Monodromic T-Branes And The $SO(10)_{GUT}$

0.14 MB·

by  Johar M. Ashfaque

7 Monodromic T-Branes And The SO(10)
1 GUT
0
2 Johar M. Ashfaque♠1

n
a
J

0 ♠Dept. of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, UK
2

] Abstract
h
t
- T-branes, which are non-Abelian bound states of branes, were first intro-
p
ducedbyCecotti, Cordova,HeckmanandVafa[1]. Theyaretherefinedversion
e
h of the monodromicbranesthat feature in the phenomenological F-theory mod-
[ els. Here, we will be interested in the T-brane corresponding to the Z mon-
3
1 odromy which is used to break the E gauge group to obtain the SO(10) .
8 GUT
v This extends the results of [1] to the case of Z monodromic T-branes used to
6 3
break the E gauge group to SO(10)×SU(3)×U(1) and compute the Yukawa
9 8
8 coupling with the help of the residue formula. We conclude that the Yukawa
5 coupling, 10 ·16 ·16 , is non-zero for E , in complete agreement with [1],
0 H M M 7
. but is zero for E8. Furthermore, the case of Z2 monodromic T-branes used to
1
break the E gauge group to E ×SU(2)×U(1), nothing interesting can bede-
0 8 6
7 duced by evaluating the Yukawa coupling 27H·27M ·27M which is dependent
1
on whether the MSSM fermion and electroweak Higgs fields can be included in
:
v the same 27 multiplet of a three-family E GUT or assign the Higgs fields to a
6
Xi different 27H multiplet where only the Higgs doublets and singlets obtain the
r electroweak scale energy.
a
1
email address: [email protected]
Contents
1 Introduction 2
2 The Z Monodromy 3
2
2.1 Review of SU(2) Field . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 The Brane Recombination . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 E → SO(10)×SU(2)×U(1) . . . . . . . . . . . . . . . . . . . . . . 5
7
3 The Z Monodromy 6
3
3.1 Review of SU(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 The Brane Recombination . . . . . . . . . . . . . . . . . . . . . . . . 7
3.3 E → SO(10)×SU(3)×U(1) . . . . . . . . . . . . . . . . . . . . . . 8
8
4 Discussion And Conclusion 9
A The F-Theory Construct 11
A.1 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
6
A.2 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
7
A.3 The Gauge Enhancements . . . . . . . . . . . . . . . . . . . . . . . . 12
A.3.1 SO(10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
A.3.2 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
7
B The Hirzebruch Surfaces F 12
r
B.1 The Semistability Condition . . . . . . . . . . . . . . . . . . . . . . . 13
B.2 The Involution Conditions . . . . . . . . . . . . . . . . . . . . . . . . 13
B.3 The Effectiveness Condition . . . . . . . . . . . . . . . . . . . . . . . 13
B.4 The Commutant Condition . . . . . . . . . . . . . . . . . . . . . . . . 13
B.5 The Three Family Condition . . . . . . . . . . . . . . . . . . . . . . . 14
1 Introduction
To breakGUTsymmetries within F-theory[6–9] inorder toconstruct phenomenolog-
ically viable models, onecan either make use of Wilson lines or introduce gaugefluxes
with the key ingredient being the seven-brane which wraps the four-dimensional in-
ternal subspace of the six internal directions of the compactification providing for
each important element. The F-theory E was discussed in [2] whilst the primary
6
focus in [3], was the E gauge group obtained in this setting.2
7
Here, however, the focus is on T-branes, or “triangular branes,” which are novel
non-Abelian bound states of branes characterized by the condition that on some loci
the Higgs field is upper triangular and indeed it can be seen to be the case in [1]
2Refer to Appendix A for details.
2
where hΦi is upper triangular on some locus. This approach deals with the spectral
equation
P (z) = det(z −Φ) = 0
Φ
which when Φ belongs to the CSA is equivalent to stating that
(z −λ ) = 0
i
Yi
where λ are the eigenvalues of Φ and they denote the directions of the intersecting
i
branes. In the case of non-diagonalizable Higgs fields, the monodromy group is now
encoded in the form of the spectral equation. Such configurations are of particular
interest when systems of 7-branes are considered [10]. When considering such a
background profile for Φ, it has to be made sure that the following equations of
motion are satisfied which read
∂ Φ = 0
A
F(0,2)5 = 0
A
for the F-term equations and for the D-term equation we have that
i
ω ∧F + [Φ†,Φ] = 0.
A
2
In this paper, the aim is to compute the Yukawa coupling with the help of the
