String theory compactifications with sources Alessandro Tomasiello - PowerPoint PPT Presentation

String theory compactifications with sources Alessandro Tomasiello PRIN Kicko ff Meeting, SNS Pisa, 19.10.2019

Introduction For de Sitter solutions in string theory, we need to break supersymmetry, and to consider… • higher - derivative operators e.g. ( Riemann ) k • orientifold - planes ( O - planes ) [ Gibbons ’84; de Wit, Smit, Hari Dass ’87, Maldacena, Nuñez ’00 ] [ Bianchi, Pradisi, Sagnotti ’91… ] • Most activity: 4d e ff ective actions [ Kachru, Kallosh, Linde, T rivedi ’03, Silverstein ’07… huge list ] furious debate! [ Bena, Graña, Halmagyi ’09, Banks ’12, Sethi ’17… ] • Finding solutions directly in 10d? still a challenge: • O - planes back - react on geometry and create singularities • when higher - derivatives get involved, they do so all at once

• it has been hard to find examples; often people have resorted to ‘smearing’ [ Acharya, Benini, V alandro ’05, Graña, Minasian, Petrini, AT ’06, Caviezel, Koerber, Körs, Lüst, Wrase, Zagermann ’08, Andriot, Goi, Minasian, Petrini ’10… ] localized smeared However, O - planes should sit at fixed loci of involutions they shouldn’t be smeared by definition. • several people tried to understand criteria for un - smearing [ Dong, Horn, Silverstein, Torroba ’10; Blåbäck, Danielsson, Junghans, V an Riet ’14… ] • But: solutions with unsmeared O - plane singularities have appeared in the last few years for supersymmetric AdS Maybe time to try again for dS?

Plan • some explicit solutions • Review: Localized sources in AdS • how to find them • why one should believe them • Ideas for supersymmetry breaking • some simple de Sitter models

AdS with sources • Sometimes solutions with sources come from near - horizon limits D4 dissolved, but O8 O8 remains D3 dissolve; no source after near - horizon N D4 N D3 AdS 5 × S 5 AdS 6 × ( top .S 4 ) [ Y oum ’99, Brandhuber, Oz ’99 ] • Unclear if all AdS are near - horizon limits • Intersecting brane solutions are rare anyway • Better strategy: work out boundary conditions corresponding to various sources

• Sources create singularities where supergravity breaks down p + 1 , . . . , 9 0 , . . . , p e φ = g s H (3 − p )/4 backreaction ds 2 10 = H � 1/2 ds 2 k + H 1/2 ds 2 ? on flat space: ⊥ = dr 2 + r 2 ds 2 ds 2 S 8 − p harmonic function in R 9 − p ⊥ • supergravity artifacts: they should be resolved in appropriate duality frame D - branes O - planes [O p − : tension=charge= − 2 p − 5 ] � r 0 � 7 − p p < 7 : H = 1 − H r H � r 0 � 7 − p p < 7 : H = 1 + r r 0 r unphysical r r 0 ‘hole’! a = 0 : p = 8 : H = a + | z / z 0 | H H H e φ → ∞ p = 8 : H = a − | z / z 0 | { z a z z

[ Apruzzi, Fazzi, Rosa, AT ’13 • Example : AdS7 in IIA. All solutions: Apruzzi, Fazzi, Passias, Rota, AT ‘15; Cremonesi, AT ’15; Bah, Passias, AT ‘17 ] r α 2 r 1 − ¨ ✓ ◆ − α α α ) 3 / 4 2 ds 2 = 8 dz 2 + e φ = 2 5 / 4 π 5 / 2 3 4 ( − α / ¨ α ds 2 α ds 2 AdS 7 + √ √ α 2 − 2 α ¨ S 2 ¨ α ˙ π α 2 − 2 α ¨ ˙ α interval ✓ α ˙ ◆ α B = π − z + vol S 2 α 2 − 2 α ¨ ˙ α ... α piecewise cubic α = F 0 ✓ ¨ π F 0 α ˙ ◆ α α , F 2 = 162 π 2 + vol S 2 α 2 − 2 α ¨ ˙ α • At endpoint, smoothness: S 2 should shrink, α α → 0 , ¨ α → 0 α finite ¨ D8s D8 • When F 0 jumps smooth endpoint z what happens with other boundary conditions?

