SO(9) supergravity. conference on the occasion of Hermann Nicolai s 60th anniversary Golm Henning Samtleben. * in two dimensions

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1 SO() supergrvity * conference on the occsion of Hermnn Nicoli s 60th nniversry Golm 0!! * in two dimensions

2 SO() supergrvity * [Thoms Ortiz, HS] 0 [HS, Mrtin Weidner] 007 [Hermnn Nicoli, HS] 000 [Hermnn Nicoli, Bernrd de Wit] 8

3 8 Domin wll / QFT correspondence hologrphy for Dp-rnes [H.J. Boonstr, K. Skenderis, P. Townsend, ] gugings of mximl supergrvity AdS / CFT cse D3 IIB AdS5 x S5 d=5, SO(6) [Günydin, Romns, Wrner, 85]

4 8 Domin wll / QFT correspondence hologrphy for Dp-rnes [H.J. Boonstr, K. Skenderis, P. Townsend, ] gugings of mximl supergrvity wrped D6 IIA AdS8 x S d=8, SO(3) D5 IIB AdS7 x S3 d=7, SO() D IIA AdS6 x S d=6, SO(5) [Pernici, Pilch, vn Nieuwenhuizen, 8] D3 IIB AdS5 x S5 d=5, SO(6) [Günydin, Romns, Wrner, 85] D IIA AdS x S6 d=, SO(7) [Hull, 8] F/D IIA/B AdS3 x S7 d=3, SO(8) [de Wit, Nicoli, 8] D0 IIA AdS x S8 d=, SO() [Slm, Sezgin, 8]??? try to construct the SO() theory...

5 8 : constructing SO() supergrvity [Hermnn Nicoli, HS]

6 : trying to construct d=, SO() supergrvity 8 two-dimensionl supergrvity is prticulrly simple :! " L = g ρ R tr[p µ Pµ ] Lferm (ψ I, ψi, χa ) coset spce sigm model E8/SO(6) coupled to dilton grvity ut hs remrkle structure : (infinite tower of) dul sclr potentils etc. ν µ YM εµν JM conserved E8 Noether current clssicl integrility, ffine Lie-Poisson symmetry E off-shell symmetry : E8 cn we guge sugroup SO()??

7 8 : trying to construct d=, SO() supergrvity 000 introducing vector fields? (non-propgting in d=) δaµ M = V M ( "I ψµj ) iiaa V M A ( "I γµ χa ) we cn introduce 8 (= dim E8) vector fields on which supersymmetry closes provided tht Fµν M = 0! (origin : d=3) guging requires Yukw couplings nd fermion shifts δψµi = Dµ #I ga γµ #J etc. with sclr tensors A (V, YM ) depending on sclrs nd dul potentils! ν Dµ YM εµν JM non-elin dulity reltion implies non-trivil consistency reltions [Fµν, Y ] = εµν (Dρ J ρ ) εµν V Vpot surprise no : consistent supersymmetric system of field equtions!

8 8 : trying to construct d=, SO() supergrvity 000 miniml couplings nd the emedding tensor Dµ µ gaµ M ΘM N tn ΘM N 3875 surprise no : group theory determines ll possile gugings! guge group G0 SO(8) SO(8) N = (nl, nr ) ckground supergroup GL GR distinguished compct guge R) group (8, 8) OSp(8, R) OSp(8, (8, 8) F () F () (8, 8) SU(, ) SU(, ) (8, 8) OSp( ) OSp( ) (8, 8) [D (, ; ) SU( )] (, ) D (, ; /3) OSp( 6) (7, )severlg(3)mximl OSp(, R) sugroups of E (6, 0) OSp( 8) SU(, ) (, ) SU(6, ) D (, ; /) (0, 6) OSp(0, R) SU(3, ) SO() singlet in 3875) no trce of SO()... (no (6, 0) SU(8, ) SU(, ) ) OSp(, R) D (, ; /3) (, (6, 0) OSp(6, R) SU(, ) 8 Tle 3: Bckground isometries of the mximlly supersymmetric ground sttes. mde explicit in (5.6) nd tle, respectively, y designting the simple fctors of H0 with the corresponding su- nd superscripts. In fct, the only guge groups for JHEP0( SO(8) SO(8) SO(7, ) SO(7, ) SO(6, ) SO(6, ) SO(5, 3) SO(5, 3) SO(, ) SO(, ) G() F() G F(0) E6(6) SL(3) E6() SU(, ) E6() SU(3) E7(7) SL() E7(5) SU() E8(8)

