<<

Euclidean Quantum

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 [1] J. Polchinski, “Dirichlet- 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 [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 : = 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 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 is with all three spatial 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 , a Casimir 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 . This results in a 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 , 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 related￿µνρF to the arising ∼ 2 from D6- instantons wrapping - 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 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, 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 and . The bulk four-dimensional 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 indices and a, b =0, 1, 2, 3aretangent space indices. Here we follow the standard notation of suppressing the indices on the gravitino, whose is given by 1 ψ = ∂ ψ + ωˆ γabψ Dν ρ ν ρ 4 abν ρ

In this formalism, the connection ωabµ that appears in the covariant derivative differs from the purely geometric 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 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 . (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 = R1,2 S1. We denote the physical radius of the circle as R. M ∼ × We choose the such that the 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 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 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 : 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 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 dynamics1 is effectively 2 2GM 2GM − three dimensional.distance The metric= degrees1 of freedomδt2 + are1 parameteriseddr2 by+ ther2 δθ familiar2 +sin2 θδφ2 − − rc2 − rc2 ￿ ￿ ￿ ￿ Kaluza-Klein ansatz, ￿ ￿ 2 2 L R 2 ds2 = ds2 + dzx2 += ARdxsini θ sin φ (2.2) (4) R2 (3) L2 i ￿ y = R sin￿θ cos φ whereAt distancesz [0, 2π largerL) is than the periodic the compactification coordinate. scaleHere RR,, theA and dynamics the 3d is metric effectivelyg(3) are ∈ z = R cosiθ ij Atthreedynamical distances dimensional. degrees larger than The of freedom, the metric compactification degrees while L ofis freedom a scale fixed,R are, fiducial the parameterised dynamics scale. is It e byff isectively the natural familiar to pick threeKaluza-Kleincoordinates dimensional. such ansatz, The that metricR(x) degreesL asymptotically of freedom are parameterised and we will eventually by the familiar do so but, for → Kaluza-Klein ansatz, 2 2 now, we leave L arbitrary. L R 2 ds2 = ds2 + dz2 + A dxi (2.2) (4) 2 2 (3)δx2 =(2 R cos θ sini φ)δθ +(R sin θ cos φ)δφ 2 L R2 R L 2 i 2 Evaluated on thisds background,(4) = 2 ds(3) the+ Einstein-Hilbert2 dz + Aidx action becomes (2.2) R δLy =(R￿ cos θ cos φ)￿δθ (R sin θ sin φ)δφ(3) where z [0, 2πL) is the2 periodic coordinate.￿ Here R￿ , Ai and− the 3d metric gij are ∈ Mpl 4 (3) wheredynamicalz [0, 2 degreesπL)S is the= of freedom, periodicd coordinate. whilex √ gLδzis= a Here fixed,(RRsin, fiducialAθi)andδθ scale.the 3d It metric is naturalg are to pick ∈ eff 2 − R(4) − ij dynamicalcoordinates degrees such of that freedom,R(x)￿ whileL asymptoticallyL is a fixed, fiducial and we scale. will It eventually is natural do to so pick but, for → 2 4 coordinates such that R(ClassicalxM) 3 L asymptotically3 Low-Energy and we willPhysics∂R eventually1 R do soij but, for now, we leave L arbitrary.= → d x g 2 F F now, we leave L arbitrary. 