Young diagrams, Brauer algebras, and bubbling geometries
aa r X i v : . [ h e p - t h ] D ec FPAUO-11-12
Young diagrams, Brauer algebras, and bubbling geometries
Yusuke Kimura and Hai Lin Departamento de Fisica, Universidad de Oviedo,33007 Oviedo, Spain Department of Particle Physics, Facultad de Fisica,Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain
Abstract
We study the 1 / / N = 4 SYM. By analyzing asymptotic structureand flux integration of the geometries, we present a mapping between dropletconfigurations arising from the geometries and Young diagrams of the Braueralgebra. In particular, the integer k classifying the operators in the Brauer basisis mapped to the mixing between the two angular directions. e-mail: [email protected] e-mail: [email protected] Introduction
In this paper we study a particular problem in the context of the gauge/gravity corre-spondence [1],[2],[3]. A family of 1/2 BPS geometries were found and they are dual toa family of 1/2 BPS operators of N =4 SYM, as described in [4],[5],[6]. On the gaugeside, they can be described by Shur polynomial operators or Young diagrams. Theycan also be described by wavefunctions of multi-body system. A droplet space on thegravity side were found and the geometries dual to the corresponding operators on thegauge side were mapped [6]. On the gravity side, the geometries can be described by thephase space of multi-body system and Young diagrams. These spacetime geometriesare nontrivial quantum states on the gravity side.There are also 1/4 BPS operators and 1/8 BPS operators that are also dual tocorresponding 1/4 BPS geometries and 1/8 BPS geometries. A general family of 1/4BPS, 1/8 BPS geometries corresponding to two-charge and three-charge geometrieswere given in [9],[10],[11],[12], pertaining to their corresponding sectors. The conditionsthat the S shrinks or S shrinks smoothly were analyzed in details in e.g. [10],[11],[13].The careful analysis of the geometries of the gravity side shows that the conditionon the droplet space which characterizes the regular geometries encodes the conditionfor having globally well-defined spacetime geometries, as emphasized by [6]. In thispaper we also analyze these conditions and characterize the geometries.Meanwhile, gauge invariant operators which are dual to geometries have scalingdimension of order N , which means that handling huge combinatoric factors arisingfrom summing up non-planar diagrams is inevitable. A new observation is that theproblem can be handled systematically with the help of group theory. Following theearlier work [4], some bases for local gauge invariant operators with two R-chargeswere given in [14],[15],[16]. These bases are labelled by Young diagrams, and thecalculations can be performed efficiently by representation theory. See [20],[21],[22],[24],for example.In this paper we study the relation between the droplet space of the two-chargegeometries or 1/4 BPS geometries and the dual two-charge operators with the variousbases. In particular we find the relation between the droplet space and the basis builtusing elements of the Brauer algebra given in [14]. A class of the BPS operators wereobtained at weak coupling in [23], in which they are labelled by two Young diagrams.We note that the size of each Young diagram is determined by the R-charge of the fieldsand an integer. We will focus on how the Young diagrams show up from the bubblinggeometries. Other bases may also be related to the droplet picture, since these basescan be related by transformations from each other. There is also a droplet descriptionfrom other method on the gauge side by [7].The Young diagrams are also convenient for describing additional excitations onthese states. In particular, starting from a large dimension BPS operator labelled byYoung diagrams, one can modify the operators by replacing some fields with other fieldsor multiplying some other fields. The presence of those other fields in the operatorsmake the states to be non-BPS and we can describe those states as new excitations1n the BPS states. See also related discussions on those viewpoints, including e.g.[8],[27],[28],[29],[36].The organization of this paper is as follows. In section 2 and 3, we introduce thegeneral metric and flux. In section 4, we study the flux integration on the dropletspace. In section 5, we analyze metric functions and mixings of metric components,for general droplet configurations. In section 6, we study the large R behavior of themixings of the metric components. In section 7, we analyze the configurations on thedroplet space and Young diagrams. In section 8, we discuss more about the operatorshandled by the Brauer algebra. In section 9, we analyze the large r asympotics of thegeometry. Finally, in section 10, we briefly discuss our results and conclusions. Wealso include several appendices. We analyze two-charge geometries with J , J of two U (1) global symmetries inside SO (6). General family of solutions have been studied in [9],[10],[11],[12],[13]. They have SO (4) × SO (2) symmetry. The geometries have been studied from various perspectives,see also e.g. [12]. It was found [10] that on the droplet space, the S shrinks smoothly,or the S shrinks smoothly, see also e.g. [11],[13].They can be written via a K¨ahler potential K ( z i , ¯ z i ; y ). We have the 1/4 BPSgeometry in the form, ds = − h − ( dt + ω ) + h (cid:18) dy + 2 ∂ i ¯ ∂ j KZ + dz i d ¯ z j (cid:19) + ye G d Ω + ye − G dψ , (2.1)and see Appendix A.We will later analyze asymptotic structure of the geometries. In order to performthe analysis, it is convenient to make a change of coordinates y = r cos θ ,z = R ( r ) sin θ cos θ e iφ , (2.2) z = R ( r ) sin θ sin θ e iφ . We also denote µ = sin θ cos θ , µ = sin θ sin θ , µ = cos θ and r i = R i ( r ) µ i , i = 1 ,
2. We often use R = r + r . We also define S ij = 2 e i ( φ i − φ j ) ∂ i ∂ ¯ j KZ + 1 / . (2.3)In the following of this section, we mainly focus on the case that K = K ( r , r , y ).This means that the S ij is symmetric: S ij = S ji . Under this condition, with the shiftof the angular variables φ i → φ i − t , the geometries can be expressed by2 s = − h − (1 + h ab M a M b − S t ) dt + h (cid:0) µ + S µ T + S µ T + 2 S µ µ T T (cid:1) dr + 2 h ( S R T µ + S T µ R − µ r ) dµ dr + 2 h ( S R T µ + S T µ R − µ r ) dµ dr + √ ∆ r d Ω + 1 √ ∆ (cid:0) dµ + H dµ + H dµ (cid:1) + 2 h (cid:16) S R R − µ µ ∆ (cid:17) dµ dµ + µ √ ∆ dψ + h − h ij ( dφ i + M i dt )( dφ j + M j dt ) , (2.4)where we have defined T i = dR i /dr . We present the details of this calculation inAppendix B.The metric functions are defined as follows,∆ = µ r Z − Z , (2.5) h − = r ∆ + µ √ ∆ , (2.6) H i = √ ∆ h (cid:18) S ii R i − µ i ∆ (cid:19) . (2.7)The mixing between time and angles, which will play an important role to determinethe angular momenta of the geometries, is given by M = − − S ω φ + N ω φ S S − N , (2.8) M = − − S ω φ + N ω φ S S − N . (2.9)The functions in the angular part are h = S , h = S , h = N , (2.10)and the function in time is S t = S + S + 2 N + 2 ω φ + 2 ω φ , (2.11)where S i = h S ii r i − ω φ i , i = 1 , , (2.12) N = h S r r − ω φ ω φ . (2.13)The AdS × S can be recovered by plugging ∆ = 1 and R = R = p r + r in(2.4): ds = − (cid:0) r (cid:1) dt + 11 + r dr + r d Ω + X i =1 (cid:0) dµ i + µ i dφ i (cid:1) , (2.14)where the above are written in unit r = 1, and we have renamed ψ = φ .3 Flux in general form
