global issues in type ii compactifications on su(3 ... · global issues in type ii...
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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)
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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)
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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.
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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?
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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
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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?
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- 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
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- 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
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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
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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)]
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- 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
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- 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)
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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
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- Need coordinates such thatz!
!! =
!
"!
"z!
"
(2,1)
!= 0
!
!z!gab =
1
||!||2!cd
a("!)cdb
- Then
A family of (almost) complex structures
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- 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]
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- 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
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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
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Why!
!!
!z!
"
(2,1)
!= 0
!
!
!!
!z!
"
"/ [!]
!
!!
!z!
"
(2,1)
!= 0By , h(3,0)
= 1
[!] = [!0] + . . . ,
!
"!
"z!
"
= [!!] + . . .
condition on expansionforms {!A, "A}
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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
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violated!
!!
!z!
"
(2,1)
!= 0
!! = "i! ! Gi= iX
i
!!
!z!= X
!("! ! i " "!)
Plug in and evaluate explicitly !!
!z!
!
!!
!z!= "i
!!
!z!
!
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Conclusions
- N=2 SUGRA imposes stringent constraints on expansion forms
- Much fun lies ahead!