PHYSICAL REVIEW D, VOLUME 70, 086007
Regular non-Abelian vacua in N 4, SO4 gauged supergravity Ali H. Chamseddine1 and Mikhail S. Volkov2 1
2
CAMS and Physics Department, American University of Beirut, Beirut, Lebanon LMPT CNRS-UMR 6083, Universite´ de Tours, Parc de Grandmont, 37200 Tours, France (Received 22 April 2004; published 13 October 2004)
We present a family of globally regular N 1 vacua in the D 4, N 4 gauged supergravity of Gates and Zwiebach. These solutions are labeled by the ratio of the two gauge couplings, and for 0 they reduce to the supergravity monopole previously used for constructing the gravity dual of N 1 super Yang-Mills theory. For > 0 the solutions are asymptotically anti- de Sitter, but with an excess of the solid angle, and they reduce exactly to anti-de Sitter for 1. Solutions with < 0 are topologically R1 S3 , and for 2 they become R1 S3 geometrically. All solutions with 0 can be promoted to D 11 to become vacua of M-theory. DOI: 10.1103/PhysRevD.70.086007
PACS numbers: 11.25.Yb, 04.65.+e, 11.25.Mj, 11.27.+d
Regular supersymmetric backgrounds in gauged supergravities (SUGRA) play an important role in the context of the AdS/CFT correspondence (see [1] for a review). Upon uplifting to higher dimensions they become vacua of string/M-theory and can be used for the dual description of strongly coupled gauge field theories. In this way, for example, the monopole solution [2] of the N 4 gauged SUGRA has given rise to the holographic interpretation of confining N 1 super Yang-Mills (SYM) theory [3]. Constructing such solutions, however, is rather involved. This is why, despite their importance, very few regular vacua of gauged SUGRA’s are known. In this paper we present a family of globally regular N 1 vacua that contains the monopole solution of Ref. [2] as special case. We work in the context of the N 4 gauged SUGRA in four dimensions. This theory exists in two inequivalent versions: the SU2 SU2 model of Freedman and Schwarz (FS) [4], whose solutions were studied in [2], and the SO4 model of Gates and Zwiebach (GZ) [5]. Both models contain in the bosonic sector the graviton g , dilaton , axion a, and two non-Abelian gauge fields Aa and Ba with gauge couplings eA and eB and with gauge group SU2 SU2. The important difference between the two models is that in the FS model the dilaton potential has no stationary points, while in the GZ model one has (when a 0) U
e2A 2 e 2 e2 4: 8
(1)
This potential does have stationary points, and, depending on the sign of eB =eA , its extremal value —the cosmological constant — can be positive or negative. If one sets a Ba 0, then the FS and GZ models coincide and admit as a solution the N 1 vacuum of Ref. [2] —the Chamseddine-Volkov (CV) monopole. If a Ba 0 but 0, then the two models are no longer the same, and we find that within the GZ model the CV monopole admits generalizations for any 0. These
1550-7998= 2004=70(8)=086007(5)$22.50
solutions are topologically different from the CV monopole, although approach the latter pointwise as ! 0. They can be uplifted to D 11, which may suggest a holographic interpretation for them. We consider the a Ba 0 truncation of the GZ model whose bosonic sector is described by the Lagrangian 1 1 1 a L R @ @ e2 F Fa U : (2) 4 2 4 a @ Aa @ Aa b c Here F abc A A with a 1; 2; 3, the scale is chosen such that eA 1, and U is given by (1). Consistency of setting the axion to zero requires that a F Fa 0. The theory also contains fermions: the gaugino and gravitino , whose supersymmetry (SUSY) variations for a purely bosonic background are
1 1 1 p @ e F e e ; 2 4 2 1 1 D p e F p e e : (3) 2 2 4 2 a and D @ 14 !; Here F 12 a F 1 a a 2 A ; the late ; and early ; Greek letters correspond to the spacetime and tangent space indices, respectively. The gamma matrices are subject to 12 diag1; 1; 1; 1. Introducing Pauli matrices of four different types, a ; b ; c ; d , which act in four different spaces, respectively, (such that, for example, a ; b 0), one can choose 0 ; a i 1 ; 2 a . The gauge group SU2 SU2 is generated by the anti-Hermitian matrices a and b , a ; b 0, a b abc c ab , and similarly for a . One can choose a i a , a i a . The generators a correspond to the field Ba that is truncated to zero. We wish to study fields that preserve some of the supersymmetries, in which case 0 for certain 0. We restrict to the static and spherically symmetric sector parametrized by coordinates t; #; #; ’ with
70 086007-1
2004 The American Physical Society
