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A certain class of Einstein-Yang-Mills systems
Vol. 39 (1997)
REPORTS
A CERTAIN
CLASS
ON MATHEMATI(‘AL
No. 3
PHYSICS
OF EINSTEIN-YANG-MILLS
SYSTEMS
GERD RUDOLPH and TORSTEN TOK Institute of Theoretical
Physics, Leipzig University, 04109 Leipzig, Augustusplatz (e-mail: [email protected]) (Received December
10, Germany
19, 1996)
A class of G-invariant Einstein-Yang-Mills (EYM) systems with cosmological constant on homogeneous spaces G/H, where G is a semisimple compact Lie group, is presented. These EYM systems can be obtained in terms of dimensional reduction of pure gravity. If G/H is a symmetric space, the EYM system on G/H provides a static solution of the EYM equations on space-time R x G/H. This way, in particular, a solution for an arbitrary Lie group F, considered as a symmetric space, is obtained. This solution is discussed in detail for the case F = SU(2). A known analytical EYM system on W x S3 is recovered and it is shown-using a relation to the BPST instanton-that this solution is of sphaleron type. Finally, a relation to the distance of Bures and to parallel transport along mixed states is shown.
1. Introduction
In recent years, there has been a lot of interest in static, globally regular, finite energy solutions (the so called soliton solutions) and black hole solutions to the Einstein equations with the Yang-Mills, Higgs and other nonlinear sources. In 1988, Bartnik and McKinnon [l] found a numerical solution to the SU(2) Einstein-Yang-Mills (EYM) system. This was quite surprising because neither the vacuum-Einstein [2] equations nor the pure Yang-Mills [3] equations on Minkowski space have nontrivial soliton solutions. Hence, the very weak gravitational interaction can change qualitatively the spectrum of soliton solutions of a theory in which gravity was neglected. Other authors discovered numerically black hole solutions [4] to the SCi(2) EYM system and the existence of both types of solutions was established rigorously [5]. Unfortunately, all solutions to EYM systems with arbitrary gauge group turned out to be unstable in the sense of linear stability [6]. Subsequently several authors have investigated other models, such as SU(2) EYM Higgs [7], EYM dilaton [8] or Einstein-Skyrme systems [9] and found in some cases linearly stable solutions. Another line of research is to look for solutions of the EYM equations in arbitrary space-time dimension, which are invariant under some symmetry group K [lo]. Then one has the possibility to apply the powerful theory of dimensional reduction and spontaneous compactification [ 11, 121. 14331
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G. RUDOLPH
and T. TOK
Some authors [13-151 discussed an analytical SU(2) EYM system in 3 + 1 dimensions with cosmological constant A. The aim of this paper is to show that this EYM system is a special case of a general construction of EYM systems with arbitrary gauge group. The paper is organized as follows: In Section 2 we discuss a genera1 construction which yields on each homogeneous space G/H a G-invariant EYM system with cosmological constant. If G/H is a symmetric space, we can use this solution to obtain a static G-symmetric EYM system on the space-time IRx G/H. This is shown in Section 3. In Section 4 we apply the construction to an arbitrary Lie group F, considered as a symmetric space, and in Section 5 we discuss the special case F = SU(2). We recover the known analytical solution mentioned above [13-1.51. Within our geometric framework we are able to give an intrinsic proof that the configuration under consideration is of sphaleron type. In the last section we show that this EYM system appears also in the theory of parallel transport along mixed states.
2. General
construction
We consider a semisimple compact Lie with Lie algebras Q5and _Q,resp. On C5let bilinear form y, for instance the negative by left transport a G-biinvariant metric g g(l,‘X,
group G and an arbitrary subgroup H c G be given a positive definite Ad(G)-invariant of the Killing form IK. The form -, defines on G
1,‘Y) = -y(X. Y).
x.
Y E 6.
We denote by 2, and r’. the left and right, respectively, multiplication in the group G with group element .r/. The prime denotes the corresponding tangent map, i.e. 1,‘X E T,G. In terms of the canonical left invariant Lie-algebra-valued l-form 0 on G we can write symbolically g = r(H.0).
(2)
Now we consider G as a principal bundle over G/H with structure group H and canonical projection rr: G + G/H. Obviously g is invariant under the right action of the structure group H. Therefore, we can use the fact that every H-invariant metric on an H-principal bundle P with base space IU defines, and is defined by, three geometrical objects: a connection r in the bundle P, a metric g,l.f on the base space A4 and, for every point :c E M, an rH-invariant metric on E,,, the fibre over .I’ [12]. We discuss these geometrical objects in our case. Let P = G, M = G/H, and g be the H-invariant metric on P. We define for every y E G the horizontal subspace Hor, of the tangent space Tq,G as the orthogonal complement of the canonical vertical subspace Ver, c T,G of the bundle P = G with respect to the metric g. The vertical subspace Ver, is obviously given by Ver, = 1,‘4. Thus, from (1) and the Ad(H)-invariance of g we see Hor, = 1,‘!7X
(3)
A CERTAIN CLASS OF EINSTEIN-YANG-MILLS
where M c @ is the orthogonal complement Ad(H) invariance of y, the decomposition
SYSTEMS
435
of fi with respect to y. Because of the
S=fi@!XI
(4)
is reductive, i.e., [fi. !J.J?]c 337. The horizontal subspaces Hor, define a connection r, which is known as the canonical connection in the bundle G(G/H. H) [16]. Its connection form ~j is given by the @component of H with respect to the decomposition (4): w = Hfi. (5) The metric gG/H on the base space M = G/H can be obtained in the following way: Take two tangent vectors UI. uZ f T,(G/H) at the point x E G/H and lift them to horizontal tangent vectors U1. i[z E Her, at an arbitrary point 9 E E,., i.e., T’(fli) = tJ, i = 1.2. We put
This definition of gG/H is, because of the ,rH-invariance of g, independent of the point !I E E,,.. If we have a (local) section s: G/H + G, then &/H reads locally gG/H = “i ((.‘i*&Q).(.ti*ti‘,,r)).
(7)
where Hm is the M-component of H and the star denotes pull back. The J’H-invariant metric on E,. is given by restriction of g to E.,.. Due to the IG-invariance of g, every such metric on E,. defines and is defined by the same Ad(H)-invariant scalar product 2h on .Fj,,namely %j(Vi. vz) = g(l,%. l,‘K). (8) where V,, V2 E _Q and 9 E G and yq is the restriction of y to the subspace 9 c @. One more consequence of the IG-invariance of g is that the connection r and the metric gG/H are invariant under the left action of G, compare with equations (5) and (7). Conversely, if we have a connection r with a connection form w in the bundle G(G/H. H) and a metric gG/H on G/H, both not necessarily IG-invariant, and if we have an Ad(H)-invariant scalar product 7fi on the Lie algebra _FJ,then these three geometrical objects define an ‘rH-invariant metric g on G by [17] g(x. y, := %j(w(x).w(Y))
+
gG/H(T’(X).
,‘ty>&
X.
Y E T,G.
(9)
In the next step we will write down the scalar curvature XC; of the Levi-Civith connection on G in terms of LJ, yq and gGfH. The Ad(H)-invariant scalar product p4 determines a biinvariant metric on H, similarly as in equation (1). The scalar curvature of H calculated with respect to this metric is constant and is denoted by RH. The curvature fj of the connection r is given by .C?=Dw=dw+i~~w.
