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?

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.

Example: SU(N) with (N − 8) F + 1 S with T ≤ 10 and N ≤ 30 (with N = 30 for T = 1) No F-theory realization for any N This theory has a ⋅ b = 1, b ⋅ b = − 1 We can choose a basis such that the bilinear form and the vectors a, b are given by: Ω = diag(1,(−1)T), a = (−3,1T), b = (0T, − 1) One can check that we may choose Q = (3,0T−1,1) in which case k = 1. N−1 ∑ Λi ≤ 1 i But symmetric matter has highest weight (2,0,⋯,0). •6d � = (1,0) Supergravity theories Inﬁnite families

• SLP would suggest that there are only ﬁnitely many consistent theories when coupled to gravity. It is interesting to investigate whether there is a UV-completion independent reason for the ﬁniteness.

Key Potential Non-abelian Inﬁnite families:

1.Non-abelian theories of fixed T with an infinite number of simple gauge group factors of finite dimension.

A more careful analysis using the gravitational anomaly shows that there can only be a ﬁnite number of such theories. Kumar, Morrison, Taylor 10’

2.Non-abelian theories with ﬁxed T and unbounded dimension. The F4 anomaly restricts the types of theories that can appear with unbounded dimension: Kumar, Morrison, Taylor 10’ 1 Gauge Factor 2 Gauge Factors1 3 Gauge Factors 0 1 Gauge group

SU(N − 8) × SU(N ) × SU(N + 8)

Sp((N − 8)/2) × SU(N ) × SO(N + 8)

For ﬁxed T < 9 it was shown that the families are ﬁnite.

H-V diverges for large N What about T ≥ 9? H − V ≤ 273 • What about T ≥ 9? T = 9 • SU(N) + 1Adj a2 = 0,b2 = 0,a ⋅ b = 0 ⟹ b = − λa with λ > 0 We can ﬁnd solutions: Ω = diag(+1,(−1)10), a = (−3,110) • SU(N) + 1S + 1A } k(N2 − 1) Unitarity Condition: ≤ c = 3Q ⋅ Q − 9Q ⋅ a + 2 k + N L

A string of charge Q needs to satisfy: • Q2 ≥ − 1 1 Levels : A minimal choice is: • ki = Q ⋅ bi, kl = (Q ⋅ Q + Q ⋅ a + 2) ≥ 0 Q = (1, − 1, − 1,0,⋯,0) 2 • Tension: Q ⋅ j ≥ 0 } λ(N2 − 1) Unitarity Condition: ≤ c = 8 λ + N L

Solutions: (k ≥ 1,N = 0,1,2,3), (4 ≥ k ≥ 1,N = 4), (2 ≥ k ≥ 1,N = 5), (k = 1,N = 6,7,8,9)

N−1 Note that for k = 1 the matter bound becomes: ∑ Λi ≤ 1 but the S, Adj do not satisfy this. i

Solutions: (k ≥ 2,N = 0,1,2,3), (4 ≥ k ≥ 2,N = 4), (k = 2 ≥ 1,N = 5)

From the F-theory perspective we have both a, b primitive and hence λ = 1 but as we saw that is not allowed by the matter bound and hence we do not expect any F-theory realizations of those remaining theories. • What about T ≥ 9? 2 Gauge Factors

For the dimension is bounded by T = 9 N ≤ 9 Kim, Shiu, Vafa 19’ using completeness of spectrum and unitarity

a2 −a ⋅ b −a ⋅ b • Consider SO(2N + 8) × Sp(N) with 1(F ⊗ F) valid for T ≤ 10: 1 2 9 − T −2 1 The charge lattice is 2 Λ = −a ⋅ b1 b1 b1 ⋅ b2 = −2 −4 2 ( ) 2 1 2 −1 −a ⋅ b2 b1 ⋅ b2 b2 -For T = 9 we have: 2 a ⋅ (b1 + 2b2) = 0, (b1 + 2b2) = 0 ⟹ b1 + 2b2 = λa ⟹ b1 = λa − 2b2 There is no that satisfy positivity conditions } b1, b2 b1 ⋅ b2 = 2 ⟹ (λa − 2b2) ⋅ b2 = − λ + 2 ⟹ λ = 0 ⟹ j ⋅ b1 = − j ⋅ 2b2

