aa r X i v : . [ h e p - t h ] N ov Gibbons-Hawking M-branes
A. M. Ghezelbash , R. Oraji Department of Physics and Engineering Physics,University of Saskatchewan, Saskatoon, Saskatchewan S7N 5E2, Canada
Abstract
We present new M2 and M5-brane solutions in M-theory based on transverseGibbons-Hawking spaces. These solutions provide realizations of fully localized typeIIA D2/D6 and NS5/D6 brane intersections. One novel feature of these solutions isthat the metric functions depend on more than two transverse coordinates (unlike allthe other previous known solutions). All the solutions have eight preserved super-symmetries and the world-volume theories of the NS5-branes are new non-local, non-gravitational, six dimensional, T-dual little string theories with eight supersymmetries.We discuss the limits in which the dynamics of the D2 and NS5-branes decouple fromthe bulk for these solutions. E-Mail: [email protected] E-Mail: [email protected]
Introduction
Fundamental M-theory in the low-energy limit is generally believed to be effectively de-scribed by D = 11 supergravity [1, 2, 3]. This suggests that brane solutions in the lattertheory furnish classical soliton states of M-theory, motivating considerable interest in thissubject. There is particular interest in finding D = 11 M-brane solutions that reduce to su-persymmetric p -brane solutions (that saturate the Bogomol’nyi-Prasad-Sommerfield (BPS)bound) upon reduction to 10 dimensions. Some supersymmetric BPS solutions of two orthree orthogonally intersecting 2-branes and 5-branes in D = 11 supergravity were obtainedsome years ago [4], and more such solutions have since been found [5].Recently interesting new supergravity solutions for localized D2/D6, D2/D4, NS5/D6and NS5/D5 intersecting brane systems were obtained [6, 7, 8, 9, 10]. By lifting a D6 (D5 orD4)-brane to four-dimensional self-dual geometries embedded in M-theory, these solutionswere constructed by placing M2- and M5-branes in different self-dual geometries. A specialfeature of this construction is that the solution is not restricted to be in the near core region ofthe D6 (or D5) brane, a feature quite distinct from the previously known solutions [11, 12].For all of the different BPS solutions, 1/4 of the supersymmetry is preserved as a resultof the self-duality of the transverse metric. Moreover, in [13], partially localized D-branesystems involving D3, D4 and D5 branes were constructed. By assuming a simple ansatz forthe eleven dimensional metric, the problem reduces to a partial differential equation that isseparable and admits proper boundary conditions.Motivated by this work, the aim of this paper is to construct the fully localized super-gravity solutions of D2 (and NS5) intersecting D6 branes without restricting to the nearcore region of the D6 by reduction of ALE geometries lifted to M-theory. Our main motiva-tion for considering ALE geometries (and specially multi-center Gibbons-Hawking spaces)is that in all previously constructed M-brane solutions [6, 7, 8, 9, 10], we have at most oneparameter in each solution. For example, NUT/Bolt parameter n for embedded transverseTaub-NUT/Bolt spaces, Eguchi-Hanson parameter a in the case of embedded transverseEguchi-Hanson geometry and a constant number with unit of length that is related to theNUT charge of metric at infinity obtained from Atiyah-Hitchin metric in the case of embed-ded transverse Atiyah-Hitchin geometry. Moreover, in all the above mentioned solutions, themetric functions depend (at most) only on two non-compact coordinates. The metric func-tions in the multi-center Gibbons-Hawking geometries depend (in general) on more physicalparameters, hence their embeddings into M-theory yield new results for the metric functionswith both non-compact and compact coordinates.We have obtained several different supersymmetric BPS solutions of interest. We shouldmention the condition of preserved supersymmetry is distinct from that of BPS which isdefined in the bosonic theory. Due to the general M2 and M5 ansatze that we consider insections 4,5 and 6, the metric functions for all M2 solutions, as well as M5 solutions are har-monic. Hence all our brane configurations are determined by solutions of Laplace equationsand so they obey the BPS property. Specifically, since in the 11 dimensional metric for anM2-brane, the M2-brane itself only takes up two of the 10 spatial coordinates, we can embed1 variety of different geometries. These include the double Taub-NUT metric, two-centerEguchi-Hanson metric and products of these 4-dimensional metrics. After compactificationon a circle, we find the different fields of type IIA string theory.In our procedure we begin with a general ansatz for the metric function of an M2 brane in11-dimensional M-theory. After compactification on a circle ( T ) , we find a solution to typeIIA theory for which the highest degree of the field strengths is four. Hence the non-compactglobal symmetry for massless modes is given by the maximal symmetry group E = R ,without any need to dualize the field strengths [14]. For the full type IIA theory, only thediscrete subgroup E ( Z ) = Z survives, in particular by its action on the BPS spectrumand as a discrete set of identifications on the supergravity moduli space. This subgroup isthe U-duality group for all type IIA theories we find in this paper.The outline of our paper is as follows. In section 2, we discuss briefly the field equationsof supergravity. In section 3 we review briefly the ALE geometries and then in section 4, weconsider the embedding of four-dimensional multi (and explicitly double) -center Gibbons-Hawking spaces in M-theory. These spaces are characterized with some (two) NUT charges.Moreover, we consider the multi (and especially two-center) Eguchi-Hanson spaces and findanalytical exact solutions for the M2-brane functions. We compare then our analyticalsolutions with the numerical solutions found a few years ago. In section 5, we present theM5-brane solutions. These solutions also are exact and analytic. In section 6, we thendiscuss embedding products of Gibbons-Hawking metrics in M2-brane solutions. All of thesolutions preserve some of the supersymmetry as we present the details in section 7. Insection 8, we consider the decoupling limit of our solutions and find evidence that in thelimit of vanishing string coupling, the theory on the world-volume of the NS5-branes is anew little string theory. Moreover, we apply T-duality transformations on type IIA solutionsand find type IIB NS5/D5 intersecting brane solutions and discuss the decoupling limit ofthe solutions. We wrap up then by some concluding remarks and future possible researchdirections. The equations of motion for eleven dimensional supergravity when we have maximal sym-metry (i.e. for which the expectation values of the fermion fields is zero), are [15] R mn − g mn R = 13 (cid:20) F mpqr F pqrn − g mn F pqrs F pqrs (cid:21) (2.1) ∇ m F mnpq = − ε m ...m npq F m ...m F m ...m (2.2)where the indices m, n, . . . are 11-dimensional world space indices. For an M2-brane, we usethe metric and four-form field strength ds = H ( y, r, θ ) − / (cid:0) − dt + dx + dx (cid:1) + H ( y, r, θ ) / (cid:0) d s ( y ) + ds ( r, θ ) (cid:1) (2.3)2nd non-vanishing four-form field components F tx x y = − H ∂H∂y (2.4) F tx x r = − H ∂H∂r (2.5) F tx x θ = − H ∂H∂θ (2.6)and for an M5-brane, the metric and four-form field strength are ds = H ( y, r, θ ) − / (cid:0) − dt + dx + . . . + dx (cid:1) + H ( y, r ) / (cid:0) dy + ds ( r, θ ) (cid:1) (2.7) F m ...m = α ǫ m ...m ∂ m H , α = ± d s ( y ) and ds ( r ) are two four-dimensional (Euclideanized) metrics, depending onthe non-compact coordinates y and r , respectively and the quantity α = ± , which corre-sponds to an M5-brane and an anti-M5-brane respectively. The general solution, where thetransverse coordinates are given by a flat metric, admits a solution with 16 Killing spinors[16].The 11D metric and four-form field strength can be easily reduced down to ten dimensionsusing the following equations g mn = (cid:20) e − / (cid:0) g αβ + e C α C β (cid:1) νe / C α νe / C β ν e / (cid:21) (2.9) F (4) = F (4) + H (3) ∧ dx . (2.10)Here ν is the winding number (the number of times the M-brane wraps around thecompactified dimensions) and x is the eleventh dimension, on which we compactify. Theindices α, β, · · · refer to ten-dimensional space-time components after compactification. F (4) and H (3) are the RR four-form and the NSNS three-form field strengths corresponding to A αβγ and B αβ .The number of non-trivial solutions to the Killing spinor equation ∂ M ε + 14 ω abM Γ ab ε + 1144 Γ npqrM F npqr ε −
118 Γ pqr F mpqr ε = 0 (2.11)determine the amount of supersymmetry of the solution, where the ω ’s are the spin connec-tion coefficients, and Γ a ...a n = Γ [ a . . . Γ a n ] . The indices a, b, ... are 11 dimensional tangentspace indices and the Γ a matrices are the eleven dimensional equivalents of the four dimen-sional Dirac gamma matrices, and must satisfy the Clifford algebra (cid:8) Γ a , Γ b (cid:9) = − η ab . (2.12)In ten dimensional type IIA string theory, we can have D-branes or NS-branes. D p -branescan carry either electric or magnetic charge with respect to the RR fields; the metric takesthe form ds = f − / (cid:0) − dt + dx + . . . + dx p (cid:1) + f / (cid:0) dx p +1 + . . . + dx (cid:1) (2.13)3here the harmonic function f generally depends on the transverse coordinates.An NS5-brane carries a magnetic two-form charge; the corresponding metric has the form ds = − dt + dx + . . . + dx + f (cid:0) dx + . . . + dx (cid:1) . (2.14)In what follows we will obtain a mixture of D-branes and NS-branes. The only instantons (in A-D-E classification) that their metrics could be written in knownclosed forms, are A k series where the metrics are given by: ds = V − ( dt + ~A · d~x ) + V γ ij dx i · dx j (3.1)where V , A i and γ ij are independent of t and ∇ V = ±∇ × ~A ; hence ∇ V = 0. The mostgeneral solution for V is then V = P ki =1 m | ~x − ~x i | . The metrics (3.1) describe the Gibbons-Hawking multi-center metrics. The k = 1 corresponds to flat space and k = 2 correspondsto Eguchi-Hanson metric. The standard form of Eguchi-Hanson metric is given by [17] ds EH = r g ( r ) [ dψ + cos( θ ) dφ ] + g ( r ) dr + r (cid:0) dθ + sin ( θ ) dφ (cid:1) (3.2)where g ( r ) = r r − a . If we change the coordinates of (3.2) to ( R, Θ , Φ , Ψ) by R = 1 a p r − a sin θ (3.3)Θ = tan − (cid:0) √ r − a r tan θ (cid:1) (3.4)Φ = ψ (3.5)Ψ = 2 φ (3.6)where a ≤ R < ∞ , ≤ Θ ≤ π, ≤ Φ ≤ π, ≤ Ψ ≤ π , then the Eguchi-Hanson metric(3.2) transforms into the two-center Gibbons-Hawking form (3.1) ds = H ( R, θ ) (cid:0) dR + R ( d Θ + sin Θ d Φ ) (cid:1) + 1 H ( R, Θ) ( a d Ψ + Y ( R, θ ) d Φ) (3.7)where H ( R, Θ) = a { R − R + 1 R − R } (3.8)and Y ( R, θ ) = a (cid:0) R cos θ − a √ R + a − Ra cos Θ + R cos θ + 2 c √ R + a + 2 Ra cos Θ (cid:1) . (3.9)4n equations (3.8) and (3.9), R = (0 , , a ) and R = − R are Euclidean position vectorsof two nut singularities.Here we consider the extension of metrics (3.1) by considering V ǫ = ǫ + k X i =1 m i | ~x − ~x i | . (3.10)The hyper-Kahler metrics (3.1) with V ǫ pose a translational self-dual (or anti-self-dual)Killing vector K µ , that means ∇ µ K ν = ± p det gǫ ρλµν ∇ ρ K λ . (3.11)This (anti-) self-duality condition (3.11) implies the three-dimensional Laplace equation for V ǫ with solutions (3.10). For ǫ = 0 in (3.10), the metrics (3.1) describe the asymptoticallylocally flat (ALF) multi Taub-NUT spaces. The removal of nut singularities implies m i = m and t a periodic coordinate of period πmk . In this section, we consider the Gibbons-Hawking space with k = 2 and metric function V ǫ with ǫ = 0, as a part of transverse space to M2 and M5-branes. The four-dimensionalGibbons-Hawking metric is ds GH = V ǫ ( r, θ ) { dr + r ( dθ + sin θdφ ) } + ( dψ + ω ( r, θ ) dφ ) V ǫ ( r, θ ) (4.1)where ω ( r, θ ) = n cos θ + n ( a + r cos θ ) √ r + a + 2 ar cos θ (4.2) V ǫ ( r, θ ) = ǫ + n r + n √ r + a + 2 ar cos θ . (4.3)The eleven dimensional M2-brane with an embedded transverse Gibbons-Hawking space isgiven by the following metric ds = H ( y, r, θ ) − / (cid:0) − dt + dx + dx (cid:1) + H ( y, r, θ ) / (cid:0) dy + y d Ω + ds GH (cid:1) (4.4)and non-vanishing four-form field components are given by eqs. (2.4), (2.5) and (2.6). Themetric (4.4) is a solution to the eleven dimensional supergravity equations provided H ( y, r, θ )is a solution to the differential equation2 ry sin θ ∂H∂r + y cos θ ∂H∂θ + r y sin θ ∂ H∂r + y sin θ ∂ H∂θ ++ ( r y sin θ ∂ H∂y + 3 r sin θ ∂H∂θ ) V ( r, θ ) = 0 . (4.5)5igure 4.1: The geometry of charges.We notice that solutions to the harmonic equation (4.5) determine the M2-brane metricfunction everywhere except at the location of the brane source. To maximize the symmetryof the problem, hence simplify the analysis, we consider the M2-brane source is placed atthe point y = 0 , r = 0. Substituting H ( y, r, θ ) = 1 + Q M Y ( y ) R ( r, θ ) (4.6)where Q M is the charge on the M2-brane, we arrive at two differential equations for Y ( y )and R ( r, θ ) . The solution of the differential equation for Y ( y ) is Y ( y ) ∼ J ( cy ) y (4.7)which has a damped oscillating behavior at infinity. The differential equation for R ( r, θ ) is2 r ∂R ( r, θ ) ∂r + r ∂ R ( r, θ ) ∂r + cos θ sin θ ∂R ( r, θ ) ∂θ + ∂ R ( r, θ ) ∂ θ = c r V ( r, θ ) R ( r, θ ) (4.8)where c is the separation constant. First, we are interested in the solutions of (4.8) farenough from the locations of NUT charges. So, we take r >> a , hence we have r ′ = r √ ( ar ) +1+2( ar ) cos θ ≈ P l =0 P l ( − cos θ ) a l r l +1 where r and r ′ are the distances to the two NUTcharges n and n , located on z -axis at (0 , ,
0) and (0 , , − a ) (figure 4.1). We keep the firsttwo terms in the expansion of 1 /r ′ , corresponding to l = 0 ,
1. The differential equation (4.8)turns to2 r ∂R ( r, θ ) ∂r + r ∂ R ( r, θ ) ∂r + cos θ sin θ ∂R ( r, θ ) ∂θ + ∂ R ( r, θ ) ∂ θ = c r { ǫ + Nr − ˜ n cos θr } R ( r, θ ) (4.9)6igure 4.2: Solutions to eq. (4.10) with different values for N.where N = n + n and ˜ n = an .By substituting R ( r, θ ) = f ( r ) g ( θ ), we find two separated second-order differential equa-tions, given by r d f ( r ) dr + 2 r df ( r ) dr − c ( ǫr + N r + M ) f ( r ) = 0 (4.10) d g ( θ ) dθ + cos θ sin θ dg ( θ ) dθ + c ( M + ˜ n cos θ ) g ( θ ) = 0 (4.11)where M is the second separation constant.We change the coordinate r to r = z , hence the differential equation (4.10) changes to d f ( z ) dz − c ( ǫz + Nz + Mz ) f ( z ) = 0 . (4.12)The solutions to equation (4.12) are z times the Whittaker functions. So, the most generalsolution to (4.10) which vanishes at infinity is f ( r ) = 1 r W W ( − cN √ ǫ , √ M c , c √ ǫr ) (4.13)where W W ( α, β, x ) is the Whittaker-Watson function, related to confluent hypergeometricfunction U , by W W ( α, β, x ) = e − / x x / β U (1 / β − α, β, x ) . (4.14)In figure 4.2, the behavior of f ( r ) is given where we choose the separation constant c = 1, ǫ = 1 and M = 0 . ξ = 1 − cos θ ,are given by g ( θ ) = H C ( ξ ) { C + C Z ξ ( ξ − H C ( ξ ) dξ } (4.15)7here H C ( ξ ) stands for H C (0 , , , n c , − c ( M + ˜ n ) , ξ ); the Heun- C function. The Heun-C differential equation and functions H C ( α, β, γ, δ, λ, x ) are reviewed briefly in appendix A.The first part of (4.15) which is proportional to H C ( ξ ), is an analytical function at ξ = 0.However the second part of (4.15) is not an analytical function at ξ = 0. To understandbetter the behavior of the second part of solution (4.15), we consider the function h ( ξ ) = 1( ξ − H C ( ξ ) (4.16)and use the Maclaurin’s theorem, we get a power series expansion as h ( ξ ) = ∞ X n =0 a n ξ n = a + a ξ + a ξ + · · · . (4.17)Here, the first few coefficients are given by a = −
