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

2. 2 AdS 4 / CFT 3 correspondence  Holographic duality between 3d supersymmetric Chern-Simons-matter theories (BLG & ABJM) and M/String Theory compactified on AdS 4 space Matter: 8 scalars  and 8 spinors  in bi-fund. rep. of U(N) k £ U(N) -k Chern—Simons gauge fields: A  and A 0  For a certain V(,) the theory has N =6 superconformal symmetry OSp(6|4) ¾ SO(6) £ Sp(4)~ SU(4) £ SO(2,3)

3. 3 AdS 4 / CFT 3 correspondence  Holographic duality between 3d supersymmetric Chern-Simons- matter theories (BLG & ABJM) with SU(N)xSU(N) and M/String Theory compactified on AdS 4 space  In a certain limit the bulk dual description of the ABJM model is given by type IIA string theory on AdS 4 £ CP 3  this D=10 background preserves 24 of 32 susy, i.e. N =6 from the AdS 4 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 AdS 4 £ CP 3 superbackground.

4. 4 AdS 4 £ CFT 3 correspondence  In a limit N 2 À 5/2, where =N/k is a `t Hooft coupling, the bulk description of the ABJM model is given by type IIA string theory on AdS 4 £ CP 3  this D=10 background preserves 24 of 32 susy, i.e. N =6 from the AdS 4 perspective, and its isometry is OSp(6|4) F 4 = k 1/2 d 4 x AdS 4, F 2 =k J CP 3, e 2  = 5/2 /N 2 = g 2 str  To study this AdS/CFT duality, it is necessary to know an explicit form of the Green-Schwarz string action in the AdS 4 £ CP 3 superbackground.

5. 5 Green-Schwarz superstring in a generic supergravity background Our goal is to get the explicit form of B 2, E A and E  as polynomials of  for the AdS 4 £ CP 3 case Type IIA sugra fields g MN, , B MN, A M, A LMN,  M ,  are contained in E A and B 2

6. 6 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 Z M (  )= (X M,   ) : 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   ()

7. 7 AdS 4 £ CP 3 superbackground  Preserves 24 of 32 susy in type IIA D=10 superspace in contrast: AdS 5 £ S 5 background in type IIB string theory preserves all 32 supersymmetries  fermionic modes of the AdS 4 £ CP 3 superstring are of different nature:  32 ()=( 24,  8 ) unbroken broken susy 24  P 24  8  P 8  P 24 + P 8 = I 24  P 24  8  P 8  P 24 + P 8 = I P 24 = 1/8 ( 6 - a’b’ J a’b’  7 ),  7 =  1   6 a’,b’=1,…,6 - CP 3 indices, J a’b’ - Kaehler form on CP 3

8. 8 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 (AdS 4 £ CP 3 ) bosons x a () (a=0,1,2,3), y a’ () (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 AdS 4 xCP 3 K 10,24 (x,y,)= e x a P a e y a’ P a’ e Q 2 OSp(6|4)/U(3)xSO(1,3) [P,P]=M,[P,M]=P,P ½ AdS 4 xCP 3 M ½ SO(1,3) £ U(3) OSp(6|4) superalgebra: {Q,Q}=P+M, [Q,P]=Q,[Q,M]=Q

9. 9 OSp(6|4) supercoset sigma model Cartan forms: K -1 dK=E a (x,y,) P a + E a’ (x,y,) P a’ + E ’ (x,y,) Q ’ +  (x,y,) M Superstring action onOSp(6|4)/U(3)xSO(1,3) – classically integrable Superstring action on OSp(6|4)/U(3)xSO(1,3) – classically integrable (Arutyunov & Frolov; Stefanskij; D'Auria, Frè, Grassi & Trigiante, 2008) S= s d 2  (- det E i A E j B  AB ) 1/2 + s E ’ Æ  E ’ J ’’ C  Reason – kappa-gauge fixing  8 P 8  = 0 is inconsistent in the AdS 4 region Reason – kappa-gauge fixing  8 = P 8  = 0 is inconsistent in the AdS 4 region [(1+  ) , P 8 ]=0 only ½ of  8 can be eliminated 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 AdS 4 only.

10. 10 Towards the construction of the complete AdS 4 £ CP 3 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 AdS 4 £ S 7 SO(2,3) £ SO(8) ½ OSp(8|4)  AdS 4 £ CP 3 sugra solution is related to AdS 4 £ S 7 (with 32 susy) by dimenisonal reduction ( Nilsson and Pope; D.S., Tkach & Volkov 1984 ) Geometrical ground: S 7 is the Hopf fibration over CP 3 with S 1 fiber: S 7 vielbein: e m a = e m’ a’ (y) A m’ (y) 0 1 CP 3 vielbein and U(1) connection F 2 ~dA=J CP 3 (does not depend on S 1 fiber coordinate z) Our goal is to extend this structure toOSp(8|4)/SO(7) £ SO(1,3) Our goal is to extend this structure to OSp(8|4)/SO(7) £ SO(1,3)

