Towards Classical De Sitter Solutions

Towards Classical de Sitter Solutions

Clay C´ordova

Kadanoﬀ Center & Enrico Fermi Institute University of Chicago

September 24, 2019 References Classical dS Solutions of 10d Supergravity [1812.04147] More on Classical dS Solutions of 10d Supergravity [1910.xxxxx]

AdS8 Solutions in Type II Supergravity [1811.06987]

In collaboration with:

De Luca (Milan/Stanford) Tomasiello (Milan) Main Results

We construct new solutions to the 10d IIA supergravity equations

Essential Features: The solutions have fully localized sources • The cosmological constant Λ is positive • The solutions have moduli at tree level, so their status in • string theory depends on quantum corrections Motivation

It is interesting to understand what kinds of de Sitter solutions exist in string theory. Recently renewed interest in this subject! [Obied-Ooguri-Spodyneiko-Vafa, 100+ more papers!] A fundamental constraint is: No-Go: Any classical solution of 10d SUGRA involving only two derivative action and sources (branes) obeying standard energy conditions (e.g. D-branes) has Λ 0 [Gibbons, Maldacena-Nunez] ≤ Thus any dS solution of string theory necessarily involves either: Quantum corrections (α , g , non-perturbative) that go • 0 s beyond the SUGRA approximation (e.g. [KKLT]) Classical sources violating standard energy conditions. Natural • candidate: orientifolds. Motivation

A wish list for classical solutions: Directly solve 10d/11d equations. Most solutions in the • literature are derived from 4d eﬀective actions (not consistent truncations). Non-linear nature of gravity makes it diﬃcult to fully justify All sources (branes, orientifolds) localized and fully • backreacted. Some literature have found solutions where orientifolds are smeared. This is encouraging but unphysical Simple. Minimize the ingredients so that the physical • properties can be easily scrutinized. This Talk: First explicit de Sitter solutions of 10d SUGRA with fully backreacted and localized sources Orientifolds Redux

Below we make use of some basic properties of orientifolds. These sit at fixed loci of involutions. They are non-dynamical i.e. unlike Dp branes they have no fluctuating worldvolume fields. There are two main varieties of orientifolds characterized by their tension τ and RR charge Q (set to 1 for a Dp) The orientifold plane Op : τ > 0 , Q = 2p 5 • + − Coincident with Dp branes this leads to SP gauge− groups The orientifold plane Op : τ < 0 , Q = 2p 5 • − Coincident with Dp branes− this leads to SO gauge− groups So any candidate dS solution must have some Op to violate the no-go assumptions. − Orientifolds Redux

In “ﬂat space” the orientifold is described by a (string frame) metric and dilaton

2 1/2 2 1/2 2 φ (3 p)/4 ds = H− ds + H ds , e = gs H − || ⊥ Here H is harmonic function of the distance r to the orientifold

r 7 p 1 0 − p < 7 H = − r (log(r/r0) p = 7

For r < r0 the metric doesn’t make sense. One might say that there is a hole in the manifold. The curvature and coupling are becoming large and supergravity breaks down near the orientifold! This makes ﬁnding solutions with backreacted orientifolds hard. We must know how string theory completes these singularities. Orientifold Redux Examples of orientifolds becoming smoothed out in string theory: O6 planes lift to M-theory on a smooth Atiyah-Hitchin space • O7 planes in IIB can be resolved by F-theory and contain two • mutually non-local D7s inside the hole [Sen] Below we will see some solutions with O8 . Here there is a single transverse coordinate z and (formally) no± hole

H = a 8gs z ∓ | | The constant a is a parameter of the solution. For a non-zero the dilaton is ﬁnite on the O8. For a = 0 the dilaton diverges on the O8. In both cases however the curvature is large there so strictly speaking the solution is outside the realm of supergravity. We won’t explain how (or if!) these solutions are completed in string theory, but instead try to at least ﬁnd some interesting starting points for analysis. An Elementary Ansatz for de Sitter In massive type IIA, assume a string frame metric of the form

ds2 = e2W ds2 + e 2W dz2 + e2λds2 dS4 − M5 The metric functions W , λ as well as the dilaton φ depend only on the coordinate z

The only non-zero ﬂux is the zero-form F0. (Sourced by 8-branes)

The compact ﬁve-manifold M5 is Einstein:

Rab = κgab

The scale of the curvature constant κ is unphysical: it can be absorbed into λ. However the sign of κ will be ﬁxed

