royal1 Ri ij = d x g Fields2 R(x ) and Ai(x ) liveFIn Fthreehere dimensions, it is often useful to dualise the gauge field in favour of a periodic 2 − (3) R(3) − R − 4 L4 ij Society. scalar σ. This is particularly true if we are interested in instanton physics [5]. The dual photon can be viewed as Lagrange multiplier which imposes the Bianchi identity, L is fiducial scale2 with M3 =2πLMpl the 3d Planck scale and Fij = ∂iAj ∂jAi the graviphoton field σ − = ijkD F (2.3) References strength. Lσ 4πL i jk With the magnetic charge quantised in integral units, σ has periodicity 2π. Integrating [1] J. Polchinski, “Dirichlet-BranesIn and three Ramond-Ramond dimensions, it is Charges often useful,” Phys. to dualise Rev. Lett. the gauge75, fieldout in the favour field ofstrength, a periodic we can write the low-energy effective action in dual form, 4724 (1995) [arXiv:hep-th/9510017].scalar σ. This is particularly true if we are interested in instanton physics [5]. The dual M ∂R 2 1 L2 ∂σ 2 S = d3x g 3 M (2.4) photon can be viewed as Lagrange multiplier which imposes the Bianchie identity,ff − (3) 2 R(3) − 3 R − M R4 2π 3 σ ijk = g exp d4x √g σ = DiFjk This action enjoys a new(2.3)U(1)J symmetry which acts by shifting the dual photon: Z D − RL 4πL σ σ + c. All other fields are left invariant under this symmetry. The symmetry is topology → With the magnetic charge quantised in integral units, σ has periodicitypreserved 2 inπ. perturbation Integrating theory but, as we will see in Section 4, is broken by instanton effects. out the field strength, we can write the low-energy effective action in dual form, Our goal in this paper is to determine the quantum corrections to the effective action 2 2(2.4). We2 describe perturbative corrections in Section 3 and instanton corrections in 3 M3 ∂R 1 L ∂σ S = d x g M Section 4. (2.4) eff − (3) 2 R(3) − 3 R − M R4 2π 3 17 This action enjoys a new U(1)J symmetry which acts by shifting the dual photon: 6 σ σ + c. This symmetry is preserved in perturbation theory but, as we will see in → Section 4, is broken by instanton effects.

Our goal in this paper is to determine the quantum corrections to the eﬀective action (2.4). We describe perturbative corrections in Section 3 and instanton corrections in Section 4.

6 At distances larger than the compactiﬁcation scale R, the dynamics is eﬀectively three dimensional. The metric degrees of freedom are parameterised by the familiar

Kaluza-Klein ansatz, 1 2 2GM 2 2GM − 2 2 2 2 2 distance = 2 1 2 δt + 1 dr + r δθ +sin θδφ 2 −L −2 rc2R 2 − irc22 ds = ds + dz +Aidx (2.2) (4) R2 (3) L2 (3) where z [0, 2πL) is the periodic coordinate. Herex = RRsin, Aθisinandφ the 3d metric g are At distances∈ larger than the compactification scale R, the dynamics is effectivelyij dynamical degrees of freedom, while L is a fixed,y = fiducialR sin θ cos scale.φ It is natural to pick three dimensional. The metric degrees of freedom are parameterised by the familiar coordinates such that R(x) L asymptotically and we will eventually do so but, for Kaluza-Klein ansatz, → z = R cos θ now,At we distances leave largerL arbitrary. than the compactification scale R, the dynamics is effectively three dimensional. The metric degrees of freedom are parameterised by the familiar 2 2 L R 2 Kaluza-KleinEvaluated ansatz, on this background,ds2 = theds2 Einstein-Hilbert+ dz2 + A dx actioni becomes (2.2) (4) R2 (3) L2 i L2 δRx2 =(R cos θ sin φ)δθ +(R sin θ cos φ)δφ M22 2 2 i 2 ds(4)pl = 42 ds(3) + 2 dz + Aidx (2.2) (3) where z [0,S2eπffL=) is the periodicRd x √ coordinate.gδLy (4)=(R cos Hereθ cosRφ),δθAi and(R sin theθ 3dsin φ metric)δφ gij are ∈ 2 − R − dynamical degrees of freedom, while L is a fixed, fiducial scale. It is(3) natural to pick where z [0, 2πL) is the periodic coordinate.δz = Here(RRsin, Aθ)andδθ the2 3d metric g are ∈ M − i ∂R 1 R4 ij dynamicalcoordinates degrees such of that= freedom,R3 (x)d while3xLLasymptoticallyisg a fixed, fiducial2 and scale. we will It is eventually naturalF F toij pick do so but, for → (3) (3) 4 ij coordinatesnow, we leave suchL thatarbitrary.R(Classical2x) L asymptotically −Low-EnergyR and− we willPhysicsR eventually− 4 L do so but, for → now, we leave L arbitrary. 2 withEvaluatedM3 =2π onLM thispl the background, 3d Planck the scale Einstein-Hilbert and Fij = ∂i actionAθ j ∂ becomesjAi the graviphoton field Evaluated on this background, the Einstein-Hilbert action becomes− strength. 2 2Mpl 4 Seff =Mpl 4 d x √ g (4) In three dimensions,Seff = 2 itd isx often√ g useful−(4) R to dualise the gaugeφ field in favour of a periodic 2 − R 2 4 scalar σ. This is particularlyM true if we are interested2 ∂R in4 instanton1 R physics [5]. The dual M3 3 3 ∂R 1 R ij = 3 d x g(3) (3) 2 ij FijF photon can be viewed= asd Lagrangex g(3) multiplier(3) 2 which imposes14,dFij1F the14 Bianchi identity, 2 2 − −R −R R− −=R4RL −− 4 SL M × σ ijk with M =2πLM 2 the2 3d Planck scaleσ and= F = ∂DAiFjk∂ A the graviphoton field (2.3) with M3 3 =2πLMpl pl the 3d PlanckL scale4π andijL Fi ijj =− ∂jiAij ∂jAi the graviphoton field strength. − strength. 1 With the magneticOr, charge if we work quantised with the dual in integral photon ∂ units,σ σ hasF periodicityνρ 2π. Integrating In three dimensions, it is often useful to dualise the gaugeµ field2 inµνρ favour of a periodic outIn the three field dimensions, strength, we it is can often write useful the to low-energy dualise the∼ eff gaugeective field action in favour in dual of form, a periodic scalar σ. This is particularly true if we are interested in instanton physics [5]. The dual photonscalar σ can. This be viewed is particularly as Lagrange true multiplier if we are which interested imposes in the2 instanton Bianchi identity,2 physics2 [5]. The dual 3 M3 Λ∂Rgrav Mpl1 L ∂σ photon canSeff be= viewedd x as Lagrangeg(3) multiplier(3) M which3 imposes the Bianchi identity, (2.4) − 2σ Rijk − R − M R4 2π σ = DiFjk 3 (2.3) 4πL L σ ijk σ = DiFjk (2.3) ThisWith action the magnetic enjoys charge a new quantisedU(1) inJL integralsymmetry4πL units, whichσ has periodicity acts by shifting 2π. Integrating2 the dual photon: 1 2 1 5 L 1 L ∂R 11 L − L2 ∂σ σout theσ + fieldc. strength, All other=Goal: we fields can MUnderstand write+ are the left low-energyquantum invariant corrections eff underectiveM actionto this this symmetry. inaction. dual form, TheM symmetry+ is With→ the magneticLeff charge2 quantised3 16π R2 in integralR(3) − units,3 −σ6hasπ R2 periodicityR − 2π. Integrating3 24π R2 R4 2π preserved in perturbation theory but, as we will see in Section4, is broken by instanton out the field strength, we can write the low-energy2 effective2 action2 in dual form, 3 M3 ∂R 1 L ∂σ effects. Seff = d x g(3) (3) M3 4 (2.4) − 2 R − R − M3 R 2π Acknowledgement 2 2 2 M3 ∂R 1 L ∂σ Our goalS in this= paperd3x is tog determine theM quantum corrections to the effective(2.4) action This action enjoyseff a new U(1) (3)symmetry(3) which acts3 by shifting the4 dual photon: My thanks− toJ Nick 2 DoreyR for− manyR useful− discussions.M3 R 2π I’m supported by the Royal (σ2.4).σ We+ c. describe This symmetry perturbative is preserved corrections in perturbation in Section theory 3 but,and as instantonwe will see in corrections in → Society. SectionSection 44,. is broken by instanton effects. This action enjoys a new U(1)J symmetry which acts by shifting the dual photon: σ σ + c. This symmetry is preserved in perturbation theory but, as we will see in Our→ goal in this paper is to determine the quantum corrections to the effective action (Section2.4). We4, describe is broken perturbative by instanton corrections effects. in Section 3 and instanton corrections in Section 4. 17 Our goal in this paper is to determine the6 quantum corrections to the effective action (2.4). We describe perturbative corrections in Section 3 and instanton corrections in

Section 4. 6

6 Perturbative Quantum Corrections If we care only about perturbative physics on R3 S1, then we can neglect these × terms. However, when we start to sum over manifolds of diﬀerent topology, they become important.

