Features of gravity-Yang-Mills hierarchies in d-dimensions

0.41 MB·English

by  Eugen Radu

Features of gravity-Yang-Mills hierarchies
in d dimensions
−
Eugen Radu†, Cristian Stelea‡ and D. H. Tchrakian†⋆
†
Department of Mathematical Physics, National University of Ireland, Maynooth, Ireland
6 ‡Department of Physics, University of Waterloo, Ontario N2L 3G1, Canada
0 ⋆
School of Theoretical Physics – DIAS,10 Burlington Road, Dublin 4, Ireland
0
2

n
a
J Abstract

3 Higher dimensional, direct analogues of the usual d = 4 Einstein–Yang-Mills (EYM) systems are
2 studied. TheseconsistofthegravitationalandYang-Millshierarchiesind=4pdimensionalspacetimes,
both consisting of 2p-form curvature terms only. Regular and black hole solutions are constructed in

1 2p+2≤d ≤4p, in which dimensions the total mass-energy is finite, generalising the familiar Bartnik-
v
McKinnon solutions in EYM theory for p=1. Ind=4p, thissimilarity is complete. Inthespecial case
8
ofd=2p+1,just beyondthefiniteenergy rangeofd,exactsolutions inclosed form arefound. Finally,
9
d=2p+1 purely gravitational systems, whose solutions generalise the static d=3 BTZ solutions, are
0
discussed.
1
0
6
0
1 Introduction
/
c
q
Regular and black hole solutions of gravitating gauge field systems have been studied for a long time since
-
r thepioneeringworkofBartnikandMcKinnon(BK)[1],whereregularsolutionsoftheYang–Mills(YM)and
g
Einstein–Hilbert (EH) systems were presented for d=4 spacetime dimensions. Other models of gravitating
:
v non-Abelian gauge fields usually share a number of common features with the BK particles, this example
i
X becoming canonical. After that discovery, there has been a great deal of numerical and analytical work on
various aspects of Einstein-Yang-Mills (EYM) theory and a variety of self-gravitating structures with non-
r
a Abelian fields have been found (for a review see [2]). These include hairy black holes solutions, which led
to certain revisions of some of the basic concepts of black hole physics based on the uniqueness and no-hair
theorems.
In the last years, gravitating solutions with nonabelian fields enjoyed renewed interest in the AdS/CFT
correspondence,and potentially in the study of Dp-branes of superstring theory. It is therefore pertinent to
extend the study of such solutions to higher dimensions other than that of d=4 spacetime.
Recently regular solutions were found in d = 6,7,8 in [3]; both regular and black hole solutions in five
dimensions were presented in [4], the d = 6,7,8 black hole solutions being discussed in [5]. These were
the classical solutions to systems consisting of higher order terms in both the YM and the gravitational
curvatures,which do in fact appear in the low energyeffective actionof string theory. Such terms employed
in[3,4]werethoseconstructedfromthetotallyantisymmetrised2p-formsinboththeYMandthe Riemann
curvature 2-forms, namely the YM and the gravitational hierarchies labelled by integers p. The p = 1
members in each case are the usual YM and the EH systems respectively, while the p = 2 gravitational
member is the familiar Gauss-Bonnet term.
These asymptotically flat configurations differ in many respects from the d=4 BK solutions. In partic-
ular, all d>4 solutions have only one node in the gauge potential. Also, the black holes exist only up to a
maximal value of the event horizon radius.
1
The salient property of these solutions [3, 4] is that they exist only for a limited range of the coupling
parameter α2 (which gives the strength of the gravitational interaction), and exhibit critical behaviours
in α2. This is not surprising since such composite models necessarily feature more than one dimensional
constant,analogoustogravitatingmonopoles[6]. (InthelattercasedimensionfulHiggsvacuumexpectation
value (VEV) plays this role.)
Morerecently,acompleteanalysisofthesecriticalbehavioursinα2 inthehigherdimensionalgravitating
YM systems was presented in [8]. The methods used in [8] were those of fixed point analysis developed
previously in [7] for proving the existence of the BK solutions [1] to the usual EYM system in d = 4.