(10)
436
G. RUDOLPH
If we choose a local coordinate
and T. TOK
system (x@) on G/H and a local section s: G/H + G,
we can write F = s* 61 = 4 F,,, dxi’ A d.cv .
(11)
where F,,,, takes values in Ji. Note that the quantity ya(F,,,, FC’u) considered as a function on G/H is independent of the section s and the coordinate system (xl’). Now the scalar curvature RG reads, see [ll, 121: RG
= ~*(RG/H)+
RH
- ~~*(Y~(F~,",F"")).
(12)
where RG,H denotes the scalar curvature on G/H and 7r* the pull back under 7~. Equation (12) looks very simple, but this is due to the Ad(H)-invariance of “ir, and the lo-invariant construction of the metric in the fibres E,., see equation (9). In general, the splitting of RG is much more complicated. It is clear that RG is constant on each fibre E,.. Therefore we can integrate the Einstein action S = \ (RG - A,) dt)C; (13) c: over the fibres E,. and we get S = VH \
(RGIH - (AI - RH) - iyb(Fpu. F/IV)) dtl~,~.
(14)
G/H
Here d,uG and d?lG,H denote the volume forms on G and G/H, respectively. VH is the volume of the structure group H, which is equal to the volume of each fibre E, and ni has the meaning of a cosmological constant. Equation (14) gives the action of a coupled Einstein-Yang-Mills system on G/H with cosmological constant A = Ai - RH. Hence, we arrive at the following result. If the metric g, given by (9), is a solution of the Einstein equations with cosmological constant Ai, then the metric go/H and the connection r form an Einstein-Yang-Mills system with cosmological constant A = fli - RH. This is clear because every variation of go/H and w yields a variation of g. But g is a solution of a variational principle with action (13). Therefore go/H and w are solutions of a variational principle with action (14). It is a known fact that the biinvariant Killing metric glc on a semisimple compact Lie group G, arising from the negative of the Killing form K, see equation (1) is a solution of the Einstein equations with some cosmological constant. One can easily verify this statement by calculating the Ricci tensor Ric. Doing this one gets [16] Ric = igI<. Another consequence of this equation is that the scalar curvature with respect to gI< is RG = $DG.
(15) &
of G calculated (16)
where DG is the dimension of G. If we choose g = r?gh_. N > 0, as the metric on G, the Einstein equations on G will be fulfilled. The corresponding cosmological constant
A CERTAIN
CLASS OF EINSTEIN-YANG-MILLS
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437
is given by ,=DG-2 -.
(17)
4cr
Thus, we can use the above construction to find G-invariant EYM systems on homogeneous spaces G/H with compact semisimple Lie group G. 3. EYM systems
on IR x G/H
Let G be a compact semisimple Lie group with Killing metric gf< and H be a subgroup of G, so that G/H is a symmetric space. Then we can construct a G-invariant static EYM system on space-time N = Ik!x G/H using the construction described in the foregoing section, with the first component of N playing the role of time. We use the Killing metric gI,- on G to obtain the EYM system (gGIII,WGIH) on G/H. Moreover, we have the projection p: N -+ G/H, which projects (t. x) E IRxG/H onto .I’ E G/H. On N we consider the static metric gx =
and on the H-bundle
-nt 6 clt +
(p*G) over G/H
pp*(gG,H).
the static connection
w.V =
(18)
/SEE&.
form
(19)
P*(dG/H).
Because the Yang-Mills equations are fulfilled on G/H, one easily shows that the Yang-Mills equations on N are fulfilled independently of 0, too. It remains to check the Einstein equations. If G/H is a symmetric space, it is easy to calculate the Ricci tensor RicGIH and the energy-momentum tensor TG,~ on G/H: R&/H
= ;gG,H.
TG/H
WV
;(4 - &/&G/H.
=
where DC;~H is the dimension of G/H.
(21)
Here we used
(22) with the components FII,, of the field strength given by equation (11). The Ricci tensor Ric.sT on N calculated with respect to gs is, because simple structure of the metric g,y (18), given by Ric.\T = +P*(gG/H). i.e., Ric.$F has no time components. the form I TX
=
The energy-momentum
$*kG/H)
DG/H - slj2tZx.
of the
(23) tensor TjV on N takes (24)
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G. RUDOLPH
and T. TOK
Now it is a matter of fine tuning the constants N. The Einstein equations on N read
to fulfill the Einstein equations
RicN - igx ( Rlv - A,V) = K,T,v.
(25)
where A.v is the cosmological constant and K is the gravitational equations (18), (23) and (24), we calculate
Ric.v -kg,dRv
-
AN) = ip*(gGIH)
=(&+
-
2-
- 1
igAvDG/H 2P DG/H 4
riT.v = (K~-$~~)
on
constant.
Using
A‘V
p*(gG,H) + dt
‘$Idt ~
--
+ dt ii dt ( K,+).
P*(&,H)
Hence, we obtain two equations, which are equivalent to /j =
DG/H
A=4-.
K,
h
Therefore, if the cosmological constant and the gravitational constant are related by equation (26), then the system (g,v.w,v) is a solution of the EYM equations on N. This solution can also be obtained by solving the equations of spontaneous compactification [ll] on R x G/H. 4. The symmetric
space (F x F)/Fdiag
Let F be a semisimple compact Lie group with Lie algebra 3 and Killing form Kz. We can consider F as a symmetric space on which G = F x F acts transitively by (.f1. .f2)
*
f
:=
f1,
.f1,f(f2)-~.
f2,
.f’
E
F.
The stabilizer of the unit element e E F is given by Fe = ((f3.f);
f E F} = Fdiag E HT
and we can write F E (F
X
F)/Fdi,g
= G/H.
The canonical splitting of the Lie algebra 8 of G into a direct sum of the Lie algebra sj of H and a vector space !?JI is given by 6 3 (X,Y) = (f(X
+ Y). ;
- Y)) .
x. Y E 5.
i.e., 4 = {(X,X); CIJI= {(X, -X);
x E 5>. x E 5).
(27) (28)
A CERTAIN CLASS OF EINSTEIN-YANG-MILLS
SYSTEMS
439
The Lie algebra 8 is the direct sum of two semisimple Lie algebras, namely 8 = 5 $5. Therefore, the Killing form K on 6 is completely defined by Kg, namely K((X1.
X2),(K,y2))
=
JG(X1.W
+ K~(x2,Y2).
(29)
One easily obtains that fi and M are orthogonal with respect to K. In the next step we use y = -_(yK, a > 0, to construct an EYM system on F as described in Section 2. We denote the projection form G = F x F onto the first and second component by prr and pr2, respectively, and we write Oi := pry HF, 1:= 1.2, where tin is the canonical left-invariant l-form on F. It is clear that G has the structure of a principal bundle over F with the structure group H = Fdiag and projection 7r: G --f F given by 74f1.f2)
=
fdf2)-',
f1,fi
(30)
E F.
