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Counting 1/8-BPS dual-giants

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A study of open strings ending on giant gravitons, chains and integrability , Diego H. Correa and Samuel E. Vázquez JHEP03(2007)031 hep032007031.pdf background March 1, 2007 March 1, 2007 March 8, 2007 5 and one non- S 5 November 22, 2006 × 5 AdS Revised: in number which share Accepted: AdS Published: to that of counting the N ns in Received: . These states can be counted by ns up to 5 S http://jhep.sissa.it/archive/papers/jhep032007031 /j in a 3-dimensional harmonic oscillator. We N by considering configurations of arbitrary number of giant Published by Institute of Physics Publishing for SISSA 5 S AdS-CFT Correspondence, D-. We count 1/8-BPS states in type IIB string on [email protected] [email protected] Department of Theoretical. Physics, Homi Bhabha Road, Mumbai, 400005,E-mail: India Perimeter Institute for , Caroline Street North, Waterloo, ON,E-mail: N2L 2Y5, Canada SISSA 2007 c ° gravitons that share at least four . Keywords: zero angular momentum on considering configurations of multiple dual-giant gravito also count 1/8-BPS states with two independent non-zero spi at least four supersymmetries.energy eigenstates We of map a this system counting of problem which carry three independent angular momenta on Counting 1/8-BPS dual-giants Gautam Mandal Nemani V. Suryanarayana Abstract: JHEP03(2007)031 ) 3 4 6 7 8 1 3 3 14 14 10 11 12 12 15 10 can , J 2 5 S , J 1 × J 5 . 5 S and ( AdS is the energy in 5 E AdS ) SYM and in type IIB tes on both sides of the view one can count either simplest case of half-BPS N ) where ( 3 ount the microstates of the U , J t to count the number of BPS ar momenta in 2 fted to 10 dimensions preserve , J = 4 1 ton states [6]. A giant is , J N 2 ,S 1 E,S ) 3 – 1 – , J 2 ) , J 1 1 . These black holes were first found by Gutowski and J 5 , J ( 2 ,S AdS 1 background with applications to black holes in mind. The S ( 5 S (1) R-. A generic state in on are the two independent angular momenta in × 2 U 5 S AdS and 1 S , 5 -BPS states -BPS states 8 1 4 1 AdS In the case of half-BPS states from the string theory point of 4.1 Supersymmetries4.2 of spinning giants 3.2 3.3 Comparison with answers 2.1 The dual-giant2.2 solutions Kappa projections2.3 Necessity of2.4 the conditions (2.11) Reduced phase2.5 space Hamiltonian and2.6 charges Interpretation of the solutions as maximal circles 9 3.1 the multi-giant graviton states or multi-dual-giant gravi are the three independent R-charges corresponding to angul global be specified by giving the quantum numbers ( 5. Conclusions states for fixed charges and supersymmetries in both 1. Introduction In the context of AdS/CFT correspondence [1] it is of interes 4. 1/8-BPS states with Contents 1. Introduction 2. 1/8-BPS dual-giant configurations 3. Counting BPS states with just 2 supersymmetries [5].AdS/CFT correspondence has This only programstates been of achieved that so counting carry far BPS one for sta the Reall [2] (and were further generalized in [3, 4]) and when li most interesting outstanding problemsupersymmetric in black holes this in context is to c string theory on JHEP03(2007)031 + 1 S time) . From = 5 possible N E ) charges. ) charges. 1 3 AdS , J , J 2 2 , J ,S (4) invariance of 1 1 J S the SYM has been SO ) and have ravitons as the inter- 1 ns which have angular a subclass of 1/8-BPS and rotating along one wn to jump from zero limit on the number of onstruct a class of dual- , J principle manifests itself 5 ant graviton can carry is 2 of dual-giant gravitons is de each dual-giant by one swas, Gaiotto, Lahiri and S this space and quantize it. tions and count them. We es with ( ,S for 1/8-BPS states suggests ations of multiple dual-giant tes with ( h is again given by ⊂ 1 limit on the number of dual- ersymmetries with the rest in one can consider an arbitrary S 3 al descriptions of the same set how that this class of solutions e a common set of at least 4 pectrum with an upper limit on S . So for counting the 1/8-BPS er half-BPS dual-giants to obtain 5 S -level equally spaced spectrum. The N and rotating along a maximal circle of with geometry that carry three independent 5 3 5 C S AdS × [7]. A dual-giant graviton is another half-BPS – 2 – ⊂ 5 5 3 S S AdS and have the energy (conjugate to the global 3 J . N continues to be valid. and 2 N J . This subclass of 1/8-BPS states preserve the part as well that share at least 4 supersymmetries. , 3 1 5 J J . We will do the counting of 1/8-BPS states by considering all 5 + 2 AdS J AdS + ⊂ . This time we consider multiple giant graviton configuratio 1 1 J 3 J The rest of this note is organized as follows. In section 2 we c We also consider 1/8-BPS states which carry non-zero ( In [11] Mikhailov described the most general 1/8-BPS giant g Progress in counting BPS states with lower in S [8, 9]. From the point of view of giants the stringy exclusion = + . The same stringy exclusion principle from the point of view (1) R-charges 5 2 In section 5 we conclude with a summary and some remarks. In section 4 we consider the problem of counting 1/8-BPS stat giant solutions which share at leastis 4 the supersymmetries. full We set s oftheir 1/2-BPS class. dual-giants that We share find a theIn given solution section 4 space 3 sup and we a give sympletic the form partition on function of the 1/8-BPS sta momenta in S E in the fact that theN maximum angular momentum that a single gi S U states one may tryhave to been quantize informed these thatMinwalla giant this [12]. graviton has configura Our been resultsthat recently agree both achieved with