We start by writing out the metric, in particular by expanding out the fibration overthe time direction: ds = − h − (cid:18) dt + 1 y (cid:0) ¯ ∂ i ∂ y K (cid:1) d ¯ z i − y ( ∂ i ∂ y K ) dz i (cid:19) + h (cid:18) dy + 2 Z + ∂ i ¯ ∂ j Kdz i d ¯ z j (cid:19) + y (cid:0) e G d Ω + e − G dψ (cid:1) , (3.1)and Z = tanh G = − y∂ y ( y ∂ y K ). The five form, is then given by F = (cid:0) − d (cid:0) y e G ( dt + ω ) (cid:1) − y dω + 2 i∂ i ¯ ∂ j Kdz i d ¯ z j (cid:1) ∧ d Ω + dual. (3.2)There are different types of components. We can split them into two types ofcomponents. The five form here may be written as F = F ∧ d Ω + F ∧ dψ. (3.3)The various components of F are F = ∂ y (cid:0) y e G (cid:1) dt ∧ dy + ∂ i (cid:0) y e G (cid:1) dt ∧ dz i + ¯ ∂ i (cid:0) y e G (cid:1) dt ∧ d ¯ z i + (cid:18) iy ( e G + 1) ∂ i Z − i y ∂ y (cid:0) y e G (cid:1) ∂ i ∂ y K (cid:19) dz i ∧ dy − (cid:18) iy ( e G + 1) ¯ ∂ i Z − i y ∂ y (cid:0) y e G (cid:1) ¯ ∂ i ∂ y K (cid:19) d ¯ z i ∧ dy + 12 iy (cid:0) ∂ i (cid:0) e G (cid:1) ∂ j ∂ y Kdz i ∧ dz j − ¯ ∂ i (cid:0) e G (cid:1) ¯ ∂ j ∂ y Kd ¯ z i ∧ d ¯ z j (cid:1) + (cid:18) i∂ i ¯ ∂ j K − iy (cid:18) ( e G + 1) (cid:0) ∂ i ¯ ∂ j ∂ y K (cid:1) + 12 (cid:0) ∂ i (cid:0) e G (cid:1) ¯ ∂ j ∂ y K + ¯ ∂ j (cid:0) e G (cid:1) ∂ i ∂ y K (cid:1)(cid:19)(cid:19) dz i ∧ d ¯ z j . (3.4)These components are multiplied by d Ω .We write the full dual field strength for the five form as [17]: F = 1(1 + 2 Z ) (cid:18) y∂ j ¯ ∂ k K∂ y (cid:18) y ∂ y ∂ i K∂ y ¯ ∂ l K (cid:19) − − Z + 2 y∂ y Zy ∂ j ¯ ∂ l K∂ i ¯ ∂ k K (cid:19) dz i ∧ dz j ∧ d ¯ z k ∧ d ¯ z l + iy Z ) ∧ (cid:18) ∂ i ¯ ∂ j K∂ y (cid:18) y ∂ y ¯ ∂ k K (cid:19) dt ∧ dz i ∧ d ¯ z j ∧ d ¯ z k − ∂ j ¯ ∂ i K∂ y (cid:18) y ∂ y ∂ k K (cid:19) dt ∧ dz j ∧ d ¯ z k ∧ d ¯ z i (cid:19) + (1 − Z )2 y (1 + 2 Z ) ∂ i ¯ ∂ k K (cid:18) − Z y ∂ j (cid:16) y e G (cid:17) − ∂ j ∂ y K (cid:19) + 2 yǫ ef ǫ gl ∂ f ¯ ∂ l K∂ y ∂ e K∂ y ∂ j ¯ ∂ g K (1 − Z ) det (cid:0) ∂ i ¯ ∂ j K (cid:1) ! dz i ∧ dz j ∧ d ¯ z k ∧ dy − ∂ i ¯ ∂ k K (cid:18) − Z y ¯ ∂ j (cid:16) y e G (cid:17) − ¯ ∂ j ∂ y K (cid:19) + 2 yǫ ef ǫ gl ∂ l ¯ ∂ f K∂ y ¯ ∂ e K∂ y ∂ g ¯ ∂ j K (1 − Z ) det (cid:0) ∂ i ¯ ∂ j K (cid:1) ! dz i ∧ d ¯ z j ∧ d ¯ z k ∧ dy ! + i (1 − Z ) y (1 + 2 Z ) ∂ i ¯ ∂ j K y − Z ∂ y ∂ i ¯ ∂ j K
16 + ǫ ek ǫ lf ∂ k ¯ ∂ l K∂ i ¯ ∂ j K∂ y ∂ e ¯ ∂ f Kdet (cid:0) ∂ i ¯ ∂ j K (cid:1) !! dt ∧ dz i ∧ d ¯ z j ∧ dy (3.5) where these should be multiplied by dψ . 4 Flux integration
We focus on the droplet space on which the S or S vanishes. These occur at y = 0.The droplet space is divided into two droplet regions, with one region where Z = − and the S vanishes, and another region where Z = and the S vanishes. The fluxintegration at y = 0 involves several different situations, depending on different typesof droplets.We look at the first term in the expression (3.5). We can find the Z = − small y behavior of this term and find this term near y = 0, with Z = − droplet region, − dz ∧ d ¯ z ∧ dz ∧ d ¯ z ∧ dψ. (4.1)The flux integration in Z = − is Z M × S F = Z M × S dz ∧ dz ∧ d ¯ z ∧ d ¯ z ∧ dψ = 2 π Z M dz ∧ dz ∧ d ¯ z ∧ d ¯ z , (4.2)with 18 π l p Z M dz ∧ dz ∧ d ¯ z ∧ d ¯ z = N i , (4.3)which are quantized, due to the quantization of the F flux. N i is the flux quantumnumber in each region. The volume of ψ is V S = 2 π. For example, for
AdS × S , when Z = − , we have r = 0, so z = r sin θ cos θ e iφ , z = r sin θ sin θ e iφ , and14 Z M dz ∧ dz ∧ d ¯ z ∧ d ¯ z = π r , (4.4)18 π l p Z M dz ∧ dz ∧ d ¯ z ∧ d ¯ z = r πl p = N, (4.5)where for the ground state, M is the compact region bounded by | z | + | z | = r . Wehave dz ∧ d ¯ z ∧ dz ∧ d ¯ z = − dx ∧ dx ∧ dx ∧ dx , where z = x + ix , z = x + ix .We may view this as a 4d droplet space.The flux quantum number would map to the lengths N i of the vertical edges ofthe Young diagram operators. Here i denotes the different Z = − regions at y = 0.Compact Z = − droplets correspond to AdS asymptotics, while non-compact Z = − droplets could give rise to other asymptotics.Now we look at the flux integration in Z = region. On the other hand, at y = 0,with Z = droplet region, the F has components2 i∂ i ¯ ∂ j K dz i ∧ d ¯ z j ∧ d Ω , (4.6)where K is defined in (A.5) in the appendix, so we have the flux integrations Z D × S F = Z D π i∂ i ¯ ∂ j K dz i ∧ d ¯ z j , (4.7) Z ˜ D × S F = Z ˜ D π i∂ i ¯ ∂ j K dz i ∧ d ¯ z j , (4.8)5here V S = 2 π , and D , ˜ D are two dimensional domains. We have that1(2 π ) l p Z D i∂ i ¯ ∂ j K dz i ∧ d ¯ z j = m i , (4.9)1(2 π ) l p Z ˜ D i∂ i ¯ ∂ j K dz i ∧ d ¯ z j = ˜ m i . (4.10)Here, we have non-contractible two-cycles, so we have several situations.For AdS × S , K = 12 aR − q log( aR ) . (4.11)In that case there is no such two-cycles. We have ∂ ¯ ∂ K | z =0 = a, ∂ ¯ ∂ K | z =0 = a, so the flux components along the z plane where z = 0 is constant, and the fluxcomponents along the z plane where z = 0 is also constant.One of the simplest situations is that there are many Z = − regions on the z planeand z plane. We can form the two-cycles between the Z = − regions on the z planefor D , and on the z plane for ˜ D . They are the white strips in figure 1. Now weconsider very thin Z = − regions on top of the Z = background. See figure 1 foran example of thin Z = − regions. In this case, on the Z = background, away fromthe very thin Z = − regions, ∂ ¯ ∂ K | z =0 , ∂ ¯ ∂ K | z =0 are approximately given bythose of the AdS expression e.g. (4.11).We now consider the D domain of Z = on z plane at z = 0, surrounded by Z = − regions, and we have1(2 π ) l p Z D i∂ ¯ ∂ K dz ∧ d ¯ z = m i . (4.12)Similarly ˜ D is the domain of Z = on z plane at z = 0 , surrounded by Z = − regions, and we have 1(2 π ) l p Z ˜ D i∂ ¯ ∂ K dz ∧ d ¯ z = ˜ m i . (4.13)According to the BPS operators in [23] expressed by the Brauer basis in [14], theoperators can be labelled by two Young diagrams. These operators are briefly sum-marized in section 8. We may identify the flux quantum numbers m i with the sizeof the horizontal edges m i of the first Young diagram, and identify the flux quantumnumber ˜ m i with the size of the horizontal edges ˜ m i of the second Young diagram. Thetwo Young diagrams have total number of boxes m − k and n − k respectively,6 i i X i ′ =1 m i ′ N i = m − k, (4.14) X i i X i ′ =1 ˜ m i ′ ˜ N i = n − k. (4.15)The flux integrations at y = 0 show how the edges of Young diagrams would be mapped,according to above discussions.Following the earlier works, we further identify the appropriate variables and func-tions to characterize the geometries, and focused on the droplet space which are dividedinto two droplet regions, with Z = − where the S vanishes and with Z = wherethe S vanishes. The flux quantum numbers are determined by the flux integrals. Thedimension of the operator is m + n . Note that these are the flux quantization in thesmall y region. These are not the same as the flux quantization in the large r region. K plays an important role in these flux integrations that involve the two-cyclesin the Z = regions. K can be solved by the coupled equations (A.7) of K , K forthese regions. Figure 1 shows a plot of K where there are several thin Z = − strips. -6 -4 -2 2 4 6-6-4-2246 2 4 6 85101520253035K0 Figure 1:
On the left side is the droplet configuration on the z plane. On the right side isthe plot of K on the z plane along the radial axis. Similarly for the z plane. There arethin Z = − regions. In the limit that the thin black strips go to zero, the function goes tothat of the AdS expression. N , M , M We now study the metric functions in the geometry as well as the metric componentsmixing time and angles, as well as components mixing the angles. The expressions in7his section are valid for all possible ranges of z i , ¯ z i . In a different section 6, we willstudy their large R behaviors, in the large R region. We denote R = r + r , and r = | z | , r = | z | . We expand K in powers of y . The entire solutions to K are determined by thefunctions K , K , K , see appendix A. The equations for K are in Appendix A.In this section we study exact expressions in all possible ranges of z i , ¯ z i , but forsmall y , since we consider K , K , K in the series expansion in powers of y .Near Z = 1 /