Usually, when working with an effective field theory, we keep all relevant and marginal terms in the action, neglecting only the irrelevant operators on the grounds that they are suppressed by some high mass scale. In the present case, there are two further four-derivative terms which come with dimensionless coefficients: 2 and µν. R RµνR However, both can be absorbed into the Einstein-Hilbert term through a redefinition of the metric [6]. For this reason, we need only consider Sα and Sθ above.

In supergravity, the Gauss-Bonnet (2.8)andPontryagin(2.9)termscanbewritten as an F-term [17, 18] (using the so-called “chiral projection operator”). This, in turn, means that the two coupling constants α and θ combine into the complex coupling

τgrav = α +2iθ (2.10) which naturally lives in a chiral multiplet.Perturbative We will see later Quantum that τgrav Correctionsappears in the instanton generated superpotential.

3. Perturbative Aspects

In this section we describe the results ofOne-Loop quantum Effects: fluctuations • UV divergences of the graviton and gravitino around the background R1,2 S1. There are two kinds• Running of effects: Gauss-Bonnet those from term × • The anomaly divergences that arise already in four dimensions; and finite corrections to the low- 2 2 energy effective action which are suppressed by the dimensionless• Finite combinationcorrections ~ 1/MplR . • Casimir forces • Other corrections 3.1 Summary We open this section by summarising the main results. The remainder of the section contains details of the computations.

Finite Corrections We start by describing finite corrections to the effective action that arise at one-loop. These only occur when the theory is compactified on R1,2 S1 and are due to loops × wrapping the spatial circle. The results depend on R, the radius of the circle and so are non-local from the four-dimensional perspective. For this reason, they do not depend on the ultra-violet details of the theory and are therefore calculable.

8 2 R

µν RµνR 2 One-Loop DivergencesR

µνρσ ‘t Hooft and Veltman ‘74 µνρσ µν R RµRνR At one-loop in pure gravity, there are three logarithmic divergences

µνρσ µνρσ 2 , Rµν R, µνρσ R RµνR RµνρσR

AcknowledgementThese two can2 be, absorbed µbyν a field, redefinitionµ νρσof the metric R RµνR RµνρσR

My thanks toThe Nick Riemann Dorey2 term can for be many massaged useful into Gauss-Bonnet. discussions. I’m supported by the Royal Society. 1 χ = d4x µνρσ 4 µν + 2 8π2 RµνρσR − RµνR R References AcknowledgementThis is purely topological. It doesn’t affect perturbative physics around flat space. [1] J. Polchinski, “Dirichlet-Branes and Ramond-Ramond Charges,” Phys. Rev. Lett. 75, 4724My thanks (1995) to [arXiv:hep-th/9510017]. Nick Dorey for many useful discussions. I’m supported by the Royal Society.

References = g exp d4x √g Z D − R topology [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

1818 2 R 2 R For ourµν purposes, the most important one-loop divergence is associated to the Gauss- Bonnetµν term (2.8). This, of course, is a total derivative in four-dimensions but will be R Rµν importantRµνR when we come to discuss gravitational instanton physics. The coefficient α is dimensionlessµνρσ and runs logarithmically at one-loop [6] µνρσ R Rµνρσ µνρσ R R M 2 α(µ)=α α log pl (3.3) 0 − 1 µ2 2 µν µνρσ , µν , µνρσ RThe2 , Gauss-BonnetR whereRµν α,0 is theR Term couplingRµνρσ at the cut-off which, for convenience, we have identified with R RµνR RµνρσR For our purposes, the most importantthe 4d one-loop Planck divergence scale M ispl. associated In general, to the for Gauss- a theory with Ns free massless spin-s fields, Bonnet term (2.8). This, of course,the is a beta-function total derivative is in given four-dimensions by [11, 12, but13] will be

important when weα come to4 discuss gravitationalµνρσ instantonµν physics.2 The coefficient α is dimensionlessSα = andα runsd logarithmically4x µνρσ µνρσ at one-loop4 µ [ν6]µν +1 2 = αχ SGB =8π2 d x Rµνρσ R −4 Rµν R + R= αχ 8π2 R R − Rα1R= R 848N2 233N3/2 52N1 +7N1/2 +4N0 48 15 − − M 2 · The coupling runs logarithmicallyα(µ)= α α log pl (3.3) The0 computation1 2 leading to this result is closely related to the trace anomaly for − µ 2 M2 α(µ)=αmasslessα log fieldsMUV inUV fixed, curved spacetime. Indeed, for spins s 1, the coefficients α(µ)=α00 α1 1log 2 ≤ where α0 is the coupling at the cut-oabove−ff−which, are for the convenience,µ same2µ as c wea haveof the identified trace anomaly. with The running coupling α(µ)results the 4d Planck scale M . In general, for a theory withN free− massless spin-s fields, pl in an RG-invariant scale,s the beta-functionwhere the beta is function given by is given [11, 12 by, 13] Christensen and Duff ’78 AcknowledgementAcknowledgement Perry ’78; Yoneya ‘78 α(µ) 1 Λ = µ exp (3.4) α1 = 848N2 233N3/2 52N1 +7N1/2 +4N0 grav MyMy thanks thanks to to Nick Nick Dorey Dorey48 for for15 many many− useful useful discussions. discussions.− I’m I’m supported supported by by the the Royal− 2 Royalα1 · Society. Society. The computation leading to this result is closely related to the trace anomaly for For our present purposes, N0 = N1/2 = N1 =0whileN3/2 = N2 =1whichgives massless fields in fixed, curved spacetime. Indeed, for spins s 1, the coefficients For us… α1 =41/48. ≤ above are the same as c a of the trace anomaly. The running coupling α(µ)results ReferencesReferences − in an RG-invariant scale, [1] J. Polchinski, “Dirichlet-Branes and Ramond-RamondIn the original discussions Charges,” of Phys. Euclidean Rev. quantumLett. 75, gravity, the suggestion seems to have [1] J. Polchinski, “Dirichlet-Branes and Ramond-Ramond Charges,” Phys. Rev. Lett. 75, been that Λαgrav(µ) (or sometimes µ)shouldbeidentifiedwiththePlanckscale.(See,for 4724 (1995) [arXiv:hep-th/9510017].Λgrav = µ exp (3.4) 4724 (1995) [arXiv:hep-th/9510017].example,− [142]).α1 In contrast, here we view Λ as a new scale which emerges in quantum grav gravity through dimensional transmutation; it dictates the length at which topological For our present purposes, N0 = N1/2 = N1 =0while4 N3/2 = N2 =1whichgives = fluctuationsg exp are unsuppressedd x √g by the Gauss-Bonnet term. Like its counterpart ΛQCD in α1 =41/48. Z D − 4 R =topology Yang-Millsg exp theory,dΛx √gcan be naturally exponentially smaller than the Planck scale. Z D − grav R topology In the original discussions of EuclideanAs we quantum will see gravity, shortly, the like suggestion its Yang-Mills seems to have counterpart, it provides the scale at which instanton effects become important. been that Λgrav (or sometimes µ)shouldbeidentifiedwiththePlanckscale.(See,for example, [14]). In contrast, here we view Λgrav as a new scale which emerges in quantum gravity through dimensional transmutation;In the it previous dictates the section, length we at whichsaw that topologicalα naturally sits in a chiral multiplet with the

ﬂuctuations are unsuppressed by thegravitational Gauss-Bonnet theta-term term. Like itsθ. counterpart These combineΛQCD intoin the complex coupling τgrav = α +2iθ. Yang-Mills theory, Λgrav can be naturally exponentially smaller than the Planckτ/2 scale.α1 This means that the scale Λgrav = µe− is also naturally complex in supergravity As we will see shortly, like its Yang-Millsand sits counterpart, in a chiral it multiplet. provides the scale at which instanton eﬀects become important.