12 (4.18) a = − M + ˜ n c −
14 (4.19) a = − M c − M c − M ˜ n c − n c − ˜ n c − . (4.20)In figure 4.3, for instance the plot of h ( ξ ) versus ξ is given where we set c = 1, ˜ n = 1 and M = 1. In this figure, h ( ξ ) is expanded up to order of ξ as h ( ξ ) = − − ξ − ξ − ξ − ξ − ξ + O ( ξ ) . (4.21)The series expansion (4.17) yields the final form of the solution g ( θ ) as g ( θ ) = H C (1 − cos θ ) { C c,M − C ′ c,M (cid:0) ln(1 − cos θ ) − ( M + ˜ n c + 14 )(1 − cos θ ) + · · · (cid:1) } (4.22)or g ( θ ) = C c,M { (1 − ( M + ˜ n c )(1 − cos θ ) + ( M c
16 +
M c ˜ n c ˜ n − M c − cos θ ) ++ O (1 − cos θ ) ) } ++ C ′ c,M { ln(1 − cos θ ) { − M + ˜ n c )(1 − cos θ ) + ( M c
16 +
M c ˜ n n c − M c − cos θ ) ++ O (1 − cos θ ) } ++ { ( 12 + ( M + ˜ n ) c )(1 − cos θ ) + ( M c − c ˜ n − n M c − M c
16 )(1 − cos θ ) ++ O (1 − cos θ ) }} . (4.23)8igure 4.3: h( ξ ) is a well defined function around origin O.Figure 4.4: The graph of g( θ ) keeping two terms of the series.9he constant C ′ c,M should be chosen zero, otherwise we get logarithmic divergence at θ = 0for r >> a . A typical functional form of g ( θ ) is shown in figure 4.4, where we set ˜ n = M = c = C c,M = 1 and C ′ c,M = 0. So, the solution to the differential equation (4.9) or theasymptotic solution to (4.8) is R ( r, θ ) = C c,M r W W ( − cN √ ǫ , √ M c , c √ ǫr ) H C (0 , , , n c , − ( M + ˜ n ) c , ξ ) . (4.24)Turning next to find the exact solution to (4.8), we change the coordinates r, θ to µ, λ ,defined by µ = r ′ + r (4.25) λ = r ′ − r (4.26)where µ > a and − a ≤ λ ≤ a . We notice that the coordinate transformations (4.25) and(4.26) are well defined everywhere except along the z-axis. The differential equation (4.8),in the new coordinates, turns out to be − λ ∂R∂λ +( a − λ ) ∂ R∂λ +2 µ ∂R∂µ +( µ − a ) ∂ R∂µ = c (cid:20) ǫ ( µ − λ ) + 12 µ ( n + n ) + 12 λ ( n − n ) (cid:21) R. (4.27)This equation is separable and yields2 λ G ∂G∂λ + ( λ − a ) 1 G ∂ G∂λ − c ( n − n ) λ − ǫc λ − M c = 0 (4.28)2 µ F ∂F∂µ + ( µ − a ) 1 F ∂ F∂µ − c ( n + n ) µ − ǫc µ − M c = 0 (4.29)upon substituting in R ( µ, λ ) = F ( µ ) G ( λ ) where M is the separation constant. The solutionto equation (4.28) is given by G ( λ ) = ˜ H C ( λ ) { ˆ g c,M + ˆ g ′ c,M Z a − λ )( a + λ ) ˜ H C ( λ ) dλ } (4.30)where ˜ H C ( λ ) stands for˜ H C ( λ ) = e c √ ǫ ( a − λ ) H C (2 ca √ ǫ, , , ac N − , −
14 ( ǫa + 2 aN − + 4 M ) c ,
12 (1 − λa )) . (4.31)In equations (4.30) and (4.31), N − = n − n and ˜ g c,M , ˜ g ′ c,M are two constants in λ . Thepower series expansion of ˜ H C ( λ ) is˜ H C ( λ ) = 1 − ( aN − c M c ǫa c − λa )++ ( ǫa c − M c ǫ a c
256 + ǫa c N −
64 + c M
16 + ǫa c M
32 + ac N − M
16 ++ a c N −
64 )(1 − λa ) + O ( λ ) . (4.32)10ence we obtain G ( λ ) = ˜ H C ( λ ) { g c,M + g ′ c,M ln (cid:12)(cid:12)(cid:12)(cid:12) − λa (cid:12)(cid:12)(cid:12)(cid:12) } + g ′ c,M ∞ X n =1 d n (1 − λa ) n (4.33)where g c,M , g ′ c,M and d n ’s are constants in λ . The first few d n ’s are d = 12 + M c + ǫa c aN − c d = M c − ǫa c
32 + 18 − ǫ a c − ǫa c N − −− c M − ǫa c M − ac N − M − a c N − . (4.34)The same approach can be used to find the solution to equation (4.29). We find F ( µ ) = ˜ H C ( µ ) { ˆ f c,M + ˆ f ′ c,M ( µ ) Z µ − a )( a + µ ) ˜ H C ( µ ) dµ } (4.35)where ˜ H C ( µ ) stands for˜ H C ( µ ) = e c √ ǫ ( a − µ ) H C (2 ca √ ǫ, , , ac N + , −
14 ( ǫa + 2 aN + + 4 M ) c ,
12 (1 − µa )) . (4.36)In equation (4.36), N + = n + n which yields the power series expansion as˜ H C ( µ ) = 1 − ( aN + c M c ǫa c − µa )++ ( ǫa c − M c ǫ a c
256 + ǫa c N +
64 + c M
16 + ǫa c M
32 + ac N + M
16 ++ a c N
64 )(1 − µa ) + O ( µ ) . (4.37)So, we obtain F ( µ ) = ˜ H C ( µ ) { f c,M + f ′ c,M ln (cid:12)(cid:12)(cid:12) − µa (cid:12)(cid:12)(cid:12) } + f ′ c,M ∞ X n =1 b n (1 − µa ) n (4.38)where b n ’s are given by (4.34) upon replacing N − by N + . In addition to the asymptoticsolution, given by (4.24) for far-zone r >> a , as well as the solution near NUT charges(near-zone), given by (4.33) and (4.38), we can obtain the solution to equation (4.8) (or(4.27)) in intermediate-zone for any values of r and θ (or any values of µ and λ ). The formof our intermediate-zone looks like the last summation term in (4.33) or (4.38). Hence, we11igure 4.5: The first bracket in (4.39) as a function of µ − a = z .find the most general solution to equation (4.27) (or equivalently to equation (4.8) aftercoordinate transformations (4.25) and (4.26)) R ( r, θ ) = ( ˜ H C ( µ ) { f c,M + f ′ c,M ln (cid:12)(cid:12)(cid:12) − µa (cid:12)(cid:12)(cid:12) } δ a,µ + f ′ c,M ∞ X n =0 b n,µ (1 − µµ ) n ) ×× ( ˜ H C ( λ ) { g c,M + g ′ c,M ln (cid:12)(cid:12)(cid:12)(cid:12) − λa (cid:12)(cid:12)(cid:12)(cid:12) } δ a,λ + g ′ c,M ∞ X n =0 d n,λ (1 − λλ ) n ) (4.39)where µ = √ r + a + 2 ar cos θ + r (4.40) λ = √ r + a + 2 ar cos θ − r (4.41)and µ ≥ a , | λ | ≤ a . In (4.39), d ,a = 0 and d n> ,a are given by (4.34). The other coefficientsare listed in appendix B. In figures 4.5 and 4.6, we plot the slices of the most general solution(4.39) at λ =const. and µ =const. respectively, for different values of separation constant c .Moreover, in addition to the general solution (4.39), we can easily obtain another inde-pendent solution by changing the separation constant c to ic in equations (4.10) and (4.11).In this case, we have d f ( z ) dz + c ( ǫz + Nz − Mz ) f ( z ) = 0 (4.42) d g ( θ ) dθ + cos θ sin θ dg ( θ ) dθ + c ( M − ˜ n cos θ ) g ( θ ) = 0 (4.43)12igure 4.6: The second bracket in (4.39) as a function of λ .where we changed M to − M for convenience and z = r . The second solution then, is givenby˜ R ( r, θ ) = 1 r ( C c,W W W ( − icN √ ǫ , √ M c , ic √ ǫr ) + C c,M W M ( − icN √ ǫ , √ M c , ic √ ǫr ) ) ×× H C (0 , , , − n c , − ( M − ˜ n ) c , ξ ) . (4.44)In figure 4.7, the solution (4.44) at a constant ξ has been plotted. The most general solutionto equation (4.8) after analytic continuation of c is given by ˜ R ( r, θ ) = ˜ F ( µ ) ˜ G ( λ ). We find˜ G ( λ ) = ˜˜ H C ( λ ) { ˆ˜ g c,M + ˆ˜ g ′ c,M Z a − λ )( a + λ ) ˜˜ H C ( λ ) dλ } (4.45)where ˜˜ H C ( λ ) stands for˜˜ H C ( λ ) = e ic √ ǫ ( a − λ ) H C (2 ica √ ǫ, , , − ac N − ,
14 ( ǫa + 2 aN − − M ) c ,
12 (1 − λa )) (4.46)and finally we obtain˜ G ( λ ) = ˜˜ H C ( λ ) { ˜ g c,M + ˜ g ′ c,M ln (cid:12)(cid:12)(cid:12)(cid:12) − λa (cid:12)(cid:12)(cid:12)(cid:12) } + ˜ g ′ c,M ∞ X n =1 ˜ d n (1 − λa ) n . (4.47)13igure 4.7: Two independent solutions in (4.44) at fixed ξ .In (4.47), ˜ g c,M , ˜ g ′ c,M and ˜ d n ’s are constants. The first few ˜ d n ’s are˜ d = 12 + M c − ǫa c − aN − c d = M c ǫa c
32 + 18 − ǫ a c − ǫa c N − − c M
16 + 3 ǫa c M