Mikhailov’s by theirs. giants Bi andof The our states agreement dual-giants like present in du the half-BPS case. that there is an upper limit on the number of dual-giants whic the multiple dual-giant gravitonsupersymmetries. configurations Since which we are preserv interested inthe putting togeth 1/8-BPS states thedual-giants argument given that by there should be an upper D3- configuration that wraps an the perspective of supergravitygiants and has probe to branes do this withunit upper the (see for way the instance 5-form [10,number of RR-flux 6]). half-BPS decreases To giants, insi count treated the as half-BPS bosons, states in an sections of holomorphic complex surfaces in same half-BPS state counting can begravitons, done treated by as counting configur bosons, inthe an dual-giants infinite given equally by spaced s made difficult by theYM fact coupling that to non-zero thestates coupling. number in of In type these this IIB states note string is we theory kno aim on to count of the transverse directions to it within a half-BPS classical D3-brane configurations wrapping an JHEP03(2007)031 (2.6) (2.3) (2.2) (2.7) (2.5) (2.8) (2.4) . 3 dξ ∧ . 2 ) τ , . The DBI and . dξ ( 3 ) θ 3 2 . ∧ 3 ξ σ ˙ ξ 1 ¤ , 9 sin 2 3 2 = = e dξ µ θ dξ 3 2 ∧ ∧ 2 + . A general dual-giant is given by 8 i 5 2 2 cos ¯ e z ˙ 5 ) (2.1) l dα, r dθ, lµ ξ S d 4 urations which preserve a 3 i i ∧ β dβ 2 2 r , ξ = = = l i ξ r µ 7 ) 1 gurations. A given half-BPS dz e e AdS τ − 5 2 8 , φ cos 3 l ( + − s any supersymmetries. Below 2 = ∧ 2 ∈ ): 2 1 2 β ξ σ ˙ t maximal circles. There is no a shares at least 4 supersymmetries lµ = 6 / ¢ ξ 3 β 1 e 2 i 2 1 3 S = = = σ sin ∆ µ ∧ dξ 2 cos h 2 1 3 2 i σ 5 dr, e α + 1 1 e α µ ) , z 4 σ , e σ 2 , e r 2 0 2 ˙ 1 ( + (4) + β σ i ξ 2 cos cos 4 2 / e sin C dφ , ξ α dξ e 1 ) 4 2 1 2 2 1 β, l . The five form RR field strength is − τ αβ ∧ σ , φ ( 2 lµ 1 2 rν lµ V 1 + 3 – 3 – sin /l ξ σ e 2 = 2 cos = = = α + cos r ∧ 2 3 = = dφ 2 4 7 1 r 2 with coordinates . As mentioned in the introduction we put half-BPS + cos , z 1 ∧ 4 e α 5 cos 1 l ( ˙ 3 2 1 S 2 ∧ 2 i ξ C l α, N e π 1 dα dφ × e ¡ 1 − 2 )=1+ 5 , ξ ∧ , θ 2 ) r dt, e ) ) ∧ , e l lµ − . ( τ ) 1 r τ 2 3 0 dθ ( ( ( r ˙ r e α dβ, e = = = ( AdS β r ∧ £ dφ dξ V 2 5 ) = (sin 1 θ, V ) / 1 L 3 l 3 4 z S 1 = = | cos ( − 2 − t/l l rν lµ V ,µ ) ( 2 onto the world-volume is r ds ( = = sin = = = = θ d ,µ V 2 (4) 6 3 9 0 1 (5) ν e e e e µ C τ, r sin , F , β θ ) θ = as a submanifold in ∆= τ ( 0 5 and ( α σ cos S 2 4 l = cos = = r 1 t = α ν = i ¯ z i (4) z C WZ lagrangian density is The pull-back of as configuration contains dual-giants rotatingpriori along guarantee differen that suchwe a will generic look for configuration a preserve classwith of single the dual-giant first solutions one. which 2.1 The dual-giant solutions We regard where dual-giants together to makedual-giant the has lower supersymmetric momentum confi along one of the maximal circles of The vielbeins are where The general embedding of a D3-brane wrapped on the 2. 1/8-BPS dual-giant configurations In this section wegiven 4 will supersymmetries find in the most general dual-giant config The 4-form potential can be written as JHEP03(2007)031 . ab η (2.9) of the (2.12) (2.11) (2.10) (2.13) (2.15) (2.14) ² = 2 234 Γ } b # geometry. Γ ) ) i , i 5 a 6+ S Γ 6+ , { Γ ˙ Γ × β i i 5 µ α brane we need the µ i 2 3 i ˙ 2 ˙ / ξ ˙ 4 ying ξ ξ 1 2 2 3 Γ / . The kappa projection AdS 3 cos 1 ∆ =1 3 1 µ =1 i X i 2 d X σ 3 3 ∆ the mn l i 2 g + l r l + re all supersymmetric. We /l metries. Different dual-giant = 0) but with fixed signs for 6 sin N r 6 N r 9) it is easy to see that setting − Γ τ r Γ e = 2 ˙ = 1 ( 2 ˙ β = = i / β | } where the effective point- = ξ 1 3 α 3 n β ˙ α ξ 3 ξ ∆ | σ , γ h dt L 3 and = m r | γ ². R + cos β 2 + cos { 4 , γ ˙ , P l − ξ 5 N , 3 5 | 2 Γ ˙ α Γ ξ Γ − = 2 , P i ². , α 1 = 2 2 / ( ˙ α ² 2 r ˙ σ | 1 = α ( ˙ µ / 9 l = 1 1 ˙ l 3 3 ξ ∆ Γ L ² – 4 – | + 2 ∆ there are 8 disjoint sets of these supersymmetric cos l + Γ 1 3 2 i · · · , N r r 1 ˙ N r l Γ ξ 0 Γ dσ ) = Γ = = ) 2 r = 0 2 ( r to get Action = 2 ˙ ( σ α 2 ˙ r β ξ dσ i / 2 ˙ r 1 / 1 σ 1 = mnpq V is dσ , γ V γ α 2 ² + , P Γ + Z = ˙ 0 2 r 0 / , P ˙ r mnpq = 1 1 ² )Γ ˙ ˙ = r geometry is ξ )Γ r 2 L 3 ( 2 1 r 1 / 5 )∆ ( 2 σ 1 mn µ r / S 2 γ g ( 1 3 / ∆ N r 1 2 × V V l V 4 5 1 " det N r l 2 = is / − 1 = = τ AdS L √ − γ r 1 ξ 4! P are the 10-dimensional tangent space gamma matrices satisf P a = ∆ Γ = To write down the kappa projection equation for the probe D3- preserve at least 4 common supersymmetries of the backgroun i ˙ matrix is: world-volume gamma matrices, which are ξ The world-volume gamma matrices of (2.2) satisfy where Γ solves them. 2.2 Kappa projections In this subsection wewill will see that show depending that on the the solutions signs in of (2.11) a dual-giant solutions which do not sharesolutions any common corresponding supersym to different values of Lagrangian We can integrate over the angles By looking at the equations of motion of the lagrangian in (2. The conjugate variables of the classical mechanics model ar where ∆ is defined as before. With this the kappa projection on background The chirality convention for JHEP03(2007)031 . , , i i i ˜ γ ˜ γ ˜ γ ying . 59 = 0 57 58 (2.18) (2.20) (2.21) (2.17) , Γ Γ 0 Γ i i 57 = 0 , , α 57 γ γ Γ i i i 0 Γ 1 M² 0 ² 12 ξ 1 γ γ ξ Γ ) (2.22) i sin # 3 4 − Γ 1234 ) l 9 ˜ i r γ − e i Γ Γ 24 Γ i e Γ ˜ γ 59 Γ l 9 − ´ ´ r . 