PeculiaritiesPeculiarities ofof StringString TheoryTheory onon 33 AdSAdS44 ×× CPCP DmitriDmitri SorokinSorokin INFN, Sezione di Padova ArXiv:0811.1566 J. Gomis, D.S., L. Wulff ArXiv:0903.5407 P.A.Grassi, D.S., L.Wulff ArXiv:0911.5228 A.Cagnazzo, D.S., L.Wulff ArXiv:1009.3498 D.S. and L. Wulff GGI, Arcetri, September 30, 2010 3 5 AdSAdS4×× CPCP versusversus AdSAdS5×× SS (peculiarities and issues) 3 Superstring theory on AdS4xCP is not maximally supersymmetric, it preserves 24 of 32 susy. The symmetry group is OSp(6|4) – The complete theory is not described by a supercoset sigma-model – The proof of it’s classical integrability turned out to be much tricky some issues have not beebeenn completely settled on the boundaboundaryry and bulk side of the AdS4/CFT3 holography: – Bethe ansatz CFT anomalous dimensions versus spinning string energy – subtleties in matching worldsheet degrees of freedom with those of S-matrix scattering theory (light and heavy worldsheet modes) – issue of the dual superconformal symmetry and fermionic T-duality 3 3 The AdS4xCP theory admits string instantons wrapping 2-cycles of CP . 2 GreenGreen--SchwarzSchwarz superstringsuperstring in a generic supergravity background S= − d2 ξ − det g− B ∫ ij ∫ 2 (ZXM= ,M ), Θα M = 0,1,...,9; α =1 ,..., 32 M α et ( ,pullbackBX )2 -Θ of worldshe the NS - NS 2 - form guage field A B gij= E- i j η EAB worldsheeinduced t metric A Μ A ( ) (EZEX ,i = ) ∂i - ξ M Θpullback ofin supevielbe theof D vector= 10 sugra 0 , 1A = ,..., 9 ; α Μ α E(= dZ ,M ) E -Θ X einspinorsupervielb of D= 10 sugra A A A BC A A BC Θ =E( +M ,X ) Ψ eM ( Γ ) XM Θ ( + ) X ωM ΘΓ BC ΓH MBC Θ + ΘΓ Γ Θ Φ A BC A BCDK + ΘΓe Γ F BC Γ ΘM +FBCDK ΘΓM Γ Γ Θ + ⋅ ⋅ ⋅ are obtained from ty superfieldconstraintty s= supergraviequationssupergravi of motion : A AA B α A β 3 T XΘ ≡ dE + Ω B E2( = i ,αβ E ) Γ E FermionicFermionic kappakappa--symmetrysymmetry Provided that the superbackground satisfies superfield supergravity aints (or, equivalently,nstrcosugra uperstring the GS s),nseld equatioif action is invariant under the following local worldsheet transformations M M α M of the string coordinates Z (ξ)=(X , Θ ), δκ Z : Μ A Μ α 1 α β ZEXδ( , )Θ 0 , = ZEX δ (,)Θ =( Ι ) +κβ Γ ( ξ ), κ M κ M 2 1 Γ = εEEij,ΓAB Γ 11 Γ2, = Ι tr Γ = 0 2− detg i j AB Due to the projector, the fermionic parameter κα(ξ) has only 16 independent components. They caf 1/2 oy gauge awan be used to 32 fermionic worldsheet fields Θα(ξ) 4 33 AdSAdS44×× CPCP superbackgroundsuperbackground Preserves 24 of 32 susy in type IIA D=10 superspace • The superstring action is not a supercoset sigma-model of OSp(6|4) • the explicit proof of the classical integrability of the complete AdS4 x CP3 superstring has been lacking until recently (D.S.