Inflation in String Theory Lecture 3

Gary Shiu From the last lecture … 120 3 Elements of String Theory where c is a constant (see [319]). Since G3 depends on the dilaton and ⌦ involves the complex structure moduli, the superpotential (3.94) leads to a non-trivial potential for these moduli. The scalar potential associated with K0 and W0 is ¯ V = eK0 KIJ D W D W 3 W 2 , (3.95) F 0 I 0 J 0 | 0| h i where I,J run over all the moduli (Ti, ⇣↵ and ⌧). Supersymmetry is pre- served if all F-terms vanish,17 i.e. if D W @ W +(@ K)W =0, (3.96) I 0 ⌘ I 0 I 0 where I runs over all the moduli. No-scale structure.—The K¨ahler potential (3.93) is of a specific form that satisfies IJ¯ K0 @I K0@J¯K0 =3. (3.97) I,JX=Ti Since the superpotential (3.94) is independent of the K¨ahler moduli, the scalar potential (3.95) is of the no-scale type, i.e. it is independent of the F-terms of the K¨ahler moduli, ¯ K0 IJ VF = e K0 DI W0DJ W0 . (3.98) I,J=T X6 i

This potential is positive semi-definite, and VF =0whenDI=Ti W0 = 0. The minimum is not necessarily supersymmetric, as in general we6 may have D W = 0. Ti 0 6 No-scale structure and D3-branes.—Thus far we have discussed them e↵ectivem action for massless closed string fields, but the positions of D-branes = provideT3 y (77) an important additional class of open string moduli. Consider a D3-branep that fills spacetime and sits at a point in a flux compactification on a Calabi- 3/2 Yau manifold. Evaluating the DBI+CS action (3.30)= inT + anT ISD background,2ln 3ln(T + T ) (78) one finds that the potential energy for D3-braneV motion vanishes) identically:V ! the complex scalars z↵, ↵ =1, 2, 3, that parameterize the D3-brane posi- tion are massless moduli. TheD3-brane four-dimensional Potential action derived from the dimensional reduction of (3.30) can be expressed in = 1 supergravityW = W (⇣ via, ⌧) (79) N 0 0 ↵ the DeWolfe-GiddingsThe Kahler potential K¨ahler potential contains:, which for a compactification with a single• K¨ahler modulus T takes the form [320]

K(T,T,z¯ , z¯ )= 3ln T + T¯ k(z , z¯ ) , k(3.99)= g (80) ↵ ↵ ↵ ↵ r↵r ↵ h i difficult to get metric for compact CY →need a simpler setting 17 When gauge multiplets are present in the e↵ective theory, D-term contributions are an importantIdea: alternative source of supersymmetry breaking, but our present discussion is confined• to the moduli sector.

approx D3

hopeless systematically at present case of interest incorporate bulk effects

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8 Compactification Effects • For a toy model: nothing! Nc D7 Scanned by CamScanner we can compute V(ϕ) in full!

D3

• Distant Σ4 can give non-negligible contribution to V(ϕ)

’ Σ4 Nc D7

we’ll parametrize these effects Scanned by CamScanner for Calabi-Yau cones ’’ D3 Σ4 w T3V4 =2⇡T (91)

aT W = W0 + e (92)

@ W = 0 (93) T 6

w DT W = 0 at T = T (94) T3V4 =2⇡T ww (91) ⇤ TT33VV44 =2=2⇡⇡TT (91)(91)

1 aT w 1 W = W0 + e T3V4 =2⇡T aTaT 3Nc N(92)f Nc (91)Nc w WW ==WWW(00m++)=ee Nc⇤ detm m(92)(92) (independent of m) (95) T3V4 =2⇡T (91) ! ⇥ aT aT @TWW==W 0+ e W = W@@T0 W+We== 0 0(92) (93) (92) (93) (93) 6 0 T 6 6 ! m W Nc V () = 0 (96) / ) / ) 6 @T W = 0D W = 0 at T = T (93) (94) DT W = 0 at T 6 = T DTT@TWW== 0 0 at T = T (94) (93)(94) ⇤ 6 ⇤⇤ aT 1 aT W = A e A f(z) Nc e (97) 1 NP 0 0 DT W = 0 at T = T 1 11 (94) 1 3Nc Nf⇤ Nc ! 31Nc Nf Nc NNc 3Nc Nf W (m)=Nc NNc⇤⇤ detdetmm mm c (independent(independent of ofmm)) (95) (95) W (m)= Nc⇤ detm mc NcDT (independentW = 0 at T!= ofTm⇥) (95) (94) ! ⇥ ! ⇤ ⇥ 1 1 3Nc Nf N W (m)= Nc⇤ detm c m Nc (independent of m) (95) ! ! ⇥ ! 1 Nc ! m W Nc 1 V () = 0Rab gab (96) (98) 3Nc mNf WN V () = 0 (96) m W(m)=W Nc⇤Nc V//(det) )m=) 0c // m N))c (independent6 6 (96) of m)/ (95) / ) / )! 6 ! ⇥ m W Nc V () = 0 (96) / ) / ) 6 1 aT 1 aT W = A e aT A f(z) Nc e aT (97) WNP 1= A0 e ! A0 f(z) Nc e (97) aT NP aT0 ! 0 + Nc 1 Nc! WNP = A0e aTmA0f(z) We aT V () = 0(97)if Y5 is Kahler (96) (99) WNP = A0e /A0f)(z) Nc e/ ) 6 (97) R ! ! ⇥ Calabi-YauR g Cones (98) Rabab gabab (98) aT // 1 aT R RWabg NPgab= A0e A0f(z) Nc e (98)(98) (97) ab / ab/ ! 2 2 2 2 • Let Y5 be a Sasaki-Einstein manifold ds10 = dr + r dsY5 (100) + if Y5 + is Kahler (99) + if Y5 R is Kahler (99) if Y5 R is Kahler⇥ R (99) +⇥ ⇥ if Y5 R is KahlerRab gab (99) (98) ⇥ / r 2 R 2 2 2 2 2 2 2 2 22 µ ⌫ 2 2 2 The cone ds = dr + r ds ds has102 a =CYdr metric2+ r 2dsY2 (100) (100) • 10 Y5ds = dr + dsr ds105 = ⌘µ⌫dx dx + (100) dr + r dsY (101) 10 Y5 R r 5 Taking2 N D3 at 2the tip2 of the2 cone, and+ in near-horizon limit, ✓ ◆ • ds10 = dr + r dsYif5 Y5 R is Kahler⇣ ⌘ (100) (99) 2 ⇥ r 2 R2 2 ds2 = ⌘ dxµ2dx⌫ + r 2 dr2 +µ r2⌫ds2 R 2 2 2(101)2 10 µ⌫ 2 r µ ⌫ Y5 R 2 2 2 R ds10 = r ⌘µ⌫dx dx + dr + r dsY5 (101) 4 2 ds10 = R✓ ◆⌘µ⌫dx dx + r dr + r dsY5 (101)4⇡ g N↵ ⇣ ⌘ R2 r 4 s 0 2 ⇣ ⌘ 2 2 2✓ 2 ◆ AdS Y with R = (102) 2 r µ ⌫ R⇣ ds⌘ 2= dr2 +2r ds✓ ◆ 5 5 (100) ds = ⌘ dx dx + 10dr + r ds Y5 (101) Vol(Y ) 10 µ⌫ 4⇡4g N↵ 2 Y5 , ⇥ 5 R r 4 s 0 4 2 AdS5 Y5 with✓ R◆= 4⇡ 4gsN↵0 2(102) ⇣ ⌘ , ⇥ Vol(Y5) 4 4⇡ g N↵ AdSAdS5 YY5 withwithRR4== s 0 (102)(102) ,, 5 ⇥⇥ 5 Vol(Vol(YY5)) r 2 4 2 R 2 5 2 4 4⇡ gsNµ ↵0⌫ 2 2 2 ds10 = ⌘µ⌫dx dx + dr + r dsY z z z z (101)= 0 (103) AdS5 Y5 withz1z2RRz3=z4 = 0r (103)5(102) 1 2 3 4 , • This⇥ is dual to an N=1 SCFTVol( (usefulY5 )later on).✓ ◆ ⇣ ⌘ z1z2 z3z4 = 0 (103) z1z2 z3z4 = 0 (103) 4 2 z1 = A1B1,z2 = A2B2,z3 = A1B2,z4 = A24B1 4⇡ gsN↵0 (104) z1z2 z3z4AdS= 05 Y5 with R = (103) (102) , ⇥ Vol(Y5) z1 = A1B1,z2 = A2B2,z3 =zA1 1=B2,zA14B=1A,z2B21 = A2B2,z3 (104)= A1B2,z4 = A2B1 (104) z1 = A9 1B1,z2 = A2B2,z3 = A1B2,z4 = A2B1 (104)

9 z1 = A1B1,z2 = A2B2,z3 = A1B2z,zz 4 =zAz29B=1 0(104) 9 (103) 1 2 3 4 9

z1 = A1B1,z2 = A2B2,z3 = A1B2,z4 = A2B1 (104)