2 − (3) R(3) − R − 4 L4 ij ￿ ￿ ￿ ￿ ￿ Evaluated on this background,￿ the Einstein-Hilbert actionθ becomes Evaluated on this background,2 the Einstein-Hilbert action becomes with M3 =2πLM the 3d Planck scale and Fij = ∂iAj ∂jAi the graviphoton field pl M 2 − 2 pl 4 strength. Seff M= d x √ g (4) S = pl 2 d4x √ g − R φ eff 2 ￿ (4) − R 2 4 In three dimensions,M￿ 3 it is often useful to dualise∂ theR gauge1 R field in favour of a periodic M= d3x g ∂2R 2 1 R4 F F ij scalar σ. This is particularly3 3 true if(3) we are(3) interested in instanton4ij ij physics [5]. The dual = 2 d x g(3)− (3)￿R 2 − R −14,dF4ij1LF 1 ￿ 2 ￿ − R − R￿ −=￿4RL − S photon can be viewed￿ as Lagrange￿ ￿ multiplier￿ whichM￿ imposes× the￿ Bianchi identity, 2 ￿ with M3 =2πLM2 pl the 3d Planck scale andσ Fij = ∂iAj ∂jAi the graviphoton field with M3 =2πLM the 3d Planck scale and Fij =ijk∂iAj ∂jA−i the graviphoton field strength. pl σ = ￿ DiF−jk (2.3) strength. L 4πL 1 νρ WithIn three the magnetic dimensions,Or, if chargewe it work is often quantised with usefulthe dual in to integralphoton dualise the units,∂µσ gaugeσ has￿ fieldµνρ periodicityF in favour of 2π a. periodic Integrating In three dimensions, it is often useful to dualise the gauge∼ field2 in favour of a periodic scalarscalaroutσ the. Thisσ. field This is particularly strength, is particularly we true can true if we write if are we the interested are low-energy interested in instanton in eff instantonective physics action physics [5]. in The [dual5]. dual The form, dual photon can be viewed as Lagrange multiplier which imposes the Bianchi identity, photon can be viewed as Lagrange multiplier which imposes the2 Bianchi2 identity,2 3 M3 ∂ΛRgrav M1 plL ∂σ Seff = d x g(3) σ σ(3) ijkM3 ￿ 4 (2.4) − σ 2= Rijk ￿− DiFjk R − M R 2π (2.3) ￿ σ =L￿ 4￿πLDiFjk ￿ ￿ 3 ￿ ￿ (2.3)￿ ￿ L 4πL WithWithThis the theaction magnetic magnetic enjoys charge charge a quantised new quantisedU(1) inJ integralsymmetry in integral units, whichunits,σ hasσ acts periodicityhas periodicity by shifting 2π. Integrating 2π the.2 Integrating dual photon: 1 2 1 5 L 1 L ∂R 11 L − L2 ∂σ outoutσ the the fieldσ + field strength,c. strength, All other=Goal: we can we fields MUnderstand can write+ are write the left low-energyquantum the invariant low-energy corrections eff underectiveM effective actionthisto this symmetry. action inaction. dual in form, dual The form, symmetryM + is → Leff 2 3 16π R2 R(3) − 3 − 6π R2 R − 3 24π R2 R4 2π preserved in perturbation￿ theory but,￿ as we will￿ see in Section￿￿4, is broken￿ by￿ instanton ￿ ￿ ￿ M M ∂R ∂2R 21 L21 L∂σ2 2∂σ 2 effects.S = d3x 3 g 3 3 M (2.4) effSeff = d x (3) g(3) (3) (3) 3 M3 4 4 (2.4) − − 2 R2 R− − R R− M3−RM R2π 2π Acknowledgement￿ ￿ ￿ ￿ ￿ ￿￿ ￿ ￿3 ￿￿ ￿ ￿ ￿ Our goal in this paper￿ ￿ is to determine the quantum corrections to the effective action This action enjoys a new U(1) symmetry which acts by shifting the dual photon: This(2.4). action We describe enjoysMy thanks a perturbative new toUJ(1) NickJ symmetry corrections Dorey for which many in Section acts useful by3 shiftingand discussions. instanton the dual I’m corrections photon: supported in by the Royal σ σSectionσ +σc+.4 This.c. This symmetry symmetry is preserved is preserved in perturbation in perturbation theory theory but, as but, we as will we see will in see in → → Society. SectionSection4, is4, broken is broken by instanton by instanton effects. effects.

OurOur goal goal in this in this paper paper is to is determine to determine the quantum the quantum corrections corrections to the to eff theective effective action action (2.4(2.4). We). We describe describe perturbative perturbative corrections corrections in Section in Section3 and3 instantonand instanton corrections corrections in in 6 17 SectionSection4. 4.

