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 Dprnes [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 Dprnes [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 twodimensionl 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 LiePoisson symmetry E offshell symmetry : E8 cn we guge sugroup SO()??
7 8 : trying to construct d=, SO() supergrvity 000 introducing vector fields? (nonpropgting 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 nonelin dulity reltion implies nontrivil 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 nonelin sclrvector dulity in d=3 surprise no : there is simple Lgrngin description! L = tr [Pµ P µ ] ΘM N AM dan... guged sigmmodel with ChernSimons 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 sigmmodel with ChernSimons coupled vector fields! SO(8) SO(8) distinguished compct guge group d=3 mximl guged supergrvity construction of threedimensionl AdS supergrvities new AdS vcu hologrphy & fluxes construction/systemtics of higherdimensionl gugings tensor hierrchies of nonelin pforms 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 sigmmodel with ChernSimons 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 offshell 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 offshell vector fields the full structure: infdim 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 offshell 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 offshell..., 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 zeromode 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 shiftsymmetries 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 infinitedimensionl symmetries) SO(8). In the e8 grding this guging thus involves numer of hidden nd zeromode symmetries. More precisely, the guge group ppering in the Lgrngin (.) is of the the theory in the E8 frme looks rther miserle nonsemisimple 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 zeromode symmetries nd hidden offshell hidden (onshell) symmetries from level, respectively. From this perspective it is thus not t ll ovious tht n SO() guge group is relized. Insted, the offshell guge group involves the go to Tdul frme in which SO() is mong the offshell symmetries mximl Aelin (36dimensionl) sulger of the zeromode 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 offshell 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 offshell 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 offshell 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 offshell symmetry SL()! T8 { 8 / / /3 808/3 8/3 85/3 87/3 Tdul frme : coset sigm model E8 /SO(6) coset sigm model! 8 SL()! T " with WZW term /SO()
21 0 : constructing SO() supergrvity Tdul 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 offshell 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 Tdul 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 nonelin 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 Yukwtype 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 Tdul 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 nerhorizon geometries hologrphy for Dprnes [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 nerhorizon 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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