We choose the following section s: F + G in this bundle f E F.
s(f) := (f. e). With these notations
the canonical left-invariant 0=
and its projection
onto fi, respectively
Moreover,
l-form 0 on G reads
(&.e,),
M, has the form
ofi = ($(& + &Jr = (@I
(31)
#2), @l
(32)
+ W).
- e,). -f
- 82)) .
(33)
we get (34) (35)
where we have used prr OS = idF and pr2 OS = e. NOW the gauge potential and the metric gF, see equations (5) and (7) can be expressed by AF =
S*W
gF = y
=
s*Or, =
((s*d,),
($OF,~I~F),
(S'e,))
= -+aK3(8~.
AF = s*ti
(36) tip).
(37)
To derive the last equation one has to take into consideration equations (35) and (29). The scalar product “ir, in the Lie algebra fi of the structure group is given by the restriction of y to 4. Identifying H = Fdiag with F and using equation (29) we get “or = -2c~K~. (38) Comparing with equation (16) we obtain
440
G. RUDOLPH and T. TOK
where RH is the scalar curvature of the structure group and DF is the dimension of F. The metric g on G = F x F fulfils the Einstein equations on G with cosmological constant A, = (DF - 1)/2tr, see equation (17). Thus, the cosmological constant A of the EYM system consisting of gF and AF reads A = Al - RH =
PDF-4
acr
.
The physical EYM action has the form
-
&(R
A) -
$(F,,,. Fbiu))
&I.
(41)
where (.. .) is the negative of the Killing form and 8rX = K is the gravitational constant, which appears in the Einstein equations (25). It is obtained by dividing the action (14) by 2cu and identifying o with the gravitational constant K. Let us summarize our results. On every compact semisimple Lie group F there exists an F-symmetric EYM system with a gauge group F consisting of gF and Ap, see equations (37) and (36). The gravitational constant (Y and the cosmological constant A are related by equation (40) (fine tuning). We considered F as a symmetric space. Therefore, we can apply the construction described in Section 3 to get an EYM system on N = R x F. We obtain the gauge potential and the metric on N from equations (19) and (la), where the EYM system on F = G/H and the scalar product in the Lie algebra of the structure group F are given by equations (36) and (37) respectively (38) with (Y= 1. The resulting relations between the occurring constants 1) , h: and A,v can be obtained from equation (26). 5. The case SU(2)
For F = X1(2) the gauge potential coordinates zj
x SU(2)/SU(2)
-
a relation
to instantons
it is easy to write down the explicit form of the metric gst.(2) and ASLTc2)in local coordinates. We parametrize SU(2) by stereographic
i,j E
R
/j = 1.2.3, (~1’ = e
(z;j)‘.
(42)
where & denote the Pauli matrices and ll is the 2 x 2 unit matrix. The Killing form Ka~rC2jon the Lie algebra so is given by K,s,,C~j(X,Y) = 4 tr(XY).
X, Y e s,u(2).
(43)
To get the metric gsrrC2) one has to use equation (37) &x7(2)
=
-_~~,s,,(2)(H,s,,(2),Bsu(2))
=
-htr
(44)
A CERTAIN CLASS OF EINSTEIN-YANG-MILLS
SYSTEMS
441
Here & denotes the symmetrized tensor product and in what follows we write % for the Hermitian conjugate of x. With H,,,Cz)= XC’dx = -rl(x-l)x = -C&X we get
(45) It is obvious that SU(2) endowed with this metric is a 3-sphere with radius 2,/G. The gauge potential A,q,,c2, = iQRuc2) takes an especially simple form if we perform a gauge transformation A’ = t~-~A,~~,(~p + U-~~IL (46) with Z-II (47)
‘U= (xAfter a simple calculation we get
where E”“? is totally antisymmetric and ~~~~= 1. The cosmological constant .1 follows from equation (40) and has the value
If we apply the construction described in Section 3, then we obtain a static EYM system on N = LRx S3 - Iw x SU(2). In local coordinates (t. z,,) this solution reads
(51) Here K is the gravitational constant. From equations product in the Lie algebra of the structure group
Y,~,,(~)(X. Y) = -8 tr(X. Y). The cosmological
(38) and (43) we get the scalar
x, Y E X/L(2).
constant A,%,we obtain from equation A, = ;.
h
(52)
(26) and D,s1;C2)= 3: (53)
The same results were obtained in [14, 151. But in these papers the geometric structure of the solution was left in the dark. We hope that our considerations clarified this point completely.
442
G. RUDOLPH
and T. TOK
In [13] there was mentioned a relation of the gauge potential Asuczj = itlsrr(2j to the BPST instanton solution [18], but only in terms of a local coordinate chart. In the bundle language, this relation looks as follows: Let us consider the principal bundle P = SU(2) x SU(2) --+ SU(2) as a subbundle of the quaternionic Hopf bundle PM: we show that the connection form w on P, see equation (5), is the pull back of the instanton connection form Wi,,t on PH. The quaternionic Hopf bundle is given by pairs (a, b), a. b E H, with cin+bb= 1. (54) and by the right action of unimodular
quaternions
?j~Ju. b) := (m, bu),
,Ilu.= 1.
(55)
Here bar denotes quaternionic conjugation. The set of unimodular quaternions is isomorphic to the group SU(2). Therefore, the bundle P = SU(2) x SU(2) is naturally embedded by a bundle homomorphism i into the bundle PM i: P 3 (a,b) + (&a:
On PM the instanton connection
is defined by Winst
Taking into account U = u-l,
&-b) E PH.
= 6da + bdb.
(57)
for ‘Uunimodular, we obtain .*
8 Winst
=
@I
+
Q2),
where 8, = pri*BsU(rJ, i = 1,2, as defined in the foregoing section. Comparing with equations (5) and (32) it is clear that w is the pull back of Wi,st under i. This gives us the possibility to calculate the Chern-Simons index Ic of the gauge potential AsrTc2), see equation (36) in a very simple geometrically intrinsic way. If we represent the base space S4 of the quaternionic Hopf bundle as the set of quaternions plus one point, we can choose the local section (59) in the bundle PH. We denote by AI the instanton gauge potential and by FI its field strength, i.e., AI = SI*Wi,,t. Notice that the section s in the bundle P, see equation (31) is the restriction of SI under the embedding 1:.Therefore, Asric2) is the pull back of AI under i. The embedding i induces an embedding of the base space of P into the base space of Pa,, which we denote by the same letter i. The image i(M) of the base space M = SU(2) of P is an equator of S 4. We denote one of the two semispheres of S4 whose boundary is i(A4) by N. Now we can calculate
k=
& \ =J(2)
tr(Ascr(2)A
dAsw(2)
+
&xq2)
A
&u(2)
A
A.m(z))
A CERTAIN
CLASS OF EINSTEIN-YANG-MILLS
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443
One easily checks that the connection AI is up to gauge transformations invariant under the natural action of SO(5) on S”. Therefore, tr(FI A 8’1) is up to a factor the volume form on S” and we get
The topological index of the basic instanton is 1, hence the Chern-Simons index k of our solution has to be i. This is an intrinsic proof that the solution found is of sphaleron type. The calculation of the Chern-Simons index in terms of local gauge potentials is much more complicated and may yield incorrect results if one chooses a singular gauge. Therefore the authors in [14] had to perform a gauge transformation before they had got the correct result. 6. A relation to Berry’s phase and the distance
of Bures
The EYM system (gsc’(a), ASLrc2)), described in the foregoing section, is well known from the study of parallel transport along 2 x 2 density matrices, see [19-211. We consider the trivial U(2) principal bundle GL(2. C) -+ GL(2. Q/11(2) -: LIZ(~). with the projection
(61)
rr given by jr(W) : =
ww*,
ui E GL(2.C).