32 + 3 ac N − M − a c N − . (4.48)By the same method, we can find the function ˜ F ( µ ), hence we get the most generalsolution as˜ R ( r, θ ) = ( ˜˜ H C ( µ ) { ˜ f c,M + ˜ f ′ c,M ln (cid:12)(cid:12)(cid:12) − µa (cid:12)(cid:12)(cid:12) } δ a,µ + ˜ f ′ c,M ∞ X n =0 ˜ b n,µ (1 − µµ ) n ) ×× ( ˜˜ H C ( λ ) { ˜ g c,M + ˜ g ′ c,M ln (cid:12)(cid:12)(cid:12)(cid:12) − λa (cid:12)(cid:12)(cid:12)(cid:12) } δ a,λ + ˜ g ′ c,M ∞ X n =0 ˜ d n,λ (1 − λλ ) n ) . (4.49)We should note that the y dependence of M2-brane metric function is˜ Y ( y ) ∼ K ( cy ) y . (4.50)So, the second M2-brane metric function is˜ H ( y, r, θ ) = 1 + Q M Z ∞ dc Z ∞ dM ˜ Y ( y ) ˜ R ( r, θ ) . (4.51)We consider now the Gibbons-Hawking space with k = 2 (4.1) with ǫ = 0 in (4.3) (orequivalently the metric (3.2)). We should mention that despite some numerical solutions for14he M-brane metric function (with embedded Eguchi-Hanson transverse metric (3.2)) havebeen found in [7], the exact closed analytic form for the M-brane function hasn’t yet beenfound. Our method in this paper allows to construct the exact solutions for the M-branefunction with embedded Eguchi-Hanson space. In the limit of r >> a , the solution to (4.10)(with ǫ = 0) is given by f ( r ) = f c,M K √ Mc (cid:16) c √ N r (cid:17) √ r (4.52)where N = n + n , in exact agreement with the numerical result of [7]. The exact M-branefunction is given by equation (4.39) where ǫ = 0 should be considered in ˜ H C ( λ ) , ˜ H C ( µ ) , f c,M ,g c,M , f ′ c,M , g ′ c,M , d n and b m . Changing c to ic generates the second set of solutions that in thelimit of r >> a yields˜ f ( r ) = ˜ f c,M J √ Mc (2 c √ N r ) + ˜ f ′ c,M Y √ Mc (2 c √ N r ) √ r . (4.53)We note that the general solution of the metric function could be written as a superpositionof the solutions with separation constants c and M . For example, the general first set ofsolution (corresponding to embedded Gibbons-Hawking space with k = 2 and ǫ = 0) is H ( y, r, θ ) = 1 + Q M Z ∞ dc Z ∞ dM J ( cy ) y ×× ( ˜ H C ( µ ) { f c,M + f ′ c,M ln (cid:12)(cid:12)(cid:12) − µa (cid:12)(cid:12)(cid:12) } δ a,µ + f ′ c,M ∞ X n =0 b n,µ (1 − µµ ) n ) ×× ( ˜ H C ( λ ) { g c,M + g ′ c,M ln (cid:12)(cid:12)(cid:12)(cid:12) − λa (cid:12)(cid:12)(cid:12)(cid:12) } δ a,λ + g ′ c,M ∞ X n =0 d n,λ (1 − λλ ) n ) . (4.54)As we notice, the solution (4.54) depends on four combinations of constants f c,M , f ′ c,M and g c,M , g ′ c,M in form of f g, f ′ g, f g ′ and f ′ g ′ which each combination has dimension ofinverse charge (or inverse length to six). Hence, the functional form of each constant couldbe considered as an expansion of the form c β M β where β ∈ Z + . Moreover we shouldmention the meaning of µ and λ in equation (4.54) that have dimensions of length. Werecall that the near-zone solutions (4.33) and (4.38) are given partly by series expansionsaround r ≃ a . The intermediate-zone solutions are given by similar power series expansions(with substitutions a → λ and d n → d n,λ in (4.33) and a → µ and b n → b n,µ in (4.38)around some fixed points, denoted by µ and λ . To calculate numerically the membranemetric function (4.54) at any µ, λ (or equivalently any r and θ ), we consider some fixedvalues for µ and λ (see appendix B).Dimensional reduction of M2-brane metric (4.4) with the metric functions (for example154.54)) along the coordinate ψ of the metric (4.1) gives type IIA supergravity metric ds = H − / ( y, r, θ ) V − / ǫ ( r, θ ) (cid:0) − dt + dx + dx (cid:1) ++ H / ( y, r, θ ) V − / ǫ ( r, θ ) (cid:0) dy + y d Ω (cid:1) ++ H / ( y, r, θ ) V / ǫ ( r, θ )( dr + r d Ω ) (4.55)which describes a localized D2-brane at y = r = 0 along the world-volume of D6-brane,for any choice of constants in the form of c β M β where β ∈ Z + . The other fields in tendimensions are NSNS fields Φ = 34 ln (cid:26) H / ( y, r, θ ) V ǫ ( r, θ ) (cid:27) (4.56) B µν = 0 (4.57)and Ramond-Ramond (RR) fields C φ = ω ( r, θ ) (4.58) A tx x = 1 H ( y, r, θ ) . (4.59)The intersecting configuration is BPS since it has been obtained by compactification alonga transverse direction from the BPS membrane solution with harmonic metric function (forexample (4.54)) [18]. Moreover, in section 7, we use the Killing spinor equation (2.11) tocalculate how much supersymmetry is preserved by M2-brane solutions in eleven dimensions.We conclude that half of the supersymmetry is removed by the projection operator that isdue to the presence of the brane, and another half is removed due to the self-dual nature ofthe Gibbons-Hawking metric. Hence embedding any Gibbons-Hawking metric into an elevendimensional M2-brane metric preserves 1/4 of the supersymmetry. The eleven dimensional M5-brane metric with an embedded Gibbons-Hawking metric hasthe following form ds = H ( y, r, θ ) − / (cid:0) − dt + dx + dx + dx + dx + dx (cid:1) ++ H ( y, r, θ ) / (cid:0) dy + ds GH (cid:1) (5.1)with field strength components F ψφry = α θ ) ∂H∂θF ψφθy = − α r sin( θ ) ∂H∂rF ψφθr = α r sin( θ ) V ( r, θ ) ∂H∂y . (5.2)16e consider the M5-brane which corresponds to α = +1; the α = − H ( y, r, θ ) is asolution to the differential equation2 r sin θV ǫ ( r, θ ) ∂H∂r + cos θV ǫ ( r, θ ) ∂H∂θ + r sin θ ∂ H∂y + sin θV ǫ ( r, θ ) { ∂ H∂θ + r ∂ H∂r } = 0 . (5.3)This equation is straightforwardly separable upon substituting H ( y, r, θ ) = 1 + Q M Y ( y ) R ( r, θ ) (5.4)where Q M is the charge on the M5-brane. The solution to the differential equation for Y ( y )is Y ( y ) = cos( cy + ς ) (5.5)and the differential equation for R ( r, θ ) is the same equation as (4.8). Hence the mostgeneral M5-brane function (corresponding to embedded Gibbons-Hawking space with k = 2and ǫ = 0) is given by H ( y, r, θ ) = 1 + Q M Z ∞ dc Z ∞ dM cos( cy + ς ) ×× ( ˜ H C ( µ ) { f c,M + f ′ c,M ln (cid:12)(cid:12)(cid:12) − µa (cid:12)(cid:12)(cid:12) } δ a,µ + f ′ c,M ∞ X n =0 b n,µ (1 − µµ ) n ) ×× ( ˜ H C ( λ ) { g c,M + g ′ c,M ln (cid:12)(cid:12)(cid:12)(cid:12) − λa (cid:12)(cid:12)(cid:12)(cid:12) } δ a,λ + g ′ c,M ∞ X n =0 d n,λ (1 − λλ ) n ) . (5.6)Similar result holds for embedded Gibbons-Hawking space with k = 2 and ǫ = 0. Thesolution (8.10) depends on four combinations of constants in form of f g, f ′ g, f g ′ and f ′ g ′ which each combination should have dimension of inverse length. Hence, the functional formof each constant could be considered as an expansion of the form c / β M β where β ∈ Z + .As with M2-brane case, reducing (5.1) to ten dimensions gives the following NSNS dilatonΦ = 34 ln (cid:26) H / ( y, r, θ ) V ǫ ( r, θ ) (cid:27) . (5.7)The NSNS field strength of the two-form associated with the NS5-brane, is given by H (3) = F φyrψ dφ ∧ dy ∧ dr + F φyθψ dφ ∧ dy ∧ dθ + F φrθψ dφ ∧ dr ∧ dθ (5.8)where the different components of 4-form F , are given by ( 5.2). The RR fields are C (1) = ω ( r, θ ) (5.9) A αβγ = 0 (5.10)17here C α is the field associated with the D6-brane, and the metric in ten dimensions is givenby: ds = V − / ǫ ( r, θ ) (cid:0) − dt + dx + dx + dx + dx + dx (cid:1) + H ( y, r, θ ) V − / ǫ ( r, θ ) dy ++ H ( y, r, θ ) V / ǫ ( r, θ ) (cid:0) dr + r d Ω (cid:1) . (5.11)From (5.8), (5.9), (5.10) and the metric (5.11), we can see the above ten dimensional metricis an NS5 ⊥ D6(5) brane solution. We have explicitly checked the BPS 10-dimensional metric(5.11), with the other fields (the dilaton (5.7), the 1-form field (5.9), and the NSNS fieldstrength (5.8)) make a solution to the 10-dimensional supergravity equations of motion. Aswe discuss in section 7, the solution (5.1) preserves 1/4 of the supersymmetry.
