The p 2 , 0 q Superconformal Bootstrap Leonardo Rastelli Yang Institute for Theoretical Physics Stony Brook Based on work with Chris Beem, Madalena Lemos and Balt van Rees YITP workshop Developments in String Theory and Quantum Field Theory Kyoto, November 13 2015
p 2 , 0 q theories Nahm’s classification: superconformal algebras exist for d ď 6 . In d “ 6 , p N , 0 q algebras. Existence of T µν multiplet requires N ď 2 . p 2 , 0 q : maximal susy in maximal d . No marginal couplings allowed. Interacting models inferred from string/M-theory: ADE catalogue. Central to many recent developments in QFT. “Mothers” of many interesting QFTs in d ă 6 . Key properties: Moduli space of vacua M g “ p R 5 q r g { W g , g “ t A n , D n , E 6 , E 7 , E 8 u . On R 5 ˆ S 1 , IR description as 5 d MSYM with gauge algebra g . At large n , A n and D n theories described through AdS/CFT: M-theory on AdS 7 ˆ S 4 and AdS 7 ˆ RP 4 . Leonardo Rastelli (YITP) p 2 , 0 q Bootstrap Nov’15 1 / 1
The p 2 , 0 q theories as abstract CFTs No intrinsic field-theoretic formulation yet. No conventional Lagrangian (hard to imagine one from RG lore). Working hypothesis: (at least) for correlators of local operators in R 6 , the p 2 , 0 q theory is just another CFT, defined by a local operator algebra ÿ OPE : O 1 p x q O 2 p 0 q “ c 12 k p x q O k p 0 q k Can symmetry and basic consistency requirements completely determine the spectrum and OPE coefficients? Leonardo Rastelli (YITP) p 2 , 0 q Bootstrap Nov’15 2 / 1
Abstract CFT Framework A general Conformal Field Theory hasn’t much to do with “fields” (of the kind we write in Lagrangians). We’ll think more abstractly. A CFT is defined by its local operators, A ” t O k p x qu , and their correlation functions x O 1 p x 1 q . . . O n p x n qy . A is an algebra. Operator Product Expansion (OPE), ÿ O 1 p x q O 2 p 0 q “ c 12 k p O k p 0 q ` . . . q , k where the . . . are fixed by conformal invariance. The sum converges. Caveat I: This definition does not capture non-local observables, such as conformal defects. (E.g., Wilson lines in a conformal gauge theory.). Leonardo Rastelli (YITP) p 2 , 0 q Bootstrap Nov’15 3 / 1
Reduce n pt to p n ´ 1 q pt, ÿ x O 1 p x 1 q O 2 p x 2 q . . . O n p x n qy “ c 12 k p x 2 q x O k p x 2 q . . . O n p x n qy . k 1pt functions are trivial, x O i p x qy “ 0 except for x 1 y ” 1 . O ∆ ,ℓ,f p x q labeled by conformal dimension ∆ , Lorentz representation ℓ and possibly flavor quantum number f . The CFT data tp ∆ i , ℓ i , f i q , c ijk u completely specify the theory. But not anything goes! Consistency conditions: Associativity: p O 1 O 2 q O 3 “ O 1 p O 2 O 3 q . Unitarity (reflection positivity): Lower bounds on ∆ for given ℓ ; c ijk P R Caveat II: In non-trivial geometries, x O y ‰ 0 Ñ additional constraints. In d “ 2 , modularity. In d ą 2 , harder to analyze, have been ignored so far. Leonardo Rastelli (YITP) p 2 , 0 q Bootstrap Nov’15 4 / 1
