The global gauge group structure of F-theory compactifications with U(1)s
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1 The global gauge group structure of F-theory compactifications with U(1)s based on arxiv: with Mirjam Cvetič Ling Lin Department of Physics and Astronomy University of Pennsylvania String Phenomenology, Virginia Tech July 4, 2017 Ling Lin Global gauge group structure of F-theory with U(1)s StringPheno 2017, 04/07/ / 10
2 Motivation Motivation Global gauge group structure of non-abelian symmetry given by Mordell Weil torsion. [Apsinwall, Morrison 98], [Mayrhofer, Morrison, Till, Weigand 14] F-theory Standard Models realize precisely physical spectrum under g SM = su(3) su(2) u(1) Y [LL, Weigand 14, 16], [Klevers et al 14], [Cvetič et al 15] = gauge group [SU(3) SU(2) U(1) Y ]/Z 6? What is the geometric origin? Ling Lin Global gauge group structure of F-theory with U(1)s StringPheno 2017, 04/07/ / 10
3 Outline Outline 1 2 Example: F-theory Standard Model 3 Global group structure as charge constraints 4 Conclusions & outlook Ling Lin Global gauge group structure of F-theory with U(1)s StringPheno 2017, 04/07/ / 10
4 Review: Shioda map and u(1)s in F-theory Sections σ k of smooth elliptic Calabi Yau π : Y B form Mordell Weil (MW) group: MW(Y ) = Z m t Z k t. [σ k ] = S k, k = 0,..., m are independent integer divisors. Codim. 1 singular fibers E i (Cartan divisors of non-ab. algebra g); Shioda Tate Wazir: H 1,1 (Y, Q) = S 1,..., S m S 0, E i π H 1,1 (B, Q). injective homomorphism (Shioda map) ϕ : MW(Y ) H 1,1 (Y, Q), s.t. image is orthogonal to S 0, E i H 1,1 (B, Q). General form: ϕ(σ) = λ (S S 0 + i l i E i (+D B )), λ a priori not constrained; for now, fix λ = 1. u(1) gauge field A arise from KK-reduction C 3 = A ϕ(σ) +... Ling Lin Global gauge group structure of F-theory with U(1)s StringPheno 2017, 04/07/ / 10
5 Fractional u(1) charges u(1) charge of matter states from codim. 2 fibral curves Γ given by Γ ϕ(σ) with ϕ(σ) = S S 0 + i l i E i. Coefficients l i are determined by orthogonality of ϕ(σ) with E i = (P 1 i {θ}); explicitly, l i = j (C 1 ) ij (S S 0 ) P 1 j, with C ij = E i P 1 j. Ling Lin Global gauge group structure of F-theory with U(1)s StringPheno 2017, 04/07/ / 10
6 Fractional u(1) charges u(1) charge of matter states from codim. 2 fibral curves Γ given by Γ ϕ(σ) with ϕ(σ) = S S 0 + i l i E i. Coefficients l i are determined by orthogonality of ϕ(σ) with E i = (P 1 i {θ}); explicitly, l i = j (C 1 ) ij (S S 0 ) P 1 j, with C ij = E i P 1 j. l i are in general fractional, depend on the fibre split type and gauge algebra g. However, there is always an integer κ s.t. i : κ l i Z. = κ ϕ(σ) has manifestly integer class. Ling Lin Global gauge group structure of F-theory with U(1)s StringPheno 2017, 04/07/ / 10
7 Fractional u(1) charges u(1) charge of matter states from codim. 2 fibral curves Γ given by Γ ϕ(σ) with ϕ(σ) = S S 0 + i l i E i. Coefficients l i are determined by orthogonality of ϕ(σ) with E i = (P 1 i {θ}); explicitly, l i = j (C 1 ) ij (S S 0 ) P 1 j, with C ij = E i P 1 j. l i are in general fractional, depend on the fibre split type and gauge algebra g. However, there is always an integer κ s.t. i : κ l i Z. = κ ϕ(σ) has manifestly integer class. Therefore, all matter charges are integral multiples of 1/κ. Ling Lin Global gauge group structure of F-theory with U(1)s StringPheno 2017, 04/07/ / 10
