1. 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: Jaume Gomis, Linus Wulff and D.S.
ArXiv: P.A.Grassi, L.Wulff and D.S.
Sakharov Conference, Moscow, 18 May 2009

2. 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.

3. 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

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():
Let us now consider a particular background of our present interest which is
Due to the projector, the fermionic parameter () has only independent components. They can be used to gauge away 1/2 of 32 fermionic worldsheet fields ()

5. 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 = 16
a’,b’=1,…,6 - CP3 indices, Ja’b’ » Fa’b’ Kaehler form on CP3

6. 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

7. 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

8. 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)
(does not depend on S1 fiber coordinate z)
Our goal is to extend this structure to OSp(8|4)/SO(7)£SO(1,3)

9. 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

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)
M10,32 is the superspace we are looking for!

11. 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:

12. 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
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

13. 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()

14. 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:
 =(24, 8),  =1/2(1+ 012) 
AdS 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
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.