Type IIA string in AdS4 £ CP3 superspace Dmitri Sorokin INFN, Sezione di Padova The motivation for this work was an explosion of activity which happened last year in studying a new type of AdS/CFT correspondence. ArXiv:0811.1566 Jaume Gomis, Linus Wulff and D.S. ArXiv:0903.5407 P.A.Grassi, L.Wulff and D.S. Sakharov Conference, Moscow, 18 May 2009
AdS4/CFT3 correspondence 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£ CP3 superbackground.
Green-Schwarz superstring in a generic supergravity background Before explaining how one can derive these quantities, let me first complete the generic overview of the superstring action by presenting its important local fermionic symmetry which is called kappa-symmetry Type IIA sugra fields gMN, , BMN, AM, ALMN, M, are contained in EA and B2 Our goal is to get the explicit form of B2 , EA and E as polynomials of for the AdS4 £ CP3 case
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(): Let us now consider a particular background of our present interest which is 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 ()
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) unbroken broken susy Less amount of supersymmetry in AdS4xCP3 case makes life more complicated but, also more interesting in many respects. 24=P24 , 8 =P8 , P24 + P8 = I P24 =1/8 (6-a’b’Ja’b’7), 7 = 16 a’,b’=1,…,6 - CP3 indices, Ja’b’ » Fa’b’ Kaehler form on CP3
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 they parametrize coset superspace OSp(6|4)/U(3)xSO(1,3) ¾ AdS4xCP3 K10,24 (x,y,)= exaPa eya’Pa’ e Q 2 OSp(6|4)/U(3)xSO(1,3) [P,P]=M, [P,M]=P, P ½ AdS4xCP3 M ½ SO(1,3)£ U(3) OSp(6|4) superalgebra: {Q,Q}=P+M, [Q,P]=Q, [Q,M]=Q
OSp(6|4) supercoset sigma model Cartan forms: K-1dK=Ea(x,y,) Pa + Ea’(x,y,) Pa’ + E’(x,y,) Q’ + (x,y,) M Superstring action on OSp(6|4)/U(3)xSO(1,3) – classically integrable (Arutyunov & Frolov; Stefanskij; D'Auria, Frè, Grassi & Trigiante, 2008) S=s d2 (- det EiAEjB AB)1/2 + s E’Æ E’ J’’ C 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. Reason – kappa-gauge fixing 8 = P8 =0 is inconsistent in the AdS4 region [(1+) , P8 ]=0 only ½ of 8 can be eliminated
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£ CP3 sugra solution is 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: I.e. we should find a coordinate realization of the OSp(8|4) supercoset space which reflects its geometry as a Hopf fibration CP3 vielbein and U(1) connection F2~dA=JCP3 S7 vielbein: ema= em’a’ (y) Am’(y) 0 1 (does not depend on S1 fiber coordinate z) Our goal is to extend this structure to OSp(8|4)/SO(7)£SO(1,3)
Hopf fibration of S7 as a coset space SU(4)£U(1) SU(3)£U(1) S = SO(8) SO(7) - symmetric space 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: KS7(y,z)=KCP3(y) ezT1=eya’Pa’ ezT1 (T1 is U(1) generator) K-1S7 dKS7 = ea’(y) Pa’ + (y) MSU(3)+A(y) T2 + dz T1 (T2 ½ U’(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
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) M10,32 is the superspace we are looking for!
1st step: Hopf fibration over K10,24 = OSp(6|4)/U(3)xSO(1,3) K11,24 = K10,24 £ S1 (locally) ¾ AdS4 x S7 SO(1,3)£SU(3)£Ud(1) OSp(6|4) £ U(1) K11,24 = OSp(6|4)£U(1) ½ OSp(8|4) K11,24 (x,y,,z)= K10,24 (x,y,) ezT1= exaPa eya’Pa’ e Q24 ezT1 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) Cartan forms:
2nd step: adding 8 fermionic directions 8 K11,32 (x,y,,z,)=K11,24 (x,y,,z) eQ8 = K10,24 (x,y,) ezT1 eQ8 Q8 are 8 susy generators which extend OSp(6|4) to OSp(8|4) OSp(6|4) superalgebra: {Q24,Q24}=MSO(2,3)+MSU(4) {Q8,Q8}=MSO(2,3)+T1 OSp(2|4) superalgebra: Extension to OSp(8|4): {Q24,Q8}=M? ½ SO(8)/SU(4)£U(1) Cartan forms of OSp(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
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 Pa P7 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()
Guage fixed superstring action in AdS4£ CP3 (upon a T-dualization), P 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+ 012) AdS4 CP3 = 012 3 The action contains fermionic terms up to a quartic order only
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 The simplest possible gauge-fixed form of the superstring action has been obtained These results can be use 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 To the best of my knowledge, this is the first example of a complete superspace which describes a supergravity background with non-maximal number of supersymmetries.