We can also embed two four dimensional Gibbons-Hawking spaces into the eleven dimen-sional membrane metric. Here we consider the embedding of two double-NUT (or twodouble-center Eguchi-Hanson) metrics of the form (4.1) with ǫ = 0 (or ǫ = 0). The M-branemetric is ds = H ( y, α, r, θ ) − / (cid:0) − dt + dx + dx (cid:1) + H ( y, α, r, θ ) / (cid:0) ds GH (1) + ds GH (2) (cid:1) (6.1)where ds GH ( i ) , i = 1 , r, θ, φ, ψ ) and( y, α, β, γ ). The non-vanishing components of four-form field are F tx x x = − H ∂H ( y, α, r, θ ) ∂x (6.2)where x = r, θ, y, α . The metric (6.1) and four-form field (6.2) satisfy the eleven dimensionalequations of motion if2 ry sin( α ) sin( θ ) { V ǫ ( r, θ ) y ∂H∂r + V ǫ ( y, α ) r ∂H∂y } ++ sin( α ) y cos( θ ) V ǫ ( r, θ ) ∂H∂θ + r sin( θ ) cos( α ) V ǫ ( y, α ) ∂H∂α ++ r sin( α ) y sin( θ ) { V ǫ ( r, θ ) ∂ H∂r + V ǫ ( y, α ) ∂ H∂y } ++ sin( θ ) sin( α ) { r V ǫ ( y, α ) ∂ H∂α + y V ǫ ( r, θ ) ∂ H∂θ } = 0 (6.3)where V ǫ ( y, α ) = ǫ + n y + n √ y + b +2 by cos( α ) . The equation (6.3) is separable if we set H ( y, α, r, θ ) = 1 + Q M R ( y, α ) R ( r, θ ). This gives two equations2 x i ∂R i ∂x i + x i ∂ R i ∂x i + cos y i sin y i ∂R i ∂y i + ∂ R i ∂ y i = u i c x i V ǫ ( x i , y i ) R i (6.4)18here ( x , y ) = ( y, α ) and ( x , y ) = ( r, θ ). There is no summation on index i and u =+1 , u = −
1, in equation (6.4). We already know the solutions to the two differentialequations (6.4) as given by (4.39) and (4.49), hence the most general solution to (6.3) is H ( y, α, r, θ ) = 1 + Q M Z ∞ dc Z ∞ dM Z ∞ d ˜ M R ( y, α ) ˜ R ( r, θ ) . (6.5)We note that changing c to ic in (6.4) makes a second solution given by replacements R ( y, α )to ˜ R ( y, α ) and ˜ R ( r, θ ) to R ( r, θ ) in (6.5). However the second solution is not independent ofthe first one.We can choose to compactify down to ten dimensions by compactifying on either ψ or γ coordinates. In the first case, we find the type IIA string theory with the NSNS fieldsΦ = 34 ln (cid:18) H / V ǫ ( r, θ ) (cid:19) (6.6) B µν = 0 (6.7)and RR fields C φ = ω ( r, θ ) (6.8) A tx x = H ( y, α, r, θ ) − . (6.9)The metric is given by ds = H ( y, α, r, θ ) − / V ǫ ( r, θ ) − / (cid:0) − dt + dx + dx (cid:1) ++ H ( y, α, r, θ ) / V ǫ ( r, θ ) − / (cid:0) ds GH (1) (cid:1) ++ H ( y, α, r, θ ) / V ǫ ( r, θ ) / (cid:0) dr + r (cid:0) dθ + sin ( θ ) dφ (cid:1)(cid:1) . (6.10)In the latter case, the type IIA fields are in the same form as (6.6), (6.7), (6.8), (6.9) and(6.10), just by replacements ( r, θ, φ, ψ ) ⇔ ( y, α, β, γ ). In either cases, we get a fully localizedD2/D6 brane system. We can further reduce the metric (6.10) along the γ direction of thefirst Gibbons-Hawking space. However the result of this compactification is not the sameas the reduction of the M-theory solution (6.1) over a torus, which is compactified type IIBtheory. The reason is that to get the compactified type IIB theory, we should compactify theT-dual of the IIA metric (6.10) over a circle, and not directly compactify the 10D IIA metric(6.10) along the γ direction. We note also an interesting result in reducing the 11D metric(6.1) along the ψ (or γ ) direction of the GH (1) (or GH (2)) in large radial coordinates. As y (or r ) → ∞ the transverse geometry in (6.1) locally approaches R ⊗ S ⊗ GH (2) (or GH (1) ⊗ R ⊗ S ). Hence the reduced theory, obtained by compactification over the circleof the Gibbons-Hawking, is IIA. Then by T-dualization of this theory (on the remaining S of the transverse geometry), we find a type IIB theory which describes the D5 defects. Thesolutions (6.1) (with ǫ = 0 or ǫ = 0) are BPS and also preserve 1/4 of the supersymmetry,as we show in the next section. 19 Supersymmetries of the Solutions
In this section, we explicitly show all our BPS solutions presented in the previous sectionspreserve 1/4 of the supersymmetry. Generically a configuration of n intersecting branespreserves n of the supersymmetry. In general, the Killing spinors are projected out byproduct of Gamma matrices with indices tangent to each brane. If all the projections areindependent, then n -rule can give the right number of preserved supersymmetries. On theother hand, if the projections are not independent then n -rule can’t be trusted. There aresome important brane configurations when the number of preserved supersymmetries is morethan that by n -rule [19, 20].As we briefly mentioned in the introduction, the number of non-trivial solutions to theKilling spinor equation ∂ M ε + 14 ω abM Γ ab ε + 1144 Γ npqrM F npqr ε −
118 Γ pqr F mpqr ε = 0 (7.1)determine the amount of supersymmetry of the solution where the indices M, N, P, ... areeleven dimensional world indices and a, b, ... are eleven dimensional non-coordinate tangentspace indices. The connection one-form is given by ω ab = Γ abc ˆ θ b , in terms of Ricci rotationcoefficients Γ abc and non-coordinate basis ˆ θ a = e aM dx M where e Ma are vielbeins. The elevendimensional M-brane metrics (2.3) and (2.7) are ds = η ab ˆ θ a ⊗ ˆ θ b in non-coordinate basis.The connection one-form ω ab satisfies torsion- and curvature-free Cartan’s structure equations d ˆ θ a + ω ab ∧ ˆ θ b = 0 (7.2) dω ab + ω ac ∧ ω cb = 0 (7.3)In (7.1), Γ a matrices make the Clifford algebra (cid:8) Γ a , Γ b (cid:9) = − η ab . (7.4)and Γ ab = Γ [ a Γ b ] . Moreover, Γ M ...M k = Γ [ M . . . Γ M n ] . A representation of the algebra isgiven in appendix C.For our purposes, we use the thirty two dimensional representation of the Clifford algebra(7.4), given by [21] Γ i = " − e Γ i e Γ i ( i = 1 . . .
8) (7.5)Γ = (cid:20) − (cid:21) (7.6)Γ ⋆ = (cid:20) (cid:21) (7.7)Γ = − Γ ⋆ (7.8)20e note Γ ⋆ = ǫ ⋆ = 1. For a given Majorana spinor ǫ , its conjugate isgiven by ¯ ǫ = ǫ T Γ . Moreover we notice that Γ Γ a a ··· a n is symmetric for n = 1 , , n = 0 , ,
4. The e Γ i ’s in (7.5), the sixteen dimensional representation ofthe Clifford algebra in eight dimensions, are given by [22] e Γ i = (cid:20) L i L i (cid:21) ( i = 1 . . .