(62)
Here star denotes Hermitian conjugation. The base space D;?(2) consists of all not normalized nonsingular 2 x 2 density matrices. On GL(2. C) we have a natural metric g which is invariant under the right action of U(2): g = !JItr(dw 6 &*). Therefore, by an analogous construction as in in the bundle (61) and a metric gB on its proposed by Uhlmann [20] and it governs the which is related to the concept of purification known as the Riemannian metric which comes is related to the transition probability between
(63)
Section 2 we obtain a connection AB base space. The connection AB was parallel transport along mixed states, of density matrices. The metric gB is from the distance of Bures [22] and mixed states.
444
G. RUDOLPH
It turns out that the connection subbundle Q2 defined by
and T. TOK
AB is reducible
Q2 := {UI E GL(2,C);
to a connection
det(w) E R+}.
[19] in the SU(2)
(64)
This corresponds to a simple property of the metric (63): If we consider GL(2. C) as a U(1) principal bundle over Q2, then every vector tangent to Q2 is orthogonal to the direction of the fibre of that bundle. Therefore, the horizontal subspaces of the connection Ag, defined as the orthogonal complements of the vector spaces tangent to the fibres of the bundle GL(2.C) + D2(2), are tangent to Qa. Another consequence of this property is that we can use the restriction of g to Q2 to construct gB. In the next step we restrict the base space of Q2 to all normalized density matrices: tr(p) = tr(‘uulw*)= 1.
(65)
We denote the resulting SU(2) bundle by $2. Obviously, there is a natural embedding j: & + GL(2, C). On $a we have the pull back j*g of the metric (63). We will show that there exists a bundle homomorphism f: Q2 + P = SU(2) x SU(2) such that j*g = f*gP, with gp being a multiple of the Killing metric on SU(2) x SU(2). Obviously, every matrix ‘UUI E y1(2. C) can be uniquely represented in the form 110 UJ= 2
i
(CL+ ib) . 0
i
(66)
1
where 0, =
zoll + 2,i(Tn,
b = yell + y,io” .
cv = 1,2,3,
x‘rry; E R.
If ~1 E &, we have det(uj) > 0 and tr(,luslj*) = 1. Hence, we obtain 0=
S(det(,ll,)) = a ($
XL.,% - 5
z=o 0 < !R(det(,u,)) = i 2
yia) (
(67)
id3
:riyi.
(68)
i=O
1 = tr(ut,/l,*) = i (t
xjZ + 5 1=0
yi2).
(69)
i=o
From equations (67) and (69) we find
&&2=-pyi2 =
1,
i=o
i=o
(70)
A CERTAIN
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OF EINSTEIN-YANG-MILLS
SYSTEMS
445
and therefore a. b E SU(2). So we can define a map f: & -+ P according to equation (66). One easily checks that this map is an injective bundle homomorphism and because of (68) the set f(&) is an open subset of P. Thus, for every (n,b) = p E f(&) with origin ‘UI= f-l(p), the tangent space T,P is isomorphic to the tangent space T,,.&. With these remarks it is a matter of a simple calculation to show j*g = ,f*gp. Let X1 and X2 be two vectors in Ttr,Q2 and (Al. &) and (AZ, &) their images in T,,P, p = (a b) E N(2) x SU(2). Taking into account equations (2) (29) and (43) as well as H(A;. B;) = (n?Ai, b+&) E s1~(2)@ SIL(~), %= 1,2. and using -(CL-lA) = (dA)*
= A’n.
(I E N(2).
we obtain gp((A1, II,). (AZ, &)) = -4cY(tr(cl,-lAln-lA2)
+ tr(b-1BIb-1B2))
= 4a(tr(A1A2*) + tr(&&*)) = 4&R tr((Ar + jB1)(A2* - i&*)) = 16~$R tr(XrX2*)),
(71)
showing that j*g = f*gp if N = &. Hence, the Uhlmann connection reduced to a connection in &, and the pull back of the Bures metric to the space of nonsingular normalized density matrices, coincide with the EYM system presented in Section 5, see equations (45) and (48). REFERENCES [I] R. Bartnik and J. McKinnon: Phys. Rev. Letf. 61 (1988) 141. [2] A. Lichnerowicz: in Theories relativistes de la gravitation et de l’electromagnetisme, Masson, Paris 1955. [3] S. Deser: Phys. Lett. B 64 (1976), 463. S. Coleman: in New Phenomenon in Subnuclear Physics, ed. A. Zichichi, Plenum, New York 1975. (41 P. Bizon: Phys. Rev. Lett. 64 (1990) 2844. M. S. Volkov and D. V. Galt’sov: Pis’ma Zh. Eksp. Teor. Fiz. 50 (1989) 312; Sov. J. Nucl. Phys. 51 (1990), 747. H. P. Kiinzle and A. K. Masoud-m-Alan: J. Math. Phys. 31 (1990), 928. [5] J. A. Smoller and A. G. Wassermann: Commun. Math. Phys. 151 (1993) 303. J. A. Smoller, A. G. Wassermann and S. T. Yau: Commun. Math. Phys. 154 (1993) 377. P. Breitenlohner, P. Forgacs and D. Maison: Commun. Math. Phys. 163 (1994), 141. J. A. Smoller, A. G. Wassermann, S. T. Yau and J. B. McLeod: Commun. Math. Phys. 143 (1992). 115. [6] N. Straumann and Z.-H. Zhou: Phys. Lett. B 237 (1990) 353. N. Straumann and Z.-H. Zhou: Phys. Lett. B 243 (1990) 33. Z.-H. Zhou and N. Straumann: Nucl. Phys. B 360 (1991) 180. 0. Brodbeck and N. Straumann: Phys. Lett. B 324 (1994) 309. P. Boschung, 0. Brodbeck, F. Moser, N. Straumann and M. S. Volkov: Phys. Rev. D 50 (1994) 3842. [7] K.-Y. Lee, V. P. Nair and E. Weinberg: Phys. Rev. Lett. 68 (1992) 1100. K.-Y. Lee, V. P. Nair and E. Weinberg: Phys. Rev. D 45 (1992), 2751. P. Breitenlohner, P. Forgacs and D. Maison: Nucl. Phys. 383 (1992), 357. P. C. Aichelburg and P. Bizon: Phys. Rev. D 48 (1993) 607. B. R. Greene, S. D. Mathur and C. M. O’Neill: Phys. Rev. D 47 (1993), 2242.
