!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!On!6d!SCFT’s! ! !! !!!!!!!!!!!!!!!!!!!!!!!Cumrun!Vafa! !!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!Strings!2014! !!!!!!!!!!!!!!!!!!!!!!!!!!!Princeton! !! !!!!!!!!!!!!!!!!!!!!!!!!June!23,!2014! !!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!! TexPoint!fonts!used!in!EMF.!! Read!the!TexPoint!manual!before!you!delete!this!box.:!AAAAAAAAAAA! ! ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Based!on:! ! arXiv:1312.5746//On/the/Classifica:on/of/6D/SCFTs/and/Generalized/ADE/Orbifolds/ Jonathan!J.!Heckman,!David!R.!Morrison,!Cumrun!Vafa! ! M.!Del!ZoPo,!J.!Heckman,!A.!Tomasiello,!C.!Vafa,!to!appear.! ! arXiv:1310.1185//On/orbifolds/of/MJStrings/ Babak!Haghighat,!Can!Kozcaz,!Guglielmo!Lockhart,!Cumrun!Vafa! ! arXiv:1406.0850!!E!+!E!!!H!! Babak!Haghighat,!Guglielmo!Lockhart,!Cumrun!Vafa!! ! 6d!SCFT’s!enjoy!a!unique!status:!!They!are!the!! highest!dimension!known!non^trivial!CFT’s.!!! ! Two!types!based!on!number!of!SUSY’s:! ! (2,0):!Believed!to!be!classified!by!an!ADE.! ! A:!!Stack!of!parallel!M5!branes! ADE:!type!IIB!at!an!ADE!singularity! ! (1,0):!More!difficult!to!classify.!!Many!examples.! Blum,Intriligator;!Seiberg,WiPen;!Hanany,Zaffaroni;!Aspinwall,Morrison;! Ganor,Hanany;!…! There!are!two!aspects:! ! 1^Classify!the!(1,0)!theories! ! Will!use!IIB/F^theory!realizakon! ! 2^Develop!tools!to!study!each! ! Tensionless!strings/suspended!M2!branes!! ! (update!with!new!results!since!strings!2013)! There are two types of deformations: Coulomb/Tensor branch: i (B¹º ; Ái ) ! String; T ension = ni Ái Higgs Branch: HyperkÄ ahler geometry Examples:!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !! M5!branes!probing!an!An!singularity! ! Higgs!branch:!!moving!away!from!singularity! Coulomb!branch:!separakng!M5!branes! ! Examples:!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !! M5!branes!probing!an!An!singularity! ! Higgs!branch:!!moving!away!from!singularity! Coulomb!branch:!separakng!M5!branes! ! Examples:!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !! M5!branes!probing!an!An!singularity! ! Higgs!branch:!!moving!away!from!singularity! Coulomb!branch:!separakng!M5!branes! ! Examples:!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !! M5!branes!probing!an!An!singularity! ! Higgs!branch:!!moving!away!from!singularity! Coulomb!branch:!separakng!M5!branes! ! ! ! ! M-theory Large N Duals Some of these theories admit simple large N duals: AdS £ S =¡ADE e.g E7: 52222….22225 AdS £ S =Z2 1222222222 AdS £ S =¡ADE £ Z2 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!M^theory!Large!N!Duals!!!!! ! Some!of!these!theories!admit!simple!large!N!duals:! ! A dS £ S =¡ A D E !! !!!!!!e.g.!!E7:!52222….22225!!!! ! A dS £ S =Z ! !!!!!!!!!!!!!1222222222!!!!!!!!! ! A dS £ S =¡ £ Z A D E ! !!!!!! ! ! Computation of the Superconformal Index ZS £S = ZR4 £T Z ZTn2 : R MC 00 Z ZR4 £T ZR £T R4 £T [Lockhart,V; Kim3] can be computed using string instantons: R4 £T = P n e ¡n¢t n ZT The elliptic genus of n strings twisted by rotations of R4 and global symmetries of CFT Alternatively ZR4 £T can be computed using topological string/Nekrasov partition function For!the!theory!of!!M5!branes!probing!A!!!!!!!:! N^1! N!!!!!!!!N!!!!!!!N!!!!!.!.!.!!!!!!!N!!!!!!!N!!!!!!!N!!!!!!!!N! There!are!M^1!different!strings.!!They!form!a! (4,0)!supersymmetric!theory!in!2d.!!If!we!denote! their!numbers!by!k!!!,…,k!!!!!!!!,!the!corresponding! 