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Peculiarities of String Theory on Peculiarities of String Theory on 3 CP CP 3 AdS 4 4 AdS Dmitri Sorokin Dmitri Sorokin INFN, Sezione Sezione di di Padova Padova INFN, ArXiv:0811.1566 ArXiv :0811.1566 J. J. Gomis Gomis, , D.S

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SLIDE 1

Peculiarities of String Theory on Peculiarities of String Theory on AdS AdS4

4 ×

× CP CP3

3

Dmitri Sorokin Dmitri Sorokin

INFN, INFN, Sezione Sezione di di Padova Padova

GGI, GGI, Arcetri Arcetri, , September September 30, 2010 30, 2010 ArXiv ArXiv: : 0903.5407

0903.5407 P.A.Grassi P.A.Grassi, D.S., , D.S., L.Wulff L.Wulff ArXiv:0911.5228 ArXiv:0911.5228 A.Cagnazzo A.Cagnazzo, D.S., , D.S., L.Wulff L.Wulff

ArXiv: ArXiv: 1009.3498 D.S. and L.

1009.3498 D.S. and L. Wulff Wulff ArXiv ArXiv:0811.1566 :0811.1566 J.

  • J. Gomis

Gomis, , D.S D.S., ., L.

  • L. Wulff

Wulff

slide-2
SLIDE 2

2 2

AdS AdS4

4×

× CP CP3

3 versus

versus AdS AdS5

5×

× S S5

5

(peculiarities and issues) (peculiarities and issues)

  • Superstring theory on AdS

Superstring theory on AdS4

4x

xCP CP3

3 is not maximally supersymmetric, it

is not maximally supersymmetric, it preserves 24 of 32 susy. The symmetry group is OSp(6| 4) preserves 24 of 32 susy. The symmetry group is OSp(6| 4) – – The The complete complete theory is not described by a supercoset sigma theory is not described by a supercoset sigma-

  • model

model – – The proof of it The proof of it ’ ’s classical integrability turned out to be much tricky s classical integrability turned out to be much tricky

  • some issues have not been completely settled on the boundary and

some issues have not been completely settled on the boundary and bulk bulk side of the AdS side of the AdS4

4/

/ CFT CFT3

3 holography:

holography: – – Bethe Bethe ansatz ansatz CFT anomalous dimensions versus spinning string energy CFT anomalous dimensions versus spinning string energy – – subtleties in matching worldsheet degrees of freedom with those subtleties in matching worldsheet degrees of freedom with those of

  • f

S S-

  • matrix scattering theory (light and heavy worldsheet modes)

matrix scattering theory (light and heavy worldsheet modes) – – issue of the dual superconformal symmetry and fermionic T issue of the dual superconformal symmetry and fermionic T-

  • duality

duality

  • The AdS

The AdS4

4xCP

xCP3

3 theory

theory admits admits string string instantons instantons wrapping wrapping 2 2-

  • cycles of CP

cycles of CP3

3.

.

slide-3
SLIDE 3

3 3

⋅ ⋅ ⋅ + Θ Γ Γ ΘΓ + Θ Γ Γ ΘΓ + Θ Γ ΘΓ + Θ Γ ΘΓ + Θ Γ Ψ + = Θ

Φ M BCDK A BCDK M BC A BC BC A MBC BC A BC M A M A M A M

F F e H X X e X E ) ( ) ( ) , ( ω

β αβ α

E E i E dE X T

A B B A A A

Γ = Ω + ≡ Θ = 2 ) , ( : motion

  • f

equations ty supergravi s constraint ty supergravi superfield from

  • btained

are

Green Green-

  • Schwarz superstring

Schwarz superstring

in a generic supergravity background in a generic supergravity background

∫ ∫

− − − =

2 2

det B g d S

ij

ξ

metric t worldshee induced

  • AB

B j A i ij

E E g η =

field guage form

  • 2

NS

  • NS

the

  • f

pullback et worldshe

  • )