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and T. TOK
[8] E. E. Donets and D. V. Gal’tsov: Phys. Lett. B 302 (1993), 411. E. E. Donets and D. V. Gal’tsov: Phys. Lett. B 312 (1993) 391. G. Lavrelashvili and D. Maison: Nucl. Whys. B 410 (1993), 407. P. Bizon: Act. Phys. Pal. B 24 (!993), 1209. [9] S. Droz, M. Heusler and N. Straumann: Phys. Left. B 268 (1991) 371. M. Heusler, S. Droz and N. Straumann: Phys. Lett. B 271 (1991), 61. M. Heusler, S. Droz and N. Straumann: Phys. Letf. B 285 (1992), 21. M. Heusler, N. Straumann and Z.-H. Zhou: Helv. Phys. Actu 66 (1993), 614. [lo] G. Rudolph, T. Tok and I. P. Volobuev: Int. J. Mod. Phys. A 11, No 28 (1996) 4999. [ll] Y. A. Kubyshin, J. M. Mourao, G. Rudolph and I. P. Volobuev: Dimensional Reduction of Gauge Theories, Spontuneow Conzpacfijkatian and Model Building, Springer, Berlin, Heidelberg 1989. [12] R. Coquereaux and A. Jadczyk: Commun. Math. Phys. 90 (1983), 79. R. Coquereaux: Ref.TH.3639-CERN. [13] Y. Hosotani: Phys. Lett. B 147 (1984), 44. [14] S. Ding and A. Hosoya: TITIHEP-242/COSMO-39, 1993. [15] P. G. Molnar: ZU-TH 7195. [16] S. Kobayashi and K. Nomizu: Foundations of Differential Geometry I/II, John Wiley & Sons, New York, London 1963. [17] A. Trautman: Rep. Math. Phys. 1 (1970), 29. [18] A. Belavin, A. Polyakov, A. Schwarz and Y. Tyupkin: Phys. Left. B 59 (1975), 85. [19] J. Dittmann and G. Rudolph: J. Geom. Phys. 10 (1992) 93. J. Dittmann and G. Rudolph: J. Math. Phys. 33 (1992), 4148. [20] A. Uhlmann: Rep. Math. Phys. 24 (1986), 229. A. Uhlmann: Lett. Math. Phys. 21 (1991), 229. [21] T. Tok: Diploma thesis, Leipzig University, 1993. [22] D. Bures: Trans. Am. Math. Sot. 135 (1969), 199.
REPORTS
A CERTAIN
CLASS
ON MATHEMATI(‘AL
No. 3
PHYSICS
OF EINSTEIN-YANG-MILLS
SYSTEMS
GERD RUDOLPH and TORSTEN TOK Institute of Theoretical
Physics, Leipzig University, 04109 Leipzig, Augustusplatz (e-mail: [email protected]) (Received December
10, Germany
19, 1996)
A class of G-invariant Einstein-Yang-Mills (EYM) systems with cosmological constant on homogeneous spaces G/H, where G is a semisimple compact Lie group, is presented. These EYM systems can be obtained in terms of dimensional reduction of pure gravity. If G/H is a symmetric space, the EYM system on G/H provides a static solution of the EYM equations on space-time R x G/H. This way, in particular, a solution for an arbitrary Lie group F, considered as a symmetric space, is obtained. This solution is discussed in detail for the case F = SU(2). A known analytical EYM system on W x S3 is recovered and it is shown-using a relation to the BPST instanton-that this solution is of sphaleron type. Finally, a relation to the distance of Bures and to parallel transport along mixed states is shown.
1. Introduction
In recent years, there has been a lot of interest in static, globally regular, finite energy solutions (the so called soliton solutions) and black hole solutions to the Einstein equations with the Yang-Mills, Higgs and other nonlinear sources. In 1988, Bartnik and McKinnon [l] found a numerical solution to the SU(2) Einstein-Yang-Mills (EYM) system. This was quite surprising because neither the vacuum-Einstein [2] equations nor the pure Yang-Mills [3] equations on Minkowski space have nontrivial soliton solutions. Hence, the very weak gravitational interaction can change qualitatively the spectrum of soliton solutions of a theory in which gravity was neglected. Other authors discovered numerically black hole solutions [4] to the SCi(2) EYM system and the existence of both types of solutions was established rigorously [5]. Unfortunately, all solutions to EYM systems with arbitrary gauge group turned out to be unstable in the sense of linear stability [6]. Subsequently several authors have investigated other models, such as SU(2) EYM Higgs [7], EYM dilaton [8] or Einstein-Skyrme systems [9] and found in some cases linearly stable solutions. Another line of research is to look for solutions of the EYM equations in arbitrary space-time dimension, which are invariant under some symmetry group K [lo]. Then one has the possibility to apply the powerful theory of dimensional reduction and spontaneous compactification [ 11, 121. 14331
434
G. RUDOLPH
and T. TOK
Some authors [13-151 discussed an analytical SU(2) EYM system in 3 + 1 dimensions with cosmological constant A. The aim of this paper is to show that this EYM system is a special case of a general construction of EYM systems with arbitrary gauge group. The paper is organized as follows: In Section 2 we discuss a genera1 construction which yields on each homogeneous space G/H a G-invariant EYM system with cosmological constant. If G/H is a symmetric space, we can use this solution to obtain a static G-symmetric EYM system on the space-time IRx G/H. This is shown in Section 3. In Section 4 we apply the construction to an arbitrary Lie group F, considered as a symmetric space, and in Section 5 we discuss the special case F = SU(2). We recover the known analytical solution mentioned above [13-1.51. Within our geometric framework we are able to give an intrinsic proof that the configuration under consideration is of sphaleron type. In the last section we show that this EYM system appears also in the theory of parallel transport along mixed states.
2. General
construction
We consider a semisimple compact Lie with Lie algebras Q5and _Q,resp. On C5let bilinear form y, for instance the negative by left transport a G-biinvariant metric g g(l,‘X,
group G and an arbitrary subgroup H c G be given a positive definite Ad(G)-invariant of the Killing form IK. The form -, defines on G
1,‘Y) = -y(X. Y).
x.
Y E 6.
We denote by 2, and r’. the left and right, respectively, multiplication in the group G with group element .r/. The prime denotes the corresponding tangent map, i.e. 1,‘X E T,G. In terms of the canonical left invariant Lie-algebra-valued l-form 0 on G we can write symbolically g = r(H.0).
(2)
Now we consider G as a principal bundle over G/H with structure group H and canonical projection rr: G + G/H. Obviously g is invariant under the right action of the structure group H. Therefore, we can use the fact that every H-invariant metric on an H-principal bundle P with base space IU defines, and is defined by, three geometrical objects: a connection r in the bundle P, a metric g,l.f on the base space A4 and, for every point :c E M, an rH-invariant metric on E,,, the fibre over .I’ [12]. We discuss these geometrical objects in our case. Let P = G, M = G/H, and g be the H-invariant metric on P. We define for every y E G the horizontal subspace Hor, of the tangent space Tq,G as the orthogonal complement of the canonical vertical subspace Ver, c T,G of the bundle P = G with respect to the metric g. The vertical subspace Ver, is obviously given by Ver, = 1,‘4. Thus, from (1) and the Ad(H)-invariance of g we see Hor, = 1,‘!7X
(3)
A CERTAIN CLASS OF EINSTEIN-YANG-MILLS
where M c @ is the orthogonal complement Ad(H) invariance of y, the decomposition
SYSTEMS
435
of fi with respect to y. Because of the
S=fi@!XI
(4)
is reductive, i.e., [fi. !J.J?]c 337. The horizontal subspaces Hor, define a connection r, which is known as the canonical connection in the bundle G(G/H. H) [16]. Its connection form ~j is given by the @component of H with respect to the decomposition (4): w = Hfi. (5) The metric gG/H on the base space M = G/H can be obtained in the following way: Take two tangent vectors UI. uZ f T,(G/H) at the point x E G/H and lift them to horizontal tangent vectors U1. i[z E Her, at an arbitrary point 9 E E,., i.e., T’(fli) = tJ, i = 1.2. We put
This definition of gG/H is, because of the ,rH-invariance of g, independent of the point !I E E,,.. If we have a (local) section s: G/H + G, then &/H reads locally gG/H = “i ((.‘i*&Q).(.ti*ti‘,,r)).