11. 11 Hopf fibration of S 7 as a coset space SU(4) £ U(1) SU(3) £ U(1) As a Hopf fibration, locally, S 7 =CP 3 £ U(1) S =S =S =S =SO(8)SO(7) - symmetric space CP 3 = SU(4)/SU(3) £ U’(1) – symmetric space SU(4) £ U(1) SU(3) £ U d (1) SU(4) £ U(1) ½ SO(8) Coset representative and Cartan forms: K S 7 (y,z)=K CP 3 (y) e zT 1 =e y a’ P a’ e zT 1 (T 1 is U(1) generator) K -1 S 7 dK S 7 = e a’ (y) P a’ +  (y) M SU(3) +A(y) T 2 + dz T 1 ( T 2 ½ U ’ (1) ) = dy m’ e m’ a’ (y) P a’ +(dz+ dy m’ A m’ (y)) P 7 + (dz - A(y)) T d +  (y) M SU(3) T d =1/2 (T 1 -T 2 ) is U d (1) generator, P 7 =1/2 (T 1 +T 2 ) – translation along S 1 fiber

12. 12 Hopf fibration of OSp(8|4)/SO(7)£SO(1,3)  K 11,32 - D=11 superspace with the bosonic subspace AdS 4 £ S 7 and 32 fermionic directions K 11,32 = M 10,32 £ S 1 (locally)  M 10,32 - D=10 superspace with the bosonic subspace AdS 4 £ CP 3 and 32 fermionic directions (it is not a coset space) M 10,32 is the superspace we are looking for! base fiber

13. 13 1 st step: Hopf fibration over K 10,24 = OSp(6|4)/U(3)xSO(1,3)  K 11,24 = K 10,24 £ S 1 (locally) ¾ AdS 4 x S 7 SO(1,3) £ SU(3) £ U d (1) OSp(6|4) £ U(1) K 11,24 = OSp(6|4) £ U(1) ½ OSp(8|4) K 11,24 (x,y,,z)= K 10,24 (x,y,) e zT 1 = e x a P a e y a’ P a’ e  Q 24 e zT 1 K -1 dK = E a (x,y,) P a + E a’ (x,y,) P a’ + E ’ (x,y,) Q ’ +(dz+ A (x,y,) ) P 7 + (dz - A) T d +  (x,y,) M SU(3) +(dz+ A (x,y,) ) P 7 + (dz - A) T d +  (x,y,) M SU(3) Cartan forms:

14. 14 2 nd step: adding 8 fermionic directions  8 K 11,32 (x,y,,z,) =K 11,24 (x,y,,z) e Q 8 = K 10,24 (x,y,) e zT 1 e Q 8 Q 8 are 8 susy generators which extend OSp(6|4) to OSp(8|4) OSp(6|4) superalgebra: {Q 24,Q 24 }=M SO(2,3) +M SU(4) {Q 8,Q 8 }=M SO(2,3) +T 1 OSp(2|4) superalgebra: Extension to OSp(8|4): {Q 24,Q 8 }=M ? ½ SO(8)/SU(4) £ U(1) Cartan forms of OSp(8|4)/SO(7)£SO(1,3): K -1 dK = [E a (x,y,)+dz V a ()] P a + E a’ (x,y,, ) P a’ +[dz (  )+ A (x,y,,) ] P 7 + E  (x,y,, ) Q  + E  (x,y,, ) Q  +[dz (  )+ A (x,y,,) ] P 7 + E  (x,y,, ) Q  + E  (x,y,, ) Q   + connection terms

15. 15 3 rd step: Lorentz rotation in AdS 4 x S 1 (fiber) tangent space K -1 dK = (E a (x,y,)+dz V a ()) P a + E a’ (x,y,, ) P a’ +(dz (  )+ A (x,y,,) ) P 7 + E  (x,y,, ) Q  + E  (x,y,, ) Q  +(dz (  )+ A (x,y,,) ) P 7 + E  (x,y,, ) Q  + E  (x,y,, ) Q   + connection terms E a (x,y,,) = (E b (x,y,)+dz V b ())  b a () E a (x,y,,) = (E b (x,y,)+dz V b ())  b a () + (dz () + A(x,y,,))  7 a () + (dz () + A(x,y,,))  7 a () Lorentz transformation of the supervielbeins: S 1 : dz()+ A (x,y,,) = (dz () + A(x,y,,))  7 7 () + (E b (x,y,) + dz V b ())  b 7 () + (E b (x,y,) + dz V b ())  b 7 () E a’ (x,y,,) = E a’ (x,y,, ) E  (x,y,, ), E  (x,y,, ) – rotated 32 fermionic vielbeins M 10,32 PaPaPaPa P7P7P7P7

16. 16 Superstring action in AdS 4 £ CP 3 superspace g ij = E i A E j B  AB – induced worldsheet metric NS-NS field B 2 is obtained by dimensional reduction of A 3 in D=11 Kappa-symmetry gauge fixing ( P.A.Grassi, L.Wulff and D.S. arXiv:0903.5407 ) (Killing spinor, supersolvable, or superconformal gauge, 1998) - parametrize 3d slice of AdS 4 - “chirality” condition on the AdS boundary

17. 17 Guage fixed superstring action in AdS 4 £ CP 3 (upon a T-dualization), P.A.Grassi, L.Wulff and D.S. arXiv:0903.5407 =( ,   ),= 1/2(1+  0  1  2 )  =( ,   ),= 1/2(1+  0  1  2 )  The action contains fermionic terms up to a quartic order only AdS 4 CP 3  0  1  2  3

18. 18 Conclusion  The complete AdS 4 £ CP 3 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 AdS 4 £ CP 3 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 AdS 4 £ CP 3, in particular, to perform higher-loop computations (involving fermionic modes) for testing AdS 4 /CFT 3 correspondence: integrability, Bethe ansatz, S-matrix etc. in the dual planar N =6 CS-matter theory