9 5.1 Inflating with Warped Branes 187

For small velocities,r ˙2 e4A(r), we can expand the square root to get ⌧ 5.1 Inflating1 with Warped Branes 187 5.1 Inflating with Warped Branes 187 (@)2 T e4A() ↵() , (5.7) L ⇡2 3 For small velocities,r ˙2 e4A(r), we can5.1 Inflating⇣ expand with the Warpedsquare⌘ Branes root to get 187 For small velocities,r ˙2 e4A(r), we can expand the square root to get⌧ 2 2 where we have defined the canonically-normalized field T3r . From ⌧ 2 4A(r) ⌘ (5.3), we see thatFor a small single1 velocities, D3-brane2 r ˙ 4 experiencesAe () , we can no expand force the in square the anti-de root to get 1 (@) T ⌧e ↵() , (5.7) (@)2 T e4A() ↵(Sitter) , background:L ⇡(5.7) electrostatic2 repulsion3 from the four-form background L ⇡2 3 1 2 4A() exactly cancels the gravitational attraction.⇣ (@) T3 ⌘e ↵() , (5.7) ⇣ ⌘ L ⇡2 2 2 where we have2 defined2 the canonically-normalized field⇣ T3⌘r . From where we have defined the canonically-normalized field T r . From ⌘ 2 2 (5.3), we see that3 awhere single we D3-branehave defined experiences the canonically-normalized no force in field the anti-deT r . From (5.3), we see that a single D3-brane experiences no force⌘ in the anti-de ⌘ 3 Sitter background:(5.3), electrostatic weD3-branes see that repulsion a on single the from D3-brane Conifold the experiences four-form no background force in the anti-de Sitter background: electrostatic repulsion from the four-form backgroundSitter background: electrostatic repulsion from the four-form background exactlyAn anti-decancels Sitter the gravitational background is attraction. not a realistic setting for D-brane inflation. exactly cancels the gravitational attraction. exactly cancels the gravitational attraction. First of all, the spacetime is not compact, but ranges from r =0tor = . Furthermore, the metric becomes singular, with infinite redshift, at r =1 0. D3-branes on theD3-branes Conifold on4 the Conifold D3-branes on the ConifoldA more promising scenario for D-brane inflation involves a D3-brane in An anti-dea finite Sitter warped background throatAn anti-de region Sitter is not of background a a flux realistic compactification is setting not a realistic for D-brane [41]. setting We for inflation. will D-brane now inflation. An anti-de Sitter background is not a realistic setting for D-brane inflation.First of all, the spacetime is not compact, but ranges from r =0tor = . Firstreview of all, a the few spacetime geometric is prerequisites not compact, for but a discussion ranges from of thisr =0to model.r = . 1 First of all, the spacetime is not compact, but ranges from r =0toFurthermore,r = . the metric becomes singular, with infinite redshift,1 at r = 0. Furthermore,Singular conifold. the metric—The1 becomes singular singular, conifold with is a infinite six-dimensional redshift,4 Calabi-Yau at r = 0. Furthermore, the metric becomes singular, with infinite redshift, atAr more= 0. promising scenario for D-brane44 inflation involves a D3-brane in A more promising scenario for D-brane inflationA more4coneinvolves promisingX6 that a D3-brane can scenarioa befinite presented in warped for D-brane throat as the region locus inflation of in aC fluxdefinedinvolves compactification by a D3-brane [41]. We in will now a finite warped throatreview region a few of geometric a flux compactification prerequisites for a discussion [41]. We of will this model. now a finite warped throat region of a flux compactification [41]. We will now The4 first of these is an S3,thelattertwoequationsgiveanS2 fibered over the S3.Asthereare review a few geometricSingular prerequisites conifold.no—The non-trivial for2 a singular discussion such fibrations, conifold of the thisis base a six-dimensional model. of the cone is S2 Calabi-YauS3.Calabi-Yauspaceswhichhavesuch review a few geometric prerequisites for a discussion of this model. zA =0, 4 (5.8)× cone X6 that canisolated beConifold presented ordinary double as the point locus singularities in C defined are known by as conifolds and the corresponding point Singular conifold.—The singular conifoldA=1 is a six-dimensional Calabi-Yau Singular conifold.—The singular conifold is a six-dimensional Calabi-Yau inX the moduli space of4 the Calabi-Yau is known as a conifold point. The ordinary double point cone4 X that can be presented assingularity the locus is also in referred4 defined to as a node. by In figure 25 we illustrate a neighbourhood of a node. cone X that can be presented as the locus in defined6 Simplest by example of CY cones is the conifold: C 2 6 C where• A 1, 2, 3, 4 . This describes a cone overz a base=0,Y5, which is topolog- (5.8) 2 { } 2 A3 ically — but not metrically —4 equivalent to SA=1 S . To see this, note that if 4 X 2 A 2 A ⇥ A A A z is a solution to (5.8) then so iszAz=0,with, C.Writingz = x +(5.8)iy , zA =0, where(5.8)A 1, 2, 3, 4 . This describes2 a cone over a base Y5, which is topolog- the complex equation (5.8)2 { may be} recast as three real equations,2 3 A=1 ically — but notA=1 metrically — equivalent to S S . To see this, note that if X A X A ⇥ A A A z is a solution to (5.8) then2 3 so is z ,with C.Writingz = xS 2+ iy , Topologically, the conifold1 2is a cone over S ×1 S 2: 2 where A 1, 2, 3, 4 . This describes a cone overwhere a baseA• Y , which1, 2, 3, is4xthe topolog-.x This complex= ⇢ describes,y equation ay (5.8)cone= may over⇢ ,x be a baserecastyY as5=0, three which. realS is3 equations, topolog-(5.9) 2 { } 25 { } 2 3 ically — but not metrically — equivalent to S2 icallyS3. To— but see this, not metrically note· that2 if — equivalent· to2S S . To· see this, note that if A A 1 ⇥ Figure1 25: TheA neighbourhoodA A around a node. zA is a solution to (5.8) then so is zA,with ⇥z is.Writing a solutionzA to= (5.8)xA + theniyA, so isxxz =,with⇢2 ,y3 Cy.Writing= ⇢2 ,xz =y x=0+. iy , (5.9) C The first equation defines a three-sphere· 2 S with2 · radius2 ⇢/p2, while· the last the2 complex equation (5.8) may beHaving recast2 described as three the real singularity equations,3 in this way we immediately discern two distinct ways of re- the complex equation (5.8) may be recast as threetwo real equations equations, describe a two-sphere S fibered over the S . More precisely, 3 solving it: either we can replace3 the apex of thep cone with an S ,knownasadeformationofthe The first equation defines a three-sphere1,1 5 S1,1with radius ⇢/2 2, while the last the baseMetrically,Y5 of the the base cone Y is the is theEinsteinsingularity, Einstein manifold or manifold T we: can replace2T the, apex which with is an theS3,knownasasmallresolutionofthesingularity. coset 1 2 1 2 • two1 equations52 describe a1 two-sphere2 S fibered over the S . More precisely, x x = ⇢ ,yy = ⇢ ,xspacey =0. x x = (5.9)⇢ ,yThesey = two possibilities⇢ ,x are illustratedy =0. in5 figure1,1 26. The(5.9) deformation simply undoes the degeneration · the2 base Y5 of the· cone2 is the Einstein· manifold3 T , which is the coset · 2 · 2 · T 1,1 =[SUby re-inflating(2) SU the(2)] shrunken/U(1) ,S to positive size. The(5.10) small resolution, on the other hand, hasa space more pronounced effect: it repairs the singularity in a mannerthatchangesthetopologyofthe ⇥1,13 The first equation defines a three-sphere S3 withThe radius first⇢ equation/p2, while defines the last a three-sphereoriginalT Calabi-Yau.S =[withSU(2) radius AfterSU all,⇢(2)] replacing/p/U2,(1) while an, S3 with the an lastS2 is a(5.10) fairly radical transformation. We 2 3 shall find2 the physical interpretation⇥ 3 of these two ways of resolving conifold singularities. 4 parametrized by coordinates θ θ Φ , Φ , Ψ (detailed form of the metric can be found e.g., two equations describe a two-sphere S fiberedtwo over equations theMirageS . cosmology More describe precisely,[629] a two-sphere is an alternative1, 2, S1 tofibered2 inflation over in which the S the. spacetime More precisely, metric is 5 1,1 in [Candelas, de4 la Ossa]). Isometry group SU(2) x SU(2)5 x1 ,U(1)1 . 3 the base Y5 of the cone is the Einstein manifoldthe baseTthe, inducedY which5 of the metric is the coneMirage on coset a is D-brane cosmology the Einstein moving[629] is through an manifold alternative a background toT inflation, whichS supergravity in which is the the solution. spacetime cosetS 2 metric is space space Discussions of miragethe cosmologies induced metric involving on a D-brane D3-branes moving in through warped a throat background regions supergravity include solution. 1,1 [630–632]. Discussions1,1 of mirage cosmologies involving D3-branes in warped throat regions include T =[SU(2) SU(2)]/U(1)5 , [630–632].T(5.10)=[SU(2) SU(2)]/U(1) , (5.10) ⇥ An Einstein manifold5 satisfies Rab g⇥ab. 2 2 An Einstein manifold/ satisfies R g . S S ab / ab S3 S3 4 Mirage cosmology [629] is an alternative to inflation4 inMirage which cosmology the spacetime[629] metric is an is alternative to inflation in which the spacetime metric is the induced metric on a D-brane moving through a backgroundthe induced supergravity metric on solution. a D-brane moving through a backgroundFigure 26: supergravity The two possible solution. resolutions of a node. Discussions of mirage cosmologies involving D3-branes inDiscussions warped throat of mirage regions cosmologies include involvingA second D3-branes piece of in background warped throat information regions is a include mathematical fact due to Lefshetz concerning [630–632]. [630–632]. monodromy. Namely, if γa for a =1,...,k are k vanishing three-cycles at a conifold point in the 5 5 An Einstein manifold satisfies Rab gab. An Einstein manifold satisfies R gmoduli. space, then another three-cycle δ undergoes monodromy / ab / ab k δ δ + (δ γa) γa (8.19) → ∩ a=1 ! upon transport around this point in the moduli space. With this background, we can now proceed to discuss the resultquotedatthebeginningofthis section. We will do so in the context of a particularly instructive example, although it will be clear

99 188 5 Examples of String Inflation with isometry group SU(2) SU(2) U(1). The metric on T 1,1 is ⇥ ⇥ 2 2 2 2 1 1 2 2 2 d⌦T 1,1 d + cos ✓idi + d✓i +sin ✓idi , (5.11) ⌘ 9 ! 6 Xi=1 Xi=1 ⇣ ⌘ 3 2 3 where ✓i [0, ⇡], i [0, 2⇡] and The first[0, 4 of⇡]. these The is an metricS ,thelattertwoequationsgivean on the conifold S fibered over the S .Asthereare 2 2 no2 non-trivial such fibrations, the base of the cone is S2 S3.Calabi-Yauspaceswhichhavesuch can then be written as × isolated ordinary double point singularities are known as conifolds and the corresponding point 2 2in the2 moduli2 space of the Calabi-Yau is known as a conifold point. The ordinary double point ds =dr + r d⌦ 1,1 , (5.12) singularityT is also referred to as a node. In figure 25 we illustrate a neighbourhood of a node. where r 3/2 ⇢2/3. To express (5.12) as a manifestly K¨ahler metric, we introduce⌘ three complex coordinates z↵, ↵ 1, 2, 3 . The Ricci-flat K¨ahler p metric on the singular conifold, 2 { }

2 ↵ S 2 ds = k ¯ dz dz , (5.13) ↵ S3 then follows from the K¨ahler potential [633] Figure 25: The neighbourhood around a node.

WarpedHaving 4Conifold described2/3 the singularity in this way we immediately discern two distinct ways of re- 3 A 2 3 k(z↵, z¯↵)= solving it:z either we, can replace the apex(5.14) of the cone with an S ,knownasadeformationofthe 2 | | 2 singularity, A=1 or! we can replace the apex with an S ,knownasasmallresolutionofthesingularity. • N D3 at the tip of the conifoldThese X→ twowarped possibilities conifold; are illustrated FT dual in is figure an N=1 26. The deformation simply undoes the degeneration by re-inflating the shrunken S3 to positive size. The small resolution, on the other hand, hasa via k↵SCFT¯ = @↵ worked@¯k. out by [Klebanov, Witten] (more later). more pronounced effect: it repairs the singularity in a mannerthatchangesthetopologyofthe 1,1 3 2 DeformedIf we conifold. have N—In D3 at the the singular tip andoriginal conifold, M D5 Calabi-Yau. wrapping the base After manifoldthe all,shrinking replacingT Sshrinks2 an, theS with an S is a fairly radical transformation. We • shall find the physical interpretation of these two ways of resolving conifold singularities. to zeroconifold size at z isA =warped 0, and and the deformed: metric on the cone has a curvature singularity. To remove the singularity, we consider a small modification of the embedding S3 2 condition (5.8), S 4 2 2 zA = " . (5.15) S 2 S 2 AX=1 S3 S3 This defines2 the deformed conifold. The3 deformation parameter " can be made realS shrinks by an appropriateto zero size phasewhile rotation.the S size Eq. remains (5.15) finite: can then be written • Figure 26: The two possible resolutions of a node. as rA =√gsMα’ SUGRA valid if gsM ≫1 A second2 piece of background information is a mathematical fact due to Lefshetz concerning x x monodromy.y y = " , Namely, if γa for a =1,...,k(5.16)are k vanishing three-cycles at a conifold point in the · · moduli space,2 then another three-cycle δ undergoes monodromy • SUGRA solution [Klebanov,x x Strassler]+ y y = is⇢ everywhere. smooth. (5.17) · · k 2 2 3 δ2 δ + (δ γa) γa (8.19) At the tip of the cone, ⇢ = " ,theS remains finite (x x = " ),→ while ∩ · a=1 the S2 shrinks to zero size (y y = 0). Suciently far from the tip, the! right-hand side of (5.15) can be· ignoredupon transport and the around metric this point of the in the deformed moduli space. conifold is well-approximated by that ofWith the this singular background, conifold. we can Most now proceed models to discuss the resultquotedatthebeginningofthis section. We will do so in the context of a particularly instructive example, although it will be clear of D-brane inflation operate in this regime.

