F-theory compactifications and model buildingstatistics.roma2.infn.it/~stringhe/doc/weigand.pdf ·...
Transcript of F-theory compactifications and model buildingstatistics.roma2.infn.it/~stringhe/doc/weigand.pdf ·...
F-theory compactifications andmodel building
Timo Weigand
ITP, University of Heidelberg
Tor Vergata Roma, November 2010 – p.1
3 good reasons for F-theoryI.) Two traditional foundations of string model building:
1) heterotic string:
orbifolds, free fermionic, het. on CY with vector bundles
• exceptional gauge groups X
• gravity and gauge fields all in bulk
2) perturbative Type II orientifolds/brane worlds:
Type IIA with D6-branes
Type IIB with D3/D7-branes
• perturbative gauge groups: U(N), Sp(N), SO(N) , but no E-groups
• localised gauge degrees of freedom X
F-theory combines both pros:
• model building based on exceptional groups
• includes ”local features”
Tor Vergata Roma, November 2010 – p.2
3 good reasons for F-theoryII.) Promising moduli stabilisation scenarios similar to Type IIB/M-Theory
well-controlled because fluxes allow for conformal Calabi-Yau
↔ power of holomorphy
III.) F-theory is fascinating beyond phenomenology:
truly non-perturbative and backreacted compactification framework
in some sense: THE way to think about 7-branes
Dynamics accessible from several different viewpoints:
• as strongly coupled Type IIB with 7-branes
• as dual to M-theory
• as dual to heterotic strings
Tor Vergata Roma, November 2010 – p.3
OverviewLecture 1: Introduction to F-theory
Lecture 2: Global Tate models and local Spectral covers for F-theory
Lecture 3: Aspects of F-theory GUT phenomenology
Tor Vergata Roma, November 2010 – p.4
Overview
Part I: Introduction to F-theory
• F-theory as Type IIB with backreaction
• Elliptic fibrations and their M-theory origin
• 7-branes and non-abelian singularities
Tor Vergata Roma, November 2010 – p.5
Reminder: Type IIB orientifoldsCompactify 10 D Type IIB theory on Calabi-Yau 3-fold X
quotient by Ω(−1)FLσ σ: Z2 hol. involution of X
⇒ Fixpoint set: O7-planes and O3-planes
charge cancellation requires 7-branes on Γa + image on Γa′
∑
a
Na(Γa + Γa′) = 8ΠO7
further constraints:
• induced
D5/D3-tadpole
cancellation
• D-term SUSY
Tor Vergata Roma, November 2010 – p.6
Probe approximationD-branes and O-planes treated in probe approximation
• neglect backreaction on metric and on sourced RR fields
• only ensure integrability conditions = tadpole conditions
For generic Dp-brane p < 7 backreaction is asymptotically negligible
• Dp brane is pointlike source in n = 9 − p normal directions
• Heuristically: need to solve Poincare equation in n dimensions
n > 2: ∆Φ(r) = δ(r) → Φ(r) = 1rn−2
more precisiely: BPS solution
ds2 = H−1/2p ηµνdxµdxν + H
1/2p dxidxi
e2φ = e2φ0H3−p2
p Cp+1 =H−1
p − 1
eφ0dx0 ∧ . . . ∧ dxp
Hp = 1 +“ rp
r
”(7−p), r
(7−p)p = #eφ0N
Note: Hp → 1 as r → ∞ XTor Vergata Roma, November 2010 – p.7
Probe approximationIn general we wish to go beyond this probe approximation:
• While above is fine for Dp-branes, O-planes have negative tension ⇒SUGRA is ill-defined unless all tadpoles are cancelled locally
• we need control of the in general varying dilaton as this gives the
string coupling
• for the interesting case of D7-branes in Type IIB orientifolds solution
does not asymptote to flat space in above sense (see later)!
F-theory = SUSY Type IIB/Ωσ(−1)FL with backreaction of
D7-branes and O7-planes on the geometry and on the varying
dilaton taken into account
Tor Vergata Roma, November 2010 – p.8
Type IIB and SL(2, Z)Consider the IIB action in Einstein frame:
SIIB =2π
ℓ8s
(∫
d10x√−gR − 1
2
∂τ∂τ
(Imτ)2+
1
ImτG3 ∧ ∗G3 +
1
2F5 ∧ ∗F5 +
+C4 ∧ H3 ∧ F3
)
,
ℓs = 2π√
α′, τ = C0 + ie−Φ, G3 = F3 − τH3, H3 = dB2,
F5 = F5 −1
2C2 ∧ H3 +
1
2B2 ∧ F3 Fp = dCp−1
Classical action is invariant under SL(2, R):
τ → aτ + b
cτ + d,
(
C2
B2
)
→(
aC2 + bB2
cC2 + dB2
)
= M
(
C2
B2
)
, detM = 1
non-perturbatively SL(2, R) → SL(2, Z) due to D(-1)-instanton
M =
(
1 b
0 1
)
: C0 → C0 + b ⇒ e2πiR
C0 invariant for b ∈ Z
Tor Vergata Roma, November 2010 – p.9
Type IIB and SL(2, Z)
SL(2, Z) : T =
(
1 1
0 1
)
: τ → τ + 1 S =
(
0 1
−1 0
)
: τ → −1
τ
same symmetry group acts on complex structure of a torus T 2
τ takes values in fundamental domain
T 2: lattice in C:
w ≃ w + 1, w ≃ w + τ
τ ∈ C ℜτ > 0: complex structure1
τ τ+1
a
b
S-duality is strong-weak duality: maps F1-string to D1-string
F1: (1,0) string, D1: (0,1) string S
(
1
0
)
=
(
0
1
)
(p,q) string: BPS-bound state of p F-strings and q D-strings
stable for p,q relatively primeTor Vergata Roma, November 2010 – p.10
7-brane backreactionD7 action: S = SDBI + SCS (string frame)
SDBI = −µ7
∫
e−Φ√
−det(G + B + 2πα′F ), SCS = µ7
∫
C8 + . . .
