Peculiarities of String Theory on Peculiarities of String Theory on - PowerPoint PPT Presentation

Peculiarities of String Theory on Peculiarities of String Theory on 3 × CP CP 3 AdS 4 4 × AdS Dmitri Sorokin Dmitri Sorokin INFN, Sezione Sezione di di Padova Padova INFN, ArXiv:0811.1566 ArXiv :0811.1566 J. J. Gomis Gomis, , D.S D.S., ., L. L. Wulff Wulff : 0903.5407 0903.5407 P.A.Grassi P.A.Grassi, D.S., , D.S., L.Wulff L.Wulff ArXiv: ArXiv ArXiv:0911.5228 A.Cagnazzo A.Cagnazzo, D.S., , D.S., L.Wulff L.Wulff ArXiv:0911.5228 ArXiv: 1009.3498 D.S. and L. 1009.3498 D.S. and L. Wulff Wulff ArXiv: GGI, Arcetri Arcetri, , September September 30, 2010 30, 2010 GGI,

3 versus 5 × CP CP 3 × S S 5 AdS 4 4 × versus AdS AdS 5 5 × AdS (peculiarities and issues) (peculiarities and issues) 3 is not maximally supersymmetric, it Superstring theory on AdS 4 xCP CP 3 is not maximally supersymmetric, it Superstring theory on AdS 4 x � � preserves 24 of 32 susy. The symmetry group is OSp(6| 4) preserves 24 of 32 susy. The symmetry group is OSp(6| 4) – The The complete complete theory is not described by a supercoset sigma theory is not described by a supercoset sigma- - model model – – The proof of it The proof of it ’ ’s classical integrability turned out to be much tricky s classical integrability turned out to be much tricky – some issues have not been completely settled on the boundary and bulk bulk some issues have not been completely settled on the boundary and � � side of the AdS 4 / CFT CFT 3 holography: side of the AdS 4 / 3 holography: – Bethe Bethe ansatz ansatz CFT anomalous dimensions versus spinning string energy CFT anomalous dimensions versus spinning string energy – – subtleties in matching worldsheet degrees of freedom with those subtleties in matching worldsheet degrees of freedom with those of of – S- S - matrix scattering theory (light and heavy worldsheet modes) matrix scattering theory (light and heavy worldsheet modes) – issue of the dual superconformal symmetry and fermionic T issue of the dual superconformal symmetry and fermionic T- - duality duality – 3 theory The AdS 4 xCP 3 theory admits admits string string instantons instantons wrapping wrapping 2 2- - cycles of CP cycles of CP 3 3 . . The AdS 4 xCP � � 2 2

Green- -Schwarz superstring Schwarz superstring Green in a generic supergravity background in a generic supergravity background ∫ ∫ = − ξ − − 2 det S d g B 2 ij Z M = Θ α = α = M ( , ), 0,1,...,9; 1 ,..., 32 X M Θ α M ( , ) - worldshe et pullback of the NS - NS 2 - form guage field B X 2 = η A B - induced worldshee t metric g E E ij i j AB Μ ξ = ∂ Θ = A A ( ) ( , ) - pullback of the vector supevielbe in of D 10 sugra E Z E X M i i = 0 , 1 ,..., 9 ; A Μ α α = Θ = ( , ) - spinor supervielb ein of D 10 sugra E dZ E X M Θ = + Ψ Γ Θ + ω ΘΓ Γ Θ + ΘΓ Γ Θ A A A BC A A BC ( , ) ( ) ( ) E X e X X H M M M M BC MBC Φ + ΘΓ Γ Γ Θ + ΘΓ Γ Γ Θ + ⋅ ⋅ ⋅ A BC A BCDK e F F BC M BCDK M = are obtained from superfield supergravi ty constraint s supergravi ty equations of motion : α β Θ ≡ + Ω = Γ A A A B A 3 3 ( , ) 2 T X dE E i E E B αβ

Fermionic kappa- -symmetry symmetry Fermionic kappa Provided that the superbackground satisfies superfield supergravity Provided that the superbackground satisfies superfield supergrav ity constraints (or, equivalently, sugra sugra field equations), the GS superstring field equations), the GS superstring constraints (or, equivalently, action is invariant under the following local worldsheet transformations rmations action is invariant under the following local worldsheet transfo Θ α δ κ M : ξ )= , Θ α ), ), δ Z M M ( ( ξ )= ( Z M ( X M , of the string coordinates Z X M : of the string coordinates κ Z 1 α β Μ Μ α δ Θ = δ Θ = Ι + Γ κ ξ A ( , ) 0 , ( , ) ( ) ( ), Z E X Z E X β M M κ κ 2 1 Γ = ε Γ Γ Γ = Ι Γ = A B 11 2 ij , , tr 0 E E i j − AB 2 det g Due to the projector, the fermionic parameter κ κ α ( ξ ξ ) α ( ) has only Due to the projector, the fermionic parameter has only 16 independent components. They can be used to gauge away 1/ 2 of 16 independent components. They can be used to gauge away 1/ 2 of Θ α ( ξ ξ ) 32 fermionic worldsheet fields Θ α ( ) 32 fermionic worldsheet fields 4 4

