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AdS 4 £ CP 3 superspace

AdS 4 £ CP 3 superspace. Dmitri Sorokin INFN, Sezione di Padova. ArXiv :0811.1566 Jaume Gomis, Linus Wulff and D.S. ArXiv:0903.5407 P.A.Grassi, L.Wulff and D.S. S QS ’09 , Dubna, 30 July 2009. AdS 4 / CFT 3 correspondence.

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AdS 4 £ CP 3 superspace

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  1. AdS4£CP3 superspace Dmitri Sorokin INFN, Sezione di Padova ArXiv:0811.1566 Jaume Gomis, Linus Wulff and D.S. ArXiv:0903.5407 P.A.Grassi, L.Wulff and D.S. SQS’09, Dubna, 30 July 2009

  2. AdS4/CFT3correspondence • Holographic duality between 3d supersymmetric Chern-Simons-matter theories (BLG & ABJM) with SU(N)xSU(N) and M/String Theory compactified on AdS4 space • In a certain limit the bulk dual description of the ABJM model is given by type IIA string theory on AdS4£ CP3 • this D=10 background preserves 24 of 32 susy, i.e. N=6 from the AdS4 perspective, and its isometry is OSp(6|4) • To study this AdS/CFT duality, it is necessary to know an explicit form of the Green-Schwarz string action in the AdS4£ CP3superbackground.

  3. Our goal is to get the explicit form ofB2 ,EAand E as polynomials of for the AdS4£ CP3 case Green-Schwarz superstring in a generic supergravity background Type IIA sugra fieldsgMN, , BMN, AM, ALMN, M, are contained inEAandB2

  4. Fermionic kappa-symmetry Provided that the superbackground satisfies superfield supergravity constraints (or, equivalently, sugra field equations), the GS superstring action is invariant under the following local worldsheet transformations of the string coordinates ZM()=(XM, ): Due to the projector, the fermionic parameter () has only 16 independent components. They can be used to gauge away 1/2 of 32 fermionic worldsheet fields ()

  5. unbroken broken susy 24=P24 , 8 =P8 , P24 +P8 = I P24 =1/8 (6-a’b’Ja’b’7), 7 = 16 a’,b’=1,…,6 - CP3 indices, Ja’b’ -Kaehler form on CP3 AdS4£CP3 superbackground • Preserves 24 of 32 susy in type IIA D=10 superspace in contrast:AdS5£S5 background in type IIB string theory preserves all 32 supersymmetries • fermionic modes of the AdS4£CP3 superstring are of different nature: 32()=(24, 8)

  6. they parametrize coset superspace OSp(6|4)/U(3)xSO(1,3) ¾AdS4xCP3 K10,24 (x,y,)= exaPaeya’Pa’e Q 2OSp(6|4)/U(3)xSO(1,3) OSp(6|4) superalgebra: [P,P]=M, [P,M]=P, P ½ AdS4xCP3 M ½ SO(1,3)£ U(3) {Q,Q}=P+M, [Q,P]=Q, [Q,M]=Q OSp(6|4) supercoset sigma model It is natural to try to get rid of the eight “broken susy” fermionic modes 8 using kappa-symmetry 8=0 - partial kappa-symmetry gauge fixing Remaining string modes are: 10 (AdS4 £CP3) bosons xa () (a=0,1,2,3), ya’ () (a’=1,2,3,4,5,6) 24 fermions () corresponding to unbroken susy

  7. Cartan forms: K-1dK=Ea(x,y,) Pa + Ea’(x,y,) Pa’ + E’(x,y,) Q’ + (x,y,) M Superstring action onOSp(6|4)/U(3)xSO(1,3)– classically integrable (Arutyunov & Frolov; Stefanskij; D'Auria, Frè, Grassi & Trigiante, 2008) 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 AdS4 only. only ½ of 8 can be eliminated [(1+), P8 ]=0 OSp(6|4) supercoset sigma model S=s d2 (- det EiAEjBAB)1/2 + s E’ÆE’ J’’ C Reason – kappa-gauge fixing 8 =P8=0is inconsistent in the AdS4 region

  8. CP3 vielbein and U(1) connection em’a’ (y) Am’(y) S7 vielbein: ema= 0 1 F2~dA=JCP3 Towards the construction of the complete AdS4£CP3 superspace (with 32  ) • We shall construct this superspace by performing the dimension reduction of D=11 coset superspace OSp(8|4)/SO(7)£SO(1,3) it has 32  and subspace AdS4£S7, SO(2,3) £ SO(8) ½ OSp(8|4) • AdS4£CP3sugra solutionis related to AdS4£S7(with 32 susy)by dimenisonal reduction (Nilsson and Pope; D.S., Tkach & Volkov 1984) Geometrical ground: S7 is the Hopf fibration over CP3 with S1 fiber: (does not depend on S1 fiber coordinate z) Our goal is to extend this structure toOSp(8|4)/SO(7)£SO(1,3)