Concerning the fairly complicated landscape of critical points here [8], we restrict our comments only to
pointing out that in addition to the two types of critical points occurring in gravitating monopoles [6],
namely the ones associated with the end point and the Reissner-Nordstrøm types, there is also a conical
fixed point, so described since at that scaling point an angular deficit appears in the gravitational metric
function. We refer the reader to [8] for a full account of these critical behaviours.
Theanalysisin[8]wascarriedoutformodelswhosegravitationalpartconsistedonlyofthep=1member
ofthegravitationalhierarchy,namelytheEinstein–Hilbert(EH)term. Thephysicalreasonforthiswasthat
the p 2 Gauss-Bonnet like terms play only a quantitative role, confirmed numerically in [3], and have no
≥
effect on the existence of finite mass solutions. By contrast in the gauge field sector, various combinations
of YM terms with p 1 essential for the existence of such solutions were employed, as required by scaling
≥
arguments. Hencehigherp 1YMtermswereemployedthere,oncedwasgreaterthan4. Thustheanalysis
≥
in [8] probed the effect of the higher p 2 YM terms.
≥
One of the two main aims of the present work is to probe the effect of higher order p 2 gravitational
≥
terms in EYM models. But we know from the numerical results of [3] that once the p = 1 (EH) term is
present, the effects of all higher p 2 terms become masked. It is therefore the case that if one wishes to
≥
study the effects of the p-th gravitationalterm, all other gravitationalterms with p <p must be excluded.
i
In practical terms, this means that we will restrict to models featuring only the p-th gravitationalterm.
Theotheroneofourtwomainaimsistoexhibitcertainqualitativesimilaritiesofthesolutionssupported
byafamilyofEYMmodels,thefirstmemberofwhichconsistsofthep=1gravitational(EH)andthep=1
YM terms, supporting the BK solutions. It follows naturally that the YM term we must choose interacting
with the (unique) p-th gravitational term, is the (unique) p-th YM term, in spacetime dimensions d = 4p.
Weexpecttoexhibitarecurring‘symmetry’inthepropertiesofthesolutionsofthisfamilyofmodelsmodulo
4p dimensions. In the models studied here the complicated critical features of the gravitating monopole [6]
and of the higher dimensional EYM solutions studied in [3, 4, 8] will be absent, since these are due to the
presence of at least one additional (to the gravitational and YM couplings) dimensionful constant, e.g. the
Higgs VEV, or, the higher curvature YM coupling constants.
In Section 2 we introduce the relevantgravitating YM models, and subject them to sphericalsymmetry.
These models will be characterised by two equal integers p = p = p specifying the model and the gauge
1 2
group, and the dimension of the spacetime d. In Section 3 we specialise the dimension d to the values
d=4p, d= 4p 1, ...,d= 2p+2, which are the only dimensions in which asymptotically flat finite energy
−
solutionsexist. Thesesolutionscanbeconstructedonlynumericallysothe resultspresentedinSection3are
mainly numerical. InSection 4 we specialise to spacetime dimensions d=2p+1 in which no asymptotically
flat, finite energy solutions exist for p = p = p. The interest in the latter, inspite of their pathological
1 2
properties,isthattheycanbegiveninclosedform,whichisanovelfeatureingravitatinggaugefieldtheory.
In addition to these we have supplied two Appendices, A and B, devoted to the extension of some the
models studied in Sections 3 and 4, to feature a non-vanishing cosmological constant Λ. Like the solutions
presented in Section 3, those in Appendices A and B are expressed in closed form. In Appendix A we
study the geometric properties of the gauge decoupled limit models in d = 2p+1 appearing in Section 3,
supplemented with acosmologicalterm, whichare the hierarchyofgravitysolutions pertaining to the static
BTZ solution [9], this last being the p=1 member. The Appendix B presents a generalisation of the exact
gravity-YMp =p =p solution in d=2p+1 dimensions for a nonzero Λ. In Section 5, we summarise our
1 2
results.