7) (7.9) e Γ = (cid:20) −
11 0 (cid:21) (7.10)in terms of L i , the left multiplication by the imaginary octonions on the octonions. Theimaginary unit octonions satisfy the following relationship o i · o j = − δ ij + c ijk o k (7.11)where c ijk is totally skew symmetric and its non-vanishing components are given by c = c = c = c = c = c = c = 1 . (7.12)We take the L i to be the matrices such that the relation (7.11) holds. In other words,given a vector v = ( v , v i ) in R , we write ˆ v = v + v j o j , where the effect of left multiplicationis o i (ˆ v ) = v o i − v i + c ijk v j o k , we then construct the 8 × L i ) ξζ by requiring o i (ˆ v ) =( L i ) ξζ o ξ v ζ , where ξ, ζ = 0 , , . . .
7. We consider first the M2-brane solutions considered insection 4, for example (4.54). Substituting ε = H − / ǫ in the Killing spinor equations (7.1)yields solutions that Γ tx x ǫ = − ǫ (7.13)and so at most half the supersymmetry is preserved due to the presence of the brane. Wenote that if we multiply all the components of four-form field strength, given in (2.4),(2.5)and (2.6), by −
1, then the projection equation (7.13) changes to Γ tx x ǫ = + ǫ . The otherremaining equations in (7.1), arising from the left-over terms from ∂ M ǫ + ω Mab Γ ab ǫ portion, In what follows in this section, we show the non-coordinate tangent space indices of Γ’s by t, x , x , · · · , φ, ψ , to simplify the notation. ∂ α ǫ −
12 Γ yα ǫ = 0 (7.14) ∂ α ǫ −
12 [sin( α )Γ yα + cos( α )Γ α α ] ǫ = 0 (7.15) ∂ α ǫ −
12 [sin( α )(sin( α )Γ yα + cos( α )Γ α α ) + cos( α )Γ α α ] ǫ = 0 (7.16) ∂ ψ ǫ + 14 r sin θ (cid:20) − V ( ∂ω∂θ Γ θφ + r ∂ω∂r Γ rφ ) + r sin θ ( ∂V∂θ Γ ψφ + r ∂V∂r Γ ψr ) (cid:21) ǫ = 0 (7.17) ∂ θ ǫ + 14 r sin θ (cid:20) − V ∂ω∂θ Γ ψφ + r sin θV ( r ∂V∂r − V )Γ rθ (cid:21) ǫ = 0 (7.18) ∂ φ ǫ + 14 (cid:20) ∂ ( V ω ) ∂r Γ ψr − rV sin θ ( V ω ∂ω∂r − r sin θ ∂V∂r + 2 rV sin θ )Γ rφ − r V sin θ ( V ω ∂ω∂θ − r sin θ ∂V∂θ + 2 r V sin θ cos θ )Γ θφ + 14 r ∂ ( V ω ) ∂θ Γ ψθ (cid:21) ǫ = 0 . (7.19)We can solve the first three equations, (7.14), (7.15) and (7.16) by using the Lorentz trans-formation ǫ = exp n α yα o exp n α α α o exp n α α α o η. (7.20)where η is independent of α , α and α .To solve equation (7.17), we note that the equation can be written as ∂ ψ η + (cid:2) f ( r, θ )(Γ θφ + Γ ψr ) + g ( r, θ )(Γ rφ − Γ ψθ ) (cid:3) η = 0 (7.21)where f ( r, θ ) = ( r + a + 2 ar cos θ ) / n + an r cos θ + n r r + a + 2 ar cos θ ) / { ( r + a + 2 ar cos θ ) / ( r + n ) + n r } (7.22) g ( r, θ ) = an r sin θ r + a + 2 ar cos θ ) / { ( r + a + 2 ar cos θ ) / ( r + n ) + n r } (7.23)So, the solution to equation (7.21) satisfiesΓ ψrθφ η = η (7.24)This equation eliminates another half of the supersymmetry provided η is independent of ψ ,too. With this projection operator, (7.18) and (7.19) can be solved to give η = exp (cid:26) − θ ˆ ψ ˆ φ (cid:27) exp (cid:26) φ ˆ θ ˆ φ (cid:27) λ (7.25)22here λ is independent of θ and φ . Finally, we conclude due to two projections (7.13) and(7.24), embedding Gibbons-Hawking space in M2 metric preserves 1/4 of supersymmetry.Next, we consider the M5-brane solutions considered in section 5, given by (5.6). Substi-tuting ε = H − / ǫ in the Killing spinor equations (7.1) yieldsΓ tx x x x x ǫ = ǫ (7.26)We note that for the anti-M5-brane α = − tx x x x x ǫ = − ǫ . Moreover, we get three equations for ǫ that are given exactly byequations (7.17), (7.18) and (7.19). The solutions to these three equations implyΓ ψrθφ ǫ = ǫ (7.27)and ǫ = exp (cid:26) − θ ˆ ψ ˆ φ (cid:27) exp (cid:26) φ ˆ θ ˆ φ (cid:27) ξ (7.28)where ξ is independent of θ and φ .So, the two projection operators given by (7.26) and (7.27) show M5-brane solutionspreserve 1/4 of supersymmetry.Finally we consider how much supersymmetry could be preserved by the solutions (6.1)with metric function (6.5), given in section 6.As in the case of M2-brane, we get the projection equationΓ tx x ǫ = − ǫ (7.29)that remove half the supersymmetry, after substituting ε = H − / ǫ into the Killing spinorequations (7.1). The remaining equations could be solved by consideringΓ ψrθφ ǫ = ǫ (7.30)Γ α yα α ǫ = ǫ (7.31)However, the three projection operators in (7.29),(7.30) and (7.31) are not independent, sincetheir indices altogether cover all the non-coordinate tangent space. Hence, we have only twoindependent projection operators, meaning 1/4 of the supersymmetry is preserved. In this section we consider the decoupling limits for the various solutions we have presentedabove. The specifics of calculating the decoupling limit are shown in detail elsewhere (seefor example [23]), so we will only provide a brief outline here. The process is the same forall cases, so we will also only provide specific examples of a few of the solutions above.At low energies, the dynamics of the D2 brane decouple from the bulk, with the regionclose to the D6 brane corresponding to a range of energy scales governed by the IR fixed23oint [24]. For D2 branes localized on D6 branes, this corresponds in the field theory to avanishing mass for the fundamental hyper-multiplets. Near the D2 brane horizon ( H ≫ g Y M = g s ℓ − s = fixed. (8.1)In this limit the gauge couplings in the bulk go to zero, so the dynamics decouple there. Ineach of our cases above, we scale the coordinates y and r such that Y = yℓ s , U = rℓ s (8.2)are fixed (where Y and U , are used where appropriate). As an example we note that thiswill change the harmonic function of the D6 brane in the Gibbons-Hawking case to thefollowing (recall that to avoid any conical singularity, we should have n = n = n , hencethe asymptotic radius of the 11th dimension is R ∞ = n = g s ℓ s ) V ǫ ( U, θ ) = ǫ + g Y M N { U + 1 √ U + A + 2 AU cos θ } (8.3)where we rescale a to a = Aℓ s and generalize to the case of N D6 branes. We notice that themetric function H ( y, r, θ ) scales as H ( Y, U, θ ) = ℓ − s h ( Y, U, θ ) if the coefficients f c,M , f ′ c,M , · · · obey some specific scaling. The scaling behavior of H ( Y, U, θ ) causes then the D2-brane towarp the ALE region and the asymptotically flat region of the D6-brane geometry. As anexample, we calculate h ( Y, U, θ ) that corresponds to (4.54). It is given by h ( Y, U, θ ) = 32 π N g Y M Z ∞ dC Z ∞ d M J ( CY ) Y ×× ( ˜ H C (Ω , g Y M ) { F C, M + F ′ C, M ln (cid:12)(cid:12)(cid:12)(cid:12) − Ω A (cid:12)(cid:12)(cid:12)(cid:12) } δ A, Ω + F ′ C, M ∞ X n =0 b n, Ω (1 − ΩΩ ) n ) ×× ( ˜ H C (Λ , g Y M ) { G C, M + G ′ C, M ln (cid:12)(cid:12)(cid:12)(cid:12) − Λ A (cid:12)(cid:12)(cid:12)(cid:12) } δ A, Λ + G ′ C, M ∞ X n =0 d n, Λ (1 − ΛΛ ) n ) . (8.4)where we rescale c = C/ℓ s and M = M ℓ s . We notice that decoupling demands rescalingof the coefficients f c,M , f ′ c,M , · · · in (4.54) by f c,M = F C, M /ℓ s , f ′ c,M = F ′ C, M /ℓ s , · · · . In (8.4),Ω = √ U + A + 2 AU cos θ + U and Λ = √ U + A + 2 AU cos θ − U and we use ℓ p = g / s ℓ s to rewrite Q M = 32 π N ℓ p in terms of ℓ s given by Q M = 32 π N g Y M ℓ s .The respective ten-dimensional supersymmetric metric (4.55) scales as ds = ℓ s { h − / ( Y, U, θ ) V − / ǫ ( U, θ ) (cid:0) − dt + dx + dx (cid:1) ++ h / ( Y, U, θ ) V − / ǫ ( U, θ ) (cid:0) dY + Y d Ω (cid:1) ++ h / ( Y, U, θ ) V / ǫ ( U, θ )( dU + U d Ω ) } (8.5)24nd so there is only one overall normalization factor of ℓ s in the metric (8.5). This is theexpected result for a solution that is a supergravity dual of a QFT. The other