99 5.1 Inflating with Warped Branes 189

D3-branes on the conifold.—Now consider placing a stack of N D3-branes at the singular tip, zA = 0, of the singular conifold. As before, the branes backreact on the geometry, producing the warped ten-dimensional line ele- ment [628]

2 2A(r) µ ⌫ 2A(r) 2 2 2 1,1 ds = e ⌘µ⌫dx dx + e dr + r d⌦T , (5.18) ⇣ ⌘ where 4 4A(r) L 4 27⇡ 2 e =1+ with L gsN(↵0) . (5.19) r4 ⌘ 4 For r L, the solution is AdS T 1,1 [634]. ⌧ 5 ⇥ Warped deformed conifold.—Finally, we describe the warped deformed coni- fold, or Klebanov-Strassler (KS) geometry [426]. This is a noncompact, smooth solution of type IIB supergravity in which warping is supported by background fluxes. The KS solution can be obtained by considering the backreaction of N D3-branes at the tip of the singular conifold, together with the backreaction of M D5-branes wrapping the collapsed S2 at the tip, but we will find it useful to give an alternative presentation in which all D-branes are replacedWarped by fluxes Deformed carrying the associated Conifold charges (cf. [635]). The geometric substrate for the solution is the deformed conifold (5.15), 3 which contains two190 independent three-cycles:5 Examples the S at the of tip, String known Inflation as Alternativethe A-cycle description, and the Poincar´edual in terms three-cycle, of fluxes: known as the B-cycle.The background three-form fluxes of the KS solution are quantized where 1 14 27⇡ 2 2 F3 = M and L 2 H3gs=NK,(↵0) ,N(5.20) MK . (5.23) (2⇡) ↵0 ZA (2⇡) ⌘↵0 ZB4 ⌘

where M 1 andHere,K r1UV areis integers. an ultraviolet These fluxes cuto give↵, rise discussed to non-trivial further below. The logarithmic warping. The line element for the KS solution takes the form 3 running3 of the warp factor corresponds to that seen in the singular warped Near tip, S x R Far from tip,A ( r) 14.1conifold Type2 IIB with2 solutionA(r 3-form) fluxesµ of⌫ [628].2A The(r) 2 warp465 factor e in (5.21) reaches a minimal ds = e ⌘µ⌫dx dx + e d˜s , (5.21) A(rIR) AIR value e e CY at the tip, and is given in terms of the1,1 flux quanta by where d˜s2 is the metric of the deformed⌘ conifold definedapproximated by (5.15). As in theby AdS5 x T S3 [295] deformed conifold, the infrared geometry is smooth:(up the A-cycle to “log is finitecorrections”) in 2⇡K F3 H3 AIR size, with radius rA = gsM↵0, so the supergravitye approximation=exp remains . (5.24) 3gsM valid near the tip providedq that gsM 1. For our purposes, it will suce ✓ ◆ to cut o↵ the radial coordinate at a minimum value rIR, and work at r rIR The~AdS5 exponential hierarchy is a consequence of the logarithmic running in (but see [426]IR for a precise descriptionUV of the tip geometry). Far from the tip, the line element(5.22). is well-approximated The KS solution by (5.8), given with in (5.21) and (5.22) is the canonical example Figure 14.1 Warped throat from theof deformed a warped conifold with throat 3-form fluxes.geometry, and provides the basis for the most explicit 4 See [GS, Underwood];[GS,studies4A(r) Underwood,L of warped3gsM D-brane Kecskemeti,3gsM inflation.r Maiden] e = 1+ + ln , for(5.22) CMB effects. The exponentially small S3 size allows for a4 useful approximate description of the throat. Beforer proceeding, 8⇡K we2⇡K shouldrUV emphasize! that the ten-dimensional KS so- The deformed conifold can be approximated by a singular conifold, with a cone structure lution, involving a noncompact warped deformed conifold, does not give rise 2 m n 2 2 2 ds gmn dy dy dr r ds , (14.46) ˜6d =˜to dynamical= + gravityT1,1 upon dimensional reduction to four dimensions: the 2 3 with T1,1 S S being the 5dcompactification base of the cone, whose details volume, are not particularly and hence relevant the four-dimensional Planck mass, are = × 3 here. Fluxes are localized aroundinfinite. the S and their For effect model-building on the supergravity solution purposes, can be one considers instead a flux compact- approximately described as a stack of Nflux D3-branes at r 0. The solution is analogous = to that of D3-branes in flat spaceification (6.33), with Z containinge 4A,bysimplyreplacingthetransverse a finite warped throat region that is well-approximated = − R6 metric by (14.46). by a finite portion of the KS solution, from the tip r = rIR to some ul- 2 2A(r) µ ν 2A(r) 2 2 2 ds e ηµtravioletν dx dx e cuto− ↵drr =r rds ., Beyond(14.47) this, the throat attaches to a bulk space, = + + UVT1,1 with the warp factor harmonic functioncorresponding! to the remainder" of the compactification (see fig. 5.2). The metric4 of the bulk is poorly characterized in general, but the influence of 4A(r) R 4 2 e− 1 , with R 4πgs Nflux α′ . (14.48) = the+ r 4 bulk supergravity= solution on dynamics in the throat region can be pa- At very large radius r R the solution becomes the conifold metric, which glues to the ≫ rameterized very e↵ectively. The validity of the finite throat approximation rest of the compact CY space. On the other hand, for r R we can neglect the constant was systematically≪ investigated in [42] — see 5.1.2. term, and obtain § 2 2 2 r µ ν R 2 2 2 ds ηµν dx dx dr R ds . (14.49) = R2 + r 2 + T1,1 The 4d Minkowski directions combine with the throat radial direction to produceA Field an AdS Range5 Bound geometry; the angular coordinates parametrize a constant size T1,1 space. This approxima- tion holds at locations r large comparedThe total with the compactificationS3 size r z1/3,atwhichthefluxescut volume is the sum of the throat volume, ˜ = off the AdS throat, with a value for the warp factor rUV 2π K 2 5 4A(r) 4 2 2 eAmin r exp , (14.50) d⌦ 1,1 r dre =2⇡ gsN(↵0) rUV , (5.25) ≃ ˜ ≃ T −3Mgs T V #⌘ $ rIR providing an exponential hierarchy of energy scalesZ of physical systemsZ localized at the bottom of the throat, due to their large redshift. 2 The system provides a stringand theory the realization volume of the Randall–Sundrumof the bulk (RS) warped space. The Planck mass (3.47), Mpl = dimensions in Section 1.4.3, see Figure2 2 14.1. The throatV (14.49)B corresponds, upon a /gs  ,isfinite,with + . Ignoring the bulk volume gives a lower boundV on the Planck mass,V ⌘ VT VB