2
2 The model and imposition of spherical symmetry
The precise form of the gravitational and non-Abelian matter content of string effective actions beyond
leadingorderisstillanevolvingresearchsubject[10]. Ourpositioninstudyinghigherdimensionalgravitating
YM solutions, in [3, 4, 8], has been to choose what we have referred to as the superposition of the p-th
members of the gravitational and YM hierarchies, consisting of all possible higher order curvature forms
allowed in any given dimension.
The said gravitationalsystem is the superposition of all possible (p,q)-Ricci scalars R
(p,q)
P
κ
(P) = p e R , (1)
Lgrav 2p (p,q)
p=1
X
where R are constructed from the 2p-form R(2p) = R R ... R resulting from the totally antisym-
(p,q)
∧ ∧ ∧
metrised p-fold products of the Riemann curvature 2-forms R. We express R in the notation of [11] as
(p,q)
eR =εµ1µ2...µ2pν1ν2...νqen1en2...enq ε Rm1m2..m2p, (2)
(p,q) ν1 ν2 νq m1m2...m2pn1n2...nq µ1µ2....µ2p
where en are the Vielbein fields, e = det(en) in (1), and Rm1m2..m2p = R(2p) is the p-fold totally antisym-
ν ν µ1µ2....µ2p
metrised product of the Riemann curvature, in component notation. This leads to the definition of the p-th
Einstein tensor
1
G a =R a ea R . (3)
(p)µ (p)µ − 2p µ (p)
One reads from (2) that
d=2p+q.
Now the minimum nontrivial value of q is q = 1, since when q = 0 (2) is manifestly a total divergence,
namely the Euler-Hirzebruch density. Thus the highest nontrivial value of P in the superposition (1) is
1
P (d 1).
max
≤ 2 −
The corresponding superposition of the members of the YM hierarchy is
P
τ
(P) = p e TrF(2p)2 , (4)
LYM 2(2p)!
p=1
X
inwhichthe2p-formF(2p)isthep-foldantisymmetrisedproductF(2p)=F F ... F oftheYMcurvature
∧ ∧ ∧
2-formF. HerethemaximumvalueofP inthesuperposition(4)issimplyP 1d.Wedefinethep-stress
max ≤ 2
tensor pertaining to each term in (4) as
1
T(p) =Tr F(2p) F(2p) λ1λ2...λ2p−1 g Tr F(2p) F(2p)λ1λ2...λ2p . (5)
µν µλ1λ2...λ2p−1 ν − 4p µν λ1λ2...λ2p
The P =1 systems (1) and (4) are the usual EH gravity and YM theories, respectively.
In [3, 4, 8], some convenient superpositions (1) and (4) were selected to be studied, taking into account
the particular properties of the solutions that were being sought.
Itisouraimheretotruncateboth(1)and(4)suchthatonlyonetermispresentineach. Thisissothat,
analogously to the BK case, the solutions not be parameterized by one (e.g. α2) or more parameters. The
other criterionstatedinSection1 is that inthe appropriatedimensions higher membersof the gravitational
hierarchy be employed. So far, subject to respecting the Derrick scaling requirements, one can choose any
P =p in (1) and any P =p in (4). The final criterion is that of symmetry and analogy with the BK case
1 2
in d=4, namely that p =p =p in d=4p dimensions 1.
1 2
1Thereisyetanotherapparentlyunrelatedcoincidencehere. InEuclideansignature,thedouble-self-dualityoftheRiemann
2p form curvature leads to the vacuum p-Einsteinequations being satisfied, and yields a self-dual SO±(4p) YM 2p-formfield
strength.
