Chiral Algebras and the Superconformal Bootstrap in Four and Six - PowerPoint PPT Presentation

Chiral Algebras and the Superconformal Bootstrap in Four and Six Dimensions Leonardo Rastelli Yang Institute for Theoretical Physics, Stony Brook Based on work with C. Beem, M. Lemos, P. Liendo, W. Peelaers and B. van Rees. Strings 2014, Princeton

SuperConformal Field Theories in d ą 2 Fast-growing body of results: Many new models, most with no known Lagrangian description. A hodgepodge of techniques: localization, integrability, effective actions on moduli space. Powerful but with limited scope. Conformal symmetry not fully used. We advocate a more systematic and universal approach. Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 1 / 27

Conformal Bootstrap Abstract algebra of local operators ÿ O 1 p x q O 2 p 0 q “ c 12 k p x q O k p 0 q k subject to unitarity and crossing constraints 1 3 1 3 � � O ′ = O O O ′ 2 4 2 4 Since 2008, successful numerical approach in any d . See Simmons-Duffin’s talk. Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 2 / 27

Two sorts of questions What is the space of consistent SCFTs in d ď 6 ? For maximal susy, well-known list of theories. Is the list complete? What is the list with less susy? Can we bootstrap concrete models? The bootstrap should be particularly powerful for models uniquely cornered by few discrete data. Only method presently available for finite N , non-Lagrangian theories, such as the 6 d (2,0) SCFT. Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 3 / 27

More technically, not clear how much susy can really help. A natural question: Do the bootstrap equations in d ą 2 admit a solvable truncation for superconformal theories? The answer is Yes for large classes of theories: Any d “ 4 , N ě 2 or d “ 6 , N “ p 2 , 0 q SCFT (A) admits a subsector – 2 d chiral algebra. (B) Any d “ 3 , N ě 4 SCFT admits a subsector – 1 d TQFT. Beem Lemos Liendo Peelaers LR van Rees, Beem LR van Rees In this talk, we’ll focus on the rich structures of (A). Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 4 / 27

Bootstrapping in two steps For this class of SCFTs, crossing equations split into (1) Equations that depend only on the intermediate BPS operators. Captured by the 2 d chiral algebra. (2) Equations that also include intermediate non-BPS operators. (1) are tractable and determine an infinite amount of CFT data, given flavor symmetries and central charges. This is essential input to the full-fledged bootstrap (2), which can be studied numerically. Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 5 / 27

Meromorphy in N “ 2 or p 2 , 0 q SCFTs Fix a plane R 2 Ă R d , 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) Superconformal Bootstrap June ’14 6 / 27

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 Ă full R symmetry R “ SU p 2 q R ˆ U p 1 q r for d “ 4 , N “ 2 R “ SO p 5 q for p 2 , 0 q : SU p 2 q R – SO p 3 q R Ă SO p 5 q . Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 7 / 27

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 ´ R ` . p p p L ´ 1 “ ¯ L 0 “ ¯ L 1 “ ¯ L 0 ´ R , z sl p 2 q “ t ◗ , . . . u ◗ -closed operators are “twisted-translated” z p L ´ 1 O p 0 q e ´ zL ´ 1 ´ ¯ z p L ´ 1 . z q “ e zL ´ 1 ` ¯ O p z, ¯ SU p 2 q R orientation correlated with position on R 2 . ´ R “ 0 ô p Chirality condition ∆ ´ ℓ L 0 “ 0 2 Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 8 / 27

By the usual formal argument, the ¯ z dependence is exact, r O p z, ¯ z qs ◗ O p z q . � Cohomology classes define left-moving 2 d operators O i p z q , with conformal weight h “ R ` ℓ. They are closed under OPE, ÿ c 12 k O 1 p z q O 2 p 0 q “ z h 1 ` h 2 ´ h k O k p 0 q . k A χ has the structure of a 2 d chiral algebra Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 9 / 27

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) Superconformal Bootstrap June ’14 10 / 27

χ 6 : 6d (2,0) SCFT Ý Ñ 2d Chiral Algebra . Global sl p 2 q Ñ Virasoro, indeed T p z q : “ r O 14 p z, ¯ z qs ◗ , with O 14 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 also play a role. Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 11 / 27

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. 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 . Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 12 / 27

General claim For the p 2 , 0 q SCFT labelled by the simply-laced Lie algebra g , the chiral algebra is W g , with c 2 d p g q “ 4 d g h _ g ` r g . Connection with the AGT correspondence. c 2 d p g q matches Toda central charge for b “ 1 . Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 13 / 27

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 W -algebra. (Gaberdiel Hartman, Campoleoni Fredenhagen Pfenninger) We find ´ ¯ ´ ¯ ´ ¯ ¨ ˛ k 123 ` 1 k 231 ` 1 k 312 ` 1 ´ α ¯ Γ Γ Γ C p k 1 , k 2 , k 3 q “ 2 2 α ´ 2 2 2 2 ˝ ‚ a Γ 3 2 Γ p 2 k 1 ´ 1 q Γ p 2 k 2 ´ 1 q Γ p 2 k 3 ´ 1 q p πN q 2 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) Superconformal Bootstrap June ’14 14 / 27

χ 4 : 4d N “ 2 SCFT Ý Ñ 2d Chiral Algebra . Global sl p 2 q Ñ Virasoro T p z q : “ r J R p z, ¯ z qs ◗ , the SU p 2 q R conserved current. c 2 d “ ´ 12 c 4 d c 4 d ” Weyl 2 conformal anomaly coefficient. Global flavor Ñ Affine symmetry J p z q : “ r M p z, ¯ z qs ◗ , the moment map operator. k 2 d “ ´ k 4 d 2 4 d Higgs branch generators Ñ chiral algebra generators. Higgs branch relations ” chiral algebra null states! Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 15 / 27

Bootstrap of the full 4pt function A I 1 I 2 I 3 I 4 p z, ¯ z q “ x O I 1 p 0 q O I 2 p z, ¯ z q O I 3 p 1 q O I 4 p8q y I = index of some SU p 2 q R irrep. Associated chiral algebra correlator z q O I p z, ¯ f p z q“x O p 0 q O p z q O p 1 q O p8qy , O p z q “ r u I p ¯ z qs ◗ . Double-OPE expansion ÿ ÿ p long G long p short G short A p z, ¯ z q “ p z, ¯ z q ` p z, ¯ z q i i k k G i = superconformal blocks = ř finite conformal blocks G ∆ ,ℓ . The short part can be entirely reconstructed from f p z q . Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 16 / 27

Symmetries & central charges c Ó Chiral algebra correlator f p z ; c q Ó Short spectrum and OPE coefficients p short p c q i (unique assuming no higher-spin symmetry) Ó A short p z, ¯ z ; c q Ó Finally, numerical bootstrap of A long p z, ¯ z q Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 17 / 27

Unitarity ñ p short p c q ě 0 ñ novel bounds on central charges. i For example, in any interacting d “ 4 , N “ 2 SCFT with flavor group G F , ě 24 h _ dim G F ´ 12 . c 4 d k 4 d c 4 d = Weyl 2 conformal anomaly, k 4 d = flavor central charge. Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 18 / 27