W arped e ff ective theories and holography Luca Martucci - PowerPoint PPT Presentation

a DIPARTIMENTO D DI FISICA F E ASTRONOMIA A Galileo Galilei W arped e ff ective theories and holography Luca Martucci University of Padova based on: 1610.02403 1411.2623 1603.04470 with Alberto Zaffaroni

Plan Part I: Effective theory of warped flux compactifications 1610.02403 1411.2623 Part II: Holographic effective field theories 1603.04470 with Alberto Zaffaroni

Part I: Effective theory of warped flux compactifications

The question F/M-theory compactifications are generically warped [Becker-Becker,Grana-Polchinski, Gubser, Giddings-Kachru-Polchinski] d s 2 10 = e 2 A ( y ) d s 2 4 + e − 2 A ( y ) d s 2 7-branes X X

The question F/M-theory compactifications are generically warped [Becker-Becker,Grana-Polchinski, Gubser, Giddings-Kachru-Polchinski] d s 2 10 = e 2 A ( y ) d s 2 4 + e − 2 A ( y ) d s 2 7-branes X warping generated by: mobile D3-branes X

The question F/M-theory compactifications are generically warped [Becker-Becker,Grana-Polchinski, Gubser, Giddings-Kachru-Polchinski] d s 2 10 = e 2 A ( y ) d s 2 4 + e − 2 A ( y ) d s 2 7-branes X warping generated by: mobile D3-branes ISD fluxes X moduli stabilisation, SUSY breaking, strongly warped throats, … e A min << e A bulk [Klebanov-Strassler]

The question (Perturbative) moduli include D3-brane moduli Kähler moduli axions (Complex structure, axion-dilaton and 7- brane moduli assumed stabilised by fluxes)

The question (Perturbative) moduli include D3-brane moduli Kähler moduli axions Fully coupled effective theory? D3-branes beyond probe approximation? impact of fluxes?

Universal modulus and Kähler potential The universal Kähler modulus: e − 4 A ( y ) = a + e − 4 A 0 ( y ) ∆ 6 e − 4 A = ∗ 6 Q D3 [Giddings-Maharana] Z D3-branes, ISD fluxes, … e − 4 A 0 dvol X = 0 UNIVERSAL MODULUS X

Universal modulus and Kähler potential The universal Kähler modulus: e − 4 A ( y ) = a + e − 4 A 0 ( y ) ∆ 6 e − 4 A = ∗ 6 Q D3 [Giddings-Maharana] Z D3-branes, ISD fluxes, … e − 4 A 0 dvol X = 0 UNIVERSAL MODULUS X Natural superconformal structure fixes [LM `14] K = − 3 log a simple but implicit!

Chiral coordinates The moduli include z i D3 positions: good chiral coordinates I universal Kähler modulus a 1 Z non-universal Kähler moduli with J = v a ω a J ∧ J ∧ J = 1 3! basis of harmonic 2-forms C 4 -moduli ( ) -moduli ignored in this talk B 2 , C 2

Chiral coordinates The moduli include z i D3 positions: good chiral coordinates I universal Kähler modulus a 1 Z non-universal Kähler moduli with J = v a ω a J ∧ J ∧ J = 1 3! basis of harmonic 2-forms C 4 -moduli Explicit form? a = 1 , . . . , h 1 , 1 chiral coordinates: ρ a

Chiral coordinates Probe SUSY D3 instantons F-terms e − S E3 ∼ D z i , must be holomorphic in ρ a I Euclidean D3-brane instanton

Chiral coordinates Probe SUSY D3 instantons F-terms e − S E3 ∼ D z i , must be holomorphic in ρ a I Euclidean D3-brane instanton S E3 = 1 Z e − 4 A J ∧ J − 1 Z ( ) Im τ F ∧ F + . . . F = F E3 − B 2 2 2 D D

Chiral coordinates Probe SUSY D3 instantons F-terms e − S E3 ∼ D z i , must be holomorphic in ρ a I Euclidean D3-brane instanton S E3 = 1 Z e − 4 A J ∧ J − 1 Z ( ) Im τ F ∧ F + . . . F = F E3 − B 2 2 2 D D Kähler moduli, mobile D3-branes, fluxes cf. [Giddings-Maharana]