9 8 : trying to construct d=, SO() supergrvity 000 miniml couplings nd the emedding tensor Dµ µ gaµ M ΘM N tn ΘM N 3875 surprise no : group theory determines ll possile gugings! SO(8) SO(8) distinguished compct guge group surprise no 3 : in fct, this is d=3 theory! ν Dµ YM εµν JM [Fµν, Y ] = εµν (Dρ J ρ ) εµν V Vpot ρ N ΘM N Fµν εµνρ JM non-elin sclr-vector dulity in d=3 surprise no : there is simple Lgrngin description! L = tr [Pµ P µ ] ΘM N AM dan... guged sigm-model with Chern-Simons coupled vector fields!

10 8 : trying to construct d=, SO() supergrvity 000 Dµ µ gaµ M ΘM N tn ΘM N 3875 L = tr [Pµ P µ ] ΘM N AM dan... guged sigm-model with Chern-Simons coupled vector fields! SO(8) SO(8) distinguished compct guge group d=3 mximl guged supergrvity construction of three-dimensionl AdS supergrvities new AdS vcu hologrphy & fluxes construction/systemtics of higher-dimensionl gugings tensor hierrchies of non-elin p-forms rigid supersymmetry, flt trget spces

11 8 : trying to construct d=, SO() supergrvity 000 Dµ µ gaµ M ΘM N tn ΘM N 3875 L = tr [Pµ P µ ] ΘM N AM dan... guged sigm-model with Chern-Simons coupled vector fields! SO(8) SO(8) distinguished compct guge group d=3 mximl guged supergrvity where is the SO() theory? yers lter... 0 : Golm, Hermnn s 60th irthdy...

12 8 0 : constructing SO() supergrvity [with Thoms Ortiz]

13 007 : guging d= supergrvity 8 Dµ µ gaµ M ΘM A TA ΘM A = (TB )M N η AB θn d= supergrvity hs n ffine symmetry group : E = E8 vector fields trnsform in the sic representtion of E the emedding tensor trnsforms in the sic representtion of E hidden symmetries θm 8 off-shell shift symmetries d=3 gugings vector fields the ingredients in

14 007 : guging d= supergrvity 8 Dµ µ gaµ M ΘM A TA ΘM A = (TB )M N η AB θn d= supergrvity hs n ffine symmetry group : E = E8 vector fields trnsform in the sic representtion of E the emedding tensor trnsforms in the sic representtion of E hidden symmetries θm 8 shift symmetries off-shell vector fields the full structure: inf-dim HW reps

15 0 : constructing SO() supergrvity 8 Dµ µ gaµ M ΘM A TA ΘM A = (TB )M N η AB θn d= supergrvity hs n ffine symmetry group : E = E8 vector fields trnsform in the sic representtion of E the emedding tensor trnsforms in the sic representtion of E hidden symmetries θm 8 off-shell shift symmetries the SO() theory cn e identified! vector fields

