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.
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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
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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.
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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
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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
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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