Chiral coordinates Probe SUSY D3 instantons F-terms e − S E3 ∼ D z i , must be holomorphic in ρ a I Euclidean D3-brane instanton S E3 = 1 Z e − 4 A J ∧ J − 1 Z ( ) Im τ F ∧ F + . . . F = F E3 − B 2 2 2 D D hidden dependence on non-universal Kähler moduli Kähler moduli, mobile [LM `16] D3-branes, fluxes δ G 3 = δ v a ∂ ¯ ∂ Λ 1 , 0 J ∧ G 3 = 0 cf. [Giddings-Maharana] a ( ) ∆ 6 Λ 1 , 0 with = − 2 ∗ 6 ( ω a ∧ G 3 ) τ = const. a

Chiral coordinates Perturbative -axionic symmetry: Im ρ a → Im ρ a + const. C 4 We can focus on the real part: Re ρ a = 1 2 a I abc v b v c + 1 X κ a ( z I , ¯ z I ; v ) + h a ( v ) 2 I Z 1 − 1 h b 0 , 1 i b 1 , 0 ∧ ¯ ∂ b 1 , 0 ∧ ∂ ¯ ¯ � � Re G 3 − 2Im τ 2 D a

Chiral coordinates Perturbative -axionic symmetry: Im ρ a → Im ρ a + const. C 4 We can focus on the real part: Re ρ a = 1 2 a I abc v b v c + 1 X κ a ( z I , ¯ z I ; v ) + h a ( v ) 2 1 I Z 1 − 1 h b 0 , 1 i b 1 , 0 ∧ ¯ ∂ b 1 , 0 ∧ ∂ ¯ ¯ � � Re G 3 − 2Im τ 2 D a 1. unwarped contribution [Grimm & Louis `04]

Chiral coordinates Perturbative -axionic symmetry: Im ρ a → Im ρ a + const. C 4 We can focus on the real part: 2 Re ρ a = 1 2 a I abc v b v c + 1 X κ a ( z I , ¯ z I ; v ) + h a ( v ) 2 1 I Z 1 − 1 h b 0 , 1 i b 1 , 0 ∧ ¯ ∂ b 1 , 0 ∧ ∂ ¯ ¯ � � Re G 3 − 2Im τ 2 D a 1. unwarped contribution [Grimm & Louis `04] 2. i ∂ ¯ ∂κ a = ω a = [ D a ] harm

Chiral coordinates Perturbative -axionic symmetry: Im ρ a → Im ρ a + const. C 4 We can focus on the real part: 2 3 Re ρ a = 1 2 a I abc v b v c + 1 X κ a ( z I , ¯ z I ; v ) + h a ( v ) 2 1 I Z 1 − 1 h b 0 , 1 i b 1 , 0 ∧ ¯ ∂ b 1 , 0 ∧ ∂ ¯ ¯ � � Re G 3 − 2Im τ 2 D a 4 1. unwarped contribution [Grimm & Louis `04] i ∂ ¯ 2. ∂κ a = ω a = [ D a ] harm e − 2 πκ a | ζ a | 2 � ✓ i G 3 ∧ ¯ ◆ Z G 3 − Q nd 3. � h a ( v ) ≡ log D3 2Im τ X G 3 = G (0) + ∂ ¯ ∂ b 1 , 0 ( v ) 4. 3

Effective theory implicit function of K = − 3 log a z i and Re ρ a I N D3 L bos = 1 1 z ¯ X 2 R 4 ⇤ 1 � G ab r ρ a ^ ⇤r ¯ z I ; v )d z i | | ( z I , ¯ I ^ ⇤ d¯ ρ b � g i ¯ I 2v 0 a I =1

Effective theory implicit function of K = − 3 log a z i and Re ρ a I N D3 L bos = 1 1 z ¯ X 2 R 4 ⇤ 1 � G ab r ρ a ^ ⇤r ¯ z I ; v )d z i | | ( z I , ¯ I ^ ⇤ d¯ ρ b � g i ¯ I 2v 0 a I =1 D3-branes kinetic terms matching probe approximation

