q -Poincar e supersymmetry in AdS 5 /CFT 4 Riccardo Borsato based - PowerPoint PPT Presentation

q -Poincar´ e supersymmetry in AdS 5 /CFT 4 Riccardo Borsato based on arXiv:1706.10265 with A. Torrielli IGST17 Paris 18 July 2017 q -Poincar´ e supersymmetry in AdS 5 / CFT 4 Riccardo Borsato

Strings on AdS 5 × S 5 and N = 4 super Yang-Mills in the planar limit Exact S-matrix governing scattering of worldhseet excitations / magnons on spin-chain ∆ op ( Q ) R = R ∆( Q ) , R = Π S Yang-Baxter equation charges on 2-particle states ⇐ ⇒ coproduct ∆ q -Poincar´ e supersymmetry in AdS 5 / CFT 4 Riccardo Borsato

• S -matrix invariant under centrally extended psu (2 | 2) [Beisert ’05] not of difference form • non linear constraint among central charges and braided coproducts • non-standard Yangian [Beisert ’07] • secret symmetry ˆ B [Matsumoto, Moriyama, Torrielli ’07] • R T T formulation [Beisert, de Leeuw ’14] universal formulation? • outer automorphisms seem to play a role see e.g. [Beisert, Hecht, de Leeuw ’16] q -Poincar´ e supersymmetry in AdS 5 / CFT 4 Riccardo Borsato

Here: new “boost” symmetry of S -matrix q -Poincar´ e supersymmetry No superimposed deformation Standard AdS 5 /CFT 4 q -Poincar´ e supersymmetry in AdS 5 / CFT 4 Riccardo Borsato

q -Poincar´ e in AdS 5 /CFT 4 � √ 1 + 4 g 2 sin 2 p H = h p 1 , P = p 1 , h p = 2 , g = λ/ 2 π Dispersion relation as Casimir of q -Poincar´ e [Gomez, Hernandez ’07] � K − K − 1 � [ J , H ] = g 2 K ≡ exp( i P ) [ J , P ] = i H , , 2 Obtained as q → 1 contraction of U q ( sl 2 ) [Celeghini et al.’90] 1 = C = H 2 + g 2 ( K 1 2 − K − 1 2 ) 2 Boost generates translations on rapidity torus J = i d 2 dz Classical limit is 2D Poincar´ e J → g J , P → P / g and g → ∞ : C = H 2 − P 2 [ J , P ] = i H , [ J , H ] = i P , q -Poincar´ e supersymmetry in AdS 5 / CFT 4 Riccardo Borsato

q -Poincar´ e in AdS 5 /CFT 4 [Young ’07] q -Poincar´ e superalgebra: reformulation and extension of psu (2 | 2) c . e . � � a α ] = ig 1 2 + K − 1 [ J , Q a K Q α , [ J , P ] = i H , 2 4 � � � K − K − 1 � α [ J , H ] = g 2 a ] = ig 1 2 + K − 1 Q α [ J , Q K a , , 2 4 2 2 ǫ αβ ǫ ab � � 1 2 − K − 1 β } = ig { Q a α , Q b K , etc. 2 Exact (fundamental) magnon repr. from boosting rest-frame repr. Coproducts (for subalgebra) 1 2 + K − 1 2 ⊗ H , ∆( P ) = P ⊗ 1 + 1 ⊗ P , ∆( H ) = H ⊗ K 1 2 + K − 1 2 ⊗ J , ∆( J ) = J ⊗ K etc. ∆ op ( H ) R � = R ∆( H ) Not symmetries of R -matrix! q -Poincar´ e supersymmetry in AdS 5 / CFT 4 Riccardo Borsato

Reminder: massless sector of AdS 3 /CFT 2 Wait for Alessandro’s talk for latest news on AdS 3 /CFT 2 ! [Stromwall, Torrielli ’16] [Fontanella, Torrielli ’16] H = h p 1 in massless case h p = 2 g sin p 2 , p ∈ [0 , 2 π ] � � 1 2 + K − 1 ⇒ 2 g sin p 1 + p 2 � = 2 g (sin p 1 2 + sin p 2 2 ⊗ H = ∆( H ) = H ⊗ K 2 ) 2 Cocommutative = ⇒ H is symmetry of R 1 2 + K − 1 2 ⊗ J + tail ∆( J ) = J ⊗ K J = i H ∂ p still not symmetry of R but annihilates it ∆( J ) R = 0 q -Poincar´ e supersymmetry in AdS 5 / CFT 4 Riccardo Borsato

q -Poincar´ e supersymmetry in AdS 5 / CFT 4 q -Poincar´ e supersymmetry in AdS 5 / CFT 4 Riccardo Borsato