16 : qudrtic constrint is utomticlly stisfied. We hve thus shown tht n emedding tensor in the 5!/3 defines consistent guging in two dimensions. This representtion cn guging d= supergrvity constructing SO() supergrvity e prmetrized y symmetric mtrix Y. By fixing prt of the SL() symmetry this mtrix cn e rought into the form off-shell..., 0,symmetries..shift.), Y = dig(symmetries,hidden...,,! "# $! "# $! "# $ p!!! 8 q (5.) r !!! vector fields 6 3 with p q r =. Such n emedding tensor guges sulger cso(p, q, r) of the zero-mode lger sl() in (5.0). The corresponding guge fields come 8from the 36/3. 8 compctifiction For r = q = 0 this is the SO() guging corresponding to the IIA S 375 mentioned ove. In ddition there is the infinite tower of shift-symmetries ccompnying 36 the SO() theory is genuine d= theory this guging, strting from the full 8/3, inside the 80, etc SO()this SO() E guging within the e8 grding of!! figure. In tht SO() in prticulr It utvisulize / E8 to is instructive! tle, the SO() singlet component of Θ which defines the guging is liner comintion of the two SO(8) singlets ppering in the (shift rnching of the 3875 nd the 750 under the full guge group is infinite-dimensionl symmetries) SO(8). In the e8 grding this guging thus involves numer of hidden nd zero-mode symmetries. More precisely, the guge group ppering in the Lgrngin (.) is of the the theory in the E8 frme looks rther miserle non-semisimple form % & 8 8 in prticulr the guge group is G = SO(8)! (R8 (5.3) R )0 (R ),! 8 8 with the (R8 R )0, nd (R ) corresponding to zero-mode symmetries nd hidden off-shell hidden (on-shell) symmetries from level, respectively. From this perspective it is thus not t ll ovious tht n SO() guge group is relized. Insted, the off-shell guge group involves the go to T-dul frme in which SO() is mong the off-shell symmetries mximl Aelin (36-dimensionl) sulger of the zero-mode e8. 3 This cn e seen s follows. According to (3.) nd (3.) the vector fields couple to genertors s N AB ENS Lyon AM η ΘN ) TA. Since η AB is invrint under L, indices in the rnge A 80 couple to B 800, µ (TB,M

17 0 : constructing SO() supergrvity ffine E8 with L0 grding shift 83 shift 8 shift 8 off-shell 80 hidden 8- hidden 8-

18 0 : constructing SO() supergrvity reking under SL(8) R ffine E8 with L0 grding shift 83 shift 8 shift 8 off-shell 80 hidden 8- hidden digonl grding : !0 L0 qr L 3

19 0 : constructing SO() supergrvity reking under SL(8) R ffine E8 with L0 grding shift 83 shift 8 shift 8 off-shell 80 hidden 8- hidden digonl grding : !0 L0 qr L 3

20 0 : constructing SO() supergrvity ffine E8 with L0 grding : decomposition under SL() 80/ /3 87/ /3 8/3 80 off-shell symmetry SL()! T8 { 8 / / /3 80-8/3 8-/3 8-5/3 8-7/3 T-dul frme : coset sigm model E8 /SO(6) coset sigm model! 8 SL()! T " with WZW term /SO()

21 0 : constructing SO() supergrvity T-dul frme : coset sigm model L0! 8 SL()! T " /SO() with WZW term /3 µ = ρr ρ P Pµ ρ Mil Mjm Mkn µ φijk µ φlmn µν ε εklmnpqrst φklm µ φnpq ν φrst 68 in fct this is the d= theory reduced on torus T... fermionic prt : i /3 I 3 µ J c c i /3 I 3 µ J cd cd i I ν I µ i ρ χ γ γ χ ϕµ ψ ν γ ψµ ρ ρ χ I γ µ Dµ χi ρ χ γ γ χ ϕµ i /3 I µ J c c i c I ν µ J ρ/3 χ I γ 3 γ ν γ µ ψνj c ϕ ρ χ γ ψ ϕ ρ χ γ γ ψ P ρ χ I γ 3 γ µ ψj Pµ µ µ ν µ! " i /3 I µ ν ν µ c c ρ/3 ψ I γ 3 γ µ ψj c ϕ ρ ψ γ γ γ γ ψνj c µ ϕµ 5 3 ρe εµν ψ I Dµ ψνi off-shell symmetry SL()! T8 SO() guging