κ < 0 , so e.g. M5 could be a quotient of hyperbolic space Reduction to ODEs

Since all functions depend only on z, the equations of motion reduce to ordinary diﬀerential equations

4φ/5 2 25κe− 4α0φ0 4 (α0) 5 2 2φ 2W 0 = + F e − α2/5 α − α2 − 2 0 2 4W 2 20W 0φ0 + 40 W 0 + 20Λe− + 4 φ0 , − 4W 2 2(W +φ) 4W 4e α0W 0 0 = F e + 4Λ 4e W 00 , 0 − − α 3/5 4W 4φ 1 4W 0 = αΛ + α κe − 5 e α00 , − 5

5λ 2φ where α = e − (signiﬁcance seen below) Equations are highly non-linear, but trivial to solve numerically. Sources

The Romans mass F0 is sourced by codimension one objects extended along all directions except z where they are localized At these loci the metric and dilaton are continuous, but have discontinuities in their ﬁrst derivatives (e.g. z ) | |

1 1 φ W ∆W 0 = ∆φ0 = e − ∆F0 , ∆α0 = 0 , 5 −4 W φ α0 F0 z (z0)− = e − φ0 W 0 + , | → − − α The first line above shows the significance of the function α. The second line above states that forces on the object are balanced (more discussion below) Note how mild the singularities are. This gives us hope that we can find them numerically. Brane Menu

The possible sources to consider are characterized by their tension τ and charge Q (= 2π∆F0) − The ordinary D8-brane (or anti-brane): τ > 0 , Q = 1 • ± The orientifold plane O8 : τ > 0 , Q = 8 • + The orientifold plane O8 : τ < 0 , Q = 8 • − − To violate the no-go assumptions we need at least one O8 − Near a source we expect the solution to asymptote to the ﬂat space form discussed earlier

2 1/2 2 2 2 1/2 2 ds H− ( dx + dx + ... dx ) + H dz , 10 ∼ − 0 1 8 and eφ H 5/4, with H a harmonic (i.e. linear) function of z ∼ − where we have used to denote tangency as in [48]. This coordinate change is illustrated || in the top part of Fig.K˜ to2.9 distinguish,intheO7 it+ fromcase. the case above, the Green–Schwarz contribution is now

2 1 p1(T ) 1 (K˜ + F˜)+(In⇤⌫(F ) D˜ + ⌫(F ) 2D˜ ) . (4.6) 2 4 · sp · 2 1 so · 2 O7+ +(n 4)D7 ✓ ◆ b m D7 u Ins where the factor 1/2infrontofD˜ 1 is due2 tom the fractionation of D3-branes on O7+, v and the factor 2 in front of D˜ 2 is due to the embedding index 2 of so2n+8 su2n+8. The ˜ ˜ ˜ ˜ ⇢ frozen divisor F is⇠=D1 + D3, where D3 is the noncompact divisor on the far right. m D7 2 2 The terms with tr Fsp and tr Fso in the two expressions (4.4)and(4.6)shouldagree, 3 4 , since they cancel the same 1-loop anomalies. Indeed, we can easily check that O7+ +(n 4)D7 = O8+ +(n 4)D8 1, 2 ⇠ K m D7 K˜ + F˜ 12 1 1 m D6 D (D1,D2)= D˜ D˜ 1, 2D˜ 2 = 11 . (4.7) 4 0 11 · 0 2 1 1 · 2 0 1 D2 2D˜ ⇣ ⌘ 1 4 3 B C B 2 C B C @ m DA7 @ A @ A In addition,1, 2 as observed earlier, K2 =(K˜ + F˜)2.