2, we have the expansion K = − y log y + K + y K + y K + y K + O ( y ) , (5.1) Z = 12 − y K − y K + O ( y ) . (5.2)These expressions will be used in the later derivations in this section.We also use ω φ i = − y ∂ y ( r i ∂ r i K ) = − r i ∂ r i K − y r i ∂ r i K + O ( y ) . (5.3)The ∆ can be expanded as∆ = µ r Z − Z = µ r − y K − y K + O ( y )4 y K + 12 y K + O ( y )= 14 r K (cid:18) − y K − y K K + O ( y ) (cid:19) . (5.4)Note that y = rµ .Around y = 0 with Z = , which is approached by µ = 0, we may expand it as Z = 12 1 − r µ r ∆ r µ r ∆ = 12 − r y r ∆ + (cid:18) r y r ∆ (cid:19) + O ( y ) . (5.5)Comparing (5.2),(5.5), we have K = r r ∆ + O ( y ) , (5.6) h = p K + O ( y ) . (5.7) We study particularly the important functions M , M , N which are the mixing be-tween time and the angles, and the mixing between the angles themselves.8e focus on the components of the geometry, h ij h ( dφ i + M i dt )( dφ j + M j dt ) . (5.8)In particular, there is also a mixing term between the angles,2 N h ( dφ + M dt )( dφ + M dt ) , (5.9)where h = N . M , M account for the mixing between t and φ , φ respectively. The function N account for the mixing of angles φ and φ . Using the small y expansions presented in section 5.1 and h = (1 / − Z ) /y , onecan expand the S i , N in section 2 as S i = h S ii r i − ω φ i = 2 K ∂ t i K − ( ∂ t i K ) + O ( y )= s i + O ( y ) (5.10)and N = h S r r − ω φ ω φ = 2 K ∂ t ∂ t K − ( ∂ t K )( ∂ t K ) + O ( y )= n + O ( y ) , (5.11)where s = 2 K ∂ t K − ( ∂ t K ) s = 2 K ∂ t K − ( ∂ t K ) n = 2 K ∂ t ∂ t K − ( ∂ t K )( ∂ t K ) (5.12)have been defined. We also have introduced t i = log r i . It is interesting to note thatonly the 2nd derivatives of K , and 1st derivatives of K , and no derivative of K appears in the expression.Hence the mixing functions M and M in (2.8),(2.9) can be expanded as M = − s ∂ t K − n ∂ t K s s − n + O ( y ) , (5.13) M = − s ∂ t K − n ∂ t K s s − n + O ( y ) . (5.14) It would be convenient to use e i ( φ i − φ j ) ∂ i ∂ ¯ j = 14 r i r j ∂ t i ∂ t j . r , r . The special case of large R (= r + r )will be discussed in section 6.The mixing function between the angles will be particularly important in our analy-sis to make a connection with the gauge theory. In general cases, it is nonzero, n = 0.But there are special cases when n = 0. In this case, the M i becomes simpler. Thespecial case that n = 0 correspond to defining N i : N i = M i | n =0 . (5.15)We find that N = − ∂ t K s + O ( y ) , (5.16) N = − ∂ t K s + O ( y ) . (5.17) We have presented general expressions for small y expansions, in section 5. The geome-tries are parameterized by y and z i , ¯ z i . We denote R = r + r , where r = | z | , r = | z | . Now we analyze these expressions in the large R region, by expansions in 1 /R .We find solutions in series expansion in powers of 1 /R . We will analyze the behaviorof the mixing components N , M , M in large R .The K¨ahler potential is given by the Monge-Ampere equation (A.4), and in thesmall y expansion, the equation gives a set of equations for K and K [13],( ∂ t ∂ t K )( ∂ t ∂ t K ) + ( ∂ t ∂ t K )( ∂ t ∂ t K ) − ∂ t ∂ t K )( ∂ t ∂ t K ) = 0 , (6.1)( ∂ t ∂ t K )( ∂ t ∂ t K ) − ( ∂ t ∂ t K ) = 4 e e t +2 t e K , (6.2)where t i = log r i .This set of the equations have some rescaling transformations: K → qK , (6.3) t i → ξ i t i + b i ( r i → e b i ( r i ) ξ i ) , (6.4) K → K + log q − X i (( ξ i − t i + b i + 12 log ξ i ) . (6.5)Overall constant shifts in K are not important because they appear with derivatives inthe metric. The ξ i = − r i with large r i . 10he most general form of solutions representing the AdS × S geometry, in a regionwith Z = 1 /
2, is K = 12 ( a ( sr m + r n )) − q a ( sr m + r n )) . (6.6)Using the rescaling transforms, this may be brought to the simplest form K = R − log R , where R = r + r . In our analysis, parameters a, q will be reserved toaccount for the possible scaling transformation for R (or z i , ¯ z i ) and K .For example, we can have the rescaling transformations: K → qK , (6.7) qa R → R , (6.8) sr → r , (6.9)together with constant shifts of K , and rescaling of y .We are interested in more general solutions. We will first consider from large R point of view. In the large R , we have that, r = aR (1 + O (1 /r )) + O ( y ), so in theregion where R is large, we have that r is also large.We denote R = r + r . Note that aR , ar , ar often appear in combinations intheir products, due to that √ a rescales the r i coordinates, as in (6.8).We find a family of expressions that satisfy the set of the equations, as follows: K is given by K = 12 aR − q log( aR ) + K (1)0 , (6.10) ∂ t K = 2 ar (cid:18) − qr aR + O (1 /r ) (cid:19) , (6.11) ∂ t K = 2 ar (cid:18) − qr aR + O (1 /r ) (cid:19) , (6.12) ∂ t ∂ t K = 2 qr r R (cid:16) qα aR + O (1 /r ) (cid:17) , (6.13)and K is K = 12 log (cid:18) a ( aR − q ) R (cid:19) + 12 + K (1)1 , (6.14) ∂ t K = qr aR (cid:16) qaR (1 + κ ) + O (1 /r ) (cid:17) , (6.15) ∂ t K = qr aR (cid:16) qaR (1 + κ ) + O (1 /r ) (cid:17) , (6.16) If we consider the following one, K = 12 ( a ( r + r )) − q a ( sr + r )) , the equations require s = 1. K is K = q aR − q ) + K (1)2 (6.17)= q a R (cid:16) qaR (2 + α ) + O (1 /r ) (cid:17) . (6.18)The expressions without K (1)0 , K (1)1 , K (1)2 is the solution for AdS (see [13] for otherrelated analysis). K (1)0 , K (1)1 , K (1)2 are deviations from AdS. The α , κ , κ , α are theeffect of turning on K (1)0 , K (1)1 , K (1)2 . In the second lines in (6.13),(6.15),(6.16),(6.18),the large r expansion with the effect of K (1)0 , K (1)1 , K (1)2 are given.From the differential equations, we find a family of K (1) , K (1)0 = q a (cid:18) dr + er R + O (1 /r ) (cid:19) , (6.19) K (1)1 = − q a R (cid:18) ( d − e )( r − r ) R + O (1 /r ) (cid:19) , (6.20) K (1)2 = q a R (cid:18) c − d + e ) + 6( d − e )( r − r ) R (cid:19) + q a R O (1 /r ) . (6.21)We note that three parameters c , d , e have come in, and they will be identified with thethree parameters m, n, k of Young diagrams in (8.2).These solutions give that in (6.13),(6.15),(6.16),(6.18), α = 2( d + e ) + 6( d − e )( r − r ) R ,α = α + c − d + e ) ,κ = α − d,κ = α − e. (6.22)Now we evaluate the N , M , M in the large r region, using the general expressions(5.11),(5.13),(5.14), as follows: N = 2 K ∂ t ∂ t K − ∂ t K ∂ t K + O ( y )= r r a R q aR ( α + α − κ − κ ) + O (1 /r ) + O ( y )= q µ µ r k + O (1 /r ) + O ( y ) , (6.23) M = − s ∂ t K − n ∂ t K s s − n + O ( y )= qaR ( − α + κ ) + O (1 /r ) + O ( y )= − qmr + O (1 /r ) + O ( y ) , (6.24)12nd M = − s ∂ t K − n ∂ t K s s − n + O ( y )= qaR ( − α + κ ) + O (1 /r ) + O ( y )= − qnr + O (1 /r ) + O ( y ) , (6.25)where we identify r = aR (1 + O (1 /r )) + O ( y ) and r = aR (1 + O (1 /r )) + O ( y ).We are making expansions in power series of y and 1 /r . More details of the abovederivations are in Appendix C.In the expression of M , M , we have identified q ( α − κ ) = qm = q , (6.26) q ( α − κ ) = qn = q . (6.27)In the large r , we have the quantization of electric charges by A i = M i dt, (6.28) F = dM dt = drdt (cid:18) qr m + O (1 /r ) (cid:19) , (6.29) F = dM dt = drdt (cid:18) qr n + O (1 /r ) (cid:19) , (6.30)14 π Z ∗ F = m, (6.31)14 π Z ∗ F = n, (6.32)where in our coordinates g tt g rr = q in large r , and where m, n are the two charges.Note that terms corresponding to O (1 /r ) in (6.24),(6.25) will not contribute to theintegral.In the expression of N , we have identified α + α − κ − κ = k. (6.33)The two angles are the angles in the z plane and z plane respectively.This k will be identified with the k parameter in the Brauer algebra representation.In other words, for nonzero k , N = k µ µ q r + O (1 /r ) . (6.34)It appears at order 1 /r with coefficients k. For the special case k = 0 , it may appearat only order 1 /r . See Appendix C for more details. The difference between M i and N i is order k/r for nonzero k , and 1 /r for zero k .13lugging these relations (6.26),(6.27),(6.33) into the solutions (6.22), we find4 e = k − m = k − q /q, d = k − n = k − q /q,c = k. (6.35)Note that in this family of solutions, the coefficients e , d k − m , k − n . Therefore the sign property of e, d , which are either negative orzero, imposes the constraints k m, n , or equivalently k min( m, n ) . In the y expansions, we have, from (6.22), qα = qk + 3 q µ + 3 q µ − q − q , (6.36) qα = 3 q µ + 3 q µ − q − q , (6.37) qκ = 3 q µ + 3 q µ − q − q , (6.38) qκ = 3 q µ + 3 q µ − q − q , (6.39)where we used that µ + µ = 1 − y /r = 1 − O ( y ) . There are several equivalent and alternative ways of writing these variables, seeAppendix C for more details. For example, qk = qα − qα + q + q = qα + qα − qκ − qκ , (6.40)which will be frequently used in the derivations. Now we turn to the analysis of the droplet configurations. The solutions in section 6are the large R expressions that result from the droplet configurations. The