Boost in AdS 5 /CFT 4 as a symmetry of R [RB, Torrielli ’17] α ⊗ K − 1 1 Usual coproducts ∆( H ) = H ⊗ 1 + 1 ⊗ H , ∆( Q a α ) = Q a 4 + K 4 ⊗ Q a α , etc. Demand [∆ a , ∆ b ] = ∆[ a , b ] in fundamental representation ∆( J ) = ∆ ′ ( J ) + T H ˆ B + T psu (2 | 2) + T 1 J = i H ∂ p , � 1 − s 12 � � 1 + s 12 � ∆ ′ ( J ) = J ⊗ 1 + 1 ⊗ J , h 1 h 2 1 + x − p x + s 12 = g sin p 1 + sin p 2 − sin( p 1 + p 2 ) 2 h p p , w p = g sin p = 2 w − 1 − w − 1 x − p + x + 2 p 1 2 B = 1 1 1 − tan p 2 ⊗ tan p � � � � H ⊗ ˆ B + ˆ B ⊗ H T H ˆ , 2 w 1 − w 2 2 T psu (2 | 2) = 1 w 1 + w 2 � K − 1 α ⊗ K − 1 α 1 α 1 4 Q a 4 Q a 4 Q 4 Q − K ⊗ K a a α 2 w 1 − w 2 � + L b a ⊗ L a b − R β α ⊗ R α . β q -Poincar´ e supersymmetry in AdS 5 / CFT 4 Riccardo Borsato

Boost in AdS 5 /CFT 4 as a symmetry of R 0 = ∆ op ( J ) R − R ∆( J ) = D R + T op R − R T = ( f 12 + T op − T 1 ) R 1 D ≡ i ( h 1 − s 12 ) ∂ p 1 + i ( h 2 + s 12 ) ∂ p 2 ( f op = − f ) f 12 is function, solve equation by T 1 = 1 2 f 12 1 + symm J is symmetry! Different scalar factor R ′ = e Φ 12 R = ⇒ shift of T 1 = 1 2 [ f 12 + D Φ 12 ] 1 It would be interesting to compute D θ BES [Beisert,Eden,Staudacher ’06] If ∆( J ) (including T 1 ) were a priori known, it would constrain Φ 12 q -Poincar´ e supersymmetry in AdS 5 / CFT 4 Riccardo Borsato

Boost in AdS 5 /CFT 4 as a symmetry of R Crossing symmetry ⇐ ⇒ antipode Hopf algebra = bialgebra + antipode S ( H ) = − H , S ( Q ) = − Q , etc. Not all Φ 12 solve crossing ⇐ ⇒ not all T 1 compatible with antipode q -Poincar´ e supersymmetry in AdS 5 / CFT 4 Riccardo Borsato

Antipode µ : µ ◦ ( S ⊗ 1) ◦ ∆( J ) = 0 ∆ : � � c (1) S ( J ) = − J − F p + d p 1 . p ���� ���� ���� ∆ ′ ( J ) T psu (2 | 2) T 1 � � c (2) From µ ◦ (1 ⊗ S ) ◦ ∆( J ) = 0 S ( J ) = − J − F p − d p 1 p ⇒ c (1) = c (2) Example : T 1 = 1 2 f 12 1 = = 0 = ⇒ not compatible p p q -Poincar´ e supersymmetry in AdS 5 / CFT 4 Riccardo Borsato

Limit g → 0 β { Q a b } = δ a b R β α + δ β α L a b + 1 2 δ a b δ β α , Q α H , [ J , P ] = i H , Coproduct remains non-trivial Boost J = i 1 ∂ p in this limit related to operators used in [Bargheer, Beisert, Loebbert ’08,’09] to generate long-range spin-chains ======================================= Limit g → ∞ ( J → g J , P → P / g ) Classical Poincar´ e superalgebra , cf. [Berenstein, Maldacena, Nastase ’02] Trivial coproducts ∆( J ) = J ⊗ 1 + 1 ⊗ J , etc. ε → 0 Obtained by contraction of d (2 , 1; ε ) ⊃ sl 2 − − − → 2D Poincar´ e [Young ’07] − − − − − − − − − a + Y a ˙ a + Y ′ ′ a ˙ a + . . . ) Local charge on the w.s. ( H = 1 a P a ˙ a Y a ˙ 4 P a ˙ a Y a ˙ � J = d σ ( σ H + τ P ) q -Poincar´ e supersymmetry in AdS 5 / CFT 4 Riccardo Borsato

