Osp(1 2M)-invariant higher-spin systems

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Corey Shields
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1 Osp M-nvarant hgher-spn systes Dtr Sorokn INFN Padova Secton Based on arv: work n progress wth Ioanns Floraks CERN and Mran Tsulaa Canerra Penn State Un August

2 Prelnares The largest nte-densonal syetry o a HS syste n D= s extended conoral syetry Sp8 t unes n nnte ultplets elds o s=0. and s=/ / nteger and hal-nteger spn elds can e urther uned nto an Osp 8 superultplet Free HS theory s a sple theory o a hyper scalar and spnor eld n a space-te extended wth 6 extra tensoral coordnates hyperspace n whch Sp8 acts as the conoral group Fronsdal 85; Bandos Lukersk D.S ; Vaslev 0; Ddenko Vaslev 0; Plyushchay Tsulaa D.S. 0; Bandos Bekaert et. al. 05; elond Vaslev 05-5; Fedoruk Ivanov 07 Descrpton o the Sp8-nvarant HS syste wth CFT ethods

3 Sp8 syetry o d HS theory Fronsdal 985 SO SO SO SO SpR Sp8R Sp8 acts on nnte spectru o d HS sngle-partcle states s=0 / / ths s a consequence o Flato-Fronsdal Theore 978: Tensor product o two d sngleton odul coprses all the assless spn-s elds n d Sngletons are d assless scalar and spnor elds whch enjoy d conoral syetry SO Sp SS Sp Sp Sp8 Can Sp8 play a role slar to Poncaré or conoral syetry actng geoetrcally on a hyperspace contanng d space-te? Whether a d HS theory can e orulated as a eld theory on ths hyperspace? `eoetrcally eans: con n x a l nx x k x k n x n x Fronsdal 85: nal denson o the Sp8 hyperspace contanng d space-te s 0

Partcles and elds n Sp8 hyperspace Bandos & Lukersk 98: Twstor-lke super partcle on a tensoral space ther otvaton was not related to HS theory ut to supersyetry Most general N= susy n lat d: { Q Q }

4 Partcles and elds n Sp8 hyperspace Bandos & Lukersk 98: Twstor-lke super partcle on a tensoral space ther otvaton was not related to HS theory ut to supersyetry Most general N= susy n lat d: { Q Q } P n n [ P nl ] 0 0d space coordnates: P x d coordnates x y n n n - y n y n 6 extra coordnates atrx coordnates Superpartcle acton: S d P - coutng twstor-lke varale possesses hdden generalzed superconoral syetry OSp 8 Sp8 Quantzaton Bandos Lukersk & D.S. 99 Curtrght 88 Are there superstrngs n D? ; 0 - descres n d ree elds o any spn s=0 / / supertwstor relatons

5 Feld theory n lat Sp8 hyperspace Feld equatons n lat hyperspace Vaslev 0: Forer transor C d e k C... C... k C 0 Free unolded equatons and are ndependent scalar and spnor hyperelds satsyng the equatons: 0 0 d content o and are Hgher-Spn curvatures Vaslev 0 Bandos et. al. 05: n x y n Integer spns: Secton condton n generalzed geoetry o M-theory Beran et. al. x y nlp x y 0 0 n n lp x y y y n n n pq x y x Fn x y Rn pq x pnq y y... n n ½ nteger spns: x y x x y... n n Eos and Banch: p 0 ; [ lfn] 0 Fn 0; R[ n p] q 0 Rn pq 0;... 5

6 Sp8 transoratons n hyperspace syetres o the eld equatons Conoral transoratons: con n x a l nx x k x k n x n x Sp8 transoratons: Sp 8 a g k g g k k g k Sp8 generators: conoral weghts o the elds P L generators o L K [ P P] 0 [ L L] L [ K K] 0 [ L P] P [ P K] L [ L K] K Hyperspace s a coset space: P Sp8 K L 6