-For we have: We can ﬁnd solutions: 10 8 and 10 and 10 T = 10 a = (−3,1 ), b1 = − 2a, b2 = (1, − 1, − 1,0 ) J = (1,0 ) Ω = diag(+1,(−1) )

k ((2N + 8)(2N + 7)/2) k (2N(2N + 1)/2) Unitarity Condition: 1 2 + ≤ cL = 3Q ⋅ Q − 9Q ⋅ a + 2 k1 + (2N + 6) k2 + (N + 1) Consider a string of charge , the minimal choice 8 with 2 Q = (q1, ⋯, q11) Q = (1, − 1,0 , − 1) Q = − 1, k2 = 0, k1 = 2

we have k1((2N + 8)(2N + 7)/2) • N = 0 SO(8) Morrison, Taylor 12’ ≤ 8 ⟹ N < = 1/2 Realized in F-theory Taylor 12’ k + (2N + 6) 1 • N = 1/2 we have SO(9) + 1F } •6d � = (1,0) Supergravity theories Inﬁnite families

} Similar analysis shows that they are all ﬁnite for a given choice of basis! • There are in fact more theories allowed by anomalies with k number of simple factors and excluded with similar tools as above: Sp((N − 8)/2) × SU(N) × SU(N + 8) × ⋯ × SO(N + 8(k − 2)) F⏟⊗ F F⏟⊗ F F⏟⊗ F Are ﬁnite! SU(N − 8) × SU(N) × SU(N + 8) × ⋯ × SU(N + 8(k − 2)) ⏟ F⏟⊗ F ⏟ A F ⊗ F F ⊗ F S SU(N − 8) × SU(N) × SU(N + 8) × ⋯ × SO(N + 8(k − 2)) ⏟ ⏟ ⏟ A F ⊗ F F ⊗ F F ⊗ F SU(N) × SU(N) × ⋯ × SU(N) F⏟⊗ F F⏟⊗ F F ⊗ F

⏟ k dimG More generally consider the unitarity condition: c = i ≤ c = 3Q ⋅ Q − 9Q ⋅ a + 2 ∑ Gi v L i ki + hi 1 The condition that k = (Q ⋅ Q + Q ⋅ a + 2) ≥ 0 implies that −Q ⋅ a ≤ Q ⋅ Q + 2 and so: l 2 k dimG c = i ≤ c ≤ 12Q2 + 20 ∑ Gi v L i ki + hi

Assumptions: There are ﬁnitely many inequivalent families of type with the same type of matter at each . • F(N) = G1(N) × G2(N) × ⋯ × Gk(N) N • The string charge lattice is generated by BPS strings.

Then the vectors are independent of and the charge lattice spanned by BPS states is also independent of because giving a a, bi N N Vev to a hypermultiplet does not aﬀect the inner product of the charge lattice.

In addition, since we can ﬁnd generators given by BPS strings then they can not all have because then they would not Qi ki = 0 generate the whole lattice.

In our examples one can show that where is the number of simple gauge factors is a consistent charge of a BPS string with Q = kJ − a k Gi 2 and . Q > 0 ki > 0

Therefore, for each family of type F(N) one can ﬁnd a minimal choice of Q2 which is independent of N. •6d � = (1,0) Supergravity theories Inﬁnite families

3. Abelian Theories U(1)N Park, Taylor 11’ The number of abelian factors is bounded by N ≤ 32 for T = 0, N ≤ 20 for T ≥ 1, unless T > 1 and b ⋅ b = 0 then N ≤ 22 i i Lee, Weigand 19’

4. Inﬁnite abelian theories with unbounded charges Taylor, Turner 19’ For example, U(1) with 54 × (±q) + 54(±r) + 54 × ( ± (q + r)) with q, r ∈ ℤ

More work is needed to exclude all inﬁnite families but the number of massless modes is ﬁnite! Thank you very much!