In the previous section, we saw that α18naturally sits in a chiral multiplet with the gravitational theta-term θ. These combine into the complex coupling τgrav = α +210iθ. τ/218α1 This means that the scale Λgrav = µe− is also naturally complex in supergravity and sits in a chiral multiplet.

10 2 R

µν RµνR For our purposes, the most important one-loop divergence is associated to the Gauss- µνρσ Bonnet term (2.8). This, of course,R isµνρσ a totalR derivative in four-dimensions but will be important when we come to discuss gravitational instanton physics. The coeﬃcient α is dimensionless and runs logarithmically at one-loop [6]

2 µν 2 µνρσ , µν , µMνρσ R α(µR)=Rα α logR pl R (3.3) 0 − 1 µ2

where α0 is the coupling at the cut-off which, for convenience, we have identified with α 4 µνρσ µν 2 the 4d PlanckS scale= MApl .New Ind x general, RG-Invariant for a theory4 withScaleNs + free massless= αχ spin-s fields, GB 8π2 RµνρσR − RµνR R the beta-function is given by [11, 12, 13] 1 α = 848N 233N 52N +7N +4N 1 48 15 2 − 3/2 − M 21 1/2 0 · α(µ)=α α log UV 0 − 1 µ2 The computation leading to this result is closely related to the trace anomaly for massless fields in fixed, curved spacetime. Indeed, for spins s 1, the coefficients ≤ AcknowledgementaboveAs are in Yang-Mills, the same as wec cana replaceof the tracethe log anomaly. running with The an running RG invariant coupling scaleα( µ)results − in an RG-invariant scale, My thanks to Nick Dorey for many useful discussions. I’m supported by the Royal α(µ) Society. Λ = µ exp (3.4) grav − 2α 1

ReferencesFor our present purposes, N0 = N1/2 = N1 =0whileN3/2 = N2 =1whichgives α1 =41/48. This scale will be associated with non-trivial spacetime topologies. [1] J. Polchinski, “Dirichlet-Branes and Ramond-Ramond Charges,” Phys. Rev. Lett. 75,

4724In (1995) the original [arXiv:hep-th/9510017]. discussions of Euclidean(We will see quantum an example) gravity, the suggestion seems to have been that Λgrav (or sometimes µ)shouldbeidentifiedwiththePlanckscale.(See,for example, [14]). In contrast, here we view Λgrav as a new scale which emerges in quantum gravity through dimensional= transmutation;g exp it dictatesd4 thex √ lengthg at which topological Z D − R fluctuations are unsuppressedtopology by the Gauss-Bonnet term. Like its counterpart Λ in QCD Yang-Mills theory, Λgrav can be naturally exponentially smaller than the Planck scale. As we will see shortly, like its Yang-Mills counterpart, it provides the scale at which instanton effects become important.

In the previous section, we saw that α naturally sits in a chiral multiplet with the

gravitational theta-term θ. These combine into the complex coupling τgrav = α +2iθ. τ/2α1 This means that the scale Λgrav = µe− is also naturally complex in supergravity and sits in a chiral multiplet.

18 10 Fermions This bosonic eﬀective action has a fermionic counterpart which is dictated by supersymmetry. Let us work for now with a Majorana basis of 4d gamma matrices,

0 γi 10 γi = 3d i =0, 1, 2 , γz = (2.5) γi 0 0 1 3d −

Upon dimensional reduction, the 4d Majorana gravitino ψµ decomposes into a 3d spin- 3/2 Dirac fermion λi and 3d spin-1/2 Dirac fermion χ. To perform this reduction, it’s µ simplest to work with the frame index, so that ψa = ea ψµ. Further, to make life easy for ourselves, we restrict to the ﬂat background R1,2 S1 with metric (2.2)andmake × the spinor ansatz,

Reλi +(γ3d)iImχ Reχ ψi = and ψz = (2.6) Imλi +(γ3d)iReχ Imχ The gravitino kinetic term in (2.1)thenbecomes, M 2 S = d4x √ g pl ψ¯ γµνρ∂ ψ fermions − 2 µ ν ρ M L 1 = d3x g 3 λ¯ ijk∂ λ χ¯ ∂χ/ (2.7) − (3) R 2 i i k − After dividing out by local supersymmetry transformations, the spin-3/2 fermion λi carries no propagating degrees of freedom. (This is the supersymmetric analog of the statement that the 3d metric carries no propagating degrees of freedom.) In contrast, the spin-1/2 fermion χ carries two propagating degrees of freedom; these are the supersymmetric partners ofAnotherR and σDivergence:. We will postpone The Anomaly a more detailed discussion of how supersymmetry relates R, σ and χ to Sections 3.4 and 4.4.

2.2 TopologicalThe classical Terms action is invariant under rotations of the phase of the fermion. There is one further one-loop divergence that will play a role in our story. This is re- µ ¯ νµρ 5 In additionsponsibleThis to for the U(1) the Einstein-HilbertR axialsymmetry anomaly does fornot action, the surviveU(1) in thereR thesymmetry quantum are two with theory. topological current J5 = termsiψνγ thatγ ψ willρ. play aroleinourstory.Botharehigherderivativeterms,withdimensionlesscoeIn general, the anomaly is given by [15, 16, 12] ﬃcients. They are the Gauss-Bonnet term1 J µ = 21N N µνρσ (3.5) ∇µ 5 24 16π2 3/2 − 1/2 Rµνρσ R · α 4 µνρσ Sα = d x√g µνρσ (2.8) For us, N1/2 =0andN3/2 = 1.32 Asπ2 usual, the anomalyR canR be compensated by shifts on The phase of the fermion can be absorbed by shifting the theta term the gravitational theta angle which means that we should view Λgrav as carrying U(1)R whichcharge. integrates to the Euler characteristic of the manifold, and the Pontryagin class,

θ 4 µνρσ 3.2 One-Loop DeterminantsSθ = d x√ g µνρσ (2.9) 16π2 − R R In this section, we present the determinants arising from one-loop ﬂuctuations of the graviton, the gravitino and their ghosts. This material is standard fare but, since we will need this for a number of subsequent calculations, we take the time to describe it in some detail. 7

The Graviton and its Ghost Throughout this paper, we use the background ﬁeld method. We work in Euclidean

space and write the metric as background gµν, which is taken to obey the Einstein

equations, and ﬂuctuation hµν,

g g + h µν → µν µν From now on, all covariant derivatives and curvatures are to be thought of with respect to the background. It’s useful to further decompose the ﬂuctuations into the trace h = gµνh and traceless parts, h¯ = h 1 g h. µν µν µν − 4 µν

We expand the Einstein-Hilbert action to quadratic order in hµν following, for example, [19]. The residual gauge freedom h h + ξ + ξ is ﬁxed by imposing µν → µν ∇µ ν ∇ν µ the condition 1 µ h g h =0 ∇ µν − 2 µν The resulting Fadeev-Popov determinants are exponentiated in the usual fashion through the introduction of ghosts which, in this context, are anti-commuting complex vectors.