µ7 =2π
lp+1s
|p=7 =2π
l8s
D7 brane is electric source for C8 ↔ magnetic source for C0 = Reτ
Task: find the backreacted solution for τ
D7 brane is pointlike source in 2 normal directions z = x + iy
• SUSY requires: τ = τ(z) → F1(z) = dC0(z)
• e.o.m. for F9 = dC8: d ∗ F9 = δ(2)(z − zi)
• duality ∗F9 = F1
• Gauss law: 1 =∫
Cd ∗ F9 =
∮
S1 F1 =∮
S1 dC0
IP1 IIB(z)
7-brane
z0
pic from: Lerche, 9910207
Solution for τ : τ(z) = τ0 + 12πi ln(z − zi) + . . . e−φ = 1
2π ln| z−zi
λ |Interpretation: non-trivial profile for τ , but still have unfixed offset
Tor Vergata Roma, November 2010 – p.11
7-brane backreactionThis is a severe backreaction on geometry:
1) deficit angle ↔ ”long-range”effect
2) monodromies
ad 1) D7-brane looks like ”cosmic string” in ambient space
metric ansatz: ds2 = −dt2 +∑7
i=1 dx2i + H7(z, z)dzdz
• pointlike source in ambient space
⇒ Expect 2 dim. Poisson equation for H7(z, z)
• Einstein equation: ∂∂logH7 = ∂τ∂ττ22
Greene,Shapere,Vafa,Yau 1989
• at long distances away from brane, space has a deficit anglesee also Bergshoeff et al. ’06, A. Braun et al. ’08
=⇒ long-range effect that does not asymptote away
• for suitable τ0 still exists regime near brane where metric can be
treated as in probe picture
but for too strong coupling, this regime might become substringy!Tor Vergata Roma, November 2010 – p.12
7-brane backreactionad 2) near D7-brane: τ(z) ≃ 1
2πi ln(z − z0) has branch cut
as we encircle z0: τ → τ + 1 ”monodromy”
consistent because Type IIB theory is SL(2, Z) invariant
More drastic monodromy for more general (p, q) branes
• D7-brane: electric charge under C8 ↔ magnetic charge under τ
• dyonic (p,q) branes = branes on which (p, q) strings can end
• SL(2, Z) transforms various (p,q) branes into one another
We had: monodromy of (1,0) brane: τ → τ + 1 ↔ T1,0
for (p, q) brane : τ → aτ + b
cτ + dT =
(
1 − pq p2
−q2 1 + pq
)
• In general, different (p, q) type branes cannot be treated perturbatively
Tor Vergata Roma, November 2010 – p.13
From branes to F-theoryF-theory geometrises the varying axio-dilaton field τ : Vafa 1996
• Interpret τ as complex structure of auxiliary torus T 2
• Kahler structure has no physical meaning
τ varies ↔ shape of T 2 varies
=⇒ fibration of T 2 → M10
pic adapted from: Denef, 0803.1194
pic adapted from: Lerche 9910207
locally near position of D-brane
τ = 12πi ln(z − z0) + . . . → i∞
↔ T 2 fiber degenerates as cycle a → 0
Tor Vergata Roma, November 2010 – p.14
F from MPrecise definition of F-theory via duality with M-theory
Idea: Compactify 11-dim SUGRA on M9 × T 2
Take limit of vanishing volume of T 2 = S1A × S1
B as follows:
• S1A: M-theory circle of radius R10
R10 → 0 ↔ weakly coupled IIA on M9 × S1B
• T-duality along S1B of radius R9 → IIB on M9 × S1
B
R9 → 0 ↔ decompactification of S1B ⇒ IIB on M9 × R
Roughly: gs = R10
R9is Im(τ) for original T 2
F-theory limit is vol(T 2) → 0 while ℓs finite
Newton’s law of string theory:
F = M |A(T 2)→0
Indeed: volume of T 2 has no physical meaning in IIB
Tor Vergata Roma, November 2010 – p.15
Elliptic fibrationsCase of phenomenological interest:
Type IIB orientifold on M10 = R1,3 × X3/σ with R1,3 filling 7-branes
• physics encoded in geometry of Y4 : T2 → B3 B3 = X3/σ
• N = 1 SUSY on M-theory side requires Y4 to be Calabi-Yau
• jargon: F-theory on elliptic fourfold Y4 = effective 4D theory obtained by
compactification of Type IIB strings with D7-branes on X3/σ
IIB language:
7-branes wrap 4-cycle Γa ∈ X3/σ
F-theory language:
Γa = locus of fiber degenerationpic adapted from: Denef, 0803.1194
Magic of F-theory:
Xall information of present D7-branes encoded in geometry of this fibration
Xbackreaction fully taken into account Tor Vergata Roma, November 2010 – p.16
Elliptic fibrationsUnderstand possible degenerations of fiber via algebraic geometry :
• elliptic curve admits various representations as hypersurfaces/complete
intersections in projective spaces
• most common: T 2 = P2,3,1[6]: (x, y, z) ≃ (λ2x, λ3y, λz) (∗)P ≡ y2 − x3 − fxz4 − gz6 = 0 f, g ∈ C with (∗) set z = 1
• This is Calabi-Yau and thus T 2 X
• T 2 singular when P = 0 and dP = 0, i.e.
1) y = 0
2) 0 = x3 − fx − g = (x − a1)(x − a2)(x − a3)
3) 0 = (x − a1)(x − a2) + (x − a2)(x − a3) + (x − a1)(x − a3)
⇒ only possible if two or more ai coincide
Tor Vergata Roma, November 2010 – p.17
Elliptic fibrationsResults from complex geometry:
• Coincidence of roots if discriminant
∆ = 27g2 + 4f3 = 0
• complex structure τ encoded in f, g via
j(τ) =4(24f)3
∆
j(τ): SL(2, Z) invariant Jacobi function
j(τ) = e−2πiτ + 744 + O(e2πiτ )
Tor Vergata Roma, November 2010 – p.18
Elliptic fibrationsNow fiber T 2 over the base B3 with coordinates ui: f, g → f(ui), g(ui)
→ Weierstrass model y2 = x3 + f(ui)xz4 + g(ui)z6
(more details on properties of B and g, f later)
position of branes on B3: ∆(ui) = 0 → holomorphic divisor X
• simple zero
only T 2 degenerates, but fourfold Y4 remains smooth
single 7-brane, possibly with U(1) gauge group
• multiple zero → genuine singularity of fourfold Y4
several coincident 7-branes with non-abelian gauge groups
Tor Vergata Roma, November 2010 – p.19
ADE gauge groupsadmissable singularities of Y analysed by Kodaira ⇒ A-D-E classification
these are the singularities of Y whose resolution in fiber does not destroy
the Calabi-Yau property of Y
ord(∆) ord(f) ord(g) sing.n 0 0 An−1 ≃ SU(n)
n + 6 2 ≥ 3 Dn+4 ≃ SO(2n + 8)8 ≥ 3 4 E6
9 3 ≥ 5 E7
10 ≥ 4 5 E8
Physical interpretation:
gauge group of brane given by corresponding A-D-E group
Bershadsky et al. 1996
Note at this stage:
on general elliptic fibration mutually non-local monodromies are present
stacks of coincident (p,q) branes of different type → exceptional gauge
symmetriesTor Vergata Roma, November 2010 – p.20
ADE groups in F-theoryExample: SU(N) from N coincident branes of same (p,q) type
as N mutually local branes approach each other the fiber develops an
AN−1 singularity obtained by shrinking N P1s in fiber
YG: singular 4-fold T 2 → B3 with ADE group G along divisor S ⊂ B3
singularities best studied by resolution YG → Y G within M-theory
• paste in tree of P1s fibered over S ΓGi i = 1, . . . , rk(G)
singular YG ↔ zero size limit of ΓGi
• resolution divisors DGi ⇐⇒
fibration ΓGi → S
• Group theory of G
⇔ extended Dynkin diagram
Tor Vergata Roma, November 2010 – p.21
ADE groups in F-theory2 sources of gauge bosons along S from M-theory reduction
• off-diagonal elements in ad(G):
M2-branes along chains of P1 ΓGi ∪ . . . ∪ ΓG
j , i ≤ j
=⇒ massless only in singular limit
• Cartan U(1)rk(G) generators:
3-form C3 expanded in ωGi = [DG
i ] ∈ H2(Y G, Z)
C3 =
rk(G)∑
i=1
Ai ∧ ωGi + . . . Ai ↔ gauge field along S
Gauge flux of Cartan U(1): G4 =∑
i Fi ∧ ωGi , Fi ∈ H2(S)
Tor Vergata Roma, November 2010 – p.22
The quest for U(1)X Non-abelian ADE gauge symmetry built in geometrically
X Cartan U(1)s easy to study in Coulomb phase in M-theory
Extra U(1) gauge symmetries not tied to non-ab. groups
harder to detect
General fact from expansion C3 =∑rk(G)
i=1 Ai ∧ ωGi :
total rank of gauge group [Morrison,Vafa I+II ’96]
nv = h1,1(Y G) − h1,1(B) − 1
nU(1) = nv − rk(G)
↔ can be computed in global models with resolution of singularity
elliptic 3-folds: [Candelas,Font ’96] [Candelas,Perevalov,Rajesh ’97]
elliptic 4-folds: [Blumenhagen,Grimm,Jurke,TW 0908.1784][Grimm,Krause,TW 0912.3524]
Tor Vergata Roma, November 2010 – p.23
Gauge groups: IIB viewpointAppearance of exceptional gauge symmetries also in strongly coupled IIB
language well-known
• origin: coincident 7-branes of mutually non-local (p,q) type
• massless degrees of freedom from string junctions:
string bound-states with multiple ends
→ graphical representation of adjoint of e.g. exceptional gauge groups
Example:
A = (1, 0), B = (3,−1), C = (1,−1) [Sen ’96/97; Gaberdiel,Zwiebach’97]
SU(N) : AN , SO(2N) : ANBC, E8 : A5BC2
Tor Vergata Roma, November 2010 – p.24
F-Theory and IIBConnection to IIB orientifold on CY 3-fold X by Sen limit: [Sen ’96/97]
Weierstrass:
y2 = x3 − f(ui) x + g(ui), ∆ = 27g2 + 4f3
General parametrisation: f = −3h2 + ǫη, g = −2h3 + ǫhη − ǫ2
12χ
Generically: ∆ is single, connected I1 object
IIB limit: ǫ → 0 ⇒ ∆ = −9ǫ2h2(η2 − hχ) + O(ǫ3)
O7 : h = 0, D7 : η2 − hχ = 0
F-theory ↔ non-perturbative recombination of O-plane and 7-branes!
