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Hee-Cheol Kim , HCT, Cumrun Vafa, 1912.06144 Sheldon Katz, Hee-Cheol Kim , HCT, Cumrun Vafa, 2004.14401

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Hee-Cheol Kim , HCT, Cumrun Vafa, 1912.06144 Sheldon Katz, Hee-Cheol Kim , HCT, Cumrun Vafa, 2004.14401 On the Finiteness of Supergravity Landscape in d = 6 Based on work with Cumrun Vafa & HCT to appear and previous work with: Hee-Cheol Kim , HCT, Cumrun Vafa, 1912.06144 Sheldon Katz, Hee-Cheol Kim , HCT, Cumrun Vafa, 2004.14401 Houri Christina Tarazi Harvard University May 11th, 2021 Swampland Program: What conditions can we use to distinguish between consistent QFT’s which cannot couple to gravity and those that can arise in the low energy limit of a quantum gravity? Swampland Consistent QFTs Landscape Consistent QFTs Coupled to Gravity Vafa 05’ Swampland Program: What conditions can we use to distinguish between consistent QFT’s which cannot couple to gravity and those that can arise in the low energy limit of a quantum gravity? Swampland Landscape Consistent QFTs Consistent QFTs Coupled to Gravity String Landscape Coming from string theory Vafa 05’ Do all the theories in the Landscape also belong in the String Landscape (SLP: YES)? Is there a way to use fundamental ideas that are believed to be true for Quantum gravitational theories to construct such condition? Unitarity Every Quantum Gravity should be Unitary Is there a way to use fundamental ideas that are believed to be true for Quantum gravitational theories to construct such condition? Unitarity Anomaly Cancellation Every Quantum All gauge- Gravity gravitational should be anomalies Unitary should cancel Is there a way to use fundamental ideas that are believed to be true for Quantum gravitational theories to construct such condition? Unitarity Anomaly Cancellation Completeness of Spectrum In a quantum Every Quantum All gauge- gravity for any p- Gravity gravitational form gauge field, should be anomalies every charged Unitary should cancel (p-1)-brane state should appear in the spectrum Polchincki 03’ Banks, Seiberg 10’ Harlow, Ooguri 18’ Is there a way to use fundamental ideas that are believed to be true for Quantum gravitational theories to construct such condition? Unitarity Anomaly Cancellation Completeness of Spectrum Cobordism Conjecture: In a quantum All the Every Quantum All gauge- gravity for any p- cobordism Gravity gravitational form gauge field, classes in a should be anomalies every charged consistent theory Unitary should cancel (p-1)-brane state of quantum should appear in gravity must the spectrum vanish Polchincki 03’ Banks, Seiberg 10’ Harlow, Ooguri 18’ McNamara,Vafa 19’ Is there a way to use fundamental ideas that are believed to be true for Quantum gravitational theories to construct such condition? Unitarity Anomaly Cancellation Completeness of Spectrum Cobordism Conjecture: Distance Conjecture: All the In a quantum In a quantum Every Quantum All gauge- gravity for any p- cobordism gravity at infinite Gravity gravitational form gauge field, classes in a distance in the should be anomalies every charged consistent theory scalar moduli space Unitary should cancel (p-1)-brane state of quantum an infinite tower