AdSAdS44 ££ CPCP33 superspace superspace

Dmitri SorokinDmitri SorokinINFN, Sezione di PadovaINFN, Sezione di Padova

ArXivArXiv:0811.1566 Jaume Gomis, Linus Wulff and D.S.:0811.1566 Jaume Gomis, Linus Wulff and D.S.

SSQSQS’09’09, Dubna, 30 , Dubna, 30 JulyJuly 2009 2009

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

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AdSAdS44//CFTCFT33 correspondencecorrespondence

Holographic duality between 3d supersymmetric Chern-Simons-Holographic duality between 3d supersymmetric Chern-Simons-matter theories (BLG & ABJM) with SU(N)xSU(N) and M/String matter theories (BLG & ABJM) with SU(N)xSU(N) and M/String Theory compactified on AdSTheory compactified on AdS44 space space

In a certain limit the bulk dual description of the ABJM model is In a certain limit the bulk dual description of the ABJM model is given by type IIA string theory on given by type IIA string theory on AdSAdS44££ CP CP33

this D=10 background preserves 24 of 32 susy, i.e. this D=10 background preserves 24 of 32 susy, i.e. NN=6 from the =6 from the AdSAdS44 perspective, and its isometry is perspective, and its isometry is OSpOSp(6|4)(6|4)

To study this AdS/CFT duality, it is necessary to know an explicit To study this AdS/CFT duality, it is necessary to know an explicit form of the Green-Schwarz string action in the form of the Green-Schwarz string action in the AdSAdS44££ CP CP33 superbackground.superbackground.

55

Green-Schwarz superstringGreen-Schwarz superstringin a generic supergravity backgroundin a generic supergravity background

22 det BgdS ij

metrict worldsheeinduced - ABBj

Aiij EEg

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

field guage form-2 NS-NS theofpullback et worldshe- ),(2

MXZ

XBM

M

M

;9,...,1,0

sugra 10D ofin supevielbe vector theofpullback - ),( )(

A

XEZE Ai

Ai M

Μ

sugra 10D ofein supervielbspinor - ),( XEdZE M

Μ

Our goal is to get the explicit form ofOur goal is to get the explicit form of BB2 2 ,, EEAA and and EE as polynomials of as polynomials of for the AdSfor the AdS44 ££ CP CP33 case case

Type IIA sugra fields gMN, , BMN, AM, ALMN, M, are contained in EA and B2

66

Fermionic kappa-symmetryFermionic kappa-symmetryProvided that the superbackground satisfies superfield supergravity Provided that the superbackground satisfies superfield supergravity constraints (or, equivalently, sugra field equations), the GS superstring constraints (or, equivalently, sugra field equations), the GS superstring action is invariant under the following local worldsheet transformationsaction is invariant under the following local worldsheet transformations

of the string coordinates of the string coordinates ZZMM(()=)=((XXMM, , ))::

0 tr , ,det2

1

),()(2

1),( ,0),(

211

ABBAij

A

EEg

XEZXEZ

ji

MΜ

MΜ

Due to the projector, the fermionic parameter Due to the projector, the fermionic parameter (()) has only 16 has only 16 independent components. They can be used to gauge away 1/2 of 32 independent components. They can be used to gauge away 1/2 of 32

fermionic worldsheet fields fermionic worldsheet fields (())

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AdSAdS44££ CPCP33 superbackground superbackground

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

in contrast:in contrast: AdSAdS55££ SS55 background in type IIB string theory background in type IIB string theory preserves all 32 supersymmetriespreserves all 32 supersymmetries

fermionic modes of the fermionic modes of the AdSAdS44££ CPCP33 superstring are of superstring are of

different nature: different nature: 3232(()=()=(2424, , 88))

unbroken broken susyunbroken broken susy

2424PP24 24 88PP8 8 PP24 24 ++ PP8 8 = I= I

PP24 24 ==1/8 1/8 ((66--a’b’a’b’JJa’b’a’b’77),), 7 7 = = 1166

a’,b’=1,…,6 - a’,b’=1,…,6 - CPCP33 indices, indices, JJa’b’ a’b’ -- Kaehler form on Kaehler form on CPCP33

88

OSpOSp(6|4) supercoset sigma model(6|4) supercoset sigma model It is natural to try to get rid of the eight “broken susy” It is natural to try to get rid of the eight “broken susy” fermionic modes fermionic modes 8 8 using kappa-symmetryusing kappa-symmetry

