1、AdS4 CP3 superspace,Dmitri Sorokin INFN, Sezione di Padova,ArXiv:0811.1566 Jaume Gomis, Linus Wulff and D.S.,SQS09, Dubna, 30 July 2009,ArXiv:0903.5407 P.A.Grassi, L.Wulff and D.S.,2,AdS4/CFT3 correspondence,Holographic duality between 3d supersymmetric Chern-Simons-matter theories (BLG & ABJM) and
2、M/String Theory compactified on AdS4 space,Matter: 8 scalars and 8 spinors in bi-fund. rep. of U(N)k U(N)-k,ChernSimons gauge fields: A and A0,For a certain V(,) the theory has N=6 superconformal symmetry OSp(6|4) SO(6)Sp(4) SU(4)SO(2,3),3,AdS4/CFT3 correspondence,Holographic duality between 3d supe
3、rsymmetric 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 CP3this D=10 background preserves 24 of 32 susy, i.e. N=6 from the AdS4 persp
4、ective, 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.,4,AdS4 CFT3 correspondence,In a limit N2 5/2, where =N/k is a t Hooft coupling, the bulk description of the ABJM model is gi
5、ven by type IIA string theory on AdS4 CP3this D=10 background preserves 24 of 32 susy, i.e. N=6 from the AdS4 perspective, and its isometry is OSp(6|4)F4= k 1/2 d4xAdS4 , F2=k JCP3, e2=5/2/N2 = g2strTo study this AdS/CFT duality, it is necessary to know an explicit form of the Green-Schwarz string a
6、ction in the AdS4 CP3 superbackground.,5,Green-Schwarz superstring,in a generic supergravity background,Type IIA sugra fields gMN, , BMN, AM, ALMN, M, are contained in EA and B2,6,Fermionic kappa-symmetry,Provided that the superbackground satisfies superfield supergravity constraints (or, equivalent
7、ly, 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 world
8、sheet fields (),7,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),8,OSp(6|4) supercoset sigma mode
9、l,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,9,OSp(6|4) supercoset
10、sigma model,S=s d2 (- det EiAEjB AB)1/2 + s E E J C,Reason kappa-gauge fixing 8 = P8 =0 is inconsistent in the AdS4 region,10,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(
11、8|4)/SO(7)SO(1,3) it has 32 and subspace AdS4 S7, SO(2,3) SO(8) 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:,(does not depend on S1 fibe
12、r coordinate z),Our goal is to extend this structure to OSp(8|4)/SO(7)SO(1,3),11,Hopf fibration of S7 as a coset space,SU(4)U(1) SU(3)U(1),As a Hopf fibration, locally, S7=CP3 U(1),CP3= SU(4)/SU(3)U(1) symmetric space,SU(4)U(1) SU(3)Ud(1),SU(4)U(1) SO(8),Coset representative and Cartan forms:,K-1S7
13、dKS7 = ea(y) Pa + (y) MSU(3)+A(y) T2 + dz T1 (T2 U(1),= dymema(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,12,Hopf fibration of OSp(8|4)/SO(7)SO(1,3),K11,32 - D=11 superspace with the bosonic subspace AdS4 S7 and
14、 32 fermionic directions,K11,32 = M10,32 S1 (locally),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!,base fiber,13,1st step: Hopf fibration over K10,24 = OSp(6|4)/U(3)xSO(1,3),K11,24 = K10
15、,24 S1 (locally) AdS4 x S7,K11,24 (x,y,z)= K10,24 (x,y,) ezT1= exaPa eyaPa e Q24 ezT1,14,2nd 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),Cartan forms of OSp(8|4)/SO(7)SO(1,3):,K-1dK = Ea(
16、x,y,)+dz Va() Pa + Ea(x,y,) Pa+dz ()+ A(x,y,) P7 + E24(x,y,) Q24 + E8(x,y, ,) Q8+ connection terms,15,3rd 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,Ea(x,y,) = (Eb(x,y,)+dz
17、Vb() ba()+ (dz () + A(x,y,) 7a(),Lorentz transformation of the supervielbeins:,S1: dz ()+ A (x,y,) = (dz () + A(x,y,) 77()+ (Eb(x,y,) + dz Vb() b7(),Ea(x,y,) = Ea(x,y,),E24(x,y,), E8(x,y, ,) rotated 32 fermionic vielbeins,M10,32,16,Superstring action in AdS4 CP3 superspace,gij=EiA EjB AB induced wor
18、ldsheet metric,NS-NS field B2 is obtained by dimensional reduction of A3 in D=11,- parametrize 3d slice of AdS4,- “chirality” condition on the AdS boundary,17,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) ,The
19、action contains fermionic terms up to a quartic order only,AdS4 CP3, = 012 3,18,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 ha
20、ve been derivedA simple gauge-fixed form of the superstring action was obtaineda 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,