Effective theory implicit function of K = − 3 log a z i and Re ρ a I N D3 L bos = 1 1 z ¯ X 2 R 4 ⇤ 1 � G ab r ρ a ^ ⇤r ¯ z I ; v )d z i | | ( z I , ¯ I ^ ⇤ d¯ ρ b � g i ¯ I 2v 0 a I =1 inverse of Z D3-branes kinetic terms e − 4 A ω a ∧ ∗ 6 ω b G ab = 4 a matching probe approximation X Z + 2 a ⇣ ⌘ Λ 1 , 0 ∧ ¯ ω a ∧ Re G 3 b Im τ X

Effective theory implicit function of K = − 3 log a z i and Re ρ a I N D3 L bos = 1 1 z ¯ X 2 R 4 ⇤ 1 � G ab r ρ a ^ ⇤r ¯ z I ; v )d z i | | ( z I , ¯ I ^ ⇤ d¯ ρ b � g i ¯ I 2v 0 a I =1 inverse of Z D3-branes kinetic terms e − 4 A ω a ∧ ∗ 6 ω b G ab = 4 a matching probe approximation X Z + 2 a ⇣ ⌘ Λ 1 , 0 ∧ ¯ ω a ∧ Re G 3 b Im τ X modifications due to warping and fluxes ( ) ∆ 6 Λ 1 , 0 = − 2 ∗ 6 ( ω a ∧ G 3 ) a [Coenden, Frey, David Marsh, Underwood `16] cf. [Frey, Roberts `14]

Effective theory implicit function of K = − 3 log a z i and Re ρ a I N D3 L bos = 1 1 z ¯ X 2 R 4 ⇤ 1 � G ab r ρ a ^ ⇤r ¯ z I ; v )d z i | | ( z I , ¯ I ^ ⇤ d¯ ρ b � g i ¯ I 2v 0 a I =1 inverse of Z D3-branes kinetic terms e − 4 A ω a ∧ ∗ 6 ω b G ab = 4 a matching probe approximation X Z + 2 a ⇣ ⌘ Λ 1 , 0 ∧ ¯ ω a ∧ Re G 3 b Im τ X Furthermore: modifications due to warping and fluxes no-scale K A ¯ B K A K ¯ B = 3 ( ) ∆ 6 Λ 1 , 0 = − 2 ∗ 6 ( ω a ∧ G 3 ) a [Coenden, Frey, David Marsh, Underwood `16] cf. [Frey, Roberts `14]

Comments Dynamical higher derivative contributions ignored cf. [Grimm, Pugh, Weissenbacher `14,`15] (Co)homological structure of unwarped theory seems lost Explicit form requires explicit knowledge of CY metric Investigate implications in non-compact models, as in [Aldazabal, Ibanez, Quevedo, Uranga`00] local pheno models … [Donagi, Wijnholt - Beasley, Heckman, Vafa `08] holography strong warping! [LM, Zaffaroni `16]

Part II: Holographic Effective Field Theories (HEFT)

IIB local holographic models Focus on 4d SCFT’s which are strongly coupled N = 1 IR fixed points of Sasaki-Einstein N D3-branes they admit a holographic dual!

IIB local holographic models Focus on 4d SCFT’s which are strongly coupled N = 1 IR fixed points of Sasaki-Einstein N D3-branes they admit a holographic dual! These SCFTs have non-trivial moduli space of susy vacua M h O i 6 = 0 spontaneous breaking Low-energy effective theory of conformal symmetry inaccessible in QFT with mass gap

IIB local holographic models Focus on 4d SCFT’s which are strongly coupled N = 1 IR fixed points of Sasaki-Einstein N D3-branes they admit a holographic dual! Use holography to determine EFT: M Φ i holographic effective field theory (HEFT) K ( Φ , ¯ Φ )

Prototype: the Klebanov-Witten model

Klebanov-Witten model [Klebanov & Witten , `98-`99] N D3-branes T 1 , 1 conifold

Klebanov-Witten model [Klebanov & Witten , `98-`99] N D3-branes T 1 , 1 conifold A 1 , A 2 UV quiver gauge theory: gauge group: SU ( N ) × SU ( N ) SU ( N ) SU ( N ) A 1 , A 2 ∈ ( N , ¯ B 1 , B 2 ∈ ( ¯ chiral matter: , N ) N , N ) B 1 , B 2 superpotential: W = h Tr( A 1 B 1 A 2 B 2 − A 1 B 2 A 2 B 1 )