(7)
where Hm is the M-component of H and the star denotes pull back. The J’H-invariant metric on E,. is given by restriction of g to E.,.. Due to the IG-invariance of g, every such metric on E,. defines and is defined by the same Ad(H)-invariant scalar product 2h on .Fj,,namely %j(Vi. vz) = g(l,%. l,‘K). (8) where V,, V2 E _Q and 9 E G and yq is the restriction of y to the subspace 9 c @. One more consequence of the IG-invariance of g is that the connection r and the metric gG/H are invariant under the left action of G, compare with equations (5) and (7). Conversely, if we have a connection r with a connection form w in the bundle G(G/H. H) and a metric gG/H on G/H, both not necessarily IG-invariant, and if we have an Ad(H)-invariant scalar product 7fi on the Lie algebra _FJ,then these three geometrical objects define an ‘rH-invariant metric g on G by [17] g(x. y, := %j(w(x).w(Y))
+
gG/H(T’(X).
,‘ty>&
X.
Y E T,G.
(9)
In the next step we will write down the scalar curvature XC; of the Levi-Civith connection on G in terms of LJ, yq and gGfH. The Ad(H)-invariant scalar product p4 determines a biinvariant metric on H, similarly as in equation (1). The scalar curvature of H calculated with respect to this metric is constant and is denoted by RH. The curvature fj of the connection r is given by .C?=Dw=dw+i~~w.
(10)
436
G. RUDOLPH
If we choose a local coordinate
and T. TOK
system (x@) on G/H and a local section s: G/H + G,
we can write F = s* 61 = 4 F,,, dxi’ A d.cv .
(11)
where F,,,, takes values in Ji. Note that the quantity ya(F,,,, FC’u) considered as a function on G/H is independent of the section s and the coordinate system (xl’). Now the scalar curvature RG reads, see [ll, 121: RG
= ~*(RG/H)+
RH
- ~~*(Y~(F~,",F"")).
(12)
where RG,H denotes the scalar curvature on G/H and 7r* the pull back under 7~. Equation (12) looks very simple, but this is due to the Ad(H)-invariance of “ir, and the lo-invariant construction of the metric in the fibres E,., see equation (9). In general, the splitting of RG is much more complicated. It is clear that RG is constant on each fibre E,.. Therefore we can integrate the Einstein action S = \ (RG - A,) dt)C; (13) c: over the fibres E,. and we get S = VH \
(RGIH - (AI - RH) - iyb(Fpu. F/IV)) dtl~,~.
(14)
G/H
Here d,uG and d?lG,H denote the volume forms on G and G/H, respectively. VH is the volume of the structure group H, which is equal to the volume of each fibre E, and ni has the meaning of a cosmological constant. Equation (14) gives the action of a coupled Einstein-Yang-Mills system on G/H with cosmological constant A = Ai - RH. Hence, we arrive at the following result. If the metric g, given by (9), is a solution of the Einstein equations with cosmological constant Ai, then the metric go/H and the connection r form an Einstein-Yang-Mills system with cosmological constant A = fli - RH. This is clear because every variation of go/H and w yields a variation of g. But g is a solution of a variational principle with action (13). Therefore go/H and w are solutions of a variational principle with action (14). It is a known fact that the biinvariant Killing metric glc on a semisimple compact Lie group G, arising from the negative of the Killing form K, see equation (1) is a solution of the Einstein equations with some cosmological constant. One can easily verify this statement by calculating the Ricci tensor Ric. Doing this one gets [16] Ric = igI<. Another consequence of this equation is that the scalar curvature with respect to gI< is RG = $DG.
(15) &
of G calculated (16)
where DG is the dimension of G. If we choose g = r?gh_. N > 0, as the metric on G, the Einstein equations on G will be fulfilled. The corresponding cosmological constant
A CERTAIN
CLASS OF EINSTEIN-YANG-MILLS
SYSTEMS
437
is given by ,=DG-2 -.
(17)
4cr
Thus, we can use the above construction to find G-invariant EYM systems on homogeneous spaces G/H with compact semisimple Lie group G. 3. EYM systems
on IR x G/H
Let G be a compact semisimple Lie group with Killing metric gf< and H be a subgroup of G, so that G/H is a symmetric space. Then we can construct a G-invariant static EYM system on space-time N = Ik!x G/H using the construction described in the foregoing section, with the first component of N playing the role of time. We use the Killing metric gI,- on G to obtain the EYM system (gGIII,WGIH) on G/H. Moreover, we have the projection p: N -+ G/H, which projects (t. x) E IRxG/H onto .I’ E G/H. On N we consider the static metric gx =
and on the H-bundle
-nt 6 clt +
(p*G) over G/H
pp*(gG,H).
the static connection
w.V =
(18)
/SEE&.
form
(19)
P*(dG/H).
Because the Yang-Mills equations are fulfilled on G/H, one easily shows that the Yang-Mills equations on N are fulfilled independently of 0, too. It remains to check the Einstein equations. If G/H is a symmetric space, it is easy to calculate the Ricci tensor RicGIH and the energy-momentum tensor TG,~ on G/H: R&/H
= ;gG,H.
TG/H
WV
;(4 - &/&G/H.
=
where DC;~H is the dimension of G/H.
(21)
Here we used
(22) with the components FII,, of the field strength given by equation (11). The Ricci tensor Ric.sT on N calculated with respect to gs is, because simple structure of the metric g,y (18), given by Ric.\T = +P*(gG/H). i.e., Ric.$F has no time components. the form I TX
=
The energy-momentum
$*kG/H)
DG/H - slj2tZx.
of the
(23) tensor TjV on N takes (24)
438
G. RUDOLPH
and T. TOK
Now it is a matter of fine tuning the constants N. The Einstein equations on N read
to fulfill the Einstein equations
RicN - igx ( Rlv - A,V) = K,T,v.
(25)
where A.v is the cosmological constant and K is the gravitational equations (18), (23) and (24), we calculate
Ric.v -kg,dRv
-
AN) = ip*(gGIH)
=(&+
-
2-
- 1
igAvDG/H 2P DG/H 4
riT.v = (K~-$~~)
on
constant.
Using
A‘V
p*(gG,H) + dt
‘$Idt ~
--
+ dt ii dt ( K,+).
P*(&,H)
Hence, we obtain two equations, which are equivalent to /j =
DG/H
A=4-.