The Einstein-Hilbert action is, famously, unbounded below. In the present context, this shows up in the negative-deﬁnite operator 2 for the trace ﬂuctuations h. We follow ∇

11 Fermions Fermions ThisThis bosonic bosonic eﬀ eﬀectiveective action action has has a a fermionic fermionic counterpart counterpart which which is is dictated dictated by by super- super- symmetry.symmetry. Let Let us us work work for for now now with with a a Majorana basis basis of of 4d 4d gamma gamma matrices, matrices,

ii i 00 γγ3d z 1010 γγi== 3d ii =0,,11,,22 ,, γγz == (2.5)(2.5) ii γγ3d 00 00 11 3d −−

UponUpon dimensional dimensional reduction, reduction, the the 4d 4d Majorana Majorana gravitino gravitinoψψµµdecomposesdecomposes into into a 3d a 3d spin- spin- 3/23/2 Dirac Dirac fermion fermionλλi iandand 3d 3d spin-1/2 spin-1/2 Dirac fermion fermionχχ.. To To perform perform this this reduction, reduction, it’s it’s µµ simplestsimplest to to work work with with the the frame frame index, index, so that that ψψaa ==eeaaψψµµ.. Further, Further, to to make make life life easy easy forfor ourselves, ourselves, we we restrict restrict to to the the flat flat background background RR11,2,2 SS11withwith metric metric (2.2 (2.2)andmake)andmake ×× thethe spinor spinor ansatz, ansatz, Reλ +(γ ) Imχ Reχ Reλii+(γ33dd)iiImχ Reχ ψi = and ψz = (2.6) ψi = and ψz = (2.6) Imλi +(γ3d)iReχ Imχ Imλi +(γ3d)iReχ Imχ The gravitino kinetic term in (2.1)thenbecomes, The gravitino kinetic term in (2.1)thenbecomes, M 2 S = d4x √ g Mpl2 ψ¯ γµνρ∂ ψ S fermions= d4x √−g 2pl ψ¯µ γµνρ∂ν ψρ fermions − 2 µ ν ρ 3 M3L 1 ¯ ijk = d x g(3) M L 1λi ∂iλk χ¯ ∂χ/ (2.7) = d3x −g R3 2 λ¯ ijk∂ λ − χ¯ ∂χ/ (2.7) − (3) R 2 i i k − After dividing out by local supersymmetry transformations, the spin-3/2 fermion λi Aftercarries dividing no propagating out by local degrees supersymmetry of freedom. (This transformations, is the supersymmetric the spin-3/2 analog fermion of the λi carriesstatement no propagating that the 3d degrees metric carries of freedom. no propagating (This is the degrees supersymmetric of freedom.) In analog contrast, of the statementthe spin-1/2 that fermion the 3dχ metriccarries carries two propagating no propagating degrees degrees of freedom; of freedom.) these are the In contrast, super- thesymmetric spin-1/2 partners fermion χ ofcarriesR and twoσ. We propagating will postpone degrees a more of freedom; detailed these discussion are the of howsuper- Ifsymmetricsupersymmetry we care only partners about relates of perturbativeRRand, σ andσ.χ Weto physics will Sections postpone on3.4R3and aS more4.41,. then detailed we can discussion neglect of these how × terms.supersymmetry However, relates whenR we, σ startand χ toto sum Sections over3.4 manifoldsand 4.4 of. different topology, they 2.2 Topological Terms become important. Topological Terms 2.2In Topological addition to the Terms Einstein-Hilbert action, there are two topological terms that will play InUsually,aroleinourstory.Botharehigherderivativeterms,withdimensionlesscoe addition when to the working Einstein-Hilbert with an effective action, field there theory, are two we topological keep all relevant terms and thatffi marginalcients. will play termsaroleinourstory.Botharehigherderivativeterms,withdimensionlesscoeThey in are the the action,One-loop Gauss-Bonnet neglecting effects tell term us only that we the should irrelevant consider operators two topological on terms the grounds thatfficients. they areThey suppressed are the Gauss-Bonnet by some high term massα scale. In the present case, there are two further S = d4x g µνρσ 2 (2.8)µν four-derivative terms whichα come with2 dimensionless√ µνρσ coefficients: and µν . 32απ R R R R R However, both can be absorbedS = into thed4 Einstein-Hilbertx√g µνρσ term through a redefinition(2.8) which integrates to the Eulerα characteristic32π2 of theRµ manifold,νρσ R and the Pontryagin class, of the metric [6]. For this reason, we need only consider Sα and Sθ above. θ which integrates to the EulerS = characteristicd4x√ ofg the manifold,µνρσ and the Pontryagin(2.9) class, In supergravity, the Gauss-Bonnetθ 16π2 (2.8)andPontryagin(− RµνρσR 2.9)termscanbewritten θ as an F-term [17, 18] (usingS = the so-calledd4x “chiral√ g projectionµνρσ operator”). This, in turn,(2.9) θ 16π2 − RµνρσR means that the twoIn supergravity coupling, these constants two couplingα and constantsθ combine sit in a chiral into multiplet the complex coupling 7 τgrav = α +2iθ (2.10) 7 which naturally lives in a chiral multiplet. We will see later that τgrav appears in the instanton generated superpotential.

3. Perturbative Aspects

In this section we describe the results of quantum fluctuations of the graviton and gravitino around the background R1,2 S1. There are two kinds of effects: those from × divergences that arise already in four dimensions; and finite corrections to the low- 2 2 energy effective action which are suppressed by the dimensionless combination 1/MplR .

3.1 Summary We open this section by summarising the main results. The remainder of the section contains details of the computations.

8 These finite corrections were first computed in the Kaluza-Klein context in [1], where they manifested themselvesFinite as a Casimir Quantum force, Corrections causing the Kaluza-Klein circle to either shrink or expand. (The analogous calculation was performed earlier in the thermal context [9]). The effective 3d potential is given by1

N N L3 Casimir Energy: V = B − F (3.1) eﬀ − 720π R6

Here NB is the number of massless bosonic degrees of freedom;Appelquist these makeand Chodos the ’83 Kaluza-

Klein circle contract. NF the number of massless fermionic degrees of freedom; these make the circle expand. Of course, in supersymmetric theories NB = NF and Kaluza- Klein compactifications are perturbatively stable. The presence of fermions with periodic boundary conditions means that the bubble-of-nothing instability is absent in this Supersymmetry means that NB=NF and this Casimir energy vanishes. theory [2], but other gravitational instantons, discussed in Section 4, will contribute. But there are other effects…. Although the perturbative potential vanishes, there are still finite one-loop effects of interest. These renormalise the kinetic terms in the effective action (2.4). Much of this section is devoted to computing these effects; we will show that the low-energy effective action becomes,

1 5 L 1 L ∂R 2 = M + M Leff 2 3 16π R2 R(3) − 3 − 6π R2 R 1 2 11 L − L2 ∂σ M + (3.2) − 3 24π R2 R4 2π This is the one-loop effective action. We certainly expect that there will be further corrections, both from higher-loops and from non-perturbative effects. Nonetheless, this will suffice for our purposes. The most important fact that we will need is the observation that the renormalisation of the R and σ kinetic terms come with different coefficients. This will prove important later when we reconcile this with supersymmetry: it results in a one-loop shift in the complex structure. We describe this in Section 3.4.

Anomalies and One-Loop Divergences Also important for our story are the divergences that appear at one-loop. It is well known that the S-matrix is one-loop finite in pure Einstein gravity [6] and two-loop finite in pure =1supergravity[7, 8]. Nonetheless, as we now review, these theories N do suffer from one-loop divergences which are related to anomalies.

1The standard Casimir potential in four dimensions scales as 1/R3.The1/R6 scaling seen here arises after a Weyl transformation to the 3d Einstein frame.

9 These finite corrections were first computed in the Kaluza-Klein context in [1], where they manifested themselves as a Casimir force, causing the Kaluza-Klein circle to either shrink or expand. (The analogous calculation was performed earlier in the thermal context [9]). The effective 3d potential is given by1

N N L3 V = B − F (3.1) eﬀ − 720π R6

Here NB is the number of massless bosonic degrees of freedom; these make the Kaluza-

Although the perturbative potential vanishes, there are still finite one-loop effects of interest. These renormalise the kinetic terms in the effective action (2.4). Much of this section is devotedOne loop to corrections computing to thesethe kinetic effects; terms we give will show that the low-energy effective action becomes,

1The standard Casimir potential in four dimensions scales as 1/R3.The1/R6 scaling seen here arises after a Weyl transformation to the 3d Einstein frame.

N N L3 V = B − F (3.1) eﬀ − 720π R6

Here NB is the number of massless bosonic degrees of freedom; these make the Kaluza-

1 5 L 1 L ∂R 2 = M + M Leff 2 3 16π R2 R(3) − 3 − 6π R2 R 1 2 11 L − L2 ∂σ M + (3.2) − 3 24π R2 R4 2π This is the one-loop effective action. We certainly expect that there will be further corrections, both from higher-loops and from non-perturbative effects. Nonetheless, this will suffice forSomething our purposes. important: The these most two importantnumbers are fact different! that we will need is the observation that the renormalisation of the R and σ kinetic terms come with different coefficients. This will prove important later when we reconcile this with supersymmetry: it results in a one-loop shift in the complex structure. We describe this in Section 3.4.

