On moduli and e ff ective theory of N=1 warped compactifications - - PowerPoint PPT Presentation

▶
on moduli and e ff ective theory of n 1 warped
SMART_READER_LITE
LIVE PREVIEW

On moduli and e ff ective theory of N=1 warped compactifications - - PowerPoint PPT Presentation

Jul 02, 2023 164 likes •896 views

Luca Martucci On moduli and e ff ective theory of N=1 warped compactifications Based on: arXiv:0902.4031 15- ti European Workshop on S ts ing Ti eor y Z rich, 7-11 Sep tf mber 2009 Motivation: fluxes and

slide-1
SLIDE 1

On moduli and effective theory

  • f N=1 warped compactifications

Luca Martucci

15-ti European Workshop on Stsing Tieory Zürich, 7-11 Septfmber 2009

Based on:

arXiv:0902.4031

slide-2
SLIDE 2

In type II flux compactifications the internal space is not CY

  • w

Motivation: fluxes and 4D physics

slide-3
SLIDE 3

In type II flux compactifications the internal space is not CY

  • w

Motivation: fluxes and 4D physics

what is the 4D effective physics?

slide-4
SLIDE 4

In type II flux compactifications the internal space is not CY

  • w

Furthermore fluxes generically generate a non-trivial warping:

with

Motivation: fluxes and 4D physics

what is the 4D effective physics?

slide-5
SLIDE 5

In type II flux compactifications the internal space is not CY

  • w

Furthermore fluxes generically generate a non-trivial warping:

with

Neglecting back-reaction:

,

4D effective theory:

(using standard CY tools)

(fluxless) CY spectrum flux induced potential

Motivation: fluxes and 4D physics

what is the 4D effective physics?

slide-6
SLIDE 6

Motivation: fluxes and 4D physics

Furthermore fluxes generically generate a non-trivial warping:

  • w

with

What can we say about 4D effective theory

  • f fully back-reacted vacua?

In type II flux compactifications the internal space is not CY

what is the 4D effective physics?

slide-7
SLIDE 7

Plan of the talk

Type II (generalized complex) flux vacua Moduli, twisted cohomologies and 4D fields Kähler potential

slide-8
SLIDE 8

Type II (generalized complex) flux vacua

slide-9
SLIDE 9

Fluxes and SUSY

  • w
slide-10
SLIDE 10

Fluxes and SUSY

  • w

NS sector:

metric dilaton 3-form ( locally)

slide-11
SLIDE 11

Fluxes and SUSY

  • w

RR sector: NS sector:

metric dilaton 3-form ( locally)

slide-12
SLIDE 12

Fluxes and SUSY

  • w

RR sector: NS sector:

metric dilaton 3-form ( locally)

( )

slide-13
SLIDE 13

Fluxes and SUSY

  • w

RR sector: NS sector:

metric dilaton 3-form ( locally)

( )

C =

  • k

Ck−1

, with

slide-14
SLIDE 14

Fluxes and SUSY

  • w

Killing spinors:

slide-15
SLIDE 15

Fluxes and SUSY

  • w

Killing spinors: Polyforms:

,

slide-16
SLIDE 16

Fluxes and SUSY

  • w

Killing spinors: Polyforms:

,

IIA

,

slide-17
SLIDE 17

Fluxes and SUSY

  • w

Killing spinors: Polyforms:

, ,

IIB IIA

,

slide-18
SLIDE 18

Fluxes and SUSY

  • w

Killing spinors: Polyforms:

, ,

IIB IIA

,

and are O(6,6) pure spinors!

slide-19
SLIDE 19

Fluxes and SUSY

  • w

Killing spinors: Polyforms:

,

they contain complete information about NS sector and SUSY

slide-20
SLIDE 20

Fluxes and SUSY

  • w

Killing spinors: Polyforms:

,

they contain complete information about NS sector and SUSY

SUSY conditions

Graña, Minasian, Petrini & Tomasiello `05

, ,

slide-21
SLIDE 21

Fluxes and SUSY

  • w

Killing spinors: Polyforms:

,

they contain complete information about NS sector and SUSY

SUSY conditions

Graña, Minasian, Petrini & Tomasiello `05

, ,

precise interpretation in terms of:

generalized calibrations

L.M. & Smyth `05

F- and D- flatness

Koerber & L.M.`07

slide-22
SLIDE 22

SUSY and GC geometry

(F-flatness)

slide-23
SLIDE 23

SUSY and GC geometry

integrable generalized complex structure

Hitchin `02

(F-flatness)

slide-24
SLIDE 24

SUSY and GC geometry

integrable generalized complex structure

Hitchin `02

e.g.

complex (IIB) symplectic (IIA)

(F-flatness)

slide-25
SLIDE 25

SUSY and GC geometry

integrable generalized complex structure

Hitchin `02

Induced polyform decomposition

Gualtieri `04

6

  • n=0

ΛnT ∗

M = 3

  • k=−3

Uk (F-flatness)

slide-26
SLIDE 26

SUSY and GC geometry

integrable generalized complex structure

Hitchin `02

Induced polyform decomposition

Gualtieri `04

6

  • n=0

ΛnT ∗

M = 3

  • k=−3

Uk (F-flatness)

slide-27
SLIDE 27

SUSY and GC geometry

integrable generalized complex structure

Hitchin `02

Induced polyform decomposition

Gualtieri `04

6

  • n=0

ΛnT ∗

M = 3

  • k=−3

Uk

complex case e.g.

(F-flatness)

slide-28
SLIDE 28

SUSY and GC geometry

integrable generalized complex structure

Hitchin `02;

Induced polyform decomposition

Gualtieri `04

6

  • n=0

ΛnT ∗

M = 3

  • k=−3

Uk

Integrability GC structure

with

(F-flatness)

slide-29
SLIDE 29

SUSY and GC geometry

integrable generalized complex structure

Hitchin `02;

Induced polyform decomposition

Gualtieri `04

6

  • n=0

ΛnT ∗

M = 3

  • k=−3

Uk

Integrability GC structure

with

Generalized Hodge decomposition

(assuming -lemma)

Cavalcanti `05

(F-flatness)

slide-30
SLIDE 30

Moduli, twisted cohomologies and 4D fields

slide-31
SLIDE 31

Moduli and polyforms

slide-32
SLIDE 32

Moduli and polyforms

The full closed string information is stored in

,

see also: Grañã, Louis & Waldram `05;`06 Benmachiche and Grimm `06 Koerber & L.M.`07

slide-33
SLIDE 33

Moduli and polyforms

`half’ of NS degrees

  • f freedom

The full closed string information is stored in

,

see also: Grañã, Louis & Waldram `05;`06 Benmachiche and Grimm `06 Koerber & L.M.`07

slide-34
SLIDE 34

Moduli and polyforms

`half’ of NS degrees

  • f freedom

information encoded in (second `half’ of NS degrees of freedom)

The full closed string information is stored in

,

see also: Grañã, Louis & Waldram `05;`06 Benmachiche and Grimm `06 Koerber & L.M.`07

slide-35
SLIDE 35

Moduli and polyforms

`half’ of NS degrees

  • f freedom

information encoded in (second `half’ of NS degrees of freedom) RR degrees of freedom

The full closed string information is stored in

,

see also: Grañã, Louis & Waldram `05;`06 Benmachiche and Grimm `06 Koerber & L.M.`07

slide-36
SLIDE 36

Moduli and polyforms

`half’ of NS degrees

  • f freedom

information encoded in (second `half’ of NS degrees of freedom) RR degrees of freedom

The and moduli are associated to twisted cohomology classes of:

,

The full closed string information is stored in

,

see also: Grañã, Louis & Waldram `05;`06 Benmachiche and Grimm `06 Koerber & L.M.`07

slide-37
SLIDE 37

Moduli and 4D fields

slide-38
SLIDE 38

Moduli and 4D fields

moduli space of

Hitchin `02;

slide-39
SLIDE 39

Moduli and 4D fields

moduli space of

Hitchin `02;

slide-40
SLIDE 40

Moduli and 4D fields

moduli space of

Hitchin `02;