K,
h
Therefore, if the cosmological constant and the gravitational constant are related by equation (26), then the system (g,v.w,v) is a solution of the EYM equations on N. This solution can also be obtained by solving the equations of spontaneous compactification [ll] on R x G/H. 4. The symmetric
space (F x F)/Fdiag
Let F be a semisimple compact Lie group with Lie algebra 3 and Killing form Kz. We can consider F as a symmetric space on which G = F x F acts transitively by (.f1. .f2)
*
f
:=
f1,
.f1,f(f2)-~.
f2,
.f’
E
F.
The stabilizer of the unit element e E F is given by Fe = ((f3.f);
f E F} = Fdiag E HT
and we can write F E (F
X
F)/Fdi,g
= G/H.
The canonical splitting of the Lie algebra 8 of G into a direct sum of the Lie algebra sj of H and a vector space !?JI is given by 6 3 (X,Y) = (f(X
+ Y). ;
- Y)) .
x. Y E 5.
i.e., 4 = {(X,X); CIJI= {(X, -X);
x E 5>. x E 5).
(27) (28)
A CERTAIN CLASS OF EINSTEIN-YANG-MILLS
SYSTEMS
439
The Lie algebra 8 is the direct sum of two semisimple Lie algebras, namely 8 = 5 $5. Therefore, the Killing form K on 6 is completely defined by Kg, namely K((X1.
X2),(K,y2))
=
JG(X1.W
+ K~(x2,Y2).
(29)
One easily obtains that fi and M are orthogonal with respect to K. In the next step we use y = -_(yK, a > 0, to construct an EYM system on F as described in Section 2. We denote the projection form G = F x F onto the first and second component by prr and pr2, respectively, and we write Oi := pry HF, 1:= 1.2, where tin is the canonical left-invariant l-form on F. It is clear that G has the structure of a principal bundle over F with the structure group H = Fdiag and projection 7r: G --f F given by 74f1.f2)
=
fdf2)-',
f1,fi
(30)
E F.
We choose the following section s: F + G in this bundle f E F.
s(f) := (f. e). With these notations
the canonical left-invariant 0=
and its projection
onto fi, respectively
Moreover,
l-form 0 on G reads
(&.e,),
M, has the form
ofi = ($(& + &Jr = (@I
(31)
#2), @l
(32)
+ W).
- e,). -f
- 82)) .
(33)
we get (34) (35)
where we have used prr OS = idF and pr2 OS = e. NOW the gauge potential and the metric gF, see equations (5) and (7) can be expressed by AF =
S*W
gF = y
=
s*Or, =
((s*d,),
($OF,~I~F),
(S'e,))
= -+aK3(8~.
AF = s*ti
(36) tip).
(37)
To derive the last equation one has to take into consideration equations (35) and (29). The scalar product “ir, in the Lie algebra fi of the structure group is given by the restriction of y to 4. Identifying H = Fdiag with F and using equation (29) we get “or = -2c~K~. (38) Comparing with equation (16) we obtain
440
G. RUDOLPH and T. TOK
where RH is the scalar curvature of the structure group and DF is the dimension of F. The metric g on G = F x F fulfils the Einstein equations on G with cosmological constant A, = (DF - 1)/2tr, see equation (17). Thus, the cosmological constant A of the EYM system consisting of gF and AF reads A = Al - RH =
PDF-4
acr
.
The physical EYM action has the form
-
&(R
A) -
$(F,,,. Fbiu))
&I.
(41)
where (.. .) is the negative of the Killing form and 8rX = K is the gravitational constant, which appears in the Einstein equations (25). It is obtained by dividing the action (14) by 2cu and identifying o with the gravitational constant K. Let us summarize our results. On every compact semisimple Lie group F there exists an F-symmetric EYM system with a gauge group F consisting of gF and Ap, see equations (37) and (36). The gravitational constant (Y and the cosmological constant A are related by equation (40) (fine tuning). We considered F as a symmetric space. Therefore, we can apply the construction described in Section 3 to get an EYM system on N = R x F. We obtain the gauge potential and the metric on N from equations (19) and (la), where the EYM system on F = G/H and the scalar product in the Lie algebra of the structure group F are given by equations (36) and (37) respectively (38) with (Y= 1. The resulting relations between the occurring constants 1) , h: and A,v can be obtained from equation (26). 5. The case SU(2)
For F = X1(2) the gauge potential coordinates zj
x SU(2)/SU(2)
-
a relation
to instantons
it is easy to write down the explicit form of the metric gst.(2) and ASLTc2)in local coordinates. We parametrize SU(2) by stereographic
i,j E
R
/j = 1.2.3, (~1’ = e
(z;j)‘.
(42)
where & denote the Pauli matrices and ll is the 2 x 2 unit matrix. The Killing form Ka~rC2jon the Lie algebra so is given by K,s,,C~j(X,Y) = 4 tr(XY).
X, Y e s,u(2).
(43)
To get the metric gsrrC2) one has to use equation (37) &x7(2)
=
-_~~,s,,(2)(H,s,,(2),Bsu(2))
=
-htr
(44)
A CERTAIN CLASS OF EINSTEIN-YANG-MILLS
SYSTEMS
441
Here & denotes the symmetrized tensor product and in what follows we write % for the Hermitian conjugate of x. With H,,,Cz)= XC’dx = -rl(x-l)x = -C&X we get
(45) It is obvious that SU(2) endowed with this metric is a 3-sphere with radius 2,/G. The gauge potential A,q,,c2, = iQRuc2) takes an especially simple form if we perform a gauge transformation A’ = t~-~A,~~,(~p + U-~~IL (46) with Z-II (47)
‘U= (xAfter a simple calculation we get
where E”“? is totally antisymmetric and ~~~~= 1. The cosmological constant .1 follows from equation (40) and has the value
If we apply the construction described in Section 3, then we obtain a static EYM system on N = LRx S3 - Iw x SU(2). In local coordinates (t. z,,) this solution reads
(51) Here K is the gravitational constant. From equations product in the Lie algebra of the structure group
Y,~,,(~)(X. Y) = -8 tr(X. Y). The cosmological
(38) and (43) we get the scalar
x, Y E X/L(2).
constant A,%,we obtain from equation A, = ;.
h
(52)
(26) and D,s1;C2)= 3: (53)
The same results were obtained in [14, 151. But in these papers the geometric structure of the solution was left in the dark. We hope that our considerations clarified this point completely.
442
G. RUDOLPH
and T. TOK
In [13] there was mentioned a relation of the gauge potential Asuczj = itlsrr(2j to the BPST instanton solution [18], but only in terms of a local coordinate chart. In the bundle language, this relation looks as follows: Let us consider the principal bundle P = SU(2) x SU(2) --+ SU(2) as a subbundle of the quaternionic Hopf bundle PM: we show that the connection form w on P, see equation (5), is the pull back of the instanton connection form Wi,,t on PH. The quaternionic Hopf bundle is given by pairs (a, b), a. b E H, with cin+bb= 1. (54) and by the right action of unimodular
quaternions
?j~Ju. b) := (m, bu),
,Ilu.= 1.
(55)
Here bar denotes quaternionic conjugation. The set of unimodular quaternions is isomorphic to the group SU(2). Therefore, the bundle P = SU(2) x SU(2) is naturally embedded by a bundle homomorphism i into the bundle PM i: P 3 (a,b) + (&a:
On PM the instanton connection
is defined by Winst
Taking into account U = u-l,
&-b) E PH.
