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

Donaldson 05, Headrick, Wiseman 05,Douglas, Karp, Lucik, Reinbacher 06Braun, Brelidze, Douglas, Ovrut 07Anderson, Braun, Karp, Ovrut 10

disadvantages: - technically quite involved - provides at one point in moduli space G

Main purpose of this talk: report progress on (approximate) analytic calculation of

as a function of the moduli

But before….

G

Results for holomorphic Yukawa couplings

CY manifold: X = {p↵ = 0} ⇢ A = Pn1 ⇥ Pn2 ⇥ · · ·

kco-dimensions of : X ⇢ A

``Wave function” is determined by ambient space forms ⌫ ⌫1, . . . , ⌫d

Then, is said to be of type ⌫ d 2 {1, . . . , k + 1}

bundle: V =M

a

La ! A V = V|X =M

a

La ! X

matter fields: ⌫a 2 ⌦a(A,V)a = 1, . . . , d k + 1

⌫ 2 H1(X,V )

line bundle models: Anderson, Gray, Lukas, Palti 11

d1 + d2 + d3 < dim(A) =) �(⌫1, ⌫2, ⌫3) = 0

Vanishing property:

�(⌫1, ⌫2, ⌫3) ⇠

Here, the relation

pb @⇣ 1

pb

⌘= 0 (4.10)

has led to the insertion of �b1b from Eq. (4.5) so that we remain with a sum over b1, as indicated above

(while the resulting factor pb1 from Eq. (4.5) cancels against 1/pb1). We can now continue integrating by

parts until all factors of the form @(1/pb) are used up. Each of these factors leads to a partial di↵erentiation

of all forms ⌫b1···ba�1i,a which appear in the integral, e↵ective replacing them by the forms ⌫b1···ba�1b

i,a+1 , which

appear one step lower down in the chain (4.6). Since there are k such partial integrations to be performed,

starting with three (0, 1)-forms, the end result is a sum which contains all product of three forms whose

degree sums up to dim(A) = 3 + k. This leads to

�(⌫1, ⌫2, ⌫3) =Ck

(2⇡)k

kX

a1,a2,a3=1a1+a2+a3=dim(A)

(�1)s(a1,a2,a3)Z

Aµ ^ ⌫1,a1 ^ ⌫2,a2 ^ ⌫3,a3 . (4.11)

where s(a1, a2, a3) = (a1 + 1)a2 + a1a3 + a2a3 determines the relative signs of the terms and Ck =

(�1)k(k+1)/2(�1)[(k+1)/2] ik is another phase. In this formula, the bundle indices have been suppressed so

the wedge product should be understood as including an appropriate tensoring of the bundle directions

to form a singlet, via anti-symmetrisation by ✏b1···bk

. The anti-symmetrisation is achieved by summing

in every case as many terms with permuted indices as required for complete anti-symmetry, each with a

factor 1 or �1 and no additional overall normalisation. This means that, for example, ⌫1,2 ^ ⌫2,2 ^ ⌫3,1 =

✏b1b2 ⌫b11,2 ^ ⌫b22,2 ^ ⌫3,1 while ⌫1,3 ^ ⌫2,1 ^ ⌫3,1 =

12✏b1b2 ⌫

b1b21,3 ^ ⌫2,1 ^ ⌫3,1.

Eq. (4.11) is our main general result for the holomorphic Yukawa couplings. All the ambient space

forms ⌫i,a can be constructed explicitly, starting with Appendix (C) in order to write down (harmonic)

representatives for ambient space cohomology for the highest degree non-trivial forms in the chain (4.6)

and then solving these equations to find all associated lower-degree forms. With these forms inserted, the

integral (4.11) can be carried out explicitly, as we will demonstrate for the examples in Section 5.

As before, it is useful to discuss some special cases. First assume, that the (0, 1)-forms ⌫i are of type

⌧i so that ⌫i,a = 0 for all a > ⌧i. If the ⌧i sum up to less than the ambient space dimension dim(A) then

all terms in Eq. (4.11) vanish due to the summation constraint. As a result the Yukawa coupling vanishes.

Let us formulate this concisely:

Theorem: Assume that the forms ⌫i which enter the integral (1.1) for the Yukawa couplings are of type

⌧i, where i = 1, 2, 3. Then

⌧1 + ⌧2 + ⌧3 < dim(A) =) �(⌫1, ⌫2, ⌫3) = 0 . (4.12)

This is the general version of the vanishing theorem we have already seen for co-dimensions one and two

in previous sections. As we have discussed, the type ⌧ of a form ⌫ 2 H1(X,K) is determined by the

cohomology H⌧ (A,^⌧�1N ⇤ ⌦ K) from which it descends via successive co-boundary maps. As a rule

14

Here, the relation

pb @⇣ 1

pb

⌘= 0 (4.10)

has led to the insertion of �b1b from Eq. (4.5) so that we remain with a sum over b1, as indicated above