In principle, all -moduli can be lifted (up to rescaling)

slide-41
SLIDE 41

Moduli and 4D fields

moduli space of

Hitchin `02;

assuming

( )

for minimal SUSY In principle, all -moduli can be lifted (up to rescaling)

slide-42
SLIDE 42

Moduli and 4D fields

moduli space of

Hitchin `02;

assuming

( )

for minimal SUSY In principle, all -moduli can be lifted (up to rescaling)

,

  • moduli

RR axionic shift

slide-43
SLIDE 43

Moduli and 4D fields

moduli space of

Hitchin `02;

assuming

( )

for minimal SUSY In principle, all -moduli can be lifted (up to rescaling)

,

  • moduli

RR axionic shift

Weyl-chiral weights:

and will be 4D chiral fields of 4D superconformal theory

see e.g.: Kallosh, Kofman, Linde & Van Proeyen`00

slide-44
SLIDE 44

Dual picture: linear multiplets

slide-45
SLIDE 45

Dual picture: linear multiplets

D-flatness condition

slide-46
SLIDE 46

Dual picture: linear multiplets

D-flatness condition

slide-47
SLIDE 47

Dual picture: linear multiplets

D-flatness condition Expand:

,

bosonic components of linear multiplets dual to

slide-48
SLIDE 48

Dual picture: linear multiplets

D-flatness condition Linear-chiral functional dependence

explicit form depends on microscopical details

Expand:

,

bosonic components of linear multiplets dual to

slide-49
SLIDE 49

Example: IIB warped CY

Grañã & Polchinski; Gubser `00 Giddings, Kachru & Polchinski `01

slide-50
SLIDE 50

Example: IIB warped CY

Grañã & Polchinski; Gubser `00 Giddings, Kachru & Polchinski `01

slide-51
SLIDE 51

Example: IIB warped CY

Grañã & Polchinski; Gubser `00 Giddings, Kachru & Polchinski `01

slide-52
SLIDE 52

Example: IIB warped CY

complex structure moduli, lifted up to conformal compensator

Grañã & Polchinski; Gubser `00 Giddings, Kachru & Polchinski `01

slide-53
SLIDE 53

Example: IIB warped CY

complex structure moduli, lifted up to conformal compensator

Grañã & Polchinski; Gubser `00 Giddings, Kachru & Polchinski `01

slide-54
SLIDE 54

Example: IIB warped CY

complex structure moduli, lifted up to conformal compensator

Grañã & Polchinski; Gubser `00 Giddings, Kachru & Polchinski `01

slide-55
SLIDE 55

Example: IIB warped CY

Chiral fields:

complex structure moduli, lifted up to conformal compensator

Grañã & Polchinski; Gubser `00 Giddings, Kachru & Polchinski `01

slide-56
SLIDE 56

Example: IIB warped CY

removed axion-dilaton

Chiral fields:

complex structure moduli, lifted up to conformal compensator

Grañã & Polchinski; Gubser `00 Giddings, Kachru & Polchinski `01

slide-57
SLIDE 57

Example: IIB warped CY

removed axion-dilaton

Chiral fields: Dual linear multiplets:

complex structure moduli, lifted up to conformal compensator

Grañã & Polchinski; Gubser `00 Giddings, Kachru & Polchinski `01

slide-58
SLIDE 58

Kähler potential

slide-59
SLIDE 59

Kähler potential

slide-60
SLIDE 60

Kähler potential

Going to the Einstein frame, one gets the Kähler potential

slide-61
SLIDE 61

Kähler potential

Going to the Einstein frame, one gets the Kähler potential

for , it reduces to Kähler potential of Grañã, Louis & Waldram `05,`06

Benmachiche and Grimm `06

slide-62
SLIDE 62

Kähler potential

what is its explicit form?

