TCS G 2 Manifolds and 4D Emergent Strings Fengjun Xu Universit at - PowerPoint PPT Presentation

TCS G 2 Manifolds and 4D Emergent Strings Fengjun Xu Universit¨ at Heidelberg arXiv: 2006.02350 Strings and Fields 2020, YITP, Kyoto Fengjun Xu TCS G 2 Manifolds and 4D Emergent Strings 18/11/2020 1 / 16

Swampland Program What kinds of (consistent) EFTs can be consistently coupled to a fundamental theory of Quantum Gravity? Fengjun Xu TCS G 2 Manifolds and 4D Emergent Strings 18/11/2020 2 / 16

Swampland Program What kinds of (consistent) EFTs can be consistently coupled to a fundamental theory of Quantum Gravity? More precisely, provided an effective QFT, what kinds of consistency conditions it should have so that � � − G [ M d − 2 S eff = R + L eqft ] (1) p be an effective description for quantum gravity? Fengjun Xu TCS G 2 Manifolds and 4D Emergent Strings 18/11/2020 2 / 16

Swampland Program What kinds of (consistent) EFTs can be consistently coupled to a fundamental theory of Quantum Gravity? More precisely, provided an effective QFT, what kinds of consistency conditions it should have so that � � − G [ M d − 2 S eff = R + L eqft ] (1) p be an effective description for quantum gravity? Swampland: all the consistent effective QFTs that CANNOT be completed to Quantum Gravity in the ultraviolet [Vafa ’05] Borrowed from [Palti ’19] Fengjun Xu TCS G 2 Manifolds and 4D Emergent Strings 18/11/2020 2 / 16

Swampland Conjectures See more details in [Palti ’19] [Brennan, Carta, Vafa ’17] No global symmetries. [Banks, Dixon ’88], [Banks,Seiberg ’13], [Harlow,Ooguri ’18] Q: What happens if g YM → 0 with M Pl finite? The Distance Conjecture: Infinite tower(s) of states becomes massless at infinite distance limits in field space of quantum gravity. [Ooguri, Vafa ’06] Typically it indicates the original effective description breaks down Q: What’s the nature of the new description at such limits? → Duality! Today’s goal: Discussing these conjectures in 4d N = 1 effective theories from M-theory on TCS G 2 manifolds Fengjun Xu TCS G 2 Manifolds and 4D Emergent Strings 18/11/2020 3 / 16

Outline Compactifications of M-theory on TCS G 2 manifolds TCS G 2 manifolds M-theory on TCS G 2 manifolds and dualities Test Infinite distance limit Emergent tensionless heterotic string Quantum corrections? Fengjun Xu TCS G 2 Manifolds and 4D Emergent Strings 18/11/2020 4 / 16

A brief intro on G 2 manifolds Definition: A G 2 manifold X is a 7D Riemannian manifold which allows a metric g µν with Hol ( g ) ⊆ G 2 Fengjun Xu TCS G 2 Manifolds and 4D Emergent Strings 18/11/2020 5 / 16

A brief intro on G 2 manifolds Definition: A G 2 manifold X is a 7D Riemannian manifold which allows a metric g µν with Hol ( g ) ⊆ G 2 X is Ricci-flat R µν = 0 → a vacuum for string/M-compactifications Fengjun Xu TCS G 2 Manifolds and 4D Emergent Strings 18/11/2020 5 / 16

A brief intro on G 2 manifolds Definition: A G 2 manifold X is a 7D Riemannian manifold which allows a metric g µν with Hol ( g ) ⊆ G 2 X is Ricci-flat R µν = 0 → a vacuum for string/M-compactifications ∃ a single covariantly constant spinor ∆ g η = 0 → 4d N = 1 theories from M-theory Fengjun Xu TCS G 2 Manifolds and 4D Emergent Strings 18/11/2020 5 / 16

A brief intro on G 2 manifolds Definition: A G 2 manifold X is a 7D Riemannian manifold which allows a metric g µν with Hol ( g ) ⊆ G 2 X is Ricci-flat R µν = 0 → a vacuum for string/M-compactifications ∃ a single covariantly constant spinor ∆ g η = 0 → 4d N = 1 theories from M-theory ∃ Two calibrated forms ( analogue of K¨ ahler two-form and canonical three-form in CY 3 ) d Φ = 0 , d ∗ Φ = 0 (2) Φ calibrates 3-cycles (associative) ∗ Φ calibrates 4-cycles (coassociative). Fengjun Xu TCS G 2 Manifolds and 4D Emergent Strings 18/11/2020 5 / 16

A brief intro on G 2 manifolds Definition: A G 2 manifold X is a 7D Riemannian manifold which allows a metric g µν with Hol ( g ) ⊆ G 2 X is Ricci-flat R µν = 0 → a vacuum for string/M-compactifications ∃ a single covariantly constant spinor ∆ g η = 0 → 4d N = 1 theories from M-theory ∃ Two calibrated forms ( analogue of K¨ ahler two-form and canonical three-form in CY 3 ) d Φ = 0 , d ∗ Φ = 0 (2) Φ calibrates 3-cycles (associative) ∗ Φ calibrates 4-cycles (coassociative). Making compact G 2 manifolds is much harder than constructing CYs, ← lacking of powerful machinery of complex algebraic geometry. Earlier compact examples: Orbifolds and their resolutions T 7 / Γ [Joyce ’96]. Fengjun Xu TCS G 2 Manifolds and 4D Emergent Strings 18/11/2020 5 / 16