compare locally with r α 2 r 1 − ¨ ✓ ◆ ds 2 10 = H � 1/2 ds 2 k + H 1/2 ds 2 − α α 2 ds 2 = 8 dz 2 + α ds 2 α ds 2 AdS 7 + √ ? α 2 − 2 α ¨ S 2 ¨ α ˙ π [ Blåbäck, Danielsson, Junghans, V an Riet, Wrase, Zagermann ’11; Apruzzi, Fazzi, Rosa, AT 13… ] H • α → 0 D6 transverse R 3 AdS 7 + z − 1/2 ( dz 2 + z 2 ds 2 ds 2 ∼ z 1/2 ds 2 z S 2 ) α ( z 0 ) ˙ α ( z 0 ) ¨ α ( z 0 ) • Other interesting 6 = 0 6 = 0 D6 0 boundary conditions: O6 6 = 0 6 = 0 0 6 = 0 regular point 0 0 O8 6 = 0 0 0

• Why should we believe this? Holographic checks: [ Cremonesi, AT ’15 ] [ Apruzzi, Fazzi ‘17 ] Examples dual quiver theory [ SU gauge groups ] D8s D8s integral over susy, grav. & [ Ohmori, Shimizu, Tachikawa, Y onekura ’14; internal dimensions R - symmetry anomalies Cordova, Dumitrescu, [ Henningson, Skenderis ’98 ] Intriligator ’15 ] 7 k 2 ( N 3 − 4 Nk 2 + 16 a = 16 5 k 3 ) O8+D8 D6 Nn 0 E 9 − n 0 ( N − 1) n 0 2 n 0 n 0 . . . a = 16 10 N 5 n 2 3 [ Bah, Passias, AT ’16 ] 0 7 [ Apruzzi, Fazzi ‘17 ]

Other examples no O - planes so far also no O - planes. Possible extension with 7 - branes? • AdS4 in IIA ( top .S 3 ) → H 3 , S 3 [ Rota, AT’15; Passias, Prins, AT ’18; Bah, Passias, W eck ’18 ] sources: ( top . S 2 ) → KE 4 , Σ g × Σ g 0 D8, D6, O8, O6 O8 [ Couzens, Lawrie, Martelli, Schäfer - Nameki ’17; • AdS 3 in F-theory Haghighat, Murthy, V andoren, V afa ’15 ]

Supersymmetry breaking • Possible way of breaking susy: consistent truncations once rare; now common, although perhaps general theory still lacking • For ex: every AdS7 solution has a non - susy ‘evil twin’ [ Passias, Rota, AT ’15 ] established via consistent truncation: some small changes 12 r r α 2 1 − ¨ ✓ ◆ − α α 2 ds 2 = 8 dz 2 + α ) 3 / 4 e φ = 2 5 / 4 π 5 / 2 3 4 ( − α / ¨ α ds 2 α ds 2 AdS 7 + √ α 2 − 2 α ¨ S 2 √ ¨ α ˙ π α 2 − 2 α ¨ ˙ α • Most are unstable [ Danielsson, Dibitetto, V argas ’17; Apruzzi, De Luca, Gnecchi, Lo Monaco, AT, in progress ] pert. instability for all solutions with part of the KK spectrum via 7d trick D8s on top of each other non - pert. instability for all solutions NS5 ‘bubbles’ with a massless region

• More general strategy? [ Legramandi, AT; in progress ] let’s start from an easy class: eg. Mink 6 × M 4 [ Imamura ’01; Janssen, Meessen, Ortin ‘99 ] ds 2 = S − 1/2 ds 2 Mink 6 + K ( S − 1/2 dz 2 + S 1/2 ds 2 R 3 ) d H ( e 3 A − φ Φ + ) = 0 [ motivated by NS5 - D6 - D8 ] d H ( e 2 A − φ Re Φ − ) = 0 z S 2 = 0 K = − 4 ⊃ ∆ 3 S + 1 2 ∂ 2 F 0 ∂ z S d H ( e 4 A − φ Im Φ − ) = e 4 A ? � ( F ) [ Lüst, Patalong, Tsimpis ’10; susy breaking Graña, Minasian, Petrini, AT ‘05 ] keep same fluxes; impose Bianchi, but not BPS d H ( e 3 A − φ Φ + ) = 0 d H ( e 2 A − φ Re Φ − ) = c e 8 A − 2 φ vol M 4 K = − 4 F 0 ∂ z S z S 2 + c z ∂ 2 ∆ 3 S + 1 2 ∂ 2 z S = 0 d H ( e 4 A − φ Im Φ − ) = e 4 A ? � ( F ) ⊃ S = e − 4 A + cz we checked that this small modification similar in spirit to adding works in several other classes primitive part to G 3 in conf. CY [ Becker, Becker ’96, Dasgupta, Rajesh, Sethi ’98, Graña, Polchinski ’00, Giddings, Kachru, Polchinski ‘01 ]

