Loop corrections to supergravity Agnese Bissi Harvard University - PowerPoint PPT Presentation

Loop corrections to supergravity Agnese Bissi Harvard University based on: 1706.02388 w/Alday 1612.03891 w/Aharony, Alday and Perlmutter

AdS/CFT correspondence: Gravitational theory in CFT in D dimensions D+1 dimensions G N N Newton Constant rank of the gauge group 1 Loops in AdS expansion N 2 Use structure and symmetries (superconformal symmetry) of the CFT to compute loop amplitudes in AdS 2

General idea: N = 4 SYM with SU(N) gauge group at large λ = g 2 N + + + + +… G = { { { 1 1 N 0 N 4 N 2 Supergravity Supergravity loops Generalised free fields 3

General idea: N = 4 SYM with SU(N) gauge group at large λ = g 2 N + + + + +… G = { { { 1 1 N 0 N 4 N 2 Supergravity Supergravity loops Generalised free fields 3

•Consider the superconformal primary with protected O 2 dimension , transforming in the of R- [0 , 2 , 0] SU (4) ∆ = 2 1 symmetry BPS operator 2 G R ( u, v ) X h O 2 ( x 1 ) O 2 ( x 2 ) O 2 ( x 3 ) O 2 ( x 4 ) i = x 4 12 x 4 34 R 4

•Consider the superconformal primary with protected O 2 dimension , transforming in the of R- [0 , 2 , 0] SU (4) ∆ = 2 1 symmetry BPS operator 2 G R ( u, v ) X h O 2 ( x 1 ) O 2 ( x 2 ) O 2 ( x 3 ) O 2 ( x 4 ) i = x 4 12 x 4 34 R cross ratios all the reps in [0 , 2 , 0] × [0 , 2 , 0] 4

•Consider the superconformal primary with protected O 2 dimension , transforming in the of R- [0 , 2 , 0] SU (4) ∆ = 2 1 symmetry BPS operator 2 G R ( u, v ) X h O 2 ( x 1 ) O 2 ( x 2 ) O 2 ( x 3 ) O 2 ( x 4 ) i = x 4 12 x 4 34 R 4

•Consider the superconformal primary with protected O 2 dimension , transforming in the of R- [0 , 2 , 0] SU (4) ∆ = 2 1 symmetry BPS operator 2 G R ( u, v ) X h O 2 ( x 1 ) O 2 ( x 2 ) O 2 ( x 3 ) O 2 ( x 4 ) i = x 4 12 x 4 34 R G R •Superconformal Ward identities imply relations among allowing the entire four point function to be expressed in terms of a non trivial function G ( u, v ) 4

OPE content O 2 × O 2 ∼ O long + O short O long O short acquire anomalous dimension protected ops (1/2 and 1/4 BPS) ∆ L = ∆ ( g Y M , N ) ∆ S = ∆ ( N ) c L = c O 2 O 2 O long ( g Y M , N ) c S = c O 2 O 2 O short ( N ) X c S c S G ( u, v ) = O L O S c L c L

h O 20 ( x 1 ) O 20 ( x 2 ) O 20 ( x 3 ) O 20 ( x 4 ) i = h O 20 ( x 1 ) O 20 ( x 2 ) O 20 ( x 3 ) O 20 ( x 4 ) i v 2 G ( u, v ) − u 2 G ( v, u ) + 4( u 2 − v 2 ) + 16( u − v ) = 0 N 2 − 1 c L c S X X c S c S = c L c L O L O S O L O S c L c S 6

c L c S X X c S c S = c L c L O L O S O L O S c L c S G ( u, v ) = G short ( u, v ) + H ( u, v ) Since are protected, O S X c 2 this function is completely determined! L g O L O L The sum runs over superconformal primaries, which are singlet of and are superconformal blocks. SU (4) R g O L 7

Properties of conformal blocks •Conformal blocks are fixed by conformal symmetry and encode the contribution of a primary and all its descendants. In 4d they are known in a closed form, for scalar external operators. •They are eigenfunctions of the quadratic Casimir operator with eigenvalue J 2 ∼ ` 2 C G ∆ , ` ( u, v ) = J 2 G ∆ , ` ( u, v ) •Note that ( ) label intermediate operators and denote the ∆ , ` conformal dimension and spin, respectively. 8