Figure 2.9: Various equivalent ways of seeing a tangential I⇤–I intersection. As in recent figures, the dot on the4.2 bottom-rightsu-su chains frame is a half-NS5. b Let us next consider the IIA configurations shown in Fig. 4.1. The top row and the An additional subtlety concerns the matter content in (2.23). One can in principle bottom row are distinguished by the type of the O8-plane; we add 16 D8-branes for work this out directly in the original setup on the left of Fig. 2.9, but it is instructive the top row to have the same Romans mass for the both rows. The configurations on to do it instead in athe dual left frame. column First contain of all tangential we change intersections coordinates, ofusing the again type (discussed2.18); in section 2.8. 1 2 3 4 only this time we takeThez configurations= x + ix , w on= thex right+ ix columnintroduced are obtained earlier, and by moving define new the half-NS5-brane at coordinatesx ˜1 + ix˜2 = zw,˜x3 = z 2 w 2, with a fourth periodic coordinate eix˜4 = zw¯ . the intersection of the 6-branes and the 8-branes away from thezw¯ intersection. Gauge 4| | | | 1 4 3 We are once againtheoretically, rewriting R thisas operation a fibration correspondsS , R to givingR . Thea vev orientifold to hypermultiplets. is 3 3 4 ! 4 !Orientifold Circle now defined by the involutionUsing the discussion:˜x inx˜ section, x˜ 2.8x˜ and; the following O7-plane [16 then, 17], sits we atfind that these con- 3 4 ! In1 our! solutions,2 z is periodic with an orientifold4 action x˜ =˜x =0,whiletheD7sareonthelocus˜figurations realize 6d quiversx whose=˜x = structures 0. (Notice are that summarized thex ˜ circle in Fig. 4.2.(Wedid 3 z z + 2z0 , z z shrinks atx ˜ =0.)IfwenowT-dualizealongdirection4,weendupwithanO8atnot explicitly write in that figure the standard∼ bifundamental↔ − matter hypermultiplets 3 Fixed loci: z = 0 an O8+, z = z0 an O8 x˜ = 0 with a half-NS5between stuck two on consecutive it, and with gauge D6s crossing factors.) it. The type of the− O8-plane is correlated to the All this is depicted on the lower part of Fig. 2.9, again for the O8+ case. At this type of the two-index tensor representation of the sun gauge algebra. Higgsing is done point we can read oby↵ the giving matter a vev content to the from hypermultiplet a perturbative in the string antisymmetric computationZ2 or symmetric similar two-index ten- to the one leading to (2.4), as already done in [16,17]; the result is that in the tangential sor representations of sun, breaking2⇡ itF0 to=+4spn/2 or son. Here2 for⇡F0 simplicity= 4 n is assumed intersection (2.23)theum has a hypermultiplet in the antisymmetric in the unfrozen to be even in the former case; if n is odd, the gauge algebra is sp n/2 and one needs to b c case, and in the symmetric in the frozen case. Z2 add a flavor to sun 8. We can deform aWe tangential note that intersection the gauge intosymmetry two transverseso32 on the intersections. O8 -plane with This 16 D8-branes on top corresponds to givingbecomes a vev a to flavor the hypermultiplet symmetry in the in theory the antisymmetric on the top right or symmetricof Fig. 4.2, as expected. The

22 33 Perturbative Analysis

Begin near the O8+. We can solve the equations analytically in series expansion:

4W F0 2 2 3 e− = c1 + z 2c1 Λz + z , √c2 − O 4 2/5 F0 c1F0Λ 5 φ 3 4 e− = c1c2 + z z + z , 10√c2 − 6 10√c2 O c1c2Λ 2 √c2F0Λ 3 4 α = c2 + z + z + z , 2 6 O c1, c2 are constants parametrizing the solution, and F0/2π = 4 depending on the sign of z ± This is a backreacted orientifold source. As a consistency check if Λ 0 the series truncates and we recover the expected behavior with→ H e 4W ∼ − Constraining the Curvature

One of the equations of motion is ﬁrst order

4φ/5 2 25κe− 4α0φ0 4 (α0) 5 2 2φ 2W 0 = + F e − α2/5 α − α2 − 2 0 2 4W 2 20W 0φ0 + 40 W 0 + 20Λe− + 4 φ0 − This is because our system has gauge constraints

Evaluating at the O8+ using the local solution we ﬁnd

5 4W 2λ Λ = κe − O8 −4 | +

Hence for Λ > 0 we must have κ < 0 so the internal M5 has negative curvature Numerical Evolution

We now evaluate the perturbative solution at small z and start numerical evolution. In an open set of the space of initial conditions (c1, c2) we ﬁnd a singularity after ﬁnite z evolution:

z 0 5 10 15

Above: eφ is solid, eW is dashed, α is dotted. Tuning to a Solution

Is the singularity a physical object i.e. a brane? Or does it signal a breakdown in our approach? One can show that by a 1-parameter tuning of the initial conditions (c1, c2) we hit precisely the strong-coupling limit of an O8 − 2 1/2 2 2 1/2 2 φ 5/4 ds H− ( dx +... dx )+H dz , e H− , H z . 10 ∼ − 0 8 ∼ ∼ | | This leads to a region of large curvature and string coupling localized near the O8 . − Nevertheless the (supersymmetric version of) the strong-coupling 4 O8 occurs in many AdS solutions (e.g. AdS6 S ) and has been tested− using holography. × Inclusion of D8 Branes?