2 2 N rUV Mpl > 3 2 . (5.26) Downloaded from University Publishing Online. This is copyrighted material 4 http://dx.doi.org/10.1017/CBO9781139018951.015 (2⇡ )gs(↵0) z1 = A1B1,z2 = A2B2,z3 = A1B2,z4 = A2B1 (91) z1 =zA1 1=B1A,z1B21=,zA22=B2A,z2B3 2=,zA31B=2,zA1B4 =2,zA42B=1 A2B1 (91) (91) z1 = A1B1,z2 = A2B2,z3 = A1B2,z4 = A2B1 (91) z1 = A1B1,z2 = A2B2,z3 = A1B2,z4 = A2B1 (91) z1 = A1B1,z2 = A2B2,z3 = A1B2,z4 = A2B1 (91) 1 Ak Ak,B` B`z,k,1 =`A=11B,12,z with2 = A2B2C,z3 = A1B2,z4 =(92)A2B1 (91) ! ! 1 1 2 Ak AAkAk,BAA`k,B,B` B`z,k,11B=B``zA,k,z,k,=1z11B===,``12A,zA=1=11B with2B,,1=22,z,zA with2 with2B==2C,zAA23B=2C,zA13B==2,zAA14BBB=2(92),z,z,zA24B===1(92)(92)AAA2BBB1 (91) (91)(91)(91) ! k! k! ` ! 1 ` 11 1 1 2 2 222C 3 11 22 44 22 11 Ak ! Ak,B` !1 B`,k,` =1, 2 with 2 C (92) Ak A!k,B` !B`z,k,1 =`A=11B,12,z with2 = A2B2C,z23 = A1B2,z4 =(92)A2B1 (91) ! ! 2 2 2 2 2 1 A1 + A2 A=k B1 A+k,BB2 ` B`,k,` =1, 2 with(93) C (92) | |2 |CFT22 |2 Dual|22!2| | 22 2| ! 22 1 1 2 A1 +AA1A2++ A=AA2 B1A==A+BB1,BB2A++ ,BBB2 B,k,1B` =1,k,,`2=1 with, 2(93) with(93)(93) (92) (92) | | | |CFT1| 2 | |CFTCFT Dualk2|| 2 A A| DualDualkkk| k|1| 2|A`kk|,B2|2`` ` B`,k,`` =1=1,,22 with withC CCC (92)(92) 2|A1| +2 |CFTA2|!= 2Dual|B!!!1| +2 !|B2| !!1 2 (93)222 A1 |+ A| 2 |A= |B1 |A+ ,BB| 2 | | B ,k,` =1, 2 with(93) (92) z KW= A CFTB ,z has= U(N)xU(N)|A B| ,z|CFT=| gaugeA BDualk| ,z group| = kA |& Bmatter|` fields:` (91) C • 1 1KW1 CFT2 has2 2U(N)xU(N)3i↵ 1 2 gauge! 4 groupi↵2 1 &! matter fields: 2 KW• KWCFTKW CFT hasCFT U(N)xU(N) hasAhask U(N)xU(N)U(N)xU(N)e A gaugek,B gauge `gaugegroupe group group& AmatterB` 2&&+ mattermatter fields:A 2 fields:=fields:B 2 + B 2 (94) (93) •z1 =••zAzz111=B==1A,zAA111B2BB1=11,z,z,zA2222B===2!,zAAA22B3B=22,z,zA313B==2,zAA1!14BB=22,z,zA244B==11AA22BB11 2 1(91) (91)(91)2 SU(N ) iSU↵(N) iSUi↵↵(2) SU(2) iU↵(1)| |Uii(1)↵↵ |U(1) | | | | | SU(N+) SU(AN)k SU(2)A+AekkSUA(2)kee,BU(1)AAB`kk,B,BU(1)eA`` U(1)BBRee`2 BBA `` 22 R2 2 2 22 2 (94)222 (94)(94) z KW= A CFTB ,z has= U(N)xU(N)A B ,z = gaugeAi↵B ,z group= A &A Bmatter1 i↵+AA11 Afields:2++ =AA2B1=(91)+BB11B2 ++ BBB222 (93) (93)(93)(93) • 1 1 1 2 2 !2 Ak3i↵!! e1 A2 k!,B4 ` !i!↵2 e1 B` (94) SU(N+) SUSUSUSU(N(((NN++)+)))SUSUSU1SUSU(N((2)N(+(NN))))SUSUSUSUSUSU(N(((2)NN+(2)(2)+)) SUSUUSUSUSUSU(1)(2)((2)(N(2)BN)) SUUSUUSUU(1)(1)(2)(1)(2)(2)ABB USUUSU(1)U|(1)(1)(2)B(2)RAA |UU2U(1)U(1)(1)|(1)|ABRBR |U||U(1)U(1)(1)R|AA2 ||UU(1)(1)|RR| | 2 | | || | 2 ||| ||| Ak e !Ak,B1 ` 1 e!1B` 1 1 (94) Ak A1,A2Ak,BN+ A`1,A2N NB+2 `,k,N1` =12, 2 with1 A1C + A2 = B1 (92)+ B2 (93) ! ! ! 2N+N !2N+N 2N2+N2 2N+N 2 SU(N+) SU(N2) SU1SU(N(2)+) 1SUSU1 2((2)N) SUU(1)(2)B 2SUU(1)(2)A U(1)| B2R |U(1)A |U(1)R| | | | | A AAAkAAAD,A,B,A=AAAB k,BA,B,BNN AA` ,A+,ANBNNA,k, NBBN`22`,k,=1,k,B,1N1`12`1 =1=1 with1,21,1122B with with1 1111 1⇠111 i=1↵11 011 (92) i↵(92)(92) (95) k AB11,A,B22kkA1k,A11 NN2 2+2+ A`k11,AkN2N+2N1++A`1`,A1 N2N+22+1` 2NN+12+2`` N21 1 122N+N 222N1N++NNC 2N+2FIN CC2N2+NN 22 ! !!! ! !!! 2−N2+NN+N 22N2N2+NN+N+NNN 2−N222+NNN2+2ANN k2N2+NN2+2N22e 22N+AN k,B2 ` e B` (94) | |2 1 |22 |2 |22 2| |22 2| !22 ! 3N+ 2N 3N+ 2N 11 11211 11111 1112111 11 11 i↵11 i↵ Ak A1,A−2ABBD1k1,B,B,BN=22+ ΛAD`D1A,ANN−=2N=++BB1A+,B,BANB2+N2N1A`,k,++NN1+11`AA=12B2N,22B withB11 1B 20 BBC⇠2 i=↵2⇠⇠M 0FI i==↵ 0(92)i↵ ii↵↵ (95) (95) (95) BΛ11,B2 B1,BN2 + B11,BN2N+1 B1,BN2N1+1 22NN+21 2 2N1N2+01N 22N12+NN1+N 222N22N2+MN+N 22N2FI+NNN2+2+N 22N+FINN 22 ! ! − 2N+N −2−N2+NN+ N − 22N2N+A+2NN k 2−−N+2NN2+AANekk 22NA+Nke,Be 2A` k,Be` Bee` BBB`` (94) (94)(94)(94) w/ a superpotentialA |2D+=|A2||A 2(see|1|=| +|B 2[Klebanov,Witten]||A2 |2+|| B| 2||2B|1|| |2 for||B2 details)|| k !⇠FI = 0k (93)` ! ` (95) 33NN++ 22N1N 3N 2N+ 32N+ 2N 1 1 2 1 112 !1 i↵!1 2 ! i!↵ 3N 2N+ 2 2 2 ˜B3N,B+ 2N 3ND+ N2−−N=B˜3N1,BA+ 2N1N Λ3N|+N−2+N1 A| 2 N2| B|1 1 | 2 2 B| 22 | ⇠2 |FI0 =!2 02M ! (95) 1 − 2 ΛΛ1 + Λ −− Λ1 − 0 2N0+N 2N2+0MN 22M 2M ΛΛ1 − Λ 1 |− Λ|11 |Λ1 1 | | | 20N0|+N |2N0+NN 20M2N2AM+ 20MN e 2MNA+ ,B2M e B (94) 1 1 − N+ − N+ k N+ N+ k ` ` w/ a superpotential2| A|222 +2 | A |2 2(see=2 |B [Klebanov,Witten]2|2 +2 B| 2−2| ! for− details) ! (93) 3N+ 2N 3N+ A2N1 3N +2N+ A2 =ikBj1` + B2 2 (93) w/ a w/superpotential a33NN Asuperpotential22NN1++ +A1A1˜23N+ (see2N=+A22B [Klebanov,Witten](see1= +B [Klebanov,Witten]1B1 2 + 2 B2 22 for2 details)2 for details)2 (93) (93) λΛ31N 2−N+ Λ3˜˜N 2−−N+ λΛ311N 2−N+ Λ˜3N −2N+ 00 2 0 20M02 2 20M 2M02 2M ˜ 1 − ˜w/Λ11− a superpotentialΛ˜ −| Λ˜|ΛW1 − tree| =| ✏(see| ✏ [Klebanov,Witten]|AiNBN+0|+NjA|k0 BN 2` NN+020 +MforN 2details)M2N 2M 2 2 (96) Λ1 Λ1 | | 12|| |||1 | 2|| ||| | 2 || 0| || | 2−N0|| ||2MNN −−N2M N −− w/ a superpotentialA + A (see= B [Klebanov,Witten]+ BDik=j` A− 1 for+ −−details)A2− −B1 (93)B2 ⇠FI = 0 (95) 3N 2N+ 1 3N 2N+ 2 1 2 2 2 22 ˜ λ ˜ − λ1 ik j` ik 2j0` | |2 20| | 2 | 0 | | | Λ − λ1 Λ1 λ1 W =0✏ ✏20 A 20BM2 A 00B 2MN+N 0 (96) 1 1 | λ| |λ | | tree| | | N i0 N+Nj kN0 2` 0 N+N 20 2 2 λ1 λ1 1 iW↵1 W= ✏ =✏0i✏↵A B✏0 AA B0 AN+0BN −N+N (96) (96) Isometry becomestree SU(2)xSU(2)tree ikiglobalN+jNj`Dk−i −symmetries=N+2N`j A−k 22` +−−2− Aand2 U(1)2 BR 22 2 B 22 ⇠ = 0 (95) • Table 1. QuantumAkTable numbers 1.eQuantum in theAkSU,B numbers(WN +`treeM) inik thee=SUSU(jN✏`D)(BNmodel;−`+=✏M) weDDAASU have−i=B=(N written)j+model;AAA1AB we+ have+ writtenAA2B B(94)1B BB⇠2 =⇠⇠ 0FI(96)== 0 0 (95) (95) (95) × 2 × 1 k112 2` 22 1 1 2 22FI FIFI λ1 λ1! for concision. ! 0 0 | | 0 | | | | | | N+ N + M for concision.N+ N + M Wtreei↵= ✏ ✏ AiBNi+↵Nj|AkB| 2`| N|+|N | 2 | | | |2 | | | |2 | | (96) ≡ Isometry≡ becomesi↵ i↵i↵ SU(2)xSU(2)i↵ D−=ii↵↵ A global|+− symmetriesA| | B| 1,1and| U(1)B| R | ⇠| = 0 (95) Isometry• IsometryTableTable 1. 1.becomesQuantumQuantum AAbecomeskTableTablek numbers numbers 1.SU(2)xSU(2)eeQuantum in in theA the kSU(2)xSU(2)SUSU,B numbers((NN ++` MM) in global theeSUSU(N)(BN model;global `+symmetries1M) weSU have symmetries(N written) 2model; and we we have have U(1)1 written writtenandR U(1)2 R(94)(94) FI • TableModuli• 1. TableQuantumIsometry space 1.AQuantumkTable numbersA 1.ofekTable Quantum numbersinbecomes vacua theA 1.kSUe,BQuantum numbersin(N theA +`=kMSU ,BSU(2)xSU(2){D=0}/U(1)) in numbers(N theeSU+`SU(MN) in(BNmodel;× thee`+SUMSU(N) we)(B Nmodel;SU ishaveglobal`+ (MNthe written)) wemodel;× SU havecoset symmetries(N we written) havemodel; space written we have T writtenand: (94)U(1)R (94) • • ! for concision. !× | | × | | | | | | N N + M for concision.N+ N!+ M ! × × The classicalN++ N field+ MThe theory!for classical concision.N+ isifor↵! wellN concision. field+ awareM theoryforfor thatconcision. concision.! is× it well represents aware!i↵ × that branes it represents moving on br aanes moving on a N+ IsometryNN++M ≡≡forN + concision.NM+ becomesforN concision.N++M ≡≡N + MSU(2)xSU(2) global symmetries and U(1)R 1,1 • ≡Table 1.≡QuantumconifoldA≡kTable numbers [6, 1.e 7].≡Quantum in Let theA uskSU,B consider numbers(N +` M the) in case theeSUSU( whereN)(BNmodel;`+ theMA) weandSU haveB(N writtenhave) model; diagonal we have expectation written (94) conifold [6, 7].Moduli Let us consider space the case of where vacua the Ai×and =B k{D=0}/U(1)have diagonali× expectationk is the coset⇠FI space=1,1 0 T1,1: (96) Moduli• Moduli space2 !space of vacuafor2 concision.of(1) vacua =2 !{D=0}/U(1)(N) = {D=0}/U(1)2 (1) is the(N) iscoset the cosetspace Tspace: T1,1 : N+ N + M for concision.N+(1) N + M(NThe) classical field(1) theory(N is) well aware that it represents branes moving on a •values,D•A=i =diag(TheTheAModuli classical classicalvalues,a ≡+, A fieldA fieldspace,ai =diag(The theory theory), B classical iaofi is=diag( isB, well wellvacua field,a awareb awarei theory,), B=B that that,bi {D=0}/U(1) is=diag( it). well represents The awareb⇠i F-term, that=,b br equationsianes 0 it). is represents Themoving the F-term oncoset br a equationsanes moving space6 on aT (95): The≡ ⟨ classical•⟩The classical1 fieldThei theory··· field⟨ classical⟩ isiThe2 theory well⟨ fieldclassical aware⟩ is theory well···1 that field aware isi it theory well represents···⟨ that aware⟩2i is it well represents that branes aware itFI represents moving··· that branes it on represents moving br aanes moving on br aanes on moving a 1,1 on a conifold [6, 7]. Let us consider the case where the Ai and Bk have diagonal expectation for aModuli supersymmetricconifoldconifold| space [6, [6,for| 7]. 7]. avacuum supersymmetric Let Let| conifold us usof consider consider| vacua [6, 7]. the| the vacuum Let case case=| us where where{D=0}/U(1) consider| the the theA|i and caseB wherek have is the diagonaltheAi and coset expectationBk have(resolved diagonalspace expectation T conifold)⇠FI: = 0 (96) • conifold [6,conifold 7].2 Let [6, us 7]. consider Let2 us the consider(1) case where2 the(N)i case the whereAk and2 theB Ahave(1) and diagonalB(Nhave) expectation diagonal expectation conifoldTheconifold [6, classical 7]. Let [6, field us 7]. considerThe Lettheory usclassicalvalues, the2 consider is(1)(1) casewell fieldA where aware the((NN=diag( theory)) case2 the that whereAa is(1)i itand well, represents theB,a aware2kA(have(1)(1)Ni )and), thatdiagonal brBBanesk(Nihave=diag( it)2 represents moving expectation diagonalk b(1)i , on br expectation a,banes(kN) ). moving The⇠ F-termFI on a = equations⇠ 0FI = 0 0 (96) (96) (96) values,D A=2i =diag(A values,2ai +2, AA,ai(1)i =diag(2), 2B(iNa(1)i)=diag(B, 2,a(bNii2) ,), BB,bi(1)i =diag(2). The(Nb⇠(1))i F-term, =,b( equationsiN) 0). The F-term equations 6 (95) values,D⟨values,A=i⟩(1)=diag(AA1values,1 ((1)Na=diag(i) +,···A⟨ A,aai(⟩N=diag(i22), ), ⟨,aBai(1)⟩ i=diag(),B, ···1B,a(N=diag((1)bi)i ),,···⟨BB,bbi(⟩2Ni=diag(,) ).,b Theb⇠i). F-termFI, The···,b= F-term equationsi 0). The equations F-term equations (95) values,D values,A=i =diag(DA1A=⟨iconifolda=diag(⟩i +A, 1A [6,,aiforaii2 7].