X : double cover of base B branched over h = 0
Simplest case: X given by equation h = ξ2, orientifold ξ → −ξ
Uplift: Reversal of Sen limit
↔ Define B = X/σ and consider Weierstrass model thereof
Tor Vergata Roma, November 2010 – p.25
7-brane tadpoleall ”extra”7-brane consistency conditions of probe IIB picture
automatically incorporated in geometry: Morrison, Vafa 1996
• Kodaira: first Chern class of tangent bundle of Y descends from
tangent bundle of B modulo singular divisor loci
c1(TY ) ≃ π∗(c1(TB) −∑iai
12δ(Γi)) ai = O(∆)|Γi
• Since N = 1 SUSY requires c1(TY ) = 0∑
i
aiδ(Γi) = 12 c1(TB)
B is no longer Calabi-Yau
• cf. 7-brane tadpole:∑
i Niδ(Γi) = 4δ(O7)
→ curvature of base B has absorbed orientifold charge
elliptic Calabi-Yau fourfold ↔ consistent, non-pert. 7-brane configuration
Example: 7-branes on S2: KS = O(−2) → need 24 branesTor Vergata Roma, November 2010 – p.26
Consistency: D3 tadpoleWhile D7-tadpole automatic in F-theory, D3-tadpole ensured extra:
Consider dual M-theory on R(1,2) × Y w/ M2 branes along R(1,2) × pointi
integrate e.o.m. for G4 field strength
0 =
∫
d ∗ G4 =
∫(
1
2G4 ∧ G4 − ℓ6MI8(R) + ℓ6M
∑
i
δ(M(i)2 )
)
Key:∫
YI8(R) = χ(Y )
24 χ(Y ) =∫
Yc4(TY ): Euler characteristic
χ24 = 1
2ℓ6M
∫G4 ∧ G4 + NM2
upon M/F-theory duality:
•χ24 : curvature dependent D3-charge of O7-planes and 7-branes
• NM2 → ND3
•1
2ℓ6M
∫G4 ∧ G4: bulk and brane flux induced D3-charge
Tor Vergata Roma, November 2010 – p.27
Consistency: D3 tadpoleReduction of M-theory 3-form:
C3 = C3 + B2 ∧ L dx + C2 ∧ L dy + B1 ∧ L dx ∧ L dy
IIB: Cy4 = C3 ∧ dy, giy = (B1)i, B2: NSNS 2-form C2: RR 2-form
Similarly: IIB 3-form flux from: G4 = H3 ∧ L dx + F3 ∧ L dy
• IIB bulk flux: reduce G4 on 4-cycle which is product of 1-cycle in
fiber and non-trival three-cycle Σ3 on B
↔ H3/F3 along Σ3
• IIB gauge flux F2 along the 7-brane on divisor S: G4 = F2 ∧ ω
ω anti-selfdual 2-form dual to 2-cycle from fibering pinched S1
between 2 branes
Tor Vergata Roma, November 2010 – p.28
Consistency: D3 tadpoleIn absence of bulk flux:
χ
24= ND3 − 1
2ℓ4s
∫
S
F2 ∧ F2
︸ ︷︷ ︸
≥0 due to F2=−∗F2
Constraint: ND3 ≥ 0 to avoid uncontrolled SUSY breaking
Important: in presence of non-abelian gauge groups, Y is singular
χ(Y ) refers to Calabi-Yau obtained after resolving these singularities
special case: no non-abelian gauge groups ⇒ only I1 degenerations
↔ Euler character: χ∗(Y ) = 12∫
Bc1(B) c2(B) + 360
∫
Bc31(B)
[Sethi,Vafa,Witten ’96]
One can give a closed expression for certain more general configurations
required for model building [Blumenhagen,Grimm,Jurke,TW ’09]
Tor Vergata Roma, November 2010 – p.29
Overview
Part II:
Global Tate models and local spectral covers
• Higher co-dimension enhancements
[Beasley,Heckman,Vafa 0802.3391, 0806.0102]
• Global Tate model
• Spectral covers as a local description of fluxes
[Donagi,Wijnholt 0802.2969, 0904.1218] , [Hayashi,Tatar,Toda,Watari,Yamazaki
0805.1057] , [Hayashi,Kawano,Tatar,Watari 0901.4941]
• global validity of spectral cover?
[Blumenhagen,Grimm,Jurke,TW ’09]; [Grimm,Krause,TW ’09]
Tor Vergata Roma, November 2010 – p.30
Previously on this show....F-theory on Calabi-Yau fourfold Y4 → B3
• B3 ↔ backreacted compactification space of IIB orientifolds with
7-branes: B3 = X3/σ
• T2 fiber: varying dilaton
• ADE group G along divisor S ⊂ B3
Tor Vergata Roma, November 2010 – p.31
Brane intersectionsgauge symmetry on compl. codimension-1 locus = divisor Γa on B
Γa, Γb intersect along curve Cab = Γa ∩ Γb
IIB picture: bifundamental matter
F-theory: singularity enhanced on Cab
Picture: collision of zero-size P1 in fiber over Γa, Γb
Example:
Γa: AN−1 sing., Γb: I1 sing. =⇒ Cab: AN sing.
along Cab: massless states from wrapped membranes fill adjSU(N+1)
extra states are localised as massless matter on Cab Katz, Vafa 1998
group theoretic decomposition:
SU(N + 1) → SU(N) × ”U(1)”
adjN+1 → (adjN )0 + 10 + (N)1 + (N)−1
Tor Vergata Roma, November 2010 – p.32
Matter at intersections• 6D hypermultiplet along Cab in ad(Gab) ⇐⇒ (b, χα), (b, χα)
• ad(Gab) → ad(Ga) ⊕ ad(Gb) ⊕⊕
i(Ria, Ri
b)
• Matter counted by cohomology groups of 6D gauge bundle
(bi, χiα) : H0(Cab, K
12
C ⊗ VRia⊗ VRi
b),
(bi, χi
α) : H1(Cab, K12
C ⊗ VRia⊗ VRi
b)
Chiral spectrum requires gauge flux Fa, Fb
(depending on embedding this will further decompose various reps.)
