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

q-Poincar´ e supersymmetry in AdS5/CFT4

Riccardo Borsato

based on arXiv:1706.10265 with A. Torrielli IGST17 Paris 18 July 2017

q-Poincar´ e supersymmetry in AdS5/CFT4 Riccardo Borsato

Strings on AdS5×S5 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 = ΠS Yang-Baxter equation charges on 2-particle states ⇐ ⇒ coproduct ∆

q-Poincar´ e supersymmetry in AdS5/CFT4 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]

• RT 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 AdS5/CFT4 Riccardo Borsato

Here: new “boost” symmetry of S-matrix q-Poincar´ e supersymmetry No superimposed deformation Standard AdS5/CFT4

q-Poincar´ e supersymmetry in AdS5/CFT4 Riccardo Borsato

q-Poincar´ e in AdS5/CFT4

H = hp1, P = p1, hp =

• 1 + 4g 2 sin2 p

2,

g = √ λ/2π Dispersion relation as Casimir of q-Poincar´ e

[Gomez, Hernandez ’07]

[J, P] = i H, [J, H] = g 2

2

• K − K−1

, K ≡ exp(iP) Obtained as q → 1 contraction of Uq(sl2)

[Celeghini et al.’90]

1 = C = H2 + g 2(K

1 2 − K− 1 2 )2

Boost generates translations on rapidity torus J = i

2 d dz

Classical limit is 2D Poincar´ e J → g J, P → P/g and g → ∞: [J, P] = i H, [J, H] = i P, C = H2 − P2

q-Poincar´ e supersymmetry in AdS5/CFT4 Riccardo Borsato

q-Poincar´ e in AdS5/CFT4

[Young ’07]

q-Poincar´ e superalgebra: reformulation and extension of psu(2|2)c.e. [J, Q a

α ] = ig 4

• K

1 2 + K− 1 2

• Q

a α ,

[J, P] = i H, [J, Q

α a ] = ig 4

• K

1 2 + K− 1 2

• Q α

a ,

[J, H] = g 2

2

• K − K−1

, {Q a

α , Q b β } = ig 2 ǫαβǫab

K

1 2 − K− 1 2

• ,

etc. Exact (fundamental) magnon repr. from boosting rest-frame repr. Coproducts (for subalgebra) ∆(P) = P ⊗ 1 + 1 ⊗ P, ∆(H) = H ⊗ K

1 2 + K− 1 2 ⊗ H,

∆(J) = J ⊗ K

1 2 + K− 1 2 ⊗ J,

etc. Not symmetries of R-matrix! ∆op(H)R = R∆(H)

q-Poincar´ e supersymmetry in AdS5/CFT4 Riccardo Borsato

Reminder: massless sector of AdS3/CFT2

Wait for Alessandro’s talk for latest news on AdS3/CFT2! [Stromwall, Torrielli ’16] [Fontanella, Torrielli ’16]

H = hp1 in massless case hp = 2g sin p

2, p ∈ [0, 2π]

∆(H) = H ⊗ K

1 2 + K− 1 2 ⊗ H =

⇒ 2g sin p1+p2

2

• = 2g(sin p1

2 + sin p2 2 )

• Cocommutative =

⇒ H is symmetry of R ∆(J) = J ⊗ K

1 2 + K− 1 2 ⊗ J + tail

J = iH∂p still not symmetry of R but annihilates it ∆(J)R = 0

q-Poincar´ e supersymmetry in AdS5/CFT4 Riccardo Borsato

q-Poincar´ e supersymmetry in AdS5/CFT4

q-Poincar´ e supersymmetry in AdS5/CFT4 Riccardo Borsato

Boost in AdS5/CFT4 as a symmetry of R

[RB, Torrielli ’17]

Usual coproducts ∆(H) = H ⊗ 1 + 1 ⊗ H, ∆(Q a

α ) = Q a α ⊗ K− 1

4 + K 1 4 ⊗ Q a

α , etc.

Demand [∆a, ∆b] = ∆[a, b] in fundamental representation

∆(J) = ∆′(J) + THˆ

B + Tpsu(2|2) + T1

J = iH∂p, ∆′(J) =

• 1 − s12

h1

• J ⊗ 1 +
• 1 + s12

h2

• 1 ⊗ J,

s12 = g 2 sin p1 + sin p2 − sin(p1 + p2) w−1

1

− w−1

2

, wp = 2 hp g sin p = 2 1 + x−

p x+ p

x−

p + x+ p

THˆ

B = 1

2 1 w1 − w2

• 1 − tan p

2 ⊗ tan p 2 H ⊗ ˆ B + ˆ B ⊗ H

• ,

Tpsu(2|2) = 1 2 w1 + w2 w1 − w2

• K− 1

4 Q a

α ⊗ K− 1

4 Q

α a

− K

1 4 Q

α a

⊗ K

1 4 Q a

α

+ L b

a ⊗ L a b − R β α ⊗ R α β

• .