, ( 32 ,..., 1 0,1,...,9; ), , (

2 α α

α Θ = = Θ =

M M

X B M X Z M

; 9 ,..., 1 , sugra 10 D

  • f

in supevielbe vector the

  • f

pullback

  • )

, ( ) ( = = Θ ∂ = A X E Z E

A i A i M Μ ξ

sugra 10 D

  • f

ein supervielb spinor

  • )

, ( = Θ = X E dZ E

α α M Μ

slide-4
SLIDE 4

4 4

Fermionic kappa Fermionic kappa-

  • symmetry

symmetry

Provided that the superbackground satisfies superfield supergrav Provided that the superbackground satisfies superfield supergravity ity constraints (or, equivalently, constraints (or, equivalently, sugra sugra field equations), the GS superstring field equations), the GS superstring action is invariant under the following local worldsheet transfo action is invariant under the following local worldsheet transformations rmations

  • f the string coordinates
  • f the string coordinates Z

ZM

M(

(ξ ξ)= )=(

(X XM

M,

, Θ Θα

α),

), δ δκ

κ Z

ZM

M :

:

tr , , det 2 1 ), ( ) ( 2 1 ) , ( , ) , (

2 11

= Γ Ι = Γ Γ Γ − = Γ Γ + Ι = Θ = Θ

AB B A ij A

E E g X E Z X E Z

j i M Μ M Μ

ε ξ κ δ δ

β β α α κ κ

Due to the projector, the fermionic parameter Due to the projector, the fermionic parameter κ

κα

α(

( ξ ξ) ) has only

has only 16 independent components. They can be used to gauge away 1/ 2 of 16 independent components. They can be used to gauge away 1/ 2 of 32 fermionic worldsheet fields 32 fermionic worldsheet fields Θ

Θα

α(

( ξ ξ) )

slide-5
SLIDE 5

5 5

AdS AdS4

4×

× CP CP3

3 superbackground

superbackground

  • Preserves 24 of 32 susy in type IIA D= 10 superspace

Preserves 24 of 32 susy in type IIA D= 10 superspace

  • fermionic modes of the

fermionic modes of the AdS AdS4

4×

× CP CP3

3 superstring are of

superstring are of different nature: different nature: Θ

Θ32

32(

( ξ ξ)= ( )= ( ϑ ϑ24

24,

, υ υ8

8)

)

unbroken broken susy unbroken broken susy

ϑ ϑ24

24=

=P P24

24 Θ

Θ, , υ υ8

8 =

=P P8

8 Θ

Θ, , P P24

24 +

+ P

P8

8 = I

= I P P24

24 =

= 1/ 8

1/ 8 (

( 6

6-

  • Γ

Γa

a’ ’b b’ ’J

Ja

a’ ’b b’ ’Γ

Γ7

7),

), Γ Γ7

7 =

= Γ Γ1

1L

LΓ Γ6

6 ε

ε1

1L L 6 6

a a’ ’,b ,b’ ’= 1, = 1,… …,6 ,6 -

  • CP

CP3

3 indices,

indices, J

Ja

a’ ’b b’ ’ -

  • Kaehler

Kaehler form on form on CP CP3

3

  • the explicit proof of the classical integrability of the complet

the explicit proof of the classical integrability of the complet e e AdS4 x CP3 superstring has been lacking AdS4 x CP3 superstring has been lacking until recently until recently (D.S.& (D.S.& L.Wulff L.Wulff, 09 , 09/2010 /2010) )

  • The superstring action is not a supercoset sigma-model of OSp(6| 4)

F F2

2

* F * F4

4= F

= F6

6

slide-6
SLIDE 6

6 6

OSp OSp(6|4) supercoset sigma model (6|4) supercoset sigma model

It is natural to try to get rid of the eight It is natural to try to get rid of the eight “ “ broken susy broken susy” ” fermionic modes fermionic modes υ υ8