χ(Ria ⊗ Ri
b) =∫
Cab
(
c1(VRia) + c1(VRi
b))
Tor Vergata Roma, November 2010 – p.33
Yukawas
at intersection of two or three matter
curves: further rank enhancement
pic from: Beasley,Heckman,Vafa 0806.0102
e.g. Γ5 : SU(5), Γa : U(1)a, Γb : U(1)b
51a,0 + c.c., 50,1b+ c.c., 11a,−1b
+ c.c.
all states incorporated at point of SU(7) enhancement
there: Yukawa couplings 〈51a,0 50,−1b1−1a,1b
〉
• Yukawas descend from cubic Yang-Mills interaction for enhanced
gauge group upon decomposition of adjoint
• Holomorphic piece given by overlap of wave functions at intersection
point
Tor Vergata Roma, November 2010 – p.34
Summary so farGauge dynamics described by singular fibers in ell. fibration Y4 → B3:
• ADE gauge group G along
4-cycle S ⊂ B3
• intersection Sa ∩ Sb along 2-cycle Cab:
extra massless matter
enhancement to Gab:
ad(Gab) → ad(Ga) + ad(Gb)
+∑
i
(Ui, Ri)
• intersection of three matter curves:
Yukawa couplings
Tor Vergata Roma, November 2010 – p.35
Tate modelRecall: elliptic fibration Y4 : T 2 → B3 given by Weierstrass model
P : y2 − x3 − f(ui) x z4 − g(ui) z6 = 0 (x, y, z) ≃ (λ2x, λ3y, λz)
f, g vary over base B3 with coordinates ui
• x, y, f, g are sections of a line bundle L on the basis
non-triviality of L responsible for non-trivial fibration
• homogeneity of P : x ↔ L2, y ↔ L3, f ↔ L4, g ↔ L6
• discriminant: ∆ = 4f3 + 27g2 ↔ L12
L determined by Calabi-Yau condition for Y4:∑
i aiδ(Γi) = 12 c1(TB3)
δ(Γi) vanishing locus of ∆ with multiplicity ai∑
i aiδ(Γi) is in same class as ∆ (because it gives full degenerate locus)
=⇒ L = K−1B3
f ↔ K−4B3
g ↔ K−6B3
Tor Vergata Roma, November 2010 – p.36
Tate modelMore convenient: Tate form
PW = x3 − y2 + x y z a1 + x2 z2 a2 + y z3 a3 + x z4 a4 + z6 a6 = 0
• local equivalence w/ Weierstrass form by completing square and cube
β2 = a21 + 4a2, β4 = a1a3 + 2 a4, β6 = a2
3 + 4a6
f = − 1
48(β2
2 − 24 β4), g = − 1
864(−β3
2 + 36β2b4 − 216 β6)
⇒ ∆ = − 14β2
2(β2β6 − β24) − 8β3
4 − 27β26 + 9β2β4β6
• precise characterisation of singularities and resulting gauge groups in
terms of Tate algorithm [Bershadsky et al. 9605200]
• Note every Weierstrass model is globally of this form (branch cuts)
Special class of models admits global Tate form and an ∈ H0(B, K−nB )
• Global Tate models: underlying E8 singularity structure: [Grimm,TW ’10]
Start with G = E8 over S and deform into G = E8/H
Tor Vergata Roma, November 2010 – p.37
Tate algorithmsing. discr. group coefficient vanishing degrees
type deg(∆) enhancement a1 a2 a3 a4 a6
I0 0 — 0 0 0 0 0
I1 1 — 0 0 1 1 1
I2 2 SU(2) 0 0 1 1 2
I ns3 3 [unconv.] 0 0 2 2 3
I s3 3 [unconv.] 0 1 1 2 3
I ns2k
2k SP (2k) 0 0 k k 2k
I s2k
2k SU(2k) 0 1 k k 2k
I ns2k+1 2k + 1 [unconv.] 0 0 k + 1 k + 1 2k + 1
I s2k+1 2k + 1 SU(2k + 1) 0 1 k k + 1 2k + 1
II 2 — 1 1 1 1 1
III 3 SU(2) 1 1 1 1 2
IV ns 4 [unconv.] 1 1 1 2 2
IV s 4 SU(3) 1 1 1 2 3
I∗ ns0 6 G2 1 1 2 2 3
stolen from: [Bershadsky,Intriligator,Kachru,Morrison,Sadov,Vafa 9605200]
Tor Vergata Roma, November 2010 – p.38
Tate algorithmsing. discr. group coefficient vanishing degrees
type deg(∆) enhancement a1 a2 a3 a4 a6
I∗ ss0 6 SO(7) 1 1 2 2 4
I∗ s0 6 SO(8)∗ 1 1 2 2 4
I∗ ns1 7 SO(9) 1 1 2 3 4
I∗ s1 7 SO(10) 1 1 2 3 5
I∗ ns2 8 SO(11) 1 1 3 3 5
I∗ s2 8 SO(12)∗ 1 1 3 3 5
I∗ ns2k−3 2k + 3 SO(4k + 1) 1 1 k k + 1 2k
I∗ s2k−3 2k + 3 SO(4k + 2) 1 1 k k + 1 2k + 1
I∗ ns2k−2 2k + 4 SO(4k + 3) 1 1 k + 1 k + 1 2k + 1
I∗ s2k−2 2k + 4 SO(4k + 4)∗ 1 1 k + 1 k + 1 2k + 1
IV∗ ns 8 F4 1 2 2 3 4
IV∗ s 8 E6 1 2 2 3 5
III∗ 9 E7 1 2 3 3 5
II∗ 10 E8 1 2 3 4 5
stolen from: [Bershadsky,Intriligator,Kachru,Morrison,Sadov,Vafa 9605200]
Tor Vergata Roma, November 2010 – p.39
SU(5) GUT from TateEngineering of concrete models by suitable choice of ai
Example of phenomenological relevance: SU(5) GUT gauge group along
divisor S : w = 0
split base coordinates ui = (w, yi) and read off from Tate:
a1 = b5w0, a2 = b4w, a3 = b3w
2, a4 = b2w3, a6 = b0w
5,
bi = bi(w, yi) but no overall factor of w
y2 = x3 + b0w5 + b2xw3 + b3yw2 + b4x
2w + b5xy
∆ = − w5︸︷︷︸
5[S]
(b45P + wb
25(8b4P + b5R) + w2(16b2
3b24 + b5Q) + O(w3)
)
︸ ︷︷ ︸
[D1]
,
P = b23b4 − b2b3b5 + b0b
25, R = 4b0b4b5 − b
33 − b
22b5
[∆] = 5[S] + [D1]
Tor Vergata Roma, November 2010 – p.40
SU(5) GUT from Tate
[∆] = 5[S] + [D1]
Generically: D1 does not factorise further → I1 locus
extra singular locus in addition to SU(5) unavoidable in global models
Tor Vergata Roma, November 2010 – p.41
SU(5) GUT from TateMatter curves are read off as follows:
• matter in 10: Enhancement SU(5) → SO(10):
45 → 24 + 1 + 10 + 10
Tate: ∆ ≃ w7, (a1, a2, a3, a4, a6) = (1, 1, 2, 3, 5)
→ P10 : w = 0 ∩ b5 = 0
Note: fits with orientifold picture, where 10 localises on O-plane
• matter in 5: Enhancement SU(5) → SU(6):
35 → 24 + 1 + 5 + 5
Tate: ∆ ≃ w6, a1 ≃ w0
P5 : w = 0 ∩ P = b23b4 − b2b3b5 + b0b
25 = 0
In phenomenological SU(5) GUT models:
10 ↔ (QL, ucR, ec
R)
5m ↔ (dcR, L) 5H ↔ (Tu, Hu), 5H ↔ (Td, Hd)
Tor Vergata Roma, November 2010 – p.42
SU(5) GUT from TateYukawa couplings from further enhancements at points:
• 〈1055〉 from enhancement SU(5) → SO(12):
〈1055〉 ⊂ 〈(66)3〉 of SO(12)
D6 point: w = 0 ∩ b5 = 0 ∩ b3 = 0
possible also perturbatively in IIB theory
• 〈10105〉 from enhancement SU(5) → E6:
〈10105〉 ⊂ 〈(78)3〉 of E6
E6 point: w = 0 ∩ b5 = 0 ∩ b4 = 0
=⇒ requires exceptional enhancements
• not possible perturbatively in IIB theory
genuine F-theory effect
• D-brane instantons do generate coupling in principle also in IIB
But different effect: E6 enhancement is not the F-theory version of
the IIB instanton effect!