M2-braneand supersymmetric ten-dimensional solutions, given by (4.51), (4.54), (6.5) and (6.10) havequalitatively the same behaviors in decoupling limit.We now consider an analysis of the decoupling limits of M5-brane solution given by metricfunction (5.6).At low energies, the dynamics of IIA NS5-branes will decouple from the bulk [25]. Nearthe NS5-brane horizon ( H >> g s → ℓ s = fixed. (8.7)In these limits, we rescale the radial coordinates such that they can be kept fixed Y = yg s ℓ s , U = rg s ℓ s . (8.8)This causes the harmonic function of the D6-brane for the Gibbons-Hawking solution (5.11),change to V ǫ ( r, θ ) = ǫ + N ℓ s { U + 1 √ U + A + 2 AU cos θ } ≡ V ǫ ( U, θ ) (8.9)where we generalize to N D6-branes and rescale a = Aℓ s g s .We can show the harmonic function for the NS5-branes (5.6) rescales according to H ( Y, U, θ ) = g − s h ( Y, U, θ ). In fact, we have H ( Y, U, θ ) = πN ℓ s g s Z ∞ dC Z ∞ d M cos( CY + ζ ) ×× ( ˜ H C (Ω , ℓ s ) { F C, M + F ′ C, M ln (cid:12)(cid:12)(cid:12)(cid:12) − Ω A (cid:12)(cid:12)(cid:12)(cid:12) } δ A, Ω + F ′ C, M ∞ X n =0 b n, Ω (1 − ΩΩ ) n ) ×× ( ˜ H C (Λ , ℓ s ) { G C, M + G ′ C, M ln (cid:12)(cid:12)(cid:12)(cid:12) − Λ A (cid:12)(cid:12)(cid:12)(cid:12) } δ A, Λ + G ′ C, M ∞ X n =0 d n, Λ (1 − ΛΛ ) n ) . (8.10)where we use ℓ p = g / s ℓ s to rewrite Q M = πN ℓ p as πN g s ℓ s . To get (8.10), we rescale c = C/ ( g s ℓ s ), M = M g s ℓ s and a = Ag s ℓ s such that h ( Y, U, θ ) doesn’t have any g s dependence.In decoupling limit, the ten-dimensional metric (5.11) becomes, ds = V − / ǫ ( U, θ ) (cid:0) − dt + dx + dx + dx + dx + dx (cid:1) + ℓ s { h ( Y, U, θ ) V − / ǫ ( U, θ ) dY ++ h ( Y, U, θ ) V / ǫ ( U, θ ) (cid:0) dU + U d Ω (cid:1) } . (8.11)25n the limit of vanishing g s with fixed l s (as we did in (8.6) and (8.7)), the decoupledfree theory on NS5-branes should be a little string theory [26] (i.e. a 6-dimensional non-gravitational theory in which modes on the 5-brane interact amongst themselves, decoupledfrom the bulk). We note that our NS5/D6 system is obtained from M5-branes by compact-ification on a circle of self-dual transverse geometry. Hence the IIA solution has T-dualitywith respect to this circle. The little string theory inherits the same T-duality from IIAstring theory, since taking the limit of vanishing string coupling commutes with T-duality.Moreover T-duality exists even for toroidally compactified little string theory. In this case,the duality is given by an O ( d, d, Z ) symmetry where d is the dimension of the compactifiedtoroid. These are indications that the little string theory is non-local at the energy scale l − s and in particular in the compactified theory, the energy-momentum tensor can’t be defineduniquely [27].As the last case, we consider the analysis of the decoupling limits of the IIB solutionthat can be obtained by T-dualizing the compactified M5-brane solution (5.1). The typeIIA NS5 ⊥ D6(5) configuration is given by the metric (5.11) and fields (5.7), ( 5.8), (5.9) and(5.10).We apply the T-duality [28] in the x − direction of the metric ( 5.11), that yields givesthe IIB dilaton field e Φ = 12 ln H ˜ f (8.12)the 10D type IIB metric, as b ds = V − / ǫ ( r, θ ) (cid:0) − dt + V ǫ ( r, θ ) dx + dx + dx + dx + dx (cid:1) ++ H ( y, r, θ ) V − / ǫ ( r, θ ) dy + H ( y, r, θ ) V / ǫ ( r, θ ) (cid:0) dr + r d Ω (cid:1) . (8.13)The metric (8.13) describes a IIB NS5 ⊥ D5(4) brane configuration (along with the dualizeddilaton, NSNS and RR fields).At low energies, the dynamics of IIB NS5-branes will decouple from the bulk. Near theNS5-brane horizon (
H >> g Y M = ℓ s = fixed (8.14)We rescale the radial coordinates y and r as in (8.8), such that their corresponding rescaledcoordinates Y and U are kept fixed. The harmonic function of the D5-brane is V ǫ ( r, θ ) = ǫ + N g Y M { U + 1 √ U + A + 2 AU cos θ } (8.15)where N is the number of D5-branes.The harmonic function of the NS5 ⊥ D5 system (8.13), rescales according to H ( Y, U, θ ) =26 − s h ( Y, U, θ ), where h ( Y, U, θ ) = πN g Y M Z ∞ dC Z ∞ d M cos( CY + ζ ) ×× ( ˜ H C ( µ, g Y M ) { F C, M + F ′ C, M ln (cid:12)(cid:12)(cid:12)(cid:12) − Ω A (cid:12)(cid:12)(cid:12)(cid:12) } δ A, Ω + F ′ C, M ∞ X n =0 b n, Ω (1 − ΩΩ ) n ) ×× ( ˜ H C ( λ, g Y M ) { G C, M G ′ C, M ln (cid:12)(cid:12)(cid:12)(cid:12) − Λ A (cid:12)(cid:12)(cid:12)(cid:12) } δ A, Λ + G ′ C, M ∞ X n =0 d n, Λ (1 − ΛΛ ) n ) . (8.16)In this case, the ten-dimensional metric (8.13), in the decoupling limit, becomes e ds = V − / ǫ ( U, θ ) (cid:0) − dt + V ǫ ( U, θ ) dx + dx + dx + dx + dx (cid:1) ++ g Y M h ( Y, U, θ ) { V − / ǫ ( U, θ ) dY + + V / ǫ ( U, θ ) (cid:0) dU + U d Ω (cid:1) } . (8.17)The decoupling limit illustrates that the decoupled theory in the low energy limit is superYang-Mills theory with g Y M = ℓ s . In the limit of vanishing g s with fixed l s , the decoupledfree theory on IIB NS5-branes (which is equivalent to the limit g s → ∞ of decoupled S-dualof the IIB D5-branes) reduces to a IIB (1,1) little string theory with eight supersymmetries. The central thrust of this paper is the explicit and exact construction of supergravity solu-tions for fully localized D2/D6 and NS5/D6 brane intersections without restricting to thenear core region of the D6 branes. Unlike all the other known solutions, the novel featureof these solutions is the dependence of the metric function to three (and four) transversecoordinates. These exact solutions are new M2 and M5 brane metrics that are presented inequations (4.39), (4.49), (4.51), (4.54), (5.6) and (6.5) which are the main results of this pa-per. The common feature of all of these solutions is that the brane function is a convolutionof an decaying function with a damped oscillating one. The metric functions vanish far fromthe M2 and M5 branes and diverge near the brane cores.Dimensional reduction of the M2 solutions to ten dimensions gives us intersecting IIAD2/D6 configurations that preserve 1/4 of the supersymmetry. For the M5 solutions, dimen-sional reduction yields IIA NS5/D6 brane systems overlapping in five directions. The lattersolutions also preserve 1/4 of the supersymmetry and in both cases the reduction yieldsmetrics with acceptable asymptotic behaviors.We considered the decoupling limit of our solutions and found that D2 and NS5 branescan decouple from the bulk, upon imposing proper scaling on some of the coefficients in theintegrands.In the case of M2 brane solutions; when the D2 brane decouples from the bulk, the theoryon the brane is 3 dimensional N = 4 SU (N ) super Yang-Mills (with eight supersymmetries)27oupled to N massless hypermultiplets [29]. This point is obtained from dual field theoryand since our solutions preserve the same amount of supersymmetry, a similar dual fielddescription should be attainable.In the case of M5 brane solutions; the resulting theory on the NS5-brane in the limit ofvanishing string coupling with fixed string length is a little string theory. In the standardcase, the system of N NS5-branes located at N D6-branes can be obtained by dimensionalreduction of N N coinciding images of M5-branes in the flat transverse geometry. In thiscase, the world-volume theory (the little string theory) of the IIA NS5-branes, in the ab-sence of D6-branes, is a non-local non-gravitational six dimensional theory [30]. This theoryhas (2,0) supersymmetry (four supercharges in