1The standard Casimir potential in four dimensions scales as 1/R3.The1/R6 scaling seen here arises after a Weyl transformation to the 3d Einstein frame.

9 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 is blurred, the obvious analogy is toz consider= R cos θ ‘initial’ and ‘final’ surfaces which are asymptotically flat hemispheres of S2 with a (necessarily trivial) S1 bundle, and require them to be glued in a locally smooth, flat manner. The non-trivial global behaviour now arises due to the possibilityδx =(R ofcos creatingθ sin φ)δθ a non-trivial+(R sin θ cos bundleφ)δφ of S1 over the whole 2 S . δy =(R cos θ cos φ)δθ (R sin θ sin φ)δφ − δz = (R sin θ)δθ We conclude that, despite the− different boundary conditions, we should be summing over Taub-NUT configurations to determine the low-energy physics on R1,2 S1.We × would reach the same conclusion by considering the low-energy world where we would expect to sum over different Dirac monopoleθ configurations provided they have a suit- able microscopic completion [5]. We also reach the same conclusion by considering the very high-energy world of string theory, where these Taub-NUT instantons can be φ viewed as D6-brane instantons wrapping manifolds of G2-holonomy [4].

The multi-Taub-NUT solution (4.2)enjoys31,d k1 bosonic1 zero modes, parameterised by = R − S M × 6 the centres Xa,and2k spin-3/2 fermionic zero modes [28] . Although this result is well known, we will provide a slightly diﬀerent derivation of the index theorem for the fermionic zero modes in Section 4.3 en route1 to calculating the one-loop determinants. ∂ σ F νρ For now, we merely note that only theµ k∼=2 1µ Taub-NUTνρ solution, with two fermionic zero modes, can contribute to the superpotential [3].

Λgrav Mpl The Action The Complex Structure The Einstein-Hilbert action evaluated on the Taub-NUT space with charge k =1is, 2 1 2 after subtracting appropriate1 counterterms,L ∂R given by11 [29, 35L], − L2 ∂σ eff = The1 two fields R2 and σ must combine+ 1+ in a complex2 number4 L − 6π M3R R 2 2 2 24π M3R R 2π STN =2π MplR where R here is interpreted as the asymptotic radius of the circle. (In the coordinates 2 2 ∂R 1 L2 2 ∂σ2 2 (4.2), we couldClassically: just as well have= written S+TN =2π MplL .) However, there are a Leff R M 2 R4 2π number of further contributions to the action. The3 first comes from the dual 3d photon 1 which, as first observed by Polyakov,= acts as∂ a∂ chemical† potential for the topological ( + )2 S S instanton charge [5]. This follows fromS theS† coupling (2.3): the 3d field strength arising from the metric (4.2)hascharge S2 F =2πL, which ensures that the single Taub-NUT instanton also comes with a factor of =2π2M 217R2 + iσ S pl 6For Yang-Mills instantons, the number of zero modes can be simply determined by integrating the anomaly. In the present case there is a mismatch between the integrated anomaly (3.5) and the number of zero modes due to the presence of boundary terms. These are known as eta-invariants [36] and will also play a role when we come to discuss the one-loop determinants around the background of the gravitational instanton.

25 1 2 2GM 2 2GM − 2 2 2 2 2 distance = 1 2 δt + 1 2 dr + r δθ +sin1 θδφ − − rc −2GMrc 2GM − distance2 = 1 δt2 + 1 dr2 + r2 δθ2 +sin2 θδφ2 − − rc2 − rc2 x = R sin θ sin φ y = R sin θ cos φ x = R sin θ sin φ z = R cos θ 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 and the classical action (2.4)forthiscomplexscalartakestheformδx =(−R cos θ sin φ)δθ +(R sin θ cos φ)δφ δz = (R sin θ)δθ − 3 δy =(R1cos θ cos φ)δθ (R sin θ sin φ)δφ S = M3 d x g(3) ∂ ∂ † − (3.25) − − ( + )2 S S δz = S (RSsin† θ)δθ − which is derived from the classical K¨ahler potentialθ

K = log( + †)(3.26) − φS S θ

The presence of the Planck mass Mpl in the complex structure (3.24)meansthatthis chiral multiplet does not survive the rigid1 limit,d 1 in1 which gravity is decoupled. (The = R − S φ distinction between rigid and gravitationalM theories× was stressed, in particular, in [24]).

This, in turn, means thatThe we cannot Complex use the Structure fact that R sits1,d in1 a chiral1 multiplet to = R − S restrict the way it appears in superpotentials1 when rigidM supersymmetric× gauge theories ∂ σ F νρ are compactiﬁed on a circle as in [38,µ39]∼4. 2 µνρ

The two fields R and σ must combine in a complex number1 νρ One-Loop Corrected Complex StructureΛgrav Mpl ∂ σ F µ ∼ 2 µνρ As we have just seen, the kinetic terms are corrected at one-loop. This in principle At one-loop affects both the complex structure and Kählerpotential. For our present purposes, we 2 Λgrav M1 pl 2 are only concerned with1 the shiftL to the∂R complex structure.11 L − L2 ∂σ eff = 1 2 + 1+ 2 4 L − 6π M3R R 24π M3R R 2π The renormalisation of the complex structure can be seen from the fact that the 2 =2 K( , †) ∂ ∂ † 2 2 1 2 (∂R) and (∂σ) terms pick up different 1/R corrections in (3.23). (Strictly speaking,− 2 We want to write thisS inS the formS S 1 L ∂R 11 L L ∂σ eff = 1 2 + 1+ 2 4 we should first perform aL conformal transformation− 6π M3R soR that we are working24π M3R in the R 2π 4 Einstein frame, but this only affects the complex2 structure at2 order 1/R and so can be eff = K∂(R, †) ∂ 1∂ L†2 ∂σ neglected at one-loop order.)L It is= simpleS S to+ checkS S that the one-loop corrected complex Leff R M 2 R4 2π structure is given by 3 1 † = 2 ∂ ∂ 2 2 2 2 2( 2+ 7†) S S2 2 ∂R 1 L ∂σ =2π MplRS +S log(M=plR )+iσ + (3.27) S 48 Leff R M 2 R4 2π 3 (Tracing the origin of this shift, we see that it depends on1 the a coefficients defined in eff = ∂ ∂ † (3.16), but is independent of the b coefficientsL defined( in+ (3.17))).2 S WeS will have use for 17 S S† this later when we compute the instanton-generated superpotential.

3.5 Divergences and the Heat Kernel The gradient expansion employed in Section 3.3 is the simplest17 approach for computing the eﬀective action at the two derivative level. However, it becomes increasingly cum- bersome as we look to higher derivatives. In particular, as described at the beginning

4We thank N. Seiberg for discussions on these issues.

20 Non-Perturbative Quantum Corrections We note that the interpretation of this “running” as a scale-dependent coupling constant comes with a caveat. In gauge theories, the running coupling g2(µ)tellsus how the strength of local interactions varies with the energy scale of the process. But, in the gravitational context, there is no local process associated to the Gauss-Bonnet term. Instead, it knows only about the global properties of the space. The real physics in this running coupling is the emergence of the infra-red scale Λgrav deﬁned in (3.4)which tells us characteristic scale at which manifolds with diﬀerent topologies contribute to the path integral.

4. Non-Perturbative Aspects Gravitational Instantons In this section we describe the instanton corrections to the low-energy eﬀective action. We will show that they generateLook for other a superpotential saddle points of the term action for the chiral multiplet . The S techniques of gravitational instanton computations were pioneered in the late 1970s [27, 28, 29] and much of this section is devoted to reviewing and extending this machinery. We start, however, with a brief introduction to gravitational instantons and the role they play in =1supergravity. N 4.1 Gravitational Instantons Gravitational instantons are saddle points of the four-dimensional path integral. In supersymmetric theories, we can restrict attention to (anti)-self-dual solutions to the Einstein equationsWe satisfying want these to contribute to the (super)potential. They must obey

= (4.1) Rµνρσ ± Rµνρσ Such backgrounds preserve half of the supersymmetry. This means that supersymmetry transformations generate only two fermionic Goldstino zero modes, which is the right number to contribute towards a superpotential in =1theories[30]. The self-duality N requirement (4.1)isanecessary,butnotsuﬃcient, condition for instantons to contribute to the superpotential; there may also be further fermionic zero modes which do not arise from broken supersymmetry which we describe below.