Tor Vergata Roma, November 2010 – p.43
Local picture: ALE fibrationsStructure of matter curves and Yukawas for states charged under SU(5)
determined only by local neighbourhood of S within Y4
Locally can think of B3 as total space of normal bundle NS/B → S
(more precisely: canonical bundle - see later)
• view normal coordinate w as parametrizing one copy of C
• T 2 fiber combines locally with w into local K3 fibration over S
Globally this is not the full story since B3 is in general not fibered over S!
Locally replace Y4 : T 2 → B3 by K3 → S
Relation to Tate model (e.g. for SU(5)):
y2 = x3 + b0w5 + b2xw3 + b3yw2 + b4x
2w + b5xy, bi = bi(w, yi)
restriction to neighbourhood of S by truncation bi → bi = bi|w=0
y2 = x3 + b0w5 + b2xw3 + b3yw2 + b4x
2w + b5xy
Tor Vergata Roma, November 2010 – p.44
Local picture: ALE fibrations
Stable degeneration limit:
• view K3 = dP9 ∪ dP9 glued over
common T 2
• focus on one of the two dP9
• blow down: dP9 → dP8 pic stolen from Donagi/Wijnholt 0802.2969
dP8 = P2 with 8 points blown up to P1 Ei
=⇒ H2(dP8) = 〈l, Ei〉, i = 1, . . . 8 l2 = 1, Ei · Ej = −δij
Dynkin diagram of E8 ↔ l − E1 − E2 − E3, Ei − Ej
⇒ maximal singularity supported by dP8 lattice is E8
Only if model has heterotic dual is the 4-fold Y globally K3 fibered over S
Tor Vergata Roma, November 2010 – p.45
Local picture: ALE fibrationsdP8 picture makes manifest that ALE fiber over S contains small P1
intersecting like the roots in E8-Dynkin diagram:
H2(ALE, Z) = 〈αE8
I 〉
• all αE8
I zero size ⇒ maximal enhancement E8 over S
• non-zero volume for subset αHi with Dynkin diagram H ⊂ E8
⇒ gauge group G = E8/H on S
To specify gauge group G:
Analyse root system of commutant H = E8/G
Tor Vergata Roma, November 2010 – p.46
Local picture: ALE fibrationsField theoretic interpretation via 8D N = 1 SYM gauge theory on S
bosonic degrees of freedom in adjoint representation of E8 on S:
• gauge field A1,0
Cartan U(1) ⊂ E8 ↔ M-theory origin: C3 = A1,0I ∧ ωI (
∫
αIωJ = δIJ)
• Higgs field = complex scalar field
deformation modes of S
Cartan part ↔ M-theory origin: δΩ4,0 = Φ2,0I ∧ ωI
geometric Higgsing of E8 → G = E8/H by deformation of ALE fiber
↔ non-zero VEV to H-valued components of Φ
ALE fibrations: non-zero VEV only for Cartan U(1) part of Higgs field Φ
This (un)Higgsing can happen even in absence of VEV for gauge flux
Analogy in brane language: displacement of coincident branes
Tor Vergata Roma, November 2010 – p.47
Spectral coversIdea of spectral covers:
describe local system by eigenvalues µi of Φ
Example: G = SU(5) → H = SU(5)⊥ Φ ↔ ad(SU(5)⊥)
• µi are the roots i = 1, . . . , 5 of det(s 15 − Φ) = 0
• Since Φ varies over S, so do the µi
• This is a degree 5 equation:
0 = s5 + e1s4 + e2s
3 + e3s2 + e4s + e5 =
∏
i
(s − µi) (∗)
• en(µi) = Tr(Φn): symmetric polynomial of degree n
• In particular: e1 =∑
i µi = Tr(φ) ≡ 0 for H = SU(5)
• More generally: Higgs field might exhibit poles
multiply by b0 to remove these poles:
0 = b0s5 + b1s
4 + b2s3 + b3s
2 + b4s + b5, en = bn
b0
Tor Vergata Roma, November 2010 – p.48
Spectral coversGeometric interpretation of
0 = b0s5 + b1s
4 + b2s3 + b3s
2 + b4s + b5 = b0
∏
i
(s − µi) (∗)
• since Φ is valued in bundle KS → S: degree 5 equation in KS → S
• s = 0 is location of S in total space KS → S
• (∗) is a degree 5 cover of S in KS called the spectral cover C• µi are the 5 intersection points of C with fiber of KS
over each p in S we have 5 points in fiber over S: (p, µi) on C → S
Significance for model building:
The intersection of C and S determines the matter curves
Tor Vergata Roma, November 2010 – p.49
Spectral covers〈φ〉 makes massive components of 248 of E8 other than ad(G) in:
248 → (ad(G), 1) + (1, ad(H)) +∑
i(Ui, Ri)
origin: interaction [〈φ〉, δφ]2 ↔ mass terms of Ui given by weights λi(Ri)
Example : 248 7→ (24, 1) + (1,24) + [(10,5) + (5,10) + h.c.]
• Generically on S: only (24, 1) massless, all others massive
• mass for (10,5) ↔λi weights for fundamental 5 of H
• Relation to roots:
λ1 = α4, λ2 = α3 + α4, λ3 = α2 + α3 + α4,
λ4 = α1 + α2 + α3 + α4, λ5 = α−θ + α1 + α2 + α3 + α4
• if λi = 0 for some i = 1, . . . 5: extra P1 shrinks
→ extra massless 10i from (10,5) ⊂ E8
• location of full 10 matter curve on S:∏
i λi = 0
Tor Vergata Roma, November 2010 – p.50
Spectral coversRelation to Tate picture
y2 = x3 + b0w5 + b2xw3 + b3yw2 + b4x
2w + b5xy :
SO(10) enhancement along b5 = 0 ⇔ Identify b5 ≃∏i λi
Tate algorithm correctly reproduced if one identifies
bn ≃ b0en(λi) → µi ≡ λi,
In particular: b1 =∑
i λi ≡ 0 for SU(n)
Non-trivial check: 5 curve ↔ weights λi + λj for 10 of SU(5)⊥
Full 5 curve P5:∏
i<j(λi + λj) = 0
together with b1 = 0 ⇒ b23b4 − b2b3b5 + b0b
25 = 0 X
fundamental weights λi = eigenvalues µi of Higgs field
Note: For gauge groups other than SU(n) Casimirs en replaced by
corresponding group theoretic invariants
Tor Vergata Roma, November 2010 – p.51
Spectral covers: FluxActual power of spectral cover: description of gauge flux G
2 types of flux:
• F2 on S with values in G = SU(5) ↔ G4 ≃ F2 ∧ ωGi ,
F2 ∈ H(1,1)(S, Q) ⇒ breaks gauge group on S
• F2 on extra branes constituting I1 locus
leaves G on S unaffected and allows for chiral matter
Locally, the I1 locus is described by spectral cover C(5)
⇒ flux on I1 ↔ line bundle N on C(5): c1(N ) ∈ H(1,1)(C(5), Z)
(1,1)-form on C(5) ↔ G4 = F2 ∧ ωHi F2 ∈ H(1,1)(S, Z)
• Within X, fibration structure of C(5) → S with fiber f = µi implies:
H2(C(5), Z) = H2(S, H0(f))
• on corresponding ALE fibration: H0(f) → 〈ωHi 〉
Tor Vergata Roma, November 2010 – p.52
Spectral covers: FluxWhy is I1 flux described as G4 = F2 ∧ ωH
i with F2 ∈ H(1,1)(S, Z)?