2 3 6+ 68 2 , 2 2 + Γ Γ i 0 Γ φ µ ˜ γ i Γ 3 i Γ Γ 2 γ i − ) iξ ξ − γ 67 0 θ − µ − − 13 = 0 (2.19) 24 M² 0 Γ ´ Γ )+ i 7 Γ ˙ Γ 0 ξ Γ 1 2 68 ´ 68 ≡ α e α e ction equation becomes φ φ (Γ Γ 134 2 ´ Γ 2 3 0 ˜ γ − 2 + =1 + sin Γ Γ ξ M² ² i 9 2 2 X e Γ sin sin 13 1 − Γ 24 24 i l µ r 3 Let us first choose the signs i i 24 Γ e 9 Γ Γ Γ 24 + i = 0 (2.16) 1 Γ 2 ) 2 iξ Γ Γ + 57 6 2 φ φ φ − − 2 ² ( − 24 − Γ . φ 3 Γ φ − − e 1 Γ µ 0 ˜ ˜ 2 − 1 γ γ 57 µ ξ ˙ ´ 2 − ² β e Γ Γ 1 θe θe φ 57 0 69 68 − 5 θ e + ˜ 2 1 Γ α γ Γ + ν Γ θ e Γ Γ 13 1 4 9 8 µ Γ ξ β e 13 13 ν sin 68 β ( Γ 1 Γ M 3 Γ Γ − 3 Γ φ ν ˜ 1 1 γ ξ 2 e 2 + sin − 1 2 2 i i ξ φ φ 2 sin + sin = sin µ 3 ν ( + cos 1 e + sin − e 1 − 1 ² µ − − ν Γ α 5 e 1 234 α + 12 Γ 1 i ) ˜ 68 γ Γ Γ Γ θ e 13 Γ Γ 7 9 Γ θ θ e µ 5 ˜ γ Γ θ 2 Γ ν 13 α Γ cos 13 1 234 cos ξ 1 2 3 4 ( ˙ Γ i Γ 69 φ 1 sin are: 2 1 i ν Γ 1 l e iξ 1 cos cos – 5 – 3 )+ − Γ µ e φ l r φ 5 ν − γ − ³ ³ − − i i 2 + − 58 − 0 S 7 57 ν + 2 θ e 0 0 8 1 l l Γ 1 Γ − β e r r Γ − (5) t Γ Γ ν θ e 1 × Γ θ e i i l ˜ γ 8 ξ i 0 2 +Γ F 234 9 24 68 5 ) 2 cos 1 Γ sin e Γ Γ Γ − − r Γ (Γ e ³ cos 3 2 2 2 cos 67 ( γ 0 4 3 i / ξ l φ ˜ γ 2 γ ˙ ³ α iξ ³ r r 68 1 1 . The full kappa projection equation then reads 6 / 0 Γ Γ − − (Γ AdS i 1920 − γ Γ 1 γ Γ Γ γ γ e V 0 ˜ e γ ) Γ 0 l 2 t 0 0 ˜ γ β h l cos Γ ξ V r i − Γ θ e + Γ Γ 9 i 2 68 l t l ( t i 57 l l t t − − 1 Γ 0 i 56789 i ² Γ e i i Γ e 3 e + 2 − in the solutions. Some useful intermediate steps in simplif µ − 1 − − ˜ ξ γ 2 − − cos ˜ iξ ξ γ e e )Γ / e e 5 9 9 − r D ³ − − 2 1 0 0 =Γ l Γ 234 Γ sinh / Γ r r 57 γ 13 24 3 1 i V ( α 2 Γ 0 i γ Γ Γ Γ β β e β e i iξ h 2 2 Γ 1 e 2 1 2 V / ξ l t / − e − φ φ h 1 i × 1 e − M − − 0 cos cos cos − e V and ˜ 3 e e ∆ Γ e M − h = 2 µ i 2 2 2 2 α α α µ / / / / 2 ² 1 1 1 1 = = − 1 µ 01234 cos cos V V V cos V 0 + iµ h h h h h h h M )Γ is the value of − =Γ r M M M M M M M 234 0 0 ( r 2 Γ γ Γ ======/ h 1 ’s to be the same and positive. On the solution the kappa proje i V M M M M M M M M ξ " 9 8 3 4 7 0 2 Γ Γ Γ Γ Γ Γ Γ Let us also register the following identity: We now show that the solutionsof in (2.11) are supersymmetric. so that we have Using these identities eq. (2.19) can be rewritten as where The Killing spinor equations of this equation are: The solution of these equations can be written as: where JHEP03(2007)031 5 S ’s to × i result (2.26) (2.24) (2.23) (2.27) (2.28) 5 λ i ˙ ξ 24 the dual- Γ AdS 2 φ 1 2 s for . the projection e , . . r 13 = 0 Γ = 0 1 = 0 0 0 = 0 φ ² 0 ² 0 and 2 1 0 ) ² ² ² e ) 9 e solutions are the full ) β ) ˜ γ Γ 9 t these are all 9 12 , 3 Γ we will assume all being positive preserve at 69 α λ θ i Γ 1 (2.25) me the solutions (2.11) to ˙ +Γ 2 1 +Γ n by redefining the tangent ξ supersymmetric subspace of i Killing spinor on tions on s are (as the second and the e 0 ± + gns (2.25)) follow uniquely if 0 f e fifth) γ ants in their respective planes. 0 − (2.24) for any arbitrary values ymmetries. ersymmetries. (Γ = 0 (Γ .24) corresponding to reversing 8 forward) s for half-BPS dual-giants which i (Γ ) Γ (Γ 3 , , ξ , + , λ 2 = 0 ξ ) = 0 . , 3 + = 0 0 0 to zero and finding the kappa projections 1 ² ² ξ 0 i , λ ) ) ˙ ² ξ 2 + = 0 = 0 ˜ γ ) ˜ γ l t ˜ ( γ 0 0 , λ 68 68 i 2 ² ² 1 68 Γ ) ) Γ e i λ i ˜ ˜ – 6 – γ γ Γ γ 2 1 − − 57 59 0 0 Γ Γ ) Γ i λ ) = ( l i i r 3 (Γ (Γ ( ˙ − 1 ξ + − − 0 , l 9 7 , , 2 ˙ (Γ . ξ sinh (Γ (Γ 0 i 2 = 0 = 0 , l , e 1 0 ˙ 0 ˆ ξ ² × M² ² , , l ) ) ( = 0 ˜ γ ˜ = γ ≡ 0 57 = 0 = 0 ² ² 0 57 ) ² Γ Γ 0 0 i ˜ γ i ˜ γ ² ² 6 ) ) 57 − planes respectively. One can further see that different sign Γ − ˜ γ 58 Γ 0 3 β 0 57 i 1 z 2 (Γ Γ , e (Γ i +Γ 2 i λ . However one should note that for fixed values of ˜ γ z r 5 − 67 , − Γ 1 0 0 α defined by (2.17),(2.26). This will therefore show that thos z (Γ i 2 (Γ and ² (Γ e β In the next subsection we will prove the converse, namely tha To begin, we rewrite the supersymmetry projections (2.24) ( , α least 4 supersymmetries consistentof with the projections in equation (2.22) can givespace 16 Γ-matrices supersymmetries. appropriately. This can be see we have in similar equations withthe different direction relative of motion signs of inThat some eqs. of is, the (2 for three the half-BPS solution dual-gi with Thus we have seen that the solutions in (2.11) with all signs o By considering the special cases of setting two of one recognizes these three projection conditions as the one rotate in to It is easy tofourth see projections that are out equivalent of and these so the are independent the projection third and th We can use these to simplify the full expression (2.17) of the be +1, the generalization to arbitrary signs being straight giant solutions which preserve at least these four2.3 given Necessity sup of the conditionsIn (2.11) this subsection,begin unlike with. the We previous willwe rather one, demand show we the that these kappa will projections solutionsspinors not (with (2.14),(2.13) assu si on the 1/8-th This equation can be solved by imposing the following projec set of solutions consistent with the specified set of 4 supers JHEP03(2007)031 sis ] 9 which (2.31) (2.32) Γ 0 3 ˙ )] (2.30) ² ξ and then 2 1. Finally 2 3 = 1. That / 0 . Using the 1 µ ² ) ± 0 ω 3 ˙ ∆ ξ + 5 = , − ˜ γ − l r ω )Γ , 1 ( ˙ 68 ˙ ] = 0 = 0 into Γ β ξ 2 5 Γ ( / 9 2 3 β β 1 , Γ 2 2 ] ˙ µ ξ 2 µ β 2 2 2 sin 1 Γ γV r 2 01 sin 24 iµ l α and Γ N is arbitrary only upto the ˜ γµ Γ ld-volume of the D3-brane Γ lete. ˜ γ i α 2 − − 0 to zero requires rators which kill 59 φ cos ² + ed ˜ γ 3 l that a linearly independent 68 0 mbination of various products does not occur anywhere else Γ − ξ 2 α i owing in terms of the canonical 57 / to zero. Doing this we see that 1 ˙ ] (2.29) ] ] 1 ,Γ r θ e Γ + sin ˜ ˜ ˜ γ γ γ ˜ γ 1 V ··· ˙ ˜ = 0. Then the first three terms in γ )+ ξ 57 ˙ 59 58 57 β , P , P + 2 2 1 + sin 56 Γ Γ Γ 0 mn σ β ˙ Γ and Γ + sin α 1 1 1 ] ’s. Since iµ 1 )Γ β ˜ = 0 = 0 γ 1 β = ··· 2 Γ iµ iµ iµ − cos / 2 2 α 1 56 1 ) 13 µ cos − − − σ mn 3 α Γ Γ ˙ ∆ 2 ξ 1 ˜ ˜ ˜ γ γ γ [(cos l α β r r φ would not occur anywhere else and so we have 2 − cos – 7 – ˆ 69 68 67 l − N − M 2 sin sin Γ Γ Γ ˙ α ξ 69 V θ e − i 2 2 2 i ( = [( 2 ˜ γ ξ − = 0 and sin + iµ iµ iµ ˆ ˆ M 6 M 5 [cos 69 τ − ] − − − γ Γ Γ and Γ appears with Γ i Γ = 0 ˙ 9 8 7 α i Γ α β r Γ Γ Γ 6+ l t 59 ˆ , P , P β has to be set to zero too and this implies i M − 3 3 3 (which does not appear anywhere else), which then implies Γ ] relations from the earlier section. Since we want to demand − i ) µ µ µ γ ˙ , Γ 6 [ [ [ [cos [cos ξ 1 ˙ = 0 = 0 234 ξ i (sin 01 58 M ˆ ˆ ˆ ˆ ˆ ˆ r 2 1 µ Γ M M M M e M M ˜ γ . Note that these are not absolute values - unlike in the analy − 2 µ P / 234 2 3 56 2 1 ======1 ˙ ξ i X r ω/l Γ ( ∆ 2 i ˆ ˆ ˆ ˆ ˆ ˆ i l N 2 32 matrices is given by Γ -matrices for the tangent space acting on a constant spinor + M M M M M M ≡ µ γ − 6 9 8 7 1 6 5 + 3 1 × − and Γ 1 ˙ ˙ Γ Γ Γ Γ Γ Γ Γ ξ β 1 Γ ˙ ξ β ˜ 2 γµ α = M P / ˙ α 2 1 0 r 2 58 ˙ ξ V Γ cos i = + 6 − 0 1 + cos ˙ ξ Γ 5 2 +Γ / Γ 1 α = 0. Next notice that Γ = 0 as the coefficient of Γ [ ˙ = 0. The coefficient of Γ V [ ˙ ˙ ˙ using the projection equations (2.27) we can convert Γ r relations obtained so far and then setting the coefficient of Γ of equations of motion. Since ˙ α to set put the coefficients ofβ the remaining independent Γ completes the proof that the set of2.4 solutions we Reduced found phase is space comp The supersymmetry constraints (2.27) translatemomenta to (see the (2.10)) foll is arbitrary but forbasis the of these projection 32 equations (2.27). Recal projections in eq. (2.27) we have to use them to eliminate ope eq. (2.18) can be simplified to Thus the full kappa projectionof equation 10-dimensional (2.18) is a linear co along with Γ the projections in (2.27)and to so be for valid convenience at we can every set point of the wor To simplify the kappa projection equations (2.18) we will ne The last five terms in eq. (2.18) can be simplified to JHEP03(2007)031 are i ξ (2.34) (2.36) (2.39) (2.33) (2.37) (2.38) P ure on the , ) 3 4 ,µ 2 5 l 3 (2.35) Nr ,µ , 1 1 2 − P B µ , 2 refers to one of the six = 0, becomes ( } l 2 Nr ij r j a r δ 2 subspace is a straightfor- ij   f = 1 P δ ,q a six-dimensional one. We ntirely by the coordinates 2 2 b j l N ckets: = 2 )= r f i = l { β Nr 2 and nt [14] gravitons. To proceed − β N l 1 P B 2 P 3 − } − = ab cos , i 1 i ,ξ = = 2 2 i ξ α M 2 , r µ α P DB 1 = 0 ,ξ P f ª i P B 1 2 cos i ¯ j P B { ζ 3 =1 r ¯ ¾ ζ } i d X 1 2 j , β, 2 a i l r N ∧ 2 ζ v u u t l ,f 2 i Nr i © l sin N −   2 q dζ i α r,α,β,ξ − 2 ξ 2 , l ⇒ 2 r P i – 8 – l − { N P B are given by r = i cos } + P ≡ 4 ab P B 2 ij i P B = α, ½ } δ ,f } M b   1 j ω 3+ 2 = ξ 4 l N ,f ,q (sin 4 a i l r and their conjugate momenta, the constraints become q f = P B − { Nr { { i } ≡ j = + , f ,ξ ) = = DB i The conclusion of this subsection is that the reduced phase 3 i 3+ } 2 2 i r are of the D3-brane action the momenta ξ 2 j µ ab , r DB = 0 P i ,f 2 i } DB ξ , r i M 1 f i } r 3 , r j ξ { =1 P i 1 X , r { r 1 ,q i ( ξ ≡ v u u t q { with the symplectic form i {   f 3 or by the complex coordinates (2.34). The symplectic struct C v u u u t , l i 1 3 iξ ,ξ e = i 2 r refers to one of the six coordinates ,ξ H 1 i = q i ζ The calculation of the Dirac brackets on the supersymmetric Thus, the original 12-dimensional is reduced to In terms of the coordinates constraints (2.32). ward generalization of thewith half-BPS the giant calculations, [13] let and us dual-gia define the complex coordinates conserved charges. The canonical hamiltonian, with The Dirac brackets are summarized by where Thus, for example, Symplectic structure: space is simply will see below thatr,α,β,ξ the reduced phase space can be described e reduced phase space can be derived by the following Dirac bra 2.5 Hamiltonian and charges Since translations along The non-zero elements of the matrix JHEP03(2007)031 (2.43) (2.42) (2.41) (second with an 5 -plane by 5 1 S z S are complex 0) invariant. , i 0 ) in the full set z 0 l, . Thus we have a r . Any such circle 2 0). We can choose (3) rotation. l πl , U 0 = V. l, = 2 2 ntersecting ] where |    t 3 π ) (2.40) tisfy the equations of mo- = ( β β z . This space has five real 2 | 3 ). It is well-known that the , ξ ~z tion) can be identified with 0 rmonic oscillator. (3) (2) nts which preserve at least 4 P + sin (0 cos U U 0 it reduces to , r ther by a 3 or 2 f a given size 2 | + iξ ∈ iξ 2 l space of solutions, viewed as a (0) 2 k e z ξ | − ) P , ξ , θ l t + , -motion of 1/8-BPS dual-giants (see β β e + + (0) k (2)) on a reference circle, for example, 2 (3)-rotation (2.43), the motion on the 5 can be parametrized by 8 parameters. 0) (0) | α 1 µ , 1 U S (0) ξ ~z U k 5 is embedded. The space of these planes sin ξ cos z 0 P | ( iθ S i ( cos 5 α l, α e e l ( 1 1 which are related to the one in S by iθ iξ 5 sin sin (0) k e = , 5 2 S 3 0 i Ue – 9 – − S r lµ iξ )) = ¯ iξ ζ i θ ≡ ζ ( 3 3 ) β e β e l ) = ) = N θ in which t t , z ( ( ( ) α ~z 6 sin cos = r k θ ( z R α α 2 2 . Hence after the sin which is 8. Let us consider a subspace of maximal circles 3 1 l , z t/l ) Nr cos cos iξ . The time-dependence of a half-BPS dual-giant on the circle θ e (2) 2 i 3 ( = z = iξ iξ 1 e SO e θ z (6) × H    SO (3) rotation (defined upto a . We will show in the next subsection that the motion in (4) to be 3 = U ) = ( C SO θ U U in which we embed the ( ~z 3 C (2) is an arbitrary matrix that leaves the column vector ( U ∈ V = 1, 2 and 3. The motion is obviously periodic with period ∆ As expected, the BPS constraint equations automatically sa passing through the origin in k (3) rotations. 