ALI H. CHAMSEDDINE AND MIKHAIL S. VOLKOV
ds24 e2V# dt2 e2(# d#2 r2 #d 2 ; i a Aa dx 1 w# T; dT ; 2
#:
PHYSICAL REVIEW D 70 086007
p BP 0 2 e( ; N
(4) (5)
p N w2 P2 ;
a
Here, with n sin# cos’; sin# sin’; cos#, one has d 2 dna dna and T a na . Imposing the isotropic gauge condition, r #e( , the spatial part of the metric becomes conformally flat, ds23 e2( dxa dxa , with xa #na . Choosing the tetrad . eV dt; e( dxa , the spin connection is obtained from d. ! ^ . 0. Setting in (3) 0 gives then the equations for the SUSY Killing spinors : p 0 2 2e( 0 2 n~ ~ 2e F e e ; p p 0 2i 2eV 1 @t 2e( V 0 2 n~ ~ e ; i w1 ~ i (0 n~ n~ ~ ~ 0 r 2 2 #
(6)
e( p 2e F e e 2 : ~ 4 2 ~ w2 1 #w0 n~ ~ n~ ~ Here F r2 #w0 ~ d and 0 d# ; also the usual operations for Euclidean three-vectors are assumed, for example n~ ~ na a and ~ @=@xa . Equations (6) comprise an overdetermined r system of 80 equations for 16 components of , whose consistency conditions we shall now study. Let A ; A ; A ; A be eigenspinors of 3 ; 3 ; 1 ; 2 , respectively, with the eigenvalues 1A , A 1; 2. We make the ansatz AB with A; B 1; 2 and AB U expit # # 2 n~ ~ 0
A
B : (7)
Here 2p1 2 1AB Q 2 where Q is a real constant, U exp 2i 3 ’ exp 2i 2 #, and 0 1 2 2 1 . In fact, AB is the most general spinor whose total angular momentum, including the orbital part plus spin plus isospin, is zero. Inserting (7) into (6), the variables decouple, and the system reduces to six linear algebraic and two ordinary differential equations for : A m Bm ;
0 A B :
(8)
Here the coefficients A m , Bm (m 1; 2; 3), A and B are functions of the background amplitudes V; (; r; w; and their derivatives. The algebraic equations can have a nontrivial solution if only their coefficients fulfill the conditions A m An Bm Bn , of which only five are independent. Introducing N #(0 1, these five conditions are equivalent to the first five of the following six relations:
P V 0 0 p e ( ; 2N
Q eV
w ; N
(9)
w0
rwB ( e ; N
r0 Ne( :
(10)
(11)
2 r p p Here P e 1w e e and B 2r 2 2 P 1 p 2r 2 e . These relations impose nonlinear differential constraints on the background functions ; w; V; ( and the parameter Q. Remarkably, although we have in (9) two equations for the same function V#, the first of these equations is in fact a differential consequence of the second one, and so the system is not overdetermined. The last equation in (11), added for the later convenience, is the identity (in the isotropic gauge used) implied by the definition of N. If Eqs. (9) –(11) are fulfilled, the algebraic equations in (8) are consistent with each other and express in terms of . Inserting this to the first differential constraint in (8) gives a linear differential equation for #, whose solution can be expressed in quadratures. The second differential constraint in (8) then turns out to be fulfilled identically, by virtue of Eqs. (9) –(11). The Bogomolnyi Eqs. (9) –(11) therefore provide the full set of consistency conditions that guarantee the existence of SUSY Killing spinors. One can now pass in these equations to an arbitrary gauge by treating (# as a free function subject to a gauge condition, while considering the second relation in (11) as the dynamical equation for the Schwarzschild radial function r#. Finding then gives the spinor AB for each choice of A; B, which finally corresponds to four independent SUSY Killing spinors, that is to N 1. Introducing y1 w, y2 , y3 V, y4 V lnr, the Bogomolnyi equations can also be written as
Yn
dyn @W Gnm m 0; @y d#
m; n 1; 2; 3; 4; (12)
where the target space metric is defined by Gmn r2 eV( diag2e2 =r2 ; 1; 1; 1 and the superpotential is W reV N with N given by (11). We note also that inserting the ansatz (4) and (5) to the Lagrangian R p (2) and integrating over the angles gives L gd#d’ m n 47Gmn Y Y total derivative. It then follows that solutions of Eqs. (12) are stationary points of the action. To integrate the Bogomolnyi Eqs. (9) –(11), the problem actually reduces to studying the closed subsystem (10) and (11) for ; w; r, since V can be obtained afterwards from (9). It seems that these equations can be resolved analytically only for some special values of , and we shall therefore resort to numerical analysis to study the generic case. First of all, we notice the following symmetry of the equations: if ; #; w#; r#; (# is a solution for some value of , then, for any ,
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REGULAR NON-ABELIAN VACUA IN N 4, SO4. . .