8 Fractional u(1) charges u(1) charge of matter states from codim. 2 fibral curves Γ given by Γ ϕ(σ) with ϕ(σ) = S S 0 + i l i E i. Coefficients l i are determined by orthogonality of ϕ(σ) with E i = (P 1 i {θ}); explicitly, l i = j (C 1 ) ij (S S 0 ) P 1 j, with C ij = E i P 1 j. l i are in general fractional, depend on the fibre split type and gauge algebra g. However, there is always an integer κ s.t. i : κ l i Z. = κ ϕ(σ) has manifestly integer class. Therefore, all matter charges are integral multiples of 1/κ. Γ w,v fibral curves with weights w, v in the same g-rep Γ v = Γ w + β k P 1 k with β k Z Ling Lin Global gauge group structure of F-theory with U(1)s StringPheno 2017, 04/07/ / 10
9 Fractional u(1) charges u(1) charge of matter states from codim. 2 fibral curves Γ given by Γ ϕ(σ) with ϕ(σ) = S S 0 + i l i E i. Coefficients l i are determined by orthogonality of ϕ(σ) with E i = (P 1 i {θ}); explicitly, l i = j (C 1 ) ij (S S 0 ) P 1 j, with C ij = E i P 1 j. l i are in general fractional, depend on the fibre split type and gauge algebra g. However, there is always an integer κ s.t. i : κ l i Z. = κ ϕ(σ) has manifestly integer class. Therefore, all matter charges are integral multiples of 1/κ. Γ w,v fibral curves with weights w, v in the same g-rep Γ v = Γ w + β k P 1 k with β k Z = l i v i = l i (E i Γ v ) = l i (E i Γ w β k C ik ) = l i w i β k (S S 0 ) P 1 k }{{} Z Ling Lin Global gauge group structure of F-theory with U(1)s StringPheno 2017, 04/07/ / 10
10 Non-trivial central element from Shioda map (a) There is κ N s.t. i : κ l i Z = q u(1) = n κ, n Z (pick smallest such κ). (b) Two weights w, v in the same g-rep R g : l i w i = l i v i mod Z =: L(R g ) ( κ L(R g ) Z). Construct non-trivial central element of U(1) G: Ling Lin Global gauge group structure of F-theory with U(1)s StringPheno 2017, 04/07/ / 10
11 Non-trivial central element from Shioda map (a) There is κ N s.t. i : κ l i Z = q u(1) = n κ, n Z (pick smallest such κ). (b) Two weights w, v in the same g-rep R g : l i w i = l i v i mod Z =: L(R g ) ( κ L(R g ) Z). Construct non-trivial central element of U(1) G: Charge of state w R g from a fibral curve Γ satisfies q(w) = (S S 0 + l i E i ) Γ = q(w) l i (E i Γ) = q(w) l i w i =: ξ(w) = (S S 0 ) Γ Z. Ling Lin Global gauge group structure of F-theory with U(1)s StringPheno 2017, 04/07/ / 10
12 Non-trivial central element from Shioda map (a) There is κ N s.t. i : κ l i Z = q u(1) = n κ, n Z (pick smallest such κ). (b) Two weights w, v in the same g-rep R g : l i w i = l i v i mod Z =: L(R g ) ( κ L(R g ) Z). Construct non-trivial central element of U(1) G: Charge of state w R g from a fibral curve Γ satisfies q(w) = (S S 0 + l i E i ) Γ = q(w) l i (E i Γ) = q(w) l i w i =: ξ(w) = (S S 0 ) Γ Z. C(w) := [e 2πiq(w) (e 2πi l i w i 1)] w (b) = [e 2πiq(w) (e 2πi L(Rg) 1)] w defines element in centre of U(1) G; (a) C κ = 1. Ling Lin Global gauge group structure of F-theory with U(1)s StringPheno 2017, 04/07/ / 10