6 6 Gravitational Instantons 7/48 41/48 1 dets = A µ2 R2 ￿ ￿ ￿ ￿

41/48 7/48 2 µ 1 S iσ τgrav − TN− − = C 2 2 2 e e W ￿Mpl ￿ ￿MplR ￿

41/48 2 Λgrav = C 2 e−S W ￿ Mpl ￿

E = p2c2 + m2c4 ￿ p2 E mc2 + + ... ≈ 2m

E = pc

8 1 c =3 10 ms− Taub￿ ×-NUT

The simplest class of gravitational instantons are the multi-Taub-NUT metrics [27],

18 2 l<10− m1 2 ds = U(x)dx dx + U(x)− (dz + A dx) (4.2) · · with L with U(x)=1+L k 1 and A = U U(x)=1+ 2(x X) and ∇× A ∇= U 2 x − X ∇× ±∇ a=1 a ￿ | − | TheAcknowledgement metric is smooth when z [0, 2πL)andtheX are distinct. For A = U, ∈ a ∇× ±∇ the Riemann tensor obeys = ￿ . My thanksFrom tothe Nicklow-energy DoreyR 3dµ forνρσ perspective, many∓ usefulRµ νρσthese discussions. look like Dirac I’m monopoles. supported by the Royal Society. The Taub-NUTThis is the gravitational metric takes verson the of same Polyakov’s form asfamous our Kaluza-Kleincalculation. ansatz (2.2)with U = L2/R2. However, because U 1asymptotically,itmeansthatwehavemadea → Gross ’84 coordinate choice in which the fiducial length L is taken to be theHartnoll physical and Ramirez asymptotic ‘13 length of the circle: R(x) L. 19 → One might wonder about 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, × the boundary of space is S2 S1. But for k = 0, the boundary of the manifold is the × ￿ S1 is fibered non-trivially over the S2. For example, with k =1,theboundaryis topologically S3. The question at hand is whether we should sum over these different boundary conditions in the path integral.

A similar question arises in gauge in flat space, where the issue becomes whether one should sum over topologically non-trivial bundles at infinity. Here the answer is certainly yes: a trivial gauge bundle can be smoothly deformed into an instanton-anti-instanton pair which are subsequently moved far apart. Such configura- tions certainly contribute to the path integral but locality and cluster decomposition then requires us to also sum over individual instanton bundles. (See, for example, [24] for a recent discussion of this topic). However, these same arguments also hold in the present case: we can equally well locally nucleate a NUT-anti-NUT pair which can then be moved far apart. This suggests that should sum over all asymptotic windings. (There is, admittedly, one loophole which is the lack of local in a theory of gravity but this does not seem to be a serious objection to the argument.)

Another way to motivate including non-trivial S1 bundles is to consider a parallel to a more familiar story with instantons. There, one imposes ‘initial’ and ‘final’ conditions in Euclidean time and 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, …. 2 R

µν RµνR 2 µνρσ R RµνρσR µν RµνR 2 µν µνρσ , µν , µνρσ µνρσ R R R R RµνρσR R

α 4 µνρσ2 µνµν 2 µνρσ Sα = d x µνρσ , 4 µν +, = αχ 8π2 R RR − RRµνRR RRµνρσR ￿

α 2 S = d4x MUV µνρσ 4 µν + 2 = αχ α(µ)=α α 2 α log µνρσ µν 80π− 1 R µ2R − R R R ￿ ￿ ￿

2 MUV 3 1 α(µ)=2 α 1 α log ∂(R S )=S S0 − 1 µ2 × × ￿ ￿

3 ∂(TNk)=S /Zk ∂(R3 S1)=S2 S1 × ×

The One-Loop Determinants3 ∂(TNk)=S /Zk 2 ∂(AHk)=RP /Zk 2Graviton ghosts ∂(AHk)=RP /Zk det(Fermions) dets = det() det Fermions dets = det BosonsGravitino ghosts dets = 1 ?