of should appear in gravity must states becomes the spectrum vanish exponentially light Polchincki 03’ Banks, Seiberg 10’ Harlow, Ooguri 18’ McNamara,Vafa 19’ Ooguri, Vafa 06’ Many more Swampland conditions exist …….. Is there a way to use fundamental ideas that are believed to be true for Quantum gravitational theories to construct such conditions? • Theories with 32 Supercharges: Low-energy limit completely determined by supersymmetry and unique except d=10 • d=11: realized through M-theory • d=10: Chiral � = (2,0): realized through IIB string theory Non-Chiral � = (1,1): realized through IIA string theory • d<10: circle compactification of d+1 All of them also belong to the String Landscape! An evidence for SLP Is there a way to use fundamental ideas that are believed to be true for Quantum gravitational theories to construct such condition? • Theories with 16 Supercharges: d=10: Anomaly cancellation gives us gauge groups , 248 496 • G = E8 × E8, SO(32) E8 × U(1) , U(1) Kim, Shiu, Vafa 19’ ⏟Adams, Dewolfe, Taylor 10’ • d<10: All non-chiral except for 6d (2,0), which is unique and realized through IIB on K3 Non-chiral: the rank of the gauge group is bounded by rank(G) ≤ 26 − d (For example N=4, d=4 with belongs to the Swampland.) rank > 22 Kim, HCT, Vafa 19’ d=9: rank(G) = 1,9,17 d=8: rank(G) = 2,10,18 d=7: rank(G) = 1 mod 2 Alvarez-Gaume, Witten 84 Montero, Vafa 20’ • M-theory on KB or IIB on DP r = 1, r = 3,5,7,11,19 CHL string On 1 • r = 9 : S Can be realized. • r = 17 : Heterotic on S1 } Can we also find restrictions on the type of gauge groups that could appear? String Theory allows only for but not 8d : SU(N), SO(2N), Sp(N), e6, e7, e8 SO(2N + 1), f4, g2 Garc´ıa-Etxebarria, Hayashi ,Ohmori, Tachikawa, Yonekura⏟ 17’ ⏟Hamada, Vafa 21’ Is there a way to use fundamental ideas that are believed to be true for Quantum gravitational theories to construct such condition? • Theories with 8 Supercharges: • d=6: Anomalies are very constraining and many theories can be ruled out. Morrison, ,Kumar, Taylor,…..09’/10’/.. -Extra conditions using extended objects (which is the approach we will follow here) Kim, Shiu, Vafa 19’ Lee,Weigand 19’ Today’s talk will mostly focus on the d=6 and on the finiteness of massless modes for these theories. • d=5: Much harder since no anomalies but we can still have some constraints: For example for theories of the form U(1) × G that can Higgs to the quintic we have: rank(G) ≤ 52 For the Higgs branch we can put a bound of the form: 3 ∑ dimRi − dimG ≤ 36D + 80 i where D is the divisor class associated with a black string. Is there a way to use fundamental ideas that are believed to be true for Quantum gravitational theories to construct such condition? Unitarity Anomaly Cancellation Completeness of Spectrum Cobordism Conjecture: Distance Conjecture: All the In a quantum In a quantum Every Quantum All gauge- gravity for any p- cobordism gravity at infinite Gravity gravitational form gauge field, classes in a distance in the should be anomalies every charged consistent theory scalar moduli space Unitary should cancel (p-1)-brane state of quantum an infinite tower of should appear in gravity must states becomes the spectrum vanish exponentially light Polchincki 03’ Banks, Seiberg 