• Spectral cover is local ”first order”description of geometry of S and
the neighbouring I1 locus
• 2 ways to break original E8 symmetry:
by gauge flux and by scalar deformations
• E8 symmetry ↔ all branes along same locus S
spectral cover C(5) = product of tilting a suitable number of copies of
branes on S
locally this is the I1 locus, i.e. the remaining branes in the system
• flux on I1 at same order of deformation as gauge flux in representation
of H, but with 2 legs on S
2 legs along I1 and in H would be ”second”order
at least for all purposes of chiral matter on curves entirely S correct
description of I1-flux
Tor Vergata Roma, November 2010 – p.53
Spectral coversExample G = SU(5):
for generic spectral cover H = SU(5)
more generally:
H can contain abelian factors
• C(5) splits into 2 or more connected components
• In Tate model: I1 splits - at least in neighbourhood of S!
Example:
H = S[U(4) × U(1)] ↔ split spectral cover C5 = C4 ∪ C1
Will become relevant for model building:
1) More flexibility for flux solutions
2) resulting U(1) may forbid unwanted couplings
Tor Vergata Roma, November 2010 – p.54
Global aspectsHow exact is the spectral cover construction (SCC)?
• A priori designed to capture only local neighbourhood of S within B
• based on a local P1 fibration over S: expect correct description of
matter curves and Yukawas on S
To some extent, SCC captures also global information
• For models with heterotic dual, SCC allows us to compute χ24 exactly
Method: Comparison of D3-tad with M5-tad in heterotic M-theory
• This formula turns out correct for global Tate models also in cases
without heterotic dual!
Blumenhagen,Jurke,Grimm,T.W.; Grimm,Krause,T.W. ’09
However: SCC is blind to physics away from non-abelian brane locus!
In particular: Insensitive to global existence of massive or massless U(1)
gauge groups
Tor Vergata Roma, November 2010 – p.55
Appendix: χ from SCCIngredients of heterotic E8 × E8 on elliptic Z:
• VEV to internal curvature Fij in subgroup H1 × H2 ⊂ E8 × E8
↔ holomorphic vector bundle V1 × V2 with structure group H1 × H2
gauge group: E8 × E8 → G1 × G2, Gi = E8/Hi
3-form H3 = dB2 + ωCS(F ) − ωCS(F ): dH3 = α′
4 (trF 2 − trR2)
• M5 branes along 2-cycle γ with dual 4-form Γ
Tadpole condition:∑
i Ni[Γi] = c2(Z) + ch2(V1) + ch2(V2)
Special case: M5 wraps T 2 fiber of Z → Γ supported on B2:
→ NM5 =
∫
B2
c2(Z) + ch2(V1) + ch2(V2)
• c2(Z) = 12σc1(B2) + 11c21(B2) + c2(B2)
• ch2(Vi) depends on concrete bundle
Tor Vergata Roma, November 2010 – p.56
Appendix: χ from SCCOn elliptic CY Z a class of such bundles can be described via SCC
For structure group Hi = SU(Ni):
• [Ci] = niσ + π∗(ηi), line bundle N•
∫
B2c2(V1) =
∫
B2η1σ − 1
24χSU(n) − 12
∫
B2πn∗(γ
2)
χSU(n) =∫
Sc21(S)(n3 − n) + 3n η
(η − nc1(S)
)↔ info of [Ci]
∫
B2πn∗(γ
2) ↔ information of line bundle c1(N )
• ⇒ η(1) = 6c1(B2) − t η(2) = 6c1(B2) + t from σ part in N5
Special case: E(2)8 broken completely via H2 = E8
χ(2)E8
= 120∫
S
(3η2
(2) − 27ηc1(S) + 62c21(S)
) ∫
B2πn∗(γ
2) = 0
N5 =
∫
B2
(11c2
1(B2) + c2(B2))
+1
24
(
χSU(1)(n) + χE
(2)8
)
+1
2
∫
B2
πn∗(γ2)
Tor Vergata Roma, November 2010 – p.57
Appendix: χ from SCCDual F-theory model:
• gauge group G1 along S t = c1(NS/B)
• M5-brane along T 2 −→ D3 brane at point on B
• N5 = N3 = χ(Y )24 − 1
2
∫
YG ∧ G
χ(Y )24 =
∫
S
(11c2
1(B2) + c2(B2))
+ 124
(
χSU(1)(n) + χE
(2)8
)
Important observation:
for H1 = E8 no non-ab. gauge group along S
→ Y is smooth and χ(Y )∗ = 360∫
Bc31(B) + 12
∫
Bc1(B) c2(B)
Eliminate∫
S
(11c2
1(B2) + c2(B2))
and find:
χ(Y) = χ∗(Y) + χSU(1)(n) − χE
(1)8
This formula holds true even in absence of heterotic dual!
Tor Vergata Roma, November 2010 – p.58
Overview
Part III:
F-theory GUT phenomenology
• SU(5) GUTS: basics
• GUT symmetry breaking
• Proton decay
• U(1) restricted Tate model
• Summary
Tor Vergata Roma, November 2010 – p.59
F-theory GUTs
Aim: construction of 4D models with realis-
tic particle phenomenology
Guideline: unification of gauge couplings at
MX ≃ 1016GeV
Strategy: Localisation of gauge sector to ex-
plain hierarchy MGUT = 10−3MPl.pic: Kazakov, 0012288
• S10D = M8∗
∫
R1,3×B3
√−gR → M2Pl. = M8
∗ Vol(B3) M∗ = ℓ−1s
• SY M = M4∗
∫
R1,3×SF 2 → α−1
GUT = M4∗ Vol(S)
• GUT breaking (see later) → M4GUT = Vol(S)−1
3 different scales Vol(S), Vol(B3), M∗ to achieve both
• MPl. = 103MGUT = 1019GeV
• α−1GUT = 24
Tor Vergata Roma, November 2010 – p.60
F-theory GUTsAchieving the hierarchy:
• In principle enough to ensure mild tuning ℓs︸︷︷︸
0.2x
< RS︸︷︷︸
2.2x
< RB3︸︷︷︸
5.6x
x = 10−16GeV−1
• Eventually this has to happen dynamically!