3
Thus we define the gravitating YM models in 2p+2 d 4p spacetime dimensions, whose static finite
≤ ≤
energy solutions will be constructed numerically in the next section, by the Lagrangians
κ τ
=e p R + p TrF(2p)2 , 2 q 2p, (6)
L(p,d) 2p (p,q) 2(2p)! ≤ ≤
(cid:18) (cid:19)
κ and τ being constants giving the strength of the gravitational and YM interactions, respectively. Note
p p
thatthegravitationalpartin(6)isdescribedbythe(p,q)-RicciscalarsR inwhichq =d 2p,asdefined
(p,q)
−
in (2). For this system, the variational equations for the YM and gravitational fields are
τ
τ D (e Fµν)=0, κ G a = p T(p)a. (7)
p µ p (p)µ 2(2p)! µ
The model (6) directly generalises the usual EYM model in d=4 spacetime for d=4p, the latter being
the p=1 case. In that case the dimensions of the gravitational part of (6) are L 2p, versus the dimensions
−
of the YM part L 4p. This choice has been made on grounds of symmetry rather than physics, since the
−
leading terms in the effective action of string theory are the p = 1 EH and YM terms, which are both
excluded. Since the p = 1 EYM system is best known for its BK solutions, the solutions to the hierarchy
defined by (6) for d=4p might be described as the BK hierarchy.
The definition of the model becomes complete on specifying the gauge groups and their representations.
Adoptingthecriterionofemployingchiralrepresentations,bothforevenandforodddin(6),itisconvenient
to choose the gauge group to be SO(d¯). We shall therefore denote our representation matrices by SO (d¯),
where d¯=d and d¯=d 1 for even and odd d respectively. ±
In this unified notat−ion (for odd and even d), the spherically symmetric Ansatz for the SO (d¯)-valued
±
gauge fields then reads [3, 4]
A0 =0 , Ai = 1−rw(r) Σi(j±)xˆj , Σi(j±) =−41 1±2Γd¯+1 [Γi,Γj] . (8)
(cid:18) (cid:19) (cid:18) (cid:19)
The Γ’s denote the d¯-dimensional gamma matrices and 1, j =1,2,...,d 1 for both cases.
−
The spherically symmetric metric Ansatz we use is parameterized by two functions N(r) and σ(r)
ds2 = N(r)σ2(r)dt2 + N(r) 1dr2 + r2dΩ2 . (9)
− − (d−2)
Inserting (8) in (6), the resulting reduced one dimensional Lagrangianis
(d 2)! κ d
L(p,d) = (d−2−p−1)!σ(22pp−1dr rd−2p−1(1−N)p
(cid:2) (cid:3)
2
τ 1dw
+ rd 2 p Wp 1 (2p)N +(d 2p 1)W , (10)
− −
2·(2p)! " (cid:18)r dr(cid:19) − − #)
where we note
(1 w2)2
W = − . (11)
r4
3 A Bartnik-McKinnon hierarchy
It follows from the Derrick-type scaling arguments that static finite energy solutions to the field equations
of the model (6) exist only in spacetime dimensions
2p+2 d 4p, (12)
≤ ≤
andcan only be constructednumerically,which we presenthere. The d=4p family is referredto as the BK
hierarchy, but we study all possible cases allowed by (12). Note that for p=1, d=4 is the only possibility
allowed by (12).
4
1.25
σ(r)
k=1
1
k=2
0.75
k=3
0.5
0.25
0
-0.25 N(r)-1
-0.5
-0.75 d=8
-1
-1.5 -1 -0.5 0 0.5 1 1.5
log10(r)
1
w(r)
k=2
0.5
0
k=3
-0.5
k=1
d=8
-1
-1 -0.5 0 0.5 1 1.5 2
log10(r)
Figure 1. The profiles of the metric functions N(r), σ(r) and gauge function w(r) are presented for k−node
globally regular solutions of the p=2 gravity-Yang-Mills model in d=8 dimensions.
When looking for numerical solutions, it is convenient to redefine
22(p−1)τp 1/(4p−2)
r r
→ (2p!)κ
(cid:18) p (cid:19)
such that for any (d, p) no free parameter appears in the field equations.