Supersymmetry Acknowledgement dets = 1 ? Hawking and Pope ‘78

My thanks to NickAcknowledgement 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 (UsingSociety. the Gibbons-Hawking-Perry prescription for rotating the conformal factor)

18 18 ∂(R3 S1)=S2 S1 × ×

3 ∂(TNk)=S /Zk

2 ∂(AHk)=RP /Zk

det(Fermions) dets = det(Bosons)

dets = 1 ? The One-Loop Determinants

1/4 1/2 − det￿ D/ † D/ det￿ D/ † D/ A somewhatdets detailed = calculation gives det D/ D/ † ￿ det D/ D/ † ￿ ￿spin 3/2 ￿spin 1/2 ￿ − ￿ − ￿ ￿ ￿ 7/48￿ 41/48 1 dets = A µ2 R2 ￿ ￿ ￿ ￿ An ugly number 41/48 7/48 2 Radius of the circle UV cut-off scale µ 1 S iσ τgrav − TN− − = C 2 2 2 e e W ￿Mpl ￿ ￿MplR ￿

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 7/48 41/48 1 dets = A µ2 R2 ￿ ￿ ￿ ￿ 41/48 7/48 2 µ 1 S iσ τgrav − TN− − = C 2 2 2 e e W ￿Mpl ￿ ￿MplR ￿

41/48 2 Λgrav = C 2 e−S W ￿ Mpl ￿

E = p2c2 + m2c4 ￿ p2 E mc2 + + ... ≈ 2m

E = pc

8 1 c =3 10 ms− ￿ ×

18 l<10− m The Superpotential L U(x)=1+ and A = U 2(x X) ∇× ∇ Putting all the pieces together gives −

41/48 7/48 we find that the superpotential is given by, 2 µ 1 2π2M 2 R2 iσ 41/24 = C e− pl − Λgrav 2 2 = CM e−S 2 W 3 M W Mpl MplR ￿ pl ￿ ￿ ￿ ￿ ￿ with the overall constant

24ζ ( 1) 1 7/48 Acknowledgement4e ￿ − − C = 3/2 Action of Taub-NUT ￿ 2(4π) ￿

Note that the superpotential is not invariant under the U(1)J symmetry which shifts the dual photon.My Further, thanks the Yukawa to Nick vertex in Dorey (4.20) explicitly for breaks many the U useful(1)R discussions. I’m supported by the Royal symmetry underSociety. which the gravitino is charge; this is manifestation of the axial (3.5). However, a combination of U(1)J and UHow(1)R symmetryto make survive. sense of this?

The Potential • It’s not holomorphic (in the naïve complex structure) The supersymmetric completion of the Yukawa term is• a potential.It’s UV divergent In three-dimensional supergravity, this is given by (see, for example, [23, 3])

K ¯ 1 2 2 V = M3 e (∂∂K)− D 4 19 | W| − |W| with D = ∂ +(∂K) . This 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 our calculation. Instead, at large , the potential 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 to decompactify to large radius R.

43 Perturbative Quantum Corrections 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

1,d 1 1 = R − S M ×

These finite corrections were first computed1 in the Kaluza-Klein context in [1], where ∂ σ ￿ F νρ they manifested themselves as a Casimirµ ∼ 2 force,µνρ causing the Kaluza-Klein circle to either shrink or expand. (The analogous calculation was performed earlier in the thermal 1 context [9]). The effective 3d potentialΛ is givenM by grav ￿ pl N N L3 V = B − F (3.1) eff − 720π R6 2 1 2 1 L ∂R 11 L − L2 ∂σ Here N iseff the= number1 Finite of massless Quantum bosonic+ Corrections degrees1+ of freedom; these make the Kaluza- BL − 6π M R2 R 24π M R2 R4 2π ￿ 3 ￿￿ ￿ ￿ 3 ￿ ￿ ￿ Klein circle contract.2 NF the number of massless fermionic degrees of freedom; these ∂ K( , †) make the circle= expand.S S Of∂ course,∂ † in supersymmetric theories NB = NF and Kaluza- Leff ∂S∂S S S Klein compactifications† are perturbatively stable. The presence of fermions with peri- The classical low-energy effective action is odic boundary conditions means that the bubble-of-nothing instability is absent in this theory [2], but other gravitational1 instantons,∂R 2 discussed1 L2 in Section∂σ 2 4, will contribute. = M M Leff 2 3R(3) − 3 R − M R4 2π Although the perturbative potential vanishes,￿ ￿ there3 are￿ still￿ finite one-loop effects of interest. These renormalise the kinetic terms in the effective action (2.4). Much of this sectionOne is devoted loop corrections to computing to the thesekinetic e fftermsects; give we will show that the low-energy effective action becomes, 17