residue formula where the required interaction terms are of the form
10 ·16 ·16 .
H M M
The present work extends the results of [1] to the case of Z monodromic T-branes
3
used to break the E gauge group to SO(10)×SU(3) ×U(1) where the case of Z
8 2
monodromic T-branes used to break the E gauge group to SO(10)×SU(2)×U(1) is
7
reviewed beforehand. Wewill showthattheYukawa coupling, 10 ·16 ·16 , isnon-
H M M
zero for E but for E is zero. Moreover, the case of Z monodromic T-branes used to
7 8 2
break the E gauge group to E ×SU(2)×U(1), nothing interesting can be deduced
8 6
by evaluating the Yukawa coupling 27 ·27 ·27 which depends on whether the
H M M
MSSM fermion and electroweak Higgs fields can be included in the same 27 multiplet
of a three-family E GUT orassign the Higgsfields to a different 27 multiplet where
6 H
only the Higgs doublets and singlets obtain the electroweak scale energy. This work
can also seen as an extension to [3] in pursuit of obtaining SO(10) GUT symmetry
especially the Flipped SO(10) from various string theoretic constructions.
2 The Z Monodromy
2
2.1 Review of SU(2) Field
Let us consider the spectral equation for an SU(2) field along the lines of [1]:
P (z) = z2 −x
Φ
3
for which there is a Z monodromy. In the holomorphic gauge the Higgs field is
2
0 1
(cid:18) x 0 (cid:19)
which is an intermediate case between a diagonal background and a nilpotent Higgs
field
0 1
.
(cid:18) 0 0 (cid:19)
We promote this to the unitary gauge by use of a positive diagonal matrix having
unit determinant
ef1 0
(cid:18) 0 ef2 (cid:19)
with the constraint
f = 0
i
Xi
where f are real. The D-term equation
i
i
ω ∧F + [Φ†,Φ] = 0
A
2
is now replaced by the SU(2) Toda equation in two complex variables
∆f = C efj
i ij
where C is the Cartan matrix of SU(2).
ij
2.2 The Brane Recombination
Infinitesimal perturbations to the holomorphic Higgs field are considered of the form
ϕ = ad (ξ)+h
Φ
and then seen which can be gauged away to zero by the U(2) transformations by way
of deforming the theory via SU(2) Higgs VEV
0 1
Φ = .
(cid:18) x 0 (cid:19)
After gauge fixing the most general perturbation that can be made is given by
1α(x,y) 0
ϕ = 2 .
(cid:18) β(x,y) 1α(x,y) (cid:19)
2
4
With such a perturbation at hand, the spectral equation can seen to be deformed by
the SU(2) Higgs VEV as
2
1
P (z) = z2 −x → z − α(x,y) −(x+β(x,y) .
Φ
(cid:18)(cid:18) 2 (cid:19) (cid:19)
Now changing coordinates yields
z˜−(x˜α(x˜2,y˜)+β(x˜2,y˜)).
This is interpreted as the three D7-branes recombining into one.
Now starting with the flat Ka¨hler metric
i
ω = (dx∧dx+dy ∧dy +dz ∧dz)
2
changing to the new coordinates and noting that x = x˜2, we have
i
ω = 1+4|x˜|2 dx˜∧dx˜+dy˜∧dy˜
2
(cid:0)(cid:0) (cid:1) (cid:1)
and therefore the recombined D7-brane is indeed curved.
2.3 E → SO(10) × SU(2) × U(1)
7
Here SU(2)×U(1) Higgs field is used that preserves an unbroken SO(10):
0 1
Φ = ⊕(y).
(cid:18) x 0 (cid:19)
The adjoint of E decomposes under this breaking as
7
133 → (1,1)0 ⊕(1,3)0 ⊕(45,1)0 ⊕(10,1)2 ⊕(10,1)−2 ⊕(16,2)−1 ⊕(16,2)1
where the U(1) generator is in parenthesis. We note that the required interaction
terms are of the form
10 ·16 ·16 .
H M M
We can readily identify (10,1)2 as the 10H and (16,2)−1 as 16M with
ϕ
ϕ16 = 16+
M (cid:18) ϕ16− (cid:19)
where the corresponding matter curve is
f = y2−x.
5
Note that the one component of the spinor doublet, namely ϕ16+ is gauge equivalent
to zero.
The trace in the adjoint of e produces the following invariant tensors
7
Tr([t10,i,tM16,α]tN16,β) ∝ (CΓi)αβǫMN
which is generated group theoretically by contraction with a Γ matrix of the SO(10)
Clifford algebra, where α, β are spinor indices, i is a vector index, and C denotes the
standard charge conjugation matrix.
The Yukawa coupling can now be evaluated simply as
Tr([η10,η16]ϕ16)
W10·16·16 = Res(0,0)(cid:20) (y)(y2−x) (cid:21)
(CΓ ) ϕα ϕβ ϕi
= Res i αβ 16− 16− 10
(0,0)(cid:20) (x)(y) (cid:21)
where
1
η10 = ϕ10
H 2 H
and
ϕ
η16 = − 16− .
(cid:18) yϕ16− (cid:19)
This coupling requires a single field, ϕ16−, to participatetwice inthetrilinear Yukawa
coupling and is known to give mass to exactly one generation of SM matter 16’s.
3 The Z Monodromy
3
3.1 Review of SU(3)
We follow [1,4] to outline some of the key ideas to paint the picture. Let us begin by
considering the spectral equation for an SU(3) field:
P (z) = z3 −x
Φ
for which there is a Z monodromy.
3
In the holomorphic gauge the Higgs field is
0 1 0
 0 0 1 
x 0 0
 