22 lm]q µ φklm Dµ φklm µ φklm 3g Ap[k, clcultion nd reds µ θpq φ 0 : constructing supergrvity klm ϕc ϕ!c SO() Vklm [c] D,A = 7 δ 5 µφ µ µ Dµ ψνi Dµ ψνi µ ψνi T-dul frme : cd cd, c c J, αβ A I= ψν, ωµ γαβ ψν Qµ c B = c, cdccd I δ c J = χi Q µ,χ Q.(.3) Dµ χi Dµ χi µ!χi ωµ αβb γ " αβ χ µ 8 0 cde cde 8 cd term cd T = /SO() guged coset sigm model SL()! C WZW δ with c, Furthermore, c c cd cd c c,c c C = c, kl kl p[k l]q 3 Fµν µ reds θ Aµ Aν, 70 µ ijk (.) µ Aν g/3 lmn 8 cd cd L = ρr clcultion ρ P nd Pµ ρ Dpq M M M D φ D φ il jm δ kn µ, 7 5 =klcd 8 cd[kl] 8 8 defines the non-elin field in (.) to n Astrength = δofthe vectors Aµ=,, coupling Aµ 0 c c D structure = wedenote this, uxiliry uxiliry field Yk l. In the resulting µνnticiption ofklm npq rst 8 8 cd φ ε εklmnpqrst Dµφc A = φ, ν 6 c c,c c field y the sme sclr potentil 68 letter s the dul E = defined in (.) ove c for the, c c B =nstz for the, unguged theory. The generl couplings is Yukw-type Yuk in 8 cd Lcd 5 (.) E =cd δ c, B generl = δiliner fermion cd, the collection of the most fermion couplings nd Yukw terms couplings cdef cdef cde c c F 0 #cde 8 " cd cd δ = δ δ δ cd cd c δ, δ c, C = I J I µ J ψ νi γ3 ψµj B i ψ 8 e LYuk = e ρ εµν ψ νi ψµj B iρ ψ γ ψµ A γν ψµ A cd F δ ccd δ cd cde cde,c δ c c c cc, c, (.6) c = c c cd c, I µc J = I 3 µ J I J I 3 J iρ χ γ ψµ C iρ ψ γ ψ D χ γ γ ψµ C 3 ψ D ρ ρ ψ 70expressed in8terms of the SO()K irreducile tensors clcultion nd reds J I J cd cd I 3 J = ρ χ Iδγ3 ψ ρ χ I ψj D E E ρ χ χ F ρ,χ γ χ F, (.5) 8 8 ρ/ T, cd cd = δ, 0 c c on thesclr nd, with tensors A, B, C, D, D E, depending uxiliry fieldscto le deter F = / cdef ghij kl kef lgh ij cd / km c c 8 8 = ρ V θ ϕ Y ρ ε T ϕ ϕ ϕ ϕ, c ml k = following. Their ppernce,6 mined A in the in (.5) implies certin symmetry properties E = c c c,c c, / cdef ghi jk jcd kef ghi / c c such sb =, = ρ V [km] θml Yk l ρ ε T ϕ ϕ ϕ, 8 cd cd 5 cd cd B = δ E, ijc ji ij, c]d ji = δ 5/ d[ == ϕ, B() = D () = 0 =B [] =D[], F FJI,ρ c TF = F (.6) JI, 0 cd cd cde cde 8 δ cd c c c 8/, efcdefe[cdef C = cd cd cd]f = F δ δ δ δ c δ, = ρ T ϕ ϕ, nd 8 c c c c cd cd 8,c!c C i, c " c = c/ i i cd cd cde cde ij ij i i i 0 = Fδc = F cc, c, (.6) C F = = C δ 3= = ρ δ.t δ T, (.7) IK IK IK IK 70 8 cd cd $ 5/ # d cd δ,,c D = d[ c]d expressed in terms of the SO() irreducile tensors c = ρ T ϕ T ϕ, (.7) As is stndrd in guged supergrvity, couplings of Kthe type (.5) induce modifiction 3 0 c c, / D = trnsformtion of the fermionic supersymmetry rules (.) y introduction of the ENS so- Lyon ρ T, where we hve defined 8 8 = clled fermion 6 shifts ccording to (kl) c c klm A =