13 Non-trivial central element from Shioda map (a) There is κ N s.t. i : κ l i Z = q u(1) = n κ, n Z (pick smallest such κ). (b) Two weights w, v in the same g-rep R g : l i w i = l i v i mod Z =: L(R g ) ( κ L(R g ) Z). Construct non-trivial central element of U(1) G: Charge of state w R g from a fibral curve Γ satisfies q(w) = (S S 0 + l i E i ) Γ = q(w) l i (E i Γ) = q(w) l i w i =: ξ(w) = (S S 0 ) Γ Z. C(w) := [e 2πiq(w) (e 2πi l i w i 1)] w (b) = [e 2πiq(w) (e 2πi L(Rg) 1)] w defines element in centre of U(1) G; (a) C κ = 1. But also: C(w) = exp(2πi ξ(w) ) w = w. }{{} Z Ling Lin Global gauge group structure of F-theory with U(1)s StringPheno 2017, 04/07/ / 10
14 Non-trivial central element from Shioda map (a) There is κ N s.t. i : κ l i Z = q u(1) = n κ, n Z (pick smallest such κ). (b) Two weights w, v in the same g-rep R g : l i w i = l i v i mod Z =: L(R g ) ( κ L(R g ) Z). Construct non-trivial central element of U(1) G: Charge of state w R g from a fibral curve Γ satisfies q(w) = (S S 0 + l i E i ) Γ = q(w) l i (E i Γ) = q(w) l i w i =: ξ(w) = (S S 0 ) Γ Z. C(w) := [e 2πiq(w) (e 2πi l i w i 1)] w (b) = [e 2πiq(w) (e 2πi L(Rg) 1)] w defines element in centre of U(1) G; (a) C κ = 1. But also: C(w) = exp(2πi ξ(w) ) w = w. }{{} Z = G global = U(1) G C = U(1) G Z κ Ling Lin Global gauge group structure of F-theory with U(1)s StringPheno 2017, 04/07/ / 10
15 Example: F-theory Standard Model Example: F-theory Standard Model Toric construction with gauge algebra su(3) su(2) u(1). [Klevers et al 14], [Cvetič et al 15] ϕ(σ) = S S E su(2) su(3) 3 (2 E1 + E su(3) 2 ) C 6 = 1, so G global = [SU(3) SU(2) U(1)]/ C = [SU(3) SU(2) U(1)]/Z 6. R su(3) su(2) (3, 2) (3, 1) (1, 2) (1, 1) L(R) 1/6 2/3 1/2 0 geometrically realized matter: (3, 2) 1/6, (1, 2) 1/2, (3, 1) 2/3, (3, 1) 1/3, (1, 1) 1 = (physical) Standard Model representations. Ling Lin Global gauge group structure of F-theory with U(1)s StringPheno 2017, 04/07/ / 10
16 Example: F-theory Standard Model Example: F-theory Standard Model Toric construction with gauge algebra su(3) su(2) u(1). [Klevers et al 14], [Cvetič et al 15] ϕ(σ) = S S E su(2) su(3) 3 (2 E1 + E su(3) 2 ) C 6 = 1, so G global = [SU(3) SU(2) U(1)]/ C = [SU(3) SU(2) U(1)]/Z 6. R su(3) su(2) (3, 2) (3, 1) (1, 2) (1, 1) L(R) 1/6 2/3 1/2 0 geometrically realized matter: (3, 2) 1/6, (1, 2) 1/2, (3, 1) 2/3, (3, 1) 1/3, (1, 1) 1 = (physical) Standard Model representations. Similar situation in models with su(3) su(2) u(1) 1 u(1) 2 [LL, Weigand 14, 16]. After identifying hypercharge: G global = SU(3) SU(2) U(1) Y U(1) Z 6 Z κ Ling Lin Global gauge group structure of F-theory with U(1)s StringPheno 2017, 04/07/ / 10
17 Global group structure as charge constraints Global group structure as charge constraints σ torsional ϕ(σ) = 0 (no u(1)), global group structure G/Z κ = not all g-reps allowed. [Mayrhofer, Morrison, Till, Weigand 14] σ free, then global group structure [U(1) G]/Z κ u(1) charges of g-reps constrained: For R (i) = (q (i), R (i) g ) we have q (i) = L(R (i) g ) mod Z. For g = su(5): [Braun, Grimm, Keitel 13], [Lawrie, Schäfer-Nameki, Wong, 15] Argument derived with normalization λ = 1 for Shioda map preferred charge normalization in F-theory: can read off global gauge group from fractional u(1) charges. Equivalently: MW-group finitely generated global gauge group structure, refined charge quantization. Ling Lin Global gauge group structure of F-theory with U(1)s StringPheno 2017, 04/07/ / 10