88=0 - =0 - partial kappa-symmetry gauge fixingpartial kappa-symmetry gauge fixing

Remaining string modes are:Remaining string modes are:10 (AdS10 (AdS4 4 ££CPCP33) bosons ) bosons xxa a (() (a=0,1,2,3), y) (a=0,1,2,3), ya’ a’ (() (a’=1,2,3,4,5,6)) (a’=1,2,3,4,5,6)24 fermions 24 fermions (() corresponding to unbroken susy) corresponding to unbroken susy

they parametrize coset superspace they parametrize coset superspace OSpOSp(6(6|4|4)/)/UU(3)x(3)xSOSO(1,3) (1,3) ¾¾ AdSAdS44xxCPCP33

KK10,24 10,24 (x,y,(x,y,)= )= eexxaaPPaa eeyya’a’PPa’a’ ee Q Q 22 OSpOSp(6(6|4|4)/)/UU(3)x(3)xSOSO(1,3)(1,3)

[[P,PP,P]]=M,=M, [[P,MP,M]]=P,=P, P P ½ ½ AdSAdS44xxCPCP33 M M ½½ SO(1,3) SO(1,3)££ U(3) U(3)

OSpOSp(6(6|4|4) superalgebra:) superalgebra:

{{Q,QQ,Q}=}=P+M,P+M, [[Q,PQ,P]=]=QQ,, [[Q,MQ,M]=]=QQ

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

Cartan forms:Cartan forms:

KK-1-1dKdK==EEaa((x,y,x,y,) ) PPaa + E + Ea’a’((x,y,x,y,) ) PPa’a’ + E + E’’((x,y,x,y,)) Q Q’’ + + ((x,y,x,y,) ) MM

Superstring action onSuperstring action on OSpOSp(6(6|4|4)/)/UU(3)x(3)xSOSO(1,3)(1,3) – – classically integrableclassically integrable(Arutyunov (Arutyunov & Frolov; Stefanskij; D'Auria, Frè, Grassi & Trigiante, 2008)& Frolov; Stefanskij; D'Auria, Frè, Grassi & Trigiante, 2008)

S=S=ss d d22 (- det (- det EEiiAAEEjj

BB ABAB))1/2 1/2 + + s s EE’’ÆÆEE’’ J J’’’’ C C

ReasonReason – kappa-gauge fixing – kappa-gauge fixing 88 = PP88 ==00 is is inconsistentinconsistent in the AdS in the AdS4 4 regionregion

[(1+[(1+)), , PP88 ]=0 ]=0 only ½ of only ½ of 88 can be eliminated can be eliminated

A problem with this model is that it does not describe all possible string A problem with this model is that it does not describe all possible string configurations. E.g., it does not describe a string moving in AdSconfigurations. E.g., it does not describe a string moving in AdS44 only. only.

1010

Towards the construction of the complete Towards the construction of the complete AdSAdS44££ CPCP3 3 superspace (with 32 superspace (with 32 ))

We shall construct this superspace by performing the dimension We shall construct this superspace by performing the dimension reduction of D=11 coset superspace reduction of D=11 coset superspace OSpOSp(8|4)/(8|4)/SOSO(7)(7)££SOSO(1,3) it (1,3) it has 32 has 32 and subspace and subspace AdSAdS44££ SS77SO(2,3) SO(2,3) £ £ SO(8) SO(8) ½ ½ OSpOSp(8|4)(8|4)

AdSAdS44££ CPCP33 sugra solutionsugra solution is related to is related to AdSAdS44££ SS77 (with 32 susy)(with 32 susy) by by dimenisonal reduction dimenisonal reduction ((Nilsson and Pope; D.S., Tkach & Volkov 1984Nilsson and Pope; D.S., Tkach & Volkov 1984))

Geometrical groundGeometrical ground: : SS7 7 is the Hopf fibration over is the Hopf fibration over CPCP33 with with SS11 fiber: fiber:

SS77 vielbein: vielbein: eemmaa==

eem’m’a’ a’ ((yy) A) Am’m’((yy))

0 10 1

CPCP33 vielbein and vielbein and UU(1) connection (1) connection

FF22~dA=J~dA=JCPCP33(does not depend on (does not depend on SS11 fiber coordinate fiber coordinate zz))

Our goal is to extend this structure toOur goal is to extend this structure to OSp(8|4)/SO(7)OSp(8|4)/SO(7)££SO(1,3)SO(1,3)

1111

Hopf fibration of Hopf fibration of SS77 as a coset space as a coset space SU(4)SU(4)££U(1)U(1)SU(3)SU(3)££U(1)U(1)

