Global issues in type II compactifications on SU(3 ... · Global issues in type II...

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Global issues in type II compactifications on SU(3) structure manifolds A.K., R. Minasian hep-th/ 0611106 KK reduction (base point dependent) vs. non-linear ansatz (base point independent)

Transcript of Global issues in type II compactifications on SU(3 ... · Global issues in type II...

Page 1: Global issues in type II compactifications on SU(3 ... · Global issues in type II compactifications on SU(3) structure manifolds A.K., R. Minasian hep-th/ 0611106 KK reduction

Global issues in type II compactificationson SU(3) structure manifolds

A.K., R. Minasianhep-th/ 0611106

KK reduction (base point dependent) vs.

non-linear ansatz (base point independent)

Page 2: Global issues in type II compactifications on SU(3 ... · Global issues in type II compactifications on SU(3) structure manifolds A.K., R. Minasian hep-th/ 0611106 KK reduction

Example: Calabi-Yau reduction

!

base point dependent: expand around fixed Ricci flat metric expand SUGRA fields in forms harmonic with regard to

g0

g0

!i

base point independent: consider family of Ricci flat metrics expansion forms vary with

{g(v)}v!V

! v!i(v)

Page 3: Global issues in type II compactifications on SU(3 ... · Global issues in type II compactifications on SU(3) structure manifolds A.K., R. Minasian hep-th/ 0611106 KK reduction

Flux compactifications

Fluxes break supersymmetry spontaneously

Calabi-Yau compactification in flux backgroundshould yield gauged N=2 SUGRA

!

Indeed: retain expansion forms, include cohomological non-trivial contribution to SUGRA fieldstrengths

4d action assembles itself into gauged N=2 SUGRA

C3 = Ai!i + . . . ! F4 = dC3 + Ni!

ie.g.

Page 4: Global issues in type II compactifications on SU(3 ... · Global issues in type II compactifications on SU(3) structure manifolds A.K., R. Minasian hep-th/ 0611106 KK reduction

Further down the rabbit hole

By mirror symmetry: what are mirror configurations to Calabi-Yau flux compactifications?

by Swampland program: can all possible N=2 SUGRA gaugings be realized in string theory?

Page 5: Global issues in type II compactifications on SU(3 ... · Global issues in type II compactifications on SU(3) structure manifolds A.K., R. Minasian hep-th/ 0611106 KK reduction

Giving up integrability

Calabi-Yau structureSU(3)

Kaehler sector structureSp(6, R)J

Complex structure ! SL(3, C) structure

Non-integrability: dJ != 0 ; d! != 0

Page 6: Global issues in type II compactifications on SU(3 ... · Global issues in type II compactifications on SU(3) structure manifolds A.K., R. Minasian hep-th/ 0611106 KK reduction

Tinkering with expansion forms

d!i= 0

d!A = eiA"i ; d#A = !miA"i

d!i = miA"A + eiA#A

d†!i = 0

What properties must these forms satisfy?

Page 7: Global issues in type II compactifications on SU(3 ... · Global issues in type II compactifications on SU(3) structure manifolds A.K., R. Minasian hep-th/ 0611106 KK reduction

- Choose basis of {!i} H2(X, Z)

- Introduce coordinates on Kaehler moduli space

vi=

!!i

J

- By Yau, choice of cplx structure and Kaehler class specifies unique Ricci flat metric with associated Kaehler form J(v)

{!i}{!i(v)}- Introduce harmonic basis dual to and expand

J(v) = vi!i

A family of Kaehler forms

Page 8: Global issues in type II compactifications on SU(3 ... · Global issues in type II compactifications on SU(3) structure manifolds A.K., R. Minasian hep-th/ 0611106 KK reduction

- Kaehler form and cplx structure determine metric

igab = Jab

- Metric variation is given by

i!gab

!vi= "i ab + vj !

!vi"j ab

Determining metric variation

Page 9: Global issues in type II compactifications on SU(3 ... · Global issues in type II compactifications on SU(3) structure manifolds A.K., R. Minasian hep-th/ 0611106 KK reduction

Constraint on variation of harmonics

- Reduction of Ricci scalar yields

VGij(v) ! (!ik + v

k "

"vi)|v=v (!j

l + vl "

"v!j)|v!=v

!X

#k(v) " ##l(v!)

- Kaehler parameters are complexified via

B = bi!i + . . .

GBij(v) !

1

V

!X

!i(v) " #!j(v)

- -model metric descends from!

!X

H ! "H

Page 10: Global issues in type II compactifications on SU(3 ... · Global issues in type II compactifications on SU(3) structure manifolds A.K., R. Minasian hep-th/ 0611106 KK reduction

Why is ?vk !

!vi"k(v) = 0

- By Lichnerowicz, is harmonic

if variation is to retain Ricci flatness

!gab

!vidza

! dzb

i!gab

!vi= "i ab + vj !