q-Poincar´ e supersymmetry in AdS5/CFT4 Riccardo Borsato

Boost in AdS5/CFT4 as a symmetry of R

0 = ∆op(J)R − R∆(J) = DR + T opR − RT = (f12 + T op

1

− T1) R D ≡ i(h1 − s12)∂p1 + i(h2 + s12)∂p2 f12 is function, solve equation by T1 = 1

2f121 + symm

(f op = −f ) J is symmetry! Different scalar factor R′ = eΦ12R = ⇒ shift of T1 = 1

2[f12 + DΦ12]1

It would be interesting to compute DθBES

[Beisert,Eden,Staudacher ’06]

If ∆(J) (including T1) were a priori known, it would constrain Φ12

q-Poincar´ e supersymmetry in AdS5/CFT4 Riccardo Borsato

Boost in AdS5/CFT4 as a symmetry of R

Crossing symmetry ⇐ ⇒ antipode Hopf algebra = bialgebra + antipode S(H) = −H, S(Q) = −Q, etc. Not all Φ12 solve crossing ⇐ ⇒ not all T1 compatible with antipode

q-Poincar´ e supersymmetry in AdS5/CFT4 Riccardo Borsato

Antipode

µ : µ ◦ (S ⊗ 1) ◦ ∆(J) = 0 ∆ : S(J) = −J − Fp

• ∆′(J)
• c(1)

p

• T1

+ dp

• Tpsu(2|2)
• 1.

From µ ◦ (1 ⊗ S) ◦ ∆(J) = 0 S(J) = −J − Fp

• c(2)

p

− dp

• 1

Example: T1 = 1

2f121 =

⇒ c(1)

p

= c(2)

p

= 0 = ⇒ not compatible

q-Poincar´ e supersymmetry in AdS5/CFT4 Riccardo Borsato

Limit g → 0

{Q a

α , Q β b } = δa bR β α + δβ αL a b + 1 2δa bδβ αH,

[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. Obtained by contraction of d(2, 1; ε) ⊃ sl2

ε→0

− − − → 2D Poincar´ e [Young ’07] − − − − − − − − − Local charge on the w.s. (H = 1

4Pa˙ aPa˙ a + Ya˙ aY a˙ a + Y ′ a˙ aY

′a˙

a + . . .)

J =

• dσ (σH + τP)

q-Poincar´ e supersymmetry in AdS5/CFT4 Riccardo Borsato

Cobracket

∆(J) − ∆op(J) = g −1δ(J) + O(g −2) δ(J) = [J ⊗ 1 + 1 ⊗ J, r] We use r of

[Beisert, Spill ’07] and assume [J, B0] = −2i B−1

and [J, qn] = ˜ qn + in

• 2qn−1 − 1

2qn+1

• ,

(˜ q0 ≡ [J, q0]) In evaluation repr. we find δ(J) = u1 + u2 u1 − u2

• Q a

α ⊗ Q α a − Q α a ⊗ Q a α + L b a ⊗ L a b − R β α ⊗ R α β

• +

1 u1 − u2 (B1 ⊗ H + H ⊗ B1) . Notice that w

g→∞

− − − − → u, like spectral parameter of Yangian

q-Poincar´ e supersymmetry in AdS5/CFT4 Riccardo Borsato

Higher partners for J?

[Beisert ’10]

Uq(psu(2|2)c.e.) R-matrix

classical limit

− − − − − − − → trigonometric classical r-matrix Deformation of loop algebra gl(2|2)[z, z−1], loop parameter z Extension of the algebra by derivation D = z d

dz rational limit

− − − − − − − → ˜ D =

d du,

• cf. boost J = i

2(4 − u2) d du

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

[RB, Torrielli ’17]

Uq( sl2) in Drinfeld’s second realisation, generated by hn, e±

n , n ∈ Z

q-affine Poincar´ e from contraction ε → 0, q = eε µ Hm = µε(e+

m + e− m),

Jm = 1

2(e+ m − e− m),

Pm = −iµε hm

q-Poincar´ e supersymmetry in AdS5/CFT4 Riccardo Borsato

q-Poincar´ e superalgebra Poincar´ e superalgebra Uq(d(2, 1; ε)) d(2, 1; ε) q-Poincar´ e Poincar´ e Uq(sl2) sl2 g → ∞ q → 1 ε → 0 ?

g → ∞ q → 1 ε → 0 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 AdS5/CFT4 Riccardo Borsato

Conclusions

Boost J is symmetry of AdS5/CFT4 R-matrix Deformed symmetry algebra where ’t Hooft coupling is def. parameter

• compute T1 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?