+ a,), Let supersymmetricB···B⟨A,a1 usiAii2⟩=diag(B considerB2),1 B⟨Bi2biAi the⟩i=diag(Bi, vacuumB1 case1···B=0,bi 2i wherebii,A)., The··· theB⟨,bi⇠1i2 F-termA⟩BFIik).andA2 The=B equationsi Ak⇠ 0i F-termhave2FIBk···A diagonal1==0 equationsi 0 expectation⇠ (resolved (10)6= 0 (95) 66conifold) (95) (96) conifold⟨ [6,forfor⟩ 7]. a a supersymmetricLet supersymmetric⟨B1A us⟩ iB consider|2|···⟨ Bfor|2|⟩A thei vacuumaB vacuum···1 supersymmetric case⟨=0|| ⟩ where,A||···−⟨ the⟩1AB|i vacuumk···Aand···⟨2 |B⟩Ak 2haveB···kA⟨| diagonal1 =0⟩ |··· expectation− ··· (10) FI(resolved conifold) for a supersymmetricfor a| supersymmetricfor|values,2| avacuum(1) supersymmetric−|forA| ( avacuumN=diag(|) supersymmetric2| a|(1) vacuum|, |,a2 (|(1)N vacuum)),| B|− (N|=diag()2 | b(1)|, ,b(N)). The F-term(resolved equations(resolved conifold) conifold) values,D A=i =diag(A1 ai +, A,ai i2 ), Bi i=diag(B1 bii , B,bi 2i ). The⇠i F-termFI = equationsi 0(r) 6 (95) ⟨ ⟩ are automatically···⟨ ⟩ ⟨ satisfied⟩ ··· in this···⟨ case,⟩ while the D-term··· eq(uationsr) 2 require a 2 + are automatically satisfiedfor a supersymmetricB inA thisB case,B A vacuumwhileB B=01 theAi,AB D-term2 B2AB eqiBAuations1 =0A,AB requireA =01Bak1A2 +A2B(resolvedkA (10)1 =01 ik conifold)j` (10) for a supersymmetric| | (r) vacuumB|11(Ari)iB|22 B(r2)2A|iiB11B=0|1Ai,AB2 | B211AB|iBk A12=0A,A2 Bk A1 =01BkA2 A2BkA (10)1 =0| | (10) (r) 2 (r) 2 (r) 2 2 2 B− A B2 BBAABB −=0B A,AB =0−B,AA A BB| AA | =0−A B A =0 (10) (10) a b Bb1Aai2B2=0.AlongwiththephasesremovedbythemaximumabelianBB1Ab21AiBiB2 1 −=0B1⇠b22Ai,AiB2=0.Alongwiththephasesremovedbythemaximumabelian1==01 21 0iB,Aik2A1 2− 2A1B2iBkkA1A21 −=01Ak2B2kA1 =012 kk W (10)21 −tree2 k (10)=1 ✏ ✏ AiB (96)jAkB` (97) 2 1 2| | − | | − | FI| − − − − (r) | | − | | − | | − − − − (r) 2 (r) 2 are automaticallysubgroupare satisfied of the automaticallyB gauge1A iniB this2 theory,6 case,B2 satisfiedAi theB while1 =0 D-terms in the,A this D-term leave case,1B onlyk eq whileAuations2 3NA the2independentB D-termkrequireA1 =0 eqa(uationsr) complex2 + require (10) vari- a1 2 + subgroupare of the automaticallyB gauge1AiB2 theory,Bare2 satisfiedA theiB automatically1 D-terms=0 in,A this leave case, satisfied1B onlyk whileA2 3 inNA the2 thisindependentB D-termkA case,1 =0 eqwhileuations complex the D-term require (10) vari- eqa1uations+ (r) require2 a1(r) 2ik+ j` (r) (r) (r)(r) 2 (r) (r) 2(r−) (r()r) 2 − (r) 2 | (1r) |2 | | are automaticallyare( automaticallyr) 2 (r)are satisfied(r) automatically2(r−) (are insatisfiedr)(ra) this(2r automatically) case, in satisfiedb( thisr) while case, inb satisfiedthe(r) this while D-term=0.Alongwiththephasesremovedbythemaximumabelian case,− in the thiseq while D-termuations case, the eqD-term whilerequireuations the eqa uations D-termrequire+ require eq|a uationsW| +a1 require+= |✏a1 |ik+✏j`A B A B (97) ables. Definea2 2n ables.b=1 a2 b Defineb2;thentheD-termandgaugeinvarianceconditionsare22 =0.Alongwiththephasesremovedbythemaximumabelian2nik =1 a2i bk⇠2FI;thentheD-termandgaugeinvarianceconditionsare2 = 01 1 treeik j` ik j` i (96)j k ` (r) ((rra))2 ik((rrb))(1r) 2i (kbr()2|r(a)r2)22|=0.Alongwiththephasesremovedbythemaximumabelian−(|r(b)r1)2 2| − |(⇠br2)FI2| =0.Alongwiththephasesremovedbythemaximumabelian= 0| (r)| | | | (r)| 2 | | (96) 2 | 22| − |aare22| automatically(−r) |ba 2| bb satisfied=0.Alongwiththephasesremovedbythemaximumabelian(r)b in this=0.Alongwiththephasesremovedbythemaximumabelian case, while the D-term eqWuations2 requireWW=tree✏a ==+✏ ✏A B✏ AAAiBBBjAAkBB` (97) (97)(97) aare2 automaticallybasubgroup12| | −bb21|satisfied satisfied of2 =0.Alongwiththephasesremovedbythemaximumabelian| the−b gauge|2 byin1subgroup|2 | this using=0.Alongwiththephasesremovedbythemaximumabelian| theory,−⇠ case,2FI| the1 of whilethe| then=−⇠ D-terms gauge|as2 0FI the coordinates.| D-term= theory,6 leave 0 eq theonlyuations These D-terms 3N 4independent requireN leavecomplexa only1 coordinates complextree+ 3N independent vari-tree satisfy1 ik complex the j` vari-i (96)j ki (96)`j k ` satisfied bysubgroup using| the of(r)|n theik− | gaugeas| subgroup(r coordinates.)| | −− theory,| | (r)| of| the−ik These| D-termsgauge| 4N theory,6 complex leave the only coordinates D-terms 3N independent leave satisfy only complex the 3N independent vari-| | complex vari- | (r|) 2− | (r)|| 2−−|| (r)||2 −2 |(r) | 2 (r) (r) 2 (r) (r) (r) | W| = ✏ ✏ A B A B (97) subgroupa subgroupb of the gaugebsubgroup ofa2 the theory,=0.Alongwiththephasesremovedbythemaximumabelian gauge(rsubgroup)b ofables.1 the the theory,(r D-terms gauge) Define⇠b(2r of) the the theory,=0.Alongwiththephasesremovedbythemaximumabelian6 =n leave D-terms gauge(r) 0= theonlya theory,6 (r D-terms leave) 3bN(r);thentheD-termandgaugeinvarianceconditionsareindependent theonly leave D-terms 3N onlyindependent complex leave 3N independent only vari- complex 3treeN independent complex vari- vari- complex vari-i (96)j k ` condition,2 ables.1 for each Define2condition,|r, | n−ik| = for|a− eachi |bk FIr;thentheD-termandgaugeinvarianceconditionsare|, ik i k | | − | ables.| −(r|) Define| (r)(nr)(ikr) ables.= (ar)i(r()b Definerk) ;thentheD-termandgaugeinvarianceconditionsare(r)n(r(ik)r) = a(ri()r)b(rk) ;thentheD-termandgaugeinvarianceconditionsare(r) ables.subgroup Defineables. of then(r Define) gauge= a ( theory,nr6 )b ;thentheD-termandgaugeinvarianceconditionsare= thea (rb D-terms) ;thentheD-termandgaugeinvarianceconditionsare leave only 3N independent complex vari- ables.subgroup Defineables.satisfied of then Defineik gauge by= a usingin theory,ikbk = the;thentheD-termandgaugeinvarianceconditionsaresatisfieda thein (ikbr D-terms)k as by;thentheD-termandgaugeinvarianceconditionsareik coordinates. usingi jik leavek` the onlyni ik Thesek 3asdetN coordinates.independentn 4ikN =0complex. These complex coordinates 4N vari-complex satisfy coordinates the (11) satisfy the satisfied by using thesatisfiednik(r)detas by coordinates.n using(ik(rr)) =0(r) the. n(r) Theseasi,k coordinates. 4N complex These coordinates 4N (11)complex satisfy coordinates the satisfy the (r)satisfiedables.W((rrtree)) Define( bysatisfiedr) using=(rn)ik✏ bythei,k= usingna✏ bas theA;thentheD-termandgaugeinvarianceconditionsare coordinates.inBikjasA coordinates.kB These` 4N Thesecomplex 4N coordinatescomplex coordinates satisfy the satisfy the (97) satisfiedables. Definesatisfied bycondition, usingnik by the= forusingnai eachbas thek condition,;thentheD-termandgaugeinvarianceconditionsare coordinates.r,n ikas coordinates. foriki These eachk r 4,Nik Thesecomplex 4N coordinatescomplex coordinates satisfy the satisfy the condition, forik eachcondition,r, ik for(r each) r, (r) satisfied(r) by using the n as coordinates.(r) These 4N(r)complex coordinates(r) satisfy the condition,Thissatisfied is thecondition, by for same using each ascondition, theThisr for, equationn eachik is theascondition,r for, coordinates. (2). same each Thus, asr for, equation eachforik These eachdetr,ik (2). 4nNr(r=1)jcomplex Thus,=0` , . for,N coordinates,thecoordinates eachdet nr(ikr=1) =0 satisfy, . ,Nn the,thecoordinates, (11) n11 , (11) (r) (r) (r) ikik j` deti,k n =0···. 11 (11) (r) (r) (r) Wtree =(r) ✏deti,k n(r✏ik) =0A···.i(rB) jAk(Bikr) ` (11) (97) condition, for eachncondition,r12, , n21 , n forW22 ,arenaturallythoughtofasthepositionofaD3-branemovin eachtreeikr, =j`✏i,kik ✏j`detAniB=0jdetAi,k. knB=0` . gona(11) (11) (97) n12 , n21 , n22 ,arenaturallythoughtofasthepositionofaD3-branemovindet nik =0det. nik =0. ik ik gona(11) (11) WtreeW=tree✏i,k =✏(r)✏Ai,k iB✏jAi,kAiBB(r) jAi,k B (r) (r) (97) (97) conifold. This is theik samej` as equationdetkn (2).ik `=0 Thus,.k for` each r =1, ,N,thecoordinates(11) n(11r) , conifold. This is theW same asThis equation= is✏ thedet (2).n sameik✏ =0 Thus,A as. equationB forA eachi,k (2).Br =1 Thus,, for,N,thecoordinates each r =1(11), ···,Nn,thecoordinates(11r), n ,(97) This(r) is(r) the( samer) tree as(r equation) (r)i,k ( (2).r) Thus,i forj eachk r =1` ,···,N,thecoordinatesn11 , (r) 11(r) This is theThisn same(r), isn the( asr), equationn same(r),arenaturallythoughtofasthepositionofaD3-branemovin as (2).equation Thus, (2). for Thus,each r for=1 each, ,Nr =1,thecoordinates(r) , ···,N,thecoordinates(r) n , gonan , This is theThisn12 same(r), isn the21( asr), n equationsame22(Therer),arenaturallythoughtofasthepositionofaD3-branemovin as areequation (2).12 various Thus,21 (2).22 forcombinations Thus, each r for=1 each, of the,Nr =1 fields,thecoordinates, ··· and,N,thecoordinates parametersn , whichgonan are, invariant11 11 Theren are, variousn ,(rn) combinations,arenaturallythoughtofasthepositionofaD3-branemovin(r) (nr(12)r) , n(r21) of, n( ther22) ,arenaturallythoughtofasthepositionofaD3-branemovin fields and parameters which are··· invariant11(r) ··· gona11 (r) gona (r) (r) (r)(12r) (r)21 This(r)22 is theconifold. same as equation (2). Thus,··· for each···r =1, ,N,thecoordinatesn11 , nThis, n isn the, conifold.n , samen,arenaturallythoughtofasthepositionofaD3-branemovin, asnn12 equation,,arenaturallythoughtofasthepositionofaD3-branemovinn21 ,nn1222 (2)., ,arenaturallythoughtofasthepositionofaD3-branemovinn21 Thus,, n22 ,arenaturallythoughtofasthepositionofaD3-branemovin for each r =1, ,N,thecoordinates···gonan11 , gonagona gona 12(r) 21(r) 12conifold.22(r) 21 22(r) (r) conifold.(r) ··· n , n , n ,arenaturallythoughtofasthepositionofaD3-branemovinn12 , n21 , n22 ,arenaturallythoughtofasthepositionofaD3-branemovingona gona conifold.12 21conifold.22Thereconifold. are variousconifold.There combinations are various of the combinations fields and parameters of the fields which and parameters are invariant which are invariant conifold. Thereconifold. are variousThere combinations are various of the combinations fields and parameters of the fields which and parameters are invariant which are invariant There areThere various areThere combinations various areThere combinations various of are the combinations various fields of the and combinations fields parameters of the and fields parameters of which andthe fields parameters are w invarianthich and parameters are which invariant are invariant which are invariant There are variousThere combinations are various of the combinations fields and parameters of the fields which and parameters are invariant which are invariant