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 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

1,d 1 1 = R − S M ×

These finite corrections were first computed1 in the Kaluza-Klein context in [1], where ∂ σ ￿ F νρ they manifested themselves as a Casimirµ ∼ 2 force,µνρ causing the Kaluza-Klein circle to either shrink or expand. (The analogous calculation was performed earlier in the thermal 1 context [9]). The effective 3d potentialΛ is givenM by grav ￿ pl N N L3 V = B − F (3.1) eff − 720π R6 2 1 2 1 L ∂R 11 L − L2 ∂σ Here N iseff the= number1 Finite of massless Quantum bosonic+ Corrections degrees1+ of freedom; these make the Kaluza- BL − 6π M R2 R 24π M R2 R4 2π ￿ 3 ￿￿ ￿ ￿ 3 ￿ ￿ ￿ Klein circle contract.2 NF the number of massless fermionic degrees of freedom; these ∂ K( , †) make the circle= expand.S S Of∂ course,∂ † in supersymmetric theories NB = NF and Kaluza- Leff ∂S∂S S S Klein compactifications† are perturbatively stable. The presence of fermions with peri- The classical low-energy effective action is odic boundary conditions means that the bubble-of-nothing instability is absent in this theory [2], but other gravitational1 instantons,∂R 2 discussed1 L2 in Section∂σ 2 4, will contribute. = M M Leff 2 3R(3) − 3 R − M R4 2π Although the perturbative potential vanishes,￿ ￿ there3 are￿ still￿ finite one-loop effects of interest. These renormalise the kinetic terms in the effective action (2.4). Much of this sectionOne is devoted loop corrections to computing to the thesekinetic e fftermsects; give we will show that the low-energy effective action becomes, 17

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,Something both important: from higher-loops these two numbers and from are non-perturbative different! 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 1 2 2GM 12 2GM − 2 2 2 2 2 2 distance2GM = 2 1 2GM δ−t +2 1 2 2 2 dr2 + r δθ +sin θδφ distance = 1 δt + 1 2 dr + r δθ +sin2 θδφ − − rc2 − −− rcrc2 − rc ￿ ￿ ￿￿ ￿￿ ￿ ￿ ￿ ￿ ￿ ￿ x = R sin θ sin φ x = R sin θ sin φ y = R sin θ cos φ y = R sin θ cos φ z = R cos θ z = R cos θ

δx =(R cos θ sin φδ)xδθ=(+(RR sincosθθcossinφφ)δφ)δθ +(R sin θ cos φ)δφ δy =(R cos θ cos φ)δθ (R sin θ sin φ)δφ δy =(− R cos θ cos φ)δθ (R sin θ sin φ)δφ and the classical action (δ2.4z =)forthiscomplexscalartakestheform(R sin θ)δθ − − δz = (R sin θ)δθ 3 −1 S = M d x g ∂ ∂ † (3.25) − 3 − (3) ( + )2 S S ￿ S S† ￿ θ which is derived from the classical K¨ahler potential θ