8

using kappa using kappa-

  • symmetry

symmetry

υ υ8

8= 0

= 0 -

  • partial kappa

partial kappa-

  • symmetry gauge fixing

symmetry gauge fixing Remaining string modes are: Remaining string modes are: 10 (AdS 10 (AdS4

4 ×

×CP CP3

3) bosons

) bosons x x a

a (

( ξ ξ) (a= 0,1,2,3), ) (a= 0,1,2,3), y y a

a’ ’ (

( ξ ξ) (a ) (a’ ’= 1,2,3,4,5,6) = 1,2,3,4,5,6) 24 fermions 24 fermions ϑ ϑ( ( ξ ξ) corresponding to unbroken susy ) corresponding to unbroken susy α α= =1, 1,… …,4 ,4 α α’ ’= 1, = 1,… …,6 ,6 they they parametrize parametrize coset coset superspace superspace OSp OSp(6 (6| 4 | 4)/ )/ U U(3)x (3)xSO SO(1,3) (1,3) ⊃ ⊃ AdS AdS4x 4xCP CP3 3 Sigma Sigma-

  • model action on

model action on OSp OSp(6 (6| 4 | 4)/ )/ U U(3)x (3)xSO SO(1,3) (1,3) ( (Arutyunov Arutyunov & & Frolov Frolov; ; Stefanskij Stefanskij; ; D'Auria D'Auria, , Fr Frè è, Grassi & , Grassi & Trigiante Trigiante, 2008) , 2008)

S= S= ∫ ∫ d d2

2ξ

ξ ( ( -

  • det

det E Ei

iA AE

Ej

j B B η

ηAB

AB)

) 1/ 2

1/ 2 +

+ ∫ ∫ E Eαα

αα’ ’∧

∧ E Eββ

ββ’ ’ J

Jα

α’ ’β β’ ’ γ

γ5

5αβ αβ

  • similar

similar to to the the AdS AdS5

5 ×

× S S5

5 string

string action on action on SU SU(2,2| 4)/ (2,2| 4)/ SO SO(1,4) (1,4) x xSO SO(5) (5)

Cartan Cartan forms: forms: K K-

  • 1

1dK

dK= = E Ea

a(

( x,y x,y, ,ϑ ϑ) ) P Pa

a +

+ E Ea

a’ ’(

( x,y x,y, ,ϑ ϑ) ) P Pa

a’ ’ +

+ E Eαα

αα’ ’(

( x,y x,y, ,ϑ ϑ) ) Q Qαα

αα’ ’ +

+ Ω Ω ( ( x,y x,y, ,ϑ ϑ) ) M M

slide-7
SLIDE 7

7 7

OSp OSp(6|4) supercoset sigma model (6|4) supercoset sigma model

Reason Reason – – kappa kappa-

  • gauge fixing

gauge fixing υ

υ8

8 = P

P8

8Θ

Θ =

= 0

0 is

is inconsistent inconsistent in the in the AdS AdS4

4 region

region [ (1+ [ (1+ Γ Γκ

κ)

) , , P

P8

8 ] = 0

] = 0

  • nly
  • nly ½

½ of

  • f υ

υ8

8 can be eliminated

can be eliminated A problem with this model is that it does not describe all possi A problem with this model is that it does not describe all possible string ble string

  • configurations. E.g., it does not describe a string moving in Ad
  • configurations. E.g., it does not describe a string moving in AdS

S4

4 only,

  • nly,
  • r the string instanton in CP
  • r the string instanton in CP3

3

and for the string instanton in and for the string instanton in CP CP3

3

To describe these string configurations the GS superstring actio To describe these string configurations the GS superstring action in n in AdS AdS4

4 x

x CP CP3

3 superspace with 32 fermionic coordinates is required

superspace with 32 fermionic coordinates is required (it is not a (it is not a coset coset superspace) superspace)