The field equations implies the relations
d 1dw 2
rd 2p 1(1 N)p = rd 2Wp 1 2pN +(d 2p 1)W , (13)
− − − −
dr − r dr − −
(cid:20) (cid:18) (cid:19) (cid:21)
(cid:2) dσ(cid:3)
= 2r2p 3σ(1 N)1 pWp 1w2, (14)
− − − ′
dr −
5
1.25
σ(r)
k=1
1
k=2
0.75
k=3
0.5
0.25
0
-0.25 N(r)-1
-0.5
-0.75 d=6
-1
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
log10(r)
1
w(r)
k=2
0.5
0
k=3
-0.5
k=1
d=6
-1
-1 -0.5 0 0.5 1 1.5 2
log10(r)
Figure 2. One,two and threenodes solutions of the p=2 gravity-Yang-Mills model in d=6 dimensions.
for the metric function, and
d 1dw
rd 4σWp 1Nw = σrd 6Wp 2w(w2 1) 2(p 1)N( )2+(d 2p 1)W . (15)
− − ′ − −
dr − − r dr − −
(cid:18) (cid:19)
(cid:0) (cid:1)
for the gauge potential.
For globally regular solutions, finite energy requirements and regularity of the metric at r =0 give
w =1 br2+O(r4), N =1 4b2r2+O(r3), σ =σ (1+4b2r2)+O(r4), (16)
0
− −
where σ , b are two positive constant. The analysis of the field equations as r gives
0
→∞
M1/p c
N =1 0 +..., w = 1+ +..., (17)
− r(d 2p 1)/p ± r(2p d+1)/p
− − −
2p(4c2)pM(1−p)/p 1
σ =1+ 0 +...,
(d 2p 1)(3p 1)+2p2r2p+(d 2p 1)(3p 1)/p
− − − − − −
6
σ(r)
k=1
1
k=2
0.75
k=3
0.5
0.25
0
N(r)-1
-0.25
-0.5
-0.75 d=12
-1
-1 -0.5 0 0.5 1 1.5
log10(r)
1
w(r)
k=2
0.5
0
k=3
-0.5
k=1
d=12
-1
-1 -0.5 0 0.5 1 1.5 2
log10(r)
Figure 3. The profiles for the metric functions N(r), σ(r) and gauge function w(r) are presented for k−node
globally regular solutions for the p=3 gravity-Yang-Mills system in d=12 dimensions.
with M , c arbitrary constants. In analogy to the Einstein gravity, it is natural to identify the constant M
0 0
that enters the asymptotic of N with the mass M of the solutions, up to a d dependent factor. To see this
−
let us notice that (13) can be written as
d
[rd 2p 1(1 N)p] = rd 2T(p)t (18)
dr − − − − t
whichinthe(p=1,d=4)caseofstandardEinsteingravityleadstothe usualdefinitionofmass. HereT(p)t
t
is the ‘energy density’ component of the stress-energy tensor defined in (5). By keeping with this analogy,
‘the total energy’ inside a (d 2)-sphere S with radius r will be given by
d 2
− −
r 1 dw 2
dr dΩ rd 2Wp 1(r ) 2pN(r ) +(d 2p 1)W(r ) =Area(S )rd 2p 1(1 N)p(19)
Z0 ′ZSd−2 (d−2) ′ − − ′ (cid:20) ′ (cid:18)r′dr′(cid:19) − − ′ (cid:21) d−2 − − −
7
1.25
σ(r)
1
k=1
0.75
k=2
N(r)
0.5 k=3
0.25
d=8
r =1
h
0
0 0.5 1 1.5 2 2.5
log10(r)
1
w(r)
k=2
0.5
0
k=3
-0.5
k=1
d=8 r =1
h
-1
0 0.5 1 1.5 2 2.5 3
log10(r)
Figure4. One,twoandthreenodesblackholessolutionsofthep=2gravity-Yang-Millsmodelind=8dimensions.
where we made use of the regularity of the function N(r) at origin r = 0 (see (16)). Taking now the
limit r we find that the total energy is finite and given precisely by M=Area(S )M as claimed.
d 2 0
→∞ −
The equations of motion (13)-(15) have been solved for p = 2, 3 and a range of d. Considering first
globally regular configurations, we follow the usual approach and, by using a standard ordinary differential
equation solver, we evaluate the initial conditions (16) at r =10 4 for global tolerance 10 12, adjusting for
− −
fixed shooting parameter b and integrating towards r .