7 Hyperspace extenson o AdS Bandos Lukersk Pretschop D.S. 99; Vaslev 0 0d group anold SpR ~ SO AdS Sp SO Sp8 Sp P K - derent Sp8 coset realzaton K L Lke Mnkowsk and AdS spaces whch are conorally lat the lat hyperspace and Sp are locally related to each other y a generalzed conoral transoraton SpM group anolds are L-lat Plyushchay D.S. & Tsulaa 0 Algera o covarant dervatves on Sp: [ ] C C r C C - - r d ; r - s Sp or AdS radus Sp Cartan or L-latness s portant or the relaton etween the eld equatons n lat and Sp hyperspace 7

8 HS eld equatons n Sp Ddenko and Vaslev 0; Plyushchay D.S. & Tsulaa 0 Flat hyperspace equatons: 0 0 Sp eld equatons Plyushchay D.S. & Tsulaa 0 : Fer: F F r C F C F Bose: B B 8r C C C B C C C C r B eneralzed conoral relatons etween lat and Sp hyperelds Floraks D.S. & Tsulaa B F r 8

9 Energy-oentu tensors o hyperelds and HS conserved currents Vaslev 0 T T Conservaton law: T T T T 0 Closed -or: J HS currents: T d d d n k k nk a n k... n... k... nk k 0 d T dj 0 Q M J Integraton easure n hyperspace Flat M 0 : Sp: Sp tr EEE tr EEEEEEE 0 tr ddd tr dddddd r 7 E d d 9

10 c c c Sp8 nvarant correlaton unctons In lat hyperspace Vaslev 0 Vaslev & akn 0 also Ddenko & Skovrtsov In Sp hyperspace Floraks D.S. & Tsulaa lat Sp lat Sp lat Sp B B B F F B B r conoral weght 0

11 Four-pont unctons Bosonc: j j Feronc:

12 Supersyetry n hyperspace oton o 8 - nvarant equatons - 0 ] [ } { auxlary elds OSp D D P D D D n n Hgher spn elds or an nnte densonsal D= superultplet Osp 8 nvarant correlaton unctons o scalar superelds j j j j k k k c c 0 ±/ ± ±/ ± ±5/ On supergroup anold Osp : 0 8 ] [ C r

Concluson Free theory o the nnte nuer o assless HS elds n d lat and AdS space has generalzed conoral Sp8 syetry and can e copactly orulated n 0d hyperspace wth the use o one scalar and one spnor eld.

13 Concluson Free theory o the nnte nuer o assless HS elds n d lat and AdS space has generalzed conoral Sp8 syetry and can e copactly orulated n 0d hyperspace wth the use o one scalar and one spnor eld. May Sp8 syetry have soethng to do wth the trvalzaton o the partton uncton o the nnte syste o s=0 Beccara & Tseytln Hgher densonal extenson to SpM nvarant hyperspaces s straghtorward Bandos Lukersk D.S. 99 Vaslev 0.. known physcally relevant cases are Bandos Bekaert de Azcarraga D.S. Tsulaa 05 M= d= M= d= M=8 d=6 M=6 d=0 descre conoral HS elds n correspondng space-tes. Supersyetrc generalzatons are avalale Bandos et. al Vaslev et. al. Ivanov et. al. P. West Floraks et. al Is there any relaton to Douled Feld Theory? Man prole: Whether one can construct an nteractng eld theory n hyperspace whch would descre HS nteractons n conventonal space-te at least wth vertces coposed o HS curvatures and ther dervatves elond & Vaslev 5: HS current nteractons reak Sp8 down to SU How can one reveal ths reakng usng conoral analyss o correlaton unctons?

total If no external forces act, the total linear momentum of the system is conserved. This occurs in collisions and explosions.

total If no external forces act, the total linear momentum of the system is conserved. This occurs in collisions and explosions. Lesson 0: Collsons, Rotatonal netc Energy, Torque, Center o Graty (Sectons 7.8 Last te we used ewton s second law to deelop the pulse-oentu theore. In words, the theore states that the change n lnear oentu