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
Fengjun Xu
Universit¨ at Heidelberg arXiv: 2006.02350
Strings and Fields 2020, YITP, Kyoto
Fengjun Xu TCS G2 Manifolds and 4D Emergent Strings 18/11/2020 1 / 16
What kinds of (consistent) EFTs can be consistently coupled to a fundamental theory of Quantum Gravity?
Fengjun Xu TCS G2 Manifolds and 4D Emergent Strings 18/11/2020 2 / 16
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 Seff = −G[Md−2
p
R + Leqft] (1) be an effective description for quantum gravity?
Fengjun Xu TCS G2 Manifolds and 4D Emergent Strings 18/11/2020 2 / 16
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 Seff = −G[Md−2
p
R + Leqft] (1) 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 G2 Manifolds and 4D Emergent Strings 18/11/2020 2 / 16
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 gYM → 0 with MPl 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 G2 manifolds
Fengjun Xu TCS G2 Manifolds and 4D Emergent Strings 18/11/2020 3 / 16
Compactifications of M-theory on TCS G2 manifolds TCS G2 manifolds M-theory on TCS G2 manifolds and dualities Test Infinite distance limit Emergent tensionless heterotic string Quantum corrections?
Fengjun Xu TCS G2 Manifolds and 4D Emergent Strings 18/11/2020 4 / 16
Definition: A G2 manifold X is a 7D Riemannian manifold which allows a metric gµν with Hol(g) ⊆ G2
Fengjun Xu TCS G2 Manifolds and 4D Emergent Strings 18/11/2020 5 / 16
Definition: A G2 manifold X is a 7D Riemannian manifold which allows a metric gµν with Hol(g) ⊆ G2 X is Ricci-flat Rµν = 0 → a vacuum for string/M-compactifications
Fengjun Xu TCS G2 Manifolds and 4D Emergent Strings 18/11/2020 5 / 16
Definition: A G2 manifold X is a 7D Riemannian manifold which allows a metric gµν with Hol(g) ⊆ G2 X is Ricci-flat Rµν = 0 → a vacuum for string/M-compactifications ∃ a single covariantly constant spinor ∆gη = 0 → 4d N = 1 theories from M-theory
Fengjun Xu TCS G2 Manifolds and 4D Emergent Strings 18/11/2020 5 / 16
Definition: A G2 manifold X is a 7D Riemannian manifold which allows a metric gµν with Hol(g) ⊆ G2 X is Ricci-flat Rµν = 0 → a vacuum for string/M-compactifications ∃ a single covariantly constant spinor ∆gη = 0 → 4d N = 1 theories from M-theory ∃ Two calibrated forms ( analogue of K¨ ahler two-form and canonical three-form in CY3) dΦ = 0, d ∗ Φ = 0 (2) Φ calibrates 3-cycles (associative) ∗Φ calibrates 4-cycles (coassociative).
Fengjun Xu TCS G2 Manifolds and 4D Emergent Strings 18/11/2020 5 / 16
Definition: A G2 manifold X is a 7D Riemannian manifold which allows a metric gµν with Hol(g) ⊆ G2 X is Ricci-flat Rµν = 0 → a vacuum for string/M-compactifications ∃ a single covariantly constant spinor ∆gη = 0 → 4d N = 1 theories from M-theory ∃ Two calibrated forms ( analogue of K¨ ahler two-form and canonical three-form in CY3) dΦ = 0, d ∗ Φ = 0 (2) Φ calibrates 3-cycles (associative) ∗Φ calibrates 4-cycles (coassociative). Making compact G2 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 G2 Manifolds and 4D Emergent Strings 18/11/2020 5 / 16
New method: twisted connected sums (TCS) [Kovalev ’03, Corti, Haskins, Nordstr¨
Key observation: ∀ CY X3 → X3 × S1 with torsion-free G2-structure but Hol(gµν) = SU(3). Having this, (approximate) construction of X(Hol(g) = G2) by gluing two building blocks X± × S1
±:
X = [X+ × S1
+] ⊔ [X− × S1 −]
(3) where X± is an asymptotically cylindrical Calabi-Yau X± : S± → X± → P1 i.e. a compact K3 fibered over an open P1 → Non-compact. At asymptotically regime: X± : S± × S1 × ∆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 G2 Manifolds and 4D Emergent Strings 18/11/2020 6 / 16
Globally, X can be viewed as an K3 fibered over S3 Figure taken from [Braun ’16 ] Idea: Fiber-wise version of 7d duality between M-theory on K3 and Heterotic on T 3. K3 − →X T 3 − → X3
S3. (4) X3 is always be the Schone CY 3-fold X19,19 with h1,1 = h2,1 = 19 but can be with different bundles [Braun, Sch¨ afer-Nameki ’17 ].
Fengjun Xu TCS G2 Manifolds and 4D Emergent Strings 18/11/2020 7 / 16
Furthermore, a chain of 4D string duality gymnastics [ Bruan, Sch¨ afer-Nameki ’17] M-theory on X ⇐ ⇒ Heterotic string on X19,19 ⇐ ⇒ F-theory on X4. (5) X4 : T 2 → X4 →π B3 = dP9 × P1. In particular, γ3 Vol(S3) = g2
het,4d = Vol(P1)
Vol(dP9) , (6) γ: the radius of the K3-fiber
Fengjun Xu TCS G2 Manifolds and 4D Emergent Strings 18/11/2020 8 / 16
We are interested in the following infinite distance limit on X: Vol(K3) → µ−1Vol(K3), Vol(S3) → µVol(S3), with µ → ∞ Vol(X) ∝ Vol(K3) × Vol(S3) ∼ finite. (7) Q: Is it at the infinite distance of the 4d N = 1 moduli space? The corresponding one in F-theory on X4 Vol(P1) → ν−1Vol(P1), Vol(dP9) → νVol(dP9), with ν → ∞ Vol(B3) ∝ Vol(P1) × Vol(dP9) ∼ finite, (8) 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 gYM → 0 with MPl finite? Q2: What’s the nature of the new description at such limits?