→∞
As expected, these solutions have many features in common with the well-known d = 4, p = 1 BK
solutions. Byadjustingthefreeparameterbappearingintheexpansion(16),we”shoot”forglobalsolutions
with the right asymptotics. The solutions are indexed by k-the number of nodes of the gauge potential w,
which is always bounded within the strip w(r) 1.
| | ≤
Forthe k-thsolution,the functionw(r) hask nodesinthe interval0<r < ,suchthatw( )=( 1)k.
∞ ∞ −
Typicalprofilesforp=2solutionsareplottedinFigures1,2(ford=8, 6)andp=3, d=12configurations
(Figure 3). It is worth noting here that for a p-gravity and p-YM theory, when the spacetime dimension is
8
4p, then the profiles of the functions N(r), σ(r) and w(r) are perfectly smooth as seen from Figures 1 (for
d=8)andFigures3 (for d=12),as is the case forthe BKsolutions ind=4,while for dimensions different
from d=4p, e.g. for d=6 depicted in Figure 2, these profiles are less smooth and the numerical analysis is
correspondingly more delicate. This is one the features highlighting the similarity of the solutions to these
theories modulo 4p.
The behaviour of the metric functions N and σ is similar for all k’s. The metric functions σ increases
with growing r from σ(0)=σ >0 at the origin to σ( )=1. As k increases, N develops a more and more
0
∞
deep minimum at some r , closely approaching the zero value for large enough values of k, as indicated in
m
Figures 1-3. At the same time, the value at the origin of the metric function σ strongly decreases with k.
Thelimitingsolutionwithk= canalsobeinvestigated[12]. SincetheSchwarzschildcoordinatesystem
∞
breaks down in this limit, we should use a different parameterization and the equations are formulated as a
dynamical system. Similar to the p=1 case, this turns out to be non-asymptotically flat.
The system (13)-(15) presents also black hole solutions. The boundary conditions at infinity are still
givenby (17), andarenowsupplemented withthe requirementthatthere isa regulareventhorizonatsome
r =r >0, with N(r)>0 for r >r . The local power series in the vicinity of the horizon reads
h h
w(r)=w +w (r r )+O(r r )2, N(r)=N (r r )+O(r r )2, σ(r)=σ +σ (r r )+O(r r )2,(20)
h h′ − h − h h′ − h − h h h′ − h − h
with
(d 2p 1) 1
Nh′ = −r − (1−r2pWhp), wh′ =wh(wh2 −1)(d−2p−1)r2N , σh′ =2σhrh2p−3Whp−1wh′2, (21)
h h′
and Wh = (1−rw4h2)2. Similar to the d=4, p=1 case, one find a sequence of global solutions in the interval
h
r <r < ,for anyvalue ofthe eventhorizonr >0. These solutionsareparameterizedagainbythe node
h h
∞
numberk ofthegaugefunctionw. Forany(r ,k)thebehaviourofthefunctionsw,σ isqualitativelysimilar
h
to that for regular solutions. The gauge function w starts from some value 0 < w < 1 at the horizon and
h
after k oscillations around zero tends asymptotically to ( 1)k. In this case againone can show that w <1
− | |
everywhere outside the horizon. In the limit r 0 the event horizon shrinks to zero and the black hole
h
→
solutions converges pointwise to the corresponding regular configuration. In Figure 4 we exhibit the one,
two and three nodes solutions of the p=2 model in eight dimensions; similar solutions have been found for
p=2,d=6 and p=3, d=9,12.
Thestabilityofsolutionsintheusualp=1, d=4casehasbeenstudiedextensively,bothperturbatively
and at the non-linear level (see the discussion in [2] and the references therein). It turns out that all
knownregularandblackholesolutionsinthatcaseareunstablewithrespecttosmallsphericallysymmetric
perturbations. Themostobviousindicationoftheinstabilitycomesfromtheabsenceofatopologicalcharge
in the YM sector. This is obvious in the d = 4 case, but when several members of the YM hierarchy are
present like in [3, 4, 8], in some of these theories a Pontryagincharge is defined, leading to stable solutions.