18 Global group structure as charge constraints An F-theory swampland criterion In preferred normalization, singlets (E i Γ = 0) have integral u(1) charges. Observation: in all u(1)-models with matter, smallest singlet charge is 1. = use singlets as measuring stick for charges. Necessary condition for field theory to be F-theory compactification: normalize u(1) such that all singlets charges are as above for all matter R (i) = (q (i), R (i) g ) we have q (i) L(R (i) g ) Z. This implies: (1) If R (1) = (q (1), R g ) and R (2) = (q (2), R g ), then q (1) q (2) Z. (2) If n i=1 R(i) g = 1 g..., then n i=1 q(i) Z. Ling Lin Global gauge group structure of F-theory with U(1)s StringPheno 2017, 04/07/ / 10
19 Global group structure as charge constraints An F-theory swampland criterion In preferred normalization, singlets (E i Γ = 0) have integral u(1) charges. Observation: in all u(1)-models with matter, smallest singlet charge is 1. = use singlets as measuring stick for charges. Necessary condition for field theory to be F-theory compactification: normalize u(1) such that all singlets charges are as above for all matter R (i) = (q (i), R (i) g ) we have q (i) L(R (i) g ) Z. This implies: (1) If R (1) = (q (1), R g ) and R (2) = (q (2), R g ), then q (1) q (2) Z. (2) If n i=1 R(i) g = 1 g..., then n i=1 q(i) Z. Problem: What if no singlets? (Non-higgsable u(1) in 6D without matter [(Martini), Morrison, (Park), Taylor 14, 16], [Wang 17]) Situation in 4D? Other manifestations of the preferred normalization in field theory? Ling Lin Global gauge group structure of F-theory with U(1)s StringPheno 2017, 04/07/ / 10
20 Conclusions & outlook Conclusions & outlook Shioda map of sections non-trivial global gauge group structure of u(1) g theory. Example: F-theory explicitly realizes Standard Model [SU(3) SU(2) U(1)]/Z 6. Equivalently: refined charge quantization condition. In preferred charge normalization: (1) Charge lattice of R g -matter integer spaced despite fractional charges. (2) Fractional charges of g-invariant matter combination sum to integer. Possible F-theory swampland criterion? What about charges of massive states? Spectrum of non-local operators? Global gauge group structure from higgsing (see paper) some unhiggsed model with non-minimal codim. 2 loci. Physical explanation? Ling Lin Global gauge group structure of F-theory with U(1)s StringPheno 2017, 04/07/ / 10
21 Conclusions & outlook Conclusions & outlook Shioda map of sections non-trivial global gauge group structure of u(1) g theory. Example: F-theory explicitly realizes Standard Model [SU(3) SU(2) U(1)]/Z 6. Equivalently: refined charge quantization condition. In preferred charge normalization: (1) Charge lattice of R g -matter integer spaced despite fractional charges. (2) Fractional charges of g-invariant matter combination sum to integer. Possible F-theory swampland criterion? What about charges of massive states? Spectrum of non-local operators? Global gauge group structure from higgsing (see paper) some unhiggsed model with non-minimal codim. 2 loci. Physical explanation? Thank you! Ling Lin Global gauge group structure of F-theory with U(1)s StringPheno 2017, 04/07/ / 10
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