Fengjun Xu TCS G2 Manifolds and 4D Emergent Strings 18/11/2020 9 / 16
Q1: What happens when gYM → 0 with MPl finite?
General expectation: The limit with gYM → 0 should be at the infinite distance! 1 4g2
YM
:= Refαβ =Re( i 2κ2
11
ωα ∧ ∗Xωβ), ωα ∈ H2(X, Z) ∝
wα ∧ wβ ∧ Φ =
Φ = Vol(Σαβ), Σαβ ∈ H3(X, Z) (9) where we have used ∗Xwα = −wα ∧ Φ. Hence Vol(Σαβ) → ∞ = ⇒ g2
YM → 0,
(10) In our case Σαβ = S3. Consistent with Weak Gravity Conjecture: Magnetic version: There exists an infinite tower of charged states with mass scale mΛ at mΛ ∝ gYMMpl, → infinite tower of charged massless states @gYM = 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 G2 Manifolds and 4D Emergent Strings 18/11/2020 10 / 16
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 C0 := P1 in X4. Topological duality twisted reduction of 4d N = 4 SYM along P1, given that C0 · KB3 = 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 G2 Manifolds and 4D Emergent Strings 18/11/2020 11 / 16
Q2: What’s the nature of the new description at such limits?
Emeregent string conjecture [Lee, Lerche, Weigand, ’19] An infinite distance limit can be either: an decompactification limit: KK tower dominant a unique critical string becomes weakly coupled and tensionless: Stringy tower dominant TEST: Are there other towers of light states competing with the heterotic stringy tower? No other stringy towers! → Uniqueness How about KK tower? Comparing their scaling level at the infinite limit: stringy towers: excitations of the tensionless critical heterotic string M2
n,Het
M2
Pl
∝ n T 2
Het
M2
Pl
= n 2πVol(K3)M2
11
M2
Pl
= n Vol(S3) ∝ n µVol(S3) , µ → ∞, (12) with
M2
Pl
M2
11 = 4πVol(X).
KK towers M2
n,kk
M2
Pl
∝ n2 Vol(S3)M2
Pl
∝ n2 µVol(S3) 1 M2
Pl
, µ → ∞, (13) Upshot: they are at the same scaling level, but the stringy tower is more dense, hence it dominants in this infinite limit. → weakly-coupled, tensionless heterotic effective theory!
Fengjun Xu TCS G2 Manifolds and 4D Emergent Strings 18/11/2020 12 / 16
So far the story is purely classical, how about quantum corrections? Do the quantum corrections
i.e. Question: VolQ(K3) = 0? Typically, one need to consider M2-brane instantons corrections + high curvature terms. ← Hard to compute! The same question in the F-theory background: VolQ(P1) = 0? Lesson in Type II: Quantum volumes of cycles in CY = The physical masses of wrapped B-branes i.e. VolQ(P1) = M(D2P1)
Fengjun Xu TCS G2 Manifolds and 4D Emergent Strings 18/11/2020 13 / 16
Key fact: F-theory on X4 × S1
A × S1 B ↔ M-theory on X4 × S1 B ↔ Type IIA theory on X4
D3-brane on P1 × S1
A
← → D2-brane on P1 Mass of the heterotic string on S1
A with the winding number w = 1 and KK momentum n = 0.
i.e. VolQ(P1) = M(D2P1) = M(D3P1×S1
A) = Mw=1,n=0
Mw,n Ms = | wRTH Ms + E0 + n RMs | = ⇒ Mw=1,n=0 Ms = 1 RMs (14) E0 := − 1
2 C · ¯
KB3 = −1, Casimir zero energy. R: the radius of S1
A
In the F-theory limit, R → ∞ → VolQ(P1) = 0, hence no quantum corrections from the world-sheet instantons. Further corrections in F-theory: The spacetime instanton corrections and α′ perturbative corrections are highly suppressed @ deeply weekly-coupled regime [Kl¨ awer, Lee, Weigand, Wiesner ’20] Conclusion: The quantum corrections do NOT obstruct our infinite distance limit! How about F-term?[Kl¨ awer, Lee, Weigand, Wiesner ’20]
Fengjun Xu TCS G2 Manifolds and 4D Emergent Strings 18/11/2020 14 / 16
We argue that at an infinite distance limit in 4d N = 1 moduli space from M-theory on TCS G2 manifold X, a tensionless weak-coupled heterotic string emerges and hence the new effective description at the limit is captured by the tensionless heterotic string, which fits with the emergent string conjecture We discuss the quantum corrections and argue that the quantum corrections do not
Fengjun Xu TCS G2 Manifolds and 4D Emergent Strings 18/11/2020 15 / 16
Fengjun Xu TCS G2 Manifolds and 4D Emergent Strings 18/11/2020 16 / 16