For theories on R3 S1, the gravitational instantons are Kaluza-Klein monopoles [31, × 32] which, in the present context are perhaps best referred to as “Kaluza-Klein instantons”. From the low-energy 3d perspective, these solutions look like Dirac monopoles and the calculation can be thought of as a gravitational completion of Polyakov’s famous computation [5]. The contribution of these “Kaluza-Klein instantons” has been discussed previously in the non-supersymmetric context in [33] and, more recently, in [34].

23 Taub-NUT Instantons

The appropriate metrics are given by the multi-Taub-NUT solutions

Gibbons and Hawking ‘78 The simplestThe simplest class class of gravitational of gravitational instantons instantons are are the the multi-Taub-NUT multi-Taub-NUT metrics metrics [27], [27],

2 2 11 2 2 ds ds= U=(xU)(dxx)dxdxdx++UU((xx))−− (dzdz++AA dxd)x) (4.2)(4.2) · · · · with with k Lk 1 with U(x)=1+L 1 and A = U 2 x X ∇× ±∇ U(x)=1+ a=1 a and A = U 2 x| −Xa | ∇× ±∇ a=1 | − | The metric is smooth when z[0, 2πL)andtheXa are distinct. For A = U, ∈ ∇× ±∇ The metricthe Riemann is smooth tensor when obeysz [0, 2=πL)andthe. Xa are distinct. For A = U, From the low-energyR∈ µ3dνρσ perspective,∓ Rµνρσ these look like Dirac monopoles.∇× ±∇ the Riemann tensor obeys = . The Taub-NUT metricµ takesνρσ the sameµνρσ form as our Kaluza-Klein ansatz (2.2)with This is the gravitationalR verson∓ R of Polyakov’s famous calculation. Polyakov ‘77 U = L2/R2. However, because U 1asymptotically,itmeansthatwehavemadea The Taub-NUT metric takes the same→ form as our Kaluza-Klein ansatz (2.2)with U = Lcoordinate2/R2. However, choice in because which theU fiducial1asymptotically,itmeansthatwehavemadea length L is taken to be the physical asymptotic length of the circle: R(x) L. → Gross ’84 coordinate choice in which the→ fiducial length L is taken to be theHartnoll physical and Ramirez asymptotic ‘13 length ofOne the might circle: wonderR(x) aboutL. the relevance of Taub-NUT spaces to the Euclidean path integral. Ultimately, we’re→ interested in physics on R1,2 S1 and, after a Wick rotation, × Onethe might boundary wonder of space about is theS2 relevanceS1. But for ofk Taub-NUT= 0, the boundary spaces of to the the manifold Euclidean is the path × integral.S1 Ultimately,is fibered non-trivially we’re interested over the inS physics2. For example, on R1,2 withS1 and,k =1,theboundaryis after a Wick rotation, 3 × the boundarytopologically of spaceS . The is S question2 S1. at But hand for isk whether= 0, the we shouldboundary sum of over the these manifold different is the × S1 is fiberedboundary non-trivially conditions in the over path the integral.S2. For example, with k =1,theboundaryis topologicallyA similarS3. question The question arises in at gauge hand theories is whether in flat we space, should where sum the over issue these becomes different boundarywhether conditions one should in the sum path over integral. topologically non-trivial bundles at infinity. Here the answer is certainly yes: a trivial gauge bundle can be smoothly deformed into an A similarinstanton-anti-instanton question arises pair in gauge which are theories subsequently in flat moved space, far where apart. theSuch issue configura- becomes whethertions one certainly should contribute sum over to topologically the path integral non-trivial but locality bundles and cluster at infinity. decomposition Here the answerthen is certainly requires us yes: to also a sum trivial over gauge individual bundle instanton can bundles. be smoothly (See, for deformed example, [ into24] an instanton-anti-instantonfor a recent discussion pair of this which topic). are However, subsequently these same moved arguments far apart. also Such hold inconfigura- the present case: we can equally well locally nucleate a NUT-anti-NUT pair which can tions certainly contribute to the path integral but locality and cluster decomposition then be moved far apart. This suggests that should sum over all asymptotic windings. then requires(There is, us admittedly, to also sum one over loophole individual which is instanton the lack of local bundles. observables (See, for in a example, theory of [24] for a recentgravity discussion but this does of not this seem topic). to be However, a serious objection these same to the arguments argument.) also hold in the present case: we can equally well locally nucleate a NUT-anti-NUT pair which can Another way to motivate including non-trivial S1 bundles is to consider a parallel then be moved far apart. This suggests that should sum over all asymptotic windings. to a more familiar story with gauge theory instantons. There, one imposes ‘initial’ (Thereand is, admittedly, ‘final’ conditions one in loophole Euclidean which time is and the boundary lack of local conditions observables at spatial in infinity a theory of gravitywhich but this require does local not decay seem everywhere, to be a serious but allow objection for non-trivial to the argument.) global behaviour of the solution. For us, where the distinction between initial and boundary conditions Another way to motivate including non-trivial S1 bundles is to consider a parallel to a more familiar story with gauge theory instantons. There, one imposes ‘initial’ and ‘final’ conditions in Euclidean time and boundary conditions at spatial infinity 24 which require local decay everywhere, but allow for non-trivial global behaviour of the solution. For us, where the distinction between initial and boundary conditions

24 2 R

2 µν R RµνR

µν µν µνρσ R R RµνρσR

µνρσ RµνρσR 2 , µν , µνρσ R RµνR RµνρσR

2 µν µνρσ , µν , µνρσ R R R R α R S = d4x µνρσ 4 µν + 2 = αχ α 8π2 RµνρσR − RµνR R α S = d4x µνρσ 4 µν + 2 = αχ α 2 µνρσThe Boundaryµν of the Space 8π R R − R R R M 2 α(µ)=α α log UV 0 − 1 µ2 M 2 α(µ)=α α log UV 0 − 1 µ2 The boundary of Taub-NUT is not the same∂( asR3 the Sboundary1)=S2 of flatS1 space. × ×

3 1 2 1 3 ∂(R S )=S S but ∂(TNk)=S /Zk × ×

3 AcknowledgementShould∂(TN wek)= includeS / Zsuchk geometries in the path integral?

My thanks to Nick Dorey for many useful discussions. I’m supported by the Royal Acknowledgement Society.

My thanks to Nick Dorey for many useful discussions. I’m supported by the Royal Society. References [1] J. Polchinski, “Dirichlet-Branes and Ramond-Ramond Charges,” Phys. Rev. Lett. 75, References 4724 (1995) [arXiv:hep-th/9510017].

[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

= g exp d4x √g Z D − R topology 18

18 2 R

2 µν R RµνR

µν µν µνρσ R R RµνρσR

µνρσ RµνρσR 2 , µν , µνρσ R RµνR RµνρσR

3 1 2 1 3 ∂(R S )=S S but ∂(TNk)=S /Zk × ×

3 AcknowledgementShould∂(TN wek)= includeS / Zsuchk geometries in the path integral?

My thanks to Nick Dorey for many useful discussions. I’m supported by the Royal Acknowledgement Society. Yes! My thanks to Nick Dorey for many useful discussions. I’m supported by the Royal Society. References [1] J. Polchinski, “Dirichlet-Branes and Ramond-Ramond Charges,” Phys. Rev. Lett. 75, References 4724 (1995) [arXiv:hep-th/9510017].

[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

= g exp d4x √g Z D − R topology 18

18 2 R

2 µν R RµνR

µν µν µνρσ R R RµνρσR

µνρσ RµνρσR 2 , µν , µνρσ R RµνR RµνρσR

3 1 2 1 3 ∂(R S )=S S but ∂(TNk)=S /Zk × ×

3 AcknowledgementShould∂(TN wek)= includeS / Zsuchk geometries in the path integral?

My thanks to Nick Dorey for many useful discussions. I’m supported by the Royal Acknowledgement Society. Yes! My thanks to Nick Dorey for many useful discussions. I’m supported by the Royal Society. References c.f. Atiyah-Hitchin with boundary a circle fibre over RP2 for which the answer is probably no! [1] J. Polchinski, “Dirichlet-Branes and Ramond-Ramond Charges,” Phys. Rev. Lett. 75, References 4724 (1995) [arXiv:hep-th/9510017].