  • This superspace was constructed by performing the dimensional re

This superspace was constructed by performing the dimensional reduction duction

  • f D= 11
  • f D= 11 coset

coset superspace superspace OSp OSp(8| 4)/ (8| 4)/ SO SO(7)x (7)xSO SO(1,3) which has 32 (1,3) which has 32 Θ Θ and and bose bose subspace subspace AdS AdS4

4 x

x S S7

7,

, SO(2,3)xSO(8)

SO(2,3)xSO(8) ⊂ ⊂ OSp OSp(8| 4) (8| 4) (

(Gomis Gomis, D.S., , D.S., Wulff Wulff, 2008) , 2008)

  • AdS

AdS4

4×

× CP CP3

3 sugra

sugra solution solution is related to is related to AdS AdS4

4×

× S S7

7 (with 32 susy) in D= 11

(with 32 susy) in D= 11 by by dimenisonal dimenisonal reduction reduction ( ( Nilsson and Pope; D.S., Tkach & Volkov 1984 Nilsson and Pope; D.S., Tkach & Volkov 1984) )

slide-8
SLIDE 8

8 8

Hopf Hopf fibration fibration of

  • f OSp

OSp(8|4)/ (8|4)/SO SO(7) (7)× ×SO SO(1,3) (1,3)

  • K

K11,32

11,32 -

  • D= 11 superspace with the bosonic subspace

D= 11 superspace with the bosonic subspace AdS AdS4

4×

× S S7

7 and 32 fermionic directions

and 32 fermionic directions

K K11,32

11,32 = M

= M10,32

10,32 ×

× S

S1

1 (locally)

(locally)

  • M

M10,32

10,32 -

  • D= 10 superspace with the bosonic subspace

D= 10 superspace with the bosonic subspace AdS AdS4

4×

× CP CP3

3, 32 fermionic directions and

, 32 fermionic directions and OSp OSp(6| 4) isometry (6| 4) isometry (but it is not a (but it is not a coset coset space) space)

M M10,32

10,32 is the superspace we are looking for

is the superspace we are looking for

base base fiber fiber

slide-9
SLIDE 9

9 9

Classical Integrability of 2d dynamical systems Classical Integrability of 2d dynamical systems

The existence of The existence of ∞ ∞ # of # of conserved conserved currents currents and and charges charges

  • The

The charges charges are are generated generated by by the the Lax Lax connection connection L

L L L (

( ξ ξ, ,z

z)

) – – 2d 2d one

  • ne-
  • form

form which which depends depends on a

  • n a spectral

spectral parameter parameter z,

z,

takes takes values values in a in a symmetry symmetry algebra and algebra and has has zero curvature: zero curvature: d dL

L + L + L ∧ ∧ L = L = 0 (on the

0 (on the mass mass-

  • shell

shell) ) The integrability The integrability is is proven proven if if one

  • ne manages

manages to to construct construct L

L (

( ξ ξ, ,z

z)

) No No generic generic prescription prescription exists exists how how to to do do this this

slide-10
SLIDE 10

10 10

Classical Integrability of 2d dynamical systems Classical Integrability of 2d dynamical systems

  • G

G/ H / H supercoset sigma supercoset sigma-

  • models with

models with Z Z 4

4-

  • grading, e.g.

grading, e.g.

  • AdS

AdS5

5 ×

× S S5

5 superstring,

superstring, SU SU(2,2|4)/ (2,2|4)/SO SO(1,4) (1,4) × × SO SO(5) (5), , Bena Bena, , Roiban Roiban, , Polchinski Polchinski ‘ ‘03 03