Can also try to include D8s extended along all directions but z. In general such branes feel two forces: gravitational attraction/repulsion mediated by DBI term •

9 φ d x e− g | | Z p Electrostatic forces from coupling to F mediated by WZ term • 0

9 d x C9 , dC9 = F0 . ∗ Z For a D8 in the background of other D8s in ﬂat space these forces exactly cancel and the probe D8 can be placed anywhere Inclusion of D8 Branes? In our more interesting background the forces do not always cancel. The force is given by:

W φ α0 F0 + e − φ0 W 0 + , − α A plot of the force as a function of z:

Only possibility is D8s on top of the O8s, but there is no way to arrange them consistent with the involution. So D8s cannot be included. O8+-O6 Solutions − If the strong-coupling O8 makes you nervous, don’t worry! We have new solutions without− this ingredient ( [1910.xxxxx]!)

ds2 = e2W ds2 + e 2W dz2 + e2λ2 ds2 + e2λ3 ds2 10 dS4 − S2 M3 2 We split the M5 into S and an Einstein three-manifold M3 with negative curvature. We assume that all functions depend only on z

In addition to F0 we now have a variety of other ﬂuxes

H3 hdz vol 2 , F2 f2vol 2 , F2 f41dz volM +f42vol ∼ ∧ S ∼ S ∼ ∧ 3 dS4

The z dependence of h(z), f2(z), f4i (z) is ﬁxed by imposing EOMs and Bianchi as well as ﬂux quantization.

As before we start an evolution at the O8+, and by suitable tuning of initial conditions can hit O6 (surrounded by the S2) − Moduli

These solutions all have moduli, i.e. the solutions all come in continuous families. The simplest solutions with only F have one modulus. • 0 (There are two initial conditions near the O8+, and reaching the O8 requires a one-parameter tuning. ) − Other solutions have more moduli • These are only moduli in the supergravity approximation. In string theory we expect quantum corrections to generate a potential on this modulus. This could either lead to a runaway (no vacuum) or stabilize the moduli somewhere perhaps at strong coupling. [Dine-Seiberg] Moduli

One reason why there are moduli is the following. The supergravity equations are invariant under the following rescalling:

2 2c 2 Nc ds e ds , φ φ c , FN e FN . 10d −→ 10d → − → For large c we therefore have weak curvature and string coupling (away from the O8 ). − In a solution with only F0, like our simplest de Sitter examples, the parameter c is the unique modulus In the examples with other branes and quantized ﬂuxes that scale with c there are still moduli, though they do not admit such a simple presentation. Scales

It is instructive to consider the parametric behavior of the scales in the problem as we dial to large c (weak coupling and small curvature). To estimate these one can go to a frame where the metric is not warped In this frame, the cosmological constant Λ is ﬁxed and the KK scale is determined by the spectrum of the Laplacian on the six-manifold

ds2 = e 4W dz2 + e2λds2 − M5 This size of this manifold is ﬁxed as c is taken large. Hence √Λ/MKK is ﬁxed. In other words there is no separation of scales. On the other hand, the 4d Planck mass grows with c so √Λ/MPlanck tends to zero at large c. “Classical” Solutions?

Regions of large string coupling and curvature mean that in a sense there is no truly classical solution with orientifolds. Somewhere in spacetime the supergravity equations of motion break down. dS solutions are not supersymmetric, so there are (currently!) no known tricks to rigorously convince us that the solutions exist. Our approach is naive. We treat the orientifolds as boundary conditions in SUGRA with a given charge and tension, and solve the SUGRA equations of motion where they are valid. It is an open question whether this is really suﬃcient to ensure that these boundary conditions are really all that is required to lift to string theory. (for some skepticism see [Cribiori-Junghans]) Thanks for Listening!

Some Open Problems

All our solutions have moduli at tree level. Perhaps there is a • theorem to this eﬀect? (If not, we should ﬁnd explicit examples with no moduli.) We have not analyzed the stability of our models. There could • be tachyons. It would be very useful to have a complete list of boundary • conditions in supergravity that ensure a solution can be lifted to string theory Some Open Problems