dS with O8 - planes same effect as O 8 + O8 − + 16 D8 • Simplest model [ Córdova, De Luca, AT ’18 ] Z 2 ds 2 = e 2 W ( z ) ds 2 dS 4 + e − 2 W ( z ) ( dz 2 + e 2 λ ( z ) ds 2 z M 5 ) compact hyperbolic O 8 − Boundary condition at O8+ Minkowski: [ Bianchi, Pradisi, Sagnotti ’91, Dabholkar, Park ’96, Witten ’97, e W � φ f 0 f i = { W, 1 5 φ , 1 i | z ! 0 + = 1 2 λ } Aharony, Komargodski, Patir ‘07 ] see also [ Silverstein, Strings 2013 talk ] Numerical evolution: e f i ∼ | z − z 0 | − 1/4 same as O8_ in flat space we manage to reach [ even the coe ffi cients work ] 30 e λ inevitably, O8_ has strongly coupled region 20 10 e W e φ z 0 5 10 15 O 8 − O 8 +

• Rescaling symmetry: g MN → e 2 c g MN , φ → φ − c 30 150 20 100 10 50 z z 0 5 10 15 10 20 30 40 50 it makes strong - coupling region small, but it doesn’t make it disappear. . . . � e − 2 φ R 4 � e − 2 φ R • In the O8_ region stringy corrections become dominant � R 4 supergravity action is least important term; ideally in this region we’d switch to another duality frame. Full string theory should then fix c • Hope that this solution is sensible comes from similarity with flat - space O8_ ( which we know to exist in string theory )

dS with O8s and O6s • W e also tried: O8 + –O6 − [ Córdova, De Luca, AT, to appear ] H = h 1 dz ∧ vol 2 + h 2 vol 3 ds 2 = e 2 W ds 2 dS 4 + e − 2 W ( dz 2 + e 2 λ 3 ds 2 M 3 + e 2 λ 2 ds 2 S 2 ) F 2 = f 2 vol 2 F 4 = f 41 vol 3 ∧ dz + f 42 vol 4 surrounds the O6 F 0 = / 0 • we already know one such solution for Λ < 0 : α = 3 k ( N 2 − z 2 ) + n 0 ( z 3 − N 3 ) from a non - susy AdS7 solution with O8+ and O6_ � � � √ π ds 2 = 12 � − α dz 2 + α 2 1 − ¨ α ds 2 α ds 2 α AdS 7 + O8+ α 2 − α ¨ S 2 ¨ ˙ α AdS 4 × H 3 O6_ compact hyperbolic

ds 2 = e 2 W ds 2 dS 4 + e − 2 W ( dz 2 + e 2 λ 3 ds 2 M 3 + e 2 λ 2 ds 2 • we slowly modified it numerically, bringing Λ up S 2 ) [ functions rescaled for clarity ] [ analytic AdS4 ] [ numeric dS4 ] 3 4 W e still obtain e λ 2 2 3 the O6 boundary. 2 1 e 4 W = e 2 λ 3 1 z e φ 10 20 30 40 z 25 50 75 100 125

Conclusions • A lot of progress in AdS solutions • often localized O - plane sources are possible • holography works even in their presence • sometimes non - supersymmetric • Time to look for de Sitter • Using numerics, we find dS solutions with O8 - planes in relatively simple setup • Also O8 - O6 solutions • There are regions where supergravity breaks down. Inevitable! If you want solutions with O - planes. W e better learn how to deal with them.