K = log(φ + †)(3.26) − S S φ The presence of the Planck mass Mpl in the1,d complex1 1 structure (3.24)meansthatthis = R − S chiral multiplet does not survive theM rigid limit× in which gravity is decoupled. (The 1,d 1 1 distinction between rigid and gravitational theories was stressed,= R in− particular,S in [24]). M × This, in turn, means thatThe we cannot Complex use the Structure fact that R sits in a chiral multiplet to 1 νρ restrict the way it appears in superpotentials∂µσ ￿ whenµνρF rigid supersymmetric gauge theories ∼4 2 are compactified on a circle as in [38, 39] . 1 νρ The two fields R and σ must combine in a complex∂µσ number￿µνρ F Λ M ∼ 2 One-Loop Corrected Complex Structuregrav ￿ pl At one-loop As we have just seen, the kinetic terms are corrected atΛ one-loop.grav M Thispl in principle 2 ￿1 2 affects both the complex1 structureL and∂R K¨ahlerpotential.11 ForL our− presentL2 ∂σ purposes, we = 1 + 1+ are only concernedLeff with− 6 theπ M shiftR2 to theR complex structure.24π M R2 R4 2π ￿ 3 ￿￿ ￿ ￿ 3 ￿ ￿ ￿ 2 1 2 2 The renormalisation= K( , of†) the∂ ∂ complex† structure1 L can be∂ seenR from the fact11 thatL the − L ∂σ S S S S= 1 + 1+ (∂RWe)2 andwant ( to∂σ write)2 terms this in pick theL form upeff di fferent− 1/R6π2McorrectionsR2 R in (3.23). (Strictly24π speaking,M R2 R4 2π ￿ 3 ￿￿ ￿ ￿ 3 ￿ ￿ ￿ we should first perform a conformal2 transformation so that we are working in the ∂ K( , †) 2 2 2 Einstein frame, but this only aeffffects= the∂R complexS S 1 structure∂L∂ †∂σ at order 1/R4 and so can be L eff = ∂S∂S+† S S neglected at one-loop order.)L It is simpleR to checkM 2 thatR4 the2π one-loop corrected complex ￿ ￿ 3 ￿ ￿ structure is given by 1 = ∂ ∂ † ( + )2 S S 2 2 2 S S7† ∂R 1 L ∂σ 2 2 2 eff 2= 2 + =2π MplR + log(LMplR )+Riσ M 2 R4 2π(3.27) S 48 ￿ ￿ 3 ￿ ￿ 1 (Tracing the origin of this shift, we see that it dependseff = on the ￿a coe∂ ffi∂cients† defined in ￿ L ( + )2 S S (3.16), but is independent of the b coefficients17 defined inS (3.17S)).† We will have use for this later when we compute the instanton-generated superpotential.

3.5 Divergences and the Heat Kernel 17 The gradient expansion employed in Section 3.3 is the simplest 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 77//4848 4141//4848 11 detsdets = = AA µµ22 RR22 ￿￿ ￿￿ ￿￿ ￿￿ 4141//4848 77//4848 µ22 µ 11 SSTN iiσσ ττgravgrav −− TN−− −− == CC 22 22 22 ee ee W ￿￿MMplpl￿￿ ￿￿MMplplRR ￿￿

4141//4848 22 ΛΛgravgrav == CC 22 ee−−SS WW ￿￿ MMplpl ￿￿

EE == pp22cc22 ++mm22cc44 ￿￿ pp22 EE mcmc22 ++ ++...... ≈≈ 22mm

EE == pcpc and the classical action (2.4)forthiscomplexscalartakestheform

3 1 † S = M3 d x g(3) 2 ∂ ∂ (3.25) − − (8 + †)1 S S ￿ c =3 108SmsS−1 c =3￿ 10 ms− which is derived from the classical K¨ahler￿￿ ×× potential

K = log( + †)(3.26) − S S 1818 l

41/48 7/48 One-Loop Corrected Complex2 41 Structure/48 7/48 µ2 1 2 2 2 µ 1 22ππ2MM2plRR2 iiσσ As we have just seen,== CC the kinetic2 terms are corrected2 atee one-loop.−− pl This−− in principle W M2 M2 R22 affects both theW complex￿ structure￿Mplpl￿￿ and K¨ahlerpotential.￿￿MplplR ￿￿ For our present purposes, we are only concerned with the shift to the complex structure.