which is an intermediate case between a diagonal background and a nilpotent Higgs
field
0 1 0
 0 0 1 .
0 0 0
 
6
We promote this to the unitary gauge by use of a positive diagonal matrix having
unit determinant
ef1 0 0
 0 ef2 0 
0 0 ef3
 
with the constraint
f = 0
i
Xi
where f are real. The D-term equation
i
i
ω ∧F + [Φ†,Φ] = 0
A
2
is now replaced by the SU(3) Toda equation in two complex variables
∆f = C efj
i ij
where C is the Cartan matrix of SU(3) which is given by
ij
2 −1
.
(cid:18) −1 2 (cid:19)
The components for the unitary transformation for the nilpotent Higgs field Φ satisfy
∂∂f = 2ef1 −ef2,
1
∂∂f = −ef1 +2ef2.
2
3.2 The Brane Recombination
Infinitesimal perturbations to the holomorphic Higgs field are considered of the form
ϕ = ad (ξ)+h
Φ
and then seen which can be gauged away to zero by the U(3) transformations by way
of deforming the theory via SU(3) Higgs VEV
0 1 0
Φ =  0 0 1 .
x 0 0
 
After gauge fixing, as was shown in [4], the most general perturbation that can be
made is given by
1α(x,y) 0 0
3
ϕ =  0 1α(x,y) 0 .
3
γ(x,y) β(x,y) 1α(x,y)
 3 
7
With such a perturbation at hand, the spectral equation can seen to be deformed by
the SU(3) Higgs VEV as
2
1 1
P (z) = z3 −x → z − α(x,y) z − α(x,y) −β(x,y) −(x+γ(x,y))
Φ
(cid:18) 3 (cid:19)(cid:18)(cid:18) 3 (cid:19) (cid:19)
which to first order reads
z3 −z2α(x,y)−zβ(x,y)−x−γ(x,y).
Now changing coordinates to
(x˜,y˜,z˜) = (z,y,P (z))
Φ
yields
z˜−(x˜2α(x˜3,y˜)+x˜β(x˜3,y˜)+γ(x˜3,y˜)).
This is interpreted as the three D7-branes recombining into one.
Now starting with the flat Ka¨hler metric
i
ω = (dx∧dx+dy ∧dy +dz ∧dz)
2
changing to the new coordinates and noting that x = x˜3, we have
i
ω = 1+9|x˜|4 dx˜∧dx˜+dy˜∧dy˜
2
(cid:0)(cid:0) (cid:1) (cid:1)
and therefore the recombined D7-brane is indeed curved.
3.3 E → SO(10) × SU(3) × U(1)
8
Here SU(3)×U(1) Higgs field is used that preserves an unbroken SO(10):
0 1 0
Φ =  0 0 1 ⊕(y).
x 0 0
 