• Stronger requirement: decoupling MPl. → ∞ while MGUT finite
While physically not absolutely necessary, appears as a sensible
organising principle of GUT model building BHV & DW ’08
• Mathematical theorem:
decoupling forces S to be Fano∫
ΓK−1
S > 0 ∀ 2-cycles Γ
• 2 kompl.-dim. Fanos: P1 × P1, P2, del Pezzo dPr, r = 1, . . . , 8
→ severely constrains possible GUT surfaces
• for del Pezzo H(0,2)(S) = 0 = H(0,1)(S)
⇒ no deformation modes or continuous Wilson lines
Welcome feature: avoiding adjoint exotics would require stabilisationTor Vergata Roma, November 2010 – p.61
F-theory GUTsdel Pezzo convenient choice, but - unfortunately - there is no
general prediction that S must be of that type
Still: Assume now that S is del Pezzo
Tor Vergata Roma, November 2010 – p.62
F-theory GUTs• Simplest realisation: SU(5) GUTs BHV & DW ’08
• SO(10) possible, but exotics arise in process of GUT breaking
Aim: SU(5) GUT group along divisor S ⊂ B3 given by locus w = 0
→ y2 = x3 + f(ui, w)xz4 + g(ui, w)z6 with f(ui, w), g(ui, w) s.t.
• ∆ = w5P (ui, w),
• P (ui, w) 6= w p(ui, w)
=⇒ SU(5)︸ ︷︷ ︸
S:w=0
× U(1)︸ ︷︷ ︸
D:P (ui,w)=0
a) matter:
enhancement of singularity type by rank one on intersection locus S ∩ D
• SU(5) × U(1) → SU(6)
35 → 24 + 1 + 5 + 5 =⇒ 5m = (dcR, L) or 5H + 5H
• SU(5) × U(1) → SO(10)
45 → 24 + 1 + 10 + 10 =⇒ 10 = (QL, ucR, ec
R)
N cR: any SU(5) singlet with suitable couplings
Tor Vergata Roma, November 2010 – p.63
F-theory GUTsb) Yukawa couplings
further enhancements at mutual intersections of curves
• 105m 5H for masses of down quarks
• 10105H for masses of up quarks
• 5H 5m 1Nc
Rfor Dirac neutrino masses
Group theory:
• 〈1055〉 ⊂ 〈(66)3〉 of SO(12) =⇒ enhancement SU(5) → SO(12)
possible also perturbativey in IIB theory
• 〈10105〉 ⊂ 〈(78)3〉 of E6 =⇒ requires exceptional enhancements
not possible perturbatively in IIB theory
(however: D-brane instantons can do it
genuine F-theory effect
• 〈551〉: SU(7) point sufficient
Tor Vergata Roma, November 2010 – p.64
SU(5) GUT breaking3 ways to break SU(5) → SU(3) × SU(2) × U(1)Y
• via GUT Higgs in adjoint 24 of SU(5) ↔ brane moduli H(0,2)(S)
problem: explicit GUT breaking potential from string theory ↔stabilisation of brane moduli?
Note: Fluxes do stabilise moduli, but hard to arrange suitable potential
• Wilson lines ↔ brane moduli H(0,1)(S)
again challenge of stabilising open moduli:
now fluxes not even avaliable
viable special case: discrete Wilson lines from discrete π1(S)
• by line bundle LY ↔ hypercharge U(1)Y
works for divisors with H(0,1)(S) = 0 = H(0,2)(S)
↔ open moduli problem avoided → Xdel Pezzo S
We take U(1)Y - approach:
first applied in F-theory context by [Beasley,Heckman,Vafa ’08]
same as in heterotic: [Blumenhagen,Honecker,TW ’06]
Tor Vergata Roma, November 2010 – p.65
SU(5) GUT breaking (II)Idea: VEV to FY on S with TY = diag(−2,−2,−2, 3, 3) ⊂ SU(5)
SU(5) −→ SU(3) × SU(2) × U(1)Y
24 → (8,1)0Y+ (1,3)0Y
+ (1,1)0Y+ (3,2)5Y
+ (3,2)−5Y
5 → (3,1)2Y+ (1,2)−3Y
10 → (3,2)1Y+ (3,1)−4Y
+ (1,1)6Y,
5H → (3,1)−2Y+ (1,2)3Y
, 5H → (3,1)2Y+ (1,2)−3Y
3 challenges:
1) U(1)Y must remain massless X
2) no exotic states (3,2)5Y+ (3,2)−5Y
from 24 X
3) preservation of unification ???
Tor Vergata Roma, November 2010 – p.66
SU(5) GUT breaking (III)Massless U(1)Y
Generically an abelian gauge factor acquires mass via Stuckelberg
mechanism:
C(2) ∧ F -type coupling in 4DU(1)c
Analyse in IIB 7-brane language:
SCS =∫
IR1,3×S
(ι∗C8 + trF ∧ ι∗C6 + tr(F 2) ∧ ι∗C4 + . . .
)
Expand: F = TY
(
F 4DY + c1(LY )
)
+ (other SU(5) generators)
2 potential sources for mixing term:
• ι∗C6 =∑
a C(a)2 ∧ ι∗αa, αa ∈ H4(Y )
• ι∗C4 =∑
i C(i)2 ∧ ι∗ωi, ωi ∈ H2(Y )
since trTY = 0: only term involving tr(F 2) relevant
⇒ F 4DY ∧ C
(i)2 trT 2
Y
∫
S(c1(LY ) ∧ ι∗ωi)
⇒ U(1)Y massless iff c1(LY ) orthogonal to ι∗H2(B, Z)
Tor Vergata Roma, November 2010 – p.67
SU(5) GUT breaking (IV)Important concept: gauge flux in relative cohomology of S ⊂ B3
gauge flux on divisor S ⇔ H2(S)
2 types of 2-forms on divisor S: Lerche, Mayr, Warner ’01/02;
pullbacks from B vs. 2-forms not inherited from B3 Jockers,Louis’05
The latter 2-forms are dual to 2-cycles in H2(S) which are a boundary of a
3-chain in B
If c1(LY ) only through such ”relative”2-cycles on S, then∫
S(LY ∧ ι∗ωi) = 0
Constraint: hypercharge flux must lie in relative cohomology of S in B3
Non-trivial constraint on compactification geometry:
rigid GUT divisor must allow for such gauge flux which is not pull-back
from ambient space
global feature - depends on compactification details, not on local data!
Tor Vergata Roma, November 2010 – p.68
MSSM matterNecessary condition: Recall: l2 = 1, Ei · Ej = −δij
if S is del Pezzo, then c1(S) = 3l −∑i Ei inherited from B3 →c1(LY ) · c1(S) = 0 ↔ c1(S) in Weyl lattice 〈l − E1 − E2 − E3, Ei − Ej〉Absence of exotics
24 → (8,1)0Y+ (1,3)0Y
+ (1,1)0Y+ (3,2)5Y
+ (3,2)−5Y
• localised on entire divisor S
• counted by combinations of Hi(S,L5Y ) = 0
for c1(LY ) in Weyl lattice of dPr, this is not possible:
Hi(Ei − Ei) = 0, but not for 5th power!
But can redefine embedding of LY such that Hi(S,L±1Y ) appears
In SCC we also have the SU(5)⊥ ⊂ E8 bundle V on S
Define S[U(5) × U(1)Y ] bundle V ⊕ LY : c1(V ) + c1(LY ) = 0
Cartan generators of the structure group of V ⊕ LY :
(1, 1, 1, 1, 1︸ ︷︷ ︸
V
, −5︸︷︷︸
LY
) ⊂ SU(6) ⊂ E8
Tor Vergata Roma, November 2010 – p.69
MSSM matter
248 7→
(24;1,1)0 + (1;1,1)0 + (1;8,1)0 + (1;1,3)0
(5;3,2)1 + (1;3,2)5 + c.c.
(10;3,1)2 + (5;3,1)−4 + c.c.
(10;1,2)−3 + (5;1,1)6 + c.c.