solution insection 6 are in the large R region of the full solution in all the ranges of z i , ¯ z i . In thissection, we also provide further duality relation with the operators labelled by Braueralgebra.We have that from section 6, K = 12 aR − q aR ) + q a (cid:18) ( d + e )2 R + ( d − e )( r − r )2 R + O (1 /r ) (cid:19) (7.1)in large R , and where − d = n − k, (7.2) − e = m − k. (7.3)As in section 4, e.g. (4.3), the flux quantization requires that the total dropletvolume to be quantized, Z D d z ′ d z ′ = Z D ( ∅ ) d z ′ d z ′ = 2 π l p N (7.4)14here D is the total Z = − droplets. Here we use the notation that d z d z = dx dx dx dx for convenience. D ( ∅ ) is the Z = − configuration such that there is noany finite Z = domains or Z = bubbles. In the large R , it was shown [13] that, ifwe expand K = 12 a ( | z | + | z | ) + e K , K = 12 + e K , (7.5)1 a ∂ ∂ ¯1 e K + ∂ ∂ ¯2 e K = e K , (7.6) ∂ ∂ ¯1 e K + ∂ ∂ ¯2 e K = 0 . (7.7)This means that a general solution to e K is − log( a | z − z ′ | + a | z − z ′ | ) with( z ′ ,¯ z ′ , z ′ ,¯ z ′ ) arbitrary, therefore we can approximately expand K as K = 12 aR − q aR ) − q π l p N Z D d z ′ d z ′ log( a | z − z ′ | + a | z − z ′ | )+ q π l p N Z D ( ∅ ) d z ′ d z ′ log( a | z − z ′ | + a | z − z ′ | ) + O (1 /R ) . (7.8)Both (7.1),(7.8) satisfy the equations (7.6),(7.7). The droplet configuration in D givesnontrivially the information of d, e in the large R , when comparing two expressions(7.1),(7.8).Suppose we consider configurations that only depend on r , r , and in the case when d and e are equal, where there are more symmetry in the droplet configuration, we canexpand (7.8), K = 12 aR − q aR ) − q ( M [2] − M [2] , ∅ )4 π l p N R − q ( M [2] − M [2] ,, ∅ )4 π l p N R + O (1 /R )(7.9)where M [2] − M [2] , ∅ = Z D d z ′ d z ′ | z ′ | − Z D ( ∅ ) d z ′ d z ′ | z ′ | , (7.10) M [2] − M [2] , ∅ = Z D d z ′ d z ′ | z ′ | − Z D ( ∅ ) d z ′ d z ′ | z ′ | , (7.11)are the second moments along the axis perpendicular to z plane and z plane respec-tively, subtracted from those of the configuration without any finite Z = domains.The crossing terms z i ¯ z ′ i cancels due to symmetric configuration in z plane and in z plane.We read off the coefficients that12 π l p N ( M [2] − M [2] , ∅ ) + 12 π l p N ( M [2] − M [2] , ∅ ) (7.12)= 14 qa [( n − k ) + ( m − k )] = − qa ( d + e ) . (7.13)15f we use the convention qa = r = (4 πl p N ) / e.g. as from (4.5), and in the unit r = 1 , ( M [2] − M [2] , ∅ ) + ( M [2] − M [2] , ∅ ) = π n − k ) + ( m − k )] . (7.14)We then identify the second moments in two directions,( M [2] − M [2] , ∅ ) = π n − k ) , (7.15)( M [2] − M [2] , ∅ ) = π m − k ) , (7.16)where we are in the unit r = 1.Note that we also have the relation, from section 4, X i i X i ′ =1 m i ′ N i = m − k, (7.17) X i i X i ′ =1 ˜ m i ′ ˜ N i = n − k. (7.18)The change in the potential K is negative, when increasing the second moments.It is analogous to the change of the potential from − log R to − log( R + δ ), dueto increased second moments. So the signs of d, e are negative or zero in the aboveexpression, that is d , e . (7.19)This sign property can be understood as that the potential in (7.8) will decrease whenthe Z = − droplets in the droplet space are more outwards. The droplet secondmoments are increased from those of the configuration when there is no any finite Z = domains. The condition (7.19) is consistent with( M [2] − M [2] , ∅ ) > , ( M [2] − M [2] , ∅ ) > . (7.20)This amounts to m − k > , n − k > k min( m, n ) (7.22)from the gravity side.The dual operator has dimension m + n, and the two Young diagrams have m − k and n − k boxes respectively. The dimension is larger than the total number of boxesby the amount 2 k . The k parameter is identified in the last section as in (6.23),(6.33)as due to the mixing of two angular directions. The two angles are the angles in the z plane and z plane respectively. We see that 2 k measures the energy excess over thetotal number of boxes, and is accounted for by the mixing of two angular directions inthe situation described here. 16 | | z z | | z (a) z (e)(d) (c)(b) Figure 2:
Illustration of the mapping between Young diagrams and the droplet configura-tions. Z = − are regions where S vanishes and are drawn in black, while Z = areregions where S vanishes and are drawn in white. In (c), there are black droplets and whitedroplets in the | z | , | z | quadrant. Figure (c) determines (a,b), where (c) is projected onto z , z planes. The white regions of (a),(b) map to the horizontal edges of (d),(e). The blackregions of (c) map to the vertical edges of (d),(e). The outward directions in the droplet planescorrespond to the upper-right directions along the edges of the Young diagrams (d),(e). Theflux quantum numbers in corresponding regions map to the lengths of the edges of the Youngdiagrams (d),(e). This is an illustration with the example of relatively few number of circles.More general configurations involve many more circles in z , z planes. The Young diagramsare filled with boxes which are not shown in the illustration. The total number of boxes match the calculation from the flux quantum numbers,as m − k and n − k for two two-planes.The Brauer algebra provides two Young diagram representations, and each cor-respond to droplet configurations in z and z planes respectively. For example, forconfigurations that have 2 l concentric circles in the droplet plane z , this configurationmaps to the 2 l edges of the first Young diagram γ + , and the flux quantization numberson each droplet region map to the lengths of the edges of the Young diagram γ + . Thereare also 2 l concentric circles in the droplet plane z , and this configuration maps tothe 2 l edges of the second Young diagram γ − , and the flux quantization numbers oneach droplet region map to the lengths of the edges of the Young diagram γ − . Seefigure 2. These two Young diagrams are depicted in figures 2(d), 2(e) in the example.17e can characterize the droplet configuration by three diagrams, for those depend-ing only on r , r . The first two diagrams are concentric ring patterns of alternative Z = and Z = − regions in z planes and in z planes, see figures 2(a), 2(b) forexample. The third diagram is a diagram in the ( r , r ) quadrant, see figure 2(c). Wefirst map the corners in two Young diagrams to ordered points along r , r axis respec-tively in figure 2(c). Then we draw diagrams connecting ordered points along r axis tothose along r axis, dividing regions in ( r , r ) space into Z = − (depicted in black)and Z = (depicted in white).We can also draw more complicated droplets. In the gravity description, when wedraw the lines, there are extra possibilities for possible lines to go in the middle regionof ( r , r ) space. These may correspond to other operators that are superpositions ofthe Brauer basis.For those configurations that depend not only on r , r , they could be the super-positions of the Young diagram operators in the above discussions. For example, onecan add small ripples on any boundaries of the droplets. See also related discussions,e.g. [18]. The configurations that correspond to ripples on the droplet boundaries,can be considered as the superpositions of the Young diagram operators in the abovediscussions.In [10], 1/2 BPS geometries were uplifted into the systems of 1/4 and 1/8 BPSgeometries, and there are disconnected droplets with various topologies in 4d or 6ddroplet space respectively, and the topology change transitions occurred in [6],[19]uplift to the topology change transitions in 4d and 6d. In general, in the 4d dropletspace we study here, the topology change transition happens commonly.The condition (7.22) is consistent with the range of k in Brauer algebra representa-tion. The droplet information are encoded in the Young diagrams. There is correspon-dence between the pair of Young diagrams and the concentric droplet configurationin the two complex planes. The operators labelled by the Young diagrams of Braueralgebra give a family of globally well-defined spacetime geometries. Other bases mayalso be related to the droplet picture, since they can be related by