!vi"j ab

harmonic by definition exact, since independent of metric variation

[!i(v)]

Page 11: Global issues in type II compactifications on SU(3 ... · Global issues in type II compactifications on SU(3) structure manifolds A.K., R. Minasian hep-th/ 0611106 KK reduction

- Choose basis of {!i} H2(X, Z)

- Introduce coordinates on Kaehler moduli space

vi=

!!i

J

- By Yau, choice of cplx structure and Kaehler class specifies unique Ricci flat metric with associated Kaehler form J(v)

{!i}{!i(v)}- Introduce harmonic basis dual to and expand

J(v) = vi!i

A family of Kaehler forms

Page 12: Global issues in type II compactifications on SU(3 ... · Global issues in type II compactifications on SU(3) structure manifolds A.K., R. Minasian hep-th/ 0611106 KK reduction

- Choose basis of {!i} H2(X, Z)

- Introduce coordinates on Kaehler moduli space

vi=

!!i

J

- By Yau, choice of cplx structure and Kaehler class specifies unique Ricci flat metric with associated Kaehler form J(v)

{!i}{!i(v)}- Introduce harmonic basis dual to and expand

J(v) = vi!i

A family of structuresSp(6, R)

Page 13: Global issues in type II compactifications on SU(3 ... · Global issues in type II compactifications on SU(3) structure manifolds A.K., R. Minasian hep-th/ 0611106 KK reduction

A family of structuresSp(6, R)

vi

- Consider a family of structures parametrized by and a family of 2-forms such that

Sp(6, R){!i(v)}

J(v) = vi!i(v)

and

vk !

!vi"k(v) = 0

Page 14: Global issues in type II compactifications on SU(3 ... · Global issues in type II compactifications on SU(3) structure manifolds A.K., R. Minasian hep-th/ 0611106 KK reduction

- Need coordinates such thatz!

!! =

!

"!

"z!

"

(2,1)

!= 0

!

!z!gab =

1

||!||2!cd

a("!)cdb

- Then

A family of (almost) complex structures

Page 15: Global issues in type II compactifications on SU(3 ... · Global issues in type II compactifications on SU(3) structure manifolds A.K., R. Minasian hep-th/ 0611106 KK reduction

- Expand in symplectic basis of! H3(X)

! = XA!A ! GA"A

Defining z!

- Cannot vary and independentlyGAXA

= GA ! XB!AGB + XBXC

!"B " !A"C + GBGC

!#B

" !A#C

0 =

!! ! !A!

exact, since independentof metric variation

[!A], ["A]

Page 16: Global issues in type II compactifications on SU(3 ... · Global issues in type II compactifications on SU(3) structure manifolds A.K., R. Minasian hep-th/ 0611106 KK reduction

- Expand in symplectic basis of! H3(X)

! = XA!A ! GA"A

Defining z!

= GA ! XB!AGB + XBXC

!"B " !A"C + GBGC

!#B

" !A#C

0 =

!! ! !A!

require vanishing of integrals as condition on expansion forms

{!A, "A}

- Cannot vary and independentlyGAXA

Page 17: Global issues in type II compactifications on SU(3 ... · Global issues in type II compactifications on SU(3) structure manifolds A.K., R. Minasian hep-th/ 0611106 KK reduction

Defining z!

- With define GA = XB

!AGB G =1

2GAX

A

- homogeneous of degree 2, homogeneous of degree 1, ...

G GA

reflection of fact that normalization of arbitrary

!

- Away from , scale such that and define

X0

= 0 X0

= 1

z!= X! , ! != 0

Page 18: Global issues in type II compactifications on SU(3 ... · Global issues in type II compactifications on SU(3) structure manifolds A.K., R. Minasian hep-th/ 0611106 KK reduction

Why!

!!

!z!

"

(2,1)

!= 0

!

!

!!

!z!

"

"/ [!]

!

!!

!z!

"

(2,1)

!= 0By , h(3,0)

= 1

[!] = [!0] + . . . ,

!

"!

"z!

"

= [!!] + . . .

condition on expansionforms {!A, "A}

Page 19: Global issues in type II compactifications on SU(3 ... · Global issues in type II compactifications on SU(3) structure manifolds A.K., R. Minasian hep-th/ 0611106 KK reduction

Naive construction attempt

- start with coclosed eigen 2-forms of Laplacian

!2!i = m2

i !i , d†!i = 0

- define 3-forms viad!i = "i , #i = !"i.

- symplectic basis, closed under , satisfyintegrality constraints

d , d† , !

violated!

!!

!z!

"

(2,1)

!= 0

Page 20: Global issues in type II compactifications on SU(3 ... · Global issues in type II compactifications on SU(3) structure manifolds A.K., R. Minasian hep-th/ 0611106 KK reduction

violated!

!!

!z!

"

(2,1)

!= 0

!! = "i! ! Gi= iX

i

!!

!z!= X

!("! ! i " "!)

Plug in and evaluate explicitly !!

!z!

!

!!

!z!= "i

!!

!z!

!

Page 21: Global issues in type II compactifications on SU(3 ... · Global issues in type II compactifications on SU(3) structure manifolds A.K., R. Minasian hep-th/ 0611106 KK reduction

Conclusions

- N=2 SUGRA imposes stringent constraints on expansion forms

- Much fun lies ahead!