  9. SO(8) SO(7) S = - symmetric space KS7(y,z)=KCP3(y) ezT1=eya’Pa’ ezT1 (T1 is U(1) generator) Hopf fibration of S7 as a coset space SU(4)£U(1) SU(3)£U(1) SU(4)£U(1) ½ SO(8) As a Hopf fibration, locally, S7=CP3£ U(1) SU(4)£U(1) SU(3)£Ud(1) CP3= SU(4)/SU(3)£U’(1) – symmetric space Coset representative and Cartan forms: K-1S7 dKS7 = ea’(y) Pa’ + (y) MSU(3)+A(y) T2 + dz T1 (T2½U’(1)) = dym’em’a’(y) Pa’ +(dz+ dym’Am’(y)) P7 + (dz - A(y)) Td+ (y) MSU(3) Td=1/2 (T1-T2) is Ud(1) generator, P7=1/2 (T1+T2) – translation along S1 fiber

  10. Hopf fibration of OSp(8|4)/SO(7)£SO(1,3) • K11,32 - D=11 superspace with the bosonic subspace AdS4£S7 and 32 fermionic directions K11,32 = M10,32£S1 (locally) base fiber • M10,32 - D=10 superspace with the bosonic subspace AdS4£CP3 and 32 fermionic directions (it is not a coset space) M10,32is the superspace we are looking for!

  11. K11,24 = OSp(6|4) £U(1) OSp(6|4)£U(1) ½OSp(8|4) SO(1,3)£SU(3)£Ud(1) Cartan forms: K-1dK = Ea(x,y,) Pa + Ea’(x,y,) Pa’ + E’(x,y,) Q’ +(dz+ A(x,y,)) P7 + (dz - A) Td + (x,y,) MSU(3) 1st step: Hopf fibration overK10,24 = OSp(6|4)/U(3)xSO(1,3) • K11,24 = K10,24£S1 (locally) ¾AdS4 x S7 K11,24 (x,y,,z)= K10,24 (x,y,) ezT1= exaPaeya’Pa’e Q24ezT1

  12. OSp(6|4) superalgebra: OSp(2|4) superalgebra: {Q24,Q24}=MSO(2,3)+MSU(4) {Q8,Q8}=MSO(2,3)+T1 Extension to OSp(8|4): {Q24,Q8}=M?½ SO(8)/SU(4)£U(1) 2nd step: adding 8 fermionic directions 8 K11,32(x,y,,z,)=K11,24 (x,y,,z) eQ8 = K10,24 (x,y,) ezT1 eQ8 Q8are 8 susy generators which extend OSp(6|4) to OSp(8|4) Cartan forms ofOSp(8|4)/SO(7)£SO(1,3): K-1dK = [Ea(x,y,,)+dz Va()]Pa + Ea’(x,y,,) Pa’ +[dz()+ A(x,y,,)] P7+ E24(x,y,,) Q24 + E8(x,y, ,) Q8 + connection terms

  13. Pa P7 3rd step: Lorentz rotation in AdS4 x S1 (fiber) tangent space K-1dK = (Ea(x,y,,)+dz Va())Pa + Ea’(x,y,,) Pa’ +(dz()+ A(x,y,,)) P7+ E24(x,y,,) Q24 + E8(x,y, ,) Q8 + connection terms Lorentz transformation of the supervielbeins: Ea(x,y,,)= (Eb(x,y,,)+dz Vb()) ba() + (dz() + A(x,y,,)) 7a() M10,32 Ea’(x,y,,)= Ea’(x,y,,) E24(x,y,,), E8(x,y, ,) – rotated 32 fermionic vielbeins S1:dz ()+ A (x,y,,) =(dz() + A(x,y,,)) 77() + (Eb(x,y,,) + dz Vb()) b7()

  14. Guage fixed superstring action in AdS4£CP3 (upon a T-dualization), P.A.Grassi, L.Wulff and D.S. arXiv:0903.5407 =(24, 8),=1/2(1+ 012)  AdS4 CP3  = 012 3 The action contains fermionic terms up to a quartic order only

  15. Conclusion • The complete AdS4 £CP3 superspace with 32 fermionic directions has been constructed. Its superisometry is OSp(6|4) • The explicit form of the actions for Type IIA superstring and D-branes in AdS4 £CP3 superspace have been derived • A simple gauge-fixed form of the superstring action was obtained • a light-cone gauge fixing of the superstring action has been recently considered by D. Uvarov arXiv:0906.4699 • These results can be used for studying various problems of String Theory in AdS4 £CP3, in particular, to perform higher-loop computations (involving fermionic modes) for testing AdS4 /CFT3 correspondence: integrability, Bethe ansatz, S-matrix etc. in the dual planar N=6 CS-matter theory

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