Generations in Flux Compactification July 23 (wed), 2014 at YITP - - PowerPoint PPT Presentation

Distribution of Number of Generations in Flux Compactification

July 23 (wed), 2014 at YITP workshop Taizan Watari (Kavli IPMU)

arXiv:1408.xxxx w/ A. Braun (King’s)

• cf. arXiv:1401.???? w/ A. Braun Y. Kimura (YITP)

flux compactification of IIB/F-theory

GVW X X

W G   



4(

;" ") { } H X G 

cpx str moduli stabilized (isolated minimum) (sub)-ensemble of low- energy eff. theories string landscape: theoretical foundation for “naturalness” gauge group, matter repr. … matter multiplicity

• eff. coupling constants

Low-energy eff. theories

algebra topology moduli

Flux

?

Specify .

3

( ,[ ], ) B S R

R: A4, D5, … unif. symmetry

• f your interest

[S] : divisor class of 3

B

:moduli space of with S = “7-brane of sym. R”

3

:

X

X B  

1

1 1 1

1,1

h

1,1

h

3,1

h

3,1

h

2,2

h

2,1

h

2,1

h

2,1

h

2,1

h

Hodge diamond of X

X : smooth (resolved) 4-fold

1,1 1,1 3

( ) 1 ( ) ran . k( ) h X h B R   

4 2,2 2,2 4

( ) ( ) ( ) ( );

V RM H

H X H X H X H X   

4 2 4,0 3,1 2,2 1,3 4,0

( ; ) { , , , } .

H X X X H

H X Span D D H H H H H         

Decomposition

• cf. Greene Morrison Plesser

cf: IIB orientifold 3-forms =

4 (

; )

H

H X

(Denef Douglas ’04)

• Observations

– Generally (be aware)

• K3 x K3
• toric hypersurface CY4: many examples

– Flux in often breaks the unif. symm. R. – Net chirality is generated by a flux in

• because the matter surface for R=SU(5) is vertical.
• We are led to a proposal of flux ensembles

2,2 (

) .

RM

H X  

A.Braun, Kimura, TW ’14 A.Braun, TW ‘14

2,2 1 2 1 2

(22 ) (22 ) .

RM

h        

2,2 (

)

RM

H X

2,2(

)

V

H X

4 4

{ | ( )} ( )

fix scan scan H

G G G H X H X   

2,2(

)

fix V

G H X 

controls N_gen constructed in Marsano et.al. ’11 (dual to Het)

• Ashok-Denef-Douglas’ theory (contin. approx)

– K = dim[ flux scanning space ], L*= D3-tadpole. – if the prefactor becomes – the distribution on

– if the scanning space covers all of non-verticals – whenever the scanning space contains

• #vac from the prefactor, copling distrib from

vacuum index density distribution

/2 * *

(2 ) ; . ( / 2)!

K I I

L d K L K    

’03, ‘04

*,

K L

*

exp[ 2 ]. KL 

I



det 1 , 2 2

I m m

R i    



        

3,1,

m h 

4 ( ) H

H X (Denef ’08) (Braun Kimura TW ’14)

• computation in examples
• more generally, whenever
• 2

2

3

[ ( )], B n  

2

S is the zero of

A.Braun TW ‘14

• prelim. result.

containing error

4

dim[ ( )].

H

K H X 

2 *

( ) 1 ( ) 24 2

fix

X L G    

3,1 1,1,

h h

2,2 2,2 2,2

, .

H V RM

h h h ( ) , X K  

max * *

(24 ) 8 . L L K   

/2 2 *

#( ) exp[ 2 ] exp[ (4 ) ].

K gen

vac KL e cN         

Gaussian distribution

algebraic topological

4 5

(10) ?

A D

K K  

( based on K3 x K3 or the examples above)