[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

= g exp d4x √g Z D − R topology 18

18 Zero Modes of Taub-NUT

The simplest class of gravitational instantons are the multi-Taub-NUT metrics [27], The simplest class of gravitational instantons are the multi-Taub-NUT metrics [27], 2 1 2 ds = U(x)dx dx + U(x)− (dz + A dx) (4.2) 2 · 1 · 2 ds = U(x)dx dx + U(x)− (dz + A dx) (4.2) with · · with k L 1 U(x)=1+ and A = U 2k x X ∇× ±∇ L a=1 1 a U(x)=1+ | − | and A = U The metric is smooth when2 z [0x, 2πLX)andthea X ∇×are distinct.± For∇ A = U, a=1∈ | − | a ∇× ±∇ the Riemann tensor obeys = . R3kµνρσ bosonic∓ Rzeroµνρσ modes The metric is smooth when z [0, 2πL)andtheXa are distinct. For A = U, The Taub-NUT metric takes∈ the same form as our Kaluza-Klein ansatz∇× (2.2)with±∇ the Riemann tensor obeys µνρσ2k fermionic= µ νρσzero. modes U = L2/R2. However, becauseR U ∓ R1asymptotically,itmeansthatwehavemadea → Thecoordinate Taub-NUT choice metric in which takes the the fiducial same length formL asis taken our Kaluza-Klein to be the physical ansatz asymptotic (2.2)with length2 2 of the circle: R(x) L. U = L /R . However, because→ U 1asymptotically,itmeansthatwehavemadea Only k=1 solution→ contributes to the superpotential coordinateOne choice might in wonder which about the fiducial the relevance length of Taub-NUTL is taken spaces to be to the the physical Euclidean asymptotic path lengthintegral. of the circle: Ultimately,R(x) we’reL interested. in physics on R1,2 S1 and, after a Wick rotation, × the boundary of space→ is S2 S1. But for k = 0, the boundary of the manifold is the × OneS might1 is fibered wonder non-trivially about the over relevance the S2. of For Taub-NUT example, with spacesk =1,theboundaryis to the Euclidean path integral.topologically Ultimately,S3. we’re The question interested at hand in physics is whether on R we1,2 shouldS1 and, sum overafter these a Wick different rotation, × the boundaryboundary of conditions space is inS2 the pathS1. But integral. for k = 0, the boundary of the manifold is the 1 × 2 S is fiberedA similar non-trivially question arises over in the gaugeS . theories For example, in flat space, with wherek =1,theboundaryis the issue becomes 3 topologicallywhetherS one. The should question sum over at topologically hand is whether non-trivial we should bundles sum at infinity. over these Here di thefferent boundaryanswer conditions is certainly in the yes: path a trivial integral. gauge bundle can be smoothly deformed into an instanton-anti-instanton pair which are subsequently moved far apart. Such configura- A similartions certainly question contribute arises in to gauge the path theories integral in but flat locality space, and where cluster the decomposition issue becomes whetherthen one requires should us tosum also over sum topologically over individual non-trivial instanton bundles. bundles (See, at for infinity. example, Here [24] the answerfor is a certainly recent discussion yes: a of trivial this topic). gauge However, bundle these can same be argumentssmoothly also deformed hold in the into an instanton-anti-instantonpresent case: we can pair equally which well are locally subsequently nucleate a moved NUT-anti-NUT far apart. pair Such which configura- can tions certainlythen be moved contribute far apart. to This the pathsuggests integral that should but locality sum over and all asymptotic cluster decomposition windings. (There is, admittedly, one loophole which is the lack of local observables in a theory of then requires us to also sum over individual instanton bundles. (See, for example, [24] gravity but this does not seem to be a serious objection to the argument.) for a recent discussion of this topic). However, these same arguments also hold in the present case:Another we way can to equally motivate well including locally non-trivial nucleateS1 abundles NUT-anti-NUT is to consider pair a parallel which can then beto moved a more far familiar apart. story This with suggests gauge theory that should instantons. sum overThere, all one asymptotic imposes ‘initial’ windings. and ‘final’ conditions in Euclidean time and boundary conditions at spatial infinity (There is, admittedly, one loophole which is the lack of local observables in a theory of which require local decay everywhere, but allow for non-trivial global behaviour of gravitythe but solution. this does For not us, seem where to the be distinction a serious between objection initial to and the argument.)boundary conditions Another way to motivate including non-trivial S1 bundles is to consider a parallel to a more familiar story with gauge theory instantons. There, one imposes ‘initial’ and ‘final’ conditions in Euclidean time and24 boundary conditions at spatial infinity which require local decay everywhere, but allow for non-trivial global behaviour of the solution. For us, where the distinction between initial and boundary conditions

24 Doing the Computation

Action, Zero Modes, Jacobians, Determinants, Propagators…. 2 R

µν RµνR 2

µνρσ R µνρσ R R µν RµνR

2 µν µνρσµνρσ , , µνρσ R RµνR RRµνρσRR

2 µν µνρσ α 4 ,µνρσ µν , µν µνρσ2 Sα = d x µRνρσ R R4 µν R+ R= αχ 8π2 R R − R R R

α 4 µνρσ µν 2 Sα = d x µνρσ M 2 4 µν + = αχ α(µ)=8π2 α αRlog R UV− R R R 0 − 1 µ2 2 MUV α(µ)=α0 α1 log 3 1 2 − 1 µ2 ∂(R S )=S S × ×

3 3 1 2 1 ∂(TNk)=∂(RS /ZSk)=S S × ×

3 The Determinants∂(TNk)= S /Zk 2 ∂(AHk)=RP /Zk

2 ∂(AHk)=RP /Zk det(Fermions) dets = det(Bosons) det Fermions dets = det Bosons dets = 1 ?

Supersymmetry dets = 1 ? Hawking and Pope ‘78 Acknowledgement Acknowledgement My thanks to Nick Dorey for many useful discussions. I’m supported by the Royal Society. My thanks to Nick Dorey for many useful discussions. I’m supported by the Royal Society.

18 18 2 R

µν RµνR

µνρσ RµνρσR

2 , µν , µνρσ R RµνR RµνρσR

α S = d4x µνρσ 4 µν + 2 = αχ α 8π2 RµνρσR − RµνR R

M 2 α(µ)=α α log UV 0 − 1 µ2

∂(R3 S1)=S2 S1 × ×

3 ∂(TNk)=S /Zk

∂(R3 S1)=S2 S1 × × 2 ∂(AHk)=RP3 /Zk ∂(TNk)=S /Zk

det(Fermions)2 ∂dets(AH =k)=RP /Zk det(Bosons) The Determinants det(Fermions) dets = detsdet(Bosons) = 1 ? In a self-dual background, you can write the determinants as

dets = 1 ? 1/4 1/2 − det D/ † D/ det D/ † D/ dets = det D/ D/ †1/4 det D/ D/ †1/2 spin 3/2 − spin 1/2 det D/ † D/ − det D/ † D/ − dets = det D/ D/ † det D/ D/ † A somewhat detailed calculation givesspin 3/2 spin 1/2 − − 7/48 41/4818 1 dets = A µ2 R2 41/48 7/48 An ugly number2 UV cut-off scale µ 1 S iσ τgrav − TN− − = C 2 2 2 e e W Mpl MplR We’ve seen these fractions before! Acknowledgement

My thanks to Nick Dorey for many useful discussions. I’m supported by the Royal Society.