4 4

4 4 4 4 4 4 4 4 4 4

9 9 9 99 9

9 9 9 9 9 9 9 9 9 9 9 Our Strategy • Non-compactification gives GN = 0 (not useful) • General SU(3) holonomy manifold is intractable (need metric) • Middle ground: inclusion of compactification effects in a finite throat:

Ψ D3 r D3 bulk warped throat

Figure 37: D3-brane inflation in a warped throat geometry. The D3-branes are spacetime-filling in four dimensions and therefore pointlike in the extra dimensions. The circle stands • Planck-scalefor effects the base manifold comeX5 fromwith angular the coordinates bulk of. The compactification brane moves in the radial (“Planck direction r.AtrUV the throat attaches to a compact Calabi-Yau space. Anti-D3-branes brane”). minimize their energy at the tip of the throat, rIR.

the tip of the throat, r = 0, is smoothed by the presence of appropriate fluxes. The tip of the throat is therefore located at a finite radial coordinate rIR, while at r = rUV the throat is glued into an unwarped bulk geometry. In the relevant regime r r

29.2.2 The Field Range Bound Before addressing the complicated problem of the shape of the inflationary potential let us ask if these models can ever source a large gravitational wave amplitude. It turns out that this question can be phrased in purely geometrical terms and does not depend on the details of inflationary dynamics [117]. By the Lyth bound we know that a large gravitational wave signal requires super- Planckian field variation. As a minimal requirement we therefore ask if super-Planckian field values are accessible in warped D-brane inflation. The inflaton kinetic term is determined by the Dirac-Born-Infeld (DBI) action for a probe D3- brane, and leads to an identification of the canonical inflaton field with a multiple of the radial 1 coordinate, 2 T r2. Here, T (2⇡)3g ↵ 2 is the D3-brane tension, with g the string ⌘ 3 3 ⌘ s 0 s ⇥ ⇤ 110 196 5 Examples of String Inflation single explicit embedding is known in which inflation can occur [218, 637]. This is the so-called Kuperstein embedding [639] f(z )=µ z . (5.45) 1 1 In this example, the scalar potential (5.40) receives a correction scaling as 3/2 ( z ), / 1 3/2 V0 2 VF () V0 + + + 2 + . (5.46) ⇡ ··· Mpl ··· The last two terms shown in (5.46) contribute to ⌘ with opposite signs. Let 0 be the point in field space where the second slow-roll parameter vanishes, ⌘(0) = 0. Near this point we have ⌘ 1. This is not even a fine-tuning, but arises dynamically. What does| | involve⌧ fine-tuning is the requirement that 0 is in the region of control (i.e. inside the warped throat) and that the potentialz1 = A1B is1,z monotonic2 = A2B2,z and3 = hasA1 aB small2,z4 = firstA2B derivative1 (small (91) ✏) at the same point. If2.2 this UV can Perturbations be arranged, then and we RG get Filtering inflation near an approximate inflectionIn an point ideal. Fig. world, 5.3 one shows would an determine example the of precise a successful D3-brane potential in a fully-specified scan in the parameter spacecompactification, of warped1 in D3-brane terms of fluxes, inflation D-brane [218]. positions, and closed string moduli vevs. With Ak Ak,B` B`,k,` =1, 2 with C (92) ! present! methods this is dicult to achieve,2 except perhaps in a toroidal orientifold setting such as [25, 33]. In this paper, our goal is to determine the general structure of the potential 10D Perspectivez1 = A1B1,z2 = A2B2,z3 = A1B2,z4 = A2B1 (91) arising from UV deformations of the background, 2 2 2 2 The example above providesA an+ existenceA = B proof+ forB inflation in warpedi throat (93) | 1| | 2| | 1| | 2| geometries, but the setup is too special to provide a goodV ()= sensec fori the 4 range, (2.12) 1 M i Ak Ak,B` B`i,k,` UV=1, 2 with C (92) of possibilities. Moreover, the above analysis! implicitly! assumedX that the 2 where is the canonically-normalized field related to the D3-brane position and M is a physics inside the throat decouplesi↵ completelyi from↵ the physics of the bulk, UV Ak e Ak,B` e B` (94) which as we stressed in UV4.2 mass is! rarely scale (related the case.! to r Finally,UV, the ultraviolet we have location modeled at which the the throat is glued into the § warped throat region bycompact a finite bulk; portion see Fig. of 2). a In noncompact terms2 of the2 parametrization warped2 Calabi- of2 Eqn. (2.12), our primary task A1 + A2 = B1 + B2 (93) Yau cone. This approximationis to compute fails the where scaling the dimensions| finite| | throati|, while is| attached leaving| | the| to coecients of individual terms, ci,2 undetermined.2 This2 undertaking2 is a necessary precursor to any calculation that does the remainder ofD the= compactification.A1 + A2 B1 In thisB2 section,⇠FI = we 0 describe a more (95) | obtain| | the| Wilson | coe| cients| | in a concrete model. For comparison, in Ref. [16] we argued general analysis that addresses these deficiencies. that in certain special circumstances,i↵ the dominant Planck-suppressedi↵ contribution to the A e A ,B e B (94) The essential idea isD3-brane that all potential ‘compactification comesUV from kPerturbations interactions! e↵ects’k — with i.e.` ! nonperturbative all infor-` e↵ects on a divisor in the mation about moduli stabilizationconifold. In the⇠FI and present= supersymmetry 0 paper, we are addressing breaking the in more the general re- situation (96) in which multiple mainder of the compactificationcompactification — can e6 ↵ects be make expressed important as contributions non-normalizable to the potential. perturbations of the noncompact solutionD = [42,A 640],2 + A 2 B 2 B 2 ⇠ = 0 (95) | 1| | 2| | 1| | 2| FI W = ✏ik✏j`A BrA B (97) (rtree)=(r ) i j k ` . (5.47) UV r ✓ UV ◆ Here, is the deviation of some supergravity field ⇠FIfrom= 0 its value in (96) 4A 6 the noncompact solution,VDr3UV= Tis3 the= radialT3 e location↵ of the ultraviolet end (98) of the throat, and is the scaling dimension of .6 By determining the probe D3-brane ik j` Wtree = ✏ ✏ AiBjAkB` (97) 6 In AdS/CFT, this corresponds to the dimension of the operator dual to the perturbation . warped conifold rUV UV perturbations 4A VD3 = T3 = T3 e ↵ (98) Picture from [Baumann, Dymarsky, Kachru, Klebanov, McAllister] rIR

Figure 2: Compactification can induce very general UV perturbations of the warped conifold solution, On the CFT side: CFT CFT + ci (99) but in the infrared only the lowest-dimensionL ! L perturbationsO contributei meaningfully to the D3-brane potential. Xi

Our interest is in the leading terms in the potential for a D3-brane that is well-separated

from the UV cuto↵ rUV. The dominant terms come from the Kaluza-Klein modes with

10

9

9 z1 = A1B1,z2 = A2B2,z3 = A1B2,z4 = A2B1 (91)

1 Ak Ak,B` B`,k,` =1, 2 with C (92) 196 ! 5 Examples! of String Inflation 2 single explicit embedding is known in which inflation can occur [218, 637]. This is the so-called Kuperstein embedding [639] 2 f(z )=µ2 z . 2 2 (5.45) A1 + 1A2 = 1 B1 + B2 (93) In this example, the scalar| | potential| | (5.40)| receives| a| correction| scaling as 3/2 ( z ), / 1 z1 = A1B1,z2 = A2B2,z3 = A1B2,z4 = A2B1 (91) 3/2 V0 2 VF () V0 + + + + . (5.46) ⇡ ···i↵ M 2 ···i↵ Ak e Ak,B` pl e B` (94) The last two terms shown! in (5.46) contribute! to ⌘ with opposite1 signs. Ak Ak,B` B`,k,` =1, 2 with C (92) Let 0 be the point in field space where the second! slow-roll! parameter 2 vanishes, ⌘( ) = 0. Near this point we have ⌘ 1. This is not even 0 | | ⌧ a fine-tuning, but arises dynamically. What doesz1 = involveA1B1,z fine-tuning2 = A2B2,z3 is= A the1B2,z4 = A2B1 (91) requirement that is in2 the region2 of control2 (i.e. inside the2 warped2 throat)2 2 2 D = A0 1 + A2 B1 B2 A1 ⇠+FIA2== 0B1 + B2 (95) (93) and that the potential| | is monotonic| | and| has| a small| first|| derivative| | | (small| | | | 1 ✏) at the same point. If this can be arranged,Ak thenA wek,B get` inflation B`,k,` near=1, 2 with C (92) an approximate inflection point. Fig. 5.3 shows an! example! of a successful 2 i↵ i↵ A e A ,B e B (94) scan in the parameter space of warped D3-brane inflation [218].k ! k ` ! ` A 2 + A 2 = B 2 + B 2 (93) ⇠FI = 0| 1| | 2| | 1| | 2| (96) 10D Perspective6 2 2 2 2 D = A1 + A2 B1 B2 ⇠FI = 0 (95) The example above provides an existence proof for inflation in warpedi↵ throati↵ | A|k e| A|k,B`| e| B|` | (94) geometries, but the setup is too special to provide a good sense! for the range! of possibilities. Moreover,W the above= ✏ analysisik✏j`A implicitlyB A B assumed that the (97) physics inside the throat decouplestree completely fromi j thek physics2 ` 2 of the2 bulk, 2 D = A1 + A2 B⇠FI1 =B 02 ⇠FI = 0 (95) (96) which as we stressed in 4.2 is rarely the case. Finally,| we| have| | modeled | | 6 the| | warped throat region by§ a finite portion of a noncompact warped Calabi- Yau cone. This approximation fails where the finite throat is attached⇠ = 0 to (96) W =FI✏6 ik✏j`A B A B (97) the remainder of the compactification. In this section,4A we describetree a morei j k ` general analysis that addressesVD3 = T these3 deficiencies.= T3 e ↵ (98) Two Dual Views ik j` The essential idea is that all ‘compactification e↵ects’Wtree —= i.e.✏ all✏ A infor-iBjAkB` (97) 4A mation about moduli stabilization and supersymmetry breakingVD3 = T3 in the= T re-3 e ↵ (98) mainder of the compactificationAdS — x canT1,1 be expressed as non-normalizableKW CFT 4A perturbations of the noncompact solution [42, 640], VD3 = T3 = T3 e ↵ (98) + c (99) CFT CFT i i L ! L r O i CFT CFT + i ci (99) (r)=(rUV) . L ! L (5.47) O rUV X + ic (99) ✓ ◆ LCFT ! LCFT OXi i Here, is the deviation of some supergravity field from its valueXi in the noncompact solution, rUV is the radial location of thevev ultraviolet describing end location of the throat, and isD3 the postion scaling dimensionD3 of .6 By determining the (100) on CoulombD3 branch (100)

6 In AdS/CFT, this corresponds to the dimension of the operator dual to the perturbation i 1 Vrenorm + ci (101) . Now in EFT, we could write: i 4 O ⇤ 1 i,i>4 V + c i X (101) renorm 1,1 i 4 Spectroscopy of T givesO the⇤ “structure”i of the potential; i,i>4 notX the Wilson coefficients ci

9

9

9 Two Dual Views

Here, the structure of operator dimensions {Δi} is not easy to get from the QFT, as the KW theory is strongly coupled.

We can obtain the spectroscopy of {Δi} in SUGRA and learn:

{Δi} ⇔ what sort of physics (fluxes, branes) in the bulk

This is a first step; next could try to understand the typical values of ci and eventually statistics of ci in the landscape. z1 = A1B1,z2 = A2B2,z3 = A1B2,z4 = A2B1 (91)

z1 = A1B1,z2 = A2B2,z3 = A11B2,z4 = A2B1 (91) Ak Ak,B` B`,k,` =1, 2 with C (92) ! ! 2