The bootstrap program Old aspiration (1970s) Ferrara Gatto Grillo, Polyakov. Associativity ” crossing symmetry of 4 pt functions 1 3 1 3 � � O ′ = O O O ′ 2 4 2 4 Vastly over-constrained system of equations for t ∆ i , c ijk u . Classification and construction of CFTs reduced to an algebraic problem. ‚ Famous success story in d “ 2 , starting from BPZZ (1984). 2 d conformal symmetry is infinite dimensional, z Ñ f p z q . In some cases, finite -dimensional bootstrap problem (rational CFTs). Many exact solutions, partial classification. Leonardo Rastelli (YITP) p 2 , 0 q Bootstrap Nov’15 5 / 1
Bootstrapping in two steps For d “ 6 , N “ p 2 , 0 q SCFTs (as well as d “ 4 , N ě 2 SCFTs) the crossing equations split into (1) Equations that depend only on intermediate BPS operators. Captured by the 2 d chiral algebra. “Minibootstrap” (2) Equations that also include intermediate non-BPS operators. “Maxibootstrap” (1) are tractable and determine an infinite amount of CFT data. This is essential input to the full-fledged bootstrap (2), which can be studied numerically. Beem Lemos Liendo Peelaers LR van Rees, Beem LR van Rees Leonardo Rastelli (YITP) p 2 , 0 q Bootstrap Nov’15 6 / 1
Meromorphy in p 2 , 0 q SCFTs Fix a plane R 2 Ă R 6 , parametrized by p z, ¯ z q . Claim : D subsector A χ “ t O i p z i , ¯ z i qu with meromorphic x O 1 p z 1 , ¯ z 1 q O 2 p z 2 , ¯ z 2 q . . . O n p z n , ¯ z n qy “ f p z i q . Rationale: A χ ” cohomology of a nilpotent ◗ , ◗ “ Q ` S , Q Poincar´ e, S conformal supercharges. z dependence is ◗ -exact: cohomology classes r O p z, ¯ z qs ◗ � O p z q . ¯ Analogous to the d “ 4 , N “ 1 chiral ring: cohomology classes r O p x qs ˜ α are x -independent. Q 9 Leonardo Rastelli (YITP) p 2 , 0 q Bootstrap Nov’15 7 / 1
Cohomology At the origin of R 2 , ◗ -cohomology A χ easy to describe. O p 0 , 0 q P A χ Ø O obeys the chirality condition ∆ ´ ℓ “ R 2 ∆ conformal dimension, ℓ angular momentum on R 2 , R Cartan generator of SU p 2 q R – SO p 3 q R Ă SO p 5 q R-symmetry. Leonardo Rastelli (YITP) p 2 , 0 q Bootstrap Nov’15 8 / 1
r ◗ , Ę r ◗ , sl p 2 qs “ 0 but sl p 2 qs ‰ 0 To define ◗ -closed operators O p z, ¯ z q away from origin, we twist the right-moving generators by SU p 2 q R , L ´ 1 ` R ´ , L ´ 1 “ ¯ p L 0 “ ¯ p L 1 “ ¯ p L 1 ´ R ` L 0 ´ R , z sl p 2 q “ t ◗ , . . . u ◗ -closed operators are “twisted-translated” z p L ´ 1 O 1 ... 1 p 0 q e ´ zL ´ 1 ´ ¯ z p e zL ´ 1 ` ¯ L ´ 1 O p z, ¯ z q “ z q O I 1 ... I k p z, ¯ “ u I 1 p ¯ z q . . . u I k p ¯ z q u I ” p 1 , ¯ z q SU p 2 q R orientation correlated with position on R 2 . Leonardo Rastelli (YITP) p 2 , 0 q Bootstrap Nov’15 9 / 1
Example: free p 2 , 0 q tensor multiplet ω ` Φ I , λ aA , ab I “ SO p 5 q R vector index. Scalar in SO p 3 q R Ă SO p 5 q R h.w. is only field obeying ∆ ´ ℓ “ 2 R Φ h.w. “ Φ 1 ` i Φ 2 ? , ∆ “ 2 R “ 2 , ℓ “ 0 . 2 Cohomology class of twisted-translated field “ ‰ z 2 Φ ˚ Φ p z q : “ Φ h.w. p z, ¯ z q ` ¯ z Φ 3 p z, ¯ z q ` ¯ h.w. p z, ¯ z q ◗ z 2 ¯ z 2 “ 1 z 2 Φ ˚ Φ p z q Φ p 0 q „ ¯ h.w. p z, ¯ z q Φ h.w. p 0 q „ z 2 . z 2 ¯ Φ p z q is an u p 1 q affine current, Φ p z q � J u p 1 q p z q . Leonardo Rastelli (YITP) p 2 , 0 q Bootstrap Nov’15 10 / 1