23 0 : constructing SO() supergrvity T-dul frme : guged coset sigm model L! 8 SL()! T " /SO() with WZW term /3 µ = ρr ρ P Pµ ρ Mil Mjm Mkn Dµ φijk Dµ φlmn µν ε εklmnpqrst φklm Dµ φnpq Dν φrst 68 vector fields couple vi LF = εµν Fµν mn Ymn with uxiliry (dul sclr) fields Ymn sclr potentil Vpot = " 5/! ρ (tr T ) tr(t ) 8 ρ/3 T d[ ϕc]d T e ϕce 6 ρ/3 T d[ ϕc]d T e ϕce 8! " T ρ3/ T c T c Yd Yd O(φ3 ) T V V eighth order polynomil in φ Y V TY V ϕ φ V the dilton powers precisely support the correct DW solution (ner horizon of AdS x S8)

24 0 : constructing SO() supergrvity different presenttions guged coset sigm model L! 8 SL()! T " /SO() with WZW term /3 µ = ρr ρ P Pµ ρ Mil Mjm Mkn Dµ φijk Dµ φlmn µν ε εklmnpqrst φklm Dµ φnpq Dν φrst εµν Fµν mn Ymn Vpot (ρ, V, φ, Y) 68 integrte out the vector fields Aµ mn LT = ρr Gij (ρ, V, φ, Y) µ Φi µ Φj εµν Bij (ρ, V, φ, Y) µ Φi ν Φj Vpot (ρ, V, φ, Y) unguged sigm model on different trget spce integrte out the uxiliry sclrs Ymn L = ρr Fµν mn F µν kl Rmn,kl (ρ, V, φ)... V pot (ρ, V, φ) guged sigm model coupled to d= SYM

25 concluding SO() supergrvity 8 Domin wll / QFT correspondence mximlly supersymmetric d= supergrvity with guge group SO() lst missing guged supergrvity round ner-horizon geometries hologrphy for Dp-rnes [H.J. Boonstr, K. Skenderis, P. Townsend, ] wrped D6 IIA AdS8 x S d=8, SO(3) [Slm, Sezgin, 8] D5 IIB AdS7 x S3 d=7, SO() [Smtleen, Weidner, 005] D IIA AdS6 x S d=6, SO(5) [Pernici, Pilch, vn Nieuwenhuizen, 8] D3 IIB AdS5 x S5 d=5, SO(6) [Günydin, Romns, Wrner, 85] D IIA AdS x S6 d=, SO(7) [Hull, 8] F/D IIA/B AdS3 x S7 d=3, SO(8) [de Wit, Nicoli, 8] D0 IIA AdS x S8 d=, SO() mission completed... hologrphy : d= supersymmetric mtrix quntum mechnics...!

26 concluding SO() supergrvity mximlly supersymmetric d= supergrvity with guge group SO() lst missing guged supergrvity round ner-horizon geometries hologrphy : d= supersymmetric mtrix quntum mechnics...! we cme here y wonderful detour vi three (nd mny other) dimensions with Hermnn vi the d=3, SO(8) x SO(8) theory which is still witing its emedding/interprettion in higher dimensions, string theory, hologrphy,... (mtrix string theory, doule field theory,...???) sufficient mteril for the next nniversries... Hppy Birthdy, Hermnn! (in ll dimensions)

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