Cobracket ∆( J ) − ∆ op ( J ) = g − 1 δ ( J ) + O ( g − 2 ) δ ( J ) = [ J ⊗ 1 + 1 ⊗ J , r ] We use r of [Beisert, Spill ’07] and assume [ J , B 0 ] = − 2 i B − 1 � � 2 q n − 1 − 1 and [ J , q n ] = ˜ q n + in 2 q n +1 , (˜ q 0 ≡ [ J , q 0 ]) In evaluation repr. we find � � δ ( J ) = u 1 + u 2 α α Q a a ⊗ Q a α + L b a ⊗ L a b − R β α ⊗ R α α ⊗ Q a − Q β u 1 − u 2 1 + ( B 1 ⊗ H + H ⊗ B 1 ) . u 1 − u 2 g →∞ − − − − → u , like spectral parameter of Yangian Notice that w q -Poincar´ e supersymmetry in AdS 5 / CFT 4 Riccardo Borsato

Higher partners for J ? [Beisert ’10] classical limit U q ( psu (2 | 2) c . e . ) R -matrix − − − − − − − → trigonometric classical r -matrix Deformation of loop algebra gl (2 | 2)[ z , z − 1 ], loop parameter z Extension of the algebra by derivation rational limit → ˜ D = z d d cf. boost J = i 2 (4 − u 2 ) d − − − − − − − D = du , dz du −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− [RB, Torrielli ’17] U q ( � sl 2 ) in Drinfeld’s second realisation, generated by h n , e ± n , n ∈ Z q = e ε µ q -affine Poincar´ e from contraction ε → 0 , H m = µε ( e + m + e − J m = 1 2 ( e + m − e − m ) , m ) , P m = − i µε h m q -Poincar´ e supersymmetry in AdS 5 / CFT 4 Riccardo Borsato

U q ( d (2 , 1; ε )) q -Poincar´ e superalgebra ? U q ( sl 2 ) q -Poincar´ e ε → 0 g → ∞ q → 1 g → ∞ q → 1 sl 2 Poincar´ e ε → 0 d (2 , 1; ε ) Poincar´ e superalgebra ε → 0 ...if it worked one may go to the affine case it would also help for the universal formulation q -Poincar´ e supersymmetry in AdS 5 / CFT 4 Riccardo Borsato

Conclusions Boost J is symmetry of AdS 5 / CFT 4 R -matrix Deformed symmetry algebra where ’t Hooft coupling is def. parameter • compute T 1 for normalisation of R with BES. Compatibility with antipode? • boost on spin-chain at weak coupling • Quantum corrections to J on worldsheet at strong coupling, non-locality • insights for universal formulation? Contraction of quantum group? see also [Beisert,Hecht,Hoare ’17] Affine case? • AdS d +1 /CFT d (e.g. AdS 3 /CFT 2 [RB, Torrielli, in preparation] ) • other manifestations of boost in AdS/CFT (cf. secret symmetry) q -Poincar´ e supersymmetry in AdS 5 / CFT 4 Riccardo Borsato

q -Poincar´ e supersymmetry in AdS 5 / CFT 4 Riccardo Borsato

A contraction of U q ( d (2 , 1; ε )) q h i − q − h i [ h i , e ± j ] = ± a ij e ± { e + i , e − [ h i , h j ] = 0 , j , j } = δ ij q − q − 1 , � � 0 ε − 1 � 2 , q = e a ij = 0 1 − ε ε − 1 1 − ε 0 ε → 0 U q ( d (2 , 1; ε )) ⊃ U q ε ( sl 2 ) − − − → 2D q -Poincar´ e � � Q 41 = 2 h 1 − iq e − 2 q − 1 1 2 h 2 e + 1 Supercharges? , etc. √ 1 2 � � q ( h 1 + h 2 ) − q − ( h 1 + h 2 ) { Q 41 , Q 32 } = − i ? 1 2 − K − 1 → − ig + rest − K , 2 2 2 q − q − 1 q ( h 1 − h 2 ) − q − ( h 1 − h 2 ) 23 } = − 1 → − R 34 + L 12 − 1 ? { Q 41 , Q − + rest 2 H . q − q − 1 2 q -Poincar´ e supersymmetry in AdS 5 / CFT 4 Riccardo Borsato