  • OSp

OSp(6|4)/ (6|4)/SO SO(1,3) (1,3) × × U(3) sigma U(3) sigma-

  • model

model Cartan Cartan forms: forms: K K-

  • 1

1dK

dK= = Ω Ω ( ( x, x,ϑ ϑ) ) M M0

0 + E

+ E2

2(

( x, x,ϑ ϑ) ) P P2

2 + E

+ E1

1(

( x, x,ϑ ϑ) ) Q Q1

1 + E

+ E3

3(

( x, x,ϑ ϑ) ) Q Q3

3

[ [ M M0

0,

,M M0

0] =

] = M M0

0, [

, [ P P2

2,

,P P2

2] =

] = M M0

0, {

, { Q Q1

1,

,Q Q1

1} =

} = P P2

2= {

= { Q Q3

3,

,Q Q3

3} , {

} , { Q Q1

1,

,Q Q3

3} =

} = M M0 Lax Lax connection: connection:

L L =

= Ω Ω ( ( x, x,ϑ ϑ)+ )+ l

l1

1 E

E2

2(

( x, x,ϑ ϑ)+ )+ l

l2

2*

* E E2

2(

( x, x,ϑ ϑ)+ )+ l

l3

3 E

E1

1(

( x, x,ϑ ϑ)+ )+ l

l4

4 E

E3

3(

( x, x,ϑ ϑ) )

Coefficients Coefficients l

li

i =

= f

fi

i(

(z z) ) are

are functions functions of the

  • f the

spectral spectral parameter parameter

d dL

L + L + L ∧ ∧ L = L = 0

  • n shell
  • n shell
slide-11
SLIDE 11

11 11

Lax connection of the Lax connection of the AdS AdS4

4×

× CP CP3

3 superstring

superstring

(D.S. & L. (D.S. & L. Wulff Wulff, ArXiv:1009.3498) , ArXiv:1009.3498)

  • Use the

Use the OSp OSp(6| 4) conserved (6| 4) conserved Noether Noether current current J= J J= JB

B+ J

+ JS

S

J JB

B(

(X

X,

,ϑ ϑ, ,υ υ) )=

= d dX XM

MK

KM

M (

(X

X)

)+

+ J J1

1A A(

(X

X,

,ϑ ϑ, ,υ υ) )K

KA

A+

+ J J2

2[ [ AB AB] ] (

(X

X,

,ϑ ϑ, ,υ υ) )K

KA

AK

KB

B

J JS

S(

( X X, ,ϑ ϑ, ,υ υ) ) –

– susy current

susy current Lax Lax connection: connection:

L L =

= α α

1 1K

K(

(X

X)

) + + α α2

2 *

* J

JB

B + (

+ ( α α2

2)

) 2

2 J

J2

2 +

+ α α

1 1α

α2

2 *

* J

J2

2 -

  • α

α2

2(

( β β1

1J

JS

S -

  • β

β2

2 *

* J

JS

S)

) + + O

O (

( X X, ,ϑ ϑ, ,υ υ4

4) +

) + L L

SO(2,3) × SU(4)

slide-12
SLIDE 12

12 12

Superstring action in Superstring action in AdS AdS4

4×

× CP CP3

3 superspace

superspace

( (up to the second order in fermions and Wick up to the second order in fermions and Wick-

  • rotated

rotated) )

AdS4 CP3 Bosonic instanton solution Bosonic instanton solution

  • In

In CP CP3

3 there is a topologically nontrivial 2

there is a topologically nontrivial 2-

  • cycle ~

cycle ~ S S2

2 associated with

associated with the the Kahler Kahler 2 2-

  • form

form J J2

2

  • In the Wick

In the Wick-

  • rotated theory, string worldsheet can wrap this

rotated theory, string worldsheet can wrap this S S2

2 ⊂

⊂ CP CP3

3,

, thus forming a stringy instanton. It is thus forming a stringy instanton. It is ½ ½ BPS BPS

Θ Θ= =0 0 and

and x xa

a=const

=const (

(AdS AdS-

  • coordinates)

coordinates);

;

  • n
  • n CP

CP3

3 complex

complex y

yI

I=

=y yI

I(z

(z) ) are

are holomorphic

holomorphic functions of the worldsheet coordinates functions of the worldsheet coordinates