• AdSd+1/CFTd (e.g. AdS3/CFT2

[RB, Torrielli, in preparation] )

• other manifestations of boost in AdS/CFT (cf. secret symmetry)

q-Poincar´ e supersymmetry in AdS5/CFT4 Riccardo Borsato

q-Poincar´ e supersymmetry in AdS5/CFT4 Riccardo Borsato

A contraction of Uq(d(2, 1; ε))

[hi, hj] = 0, [hi, e±

j ] = ±aije± j ,

{e+

i , e− j } = δij

qhi − q−hi q − q−1 , q = e

• 2 ,

aij =

• ε

−1 ε 1 − ε −1 1 − ε

• Uq(d(2, 1; ε)) ⊃ Uqε(sl2)

ε→0

− − − → 2D q-Poincar´ e Supercharges? Q41 =

1 √ 2

• e−

2 q− 1

2 h1 − iq 1 2 h2e+

1

• , etc.

{Q41, Q32} = − i 2 q(h1+h2) − q−(h1+h2) q − q−1 + rest

?

− → − ig

2

• K

1 2 − K− 1 2

• ,

{Q41, Q

23} = −1

2 q(h1−h2) − q−(h1−h2) q − q−1 + rest

?

− → −R34 + L12 − 1

2H.

q-Poincar´ e supersymmetry in AdS5/CFT4 Riccardo Borsato

Antipode

µ : µ ◦ (S ⊗ 1) ◦ ∆(J) = 0 ∆ : µ ◦ (S ⊗ 1) ◦ THˆ

B = lim p2→p1

1 w1 − w2

• −w1

2

• 1g,

µ ◦ (S ⊗ 1) ◦ Tpsu(2|2) = lim

p2→p1

1 w1 − w2

• +w1

2

• 1g + finite,

− − − − − − − − − S(J) = −J − Fp

• ∆′(J)
• dp
• Tpsu(2|2)

+ c(1)

p

• T1
• 1.

− − − − − − − − − From µ ◦ (1 ⊗ S) ◦ ∆(J) = 0 S(J) = −J − Fp

• − dp + c(2)

p

• 1

Example: T1 = 1

2f121 =

⇒ c(1)

p

= c(2)

p

= 0 = ⇒ not compatible

q-Poincar´ e supersymmetry in AdS5/CFT4 Riccardo Borsato

Cobracket

R = 1+g −1r +O(g −2), ∆(J) = J⊗1+1⊗J+g −1∆(1)(J)+O(g −2) 0 = ∆op(J)R − R∆(J) = g −1 [J ⊗ 1 + 1 ⊗ J, r] − δ(J)

• + O(g −2)

where δ(J) ≡ ∆(1)(J) − ∆op

(1)(J).

We use r of

[Beisert, Spill ’07] and assume [J, B0] = −2i B−1

and [J, qn] = ˜ qn + in

• 2qn−1 − 1

2qn+1

• ,

(˜ q0 ≡ [J, q0]) In evaluation repr. we find δ(J) = u1 + u2 u1 − u2

• Q a

α ⊗ Q α a − Q α a ⊗ Q a α + L b a ⊗ L a b − R β α ⊗ R α β

• +

1 u1 − u2 (B1 ⊗ H + H ⊗ B1) . Notice that w

g→∞

− − − − → u, like spectral parameter of Yangian

q-Poincar´ e supersymmetry in AdS5/CFT4 Riccardo Borsato

Limit g → ∞

J → g J, P → P/g, g → ∞ Classical Poincar´ e superalgebra, cf. [Berenstein, Maldacena, Nastase ’02] {Q a

α , Q β b } = δa bR β α + δβ αL a b + 1 2δa bδβ αH,

[J, P] = i H {Q a

α , Q b β } = − 1 2ǫαβǫabP,

[J, H] = i P [J, Q a

α ] = i 2 Q a α ,

etc. Trivial coproducts ∆(J) = J ⊗ 1 + 1 ⊗ J, etc. Obtained by contraction of d(2, 1; ε) ⊃ sl2

ε→0

− − − → 2D Poincar´ e [Young ’07] − − − − − − − − − Local charge on the w.s. (H = 1

4Pa˙ aPa˙ a + Ya˙ aY a˙ a + Y ′ a˙ aY

′a˙

a + . . .)

J =

• dσ (σH + τP)

q-Poincar´ e supersymmetry in AdS5/CFT4 Riccardo Borsato

q-affine Poincar´ e

[Pm, Pn] = 0, [Pm, Hn] = 0, [Hm, Hn] = 0, [Jm, Pn] = i Hm+n, [Jm, Hn] = 1 2

• ψ+

m+n − ψ− m+n

• + 1

2 sign(m − n) m−n

• ℓ=0

Hm−ℓHn+ℓ − HmHn

• ,

[Jm, Jn] = 1 4 sign(m − n) m−n

• ℓ=0

(Hm−ℓJn+ℓ + Jm−ℓHn+ℓ + Hn+ℓJm−ℓ + Jn+ℓHm−ℓ) − (HnJm + JnHm + HmJn + JmHn)

• .

q-Poincar´ e supersymmetry in AdS5/CFT4 Riccardo Borsato