As a Hopf fibration, locally, As a Hopf fibration, locally, SS77==CPCP33 ££ U(1) U(1)

SS = = SO(8)SO(8)SO(7)SO(7)

- symmetric space- symmetric space

CPCP33= = SU(4)/SU(3)SU(4)/SU(3)££U’(1) – symmetric spaceU’(1) – symmetric space

SU(4)SU(4)££U(1)U(1)SU(3)SU(3)££UUdd(1)(1)

SU(4)SU(4)££U(1) U(1) ½½ SO(8) SO(8)

Coset representative and Cartan forms:Coset representative and Cartan forms:

KKSS77(y,z)=K(y,z)=KCPCP33(y)(y) e ezTzT11=e=eyya’a’PPa’a’ e ezTzT11 ((TT11 is U(1) generator) is U(1) generator)

KK-1-1SS7 7 dKdKSS7 7 = e= ea’a’((yy) ) PPa’a’ + + ((yy) ) MMSU(3)SU(3)+A+A((yy) ) TT22 + + ddz z TT1 1 ((TT22 ½½ UU’’(1)(1)))

= dy= dym’m’eemm’’a’a’((yy) ) PPa’a’ + +((dzdz+ + dydym’m’ AA

m’m’ ((yy)) )) PP77 + ( + (ddz - z - AA((yy)) )) TTdd+ + ((yy) ) MMSU(3)SU(3)

TTdd=1/2 (T=1/2 (T11-T-T22) is U) is Udd(1) generator, (1) generator, PP77 ==1/2 (T1/2 (T11+T+T22) – translation along ) – translation along SS11 fiber fiber

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Hopf fibration of Hopf fibration of OSpOSp(8|4)/(8|4)/SOSO(7)(7)££SOSO(1,3)(1,3)

KK11,3211,32 - D=11 superspace with the bosonic subspace - D=11 superspace with the bosonic subspace AdSAdS44££ SS77 and 32 fermionic directions and 32 fermionic directions

KK11,32 11,32 = M= M10,3210,32 ££ SS11 (locally) (locally)

MM10,3210,32 - D=10 superspace with the bosonic subspace - D=10 superspace with the bosonic subspace AdSAdS44££ CPCP33 and 32 fermionic directions and 32 fermionic directions (it is not a coset (it is not a coset space)space)

MM10,3210,32 is the superspace we are looking foris the superspace we are looking for!!

basebase fiber fiber

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11stst step: step: Hopf fibration over Hopf fibration over KK10,2410,24 = = OSpOSp(6(6|4|4)/)/UU(3)x(3)xSOSO(1,3)(1,3)

KK11,2411,24 = = KK10,2410,24 ££ SS11 (locally) (locally) ¾¾ AdSAdS4 4 xx S S77

SOSO(1,3)(1,3)££SUSU(3)(3)££UUdd(1)(1)

OSpOSp(6(6|4) |4) ££ UU(1)(1)KK11,24 11,24 == OSpOSp(6(6|4)|4)££UU(1) (1) ½½ OSpOSp(8(8|4)|4)

KK11,24 11,24 ((x,yx,y,,,,zz)=)= K K10,24 10,24 ((x,yx,y,,) ) eezTzT11= = eexxaaPPaa eeyya’a’PPa’a’ eeQQ2424 eezTzT11

KK-1-1dK dK = = EEaa((x,y,x,y,) ) PPaa + E + Ea’a’((x,y,x,y,) ) PPa’a’ + E + E’’((x,y,x,y,)) Q Q’’

+(+(dzdz+ A+ A((x,y,x,y,))) ) PP77 + ( + (ddz - z - AA) ) TTd d + + ((x,y,x,y,) ) MMSU(3)SU(3)

Cartan forms:Cartan forms:

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22ndnd step: step: adding 8 fermionic directions adding 8 fermionic directions 88

KK11,3211,32 ((x,yx,y,,,,z,z,))=K=K11,24 11,24 ((x,yx,y,,,,zz) ) eeQQ8 8 == K K10,24 10,24 ((x,yx,y,,) ) eezTzT1 1 eeQQ88

QQ88 are 8 susy generators which extend OSp(6|4) to OSp(8|4)are 8 susy generators which extend OSp(6|4) to OSp(8|4)

OSp(6|4) superalgebraOSp(6|4) superalgebra::