Thestabilityquestioninsuchmodelshasbeenstudiedin[13]. Inthepresentworkhoweverwehaverestricted
to a single (the p-th) member of the YM hierarchy, so that no topological charge can be accommodated in
the case of static solutions. Indeed, the stability analysis in [13] leading to the conclusionthat the solutions
in dimensions in which a Pontryagin charge is not defined are unstable (sphalerons) applies to the models
studied in the present Section, and is not repeated here.
AnotheraspectofthestabilityofgravitatingYMfieldsistheinterestingcasewhenanegativecosmological
constant is introduced, in the usual EYM model with p = 1, d=4 [14, 15]. In that case, one can see from
the asymptotic analysis that the value of the gauge field function w(r) at infinity is not restricted to 1,
±
and in particular lim w(r) = 0 is allowed. This is the asymptotic value for a static monopole, i.e.
r
→∞
the particular solution in question is a finite energy lump with (topological) monopole charge, rendering
it stable. Unfortunately this property does not persist 2 in the p 2 models with negative cosmological
≥
constant, implying that these would be exclusively sphalerons. As a result we have eschewed an analysis of
these.
2This can be seen from the large r asymptotic analysis in the case of p ≥ 2 models with negative cosmological constant,
whichleadsexclusivelytolimr→∞w(r)=±1.
9
WeclosethisSectionwithseveralcommentsonthethermodynamicpropertiesoftheblackholesolutions.
The Hawking temperature of these configurations can easily be found by using the standard Euclidean
method. For the line element (9), if we treat t as complex, then its imaginary part is a coordinate for a
non-singular Euclidean submanifold iff it is periodic with period
4π
β = . (22)
N (r )σ(r )
′ h h
Then continuous Euclidean Green functions must have this period, so by standard arguments the Hawking
temperature is T =1/β. For any (p,d) this is found to decrease with the node number k. To compute the
entropyof these solutions,one may use the standardrelationshipbetween the temperature andthe entropy,
S = dM/T +S , with S a mass-independent constant. Although further study is necessary, we expect
0 0
this expression to differ from the standard one quarter event horizon area value, which holds for p = 1 EH
R
gravity only [16, 17].
4 Exact gravity-Yang-Mills solutions in d = 2p + 1
In the particular dimensions d=2p+1, the analogues of (6),
κ τ
=e p R + p TrF(2p)2 . (23)
L(2p+1) 2p (p,1) 2(2p)!
(cid:18) (cid:19)
do not support static finite energy solutions, however, their solutions can be constructed in closed form.
Since to the best of our knowledge no exact (nontrivial) solution is known in the literature for the coupled
gravity-Yang-Millsequations 3, we discuss here the basic properties of these d=2p+1 configurations. The
straightforwardgeneralisationin the presence of a cosmological constant is presented in Appendix B.
After severalredefinitions of the theory’s constants, the effective Lagrangeanof this theory reads
2
d dw
L=σ (N 1)p cr1 2pN(w2 1)2p 2 , (24)
− −
dr − − − dr
(cid:20) (cid:18) (cid:19) (cid:21)
where c τ/κ is a free parameter (in this general case, we do not fix the sign of c). It is also convenient to
∼
redefine the gauge potential according to
1 3
a(w)= (w2 1)p 1dw =( 1)p+1 F ( ,1 p, ,w2), (25)
− 2 1
− − 2 − 2
Z
( F (a,b,c,z) being the hypergeometric function), such that the system (24) admits the first integral
2 1
α
a′ (w2 1)p−1w′ = , (26)
≡ − r1 2pNσ
−
where α is an arbitrary real constant.
The metric variables σ and N satisfy the equations:
dXp cα2
= r2p 1 , (27)
−
dr (X +1)Y
(cid:20) (cid:21)
dY 2cα2 r2p 1
−
= , (28)
dr − p (X +1)2Xp 1
−
where X =N 1, Y =σ2. This implies the relation:
−
dY 2Y
= , (29)
dX −(X +1)
which gives
Y(X +1)2 =C, (30)
3See,however,theexactsolutionwithplanarsymmetryofthed=4EYMequations withanegativecosmologicalconstant
Λ=−3presentedin[18].
10