2 has a dimension of R To see this note that every maximal circle can be obtained by i 2.6 Interpretation of the solutionsLet as us maximal now circles try to understand the full solution set of dual-gia U common supersymmetries. A maximal circle on line of (2.41)) are all in maximal circles in space of solutions ofits a phase dynamical space. system Folowingsymplectic (modulo section space, time-evolu 2.4, is this six-dimensiona 6-dimensional space of solutions parametrized by: ( Here where we can take Therefore the space of these circles can be identified with of 1/8-BPS dual-giant solutions is simply related to every o can be obtained by a which is simply the Hamiltonian of a 3-dimensional simple ha tion. The solutions to (2.32) and (2.40) are second line of (2.41)). We conclude that every dual-giant (o (2.42) is given by putting After imposing the remaining three constraints from (2.32) coordinates in for dimensions and gives the parameterization (2.1) of the vect parametrized as generic maximal circle coincides with the generic the representation (2.1) for JHEP03(2007)031 j ¯ ζ ) ) by (3.1) (3.2) (3.4) (3.3) 3 N ξ √ , P l/ 2 ξ apply. This , ( bosons with 1 i , P a − 1 (6). In what ξ ) N 3 P → n 3 SO q i 2 ζ harmonic oscillator neralizing from the n 2 erefore, ) this reduced Hilbert q 1 ) of N n 1 3 menta ( √ ro point energy which is ζq , J of the configurations with l/ 2 scillator is exact, quantum bosons in a 3-dimensional Since each dual-giant being iants with exactly the same − PS objects (with respect to ual energies (like in the case ts: particle phase space is sim- , J e dual-giant graviton system, ns with dual-giant gravitons. N 1 . (1 J i 3 3-dimensional simple harmonic 3 0 =0 , | , ∞ 3 Y 2 i 2 ! n , n i , ) n † i =0 ∞ . The total angular momenta should 2 a Y √ = 1 ( n = 1 N 3 . i 3 , i =0 2 n , ξ ∞ 1 Y ) n P Y 3 are quantized as ( + =1 i , i for j 2 i ,J ¯ ζ ]= n n = 2 i – 10 – , ) i J + k = i ζ i ,J ( i 3 1 1 β i J J n J ,n − ( 2 =1 N ≡ = in stead of k X i ,n i µN J 1 = ξ − n lH i P | J exp[ T r ≡ = 0 state we may simply count all states with a total of ) acts as a domain wall and therefore considerations of [10, 6] 3 i J 5 ,q 2 ,q AdS 1 dual-giants. The grand canonical partition function is, th N ζ,q ( , with the symplectic form (2.38) and a 3-dimensional simple Z inside -BPS states 3 . The operator ordering of the charges has been obtained by ge 1 3 8 † j As we have argued earlier, putting any number of in Since the semiclassical quantization of a simple harmonic o C a S construction correspond to the three angular momenta ( 3.1 restricts the maximum number of dual-giants to follows we will use the notation space is consistent witheach 1/8 other) supersymmetry. the Since total they energyof are is half-BPS given B states). by Further the wequantum sum can number of and have the so more they individ than can onean be dual-g treated as bosonic objects. mechanically the single-particle Hilbertoscillator space eigenstates, is viz. given by Hamiltonian, (2.40). also be given by the sum of those for the individual dual-gian half-BPS case [15]; innot the important Hamiltonian for our we have purposes. dropped We the note ze that the conserved mo → The hamiltonian and the charges are given by We found in section 2.4 that classically the reduced single- 3. Counting BPS states with ply Here the classical phase space variables From (3.2) it is easyconsistent to with see 1/8 that the supersymmetries,simple partition harmonic is function oscillator. given for th byBy Here including that we the can for identify these boso some of them sittingless at than the zero energy state. This takes care JHEP03(2007)031 3 µ ), ), 2 2 − ,n e 2 (3.6) (3.7) (3.8) , J ,n 1 1 = should ,n J n l-giants 1 ζ ( α n N = 0 plane 3 µ ). Counting the identical bosonic 2 N , J s). In terms of the useful way to think . 1 . r of ways, Ω J 3 2 e. The conserved mo- N d to have at least two l J 3 2 q 2 and the ‘fugacity’ r-dimensional phase space J 2 ich eight supersymmetries = ne to be the i q ) (3.5) α. 1 J 3 metries to either 1/4 or half- J 1 2 ns for which all bosons lie on DB arately in the next subsection. q configurations of points on this ,q pecified by ) ª 2 . The specific values of 3 cos ttice point coordinates α dual-giants have the same specific ,q α 2 2 1 , J 2 q 2 N l ( cos Nr , J N 2 1 (and therefore fixes a plane in the 3-d Z = J , r ( N 2 2 β ξ ζ N ξ . The other constraint is that one can have 2 © Ω J =0 ∞ X and N =0 , 3 – 11 – points fall on a single 2-plane going through the α, P 2 α ,J N and l ∞ 2 2 2 )= X 1 N ,J 3 J 1 sin = J ,q 2 , expressed by the following Dirac brackets 2 = 0. Classically, 2 2 points fall on a single line going through the origin. The are conjugate to the charges l ,q DB 3 C )= Nr 1 i ξ ª 3 β N P α − = ,q identical bosons on a 2-dimensional lattice of integers ( ζ,q 2 e 2 ( 1 ξ dual-giant gravitons. N of dual-giants: ,q ≡ Z sin P 1 i 2 that configuration preserves only 8 supersymmetries. q N q N ( , r α N 1 they preserve 16 supersymmetries. But if we put together dua ξ Z © α ) are identified as the quantized angular momenta ( 2 ξ , P 1 preserve 1/4 of the supersymmetries (eight supersymmetrie ξ P 2), which in turn fixes β -BPS states π/ 1 4 Quantization proceeds in a manner similar to the 1/8-BPS cas So the 1/4-BPS states can be counted by considering the numbe = and β in which one can distribute menta ( 1/4-BPS states goes asdual-giants in follows. that configuration To with make different a values of 1/4-BPS state we nee with various values of a total of no more than be such that we get integer values for which take integer values. The half-BPS3-dimensional states lattice again are where the all 1/4-BPS states are those for which all origin