{{QQ2424,,QQ2424}=}=MMSOSO(2,3)(2,3)++MMSUSU(4)(4) {{QQ88,,QQ88}=}=MMSOSO(2,3)(2,3)++TT11

OSp(2|4) superalgebraOSp(2|4) superalgebra::

Extension to OSp(8|4)Extension to OSp(8|4):: {{QQ2424,,QQ88}=M}=M?? ½½ SO SO(8)/(8)/SUSU(4)(4)££UU(1)(1)

Cartan forms ofCartan forms of OSpOSp(8|4)/(8|4)/SOSO(7)(7)££SOSO(1,3):(1,3):

KK-1-1dK dK = [= [EEaa((x,y,x,y,)+)+dzdz V Vaa(())] ] PPaa + E + Ea’a’((x,y,x,y,,,) ) PPa’a’

+[+[dzdz (()+ A)+ A((x,y,x,y,,,))] ] PP77 + E+ E((x,y,x,y,,,)) Q Q+ E+ E((x,y,x,y,,,)) Q Q

+ connection terms+ connection terms

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33rdrd step step: Lorentz rotation in : Lorentz rotation in AdSAdS4 4 x x SS11 (fiber) tangent space (fiber) tangent space

KK-1-1dK dK = (= (EEaa((x,y,x,y,)+)+dzdz V Vaa(())) ) PPaa + E + Ea’a’((x,y,x,y,,,) ) PPa’a’

+(+(dzdz (()+ A)+ A((x,y,x,y,,,))) ) PP77 + E+ E((x,y,x,y,,,)) Q Q+ E+ E((x,y,x,y,,,)) Q Q

+ connection terms+ connection terms

EEaa((x,y,x,y,,,) ) = (= (EEbb((x,y,x,y,)+)+dzdz V Vbb(())) ) bbaa(())

+ (+ (dzdz (() + A) + A((x,y,x,y,,,)) )) 77aa(())

Lorentz transformation of the supervielbeins:Lorentz transformation of the supervielbeins:

SS11:: dzdz(())+ + A A ((x,y,x,y,,,) ) == ((dzdz (() + A) + A((x,y,x,y,,,)) )) 7777(())

+ (+ (EEbb((x,y,x,y,) + ) + dzdz V Vbb(())) ) bb77(())

EEa’a’((x,y,x,y,,,)) == E Ea’a’((x,y,x,y,,,))

EE((x,y,x,y,,,)), , EE((x,y,x,y,,,) – rotated 32 fermionic vielbeins) – rotated 32 fermionic vielbeins

MM10,3210,32

PPaa

PP77

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Guage fixed superstring action in Guage fixed superstring action in AdSAdS44££ CPCP3 3

(upon a T-dualization), (upon a T-dualization), P.A.Grassi, L.Wulff and D.S. arXiv:0903.5407P.A.Grassi, L.Wulff and D.S. arXiv:0903.5407

11''''

11''

112

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2

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baba

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aa

mm

aj

aijimj

mi

ij

yeyieyedrr

idxdx

r

i

dyieddrr

iddx

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==((, , ),),==1/2(1+ 1/2(1+

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The action contains fermionic terms up to a quartic order onlyThe action contains fermionic terms up to a quartic order only

AdS4 CP3

00112 2 33

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ConclusionConclusion The complete The complete AdSAdS4 4 ££CPCP33 superspace with 32 fermionic directions superspace with 32 fermionic directions

has been constructed. Its superisometry is OSp(6|4)has been constructed. Its superisometry is OSp(6|4)

The explicit form of the actions for Type IIA superstring and D-The explicit form of the actions for Type IIA superstring and D-branes in branes in AdSAdS4 4 ££CPCP33 superspace have been derived superspace have been derived

A simple gauge-fixed form of the superstring action was obtainedA simple gauge-fixed form of the superstring action was obtained

a light-cone gauge fixing of the superstring action has been recently a light-cone gauge fixing of the superstring action has been recently considered by D. Uvarov considered by D. Uvarov arXiv:0906.4699arXiv:0906.4699

These results can be used for studying various problems of String These results can be used for studying various problems of String Theory in Theory in AdSAdS4 4 ££CPCP33, in particular, to perform higher-loop , in particular, to perform higher-loop computations (involving fermionic modes) for testing computations (involving fermionic modes) for testing AdSAdS4 4 //CFTCFT33 correspondence: integrability, Bethe ansatz, S-matrix etc. in the correspondence: integrability, Bethe ansatz, S-matrix etc. in the dual planar dual planar NN=6 CS-matter theory=6 CS-matter theory