TCS G 2 manifold New method: twisted connected sums (TCS) [Kovalev ’03, Corti, Haskins, Nordstr¨ om, Pacini ’13] Key observation: ∀ CY X 3 → X 3 × S 1 with torsion-free G 2 -structure but Hol ( g µν ) = SU (3). Having this, (approximate) construction of X ( Hol ( g ) = G 2 ) by gluing two building blocks X ± × S 1 ± : X = [ X + × S 1 + ] ⊔ [ X − × S 1 − ] (3) where X ± is an asymptotically cylindrical Calabi-Yau X ± : S ± → X ± → P 1 i.e. a compact K 3 fibered over an open P 1 → Non-compact. At asymptotically regime: X ± : S ± × S 1 × ∆ cyl , one glue these two building blocks Figure taken from [Braun ’16 ] Physics literatures: [Halverson, Morrison ’15] [Guio, Jockers, Klemm, Yeh ’17], [ Bruan, Sch¨ afer-Nameki ’17], [ Bruan, Del Zotto ’17], [ Guio, Jockers, Klemm, Yeh ’2017] [Braun, Del Zotto, Halverson, Larfors, Morrison, Sch¨ afer-Nameki ’18 ] and many more.... Fengjun Xu TCS G 2 Manifolds and 4D Emergent Strings 18/11/2020 6 / 16

TCS G 2 compactifications Globally, X can be viewed as an K3 fibered over S 3 Figure taken from [Braun ’16 ] Idea: Fiber-wise version of 7d duality between M-theory on K 3 and Heterotic on T 3 . T 3 − K 3 − → X → X 3     (4) � � S 3 S 3 . X 3 is always be the Schone CY 3-fold X 19 , 19 with h 1 , 1 = h 2 , 1 = 19 but can be with different bundles [Braun, Sch¨ afer-Nameki ’17 ]. Fengjun Xu TCS G 2 Manifolds and 4D Emergent Strings 18/11/2020 7 / 16

M-theory on TCS G 2 compactifications and dualities Furthermore, a chain of 4D string duality gymnastics [ Bruan, Sch¨ afer-Nameki ’17] M-theory on X ⇐ ⇒ Heterotic string on X 19 , 19 ⇐ ⇒ F-theory on X 4 . (5) X 4 : T 2 → X 4 → π B 3 = dP 9 × P 1 . In particular, γ 3 het , 4 d = Vol( P 1 ) Vol( S 3 ) = g 2 Vol( dP 9 ) , (6) γ : the radius of the K 3-fiber Fengjun Xu TCS G 2 Manifolds and 4D Emergent Strings 18/11/2020 8 / 16

Infinite Distance Limit We are interested in the following infinite distance limit on X : Vol( K 3) → µ − 1 Vol( K 3) , Vol( S 3 ) → µ Vol( S 3 ) , with µ → ∞ (7) Vol( X ) ∝ Vol( K 3) × Vol( S 3 ) ∼ finite . Q: Is it at the infinite distance of the 4d N = 1 moduli space? The corresponding one in F-theory on X 4 Vol( P 1 ) → ν − 1 Vol( P 1 ) , Vol( dP 9 ) → ν Vol( dP 9 ) , with ν → ∞ (8) Vol( B 3 ) ∝ Vol( P 1 ) × Vol( dP 9 ) ∼ finite , which can be proved at infinite distance [Lee, Lerche, Weigand, ’19]. It is weakly coupled regime in the dual heterotic framework → general theme in string theory The next steps: Testing the conjectures Q1: What happens when g YM → 0 with M Pl finite? Q2: What’s the nature of the new description at such limits? Fengjun Xu TCS G 2 Manifolds and 4D Emergent Strings 18/11/2020 9 / 16

Infinite distance limits as weak gauge coupling limit Q1: What happens when g YM → 0 with M Pl finite? General expectation: The limit with g YM → 0 should be at the infinite distance! � 1 i ω α ∈ H 2 ( X , Z ) := Re f αβ =Re( ω α ∧ ∗ X ω β ) , 4 g 2 2 κ 2 11 YM X (9) � � ∝ w α ∧ w β ∧ Φ = Φ = Vol (Σ αβ ) , Σ αβ ∈ H 3 ( X , Z ) X Σ αβ where we have used ∗ X w α = − w α ∧ Φ. Hence ⇒ g 2 Vol(Σ αβ ) → ∞ = YM → 0 , (10) In our case Σ αβ = S 3 . Consistent with Weak Gravity Conjecture: Magnetic version: There exists an infinite tower of charged states with mass scale m Λ at m Λ ∝ g YM M pl , → infinite tower of charged massless states @ g YM = 0 (11) [Arkani-Hamed, Motl, Nicolis, Vafa ’06 ] Lesson: Such a limit should NOT be attained within the original effective theory (quantum gravity censorship) Fengjun Xu TCS G 2 Manifolds and 4D Emergent Strings 18/11/2020 10 / 16

Emergence of Tensionless Heterotic String Q2: What’s the nature of the physics at such limits? Tensionless solitonic string: a wrapped M5-brane on the shrunk K3. Key Fact: Tensionless solitonic string= weakly coupled, critical heterotic string [Harvey,Strominger ’95][Maldacena,Strominger,Witten ’97] In F-theory, the same from a D3-brane wrapping on the shrunk C 0 := P 1 in X 4 . Topological duality twisted reduction of 4d N = 4 SYM along P 1 , given that C 0 · K B 3 = 2 and trivial normal bundle [Lawrie,Sch¨ afer-Nameki,Weigand ’16] → 2d N = (0 , 2) effective theory with Eight left-moving scalars+ Eight right-moving scalars + fermionic partners Sixteen left-moving fermions, Exactly reproduce the massless spectra of a critical heterotic string! The light excitations of this tensionless Heterotic string satisfies the SDC! Fengjun Xu TCS G 2 Manifolds and 4D Emergent Strings 18/11/2020 11 / 16