transformations,and it might be interesting to see how k is produced in other bases. The system ofgeometries are dual to the system of the corresponding operators. Now we will briefly summarize the operators based on the Brauer algebra [14],[22]. Seealso related discussion in [30],[31].We take two complex fields X , Y out of three complex fields and consider gaugeinvariant operators constructed from m X s and n Y s. The U ( N ) gauge group isconsidered. The operators are conveniently expressed in the notation of [14],[22] by O γA,ij ( X, Y ) = tr m,n (cid:0) Q γA,ij X ⊗ m ⊗ ( Y T ) ⊗ n (cid:1) , (8.1)where Q γA,ij is given by a linear combination of elements in the Brauer algebra. Irre-ducible representations of the Brauer algebra are denoted by γ , which are given by two18oung diagrams: γ = ( γ + , γ − ) , (8.2)where γ + is a Young diagram with m − k boxes and γ − is a Young diagram with n − k boxes. k is an integer satisfying 0 k min( m, n ). A = ( R, S ) is an irreduciblerepresentation of S m × S n , labelled by two Young diagrams with m and n boxes. i, j are multiplicity indices with respect to the embedding of A into γ .An advantage of this basis is that the free two-point functions are diagonal. Wealso note that non-planar corrections are fully taken into account.Because the number of boxes in γ is characterized by k , it is convenient to classifythe operators by the integer k .The labels can be simplified when k = 0, because i, j are trivial, and we have γ + = R , γ − = S . Hence the operators in k = 0 are labelled by two Young diagrams.Denoting P R,S = Q γ ( k =0) A,ij , the expression in (8.1) becomes O R,S ( X, Y ) = tr m,n (cid:0) P R,S X ⊗ m ⊗ ( Y T ) ⊗ n (cid:1) . (8.3)In this sector, the operators have the nice expansion with respect to 1 /N , whose leadingterm is given by O R,S ( X, Y ) = O R ( X ) O S ( Y ) + · · · . (8.4)Here O R ( X ) is the Schur polynomial built from the X fields labelled by a Youngdiagram R with m boxes, and O S ( Y ) is the Schur polynomial built from the Y fieldslabelled by a Young diagram S with n boxes.In general, including the case k = 0, the leading term of the operators (8.1) looksschematically like O γA,ij ( X, Y ) ∼ tr m,n (cid:0) σC k X ⊗ m ⊗ ( Y T ) ⊗ n (cid:1) + · · · , (8.5)where σ is an element in S m × S n and C is an operation contracting the upper index ofan X and the upper index of a Y T . Each term in the dots in the above expression (8.5)contains more contractions. In other words, k is the minimum number of contractionsinvolved in the Q γA,ij of an operator. Therefore the k can be given the intuitive meaningthat it measures the degree of the mixing between the two fields. For example, the k = 0 has no mixing in the sense of (8.4). The opposite case is the case k takes themaximum value. With the condition m = n , operators in k = m = n are found to beexpressed by operators of the combined matrix XY [31].In the paper [23], a class of the 1 / O R,S ( X, Y ) and P R,S,i O γ ( k =0) A,ii ( X, Y ) are annihilated by the one-loop dilatation oper-ator, for any m , n and N . Defining P γ = P A,i Q γA,ii for k = 0, the BPS operators arepresented by O γ ( X, Y ) = tr m,n ( P γ X ⊗ m ⊗ ( Y T ) ⊗ n ) (8.6)for both k = 0 and k = 0. Note that P γ is the projector associated with an irreduciblerepresentation γ of the Brauer algebra. 19t is interesting to find that they are labelled by two Young diagrams with boxeswhose total number depends on the integer k for fixed m and n . In k = 0, the number ofboxes involved in a representation γ is equal to the sum of the R-charges. On the otherhand, when k is non-zero, the Young diagrams γ = ( γ + , γ − ) have a smaller number ofboxes than the sum of the R-charges. The number of the deficit boxes in the γ is 2 k .When k is equal to n for m > n , the operators are labelled by a single Young diagramwith m − k boxes. A special case happens for k = m with m = n . The operator doesnot have any Young diagrams, but this is different from the case m = n = 0. For agiven m = n , there is only one 1/4 BPS operator labelled by the trivial representation.We now have a remark on a constraint for representations of the Brauer algebra.The representations γ have the constraint c ( γ + ) + c ( γ − ) ≤ N , where c denotes thelength of the first column of the Young diagram. More explanations are provided in[14]. This constraint is consistent with the identifications in sections 4 and 7.The Brauer basis can be used in various ways. One of the motivations of construct-ing the Brauer basis in [14] was to construct an operator describing a set of D-branesand anti-D-branes, which is realized as the k = 0 sector. In the application of theBrauer basis to the su (2) sector, which is relevant for the present work, the k = 0sector realizes natural operators dual to the objects with two fields. Furthermore theconstruction of the basis introduces the operators labelled by k = 0 as well. In themapping proposed in this paper, the quantum number k has been given a meaning asthe mixing between the two angular directions from the gravity point of view. The useof the Brauer algebra may be rephrased as the manifestation of such a good quantumnumber.On the other hand, the BPS operators may be described by other bases diagonaliz-ing free two-point functions, as in [15],[16], see also related discussions, [36],[32],[34],[35],[33]. Because other bases respect other quantum numbers, understanding a map be-tween the two sides for other bases could also be useful for getting a complete dualityof this sector. In this section we analyze the large r asymptotics of the geometry. We start from theexpression (2.4). The large r region includes the large R region of the droplet space.We first expand in small y , and have K , K , K . We then expand these functions inpowers of 1 /r . In the large r , we have r R = µ (cid:0) O (1 /r ) (cid:1) , r R = µ (cid:0) O (1 /r ) (cid:1) , (9.1) R = 1 √ a p r + qC (cid:0) O (1 /r ) (cid:1) , R = 1 √ a p r + qC (cid:0) O (1 /r ) (cid:1) . (9.2)20ear y = 0, but large R , from (5.4),∆ = 14 r K + O ( y )= a R r q (cid:18) − q (2 + α ) aR + O (1 /r ) (cid:19) + O ( y )= 1 q (cid:18) q (2 C i µ i − − α ) r (cid:19) + O (1 /r ) + O ( y )= 1 q (cid:18) q ( q + q − q µ − q µ ) r (cid:19) + O (1 /r ) + O ( y ) (9.3)where we have used (6.37) and have identified qC = q + q , qC = q + q . (9.4)We find that ∆ has the same asymptotic form as the two-charge superstar.The asymptotic form of the metric can be rewritten in the following form ds = √ ∆ ds + 1 √ ∆ ds (9.5)analogous to the gauged supergravity ansatz. We have that ds = − (cid:18) r + q − q ( C µ + C µ − α − κ µ − κ µ ) r + O (1 /r ) (cid:19) dt + qr (cid:18) − q (3 C µ + 3 C µ − α − r + O (1 /r ) (cid:19) dr + r d Ω (9.6)and ds = dµ + µ dψ + H (cid:2) dµ + µ (cid:0) O (1 /r ) (cid:1) ( dφ + M φ dt ) (cid:3) + H (cid:2) dµ + µ (cid:0) O (1 /r ) (cid:1) ( dφ + M φ dt ) (cid:3) + O (1 /r ) dµ dr + O (1 /r ) dµ dr +2 q r ( C + C + α − α − O (1 /r )) µ µ dµ dµ +2 q µ µ r ( α + α − κ − κ )(1 + O (1 /r ))( dφ + M φ dt )( dφ + M φ dt )(9.7)where the M i were calculated in the section 6, M = qr ( − α + κ ) + O (1 /r ) + O ( y ) , (9.8) M = qr ( − α + κ ) + O (1 /r ) + O ( y ) . (9.9)See Appendix D for detailed derivation. 21hen we use (see section 6 and Appendix C, e.g. equation (6.37)), qκ = qα − q , (9.10) qκ = qα − q , (9.11) qα = 3 q µ + 3 q µ − q − q , (9.12)for the g tt and g rr , we get ds = − (cid:18) r + 1 − q + q r + O (1 /r ) (cid:19) dt + 1 r (cid:18) − q + q + 1 r + O (1 /r ) (cid:19) dr + r d Ω (9.13)and we have rescaled r → qr . By a rescaling r → qr , q appears as an overallfactor of the metric. The metric has the same asymptotic form as two-charge superstar[25],[26].When we use qk = qα − qα + q + q = qα + qα − qκ − qκ , (9.14)then the last two terms in ds becomes2 µ µ r kdµ dµ + 2 µ µ r k ( dφ + M φ dt )( dφ + M φ dt ) , (9.15)and we have rescaled r → qr .The asymptotic geometry is similar in form to the U (1) gauged supergravity ansatz.One of the differences is that we have additional mixing of µ , µ , and mixing of φ , φ , as in (9.7) or (9.15). These mixing terms correspond to the k parameter in Braueralgebra.