again. We will discuss counting of 1/4-BPS3.2 states sep ( with Some special configurations however haveBPS enhanced supersym states and soabout we this counting may is choose the following. to A subtract generic them state can out. be s For this a is conjugate to number particles sitting on the 3-dimensional lattice with each la Repeating the same steps aswith in the section 2.4, symplectic we structure can of now find a fou As stated above, dual-giant configurations for whichα all bosons on a 3-dimensionalthe lattice same these 2-plane are passing thewe through configuratio want the to origin. preserve The fixes choice the of values wh of lattice). For the purpose of counting, we may choose this pla The chemical potentials As before, for fixed JHEP03(2007)031 . A 2 (4.1) (3.9) (4.2) J ’s is 2 ) (4.3) , n 2 1 ˙ ) ξ τ ( α . Then the D3- 2 3 2 1 ˙ σ φ ξ ] representation of N = 4 Yang-Mills the- 2 ζ = = ) σ + sin 2 3 2 N 2 p,q,p ,q cos ier sections and consider α 1 ( ˙ ely that of configurations q 1 2 ( ) corresponds to the points l 3.9) agree with their result. ntribution of configurations σ he [ , φ ered in the literature before N i /4 of the supersymetries. To ) volve the fermionic fields. 1 ) gauge group have been con- − s we constructed above are to r, s ˙ Z mmetries one needs to exclude τ ) defined in (3.9). ) charges and preserve at least ξ sin ( ) , ξ assing through the origin. Each 2 1 N 1 2 2 2 α h counts 1/8-, 1/4- and 1/2-BPS ( an be uniquely specified by a pair =0 ˙ α ∞ σ φ φ ,q , J X 4 N 1 2 θ q cos SU = = 2 ( = ,S l cos 2 1 N 1 1 4 S Z − − l ) 2 + sin 2 − / and the sum of the individual n 2 2 1 1 ). This line ( ˙ , ξ q 1 φ = , φ ) 1 ∆ J ) θ n h τ 1 r, s 3 τ ( 2 σ ( – 12 – 1 α 2 ζq ξ θ σ 3 ) ’s is 1 1 − ) = ( 1 l = = = 4 SYM with (4) 2 τσ n cos + cos ,J 1 (1 C ). More generally, an index for 2 2 ,n N ˙ θ 1 N =0 ( 2 n ,S . So to eliminate the contribution from the states with q,p − 2 ∞ 1 ,n Y r , θ 1 + ∞ S = ) n ( , ξ − p τ 1 ( L ) σ r r 2 )= ,..., ( ˙ r 2 = = ) = ( V 2 ,q 1 = 1 − , J 1 ) ζ,q J r , k ( ( ) , β Z V ) particles lie on the same line from τ r, s ( ( N α τ, ,r k ∆= = = t α ) = 2 (4) which has ( ,n 1 n 3.3 Comparison with gauge theory answers Single trace 1/4-BPS operators of brane Lagrangian becomes structed in the literature [16 – 18]. These states belong to t in which all enhanced supersymmetries one should simply subtract the co The pull-back of the RR 4-form is The above counting includes statesexclude the which special preserve states at which leastconfigurations preserve in 1 1/2 which of all the particlesline supersy lie passing through on the a origin straight inof line a relatively p 2-dimensional prime lattice integers, c say ( partition function which generates this number is SU 4 supersymmetries. Similarin configurations [21, have 22]. been Weconfigurations consid work of in D3-branes the of the coordinates type: that we used in the earl where such that the sum of their individual ( In this section weof multiple consider giant a gravitons which different carry counting non-zero problem, ( nam ories has recently been calculatedstates [19] of (see the also kind [20])In we whic case discussed of above. the Ourbe 1/8-BPS results identified states, (3.4) with the and gauge-invariant dual-giant operators ( graviton which state do not in 4. 1/8-BPS states with JHEP03(2007)031 (4.6) (4.7) (4.4) (4.9) (4.8) (4.10) , . . So the l and their i = 0 = 0 . Note that ¯ ζ 3 |≤ , α α i 3 C ζ ζ 2 θ. | 1 2 i ξ cos sin α e N 1 ξ ) − 2 P cos | α ) e is that 1 3 l ξ 2 ζ it should be. ution with positive sign of he Hamiltonian reads (for | .4) share at least 4 super- /l = cos /l on for this is we have chosen 2 2 e previous case and we get he equations of motion l 3 , P N , P pace. The problem of finding r fine se space of the onal simple harmonic oscillator θ, l aints = 1 = , ζ = ( = 0 = 0 2 1 1 ˙ ) (4.5) ξ sin ξ 2 iφ θ θ 1 φ ξ ij 2 α = δ θ e P i | , P ¯ 2 ζ 2 θ, P sin 2 l + i N ˙ | cos sin φ 2 ζ 2 2 i | α ζ φ | α 2 − 3 . In terms of the canonical variables, the =1 = P cos i θ, r X 2 | α l = N 1 1 cos + cos 2 ξ ˙ l N φ 1 cos P | r = – 13 – φ N ) DB and 2 = α 2 P 2 ª 2 , = φ ( l r j θ /l l 2 ¯ 1 ζ 2 , cos , lH − r = 0 i r , P , ζ = ζ 2 r ˙ θ 2 1 φ © | = ( 1 iφ for quantization will be discussed shortly). H = ζ 1 | | φ r , P , P 3 ) = ( θ e 2 3 ζ l with the symplectic form (2.38) inherited from | N = ˙ , r cos = 0 = 0 α, P ˙ 2 ) D = α 2 2 r θ α 1 ˙ , r × φ P φ 1 2 l 2 r cos P ( , cos C 1 cos ˙ r N φ α l = = 2 1 1 ξ ζ cos P N 2 2 l r − 1 In terms of the new variables, (4.6) reads The main difference of this solution space from the earlier on φ . It is easy to see that the following configurations satisfy t P 1 ˙ the boundary is a null curve of the symplectic form which is as phase space is really conjugate momenta gets reduced tothe a Dirac six-dimensional brackets phase in s the reduced phase space is similar to th We will show insymmetries the for next subsection arbitrary that values the of configurations (4 As for the case of the dual-giant gravitons earlier, we can de In fact these configurationspositive saturate values a of Bogomolny bound and t where supersymmetry conditions (4.4) imply the following constr and the Hamiltonian written in these coordinates becomes which gives the same symplectic form, eq. (2.38), as before. Notice the change of sign forhere the an Chern-Simons anti-D3-brane term. rather than The a reas D3-brane so as to find a sol ξ Eqs. (4.10), (4.8) imply that(the the implication system is of again bounded a 3-dimensi Because of the six constraints (4.7) the 12-dimensional pha and JHEP03(2007)031 , ) 7 (4.14) (4.16) (4.15) (4.13) (4.11) Γ 1 ˙ . ξ γ ltonian) is α ) = 0 24 . . 