Chiral algebras and the superconformal

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 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 / 27 Conformal Bootstrap Abstract algebra of local operators ÿ O1 pxqO2 p0q “ c12k pxqOk p0q k subject to unitarity and crossing constraints 1 O = O O O 4 Since 2008, successful numerical approach in any d See Simmons-Duffin’s talk Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 / 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 6d (2,0) SCFT Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 / 27 More technically, not clear how much susy can really help A natural question: Do the bootstrap equations in d ą admit a solvable truncation for superconformal theories? The answer is Yes for large classes of theories: (A) Any d “ 4, N ě admits a subsector (B) Any d “ 3, N ě admits a subsector or d “ 6, N “ p2, 0q SCFT – 2d chiral algebra SCFT – 1d 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 / 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 2d 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 / 27 Meromorphy in N “ or p2, 0q SCFTs Fix a plane R2 Ă Rd , parametrized by pz, z¯q Claim : D subsector Aχ “ tOi pzi , z¯i qu with meromorphic xO1 pz1 , z¯1 q O2 pz2 , z¯2 q On pzn , z¯n qy “ f pzi q Rationale: Aχ ” cohomology of a nilpotent ◗ , ◗ “Q`S, Q Poincar´e, S conformal supercharges z¯ dependence is ◗ -exact: cohomology classes rOpz, z¯qs◗ Analogous to the d “ 4, N “ chiral ring: cohomology classes rOpxqsQ˜ α9 are x-independent Leonardo Rastelli (YITP) Superconformal Bootstrap Opzq June ’14 / 27 Cohomology At the origin of R2 , ◗ -cohomology Aχ easy to describe Op0, 0q P Aχ Ø O obeys the chirality condition ∆´ “R ∆ conformal dimension, angular momentum on R2 , R Cartan generator of SU p2qR Ă full R symmetry R “ SU p2qR ˆ U p1qr for d “ 4, N “ R “ SOp5q for p2, 0q: SU p2qR – SOp3qR Ă SOp5q Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 / 27 r◗ , slp2qs “ Ę ‰0 but r◗ , slp2qs To define ◗ -closed operators Opz, z¯q away from origin, we twist the right-moving generators by SU p2qR , p´1 “ L ¯ ´1 ` R´ , L p0 “ L ¯0 ´ R , L p1 “ L ¯ ´ R` L z “ t◗ , u slp2q ◗ -closed operators are “twisted-translated” Opz, z¯q “ ezL´1 `¯zL´1 Op0q e´zL´1 ´¯zL´1 p p SU p2qR orientation correlated with position on R2 Chirality condition Leonardo Rastelli (YITP) ∆´ p0 “ ´R“0 ô L Superconformal Bootstrap June ’14 / 27 By the usual formal argument, the z¯ dependence is exact, rOpz, z¯qs◗ Opzq Cohomology classes define left-moving 2d operators Oi pzq, with conformal weight h“R` They are closed under OPE, O1 pzqO2 p0q “ ÿ k c12k h z `h2 ´hk Ok p0q Aχ has the structure of a 2d chiral algebra Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 / 27 General claim For the p2, 0q SCFT labelled by the simply-laced Lie algebra g, the chiral algebra is Wg , with c2d pgq “ 4dg h_ g ` rg Connection with the AGT correspondence c2d pgq matches Toda central charge for b “ Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 13 / 27 Half-BPS 3pt functions of p2, 0q SCFT OPE of Wg generators ñ half-BPS 3pt functions of SCFT Let us check the result at large N WN Ñ8 with c2d „ 4N Ñ a classical W -algebra (Gaberdiel Hartman, Campoleoni Fredenhagen Pfenninger) We find ¯ ´ ¯ ´ ¯˛ ´ Γ k1232 `1 Γ k2312 `1 Γ k3122 `1 ‚ ˝a Cpk1 , k2 , k3 q “ Γ Γp2k ´ 1qΓp2k ´ 1qΓp2k ´ 1q pπN q ¨ 22α´2 ´α¯ kijk ” ki ` kj ´ kk , α ” k1 ` k2 ` k3 , in precise agreement with calculation in 11d sugra on AdS7 ˆ S ! (Corrado Florea McNees, Bastianelli Zucchini) 1{N corrections in WN OPE ñ quantum M-theory corrections Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 14 / 27 χ4 : 4d N “ SCFT ÝÑ 2d Chiral Algebra Global slp2q Ñ Virasoro T pzq :“ rJR pz, z¯qs◗ , the SU p2qR conserved current c2d “ ´12 c4d c4d ” Weyl2 conformal anomaly coefficient Global flavor Ñ Affine symmetry Jpzq :“ rM pz, z¯qs◗ , the moment map operator k2d “ ´ k4d 4d 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 AI1 I2 I3 I4 pz, z¯q “ x OI1 p0q OI2 pz, z¯q OI3 p1q OI4 p8q y I= index of some SU p2qR irrep Associated chiral algebra correlator f pzq“xOp0qOpzqOp1qOp8qy , Opzq “ ruI p¯ z qOI pz, z¯qs◗ Double-OPE expansion ÿ ÿ long long short Apz, z¯q “ pshort G pz, z ¯ q ` pk Gk pz, z¯q i i Gi = superconformal blocks = ř finite conformal blocks G∆, The short part can be entirely reconstructed from f pzq Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 16 / 27 Symmetries & central charges c Ó Chiral algebra correlator f pz; cq Ó Short spectrum and OPE coefficients pshort pcq i (unique assuming no higher-spin symmetry) Ó short A pz, z¯; cq Ó Finally, numerical bootstrap of Along pz, z¯q Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 17 / 27 Unitarity ñ pshort pcq ě ñ novel bounds