χ 6 : 6d (2,0) SCFT Ý Ñ 2d Chiral Algebra . Global sl p 2 q Ñ Virasoro, indeed T p z q : “ r Φ p IJ q p z, ¯ z qs ◗ , with Φ p IJ q the stress-tensor multiplet superprimary. c 2 d “ c 6 d in normalizations where c 6 d (free tensor) ” 1 . All 1 2 -BPS operators p ∆ “ 2 R ) are in ◗ cohomology. Generators of the 1 2 -BPS ring Ñ generators of the chiral algebra. Some semi-short multiplets with non-zero spin also play a role. Leonardo Rastelli (YITP) p 2 , 0 q Bootstrap Nov’15 11 / 1
Chiral algebra for p 2 , 0 q theory of type A N ´ 1 One 1 2 -BPS generator each of dimension ∆ “ 4 , 6 , . . . 2 N ó One chiral algebra generator each of dimension h “ 2 , 3 , . . . N. Most economical scenario: these are all the generators. Check: the superconformal index computed by Kim 3 is reproduced: I p q, s q : “ Tr p´ 1 q F q E ´ R s h 2 ` h 3 « ff q 2 ` ¨ ¨ ¨ ` q n ź n ź 8 1 I p q, s ; n q “ 1 ´ q k ` m “ PE . 1 ´ q m “ 0 k “ 2 Plausibly a unique solution to crossing for this set of generators. The chiral algebra of the A N ´ 1 theory is W N , with c 2 d “ 4 N 3 ´ 3 N ´ 1 . Generalization to all ADE cases: W g with c 2 d “ 4 d g h _ g ` r g . Leonardo Rastelli (YITP) p 2 , 0 q Bootstrap Nov’15 12 / 1
Half-BPS 3pt functions of p 2 , 0 q SCFT OPE of W g generators ñ half-BPS 3pt functions of SCFT. Let us check the result at large N . W N Ñ8 with c 2 d „ 4 N 3 Ñ a classical Poisson algebra. We can use results on universal Poisson algebra W 8 r µ s , with µ “ N . (Gaberdiel Hartman, Campoleoni Fredenhagen Pfenninger) We find ¯ ˜ ¸ ` k 123 ` 1 ˘ ` k 231 ` 1 ˘ ` k 312 ` 1 ˘ ´ α C p k 1 , k 2 , k 3 q “ 2 2 α ´ 2 Γ Γ Γ 2 2 2 a 2 Γ 3 2 p πN q Γ p 2 k 1 ´ 1 q Γ p 2 k 2 ´ 1 q Γ p 2 k 3 ´ 1 q k ijk ” k i ` k j ´ k k , α ” k 1 ` k 2 ` k 3 , in precise agreement with calculation in 11 d sugra on AdS 7 ˆ S 4 ! (Corrado Florea McNees, Bastianelli Zucchini) 1 { N corrections in W N OPE ñ quantum M-theory corrections. Leonardo Rastelli (YITP) p 2 , 0 q Bootstrap Nov’15 13 / 1
p 2 , 0 q maxibootstrap Beem Lemos LR van Rees Universal 4pt function of Φ p IJ q , superprimary of T µν multiplet. Unique structure in superspace. Only input: 6 d Weyl anomaly coefficient c . For ADE theories, c “ 4 d g h _ g ` r g , but we keep it general. Leonardo Rastelli (YITP) p 2 , 0 q Bootstrap Nov’15 14 / 1
Double OPE expansion ÿ f 2 ΦΦ O G Φ x ΦΦΦΦ y “ O O P Φ ˆ Φ We impose the absence of higher-spin currents. The O s P Φ ˆ Φ are: Infinite set t O χ u of ◗ -chiral BPS multiplets, fixed from χ -algebra. Infinite tower of BPS multiplet t D , B 1 , B 3 , . . . u , not in χ -algebra. Infinite set of non-BPS multiplets L ∆ ,ℓ , so p 5 q R singlets. Bose symmetry Ñ ℓ is even . Unitarity bound ∆ ě ℓ ` 6 . Unfixed BPS multiplets correspond to long multiplets at threshold, ∆ Ñ ℓ ` 6 G Φ L ∆ ,ℓ “ G Φ lim p D ” B ´ 1 q , B ℓ ´ 1 Leonardo Rastelli (YITP) p 2 , 0 q Bootstrap Nov’15 15 / 1
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