Going to the Einstein frame, one gets the Kähler potential

for , it reduces to Kähler potential of Grañã, Louis & Waldram `05,`06

Benmachiche and Grimm `06

slide-63
SLIDE 63

Kähler potential

Going to the Einstein frame, one gets the Kähler potential does not seem topological! However

topologically well defined & in agreement with 4D interpretation

Lindstrøm & Rocek; Ferrara, Girardello, Kugo & Van Proeyen `83

what is its explicit form?

slide-64
SLIDE 64

Kähler potential

Going to the Einstein frame, one gets the Kähler potential does not seem topological! However

topologically well defined & in agreement with 4D interpretation

Lindstrøm & Rocek; Ferrara, Girardello, Kugo & Van Proeyen `83

Freezing the -moduli, knowing one can obtain by integration

la(t + ¯ t)

what is its explicit form?

slide-65
SLIDE 65

In general, dependence of linear multiplets on chiral multiplets cumbersome!

Example: IIB warped CY

Giddings, Kachru & Polchinski `01 Grañã & Polchinski; Gubser `00

slide-66
SLIDE 66

In general, dependence of linear multiplets on chiral multiplets cumbersome!

Example: IIB warped CY

Giddings, Kachru & Polchinski `01

However, if ( )

universal modulus

,

v ≃ 1 = ∂ exp(−K/3) ∂Reρ

la ≃ Iab Re φb = ∂ exp(−K/3) ∂Re φa

Grañã & Polchinski; Gubser `00

slide-67
SLIDE 67

In general, dependence of linear multiplets on chiral multiplets cumbersome!

Example: IIB warped CY

Giddings, Kachru & Polchinski `01

However, if ( )

universal modulus

,

v ≃ 1 = ∂ exp(−K/3) ∂Reρ

la ≃ Iab Re φb = ∂ exp(−K/3) ∂Re φa

Grañã & Polchinski; Gubser `00

slide-68
SLIDE 68

In general, dependence of linear multiplets on chiral multiplets cumbersome! These equations can be integrated

Example: IIB warped CY

Giddings, Kachru & Polchinski `01

However, if ( )

universal modulus

,

v ≃ 1 = ∂ exp(−K/3) ∂Reρ

la ≃ Iab Re φb = ∂ exp(−K/3) ∂Re φa

Grañã & Polchinski; Gubser `00

slide-69
SLIDE 69

In general, dependence of linear multiplets on chiral multiplets cumbersome! These equations can be integrated

Example: IIB warped CY

Giddings, Kachru & Polchinski `01

However, if ( )

universal modulus

,

v ≃ 1 = ∂ exp(−K/3) ∂Reρ

la ≃ Iab Re φb = ∂ exp(−K/3) ∂Re φa

Grañã & Polchinski; Gubser `00

slide-70
SLIDE 70

In general, dependence of linear multiplets on chiral multiplets cumbersome! These equations can be integrated

if , in agreement with

Frey,Torroba, Underwood & Douglas `08

Example: IIB warped CY

Giddings, Kachru & Polchinski `01

However, if ( )

universal modulus

,

v ≃ 1 = ∂ exp(−K/3) ∂Reρ

la ≃ Iab Re φb = ∂ exp(−K/3) ∂Re φa

Grañã & Polchinski; Gubser `00

slide-71
SLIDE 71

In general, dependence of linear multiplets on chiral multiplets cumbersome! These equations can be integrated

if , in agreement with

Frey,Torroba, Underwood & Douglas `08

redefining unwarped Kähler potential Grimm & Louis`04

Example: IIB warped CY

Giddings, Kachru & Polchinski `01

However, if ( )

universal modulus

,

v ≃ 1 = ∂ exp(−K/3) ∂Reρ

la ≃ Iab Re φb = ∂ exp(−K/3) ∂Re φa

Grañã & Polchinski; Gubser `00

slide-72
SLIDE 72

Conclusions

Under some assumptions (e.g. -lemma), the 4D spectrum has been identified with -twisted cohomologies The Kähler potential determined only implicitly. However, 4D chiral-linear duality can help in reconstructing it. The 4D couplings of probe D-branes (space-filling, instantons, DW’s and strings) depend only on the cohomology classes