Properties of conformal blocks II where 2 ˜ τ G ∆ , ` ( u, v ) = u g ∆ , ` ( u, v ) ⌧ = ∆ − ` •Small u limit 2 ˜ τ g coll G ∆ , ` ( u, v ) → u ∆ , ` ( v ) + . . . •Small v limit g ∆ , ` ( u, v ) → log( v ) ˜ 2 ˜ τ •Superconformal blocks g ∆ , ` ( u, v ) = u g ∆ +4 , ` ( u, v ) 9

Large N expansion H ( u, v ) = H (0) ( u, v ) + 1 N 2 H (1) ( u, v ) + 1 N 4 H (2) ( u, v ) ∆ = ∆ (0) + 1 N 2 γ (1) + 1 N 4 γ (2) ∆ , ` + 1 ∆ , ` + 1 ∆ , ` = a ∆ , ` = a (0) N 2 a (1) N 4 a (2) c 2 ∆ , ` ( u, v ) 10

Large N expansion H ( u, v ) = H (0) ( u, v ) + 1 N 2 H (1) ( u, v ) + 1 N 4 H (2) ( u, v ) ∆ = ∆ (0) + 1 N 2 γ (1) + 1 N 4 γ (2) ∆ , ` + 1 ∆ , ` + 1 ∆ , ` = a ∆ , ` = a (0) N 2 a (1) N 4 a (2) c 2 ∆ , ` ( u, v ) 10

Order N 0 For large and large the result reduces to the one of λ N generalised free fields: ✓ ◆ 1 + 1 G (0) ( u, v ) = 4 v 2 From the OPE content of protected operators H (0) ( u, v ) G (0) ( u, v ) = G (0) a (0) X n, ` u n +2 g n, ` ( u, v ) short ( u, v ) + n, ` Double trace operators! a (0) ∆ (0) = 2 n + ` + 4 O 2 ⇤ n ∂ ` O 2 n, ` 11

Comments on large spin limit ∆ − ` X H (0) ( u, v ) = 2 g ∆ , ` ( u, v ) a ∆ , ` u ∆ , ` For small u and v: LHS RHS 12

Comments on large spin limit ∆ − ` X H (0) ( u, v ) = 2 g ∆ , ` ( u, v ) a ∆ , ` u ∆ , ` For small u and v: LHS RHS u 2 H (0) ( u, v ) → 2 v 2 + . . . 3 12

Comments on large spin limit ∆ − ` X H (0) ( u, v ) = 2 g ∆ , ` ( u, v ) a ∆ , ` u ∆ , ` For small u and v: LHS RHS u 2 H (0) ( u, v ) → 2 v 2 + . . . 3 Power law divergence! 12

Comments on large spin limit ∆ − ` X H (0) ( u, v ) = 2 g ∆ , ` ( u, v ) a ∆ , ` u ∆ , ` For small u and v: LHS RHS u 2 H (0) ( u, v ) → 2 • Each conformal block v 2 + . . . 3 diverges as log( v ) Power law divergence! • Need of an infinite sum on the spin, of operators whose twist approaches . ∆ − ` = 4 12

Comments on large spin limit ∆ − ` X H (0) ( u, v ) = 2 g ∆ , ` ( u, v ) a ∆ , ` u ∆ , ` For small u and v: LHS RHS u 2 H (0) ( u, v ) → 2 • Each conformal block v 2 + . . . 3 diverges as log( v ) Power law divergence! • Need of an infinite sum on the spin, of Fix dimensions and operators whose twist OPE coe ffi cients to all approaches . ∆ − ` = 4 orders in 1 / ` 12

Large N expansion H ( u, v ) = H (0) ( u, v ) + 1 N 2 H (1) ( u, v ) + 1 N 4 H (2) ( u, v ) ∆ = ∆ (0) + 1 N 2 γ (1) + 1 N 4 γ (2) ∆ , ` + 1 ∆ , ` + 1 a ∆ , ` = a (0) N 2 a (1) N 4 a (2) ∆ , ` 13

Large N expansion H ( u, v ) = H (0) ( u, v ) + 1 N 2 H (1) ( u, v ) + 1 N 4 H (2) ( u, v ) ∆ = ∆ (0) + 1 N 2 γ (1) + 1 N 4 γ (2) ∆ , ` + 1 ∆ , ` + 1 a ∆ , ` = a (0) N 2 a (1) N 4 a (2) ∆ , ` 13