1!!!!!!!!M^1! theory!is!a!quiver!theory:! The!ellipkc!genus!can!be!computed! [Gadde,Gukov;!Benini,Eager,Hori,Tachikawa]! One learns unlike fundamental strings, tensionless strings form bound states which is reflected in: Z2 (¿ ) 6= 12 [(Z1 (¿ )2 + Z1 (2¿ ) + Z1 ( ¿2 ) + Z1 ( ¿ +1 )] !!!!!!!!!!!!!!!!!!!!!!!!!!A!Dual!Descripkon:!QM! ! In!M^theory,!strings!arise!as!M2!branes!suspended! between!M5!branes.!Leads!to!a!dual!perspeckve:! !!!!!!!!!!!!!!!!!!!!!!3=2+1;!!i.e.!string,!versus!QM! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! A Dual Description: QM In M-theory, strings arise as M2 branes suspended between M5 branes Leads to a dual perspective: 3=2+1; i.e string, versus QM h0j:::D exp(¡¯1 H)D exp(¡¯2 H)D :::j0i E-string: hM 9jexp(¡¯1 H)Dj0i Z2E-str = D M 9;L DM M 9;L M + D D ; ; Using recent results for topological string on K3/2 [Huang,Klemm,Poretschkin] extract Z2 ! hM 9(2) j n n n hM 9jexp(¡¯1 H)D exp(¡¯2 H)jM 9i E+E H hM 9jexp(¡¯1 H) exp(¡¯2 H)jM 9i = ZHeterotic !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!E+E!!!!!H! hM 9jexp(¡ * H ) exp(¡ * H )jM 9i = Z H eter oti c ~ E8 ;R ) = Z2het (¿; ²; m ~ E8 ;L ; m + N (¿; m ~ E8 ;L ; ²)N (¿; m ~ E8 ;R ; ²) ´(¿ )24 µ1 (¿ ; ¡3²)µ1 (¿ ; ¡2²)2 µ1 (¿ ; ¡²)µ1 (¿ ; ²)2 µ1 (¿ ; 2²)2 N (¿; m ~ E8 ;L ; ²)N (¿; m ~ E8 ;R ; ²) : ´(¿ )24 µ1 (¿ ; ¡²)µ1 (¿ ; ²)2 µ1 (¿ ; 2²)2 µ1 (¿ ; 3²)µ1 (¿ ; 4²)2 (1) Comparing to the usual expression Z2het (¿;~²; m ~ E8 £E8 ) = Z1het = µ · ´2 ³ het ~ E8 £E8 ) ~ E8 £E8 ) + Z1het (2¿; 2~²; 2m Z1 (¿;~²; m ¸ ¿ + ¿ + Z1het ( ;~²; m ;~²; m ~ E8 £E8 ) : ~ E8 £E8 ) + Z1het ( 2 ~ E8 ;L ) £ #E8 (m ~ E8 ;R ) #E8 (m ´ 16 ¶ ´4 ; µ1 (²1 )µ1 (²2 )µ1 (²3 )µ1 (²4 ) (1) leads to highly non-trivial identities, that we have checked to high order !!!!!!!!!!!!!!!!!!!!!!!!!!!!Conclusions!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! 1^Classificakon!of!maximally!Higgsed!branches! of!(1,0)!theories!completed!(within!F^theory)! leading!to!a!generalized!ADE!classificakon.! ! 2^Strings!in!the!Coulomb!branch!of!some!of!! these!theories!are!being!understood!and!used! to!compute!supersymmetry!protected!quankkes! for!the!(1,0)!theory.! ... !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!and!each!other! M5!branes!at !the! Hořava^WiPen!wall! ! Higgs!branch:!!Dissolving!branes !in! the! wall! !!!!!!!!!!!!!!!!!!!!!!!!!!!giving!finite!size!E8!instantons! Coulomb!branch:!!Moving!away!from !the! wall!... M5!branes!at !the! Hořava^WiPen!wall! ! Higgs!branch:!!Dissolving!branes !in! the! wall! !!!!!!!!!!!!!!!!!!!!!!!!!!!giving!finite!size!E8!instantons! Coulomb!branch:!!Moving!away!from !the! wall! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!and!each!other!... Higgs!branch:!!Dissolving!branes !in! the! wall! !!!!!!!!!!!!!!!!!!!!!!!!!!!giving!finite!size!E8!instantons! Coulomb!branch:!!Moving!away!from !the! wall! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!and!each!other! ! !