References

[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

18 ∂(R3 S1)=S2 S1 × ×

3 ∂(TNk)=S /Zk

is blurred, the obvious analogy is to consider ‘initial’ and ‘final’ surfaces which are 2 asymptotically flat hemispheres∂(AHk)=RP of S2 /withZk a (necessarily trivial) S1 bundle, and require them to be glued in a locally smooth, flat manner. The non-trivial global behaviour now arises due to the possibility of creating a non-trivial bundle of S1 over the whole S2. det(Fermions) and the classical action (2.4)forthiscomplexscalartakestheformdets = det(Bosons) We conclude that, despite the different boundary conditions, we should be summing 3 1 1,2 1 S = M overd Taub-NUTx g configurations∂ ∂ to† determine the low-energy(3.25) physics on R S .We − 3 − (3) ( + )2 S S × would reach the sameS conclusionS† by considering the low-energy world where we would dets = 1 ? which is derived from the classicalexpect Kähler to sum potential over different Dirac monopole configurations provided they have a suit- able microscopic completion [5]. We also reach the same conclusion by considering the very high-energy world1/ of4 string theory, where1/2 these Taub-NUT instantons can be K = log( + †)(3.26)− viewed as− D6-braneSdetSD/ instantons† D/ wrappingdet D/ manifolds† D/ of G2-holonomy [4]. dets = The presence of the Planck mass M in the complex/ / † structure (3.24/)meansthatthis/ † Thepl multi-Taub-NUTdet D D solutionspin 3/2 (4.2det)enjoys3D D kspinbosonic1/2 zero modes, parameterised by − − chiral multiplet does not survive the rigid limit in which gravity is decoupled. (The 6 the centres Xa,and2The Superpotentialk spin-3/2 fermionic zero modes [28] . Although this result is distinction between rigid and gravitationalwell known, we theories will provide was stressed, a slightly in di particular,fferent7/48 derivation in [24]). of the index theorem for the 41/48 1 This, in turn, means that we cannotfermionic use zero thedets modes fact = that inA SectionµR2 sits4.3 inen a route chiral to multiplet calculating to the one-loop determinants. The calculation gives R2 restrict the way it appears in superpotentialsFor now, we merely when note rigid that supersymmetric only the k= 1 gauge Taub-NUT theories solution, with two fermionic are compactified on a circle aszero in [38 modes,, 39]4. can contribute to the superpotential [3]. we find that the superpotential is given by, 41/48 7/48 µ2 1 S7/TN48 iσ τgrav Λ The41/24 Action= C e− − e− One-Loop Corrected Complexgrav Structure2 22 412/48 1 = CM3 W e−S M M R W Mpl pl dets = A µpl The Einstein-Hilbert action evaluated onR2 the Taub-NUT space with charge k =1is, As we have just seen, the kinetic terms are corrected at one-loop. This in principle with the overall constant after subtracting appropriate counterterms, given by [29, 35], affects both the complex structure24ζ ( 1) 1 7/ and48 Kählerpotential. For our present purposes, we Acknowledgement4e − − 2 2 2 Topological terms C = 41/48 S =27/48π M R are only concerned with the shift3/2 to the complex2 structure. TN pl 2(4π) µ 1 S iσ τgrav = C e− TN− e− Note that the superpotentialMy thanks is not invariant to Nickwhere under Dorey theR Uhere(1)J forsymmetry is interpreted many2 which useful shifts as the discussions.2 asymptotic2 radius I’m supported of the circle. by (In the the Royal coordinates The renormalisation of the complexW structureMpl can be seenMpl fromR the fact that the the dual photon. Further, the Yukawa vertex in (4.20) explicitly breaks the U(1)R 2 2 2 2 Society.2 All the pieces(4.2 now), we fit together could2 just as well have written STN =2π MplL .) However, there are a (symmetry∂R) and under ( which∂σ) theterms gravitino pick is charge; up di thisff iserent manifestation 1/R ofcorrections the axial anomaly in (3.23). (Strictly speaking, number of further contributions to the action. The first comes from the dual 3d photon we(3.5). should However, firsta combination perform of U(1) aJ conformaland U(1)R symmetry transformation survive. so that we are working in the which, as first observed by Polyakov,41 acts/48 as a chemical potential for the topological 2 4 EinsteinThe Potential frame,References but this only affects the complex structure atΛgrav order 1/R and so can be instanton charge [5]. This= C follows from thee coupling−S (2.3): the 3d field strength arising The supersymmetric completion of the Yukawa term is a potential. In three-dimensional 2 neglected at one-loop order.) It is simple to checkW that the one-loopMpl corrected complex supergravity, this is given[1] by J. (see, Polchinski, for example,from “ [23Dirichlet-Branes, the3]) metric (4.2)hascharge and Ramond-RamondS2 F =2πL, Charges which ensures,” Phys. that Rev. the single Lett. Taub-NUT75, structure is given by instanton also comes with a factor of K 1 2 2 V =4724M e (1995)(∂∂¯K)− [arXiv:hep-th/9510017].D 4 3 | W| − |W| Acknowledgement7 =2π2M 2 R2 + iσ with D = ∂ +(∂K) .with This potential includes2 2 some2 critical points at2 2 (1). pl W W W =2π MplR + log(MplRS ∼)+O iσ S (3.27) They are not to be trusted as they lieS outside the6 semi-classicalFor Yang-Mills48 regime instantons, of large where the number of zero modes can be simply determined by integrating My thanks to Nick Dorey for manyS useful discussions.4 I’m supported by the Royal we performed our calculation. Instead, at largethe anomaly., the potential In= the is dominated present case by the thereg exp is a mismatchd x between√g the integrated anomaly (3.5) and the 2 S Z D − R (Tracing term and the takes origin the runawaySociety. of this form shift,number we see of zero that modes ittopology depends due to the on presence the a ofcoe boundaryfficients terms. defined These in are known as eta-invariants [36] |W | (3.16), but is independent3 of41/24 theandb willcoe2 alsoffi2 cients2 play a defined role when in we ( come3.17)). to discuss We will the haveone-loop use determinants for around the background V M3 (RΛgrav) exp 4π MplR this later when we∼ compute theof instanton-generated the− gravitational instanton. superpotential. We learn that the Kaluza-KleinReferences compactification of =1supergravityon R3 S1 is N × not a ground state of the theory. This instanton-generated potential causes the circle 3.5to decompactify Divergences to large radius and[1]R. J. the Polchinski, Heat Kernel “Dirichlet-Branes and Ramond-Ramond Charges,” Phys. Rev. Lett. 75, 18 The gradient expansion4724 employed (1995) in [arXiv:hep-th/9510017]. Section 3.3 is the simplest approach25 for computing the effective action at the two derivative level. However, it becomes increasingly cum- bersome as we look to higher derivatives. In particular, as described at the beginning = g exp d4x √g Z D − R 4We thank N. Seiberg for discussions on these issues. topology

20 43

19 we ﬁnd that the superpotential is given by,

41/24 Λgrav = CM e−S W 3 M pl with the overall constant

24ζ ( 1) 1 7/48 4e − − C = 3/2 2(4π)

Note that the superpotential is not invariant under the U(1)J symmetry which shifts the dual photon. Further, the Yukawa vertex in (4.20) explicitly breaks the U(1)R symmetry under which the gravitino is charge; this is manifestation of the axial anomaly

(3.5). However, a combination of U(1)J and U(1)R symmetry survive.

The Potential The supersymmetric completion of the Yukawa term is a potential. In three-dimensional supergravity, this is given by (see, for example, [23, 3])

K 1 2 2 V = M e (∂∂¯K)− D 4 3 | W| − |W| with D = ∂ +(∂K) . ThisThe potential Potential includes some critical points at (1). W W W S ∼ O They are not to be trusted as they lie outside the semi-classical regime of large where S we performed ourKaluza calculation.-Klein compactification Instead, at of largeN=1 supergravity, the potential is unstable is dominated by the 2 S term and takes the runaway form |W | V M 3(RΛ )41/24 exp 4π2M 2 R2 ∼ 3 grav − pl We learn that the Kaluza-Klein compactiﬁcation of =1supergravityon R3 S1 is N × not a ground state of the theory. This instanton-generated potential causes the circle The ground state has R ∞ to decompactify to large radius R.

43 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 −

1,d 1 1 = R − S M × Open Questions 1 ∂ σ F νρ µ ∼ 2 µνρ

Λ M grav pl

What is this good for?

2 1 2 2 1 L What is it∂ inR our Universe? 11 L − L ∂σ = 1 + 1+ Leﬀ − 6π M R2 R 24π M R2 R4 2π 3 3 = K( , †) ∂ ∂ † Leﬀ S S S S

∂R 2 1 L2 ∂σ 2 = + Leﬀ R M 2 R4 2π 3 1 = ∂ ∂ † Leﬀ ( + )2 S S S S†

17 Thank you for your attention