e2 ; # ; w#; e r#; (#
PHYSICAL REVIEW D 70 086007
(13)
is also a solution. To fix this symmetry, we impose the condition 0 0, with 0 being the value of at r 0. Since the sign of is invariant under (13), there are three separate cases to study: > 0, < 0, and 0. 0. Eqs. (10) and (11) reduce in this case to the system previously studied in the context of the halfgauged FS model [2]. Its solution is the CV monopole: # re #e2 q p w 2# coth# w2 1; ; w sinh# 2 (14) p ( and e 2e . In this case one has # as # ! 1. > 0. Choosing the Schwarzschild gauge, # r, the essential equations are given by (10) and (11) with e( 1=N. They determine r, wr, while (9) gives e2V Q2 N 2 e2 =w2 . We are interested in everywhere regular solutions, in which case Or2 , w 1 Or2 , N 1 Or2 for r ! 0. For r ! 1 one has ln b Or2 ; 2 r w b Or2 ; w w r 1 2bw2 1 r2 N 2 1 b2 Or2 ; 2 r 2
FIG. 1. Globally regular solutions with > 0. For 0:001 and 0 the amplitudes w are almost identical.
(15)
where b and w are integration constants. The numerical integration of the equations reveals for every value of > 0 a global solution r, wr with such boundary conditions in the interval r 2 0; 1; see Fig. 1. For all these solutions the dilaton varies in the finite range and runs into the stationary point of U for # ! 1. As ! 0, the asymptotic value of tends to infinity and the solutions approach pointwise the CV monopole (14). For 1 the solution can be obtained analytically: r 0, wr 1. Choosing Q 1 (the value of Q can be adjusted by rescaling the time), the metric assumes the standard anti-de Sitter (AdS) form, ds2 N 2 dt2 2 dr2 =N 2 r2 d 2 with N 2 1 r2 , while the gauge field vanishes. This solution actually has N 4 supersymmetry, since in this case there are additional SUSY Killing spinors not contained in the ansatz (7). Solutions with 1 describe globally regular N 1 deformations of the AdS. Their asymptotic form is determined by (15). Choosing the new radial coordinate r~ p r= 1 with 12 b2 and setting Q p w = 1 , the metric asymptotically approaches ds2 N 2 dt2 2
d~ r2 1 ~ r2 d 2 ; N2
(16)
~ r 2 3=2 where N 2 1 2M . r~ 2 and M bw 1=1 This is the Schwarzschild-AdS metric with an excess of
the solid angle —the area of the two-sphere of constant r~ being 471 ~ r2 in this geometry. The excess parameter and the ‘‘mass’’ M vanish only for 1, and they tend to infinity as ! 0. < 0. Solutions in this case are of the ‘‘bag of gold’’ type, since they have compact spatial sections with the S3 topology. The range of the Schwarzschild function r# is finite: it starts from zero at # 0 (‘‘north pole’’), increases up to a maximal value at some #e > 0 (‘‘equator’’), and then decreases down to zero at some # > #e (‘‘south pole’’). Since N r0 0 at the equator, Eqs. (9) – (11) become singular at this point. To desingularize them, p 2 we set P wS, thus obtaining N w 1 S weV [having chosen Q 1 in (9)]. Using this in (9) –(11) reduces the system to p r0 weV ( ; 0 2BSe(V ; (17) w0 rBe(V2 ; V 0 0 p Se(V ; 2 p 12 e . In addiwith S e2V2 1 and B pwS 2r tion, the relation P wS with P defined after Eqs. (11) gives the first integral for these equations. Equations (17) are completely regular at the equator, whose position coincides with zero of w. Imposing the gauge condition ( 0 and demanding the solution to be regular at the north pole gives r # O#3 , w 1 O#2 , O#2 , V O#2 for small #. At the south pole we find the formal power series solution to be generically
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ALI H. CHAMSEDDINE AND MIKHAIL S. VOLKOV
w : 4 x Ox6 ; 8 (18) : x2 Ox4 ; eV 4
r 3w x Ox3 ; j:jx e p Ox3 ; 3 2
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w w