3 superbackground × CP CP 3 AdS 4 4 × superbackground AdS � � Preserves 24 of 32 susy in type IIA D= 10 superspace 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 complet the explicit proof of the classical integrability of the complet e e • AdS4 x CP3 superstring has been lacking AdS4 x CP3 superstring has been lacking until recently until recently (D.S.& (D.S.& L.Wulff L.Wulff, 09 , 09/2010 /2010) ) 3 superstring are of � fermionic modes of the � CP 3 × CP fermionic modes of the AdS AdS 4 4 × superstring are of Θ 32 ( ξ ξ )= ( )= ( ϑ ϑ 24 υ 8 different nature: Θ , υ ) 32 ( 24 , 8 ) different nature: unbroken broken susy unbroken broken susy ϑ 24 = P Θ, , υ 8 = P Θ, , ϑ 24 = P 24 24 Θ υ 8 = P 8 8 Θ P 24 + P P 8 P = I * F 4 = F 6 8 = I * F 4 = F 24 + 6 - Γ Γ a Γ 7 Γ 7 Γ 1 Γ 6 ε 1 P 24 ’ Γ Γ = Γ L Γ 6 ε L 6 6 P a’ ’b b’ ’ J 1 L = 1/ 8 ( 6 J a ), 1 L 24 = 1/ 8 ( 6 - 7 ), 7 = a’ ’b b’ 3 indices, indices, J J a a’ ’,b ,b’ ’= 1, = 1,… …,6 ,6 - - CP CP 3 - Kaehler Kaehler form on form on CP CP 3 3 a ’ - a’ ’b b’ 5 5 F 2 F 2

OSp (6|4) supercoset sigma model (6|4) supercoset sigma model OSp υ 8 fermionic modes υ It is natural to try to get rid of the eight “ “ broken susy broken susy” ” fermionic modes It is natural to try to get rid of the eight 8 using kappa- - symmetry symmetry using kappa υ 8 υ = 0 - 8 = 0 - partial kappa- - symmetry gauge fixing symmetry gauge fixing partial kappa Remaining string modes are: Remaining string modes are: 3 ) bosons CP 3 a ( ( ξ ξ ) (a= 0,1,2,3), ’ ( ( ξ ξ ) (a × CP a’ 10 (AdS 4 4 × ) bosons x x a ) (a= 0,1,2,3), y y a ) (a’ ’= 1,2,3,4,5,6) = 1,2,3,4,5,6) 10 (AdS ϑ ( ( ξ ξ ) corresponding to unbroken susy 24 fermions ϑ 24 fermions ) corresponding to unbroken susy ⊃ AdS they parametrize they parametrize coset coset superspace superspace OSp OSp (6 (6| 4 | 4)/ )/ U U (3)x (3)x SO SO (1,3) (1,3) ⊃ AdS 4x 4x CP CP 3 3 5 string × S - similar similar to to the the AdS AdS 5 5 × S 5 string action on action on SU SU (2,2| 4)/ (2,2| 4)/ SO SO (1,4) (1,4) x x SO SO (5) (5) Cartan forms: Cartan forms: α= = 1, α ’ α ,4 α 1,… …,4 ’= 1, = 1,… …,6 ,6 ϑ ) ϑ ) ϑ ) Ω ( ϑ ) , ϑ , ϑ αα ’ , ϑ + Ω , ϑ E αα - 1 1 dK a ( a’ ’ ( ’ ( K - dK = = E E a ( x,y x,y, ) P P a + E E a ( x,y x,y, ) P P a + E ( x,y x,y, ) Q Q αα ( x,y x,y, ) M M K a + ’ + ’ + αα ’ a’ Sigma- - model action on model action on OSp OSp (6 (6| 4 | 4)/ )/ U U (3)x (3)x SO SO (1,3) (1,3) Sigma (Arutyunov Arutyunov & & Frolov Frolov; ; Stefanskij Stefanskij; ; D'Auria D'Auria, , Fr Frè è, Grassi & , Grassi & Trigiante Trigiante, 2008) , 2008) ( B η ξ ( η AB ’ J γ 5 2 ξ αα ’ ββ ’ ’ γ E αα E ββ 5αβ A E 1/ 2 + ’ ∧ S= ∫ ∫ d d 2 iA j B ) 1/ 2 ∫ E ∧ E ( - - det det E E i E j + ∫ J α S= AB ) α ’ β ’ αβ ’ β 6 6