F F4

4 and

and F F2

2 fluxes

fluxes the the Virasoro Virasoro constraints are identically satisfied: constraints are identically satisfied: [ Cvetic’ et al. 1999]

slide-13
SLIDE 13

13 13

String instanton on String instanton on CP CP3

3

A.Cagnazzo A.Cagnazzo, , L.Wulff L.Wulff and D.S. and D.S. ArXiv:0911.5228 ArXiv:0911.5228

Instanton action Instanton action Twelve Twelve fermionic zero modes, i.e. solutions of the 2d Dirac equation: fermionic zero modes, i.e. solutions of the 2d Dirac equation:

υ υ=

=0

Θ Θ= =1/2(1 1/2(1-

  • Γ

Γ) )Θ Θ = ( = (ϑ ϑ, , υ υ) ) – – gauged fixed kappa

gauged fixed kappa-

  • symmetry, 16 physical fermions

symmetry, 16 physical fermions ∫ ∫ B B2

2,

, B B2

2 ∼

∼ J J2

2,

, d J d J2

2=

= d d*

* J

J2

2= 0

= 0 – – Kaehler Kaehler form on form on CP CP2

2

8 zero modes 8 zero modes are 4 pairs of Killing spinors on are 4 pairs of Killing spinors on S S2

2

4 zero modes 4 zero modes are massless charged fermions are massless charged fermions interacting with a monopole field on interacting with a monopole field on S S2

2

ABJ ABJ

slide-14
SLIDE 14

14 14

Fermionic zero modes Fermionic zero modes

The 12 fermionic zero modes are goldstinos which manifest breaki The 12 fermionic zero modes are goldstinos which manifest breaking of ng of the the ½ ½ susy of the AdS susy of the AdS4

4 x CP

x CP3

3 background by the string instanton

background by the string instanton

fermions 4 ) ( fermions 8 ) (

3

2 2

= + ∇ = + ∇

− +

ϑ γ ϑ γ

i S i i S i

iA R i

projected on the instanton projected on the instanton

) (

5

= Γ + ∇ ϑ γ

M M

R i

AdS AdS4

4 x

x CP CP3

3 Killing spinor equation

Killing spinor equation

(1+ (1+ Γ Γκ

κ)

) ϑ

ϑ= 0

= 0

This fermionic zero modes solve also the complete non This fermionic zero modes solve also the complete non-

  • linear string

linear string e.o.m e.o.m 24 target 24 target -

  • space susy:

space susy:

δ δϑ ϑ =

= ε

εKilling

Killing ,

, δ

δ X XM

Me

eM

MA A(X

(X) ) =

= i

iϑ ϑ Γ ΓA

A ε

εKilling

Killing

slide-15
SLIDE 15

15 15

Discussion Discussion

  • There also exists an NS5

There also exists an NS5-

  • brane instanton wrapping the

brane instanton wrapping the whole whole CP CP3

3

  • What are the 3d CFT counterparts of the string/ brane

What are the 3d CFT counterparts of the string/ brane instantons? instantons?

  • What are possible effects of the stringy instantons on the

What are possible effects of the stringy instantons on the structure of the supergravity and superstring theory on structure of the supergravity and superstring theory on AdS AdS4

4 ×

×CP CP3

3? May their existence result in peculiarities of

? May their existence result in peculiarities of the the AdS AdS4

4 /

/ CFT CFT3

3 correspondence?

correspondence?

Drukker Drukker, Mari , Mariñ ño &

  • & Putrov

Putrov ( (arXiv arXiv:1007.3837) :1007.3837) found found contributions contributions coming coming from from world world-

  • sheet

sheet instanons instanons to to the the partition partition function function and and Wilson Wilson loop loop observables

  • bservables computed

computed in a in a matrix matrix model model description description of

  • f

ABJ(M) ABJ(M) theory theory