0 Γ ² is embedded as ] θ + sin 2) rotations act- = 0 = 0 5 ˜ γ , 2 = 0 (4.12) 5 ² 0 Γ (1 ) 57 ² ² . ¸ θ ¸ ) Γ 9 α U AdS α ( ˙ i ˜ γ Γ + sin y show that the full set l 689 sin 3 ˙ − 57 ξ Γ 13 cos 4 Γ 3 2 Γ d by 24 1. )+ Γ i / ˜ γ Γ θ θ 1 2 689 lµ 2 ˙ 2 57 − emiclassical quantization cor- φ um mechanics is described by Γ φ ∆ lutions we have found share at |≤ i Γ 0 giant gravitons with generically sin = i dered above. − 3 (in which + this reduces to ζ 3 4 13 − | (cos (Γ − 2 θ σ , Γ Γ 2 2 i M ) 1 α /l 1 l r 2 , 7 φ ˙ i C − φ cos ) Γ , γ − sin 8 γ = 1 1 3 − = ˙ = 0 + ξ Γ 7 1 ) Γ γ e ˙ 01 0 θ ) of ξ 2 α γ 1 ) ˙ ² 2 M Γ ξ ˙ ) φ ) i 14 = 2 Φ 02 ( cos γ , )+Γ 2 Γ r + 1 689 ˙ 3 lµ θ 24 i φ + sin 3 Γ +Γ Φ Γ + Γ – 14 – , 5 = + i 1 = α 2 0 sin (Γ 23 ˙ Γ 2 4 φ γ Γ 1 4 σ − ( ˙ ˙ 0 α φ (Γ θ cos (Γ ( ˙ r 0 represents the Disc Γ θ r l γ t l i i 0 +Γ , γ + (Γ + D 6 + Γ − − 2 θ l sin t = 0, Γ 1 13 , i 7 Γ θ Γ Γ ˙ − α α Γ θ 1 cos 2 24 r φ = 0 α ) on the original half-BPS giant graviton. / 3 Γ ˙ cos 2 where 1 − r cos 0 l 1 + 2 ² 2 (Γ φ = 0, ˙ V l l − 1 r D θ e ) we may impose the projections ) l (sin ˙ r − θ Γ i γ = + = × 2 + θ e − 0 1 13 cos / 2 2 σ ˙ | 0 1 Γ r Γ 2 C γ 2 r, θ, α i = 0, Γ 2 / sin V / 1 Φ 2 r | 1 2 − / V / + 1 V 0 1 l + r 0 · V 2 V Γ (Γ | · + part of the phase space (with the symplectic form and the Hami l r 1 2 0 / Φ + 1 | Γ D V + V [ 2 = | The Using methods similar to the ones used in section (2.3) one ma Further one can interpret the solutions as all those obtaine 0 M τ γ Φ identical to that ofresponds the to half-BPS a giant gravitons fuzzy [15]. disc and The the s exact single-particle quant Then the killing spinor equation can be rewritten as with which the killing spinorleast equation 4 is supersymmetries. satisfied. So the so of solutions given the projections in (4.16) is the one consi 4.2 Quantization The phase space is Since we are interested in havingdifferent configurations values of of multiple ( ing on the homomorphic coordinates (Φ 4.1 Supersymmetries of spinning giants The world-volume gamma matrices are −| On the solutions ˙ To simplify this equation we use the following identity The kappa projection equations for an anti-D3-brane are JHEP03(2007)031 ) 1 . 0 um , J 2 (4.19) (4.18) (4.17) . Two ×H ,S 1 ne wishes S ,N 1 1 H J 3 q = 2 S 1 2 d to preserve 16 levels of a simple q - system in a H 1 N S are the two indepen- 1 N e charges ( q i correspond to angular ) st ingle-particle quantum S 1 1 is their energy conjugate J , J E tates in the type IIB string 2 ies. For example whenever ) one has configurations in 3 ates from (4.19) to get the s of an } and n be obtained by an isometry ,S where ∞} , J 1 5 giants in a given state have the 1 2 S where J ( ns is given by are the three independent angular , J 3 N 1 + i ,..., J J Ω J 2 ,...,N 1 + , 2 S 2 , =0 J 1 + = 0 1 ,J + ∞ 2 = 1 1 S X ,S J 1 = ,n S i = ,l,m . i n E 5 = – 15 – E {| n 3 l,m q AdS {| m 2 ) with q ⊂ ) with -particle states of dual-giant gravitons rotating along 1 1 l 1 3 3 = Span q N , J that share at-least four supersymmetries. The result can S is given by the states of a 2-dimensional simple harmonic , J 2 − 2 ,N 5 2 1 = Span 1 ,S S 0 , J C 1 H 1 S H =1 J N Y n = 0 state from the spectrum of giant gravitons since the minim =0 background that preserve at least 4 (1/8-BPS) ∞ n , Y 5 5 l,m S S . that state is expected to preserve 8 supercharges. Again if o × 5 )= 3 5 r S ,q 2 AdS ,q acting on the standard half-BPS giant graviton it is expecte 1 q 5 ( which is an N-dimensional Hilbert space consisting of the fir those with non-zero ( dent angular momenta on momenta on to the time coordinate in global coordinates and those with non-zero ( Z The three angular momenta (4.6) are identified now as the thre • • AdS ,N 1 where the first two correspond to angular momenta in AdS momentum in The full single-particle Hilbert space of the giant gravito oscillator of arbitrarily high quantum numbers: We have omitted the angular momentum of amechanics corresponding half-BPS to giant graviton is unity. The s this partition function whichthere have is more just than one 4 giantof graviton supersymmetr making up asupercharges state, since [22] it (see ca also [21]).same Further value whenever of all the Similar to the situation in the 1/8-BPS states with ( be expressed quite simply in terms of the degeneracy of state one can systematicallydegeneracies exclude of the exactly 1/8 contribution or of 1/4 supersymmetric these states. st 5. Conclusions In this paper we consideredtheory the on counting problem of quantum s three-dimensional harmonic oscillator potential. H types of states have been considered: For the first set we considered arbitrary maximal circles of harmonic oscillator [15]. JHEP03(2007)031 , (0) 1 ζ (5.1) = evolving . 1 2 ) quantum l Z N 3 are arbitrary = , J . Remarkably, 2 2 5 | and i directions which (0) 1 S , J e very interesting Z 5 1 | (0) 0 J , ζ ϕ =1 3 i describe a dual-giant (0) 0 ions containing an ar- fication of giant gravi- AdS alf-BPS states there is denotes the number of ϕ resting to see if such a = P i illator, but this time with -particle energy levels for 0 r graviton configurations by = 0 (5.2) ates. This points to some and giants’ [26] 3 n the and oo. It is possible that such a the system [23, 24] in terms ounded above by giant graviton states. If we tes with ( (counting from the dual-giant ons and with one of the three Z 3 2 g problem for this set of states given time, is a three-sphere of ) are holomorphic coordinates ton descriptions. e the giant gravitons wrap dif- l BPS sector the duality between ting problem in terms of giant -giant gravitons, however, have