10 Discussions
We studied the characterization of the droplet configurations of the 1/4 BPS geome-tries which are dual to a family of 1/4 BPS operators with large dimensions in N =4SYM. We characterized the 4d droplet configurations underlying the 1/4 BPS geome-tries, following early works. The droplet space is enlarged from the 2d droplet spaceobserved in the 1/2 BPS case. The droplet regions are divided into two regions, and weprojected the droplet configuration into two complex planes. We map the concentriccircle patterns in the z plane and z plane to two Young diagrams. We identify thesetwo Young diagrams as the two Young diagrams in the operators of the Brauer basis[14],[23], which have total number of boxes m − k and n − k . The flux quantum num-bers on the droplets map to the edges of the Young diagrams and the radial directionsin the two-planes correspond to the upper-right directions along the edges of the twoYoung diagrams. 22e simplified the droplet configurations by projecting it to two two-planes, anddraw three diagrams. The first two diagrams are black/white coloring on the two-planes, and the third diagram is the black/white coloring on the ( r , r ) space. Wesimplified droplet configurations in particular by the third diagram. These includegeneral radially symmetric configurations in two two-planes.There is also a droplet description from other method on the gauge side by [7]. Wesee more consistency suggesting it to be the 4d droplet space related to the multi-bodysystem. There are some subtleties in this droplet space.We studied more about the small y expansion of the geometries. In particular, K can be viewed as a potential on the droplet space, and itself is determined by the totaldroplet configurations. We also performed the large R analysis in the droplet space,and find their connections with the large r asymptotics of the geometries. The large R in the droplet space encodes information of the large r asymptotics. These informationare encoded in the K , K , K in the small y expansion. The large R expansion capturesthe large r expansion. The asymptotics knows m, n, k , since J = ( m − k ) + k , J =( n − k ) + k . The droplet configuration captures m − k, n − k, k and the details of allthe information of the shapes of two Young diagrams.We identified families of geometries that have mixing between two angular directionswhich are the two angles φ , φ in the z plane and z plane. The geometries in theasymptotic region have mixing metric-component in h φ φ with a family of parameter k . An interesting observation was given that the parameter k have to satisfy k min( m, n ). We gave the interpretation that this parameter is identified with the k parameter in the Brauer algebra representations.Performing a similar analysis for the 1/8 BPS sector would raise an interestingquestion. A proper basis using Brauer algebras has not been constructed to dealwith 1/8 BPS operators. One may guess that three integers would be involved asthe coefficients of the mixing among the three angular directions, generalizing the oneinteger k in the present case. Such analysis may give rise to a hint to apply Braueralgebra for gauge invariant operators involving more kinds of fields than two.We mainly provided a mapping for the operators built from the projector of theBrauer algebra in (8.6). However, this above-mentioned class of expressions do notcover all types of 1/4 BPS operators. In the droplet picture, there are also other morecomplicated droplet configurations. It would be nice to understand other types of BPSoperators from the Brauer algebra, as well as in other bases. The droplet configurationson gravity side would be helpful to get a complete list of the BPS operators manipulatedby the Brauer algebra.We can also study other excitations on these states. One can consider the su-pergravity field excitations on them. We can also consider strings excited on them.One can also see the emergence of the other local excitations on the geometries, e.g.[27],[28],[29].We may view the non-BPS states as the excitation above the BPS states. Startingfrom these heavy BPS states, one can add additional non-BPS excitations on them.These can be done by modifying the operators by adding other fields or multiplying23ther fields. These studies will also provide another view on the physical meaning ofthe parameters of the bases. Acknowledgments
The work of Y.K. is supported in part by the research grants MICINN FPA2009-07122and MEC-DGI CSD2007-00042. We thank Jonathan P. Shock for collaboration at anearly stage. The work of H.L. is supported in part by Xunta de Galicia (Conselleria deEducacion and grants PGIDIT10PXIB 206075PR and INCITE09 206121PR), by theSpanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042), by the Juan dela Cierva of MICINN, and by the ME, MICINN and Feder (grant FPA2008-01838).We also would like to thank University of Valencia for hospitality. H. Lin also wouldlike to thank the UNIFY International Workshop on Frontiers in Theoretical Physics,University of Porto, and University of Cambridge for hospitalities.
A Review of gravity ansatz
We review the geometry and the ansatz for the 1/4 BPS configurations. These back-grounds have an additional S isometry compared with the 1/8 BPS geometries, andhave a ten-dimensional solution of the form, in the conventions of [9],[10],[11], ds = − h − ( dt + ω ) + h (( Z + 12 ) − ∂ i ¯ ∂ j Kdz i d ¯ z ¯ j + dy ) + y ( e G d Ω + e − G ( dψ + A ) ) ,F = (cid:8) − d [ y e G ( dt + ω )] − y ( dω + η F ) + 2 i∂ ¯ ∂K (cid:9) ∧ d Ω + dual ,h − = 2 y cosh G,Z = 12 tanh G = − y∂ y ( 1 y ∂ y K ) ,dω = i d [ 1 y ∂ y ( ¯ ∂ − ∂ ) K ] = iy ( ∂ i ∂ ¯ j ∂ y Kdz i d ¯ z ¯ j + ∂ ¯ i Zd ¯ z i dy − ∂ i Zdz i dy ) , η F = − i∂ ¯ ∂D. (A.1) K = K ( z i , ¯ z i ; y ) , where i = 1 , , is the K¨ahler potential for the 4d base, which alsovaries with the y direction. D can be set to a constant, if the fibration of the S is adirect product. The volume of the 4d base is constrained by a Monge-Ampere equation,as well as an equation for function D ,log det h i ¯ j = log( Z + 12 ) + nη log y + 1 y (2 − nη ) ∂ y K + D ( z i , ¯ z ¯ j ) , (A.2)(1 + ∗ ) ∂ ¯ ∂D = 4 y (1 − nη ) ∂ ¯ ∂K. (A.3)In other words, det ∂ i ∂ ¯ j K = ( Z + 12 ) y nη e y (2 − nη ) ∂ y K e D . (A.4)24ccording to the analysis of [10],[11],[13] a family of geometries have the expansionfrom the droplet space as: K = − y log( y )+ K ( z i , ¯ z i )+ y K ( z i , ¯ z i )+( y ) K ( z i , ¯ z i )+ X n > ( y ) n K n ( z i , ¯ z i ) (A.5)for Z = and K = 14 y log( y ) + K ( z i , ¯ z i ) + y K ( z i , ¯ z i ) + ( y ) K ( z i , ¯ z i ) + X n > ( y ) n K n ( z i , ¯ z i ) (A.6)for Z = − , with ∂ i ∂ ¯ j K = 0. The last terms above correspond to expansion withhigher order terms ( y ) n K n ( z i , ¯ z i ). The K ( z i , ¯ z i ) is a function defined on the dropletspace.As was shown in [13], all other higher K n , with n >
3, are expressed in terms of K , K , K , and thus the entire solutions to K are given uniquely by K , K , K .K , K are determined by the coupled equations on the Z = droplet region [13]:( ∂ ∂ ¯1 K )( ∂ ∂ ¯2 K ) + ( ∂ ∂ ¯1 K )( ∂ ∂ ¯2 K ) − ∂ ∂ ¯2 K )( ∂ ¯1 ∂ K ) = 0 , det ∂ i ∂ ¯ j K = 14 e e K . (A.7)For the AdS × S , K = (cid:26) aR − ar log( R /r ) , R > r ar , R r (A.8)as analyzed in [13], where Z = − droplets are in 0 R r and Z = droplets are in R > r . Here, the K is constant in Z = − droplets. In the above, R = | z | + | z | ,in this appendix. The plot in figure 1 is a more general situation when there are alsoother three Z = − strips; and in the limit when these three strips go to zero, itrecovers the expression (A.8). One can also introduce an overall constant shift in K ,since only its derivatives appear. One can also rescale the above expression by therescaling transformations as in section 6. B Derivation of the metric