with x # # 1=3 . Here # , w < 0, > 0 and are integration constants, and : 1 w2 =w2 2 . Solutions of Eqs. (17) in the interval # 2 0; # comprise a one-parameter family labeled by . These solutions are regular for # < # , while at # # the dilaton diverges and the curvature is singular too. For any , the profile of these solutions is qualitatively similar to the one shown in Fig. 2. As ! 0, one has # ! 1, r#e ! 1, and the solutions approach pointwise the CV monopole. For one special value, 2, one has w 1 and the expansions (18) are no longer valid. However, the solution can then be obtained p analytically: p V P S 0, w cos#, r 2 sin#, e( 2. This solution is globally regular, also at the south pole, the spatial geometry being that of the round S3 . One can write down the metric and gauge field as ds2 dt2 2.a .a ;
Aa .a ;
We have thus obtained the generalizations of the CV monopole (14) that comprise a two-parameter family labeled by and 0 . Although we have described explicitly only solutions with 0 0, those with 0 0 can be obtained by using the symmetry (13). The solutions generically have N 1, while for 1 the sypersymmetry is enhanced up to N 4. We know that the 0 solution can be uplifted to D 10 to become a vacuum of string theory [2]. It turns out that solutions with 0 can be uplifted to D 11 to become vacua of M-theory. The derivation of the GZ model via dimensional reduction of D 11 SUGRA was considered in Ref. [6]. Using formulas given there combined with the symmetry (13), every D 4 vacuum ds24 ; Aa ; considered above maps to the M-theory solution ds211 ; F 4 . The metric in D 11 is given by ds211 jj 2=3 ds24 8 1=3 ds27 with
ds27 d 2
(20)
(19)
where .a are invariant forms on S3 , d.a abc .b ^ .c 0. However, there is no SUSY enhancement in this case, and so N 1.
2 X c2 X a 1 .1 Aa s2 X .a2 2 : 2 X a a
p Here s2 =X c2 X with 1=X jje , and .a< < 1; 2 are invariant forms on two different three-spheres, d.a< abc .b< ^ .c< 0. The case > 0 corresponds to the reduction on S7 ; one has then c cos , s sin with 2 0; 7=2 , while for < 0 one reduces on the H 2;2 hyperbolic space [6], in which case c cosh , s sinh and 2 0; 1. The four-form in D 11 reads p 4 2 2 s2 c X 2 2 p ; F 4 2sc d ^ d jj X 2 (21) where 4 is the four-volume form and is the Hodge dual in the four-space. These formulas suggest a holographic interpretation for our solutions. For 0, according to [3], the dual theory is D 4, N 1 SYM. For 1 we have Aa 0, and the D 11 geometry is AdS4 S7 . This is the near-horizon limit of the M2 brane, and the dual theory is therefore D 3, N 4 SYM. This suggests that other solutions with positive 1 may describe some N 1 deformations of this theory. It would also be interesting to work out an interpretation for the compact solutions with < 0, especially for the one given by Eq. (19).
FIG. 2. The compact solutions with < 0. Unless 2, they are singular at the south pole where r vanishes and diverges.
M. S.V. thanks Michaela Petrini for discussions. Research of A. H. C. is supported in part by the National Science Foundation Grant No. Phys-0313416.
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REGULAR NON-ABELIAN VACUA IN N 4, SO4. . . [1] O. Aharony, S. Gubser, J. Maldacena, H. Ooguri, and Y. Oz, Phys. Rep., 323, 183 (2000). [2] A. H. Chamseddine and M. S. Volkov, Phys. Rev. Lett., 79, 3343 (1997); Phys. Rev. D 57, 6242 (1998). [3] J. Maldacena and C. Nunez, Phys. Rev. Lett., 86, 588 (2001).
PHYSICAL REVIEW D 70 086007 [4] D. Z. Freedman and J. H. Schwarz, Nucl. Phys., B137, 333 (1978). [5] S. J. Gates and B. Zwiebach, Phys. Lett., 123B, 200 (1983); B. Zwiebach, Nucl. Phys., B238, 367 (1984). [6] M. Cvetic, H. Lu, and C. N. Pope, Nucl. Phys., B574, 761 (2000); M. Cvetic, G.W. Gibbons, and C. N. Pope, hep-th/ 0401151.
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