Acknowledgement 41/48 The renormalisation of the complex structure2 can be seen from the fact that the ∂R 2 ∂σ 2 /Rµ2 ( ) and ( ) terms pick up diff=erentC 1 correctionse− inS (3.23). (Strictly speaking, My thanks to Nick Dorey for many useful2 discussions. I’m supported by the Royal we should first perform a conformalW transformation￿Mpl ￿ so that we are working in the Society.Einstein frame, but this only affects the complex structure at order 1/R4 and so can be neglected at one-loop order.) It is simple to check that the one-loop corrected complex structureWith is giventhe one-loop by corrected complex structure 7 =2π2M 2 R2 + 19log(M 2 R2)+iσ (3.27) S pl 48 19 pl (Tracing the origin of this shift, we see that it depends on the ￿a coefficients defined in (3.16), but is independent of the ￿b coefficients defined in (3.17)). We will have use for 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 simplest 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 2 R

µν RµνR 2 Divergences Rat One-Loop

µνρσ ‘t Hooft and Veltman ’74 µνρσ µν Deser, Kay and Stelle ‘77 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 , 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 7/48 41/48 1 dets = A µ2 R2 ￿ ￿ ￿ ￿ 41/48 7/48 2 µ 1 S iσ τgrav − TN− − = C 2 2 2 e e W ￿Mpl ￿ ￿MplR ￿

41/48 2 Λgrav = C 2 e−S W ￿ Mpl ￿

E = p2c2 + m2c4 ￿ p2 E mc2 + + ... ≈ 2m

E = pc

2 8 1 c =3 10 ms− ￿ ×R

For ourµν purposes, the most important one-loop divergence is associated to the Gauss- µν Bonnet term18 (2.8). This, of course, is a total derivative in four-dimensions but will be l

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 α naturally 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 1 2GM 2GM − distance2 = 1 δt2 + 1 dr2 + r2 δθ2 +sin2 θδφ2 − − rc2 − rc2 ￿ ￿ ￿ ￿ ￿ ￿ x = R sin θ sin φ For our purposes, the most important one-loop divergence is associated to the Gauss- Bonnet term (2.8). This,y = ofR course,sin θ cos is aφ total derivative in four-dimensions but will be important when we comez = toR discusscos θ gravitational instanton physics. The coefficient α is dimensionless and runs logarithmically at one-loop [6]

M 2 α(µ)=α α log pl (3.3) δx =(R cos θ sin φ)δθ +(R0 −sin1θ cos φµ)δφ2 ￿ ￿ δy =(R cos θ cos φ)δθ (R sin θ sin φ)δφ where α0 is the coupling at the cut-o− ff which, for convenience, we have identified with δz = (RThesin θ )Superpotentialδθ is now Finite the 4d Planck scale− Mpl. In general, for a theory withµ2 Ns free massless spin-s fields, the beta-function is given byα(µ [11￿)=, 12α, (13µ)] α log − 1 µ 2 ￿ ￿ ￿ If we include the1 running of the Gauss-Bonnet term α1 = 848θ N2 233N3/2 52N1 +7N1/2 +4N0 48 15 − − · 41/48 41/48 2￿ 2 ￿ µ α(µ) Λgrav The computation= leadingC to this resulte−S e− is closely= C related to thee trace−S anomaly for 2 φ 2 massless fieldsW in fixed,￿M curvedpl ￿ spacetime. Indeed,￿ forM spinspl ￿ s 1, the coefficients ≤ above are the same as c a of the trace anomaly. The running coupling α(µ)results − 1,d 1 1 in an RG-invariant scale, = − S R 41/48 With the RG invariantM scale × 2 Λgrav = C 2 α(µ)e−S WΛgrav =￿µ expMpl ￿ (3.4) 1 − 2α1 ∂ σ ￿ F νρ ￿ ￿ µ ∼ 2 µνρ AcknowledgementFor our present purposes, N0 = N1/2 = N1 =0whileN3/2 = N2 =1whichgives α1 =41/48. Note: in supergravity Λ is naturallyM complexified by the gravitational theta angle. My thanks to Nick Dorey forgrav many￿ pl useful discussions. I’m supported by the Royal Society.In the original discussions of Euclidean quantum gravity, the suggestion seems to have been that Λ (or sometimes µ)shouldbeidentifiedwiththePlanckscale.(See,for grav 2 1 2 1 L ∂R 11 L − L2 ∂σ =example,1 [14]). In contrast, here+ we view1+ Λgrav as a new scale which emerges in quantum LReferenceseff − 6π M R2 R 24π M R2 R4 2π gravity￿ through3 dimensional￿￿ ￿ transmutation;￿ it dictates3 ￿ the length￿ ￿ at which topological =fluctuationsK( , †) are∂ unsuppressed∂ † by the Gauss-Bonnet term. Like its counterpart ΛQCD in L[1]eff J. Polchinski,S S S “Dirichlet-BranesS and Ramond-Ramond Charges,” Phys. Rev. Lett. 75, Yang-Mills4724 (1995) theory, [arXiv:hep-th/9510017].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. ∂R 2 1 L2 ∂σ 2 = + 4 Leff R = M 2 R4 g exp2π d x √g In the previous section,￿ Z we￿ saw that3 Dα￿naturally￿ − sits in a chiralR multiplet with the 1 topology￿ ￿ ￿ ￿ ￿ gravitational theta-term θ. These combine† into the complex coupling τgrav = α +2iθ. eff = 2 ∂ ∂ L ( + †) S S τ/2α1 This means that the scale Λgrav = µe− is also naturally complex in supergravity S S p2 and sits in a chiral multiplet. E = 2m