In this appendix we derive the general form of the metric, using new variables in thesection 2.For the situation that the K¨ahler potential does not depend on the angular coordi-25ates φ , φ , that K = K ( r , r , y ), the metric (2.1) can be expressed by ds = − h − (cid:0) dt + 2( ω φ dφ + ω φ dφ ) dt + ω φ dφ + ω φ dφ + 2 ω φ ω φ dφ dφ (cid:1) + h (cid:0) µ dr + r dµ + S R ( dµ + µ dφ ) + S R ( dµ + µ dφ )+ (cid:0) S µ T + S µ T + 2 S µ µ T T (cid:1) dr +2 ( S R T µ + S T µ R − µ r ) dµ dr +2 ( S R T µ + S T µ R − µ r ) dµ dr +2 S R R dµ dµ + 2 S R R µ µ dφ dφ )+ ye G d Ω + ye − G dψ , (B.1)where we have used ω = ω φ dφ + ω φ dφ and S ij = S ji , which are the consequence ofthe assumption. We have defined T i = dR i /dr .Performing the shift of the angular variables φ i → φ i − t , i = 1 ,
2, the metric canbe further written to be the form ds = − h − (1 + h ab M a M b − S t ) dt + h (cid:0) µ + S µ T + S µ T + 2 S µ µ T T (cid:1) dr + 2 h ( S R T µ + S T µ R − µ r ) dµ dr + 2 h ( S R T µ + S T µ R − µ r ) dµ dr + √ ∆ r d Ω + 1 √ ∆ (cid:0) dµ + H dµ + H dµ (cid:1) + 2 h (cid:16) S R R − µ µ ∆ (cid:17) dµ dµ + µ √ ∆ dψ + h − h ab ( dφ a + M a dt )( dφ b + M b dt ) . (B.2)The more detailed computations are given below. B.1 Metric functions
We introduce ∆ by the equation e G = r √ ∆ µ . (B.3)Then Z and h − may be expressed as Z = 12 tanh G = 12 r ∆ − µ r ∆ + µ , (B.4) h − = 2 y cosh G = r ∆ + µ √ ∆ . (B.5)(B.4) may be used to give ∆ in terms of Z as∆ = µ r Z − Z . (B.6)26 .2 Terms with dµ i dµ j We first calculate the following, h r dµ = 1 √ ∆ r ∆ r ∆ + µ dµ = 1 √ ∆ dµ − √ ∆ 1 r ∆ + µ ( µ dµ + µ dµ ) = 1 √ ∆ dµ − h ∆ ( µ dµ + µ dµ + 2 µ µ dµ dµ ) . (B.7)Using this, we have h (cid:0) r dµ + S R dµ + S R dµ + 2 S R R dµ dµ (cid:1) = 1 √ ∆ (cid:0) dµ + H dµ + H dµ (cid:1) + 2 h (cid:16) S R R − µ µ ∆ (cid:17) dµ dµ , (B.8)where we have defined H i = √ ∆ h (cid:18) S ii R i − µ i ∆ (cid:19) (B.9)for i = 1 ,
2. For two-charge superstar, this becomes the H i of the superstar.When the following condition is satisfied, S R R − µ µ ∆ = 0 , (B.10)the metric does not have the mixing term dµ dµ . For superstar, this is the case. B.3 Terms with dφ i dφ j The relevant terms in the metric (B.1) are X i =1 , ( − h − ω φ i dtdφ i − h − ω φ i dφ i + h S ii r i dφ i ) − h − ω φ ω φ dφ dφ + 2 h S R R µ µ dφ dφ = h − (cid:0) S dφ − ω φ dtdφ + S dφ − ω φ dtdφ + 2 N dφ dφ (cid:1) , (B.11)where we have defined S i = h S ii r i − ω φ i ( i = 1 , , (B.12) N = h S r r − ω φ ω φ . (B.13)Making the shift φ i → φ i − t , (B.11) becomes h − S dφ − h − ( S + ω φ + N ) dtdφ + h − S dφ − h − ( S + ω φ + N ) dtdφ + 2 h − N dφ dφ + h − ( S + S + 2 ω φ + 2 ω φ + 2 N ) dt = h − (cid:0) S dφ + 2 S t dtdφ + S dφ + 2 S t dtdφ + 2 N dφ dφ + S t dt (cid:1) . (B.14)27here we have defined S t = − S − ω φ − N ,S t = − S − ω φ − N ,S t = S + S + 2 ω φ + 2 ω φ + 2 N . (B.15)Finally, (B.14) can be written as S dφ + 2 S t dtdφ + S dφ + 2 S t dtdφ + 2 N dφ dφ + S t dt = h ij ( dφ i + M i dt )( dφ j + M j dt ) − h ij M i M j dt + S t dt , (B.16)where h = S , h = S , h = N , (B.17)and (cid:18) M M (cid:19) = 1 S S − N (cid:18) S S t − N S t S S t − N S t (cid:19) = − S S − N (cid:18) − S ω φ + N ω φ − S ω φ + N ω φ (cid:19) . (B.18)When h = N = 0, the metric does not have the mixing term dφ dφ , and M i will get a simple expression. Defining N i = M i | N =0 , we have N i = − − S − i ω φ i = − − ω φ i h S ii r i − ω φ i . (B.19)For superstar, N = 0, and N i are calculated N i = − q i r + q i . (B.20)See Appendix D.3 for more details. B.4 Terms with dµ i dr Here we will analyze the condition under which the mixing terms dµ i dr vanish.The mixing terms vanish when the following conditions are satisfied S R T µ + S T µ R = rµ ,S R T µ + S T µ R = rµ , (B.21)One can show that these are satisfied for two-charge superstar with the help of theequations in Appendix D.3. 28he set of the equations can be summarized as (cid:18) S R µ S R µ S R µ S R µ (cid:19) (cid:18) T T (cid:19) = r (cid:18) µ µ (cid:19) (B.22)Note that the determinant of the matrix is ( S S − S ) R R µ µ = (det S ij ) r r .Using T i = R ′ i , (cid:18) R ′ R ′ (cid:19) = r (det S ij ) r r (cid:18) S R µ − S R µ − S R µ S R µ (cid:19) (cid:18) µ µ (cid:19) = r (det S ij ) r r (cid:18) µ ( S R µ − S R µ ) µ ( S R µ − S R µ ) (cid:19) = r (det S ij ) (cid:18) S R − − R R − µ µ − S R − − R R − µ µ − (cid:19) (B.23)It can be further simplified if we use (B.10), which is the equation for vanishing dµ dµ term. We then have (cid:18) R ′ R ′ (cid:19) = r (det S ij ) (cid:18) S R − − R − R − µ S R − − R − R − µ (cid:19) = r (det S ij ) √ ∆ h − H R − R − √ ∆ h − H R − R − ! . (B.24)where we have used the definition of H i (B.9). Dividing the first equation by the secondequation, we obtain (log R ) ′ H = (log R ) ′ H (B.25)We have that for superstar, (B.25) is equal to 2 rH H f − . C Derivation of the N , M , M in large r In this appendix, we present the details of the calculations to derive the asymptoticforms of the mixing functions N , M , M .Using (6.13),(6.15),(6.16),(6.18), the expressions (5.12) are evaluated as n = 2 K ∂ t ∂ t K − ∂ t K ∂ t K = q a R (cid:16) qaR (2 + α ) + O (1 /r ) (cid:17) qr r R (cid:16) qα aR + O (1 /r ) (cid:17) − r r (cid:16) qaR (cid:17) (cid:16) qaR (1 + κ ) + O (1 /r ) (cid:17) (cid:16) qaR (1 + κ ) + O (1 /r ) (cid:17) = r r a R (cid:18) q aR ( α + α − κ − κ ) + O (1 /r ) (cid:19) = q r r a R k + O (1 /r ) , (C.1)29nd s = 2 K ∂ t K − ( ∂ t K ) = q a R (cid:18) q ( α + 2) aR (cid:19) (cid:18) ar − qr r R + O (1 /r ) (cid:19) − (cid:18) qr aR (cid:19) (cid:16) qaR (1 + κ ) (cid:17) = qr aR (cid:18) q ( α + 2) aR − qr aR − qr aR + O (1 /r ) (cid:19) = qr aR (cid:18) q ( α + 1) aR + O (1 /r ) (cid:19) ,s = qr aR (cid:18) q ( α + 1) aR + O (1 /r ) (cid:19) . (C.2)The mixing between time and the angles are given by M i M = − s ∂ t K − n ∂ t K s s − n , (C.3) M = − s ∂ t K − n ∂ t K s s − n . (C.4)Since s , s , ∂ t K , ∂ t K = O (1 /r ) , and n = O ( k/r ), the effect of n in the denom-inator is to give at most O (1 /r ) terms in M , M . So the following expressions are upto O (1 /r ), M = − ∂ t K s − n ∂ t K s s + O (1 /r ) , (C.5) M = − ∂ t K s − n ∂ t K s s + O (1 /r ) , (C.6)where the last terms will be evaluated as − n ∂ t K s s = − q r a R k + O (1 /r ) , (C.7) − n ∂ t K s s = − q r a R k + O (1 /r ) . (C.8)In other words, the difference between M i , N i is M − N = − n ∂ t K s s + O (1 /r ) = (cid:18) − k q r a R + O (1 /r ) (cid:19) + O (1 /r ) , (C.9) M − N = − n ∂ t K s s + O (1 /r ) = (cid:18) − k q r a R + O (1 /r ) (cid:19) + O (1 /r ) . (C.10)The − k q r a R is one of the O (1 /r ) terms in N and in M , but it is the only O (1 /r )term in M − N . Therefore M i − N i = O ( k/r ) , for k = 0; and M i − N i = O (1 /r ) , for k = 0 . M i in the large r , M = − ∂ t K s + O (1 /r )= − (cid:16) qaR (1 + κ ) (cid:17) (cid:18) q ( α + 1) aR + O (1 /r ) (cid:19) − + O (1 /r )= qaR ( − α + κ ) + O (1 /r )= qr ( − α + κ ) + O (1 /r ) . (C.11)Similarly M = − ∂ t K s + O (1 /r )= − (cid:16) qaR (1 + κ ) (cid:17) (cid:18) q ( α + 1) aR + O (1 /r ) (cid:19) − + O (1 /r )= qaR ( − α + κ ) + O (1 /r )= qr ( − α + κ ) + O (1 /r ) . (C.12)Now we focus on the family of solutions K (1)0 = q a (cid:18) dr + er R + f r r R (cid:19) . (C.13)1. when f = 0, κ = 4( e − d ) R (2 r − r ) ,κ = 4( d − e ) R (2 r − r ) ,α = 4 R (cid:0) d (2 r − r ) + e (2 r − r ) (cid:1) . (C.14)Note that κ = κ = 0, α = 0 when d = e .2. when d = e = 0, κ = − fr r R ( r + 13 r r − r r + 3 r ) ,κ = − fr r R ( r + 13 r r − r r + 3 r ) ,α = − fr r R ( r − r r + r ) . (C.15)Although it is possible to have the third parameter f, we think that it does not havea relevant physical meaning, and we have chosen f = 0. We have set therefore f = 0 . (C.16)31hese are the solutions presented in the text in (6.19)-(6.21),(6.22).From the expressions of N , M , M ,q ( α − κ ) = qm = q ,q ( α − κ ) = qn = q ,α + α − κ − κ = k. (C.17)Plugging these conditions into the solutions (6.22), we find4 e = k − m = k − q /q, d = k − n = k − q /q,c = k. (C.18)and we have κ − κ = n − m. There are several equivalent and alternative ways of writing these variables. Forexample, qα = qα + qk − q − q ,qα = qα − qk + q + q ,qκ = qα − q = qα − qk + q ,qκ = qα − q = qα − qk + q ,qk = qα − qα + q + q = qα + qα − qκ − qκ , qα = qκ + qκ + q + q ,q + q = qα − qα + qk = 2 qα − qκ − qκ ,qκ − qκ = q − q ,qκ + qκ = qα + qα − qk = 2 qα − q − q . (C.19) D Derivation of asymptotic metric
In this appendix, we present the details of the derivation of (9.6) and (9.7).