1 Ak Ak,B` B`,k,` =1, 2 with C (92) ! ! A 2 + A 2 = B 2 +2 B 2 (93) | 1| | 2| | 1| | 2| 198 5 Examples of String Inflation A 2 + A 2 = B 2 + B 2 (93) | 1| | 2| | 1| | 2| i↵ i↵ Taking theA metrick e ansatzAk,B` e B` (94) 198 5 Examples! of String! Inflation i↵ i↵ Ak e Ak,B` 2e B2` A(y) µ ⌫ 2(94)A(y) m n ! d!s = e gµ⌫dx dx + e gmndy dy , (5.49) D = A 2 + A 2 B 2 B 2 ⇠ = 0 (95) Taking the metric ansatz| 1| | 2| | 1| | 2| FI where gµ⌫ is the metric of a maximally symmetric four-dimensional space- 198 D = A 52 + ExamplesA 2 B 2 ofB String2 ⇠ = Inflation 0 (95) time,| 12| the| 2|A field(y|) 1 equations| | 2µ| ⌫FI of ten-dimensional2A(y) m typen IIB supergravity imply the ds = e gµ⌫7dx dx + e gmndy dy , (5.49) master equation ⇠ = 0 (96) Taking the metric ansatz FI 6 ⇠ = 0 (96) where gµ⌫ is the metric ofFI 6 a maximally2 symmetricgs 2 four-dimensional4A 2 space- 2 2A(y) µ ⌫ = R2A4 (+y) ⇤ m+ en + loc , (5.50) time, the fieldds = equationse gµ⌫ ofdx ten-dimensionalrdx +ik ej` g96mn type| d|y IIBdy supergravity, |r | (5.49)S imply the 7 Wtree = ✏ ✏ AiBjAkB` (97) master equation 2 ik j` where g is the metricwhereConifoldWtree of= a✏is maximally the✏ A iSpectroscopy LaplacianBjAkB` symmetric constructed four-dimensional using(97) the conifold space- metric (5.12), µ⌫ r Sloc is a2 localized sourceg due2 to anti-D3-branes,4A 2 and time, the field equations of ten-dimensionals type4A IIB supergravity imply the 7 = RVD43 += T3 ⇤= T3+ee ↵ + loc , (98) (5.50) D3-potential is given by Φ−, 4 Awhose solutions are given by: master equation198 rVD3 = T3 5= ExamplesT3 e96| ↵| of String Inflation|r | (98)S ⇤ +G + G+ , (5.51) 2 ⌘ 2 gs 2 4A 2 where Takingis the the Laplacian metric= R ansatz+ constructed⇤ + e using the+ conifold, metric(5.50) (5.12), loc r with 4 CFT CFT + i ci loc (99) S r CFT CFTL96+| !| Li ci |rO | S (99) is a localized sourceL due! L to anti-D3-branes,O i and 4A 2 2A(yG) i µ(? ⌫ i)G2A(y) andm n e ↵ . (5.52) ds = e gX dx dx6 X+ e 3 g dy dy , (5.49) where 2 is the Laplacian constructed±µ⌫ ⌘ using± the conifoldmn metric± (5.12),⌘ ± At the same time,⇤ theG three-form+ G flux, must satisfy the equationloc (5.51) of motion r + + (100)S is a localizedwhere sourcegµ⌫ dueis the to metric anti-D3-branes,D3 of⌘ a maximallyD3 and symmetric four-dimensional(100) space- time, the field equations of ten-dimensionali typed⌧ IIB supergravity imply the withThe 3-form flux in7 turn satisfies: d⇤ + (⇤ + ⇤¯)=0. (5.53) master equation ⇤ +G + G+ , 4A (5.51) 1 21Im⌧ ^ G (? ⌘ i)G i and i e ↵ . (5.52) Vrenorm +6Vrenormci +3 4 ci (101) (101) ± ⌘ 2 ± O ⇤gsi 2 O ⇤4Ai 4 ± ⌘2 ± i,i=>4R + ⇤ + e + , (5.50) with Consider first:The solutions X 4 toi, (5.50)i>4 can be organizedloc as follows: At the same time, ther three-form 96X flux| | must|r satisfy| theS equation of motion G (? i)G and e4A ↵ . (5.52) 2 26 3 2 2 whereR±4 ⌘12isH the± LaplacianV = T32 =V constructedV(0x,+ H )= + usingV...0±+⌘V the2 (2x conifold)+± V (102) metric(x)+ (5.12),V (x, loc) , (5.54) In quasi-dS:r⇡ R)4 12H Vi =dT⌧3 = V0 + H + ... (102)S At the sameis time, a localized the three-form source⇡ d due⇤ +) to flux anti-D3-branes, must (⇤ satisfy+ and⇤¯)=0C the equation. R ofB motion (5.53) where x r/rUV2andIm⌧ ^stands collectively for all five angular coordinates. ⌘2 V 00 2⇤ Gcan+ thisG contribution, be (5.51) We will⌘ = describeMP i d=⌧ each+ ... + of the terms+ in (5.54)(103) in turn. ⌘2 V 00 2¯ The solutions tod (5.50)⇤ + V can⌘ =3 M be( organized⇤canceled=+ ⇤+)=0... asby. follows:other sources? (5.53)(103) 2 Im⌧ ^P V 3 with Constant contributions.—The constant V0 represents possible contributions 4A from distantG (? sources6 i)G3 of supersymmetryand e breaking↵ . — in(5.52) the bulk of the com- The solutions to (5.50)V (x, can± )=⌘ beV±0 organized+ V (x)+ asV follows:(x±)+⌘ V (±x, ) , (5.54) pactification, or in otherC throatsR — thatB exert negligible forces on the D3- At the same time, the9 three-form flux must satisfy the equation of motion where x r/rV (x,brane, and)= andV onlystands+ V contribute(x)+ collectivelyV (x to)+ the forV inflationary(x, all five) , angular vacuum coordinates.(5.54) energy. This situation UV 0 i d9⌧ ⌘ corresponds tod⇤C maximal+ R decoupling(⇤ + ⇤¯)=0B of. the source of supersymmetry(5.53) break- We will describe each of the terms2 Im⌧ in^ (5.54) in turn. where x r/rUV anding fromstands the D3-branecollectively action: for all the five two angular sectors coordinates. communicate only through ⌘ We willConstant describeThe contributions. each solutionsfour-dimensional of the to (5.50)—The terms can in constant curvature. be (5.54) organized inV0 As turn.represents as explained follows: possible in 4.2, contributionscomplete decoupling of from distant sources of supersymmetry breaking — in the§ bulk of the com- Constant contributions.this—The sort is constantvery rare.V Werepresents have in factpossible made contributions an artificial but convenient pactification, or in otherV (x, throats)=V0 + —V (0x that)+V exert(x)+V negligible(x, ) ,8 forces(5.54) on the D3- from distant sourcesdivision, of supersymmetry using V0 to representC breakingR the — sum inB the of allbulkconstant of the com- contributions to the brane, andwhere onlyx potential, contributer/r and from tostands diverse the inflationary collectively sources, for each vacuum all of five which angular energy. will coordinates. in This general situation also contribute pactification, or in otherUV throats — that exert negligible forces on the D3- We will describe⌘ each of the terms in (5.54) in turn. brane,corresponds and only contribute tonon-constant maximal to the decoupling terms inflationary in other of the vacuum categories source energy. of described supersymmetry This below. situation break- correspondsing fromConstant to the maximal D3-brane contributions. decoupling action:—The of the constant the two sourceV sectors0 represents of supersymmetry communicate possible contributions only break- through from distant sources of supersymmetry breaking — in the bulk of the com- ing fromfour-dimensional the D3-brane7 curvature. action: the As two explained sectors communicate in 4.2, complete only through decoupling of pactification,In comparison or in other to throats (3.88), — we that have exert now allowed negligible§ the forces four-dimensional on the D3- curvature R4 to be four-dimensionalthis sortbrane, is very curvature. andnonvanishing: only rare. contribute We As compare have explained to the in inflationary (3.82) fact in and made4.2, (5.49). vacuum complete an energy. artificial decoupling This but situation convenient of 8 § 8 division, using VInto fact, represent one constant the contribution sum of all is groupedconstant in V rather contributions than in V0:thisisthevacuum to the this sort is verycorresponds rare.0 to We maximal have decoupling in fact made of the an source artificial of supersymmetryC but convenient break- energy contributed by the brane-antibrane8 pair, denoted D0 in (5.55). division,potential, usinging fromV from0 to the diverse represent D3-brane sources, the action: sum each the of two ofall which sectorsconstant will communicate contributions in general only also through to contribute the potential,non-constant fromfour-dimensional diverse terms sources, in curvature. other each categories As of which explained described will in in4.2, general complete below. also decoupling contribute of this sort is very rare. We have in fact made§ an artificial but convenient non-constant terms in other categories described8 below. division, using V0 to represent the sum of all constant contributions to the 7 potential, from diverse sources, each of which will in general also contribute In comparison to (3.88), we have now allowed the four-dimensional curvature R4 to be 7 non-constant terms in other categories described below. In comparisonnonvanishing: to (3.88), compare we have (3.82) now and allowed (5.49). the four-dimensional curvature R4 to be 8 nonvanishing:In fact, one compare constant (3.82) contribution and (5.49). is grouped in V rather than in V0:thisisthevacuum 8 7 C In fact,energy one constant contributedIn comparison contribution by to the (3.88), brane-antibrane is we grouped have now in allowedV pair,rather the denoted four-dimensional than inDV00:thisisthevacuumin curvature (5.55). R4 to be energy contributednonvanishing: by the compare brane-antibrane (3.82) and (5.49). pair,C denoted D in (5.55). 8 0 In fact, one constant contribution is grouped in V rather than in V0:thisisthevacuum C energy contributed by the brane-antibrane pair, denoted D0 in (5.55). 198 5 Examples of String Inflation

Taking the metric ansatz

2 2A(y) µ ⌫ 2A(y) m n ds = e gµ⌫dx dx + e gmndy dy , (5.49) Conifold Spectroscopy where gµ⌫ is the metric of a maximally symmetric four-dimensional space- time, the field equations of ten-dimensional type IIB supergravity imply the master equationLocalized7 sources:

2 gs 2 4A 2 = R4 + ⇤ + e + loc , (5.50) 5.1r Inflating with Warped96| Branes| |r | 199S 2 Localwhere sources.—Asis before, the LaplacianV (x)⇓ is the constructed Coulomb potential using sourced the by conifoldlocal, metric (5.12), r C S Sloc is a localized source due to27 anti-D3-branes,D0 1 and V (x)=D0 1 2 2 4 4 . Brane Inflation(5.55) [Dvali-Tye] C 64⇡ T r x ✓ 3 UV ◆ ⇤ +G + G+ , (5.51) In the inflationary regime (far from the⌘ tip), the dependence on the D3- brane position x is a subdominant e↵ect. This is a restatement of the fact with that warping — captured by the smallnessWarping of flattensD0 in (5.55) the potential — makes4 theA Coulomb potential extremelyG flat.(?6 i)G3 and e ↵ . (5.52) [Kachru,± ⌘Kallosh,± Linde, Maldacena, McAllister,± ⌘ Trivedi]± (KKLMMT) The eta problem revisited.—The Friedmann equation relates the Ricci cur- At the same time, the three-form2 flux must satisfy the equation of motion vature in four dimensions, R4 = 12H , to the inflationary energy density, V V0 + D0. Integrating (5.50), we find a curvature-induced mass term ⇡ i d⌧ ¯ d⇤ + (⇤ + ⇤2 )=0. (5.53) 1 4 2 24 Im⌧ ^ T3rUV V (x)= µ x + , where µ (V0 + D0) 2 . (5.56) R 3 ··· ⌘ M The solutions to (5.50) can be organizedpl as follows: This is how the curvature-coupling aspect of the eta problem arises in ten- dimensional supergravity.V (x, )=V0 + V (x)+V (x)+V (x, ) , (5.54) Bulk contributions.—Finally, we have a term thatC characterizesR all possibleB contributionswhere x fromr/r stress-energyand in thestands bulk of collectively the compactification, for all five angular coordinates. ⌘ UV We will describe each4 of the terms(L) in (5.54) in turn. V (x, )=µ cLM x fLM ( ) , (5.57) B Constant contributions.XLM—The constant V0 represents possible contributions wherefromc distantare constant sources coecients, of supersymmetryL (j ,j ,R) and breakingM (m ,m —) label in the bulk of the com- LM ⌘ 1 2 ⌘ 1 2 the SUpactification,(2) SU(2) orU(1) in quantum other throats numbers under — that the isometries exert negligible of T 1,1, forces on the D3- ⇥ ⇥ 1,1 andbrane, the functions and onlyfLM ( contribute) are angular to harmonics the inflationary on T (whose vacuum explicit energy. This situation formscorresponds can be found to in maximal [42]). The decoupling exponents (L of) have the source been computed of supersymmetry break- in detail in [42], building on a spectroscopic analysis of perturbations on AdSingT from1,1 [641]. the We D3-brane briefly summarize action: the the results. two sectors We split communicate the bulk only through four-dimensional5 ⇥ curvature. As explained in 4.2, complete decoupling of contributions into homogeneous solutions of the six-dimensional§ Laplace equationthis [317] sort is very rare. We have in fact made an artificial but convenient 2 =0, 8 (5.58) division, using V0 to representr h the sum of all constant contributions to the andpotential, inhomogeneous from contributions diverse sources, sourced by each flux [640], of which will in general also contribute non-constant terms in otherg categories described below. 2 = s ⇤ 2 . (5.59) r f 96| |

The solutions7 are characterized by their scaling dimensions (L). Solutions to (5.58)In satisfycomparison to (3.88), we have now allowed the four-dimensional curvature R4 to be nonvanishing: compare (3.82) and (5.49). 8 h(L) 2 H(j1,j2,R)+4, (5.60) In fact, one constant⌘ contribution is grouped in V rather than in V0:thisisthevacuum where p C energy contributed by the brane-antibrane pair,1 denoted D0 in (5.55). H(j ,j ,R) 6 j (j + 1) + j (j + 1) R2 . (5.61) 1 2 ⌘ 1 1 2 2 8  5.1 Inflating with Warped Branes 199

Local sources.—As before, V (x) is the Coulomb potential sourced by local, C S

27 D0 1 V (x)=D0 1 2 2 4 4 . (5.55) C 64⇡ T r x ✓ 3 UV ◆ In the inflationary regime (far from the tip), the dependence on the D3- brane position x is a subdominant e↵ect. This is a restatement of the fact that warping — captured by the smallness of D0 in (5.55) — makes the Coulomb potential extremely flat. The eta problem revisited.—The Friedmann equation relates the Ricci cur- 2 vature in four dimensions, R4 = 12H , to the inflationary energy density, V V + D . Integrating (5.50), we find a curvature-induced mass term ⇡ 0 0 2 1 4 2 4 T3rUV V (x)= µ x + , where µ (V0 + D0) 2 . (5.56) R 3 ··· ⌘ Mpl This is how the curvature-coupling aspect of the eta problem arises in ten- dimensional supergravity. Bulk contributions.Conifold—Finally, Spectroscopy we have a term that characterizes all possible contributions from stress-energy in the bulk of the compactification,