the representation of Lorentz symmetry Spin (5 , Spin (4) remnant of the original ten dimensional Lorentzsymmetry. The presence of the D6-branes breaks the supersymmetry down to (1,0), witheight supersymmetries. Since we found that some of our solutions preserve 1/4 of super-symmetry, we expect that the theory on NS5-branes is a new little string theory. ByT-dualization of the 10D IIA theory along a direction parallel to the world-volume of theIIA NS5, we find a IIB NS5 ⊥ D5(4) system, overlapping in four directions. The world-volumetheory of the IIB NS5-branes, in the absence of the D5-branes, is a little string theory with(1,1) supersymmetry. The presence of the D5-brane, which has one transverse direction rel-ative to NS5 world-volume, breaks the supersymmetry down to eight supersymmetries. Thisis in good agreement with the number of supersymmetries in 10D IIB theory: T-dualitypreserves the number of original IIA supersymmetries, which is eight. Moreover we con-clude that the new IIA and IIB little string theories are T-dual: the actual six dimensionalT-duality is the remnant of the original 10D T-duality after toroidal compactification.A useful application of the exact M-brane solutions in our paper is to employ them assupergravity duals of the NS5 world-volume theories with matter coming from the extrabranes. More specifically, these solutions can be used to compute some correlation functionsand spectrum of fields of our new little string theories.In the standard case of A k − (2,0) little string theory, there is an eleven dimensional holo-graphic dual space obtained by taking appropriate small g s limit of an M-theory backgroundcorresponding to M5-branes with a transverse circle and k units of 4-form flux on S ⊗ S . Inthis case, the supergravity approximation is valid for the (2,0) little string theories at large k and at energies well below the string scale. The two point function of the energy-momentumtensor of the little string theory can be computed from classical action of the supergravityevaluated on the classical field solutions [26].Near the boundary of the above mentioned M-theory background, the string couplinggoes to zero and the curvatures are small. Hence it is possible to compute the spectrumof fields exactly. In [27], the full spectrum of chiral fields in the little string theories wascomputed and the results are exactly the same as the spectrum of the chiral fields in thelow energy limit of the little string theories. Moreover, the holographic dual theories can beused for computation of some of the states in our little string theories.We conclude with a few comments about possible directions for future work. Investiga-tion of the different regions of the metric (5.1) or alternatively the 10D string frame metric288.11) with a dilaton (also for other considered EH and TB cases) for small and large Higgsexpectation value U would be interesting, as it could provide a means for finding a holo-graphical dual relation to the new little string theory we obtained. Moreover, the Penroselimit of the near-horizon geometry may be useful for extracting information about the highenergy spectrum of the dual little string theory [31]. The other open issue is the possibilityof the construction of a pp-wave spacetime which interpolates between the different regionsof the our new IIA NS5-branes. Moreover, it would be interesting (and of course very com-plicated) to find the exact analytic solutions for the brane functions with the embeddedGibbons-Hawking spaces with k > Acknowledgments
This work was supported by the Natural Sciences and Engineering Research Council ofCanada.
A The Heun-C functions
The Heun-C function H C ( α, β, γ, δ, λ, z ) is the solution to the confluent Heun’s differentialequation [32] H ′′ C + ( α + β + 1 z + γ + 1 z − H ′ C + ( µz + νz − H C = 0 (A.1)where µ = α − β − γ + αβ − βγ − λ and ν = α + β + γ + αβ + βγ + δ + λ . The equation (A.1) has two regularsingular points at z = 0 and z = 1 and one irregular singularity at z = ∞ . The H C function isregular around the regular singular point z = 0 and is given by H C = Σ ∞ n =0 h n ( α, β, γ, δ, λ ) z n ,where h = 1. The series is convergent on the unit disk | z | < h n aredetermined by the recurrence relation h n = Θ n h n − + Φ n h n − (A.2)where we set h − = 0 andΘ n = 2 n ( n −
1) + (1 − n )( α − β − γ ) + 2 λ − αβ + βγ n ( n + β ) (A.3)Φ n = α ( β + γ + 2( n − δ n ( n + β ) . (A.4)29 Coefficients of Series in (4.39)
Here we list some coefficients that appear in (4.39) b ,µ >a = 1 b ,µ >a = − µ b ,µ >a = {− µ ( µ − a ) + c ( ǫµ + 4 M + 2 N + µ )8( µ − a ) } µ b ,µ >a = { c ( ǫµ + 8 µ M + 3 N + µ + N + a + ǫµ a )12( µ − a ) + − µ − a − c ǫµ + c ǫµ a − c M µ + 4 c M a − c N + µ + 2 c N + µ a µ − a ) } µ (B.1) d , | λ | a = ( − µ ) n Q n (B.4)and the functions Q depend on ǫ, µ , n, c, a, M, N + . For (B.2), the relation to Q ’s is d n, | λ |
75, the series isdivergent for 0 . < µ < . Representation of Clifford Algebra The gamma matrices satisfy the Clifford Algebra { Γ a , Γ b } = − η ab (C.1)where we are using the Lorentzian signature [ − , +1 , . . . , +1]. A representation of the algebra(C.1) is given by Γ ξ = γ ξ ⊗ Ξ+4 = γ ⊗ b Γ Ξ (C.3)where ξ = 0 , , , , , ..., SO (1 , 3) and SO (7). The Γ Ξ+4 (and b Γ Ξ ) satisfy the anticommutation relations { Γ Ξ+4 , Γ Ψ+4 } = { b Γ Ξ , b Γ Ψ } = − δ ΞΨ (C.4)where the b Γ Ξ ’s are given by b Γ = iγ ⊗ b Γ i = γ i ⊗ b Γ i +3 = iγ ⊗ σ i (C.5)in terms of the Pauli matrices σ i ( i = 1 , , γ = (cid:18) (cid:19) , and γ = iγ γ γ γ . References [1] E. Witten, Nucl. Phys. B443 (1995) 85.[2] M.J. Duff, J.T. Liu and R. Minasian, Nucl. Phys. B452 (1995) 261.[3] J.H. Schwarz, Phys. Lett. B367 (1996) 97.[4] A.A. Tseytlin, Nucl. Phys. B475 (1996) 149.[5] A. Loewy, Phys. Lett. B463 (1999) 41.[6] S.A. Cherkis and A. Hashimoto, JHEP (2002) 036.[7] R. Clarkson, A.M. Ghezelbash and R.B. Mann, JHEP (2004) 063; (2004)025.[8] A.M. Ghezelbash and R.B. Mann, JHEP (2004) 012.319] A.M. Ghezelbash, Phys. Rev. D74 (2006) 126004.[10] A.M. Ghezelbash, Phys. Rev. D77 (2008) 026006.[11] N. Itzaki, A.A. Tseytlin and S. Yankielowicz, Phys. Lett. B432 (1998) 298.[12] A. Hashimoto, JHEP (1999) 018.[13] S. Arapoglu, N.S. Deger and A. Kaya, Phys. Lett. B578 (2004) 203.[14] E. Cremmer, B. Julia, H. Lu and C.N. Pope, Nucl. Phys. B523 (1998) 73.[15] M.J. Duff, B.E.W. Nilsson and C.N. Pope, Phys. Rep. (1986) 1.[16] D.J. Smith, Class. Quant. Grav. (2003) R233.[17] T. Eguchi and A.J. Hanson, Phys. Lett. B74 (1978) 249.[18] A.A. Tseytlin, Class. Quant. Grav. (1997) 2085; Nucl. Phys. B487 (1997) 141.[19] I.R. Klebanov and A.A. Tseytlin, Nucl. Phys. B475 (1996) 179.[20] J.P. Gauntlett, D.A. Kastor and J. Traschen, Nucl. Phys. B478 (1996) 544.[21] J.P. Gauntlett, J.B. Gutowski and S. Pakis, JHEP (2003) 049.[22] J.M. Figueroa-O’Farrill, , Class. Quant. Grav. (2000) 2925.[23] J. Maldacena, Adv. Theor. Math. Phys. (1998) 231.[24] J. Maldacena, Int. J. Theor. Phys. (1999) 1113.[25] N. Itzhaki, J.M. Maldacena, J. Sonnenschein and S. Yankielowicz, Phys. Rev. D58 (1998) 046004.[26] S. Minwalla and N. Seiberg, JHEP (1999) 007.[27] O. Aharony, M. Berkooz, D. Kutasov and N. Seiberg, JHEP (1998) 004.[28] J.F.G. Cascales and A.M. Uranga, JHEP (2004) 021.[29] O. Pelc and R. Siebelink, Nucl. Phys. B558 (1999) 127.[30] N. Seiberg, Phys. Lett. B408 (1997) 98.[31] J. Gomis and H. Ooguri, Nucl. Phys.