(while the resulting factor pb1 from Eq. (4.5) cancels against 1/pb1). We can now continue integrating by

parts until all factors of the form @(1/pb) are used up. Each of these factors leads to a partial di↵erentiation

of all forms ⌫b1···ba�1i,a which appear in the integral, e↵ective replacing them by the forms ⌫b1···ba�1b

i,a+1 , which

appear one step lower down in the chain (4.6). Since there are k such partial integrations to be performed,

starting with three (0, 1)-forms, the end result is a sum which contains all product of three forms whose

degree sums up to dim(A) = 3 + k. This leads to

�(⌫1, ⌫2, ⌫3) =Ck

(2⇡)k

kX

a1,a2,a3=1a1+a2+a3=dim(A)

(�1)s(a1,a2,a3)Z

Aµ ^ ⌫1,a1 ^ ⌫2,a2 ^ ⌫3,a3 . (4.11)

where s(a1, a2, a3) = (a1 + 1)a2 + a1a3 + a2a3 determines the relative signs of the terms and Ck =

(�1)k(k+1)/2(�1)[(k+1)/2] ik is another phase. In this formula, the bundle indices have been suppressed so

the wedge product should be understood as including an appropriate tensoring of the bundle directions

to form a singlet, via anti-symmetrisation by ✏b1···bk

. The anti-symmetrisation is achieved by summing

in every case as many terms with permuted indices as required for complete anti-symmetry, each with a

factor 1 or �1 and no additional overall normalisation. This means that, for example, ⌫1,2 ^ ⌫2,2 ^ ⌫3,1 =

✏b1b2 ⌫b11,2 ^ ⌫b22,2 ^ ⌫3,1 while ⌫1,3 ^ ⌫2,1 ^ ⌫3,1 =

12✏b1b2 ⌫

b1b21,3 ^ ⌫2,1 ^ ⌫3,1.

Eq. (4.11) is our main general result for the holomorphic Yukawa couplings. All the ambient space

forms ⌫i,a can be constructed explicitly, starting with Appendix (C) in order to write down (harmonic)

representatives for ambient space cohomology for the highest degree non-trivial forms in the chain (4.6)

and then solving these equations to find all associated lower-degree forms. With these forms inserted, the

integral (4.11) can be carried out explicitly, as we will demonstrate for the examples in Section 5.

As before, it is useful to discuss some special cases. First assume, that the (0, 1)-forms ⌫i are of type

⌧i so that ⌫i,a = 0 for all a > ⌧i. If the ⌧i sum up to less than the ambient space dimension dim(A) then

all terms in Eq. (4.11) vanish due to the summation constraint. As a result the Yukawa coupling vanishes.

Let us formulate this concisely:

Theorem: Assume that the forms ⌫i which enter the integral (1.1) for the Yukawa couplings are of type

⌧i, where i = 1, 2, 3. Then

⌧1 + ⌧2 + ⌧3 < dim(A) =) �(⌫1, ⌫2, ⌫3) = 0 . (4.12)

This is the general version of the vanishing theorem we have already seen for co-dimensions one and two

in previous sections. As we have discussed, the type ⌧ of a form ⌫ 2 H1(X,K) is determined by the

cohomology H⌧ (A,^⌧�1N ⇤ ⌦ K) from which it descends via successive co-boundary maps. As a rule

14

+1

Wave function normalization from localisationFocus on a single line bundle , and type 1 L ! A L = L|X ! X

ambient space :A Calabi-Yau :Xrestriction

Kahler form: J = tiJi J = tiJi Ricci-flat

[J ] = [J |X ]

wave fct: ⌫ 2 H1(A,L) ⌫ 2 H1(X,L)

[⌫] = [⌫|X ]

@⌫ = 0 J ^ J ^ @(H⌫) = 0harmonic: ⌫

connection: F = � i

2⇡@@H

= kiJi

F = � i

2⇡@@H= kiJi

HYM : J2 ^ F!= 0

Suppose that is localised on a patch H|⌫|2 U ⇢ X

h⌫, ⇢i :=Z

X⌫ ^ (H ? ⇢) = � i

2

Z

XJ ^ J ^ ⌫ ^ (H⇢)

inner product:

On we can use (approximately) flat and U J H

Does the wave function localise on projective spaces?