d in [25] to capture the physics 2 C th = − Φ 2 , ) where 1 = N Φ 2 | , , Z 2 0 , r Φ (0) 1 | k and a multi-dual-giant graviton state by ζ r · · · i + i , = 2 2 = | = k X 1 1 , r ) of dual-giants get related (5.1) to occupation 1 1 Φ i = where (Φ – 16 – | J r s i l , Z t + s i e 2 | 1 0 Z ) = 0 Φ 2 −| Φ → , which can have complicated intersections and quantization 1 1 5 Z Φ and S , 0 3 C is the angular momentum of the i and their motion is that of point particles on the and (Φ ) are holomorphic coordinates on i g 5 × l 3 t s i 2 , e , Z 1 0 2 AdS C Φ , Z 1 ) then the duality map is given by → ) where 1 Z 0 J N ) of individual levels in the giant graviton system. It will b k ,s r = 0 with and ( 3 · · · 2 , in a harmonic oscillator potential. It will be inte , Z 2 1 Similar to the description of giant gravitons in [11] one can It is pertinent to recall at this point [6, 15] that in case of h The first type of 1/8-BPS states can also be given by the classi For the second set of 1/8-BPS states we considered configurat C = ,s N 1 2 s in time as Φ to explore if this type of a duality exists for the 1/8-BPS sta numbers ( numbers considered here and in [11,giants 12]. and Further in dual-giants the half- followsof form a unifiedunified description picture of can bedescription given for arises lower in supersymmetric the cases solutionof t of 1/8-BPS matrix states. models propose graviton as a three surface obtained by the intersection of Φ Z giant gravitons with angular momentum ( Thus, the number ofgiants. dual-giants Also, becomes the the occupied number energy of levels ( single with largest both these descriptions giveduality the between the same giant counting graviton of and the 1/8-BPS dual-giant st gravi an explicit duality betweenspecify giant a graviton multi-giant states graviton and state dual- by ( complex numbers. This can be generalized to ‘wobbling dual- involves quantizing the space ofa these much simpler 3-surfaces. description The since their dual a world-volume, given at size any inside share at least 4 supersymmetries.can The again result be for mapped thean onto countin that arbitrary of number a of 3-dimensionalquantum bosons harmonic numbers representing osc of the the giant 3-dimensional gravit harmonic oscillator b tons of [11].quantizing Recently Mikhailov’s [12] solutions have (See countedgravitons also these appears [28]). 1/8-BPS to be giant The significantlyferent coun 3-surfaces more within complicated the sinc bitrary numbers of giant gravitons having angular momenta i on JHEP03(2007)031 ) 5 ic S 01 N ( ]. , ⊂ U Class. 3 , Adv. 5 S , JHEP S = 4 , × rs N 5 black holes 5 upersymmetries (2004) 006 AdS (2004) 048 02 hep-th/9711200 AdS 04 (4) (coming d-volume of the probe (2005) 161301 n the dual ul discussions. We thank nts on the dual operators JHEP SO sect we suggest that their ]. It will be interesting to , ce of these ‘wobbling dual- 95 tes an orthonormal basis of ages of this work. to publication. NVS would JHEP iven charges in our language ) charges and from y actually coincide). Usually gurations and these wobbling , 3 d therefore the degeneracies of t will be interesting to see if one . These states carry non-zero tion for the dual-giant counting. (1999) 1113] [ , J l t 2 A deformation of i e 38 , J 1 black holes 1 Z J 5 ]. black holes 5 → General non-extremal rotating black holes in AdS Phys. Rev. Lett. 1 , AdS Z Supersymmetric multi-charge – 17 – ]. and ). l t 1 i hep-th/0406188 Int. J. Theor. Phys. e , J i 2 Φ Supersymmetric limit of superconformal field and ,S for the states with ( ]. 1 N 5 → S i (4) invariance but not supersymmetry. AdS ) charges). So different giants in a given state form concentr (1998) 231 [ hep-th/0601156 (2004) 5021 [ 1 Half-BPS giants, free fermions and microstates of supersta 2 ⊂ SO General supersymmetric , J 21 2 3 ]. ]; ]. The large- S ,S 1 S hep-th/0411145 (2006) 036 [ 04 ) generically and can be shown to preserve at least 1/8 of the s 1 , J 2 hep-th/0401129 hep-th/0401042 hep-th/0506029 and Quant. Grav. (2006) 082 [ JHEP minimal five- dimensional [ Theor. Math. Phys. [ [ In both types of 1/8-BPS states we considered here preserve It is of interest to find an exact orthonormal basis of states i ,S 1 [6] N.V. Suryanarayana, [5] J.P. Gauntlett, J.B. Gutowski and N.V. Suryanarayana, [1] J.M. Maldacena, [3] Z.W. Chong, M. H. Cvetiˇc, Lu and C.N. Pope, [4] H.K. Kunduri, J. Lucietti and H.S. Reall, [2] J.B. Gutowski and H.S. Reall, S Acknowledgments We would like to thankShiraz Minwalla and for Rob sharing Myerslike with for to helpf us thank TIFR, the Mumbai results for in hospitality [12] during the prior References final st SYM for the 1/8-BPS statesoperators considered in here. SYM For was half-BPS providedfor sta in the [27]. states See with [22] non-zero for ( some comme ( of the background [26].giants’ One and should be count able thesee to 1/8-BPS quantize if states the there using spa exists thedual-giants a in methods duality which of between case [12 our our counting giant results graviton give confi a predic with the time evolution Φ from isometries of three-spheres and never intersect (andD-branes which whenever intersect they are do expectedsuch the to states split can and change. rejoindegeneracies an But should in not receive our any case quantumcan corrections. since give they a I description do of notby the inter turning full on set of somebranes 1/8-BPS that bosonic states break or with the g fermionic zero modes on the worl for those with ( JHEP03(2007)031 , D , ]. , JHEP D 71 , JHEP , Yang-Mills super ]. Phys. Rev. , =4 (2006) 125 ]. =4 ]. N Phys. 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