17 10

20 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 −

µ2 θ α(µ￿)=α(µ) α log − 1 µ 2 ￿ ￿ ￿ Conclusions φ 41/48 41/48 2 2 µ α(µ) Λgrav = C 2 e−S e− = C 2 1,d e1−S 1 W ￿Mpl ￿ ￿ Mpl=￿R − S Kaluza-Klein compactification of N=1 supergravityM is unstable ×

41/48 2 Λgrav 1 = C e−S νρ M 2 ∂µσ ￿µνρF W ￿ pl ￿ ∼ 2

The superpotential depends on a “hidden” scale Λgrav Mpl Λgrav ￿

AcknowledgementWhere else does this scale show up? 2 1 2 2 1 L ∂R 11 L − L ∂σ eff = 1 + 1+ My thanks to Nick DoreyL for many− useful6π M discussions.R2 R I’m supported24 byπ theM R Royal2 R4 2π ￿ 3 ￿￿ ￿ ￿ 3 ￿ ￿ ￿ Society. = K( , †) ∂ ∂ † Leff S S S S References 2 2 2 [1] J. Polchinski, “Dirichlet-Branes and Ramond-Ramond∂ ChargesR ,” Phys.1 L Rev.∂σ Lett. 75, eff = + 4724 (1995) [arXiv:hep-th/9510017]. L R M 2 R4 2π ￿ ￿ 3 ￿ ￿ 1 = ∂ ∂ † Leff ( + )2 S S = g exp d4x √g † Z D − S SR topology￿ ￿ ￿ ￿ ￿

p2 E = 17 2m

20 Happy Birthday Gary! Extra Material 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

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. 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

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. 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 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 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

2 ∂(AHk)=RP /Zk

The Determinantsdet(Fermions) dets = det(Bosons)

In a self-dual background, you can writedets the = 1determinants ? as

1/4 1/2 − det￿ D/ † D/ det￿ D/ † D/ dets = det D/ D/ † ￿ det D/ D/ † ￿ ￿spin 3/2 ￿spin 1/2 ￿ − ￿ − ￿ ￿ ￿ ￿

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 K¨ahler 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

theK very= high-energylog( + † world)(3.26)1/ of4 , where1/2 these Taub-NUT instantons can be − viewed as− D6-braneSdet￿SD/ instantons† D/ wrappingdet￿ D/ manifolds† D/ of G2-holonomy [4]. dets = The presence of the Planck mass Mpl in the complex/ / † structure (3.24/)meansthatthis/ † The multi-Taub-NUTdet D D solution￿spin 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 K¨ahlerpotential. 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 structure￿Mpl 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, afirst 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.) Itfrom is simple the metric to check (4.2W)hascharge that the￿ one-loopMpl F￿=2 correctedπL, which complex ensures that the single Taub-NUT supergravity, this is given[1] by J. (see, Polchinski, for example, “ [23Dirichlet-Branes, 3]) and Ramond-RamondS2 Charges,” Phys. Rev. Lett. 75, 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 boundary￿fficients 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 First…to

Henriette Elvang Don Marolf

Thank you!