on central charges i For example, in any interacting d “ 4, N “ SCFT with flavor group GF , dim GF 24h_ ě ´ 12 c4d k4d c4d = Weyl2 conformal anomaly, k4d = flavor central charge Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 18 / 27 Bootstrap Sum Rule ÿ long super primaries p∆, F∆, pz, z¯q ` F short pz, z¯; cq “ ˆ F∆, ” G∆, ´ G∆, is the superconformal block minus its crossing Contrast with sum rule from Rattazzi Rychkov Tonni Vichi ÿ p∆, F∆, pz, z¯q ` F identity pz, z¯q “ primaries F∆, ” G∆, ´ Gˆ ∆, is the conformal block minus its crossing Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 19 / 27 Three paradigmatic cases d “ 6, p2, 0q: stress-tensor-multiplet 4pt function Beem Lemos LR van Rees, to appear d “ 4, N “ 4: stress-tensor multiplet 4pt function Beem LR van Rees Alday Bissi d “ 4, N “ 2: moment-map 4pt function Beem Lemos Liendo LR van Rees, to appear Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 20 / 27 Bootstrap of stress-tensor multiplet 4pt in p2, 0q 9.0 8.5 8.0 7.5 7.0 6.5 6.0 22 derivatives 21 derivatives 20 derivatives 19 derivatives 18 derivatives 17 derivatives 16 derivatives 15 derivatives 14 derivatives 10 20 A1 theory c1 Figure : Upper bound for the dimension ∆0 of the leading-twist unprotected operator of spin “ 0, as a function of the anomaly c Within numerical errors, the bound at large c agrees with the dimension (=8) of the “double-trace” operator : O14 O14 : Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 21 / 27 Bootstrap of stress-tensor multiplet 4pt in p2, 0q 11.0 10.5 10.0 9.5 9.0 8.5 8.0 22 derivatives 21 derivatives 20 derivatives 19 derivatives 18 derivatives 17 derivatives 16 derivatives 15 derivatives 14 derivatives 10 20 A1 theory c1 Figure : Upper bound for the dimension ∆2 of the leading-twist unprotected operator of spin “ 2, as a function of the anomaly c Within numerical errors, the bound at large c agrees with the dimension (=10) of the “double-trace” operator : O14 B O14 : Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 22 / 27 Bootstrap of stress-tensor multiplet 4pt in p2, 0q 13.0 12.5 22 derivatives 21 derivatives 12.0 20 derivatives 19 derivatives 11.5 18 derivatives 17 derivatives 11.0 16 derivatives 15 derivatives 10.5 10.0 14 derivatives A1 theory 10 20 13 c Figure : Upper bound for the dimension ∆4 of the leading-twist unprotected operator of spin “ 4, as a function of the anomaly c Within numerical errors, the bound at large c agrees with the dimension (=12) of the “double-trace” operator : O14 B O14 : Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 23 / 27 Bootstrap of stress-tensor multiplet 4pt in N “ 4.5 SU 6.0 U1 4.0 5.5 3.5 3.0 U 5.0 4.5 SU 2.5 4.0 a a Figure : Bounds for the scaling dimension of the leading-twist unprotected operator of spin “ 0, 2, as a function of the anomaly a For a Ñ 8, saturated by AdS5 ˆ S sugra, including 1{a corrections In planar N “ SYM for large ’t Hooft coupling, leading-twist unprotected operators are double-traces of the form Os “ O201 B s O201 Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 24 / 27 Bootstrap of moment map 4pt in d “ 4, N “ Input: flavor group GF , flavor central charge k, conformal anomaly c 0.10 0.09 0.08 0.07 c 0.06 0.05 0.04 0.03 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 30derivatives, 32spins 26derivatives, 28spins 22derivatives, 24spins 18derivatives, 22spins 14derivatives, 20spins 10derivatives, 16spins Free hypermultiplet k Figure : Exclusion plot in the plane p k1 , cq for a general N “ SCFT with GF “ SU p2q flavor symmetry Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 25 / 27 Outlook: miniboostrap Chiral algebras of the N “ SCFTs of class S Generalized TQFT structure Interesting purely mathematical conjectures Beem Peelaers LR van Rees, to appear For a given SCFT T , develop systematic tools to characterize χrT s in terms of generators Classification of SCFTs related to classification of “special” chiral algebras Add non-local operators Particularly interesting in d “ 6: a derivation of AGT? Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 26 / 27 Outlook: numerical boostrap (2, 0) bootstrap: in progress Stay tuned Exploration of landscape of N “ SCFTs, especially non-Lagrangian ones More d “ 4, N “ Intriguing interplay of mathematical physics and numerical experimentation Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 27 / 27 ... generators The chiral algebra of the AN ´1 theory is WN , with c2d “ 4N ´ 3N ´ Leonardo Rastelli (YITP) Superconformal Bootstrap June ’14 12 / 27 General claim For the p2, 0q SCFT labelled by the simply-laced... of the 12 -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 p2, 0q theory... 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