Heterotic Wave Function Normalisation from Localization Andre Lukas University of Oxford “String Phenomenology 2017" , July 3 - 7, 2017, Virginia Tech, Blacksburg based on: 1512.05322, 1606.04032, 1607.03461, 1707.nnnn to appear in collaboration with: Stefan Blesneag, Evgeny Buchbinder, Andrei Constantin, Eran Palti
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Heterotic Wave Function Normalisation from Localization
Andre Lukas
University of Oxford
“String Phenomenology 2017" , July 3 - 7, 2017, Virginia Tech, Blacksburg
based on: 1512.05322, 1606.04032, 1607.03461, 1707.nnnn to appear
in collaboration with: Stefan Blesneag, Evgeny Buchbinder, Andrei Constantin, Eran Palti
• Introduction
Outline
• Wave function normalization from localisation
• An explicit example
• Conclusion
• Result for holomorphic Yukawa couplings
Introduction
Yukawa couplings in 4d supergravity:
K = Kmod
+GIJ
(T, T , S, S, Z, Z)CICJ
matter field Kahler metric
holomorphic Yukawa couplings
W = �IJK(Z)CICJCK
Metric needs to be diagonalized for physical Yukawa couplings GIJ
-> Both and are required for phenomenology GIJ�IJK
Yukawa couplings in heterotic CY models:
Model specified by a CY manifold and a bundle .X V ! X
Matter fields: C ! ⌫ 2 H1(X,V ) , harmonic
holomorphic Yukawa couplings given by:
�(⌫1, ⌫2, ⌫3) =
Z
X⌦ ^ ⌫1 ^ ⌫2 ^ ⌫3
invariant under ⌫i ! ⌫i + @↵i
Holomorphic Yukawa couplings are quasi-topological:Harmonic representatives and CY metric not required
Matter field Kahler metric given by:
G(⌫, ⇢) =1
V
Z
X⌫ ^ (?V ⇢)
not invariant under ⌫ ! ⌫ + @↵ , ⇢ ! +@�
Matter field Kahler metric is not quasi-topological.Its calculation requires the CY metric and the HYM connection.
So far, only known method to work out : numerical G
We would like to consider line bundles L = OA(k1, k2) and their restrictions L = L|X
= OX
(k1, k2). Thehermitian bundle metric is given by
H = �k11 �k2
2 (72)
Provided that k1 �2 and k2 > 0, which I assume in the following, the only non-zero cohomology of L isH1(A,L) ⇠= H1(X,L), so all relevant (0, 1)-forms are of type 1. Explicitly, ⌫ 2 H1(A,L) can be written as
⌫ = k11 P (z1, z↵)dz1 . (73)
Taking the flat limit on U of all the relevant quantities gives
J1 = i
2⇡dz1 ^ dz1 J2 = i
2⇡
P4↵=2 dz↵ ^ dz
↵
J = t1J1 + t2J2
H = e�k1|z1|2�k2P4
↵=2 |z↵|2 ⌫ = ek1|z1|2P (z1, z↵)dz2 .
(74)
We should now restrict those various quantities to the CY patch U = U \ X, by using the approximatedefining equation (67). For the Kahler forms this leads to
We would like to consider line bundles L = OA(k1, k2) and their restrictions L = L|X
= OX
(k1, k2). Thehermitian bundle metric is given by
H = �k11 �k2
2 (72)
Provided that k1 �2 and k2 > 0, which I assume in the following, the only non-zero cohomology of L isH1(A,L) ⇠= H1(X,L), so all relevant (0, 1)-forms are of type 1. Explicitly, ⌫ 2 H1(A,L) can be written as
⌫ = k11 P (z1, z↵)dz1 . (73)
Taking the flat limit on U of all the relevant quantities gives
J1 = i
2⇡dz1 ^ dz1 J2 = i
2⇡
P4↵=2 dz↵ ^ dz
↵
J = t1J1 + t2J2
H = e�k1|z1|2�k2P4
↵=2 |z↵|2 ⌫ = ek1|z1|2P (z1, z↵)dz2 .
(74)
We should now restrict those various quantities to the CY patch U = U \ X, by using the approximatedefining equation (67). For the Kahler forms this leads to
For the lowest mode, I = 0 the corresponding number is given by
N0,0 = 6⇡|k1 + k2/6|
k22. (94)
For illustration, let me consider the simplest example k1 = �2 and k2 = 1 with four families ordered asI 2 {(0, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)}. With this ordering, the matrix N is given by
N =11⇡
2diag(1, 1, 1, 0) . (95)
The zero eigenvalue is a concern and this is something that happens for other line bundles as well. Have I madea mistake or is it possible we can’t see some normalizations at the order we are calculating?
4 Bi-cubic
Next, I consider the bi-cubic, defined as a zero-locus of a bi-degree (3, 3) polynomial p in the ambient spaceA = P2 ⇥ P2. Homogeneous coordinates are denoted by x
i,↵
, where i = 1, 2 and ↵ = 0, 1, 2 and we have a�necoordinates
z1 =x1,1x1,0
, z2 =x1,2x1,0
, z3 =x2,1x2,0
, z4 =x2,2x2,0
. (96)
Near zµ
= 0 the defining polynomial is expanded as
p = p0 +4X
µ=1
pµ
zµ
+O(z2) . (97)
11
Find harmonic wave fcts. with and work out ⌫I [⌫I] = [⌫I|U ]
GI,J =1
Vh⌫I, ⌫Ji
Conclusion• For heterotic line bundle models, we have a fairly good understanding of how to calculate holomorphic Yukawa couplings.
• For large flux, the matter field Kahler metric can be approximately calculated due to localisation.
• The local calculation can be linked to the global properties, so the result is obtained as a function of global moduli!
• Using localisation, we have calculated the matter field Kahler metric explicitly for simple examples.
• Analogous methods may well apply to F-theory in suitable global models.
• However, calculation becomes very involved for more complicated CY manifolds.
• The large flux required for localisation typically implies a large family number. Can the method be applied to realistic models?
The global-local link allows us to better understand the limitations of this method. Some tensions emerge:
• Large flux quickly runs up agains anomaly cancellation.