= 1 + |z|2

J =i

2⇡2dz ^ dzKahler form:

Toy example with affine coordinate , P1 z L = OP1(k), k �2

bundle metric: H = �k

wave fct: ⌫ = k P (z) dz

�! J ' i

2⇡dz ^ dz

�! H ' e�k|z|2

�! ⌫ ' P (z) ek|z|2

dz

Palti 12

H|⌫|2 ⇠ |P |2k ⇠ |P |2ekz2 for large localised near z ' 0|k|

An explicit example

ambient space: A = P1 ⇥ P3

CY manifold: X ⇠

P1 2P3 4

� z1

z2, z3, z4

line bundle: L = OA(k1, k2) , k1 �2 , k2 > 0

for some constant c to be fixed later. The Kahler potentials on the standard patches in P1 and P3 are

1 = 1 + |z1|2 , 2 = 1 +4X

↵=2

|z↵

|2 (68)

with associated Kahler forms

J1 =i

2⇡@@ log 1 =

i

2⇡21dz1 ^ dz1 , J2 =

i

2⇡@@ log 2 =

i

2⇡22(2�

↵�

� z↵

z�

) dz↵

^ dz�

. (69)

Kahler forms J on A can then be parametrized as

J = t1J1 + t2J2 , (70)

where t1 > 0 and t2 > 0. The non-zero intersection numbers, relative to this basis, are given by

d122 = d212 = d221 = 4 , d222 = 2 ) K = 6V = 12t1t22 + 2t32 (71)

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

J1 = J1|X

=i

2⇡dz1 ^ dz1 , J2 = J2|

X

=i

2⇡

�|c|2dz1 ^ dz1 + dz2 ^ dz2 + dz3 ^ dz3�. (75)

and

J =i

2⇡

�t1dz1 ^ dz1 + t2(|c|2dz1 ^ dz1 + dz2 ^ dz2 + dz3 ^ dz3)

�. (76)

The benefit of writing the defining equation of the CY in the form (67) is that

Ji

^ Jj

^ Jk

= � 1

16⇡3dijk

3

a=1

dza

^ dza

, (77)

provided that we choosec = 1/

p6 . (78)

9

2 Kahler moduli

defining eqn. for CY:

for some constant c to be fixed later. The Kahler potentials on the standard patches in P1 and P3 are

1 = 1 + |z1|2 , 2 = 1 +4X

↵=2

|z↵

|2 (68)

with associated Kahler forms

J1 =i

2⇡@@ log 1 =

i

2⇡21dz1 ^ dz1 , J2 =

i

2⇡@@ log 2 =

i

2⇡22(2�

↵�

� z↵

z�

) dz↵

^ dz�

. (69)

Kahler forms J on A can then be parametrized as

J = t1J1 + t2J2 , (70)

where t1 > 0 and t2 > 0. The non-zero intersection numbers, relative to this basis, are given by

d122 = d212 = d221 = 4 , d222 = 2 ) K = 6V = 12t1t22 + 2t32 (71)

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

J1 = J1|X

=i

2⇡dz1 ^ dz1 , J2 = J2|

X

=i

2⇡

�|c|2dz1 ^ dz1 + dz2 ^ dz2 + dz3 ^ dz3�. (75)

and

J =i

2⇡

�t1dz1 ^ dz1 + t2(|c|2dz1 ^ dz1 + dz2 ^ dz2 + dz3 ^ dz3)

�. (76)

The benefit of writing the defining equation of the CY in the form (67) is that

Ji

^ Jj

^ Jk

= � 1

16⇡3dijk

3

a=1

dza

^ dza

, (77)

provided that we choosec = 1/

p6 . (78)

9

=)

How to choose ? Ji

p = p0 +4X

a=1

paza +O(z2) near U = {za ' 0}!= 0

U = {z4 ' z1/p6}Ji = Ji|U for

(HYM eqs locally satisfied)=) J2 ^ F ⇠ µ(L) = 0

Families: ⌫I = ek1|z1|2 zI11 zI22 zI33 zI44 dz1

I1 2 {0, . . . ,�k1 � 2} I2 + I2 + I3 2 {0, . . . , k2}

This leads to:

GI,J =NI,J

6t1 + t2

with numbers

and

Sq,p := (MT )q,p := h⌫q, ⌫pi = 2⇡t22 �q1,p1�p4�q2,p2�q3,p3

3Y

a=1

pa

! |ka

|�pa�1 (89)

Combining the last two results the matrix T is then given by

Tq,p = �q1,p1�p4�q2,p2�q3,p3

p1! |k1|�p1�1

q1! |K1|�q1�1. (90)

and I find

(T †MT )q,p = 2⇡t22✓(q1 � q4)�q1�q4,p1�p4�q2,p2�q3,p3p1! q1! q2! q3!|K1|q1�q4+1

(q1 � q4)! |k1|p1+1kq2+q3+22

. (91)

For simplicity of notation, I write hated indices as I = q. With this notation, the prospective matter fieldKahler metric reads

GI,J :=1

V(T †T †MTT )I,J =

NI,J

6t1 + t2, (92)

where the numbers NI,J are defined by

NI,J = ⇡J1! I1! I2! I3! |k1 + k2/6|I1�I4+1 6I4/2+J4/2+1

(I1 � I4)! |k1|J1+1kI2+I3+22

✓(I1 � I4)�I1�I4,J1�J4�I2,J2�I3,J3 . (93)

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.

Thanks