4 (L) Bulk contributions: V (x, )=µ cLM x fLM ( ) , (5.57) B XLM where c are constant coecients, L (j ,j ,R) and M (m ,m ) label LM ⌘ 1 2 ⌘ 1 2 Afterthe someSU(2) work,SU one(2) obtainsU(1) in quantum this example numbers [Baumann under et the al] isometries of T 1,1, ⇥ ⇥ and the functions f ( ) are angular harmonics on T 1,1 (whose explicit LM 1 3/2 formsV = canV0 be+ foundcij( in) [42]).+ a3 The/2h3/ exponents2( ) (L) have been computed in detail in [42], building on a spectroscopic2 analysis of perturbations on +[b2 + c2 + a2h2( )] AdS T 1,1 [641]. We briefly summarize the results. We split the bulk 5 ⇥ p28 3 contributions+ intocp homogeneous28 3jp28 3( solutions) of the six-dimensional Laplace equation [317]+ ... (104) 2 =0, (5.58) r h and inhomogeneous contributions sourced by flux [640], Tracing the origin: a-terms: fluxes; b-terms: effect of R4 ; 2 gs 2 Φ c-terms: harmonic= ⇤ parts. of − (5.59) r f 96| | The solutions are characterized by their scaling dimensions (L). Solutions to (5.58) satisfy (L) 2 H(j ,j ,R)+4, (5.60) h ⌘ 1 2 where p 1 H(j ,j ,R) 6 j (j + 1) + j (j + 1) R2 . (5.61) 1 2 ⌘ 1 1 2 2 8 

10 potential can be generated if the CFT Lagrangian is perturbedbyoperatorscomposedof the scalar fields that parameterize the Coulomb branch. In particular, we are interested in perturbations that do notexplicitlybreaksuper- symmetry and that incorporate the effects of bulk Calabi-Yau fields. Consider an operator of the form of Eq. (4.54), with

d4θ X†X , (4.55) Oδ ≡ O∆ ! where X is a bulk moduli field. This term, being an integral over superspace, is allowed in a supersymmetric Lagrangian, but will break supersymmetry spontaneously if X obtains an F-term vacuum expectation value. Notice that is a composite operator, containing Oδ both bulk and CFT fields, with total dimension δ =4+∆.Suchaperturbationyieldsa term in the D3-brane potential of the form [154]

∆V φ∆ . (4.56) ∝ In the following we will identify the CFT operators that correspond to perturbations O∆ of Φ− and hence induce a D3-brane potential; the operator dimensions ∆ will then dictate the structure of possible terms in the D3-brane potential.

4.5.3 Classification of Operators To enumerate the lowest-dimension contributing operators,wemustgiveafewmoredetails of the structure of the gauge theory. (More background on the gauge theory dual to the KS throat may be found in Ref. [160].) The approximate CFT thatisdualtotheKS throat is an SU(N + M) SU(N)gaugetheorywithbi-fundamentalfieldsA ,B (i, j = × i i 1, 2). These fields parameterize the Coulomb branch and, in particular, contain the data specifying the D3-brane position. The single-trace operators built out of the fields Ai,Bi and their complex conjugates are labeled by their SU(2)A SU(2)B U(1)R quantum where × × numbers (J1,J2,R); this symmetry group corresponds to the isometries of the base manifold T 1,1.UsingtheAdS/CFTcorrespondence,thedimensionsofthesefL(Ψ) ΦLM YLM (Ψ)+c.c. , operators are given by(4.45) ≡ Eq. (4.46). In fact, the leadingDual contributions CFT!M Operators to the D3-brane potential involve either chiral and Φoperatorsare constant whose dimensions coefficients. are dictated The quantities by their U(1)∆(RL)arerelatedtotheeigenvaluesofcharges, or operators related by LM Δ the angularsupersymmetryOn Laplacian the gravity to the Noether side, currents is related of the to global the eigenvalue symmetries, andof Laplacian: in either case the dimensions are protected and could be computed directly in the gauge theory. However, this will not be true of the operators that induce subleading corrections.2 ∆(L) 2+ 6[J1(J1 +1)+J2(J2 +2) R /8] + 4 . (4.46) Chiral operators.≡−For J1 = J2 = R/2, one has chiral operators− of the form To determine the leading perturbations" to the brane potential we are interested in the lowest =TrA(i1 B Ai2 B ...AiR)B + c.c. (4.57) Δ should correspond∆ to conformal(j1 j2 dim. ofjR )operators in CFT. eigenvalues, since via Eq.O (4.44) these correspond to the perturbations that diminish most " # The dimensions of these operators, ∆ =3R/2, are fixed by =1superconformalinvari- 11 slowly towardsStrongly the tipcoupled of the CFT, throat. but IncorporatingΔ of some operators the grouN p-theoretic are protected. selection rules ance. The lowest-dimension such operators are that restrict the allowed quantum numbers [155]: Δ determined Chiral operators 3/2 =Tr(AiBj)+c.c. , (4.58) O J1 and J2 are both integers or both half-integers. by R-charge • which have J Non-chiral,J ,R = operators.1 , 1 , 1 .ThesechiraloperatorshaveThere are a number of operators∆ =3/2, which and have in generic the next lowest { 1 2 } { 2 2 } J = J = R/2 = 1/4 situationsm J theydimension,, contribute,J ∆and=2.Forexample,thereareoperatorswith them leading termJ , in,J the inflaton. potential viaJ1,J Eq.2,R (4.56).= 1, 0, 0 : • 1 ∈ {− 1 ··· 1} 2 ∈ {− 2 ··· 2} { } { } Non-chiralZ R R 1 R sin with J1,2 =Tr,J1 andA1A¯2 , Tr JA2,2A¯1 ,J, 2 . Tr A1A¯1 A2A¯2 , (4.59) • operators2 ∈ {− O ··· } 2 ∈ {− ··· }√2 − ! " ! " ! " one finds that theand smallest the corresponding eigenvaluesJ ,J corresponding,R–31–= 0, 1, 0 tooperators nontrivial made perturbations out of the fields areB .These same multiplet as SU(2)xSU(2){ 1 2 } global{ }sym, current, Δ=2 j non-chiral operators3 are in the same supermultiplets as SU(2) SU(2) global symmetry ∆ = for (J ,J ,R)=(1/2, 1/2, 1) , × (4.47) currents, and so2 their dimension1 2 is exactly 2. ∆ =2 for (J1,J2,R)=(1, 0, 0), (0, 1, 0) . (4.48) Table 1: AdS/CFT Dictionary for Warped D-brane Inflation. For simplicity, we now assume that a single mode dominates theexpansioninEq.(4.44), Gravity Side Gauge Theory Side 4 † ∆ ∆V = T Φ− ∆(L) ∆ = d θ X X L 3 r L O∆ 2 (i1 iR) Φ− Φf−L=0(Ψ) . ∆ =Tr A B(j ...A Bj )(4.49) ∇≈ rUV∆ O % 1 ∆ R ∆V = #φ $ ∆V = φ − φUV !− φUV " (If more than one angular mode is relevant# during$ inflation, then the# dynamics$ is signifi- cantly more complicated∆ thaneigenvalue what is describedof Laplacian below.)operator To isolate dimension the radial dynamics, we first minimize the potentialφ inradial the location angular directions.energy When scalethe angular coordinates have φ maximal UV radius UV cutoff relaxed to their minima,UV the potential reduces to an effectivesingle-fieldpotentialforthe radial direction r.Inthesingle-perturbationcaseofEq.(4.49),theΦ− perturbation then always leads to a repulsive force, i.e. the effect of the perturbation is to push the brane out of the throat. The proof is straightforward: any non-constant spherical harmonic is orthogonal to the constant (L =0)harmonic,andhenceanynontrivialharmonicneces- sarily attains both positive and negative values. Therefore, there always exists an angular location Ψ⋆ where fL(Ψ⋆)isnegative.Itfollowsthatatfixedradiallocation,theD3-brane potential induced by the term in Eq. (4.49) is minimized at an angular location where the contribution of Eq. (4.49) to the radial potential is negative. This contribution to the potential is minimized at r .Thepotentialinducedbyanyindividualperturbation →∞ of the form of Eq. (4.49) therefore produces a radially-expulsive force. This is fortunate, since only a repulsive force allows cancellation with the mass term in Eq. (4.8) to alleviate the eta problem.

11so as to ensure the existence of single-valued regular solutions

–28–

–32– 5.1 Inflating with Warped Branes 195 where 4 is the ‘warped volume’ (3.107) wrapped by the D7-branes. Chang- ing theV position of a spacetime-filling D3-brane alters the warp factor A(), and hence 4(), so that W = W (). To quantify this e↵ect, one computes the backreactionV of the D3-brane on the four-cycle wrapped by the D7-branes [335]. For a four-cycle defined by a holomorphic embedding

f(z↵)=0, (5.42) the result can be written as

aT 2⇡ W (T,z↵)=W0 + (z↵)e ,a, (5.43) A ⌘ Nc where the function (z ) is defined in terms of the embedding (5.42), A ↵ f(z ) 1/Nc (z )= ↵ . (5.44) A ↵ A0 f(0) Some Features✓ of D3-brane◆ Inflation • Small field inflation (undetectable r) → next 15.71 7.85 • Kinetic term of DBI form: for “relativistic” motion, inflaton 15.70 reaches “speed limit” (strong7.84 and distinctive NG). 15.69 7.83 15.68 • V(ϕ) receives crucial contribution7.82 from moduli stabilization, e.g., 15.67 WNP & bulk effects. 7.81 0.2• Wilson 0.4 coefficients 0.6 0.8 depend on UV0.2 completion 0.4 0.6 (string 0.8 theory). 5.47 • Phenomenology: 2.29 5.46 2.28

5.45 2.27 inflection point 5.44 inflation 2.26 5.43 2.25

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8

Fig. 5.3. Example scan through the parameter space of warped D3-brane inflation (figure adapted from [218]). The scan parameter s is the ratio of the antibrane energy to the F-term energy before uplifting. Successful inflation occurs in the gray shaded region.

Fine-tuning to produce a flat potential.—Which embedding functions f(z↵) lead to forces that can balance the curvature coupling? This question was addressed in a number of papers [217, 218, 636, 637]. An important no-go result was proven in [218, 636]. The infinite class of embeddings studied in [638] does not allow any inflationary solutions. In fact, to date only a Lecture 3 Primordial B-mode

7 1 1 B-mode signal 0.3 BK+P B+P 6 ]

−50 K+P 2 0.8 0.8 K µ 5 −55 0.6 0.6 4

peak 0 peak

−60 L/L L/L 3 0.4 0.4 2 @ l=80 & 353 GHz [ d Declination [deg.] −65 Joint −0.20.3 0.2 A BICEP2 Collaboration BICEP-Planck 1 0 0 0 50 0 −50 0 0.05 0.1 0.15 0.2 0.25 0.3 0 1 2 3 4 5 6 7 0 0.05 0.1 0.15 0.2 0.25 0.3 2 A @ l=80 & 353 GHz [µK ] Right ascension [deg.] r d r

Many experiments including BICEP/KECK, PLANCK, ACT, PolarBeaR, SPT, SPIDER, QUEIT, Clover, EBEX, QUaD… can potentially detect such primordial B-mode with sensitivity Δr≾10-2.

LiteBIRD (& CMBS4) may even have the sensitivity of Δr ~ 10-3. Any massless field experiences quantum fluctuations during inflation:

Inflation stretches these to macroscopic scales:

classical 380,000 yrs

10-33 sec

expansion by e60

t=0 quantum Any massless field experiences quantum fluctuations during inflation:

Inflation stretches these to macroscopic scales:

classical 380,000 yrs

10-33 sec

expansion by e60

t=0 quantum Two massless fields that are guaranteed to exist are:

Goldstone boson graviton ζ of broken time translations hij Two massless fields that are guaranteed to exist are:

Goldstone boson graviton ζ of broken time translations hij expansion

symmetry breaking ( for slow-roll inflation) Two massless fields that are guaranteed to exist are:

Goldstone boson graviton ζ of broken time translations hij expansion

symmetry breaking ( for slow-roll inflation)

E-modes:

B-modes: ζ hij B-mode and Inflation

If primordial B-mode is detected, natural interpretations:

✦ Inflation took place at an energy scale around the GUT scale

1/4 r 2 E 0.75 10 M inf ' ⇥ 0.1 ⇥ Pl ⇣ ⌘ ✦ The inflaton field excursion was super-Planckian

r 1/2 M 5Sth W96 & 0.01 Pl ⇣ ⌘ ✦ Great news for string theory due to strong UV sensitivity! Assumptions in the Lyth Bound single field slow-roll

Bunch-Davies initial conditions vacuum fluctuations Assumptions in the Lyth Bound single field slow-roll

Bunch-Davies initial conditions

For counterexamples, see, e.g., vacuum fluctuations Cook and Sorbo Senatore, Silverstein, Zaldarriaga Barnaby, Moxon, Namba, Peloso, GS, Zhou Mukohyama, Namba, Peloso, GS