10’ Harlow, Ooguri 18’ McNamara,Vafa 19’ Ooguri, Vafa 06’ • 6d � = 1 Supergravity theories Super-multiplets: Chiral fields contribute to gauge/gravitational anomalies Supergravity-multiplet − (gμν, Bμν, ψμ ) Cancelled by the Green-Schwarz-Sagnotti Mechanism Tensor-multiplet(T) + (Bμν, ϕ, χ ) Vector-multiplet(V) − (Aμ, λ ) Hyper-multiplet(H) + } (4h, ψ ) Anomaly polynomial factorizes as: 1 α β symmetric of signature (1,T) I = Ω X X Ωαβ 8 2 αβ 4 4 α 1 α 2 a 2 2 α α 1,T X = a trR + b ( trF ) a , bi ∈ ℝ 2 ∑ i λ i Anomaly Cancellation: i i 1 4 2 2 i i i group theory coefficients R : H − V = 273 − 29T F R : a ⋅ bi = λi(AAdj − ∑ nRAR) ∈ ℤ AR, BR, CR 6 R 4 i i i 2 2 4 4 2 2 F : 0 = BAdj − ∑ nRBR 2 2 1 2 i i i trRF = ARtrF , trRF = BRtrF + CR(trF ) (F ) : bi ⋅ bi = λi (∑ nRCR − CAdj) ∈ ℤ 3 R (R2)2 : a ⋅ a = 9 − T ∈ ℤ ni = hypers in number of in R F2F2 : b ⋅ b = λ λ nij Ai Aj ∈ ℤ, i ≠ j R i j i j ∑ i j RS R S R,S Taylor, ,Kumar, Morison,…. The number of theories that satisfy these anomaly conditions are infinite. SLP would though suggest that only a finite number of those can truly be UV-completed in a quantum gravity. • 6d � = 1 Supergravity theories Super-multiplets: Completeness of Spectrum Supergravity-multiplet (g , B+ , ψ−) μν μν μ • For 6d � = 1 supergravity there exist Tensor-multiplet(T) − + (anti-)shelf dual BPS strings charged under B (Bμν, ϕ, χ ) 2 } (0,4) SCFT at low energy Vector-multiplet(V) − (Aμ, λ ) Unitarity of the string worldsheet + Hyper-multiplet(H) (4h, ψ ) ki dimG c = ≤ c ∑ Gi v L i ki + hi Anomaly cancellation for string of charge Q Worldsheet inflow WS −I4 + I4 = 0 Bulk Central charges: • cL = 3Q ⋅ Q − 9Q ⋅ a + 2, cR = 3Q ⋅ Q − 3Q ⋅ a ≥ 0 1 Levels of : • Gi, SU(2)l ki = Q ⋅ bi, kl = (Q ⋅ Q + Q ⋅ a + 2) ≥ 0 Kim, Kim, Park 16’ 2 Kim, Shiu, Vafa, 19’ Shimizu, Tachikawa 16’ • 6d � = 1 Supergravity theories Super-multiplets: Completeness of Spectrum Supergravity-multiplet (g , B+ , ψ−) μν μν μ • For 6d � = 1 supergravity there exist Tensor-multiplet(T) − + (anti-)shelf dual BPS strings charged under B (Bμν, ϕ, χ ) 2 Other consistency conditions: (0,4) SCFT at low energy } Moduli space parametrized by j ∈ ℝ1,T − Vector-multiplet(V) (A , λ ) 2 μ Unitarity of the string worldsheet • j ⋅ bi trF ⟹ j ⋅ bi > 0 + Hyper-multiplet(H) (4h, ψ ) ki dimG • j ⋅ j > 0 c = ≤ c ∑ Gi k + hv L i i i • Tension: Q ⋅ j ≥ 0 • −j ⋅ a trR2 ⟹ j ⋅ a > 0 Anomaly cancellation for string of charge Q Hamada, Noumi, Shiu 18’ Worldsheet Cheung, Remmen 16’ inflow WS −I4 + I4 = 0 Bulk Central charges: • cL̂ = 3Q ⋅ Q − 9Q ⋅ a + 2, cR̂ = 3Q ⋅ Q − 3Q ⋅ a ≥ 0 1 Levels of : • Gi, SU(2)l ki = Q ⋅ bi, kl = (Q ⋅ Q + Q ⋅ a + 2) ≥ 0 2 New Ingredient: Bound on the matter • Q2 ≥ − 1 • The vertex operator of the massless modes with 1 representation R of G with conformal weight Levels : • ki = Q ⋅ bi, kl = (Q ⋅ Q + Q ⋅ a + 2) ≥ 0 C2(R) 2 ΔR = where C (R) is the second Casimir of the 2(k + hv) 2 • Tension: Q ⋅ j ≥ 0 R must obey: ΔR ≤ 1 • j ⋅ bi > 0 • The representation R of a primary with highest weight where is the rank of the Lie • j ⋅ j > 0 Λ = (Λi, ⋯, Λr) r algebra must satisfy : r • j ⋅ a > 0 ∑ Λi ≤ k i where k is the level of the current algebra of G on the worldsheet.
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