& L.Wulff, 09/2010) 3 fermionic modes of the AdS4× CP superstring are of different nature: Θ32(ξ)=(ϑ24, υ8) unbroken broken susy *F =F ϑ24=P24 Θ, υ8 =P8 Θ, P24 + P8 = I 4 6 a’b’ 1L 6 P24 =1/8 (6-Γ Ja’b’Γ7), Γ7 = Γ1LΓ6 ε 3 3 a’,b’=1,…,6 - CP indices, Ja’b’ - Kaehler form on CP 5 F2 OSpOSp(6|4)(6|4) supercosetsupercoset sigmasigma modelmodel It is natural to try to get rid of the eight “broken susy” fermionic modes υ8 using kappa-symmetry υ8=0 - partial kappa-symmetry gauge fixing Remaining string modes are: 3 a a’ 10 (AdS4 ×CP ) bosons x (ξ) (a=0,1,2,3), y (ξ) (a’=1,2,3,4,5,6) 24 fermions ϑ(ξ) corresponding to unbroken susy they parametrize coset superspace OSp(6|4)/U(3)xSO(1,3) ⊃ AdS4xCP3 5 - similar to the AdS5 × S string action on SU(2,2|4)/SO(1,4)xSO(5) Cartan forms: α=1,…,4 α’=1,…,6 -1 a a’ αα’ K dK=E (x,y,ϑ) Pa + E (x,y,ϑ) Pa’ + E (x,y,ϑ) Qαα’ + Ω (x,y,ϑ) M Sigma-model action on OSp(6|4)/U(3)xSO(1,3) (Arutyunov & Frolov; Stefanskij; D'Auria, Frè, Grassi & Trigiante, 2008) 2 A B 1/2 αα’ ββ’ 5 S=∫ d ξ (- det Ei Ej ηAB) + ∫ E ∧ E Jα’β’ γ αβ 6 OSpOSp(6|4)(6|4) supercosetsupercoset sigmasigma modelmodel 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, or the string instanton in CP3 Reason – kappa-gauge fixing υ8 = P8Θ =0 is inconsistent in the AdS4 region and for the string instanton in CP3 only ½ of υ can be eliminated [(1+Γκ) , P8 ]=0 υ8 To describe these string configurations the GS superstring action in 3 AdS4 x CP superspace with 32 fermionic coordinates is required (it is not a coset superspace) 3 7 AdS4× CP sugra solution is related to AdS4× S (with 32 susy) in D=11 by dimenisonal reduction (Nilsson and Pope; D.S., Tkach & Volkov 1984) This superspace was construconstructedcted by performing the dimensional reduction of D=11 coset superspace OSp(8|4)/SO(7)xSO(1,3) which has 32 Θ and 7 bose subspace AdS4 x S , SO(2,3)xSO(8) ⊂ OSp(8|4) (Gomis, D.S., Wulff, 2008) 7 HopfHopf fibrationfibration ofof OSpOSp(8|4)/(8|4)/SOSO(7)(7)××SOSO(1,3)(1,3) KK11,32 - D=11 superspace with the bosonic subspace 7 AdS4× S and 32 fermionic directions 1 KK11,32 == MM10,32 ×× S (locally) base fiber MM10,32 - D=10 superspace with the bosonic subspace 3 AdS4× CP , 32 fermionic directions and OSp(6|4) isometry (but it is not a coset space) MM10,32 is the superspace we are looking for 8 ClassicalClassical IntegrabilityIntegrability ofof 2d2d dynamicaldynamical systemssystems The existence of ∞ # of conserved currents and charges The charges are generated by the Lax connection L L (ξ,z) – 2d one-form which depends on a spectral parameter z,z, takes values in a symmetry algebra and has zero curvature: dL + L ∧ L = 0 (on the mass-shell) The integrability is proven if one manages to construct L (ξ,z) No generic prescription exists how to do this 9 ClassicalClassical IntegrabilityIntegrability ofof 2d2d dynamicaldynamical systemssystems G/H supercoset sigma-models with Z4-grading, e.g. 5 - AdS5 × S superstring, SU(2,2|4)/SO(1,4) × SO(5), Bena, Roiban, Polchinski ‘03 - OSp(6|4)/SO(1,3) × U(3) sigma-model Cartan forms: -1 2 1 3 K dK=Ω (x,ϑ) M0 + E (x,ϑ) P2 + E (x,ϑ) Q1 + E (x,ϑ) Q3 [M0,M0]=M0, [P2,P2]=M0, {Q1,Q1}=P2={Q3,Q3}, {Q1,Q3}=M0 Lax connection: 2 2 1 3 L = Ω (x,ϑ)+l1 E (x,ϑ)+l2*E (x,ϑ)+l3 E (x,ϑ)+l4 E (x,ϑ) Coefficients l =f (z) are functions of the dL + L ∧ L = 0 i i spectral parameter on shell 10 3 LaxLax connectionconnection ofof thethe AdSAdS4×× CPCP superstringsuperstring (D.S. & L. Wulff, ArXiv:1009.3498) Use the OSp(6|4) conserved Noether current J=JB+JS SO(2,3) × SU(4) M A [AB] JB(X,ϑ,υ)= dX KM (X)+J1 (X,ϑ,υ)KA+J2 (X,ϑ,υ)KAKB JS(X,ϑ,υ) – susy current Lax connection: L = α K(X) + α *J + (α )2 J + α α *J - α (β J - β *J ) L 1 2 B 2 2 1 2 2 2 1 S 2 S 4 + O (X,ϑ,υ ) + L 11 3 SuperstringSuperstring actionaction inin AdSAdS4×× CPCP superspacesuperspace (up to the second order in fermions and Wick-rotated) [Cvetic’ et al. 1999] 3 AdS4 CP F4 and F2 fluxes Bosonic instanton solution In CP3 there is a topologically nontrivial 2-cycle ~ S2 associated with the Kahler 2-form J2 In the Wick-rotated theory, string worldsheet can wrap this S2 ⊂ CP3, thus forming a stringy instanton. It is ½ BPS Θ=0 and xa=const (AdS-coordinates); on CP3 complex yI=yI(z) are holomorphic functions of the worldsheet coordinates the Virasoro constraints are identically satisfied: 12 StringString instantoninstanton onon CPCP33 A.Cagnazzo, L.Wulff and D.S. ArXiv:0911.5228 ABJABJ Instanton action * 2 ∫ B2, B2 ∼ J2, d J2=d J2=0 – Kaehler form on CP Θ=1/2(1-Γ)Θ = (ϑ, υ) – gauged fixed kappa-symmetry, 16 physical fermions Twelve fermionic zero modes, i.e. solutions of the 2d Dirac equation: υ=0 8 zero modes are 4 pairs of Killing spinors on S2 4 zero modes are massless charged fermions interacting with a monopole field on S2 13 FermionicFermionic zerozero modesmodes The 12 fermionic zero modes are goldstinos which manifest breaking of 3 the ½ susy of the AdS4 x CP background by the string instanton M A A 24 target-space susy: δϑ =εεKilling , δ X eM (X) =iϑ Γ εεKilling 3 AdS4 x CP Killing spinor equation projected on the instanton 2 i (1+Γ ) ϑ=0 S κ ( )∇i 0 +γ i ϑ + = 8 fermions i R ∇( M +γM Γ)5 ϑ 0 = S 2 R ( )∇i 0 +iAγi 3 ϑ − =4 fermions This fermionic zero modes solve also the complete non-linear string e.o.m 14 DiscussionDiscussion There also exists an NS5-brane instanton wrapping the whole CP3 What are the 3d CFT counterparts of the string/brane instantons? What are possible effects of the stringy instantons on the structure of the supergravity and superstring theory on 3 AdS4 ×CP ? May their existence result in peculiarities of the AdS4 /CFT3 correspondence? Drukker, Mariño & Putrov (arXiv:1007.3837) found contributions coming from world-sheet instanons to the partition function and Wilson loop observables computed in a matrix model description of ABJ(M) theory 15.