D.1 Metric functions
Here we collect equations which would be helpful to calculate the asymptotic form ofthe metric. h = 1 r √ ∆ + O ( y ) , (D.1)∆ = 14 r K + O ( y ) . (D.2)32 = q a R (cid:18) q ( α + 2) aR (cid:19) + O (1 /r )= q r (cid:18) − qC µ + qC µ r (cid:19) (cid:18) q ( α + 2) r (cid:19) + O (1 /r )= q r (cid:18) q ( α + 2 − qC µ − qC µ ) r (cid:19) + O (1 /r ) . (D.3) R = 1 √ a p r + qC (cid:0) O (1 /r ) (cid:1) , R = 1 √ a p r + qC (cid:0) O (1 /r ) (cid:1) , (D.4) aR = a ( r + r ) = r + qC µ + qC µ + O (1 /r ) − O ( y ) . (D.5) S ii = 2 ∂ i ∂ ¯ ı KZ + = 12 1 r i ∂ t i K + O ( y )= a − q ( R − r i ) R + O (1 /r ) + O ( y ) , (D.6) S = 12 ∂ r ∂ r KZ + = 12 ∂ r ∂ r K + O ( y )= qr r R (cid:16) qα aR + O (1 /r ) (cid:17) + O ( y ) . (D.7) s i = qr i aR (cid:18) q ( α + 1) aR + O (1 /r ) (cid:19) ,h = p K + O ( y ) = √ qaR (cid:18) q ( α + 2)2 aR (cid:19) + O (1 /r ) + O ( y ) . (D.8)Comparing with (5.2),(5.5), K = r r ∆ . (D.9)Because AdS is given by ∆ = 1, K AdS ) = r r . We also have, K = q aR − q ) , (D.10)for AdS. So we have that for AdS, C = C = 1. Note that R = r + r .33 .2 Calculation of metric The factor in front of dt is now calculated. Taking account of M i = O (1 /r ) and N = O (1 /r ), we have h − (1 + h ab M a M b − S t )= h − (cid:0) S M + S M + 2 N M M − S − S − ω − ω − N (cid:1) = h − (cid:0) − S − S − ω − ω + O (1 /r ) (cid:1) = h − (cid:18) − qr aR (cid:18) q ( α + 1) aR (cid:19) − qr aR (cid:18) q ( α + 1) aR (cid:19) +2 qr aR (cid:18) q (1 + κ ) aR (cid:19) + 2 qr aR (cid:18) q (1 + κ ) aR (cid:19) + O (1 /r ) + O ( y ) (cid:19) = √ ∆ r (cid:18) qaR + q (1 − α ) a R + 2 q ( r κ + r κ ) a R (cid:19) + O (1 /r ) + O ( y )= √ ∆ (cid:18) r + q − q C µ + C µ r + q (1 − α ) r + 2 q ( κ µ + κ µ ) r (cid:19) + O (1 /r ) + O ( y ) . (D.11)We next calculate the factor in front of dr in large r . We first show µ + S µ T + S µ T + 2 S µ µ T T = µ + a (cid:18) − qr aR (cid:19) µ T + a (cid:18) − qr aR (cid:19) µ T + 2 qr r R µ µ T T + O (1 /r )= µ + aµ T + aµ T − qR ( r µ T − r µ T ) + O (1 /r )= µ + µ (cid:18) − qC r (cid:19) + µ (cid:18) − qC r (cid:19) + O (1 /r )= 1 − qC µ + qC µ r + O (1 /r ) . (D.12)With the help of (D.1), (D.2) and (D.3), we obtain h (cid:0) µ + S µ T + S µ T + 2 S µ µ T T (cid:1) = √ ∆4 r K (cid:18) − qC µ + qC µ r (cid:19) + O (1 /r ) + O ( y )= √ ∆ qr (cid:18) q ( α + 2) − q ( C µ + C µ ) r (cid:19) + O (1 /r ) + O ( y ) . (D.13)34he factor H i is evaluated at large r , H = √ ∆ h (cid:18) S R − µ ∆ (cid:19) = 1 r (cid:18)(cid:18) a − qr R (cid:19) R − r K µ (cid:19) + O (1 /r ) + O ( y )= 1 r (cid:18)(cid:18) a − qr R (cid:19) R − r qa R µ (cid:19) + O (1 /r ) + O ( y )= 1 r (cid:18)(cid:18) − qµ r (cid:19) ( r + qC ) − qµ (cid:19) + O (1 /r ) + O ( y )= 1 + q ( C − r + O (1 /r ) + O ( y ) , (D.14) H = 1 + q ( C − r + O (1 /r ) + O ( y ) . (D.15)We calculate the factor in front of dµ dµ in large r :2 h (cid:16) S R R − µ µ ∆ (cid:17) = 2 r √ ∆ (cid:16) qr r R (cid:16) qα aR + O (1 /r ) (cid:17) R R − µ µ r K (cid:17) + O ( y )= 2 r √ ∆ (cid:16) qR (cid:16) qα aR + O (1 /r ) (cid:17) R R − r K (cid:17) µ µ + O ( y )= qa R r √ ∆ (cid:18) a (cid:16) qα aR (cid:17) R R − r (cid:18) q ( α + 2) r (cid:19) + O (1) (cid:19) µ µ + O ( y )= 2 √ ∆ q r ( C + C + α − α − µ µ + O (1 /r ) + O ( y ) . (D.16)In order to calculate the factor in front of dµ i dr , we calculate S R T µ + S T µ R = (cid:18) a − qr R + O (1 /r ) (cid:19) R T µ + (cid:16) qr r R + O (1 /r ) (cid:17) T µ R + O ( y )= aR T µ − qr R R T µ + qr r R T µ R + O (1 /r ) + O ( y )= aR T µ + qµ µ R ( − R T R + R T R ) + O (1 /r ) + O ( y )= rµ + qµ µ R (cid:16) − ra R + ra R (cid:17) + O (1 /r ) + O ( y )= rµ + q µ µ r ( C − C ) + O (1 /r ) + O ( y ) . (D.17)35ote that R T = r/a + O (1 /r ). Therefore, we have h ( S R T µ + S T µ R − rµ )= 1 r √ ∆ O (1 /r ) + O ( y )= 1 √ ∆ O (1 /r ) + O ( y ) . (D.18)The functions in front of dφ i dφ j will be evaluated. We first calculate the functionin the diagonal part: h − S i = H i √ ∆ H − i r ∆ s i + O ( y )= H i √ ∆ (cid:18) − q ( C i − r (cid:19) r a R r q (cid:18) − q ( α + 2) aR (cid:19) qr i aR (cid:18) q ( α + 1) aR (cid:19) + O (1 /r ) + O ( y )= H i √ ∆ ar i r (cid:18) − q ( C i − r (cid:19) (cid:18) − q ( α + 2) aR (cid:19) (cid:18) q ( α + 1) aR (cid:19) + O (1 /r ) + O ( y )= H i √ ∆ µ i (cid:18) qC i r (cid:19) (cid:18) − q ( C i − r (cid:19) (cid:18) − q ( α + 2) r (cid:19) (cid:18) q ( α + 1) r (cid:19) + O (1 /r ) + O ( y )= H i √ ∆ µ i + O (1 /r ) + O ( y ) . (D.19)We next calculate the mixing part: h − h = 1 √ ∆ √ ∆ h − N = 1 √ ∆ 14 r K N + O ( y )= 1 √ ∆ 1 r a R q r r a R q aR ( α + α − κ − κ ) + O (1 /r ) + O ( y )= 1 √ ∆ 1 r q r r aR ( α + α − κ − κ ) + O (1 /r ) + O ( y )= 1 √ ∆ q µ µ r ( α + α − κ − κ ) + O (1 /r ) + O ( y ) . (D.20)The metric in the angles is then expressed by √ ∆ h − h ab ( dφ a + M φ a dt )( dφ b + M φ b dt )= H (cid:2) µ (cid:0) O (1 /r ) (cid:1) ( dφ + M φ dt ) (cid:3) + H (cid:2) µ (cid:0) O (1 /r ) (cid:1) ( dφ + M φ dt ) (cid:3) + (cid:18) q µ µ r ( α + α − κ − κ ) + O (1 /r ) (cid:19) ( dφ + M φ dt )( dφ + M φ dt ) . (D.21)36 .3 Related formulas In this appendix, we will summarize some formulas. These formulas are also related tosome useful expressions in [10]. (The conventions here are obtained from the conven-tions in [10] by ∆ → ( H H ) − / ∆, the subscript change (1 , , → , , ρ i → R i .)We also summarize equations which correspond to the two-charge superstar. h − = r ∆ + µ √ ∆ , (E.1) ω φ i = − h µ i √ ∆ , ( i = 1 ,
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