The adjoint of E , following [5], decomposes as
8
248 → (1,1)0 ⊕(1,8)0 ⊕(45,1)0 ⊕(1,3)−4 ⊕(1,3)4 ⊕(10,3)2 ⊕(10,3)−2 ⊕
(16,1)3 ⊕(16,1)−3 ⊕(16,3)−1 ⊕(16,3)1
where the U(1) generator is in parenthesis. We note that the required interaction
terms are of the form
10 ·16 ·16 .
H M M
8
We can readily identify (10,3)2 as the 10H and (16,3)−1 as 16M. The 16M is in
the fundamental of the SU(3)
ϕ1
16
M
ϕ16 =  ϕ216 .
M M
ϕ3
16
 M 
The torsion equation can be solved using the adjugate matrix
4y2 x −2xy ϕ1 4y2ϕ1
16 16
M M
η16 =  −2y 4y2 x  0  =  −2yϕ116 .
M M
1 −2y 4y2 0 ϕ1
16
    M 
A
| {z }
The 10 transforms in the fundamental of SU(3) as well and as a result replacing y
H
with -2y the solution of the torsion equation is found to be
16y2 x 4xy ϕ1 16y2ϕ1
10 10
H H
η10 =  4y 16y2 x  0  =  4yϕ110 .
H H
1 4y 16y2 0 ϕ1
10
    H 
B
| {z }
Using that the matter curve corresponding to A and B are
f = 8y3 +x, f = 64y3−x
A B
and again using the fact that the trace in the adjoint of e is
8
Tr([tA ,tB ]tC ) ∝ (CΓ ) ǫABC
10,i 16,α 16,β i αβ
allowing the Yukawa coupling to be evaluated simply as
W10·16·16 = 0
as was expected since it was noted in [4] that for n ≥ 2 and arbitrary a and b, matter
curves of the form
f = ayn +bx
will always yield zero for the computation of the trilinear Yukawa coupling.
4 Discussion And Conclusion
The low-energy string-derived model of [11] was constructed in the free fermionic for-
mulation [12] of the four-dimensional heterotic string in which the space-time vector
9
bosons are obtained solely from the untwisted sector and generate the observable and
hidden gauge symmetries:
3
observable : SO(6)×SO(4)× U(1)
i
Xi=1
hidden : SO(4)2×SO(8) .
where the E combination being
6
3
U(1) = U(1) ,
ζ i
Xi=1
which is anomaly free whereas the orthogonal combinations of U(1) are anoma-
1,2,3
lous.
Motivated by such string-derived low-energy effective models the Flipped SO(10)
was derived from the F-theory E as was discussed in [2] and investigated further
6
in [3], where the gauge group was E . Another possibility, that was explored in [3],
7
was that of nonperturbative heterotic vacua arising from the Horaˇva-Witten theory.3
The space of solutions of type A contains exactly one vacua over the Hirzebruch
surfaces for any allowed value of r for 6 ≤ s ≤ 24 and the corresponding appropriate
choice for λ with
s even, e−r even, λ = ±1,±3.
s e(r;λ)
6 3r +6+ 1 ∈ Z
λ
λ = ±1
whereas the space of solutions of Type B
1 3
r even, λ = ± ,± .
2 2
are given by
1 1
s = 6, e r;λ = ± = 3r +6+ = 3r +6±2.
(cid:18) 2(cid:19) λ
3See Appendix B for the rules which allow the construction of realistic, viable vacua with E6
GUT symmetry where the base manifold is taken to be the Hirzebruch surfaces.
10