Order N − 2 Supergravity result: G (1) ( u, v ) = − 16 u 2 ¯ D 2422 ( u, v ) + = G (1) short ( u, v ) + H (1) ( u, v ) Absence of new operators only corrections to the dimensions and the OPE coefficients of double trace operators ✓ ◆ n, ` + 1 n, ` (log u + ∂ a (1) 2 a (0) n, ` γ (1) X H (1) ( u, v ) = u 2+ n ∂ n ) g 4+2 n + ` , ` ( u, v ) n, ` ∂ n, ` = 1 ⇣ ⌘ γ (1) a (1) a (0) n, ` γ (1) n, ` n, ` ∂ n 2 14

Alternative method Crossing equation: v 2 G ( u, v ) − u 2 G ( v, u ) + 4( u 2 − v 2 ) + 16( u − v ) = 0 N 2 − 1 G (1) ( u, v ) = G (1) short ( u, v ) + H (1) ( u, v ) • protected operators • double trace operators • no n ≥ 0 τ = 4 + 2 n log u 15

Equation for H (1) ( u, v ) H (1) ( u, v ) = u 2 v 2 H (1) ( v, u ) − 4( u 2 − v 2 ) 16 − v 2 v 2 ( N 2 − 1) + u 2 v 2 G (1) short ( v, u ) − G (1) short ( u, v ) 16

Equation for H (1) ( u, v ) H (1) ( u, v ) = u 2 v 2 H (1) ( v, u ) − 4( u 2 − v 2 ) 16 − v 2 v 2 ( N 2 − 1) + u 2 v 2 G (1) short ( v, u ) − G (1) short ( u, v ) 16

Equation for H (1) ( u, v ) H (1) ( u, v ) = u 2 v 2 H (1) ( v, u ) − 4( u 2 − v 2 ) 16 − v 2 v 2 ( N 2 − 1) + u 2 v 2 G (1) short ( v, u ) − G (1) short ( u, v ) = u 2 u 2 a (0) n, ` v 2+ n γ (1) X v 2 H (1) ( v, u ) n, ` g 4+2 n + ` , ` ( v, u ) log v + . . . v 2 n, ` •no divergence! •all the divergences in v come from the protected part 16

Equation for H (1) ( u, v ) H (1) ( u, v ) = u 2 v 2 H (1) ( v, u ) − 4( u 2 − v 2 ) 16 − v 2 v 2 ( N 2 − 1) + u 2 v 2 G (1) short ( v, u ) − G (1) short ( u, v ) a (0) n, ` u 2+ n γ (1) X H ( u, v ) | log u = n, ` g 4+2 n + ` , ` ( u, v ) n, ` = u 2 v 2 G (1) short ( v, u ) | log u 16

Equation for H (1) ( u, v ) H (1) ( u, v ) = u 2 v 2 H (1) ( v, u ) − 4( u 2 − v 2 ) 16 − v 2 v 2 ( N 2 − 1) + u 2 v 2 G (1) short ( v, u ) − G (1) short ( u, v ) a (0) n, ` u 2+ n γ (1) X H ( u, v ) | log u = n, ` g 4+2 n + ` , ` ( u, v ) n, ` = u 2 v 2 G (1) short ( v, u ) | log u γ (1) •match the divergences on both sides to all orders in 1 / ` n, ` 16

Large N expansion H ( u, v ) = H (0) ( u, v ) + 1 N 2 H (1) ( u, v ) + 1 N 4 H (2) ( u, v ) ∆ = ∆ (0) + 1 N 2 γ (1) + 1 N 4 γ (2) ∆ , ` + 1 ∆ , ` + 1 a ∆ , ` = a (0) N 2 a (1) N 4 a (2) ∆ , ` 17

Large N expansion H ( u, v ) = H (0) ( u, v ) + 1 N 2 H (1) ( u, v ) + 1 N 4 H (2) ( u, v ) ∆ = ∆ (0) + 1 N 2 γ (1) + 1 N 4 γ (2) ∆ , ` + 1 ∆ , ` + 1 a ∆ , ` = a (0) N 2 a (1) N 4 a (2) ∆ , ` 17

Order N − 4 γ (2) Idea: extract information about using n, ` • CFT data from previous orders •crossing symmetry General picture: n, ` + 1 n, ` ∂ n + 1 a (2) 2 a (0) n, ` γ (2) 2 a (1) n, ` γ (1) X H (2) ( u, v ) = n, ` ∂ n n, ` ! + 1 8 a (0) n, ` ( γ (1) n, ` ) 2 ∂ 2 u 2+ n g n, ` ( u, v ) n 18