SU(3) × SU(2) × U(1)Y bundle Standard Model particles
(3,2)1 V qL L-handed quark
(3,2)5 L−1Y − (exotic matter)
(3,1)2∧2
V dL = dcR L-handed down antiquark
(3,1)−4 V ⊗ LY uL = ucR L-handed up antiquark
(1,2)−3
∧2V ⊗ LY lL L-handed lepton
(1,1)6 V ⊗ L−1Y eL = ec
R L-handed antielectronTor Vergata Roma, November 2010 – p.70
MSSM matter - Higgsappearance of full GUT matter multiplets:
c1(LY ) · P10 = 0 = c1(LY ) · P5m
Higgs sector:
5H → (3,1)−2Y+ (1,2)3Y
, 5H → (3,1)2Y+ (1,2)−3Y
controlled mechanism for stringy doublet-triplet splitting via U(1)Y
[Beasley,Heckman,Vafa ’08]
PH splits into Pu ∪ Pd such that 5H along Pu, 5H along Pd
net U(1)Y flux through Pu and Pd such that:
H∗(Pu,∧V ⊗ LY ) = (1, 0), H∗(Pd,∧V ⊗ LY ) = (0, 1)
↔ one Hu and Hd
H∗(Pu,∧V ) = (0, 0), H∗(Pd,∧V ) = (0, 0)
↔ massless triplets projected out
topological solution to doublet triplet splitting
Tor Vergata Roma, November 2010 – p.71
Proton DecayConventional SU(5) GUTs suffer from too large proton decay
Dimension 4:
• R-parity violating terms ucR dc
R dcR , L L ec
R, Q L dcR ↔ 105m 5m
• must be absent due to experimental bound
• Necessary condition: Pm, PH on different curves
otherwise: 105m 5H implies 105m 5m
• Note: This is not sufficient - see later
Dimension 6:
• From exchange of heavy gauge bosons in (3,2) + (3,2)
• leads to decay of type p → π0 + e+
• decay rate is well within experimental bounds → no problem X
Tor Vergata Roma, November 2010 – p.72
Proton decayDimension 5:
• effective terms of type c2
Meff(QQQL + uc
RucRdc
RecR)
• lead to decay of type p → K+ν
• strongly constrained by proton lifetime
• Generation via triplet exchange: 5H = (Tu, Hu),5H = (Td, Hd):
QQTu + QLTd + MKKTuTd → 1MKK
Q Q Q L
• realised if triplets acquire mass by pairing up with each other
↔ Hu, Hd on same curve PH
• Remedy: Missing partner mechanism:
QQTu + QLTd + MKKTuTd + MKK TuTd
=⇒ no integration to QQQL possible
• Necessary: need Hu and Hd on two separate curves
• Note: In this case there is also no O(1) µ term µHuHd X
Tor Vergata Roma, November 2010 – p.73
U(1) selection rules• Generic Tate model: single 5-matter curve
=⇒ 105m 5H ↔ 105m 5m: dimension 4 proton decay
• Remedy: split P5 → Pm + PH + effective U(1) selection rule
[Beasley,Heckman,Vafa II ’08] [Hayashi,Kawano,Tatar,Watari 0901.4941]
Group theory: E8 → G × H, G = SU(5)GUT
• H = SU(5)⊥ → S[U(4) × U(1)X ]
• U(1)X charges: 101 (5m)−3 (5H)−2 + (5H)2 1−5
• If U(1)X unbroken in fully fledged model, then 105m 5m forbidden X
• Necessary condition for U(1)X : split P5 → Pm + PH
achieved by split spectral cover → realises S[U(4) × U(1)X ] ”locally”
[Marsano,Saulina,Schafer-Nameki 0906.4672][Tatar,Tsuchiya,Watari 0905.2289]
Tor Vergata Roma, November 2010 – p.75
Global caveatsApplied to construction of 3-generation models:
[Marsano,Saulina,Schafer-Nameki 0906.4672][Blumenhagen,Grimm,Jurke,TW 0908.1784]
[Marsano,Saulina,Schafer-Nameki 0912.0272] [Grimm,Krause,TW 0912.3524]
[Chen,Knapp,Kreuzer,Mayrhofer 1005.5735]
Caveat:
• U(1)X might be higgsed by GUT singlets Φ
[Tatar,Tsuchiya,Watari 0905.2289],[Grimm, TW 1006.0226]
• happens away from GUT brane ⇐⇒ beyond regime of spectral cover
• This is a global question of direct physical relevance:
If U(1)X higgsed, effective proton decay operators generated
W ⊃ 1
M105m 5m Φ =⇒ 〈Φ〉
M105m 5m
Independent analysis via monodromies: [Hayashi,Kawano,Tsuchiya,Watari
1004.3870]
Tor Vergata Roma, November 2010 – p.76
U(1) restricted Tate modelConstructive method to ensure presence of abelian gauge symmetries:
[Grimm, TW 1006.0226]
• In generic models U(1) absent due to maximal Higgsing compatible
with non-abelian gauge symmetries ↔ VEV for gauge singlets 〈1−5〉• U(1) symmetries recovered by unhiggsing
→ massless gauge singlets away from GUT brane
• requires singular matter curves away
from GUT brane
• self-intersection of I1 locus
enhancement I1 → A1 ≃ SU(2)
• further specification of complex
structure:
U(1) restricted Tate model
Tor Vergata Roma, November 2010 – p.77
U(1) restricted Tate modelSafe way to check for U(1)X : [Grimm,TW 1006.0226]
Resolve space YG → Y G and count h1,1(Y G)
Result: nU(1) = h1,1(Y G) − h1,1(B) − 1 − rk(G) = 1
Reason: Resolution divisor DC for singular curve C ↔ dual 2-form ωC
C3 = AX ∧ ωX +∑
i
Ai ∧ ωGi + . . . ωX ↔ ωC
• presence of U(1) does not hinge on any factorisation of the
discriminant
• Compatible with appearance of U(1) symmetries in IIB limit
F-theory lift does not destroy U(1) - only affects split
O7-plane/D7-brane intersection
• U(1) are non-generic both in IIB and in F-theory
Tor Vergata Roma, November 2010 – p.78
U(1) restricted Tate modelPractical consequence of U(1) restriction: χ(Y ) decreases drastically
GUT example of [Grimm,Krause,TW 0912.3524]
Generic SU(5) Tate model: χ = 5718 ↔ U(1)-restricted model: χ = 2556
What about U(1)X flux?
• bundle is now of type S[U(4) × U(1)X ]
• global description in terms of G4 complicated
tempting: G = FX ∧ ωX + . . . ? details still under investigation
• U(1)X flux =⇒ Fayet-Iliopoulos D-term
↔ U(1)X acquires Stuckelberg mass in presence of suitable gauge flux
=⇒ global selection rule ↔ instantons → work in progress
Tor Vergata Roma, November 2010 – p.79
Summary of approachCharacteristic features of F-theory model building:
• localisation of gauge d.o.f.
• emergence of exceptional gauge groups
=⇒ good for GUTs
We have seen interesting mechanisms to accomodate:
• all required Yukawas - in principle X
• GUT breaking without GUT Higgs X
• suppress proton decay X
• achieve doublet-triplet splitting X
Many further phenomenological directions have been explored, inlcuding:
• hierarchy in flavour sector/structure of Yukawa couplings
• neutrino sectorsTor Vergata Roma, November 2010 – p.80
Summary of approachWhat about explicit constructions?
Many of the above realised in explicit compact models - thanks to
• spectral cover construction
• detailed understanding of Calabi-Yau fourfolds
Urgent directions:
• no stabilisation of moduli achieved
• no dynamical argument for specific choice of brane moduli
• coupling to, say, inflationary cosmology...
Tor Vergata Roma, November 2010 – p.81