= 6da + bdb.
(57)
for ‘Uunimodular, we obtain .*
8 Winst
=
@I
+
Q2),
where 8, = pri*BsU(rJ, i = 1,2, as defined in the foregoing section. Comparing with equations (5) and (32) it is clear that w is the pull back of Wi,st under i. This gives us the possibility to calculate the Chern-Simons index Ic of the gauge potential AsrTc2), see equation (36) in a very simple geometrically intrinsic way. If we represent the base space S4 of the quaternionic Hopf bundle as the set of quaternions plus one point, we can choose the local section (59) in the bundle PH. We denote by AI the instanton gauge potential and by FI its field strength, i.e., AI = SI*Wi,,t. Notice that the section s in the bundle P, see equation (31) is the restriction of SI under the embedding 1:.Therefore, Asric2) is the pull back of AI under i. The embedding i induces an embedding of the base space of P into the base space of Pa,, which we denote by the same letter i. The image i(M) of the base space M = SU(2) of P is an equator of S 4. We denote one of the two semispheres of S4 whose boundary is i(A4) by N. Now we can calculate
k=
& \ =J(2)
tr(Ascr(2)A
dAsw(2)
+
&xq2)
A
&u(2)
A
A.m(z))
A CERTAIN
CLASS OF EINSTEIN-YANG-MILLS
SYSTEMS
443
One easily checks that the connection AI is up to gauge transformations invariant under the natural action of SO(5) on S”. Therefore, tr(FI A 8’1) is up to a factor the volume form on S” and we get
The topological index of the basic instanton is 1, hence the Chern-Simons index k of our solution has to be i. This is an intrinsic proof that the solution found is of sphaleron type. The calculation of the Chern-Simons index in terms of local gauge potentials is much more complicated and may yield incorrect results if one chooses a singular gauge. Therefore the authors in [14] had to perform a gauge transformation before they had got the correct result. 6. A relation to Berry’s phase and the distance
of Bures
The EYM system (gsc’(a), ASLrc2)), described in the foregoing section, is well known from the study of parallel transport along 2 x 2 density matrices, see [19-211. We consider the trivial U(2) principal bundle GL(2. C) -+ GL(2. Q/11(2) -: LIZ(~). with the projection
(61)
rr given by jr(W) : =
ww*,
ui E GL(2.C).
(62)
Here star denotes Hermitian conjugation. The base space D;?(2) consists of all not normalized nonsingular 2 x 2 density matrices. On GL(2. C) we have a natural metric g which is invariant under the right action of U(2): g = !JItr(dw 6 &*). Therefore, by an analogous construction as in in the bundle (61) and a metric gB on its proposed by Uhlmann [20] and it governs the which is related to the concept of purification known as the Riemannian metric which comes is related to the transition probability between
(63)
Section 2 we obtain a connection AB base space. The connection AB was parallel transport along mixed states, of density matrices. The metric gB is from the distance of Bures [22] and mixed states.
444
G. RUDOLPH
It turns out that the connection subbundle Q2 defined by
and T. TOK
AB is reducible
Q2 := {UI E GL(2,C);
to a connection
det(w) E R+}.
[19] in the SU(2)
(64)
This corresponds to a simple property of the metric (63): If we consider GL(2. C) as a U(1) principal bundle over Q2, then every vector tangent to Q2 is orthogonal to the direction of the fibre of that bundle. Therefore, the horizontal subspaces of the connection Ag, defined as the orthogonal complements of the vector spaces tangent to the fibres of the bundle GL(2.C) + D2(2), are tangent to Qa. Another consequence of this property is that we can use the restriction of g to Q2 to construct gB. In the next step we restrict the base space of Q2 to all normalized density matrices: tr(p) = tr(‘uulw*)= 1.
(65)
We denote the resulting SU(2) bundle by $2. Obviously, there is a natural embedding j: & + GL(2, C). On $a we have the pull back j*g of the metric (63). We will show that there exists a bundle homomorphism f: Q2 + P = SU(2) x SU(2) such that j*g = f*gP, with gp being a multiple of the Killing metric on SU(2) x SU(2). Obviously, every matrix ‘UUI E y1(2. C) can be uniquely represented in the form 110 UJ= 2
i
(CL+ ib) . 0
i
(66)
1
where 0, =
zoll + 2,i(Tn,
b = yell + y,io” .
cv = 1,2,3,
x‘rry; E R.
If ~1 E &, we have det(uj) > 0 and tr(,luslj*) = 1. Hence, we obtain 0=
S(det(,ll,)) = a ($
XL.,% - 5
z=o 0 < !R(det(,u,)) = i 2
yia) (
(67)
id3
:riyi.
(68)
i=O
1 = tr(ut,/l,*) = i (t
xjZ + 5 1=0
yi2).
(69)
i=o
From equations (67) and (69) we find
&&2=-pyi2 =
1,
i=o
i=o
(70)
A CERTAIN
CLASS
OF EINSTEIN-YANG-MILLS
SYSTEMS
445
and therefore a. b E SU(2). So we can define a map f: & -+ P according to equation (66). One easily checks that this map is an injective bundle homomorphism and because of (68) the set f(&) is an open subset of P. Thus, for every (n,b) = p E f(&) with origin ‘UI= f-l(p), the tangent space T,P is isomorphic to the tangent space T,,.&. With these remarks it is a matter of a simple calculation to show j*g = ,f*gp. Let X1 and X2 be two vectors in Ttr,Q2 and (Al. &) and (AZ, &) their images in T,,P, p = (a b) E N(2) x SU(2). Taking into account equations (2) (29) and (43) as well as H(A;. B;) = (n?Ai, b+&) E s1~(2)@ SIL(~), %= 1,2. and using -(CL-lA) = (dA)*
= A’n.
(I E N(2).
we obtain gp((A1, II,). (AZ, &)) = -4cY(tr(cl,-lAln-lA2)
+ tr(b-1BIb-1B2))
= 4a(tr(A1A2*) + tr(&&*)) = 4&R tr((Ar + jB1)(A2* - i&*)) = 16~$R tr(XrX2*)),
(71)
showing that j*g = f*gp if N = &. Hence, the Uhlmann connection reduced to a connection in &, and the pull back of the Bures metric to the space of nonsingular normalized density matrices, coincide with the EYM system presented in Section 5, see equations (45) and (48). REFERENCES [I] R. Bartnik and J. McKinnon: Phys. Rev. Letf. 61 (1988) 141. [2] A. Lichnerowicz: in Theories relativistes de la gravitation et de l’electromagnetisme, Masson, Paris 1955. [3] S. Deser: Phys. Lett. B 64 (1976), 463. S. Coleman: in New Phenomenon in Subnuclear Physics, ed. A. Zichichi, Plenum, New York 1975. (41 P. Bizon: Phys. Rev. Lett. 64 (1990) 2844. M. S. Volkov and D. V. Galt’sov: Pis’ma Zh. Eksp. Teor. Fiz. 50 (1989) 312; Sov. J. Nucl. Phys. 51 (1990), 747. H. P. Kiinzle and A. K. Masoud-m-Alan: J. Math. Phys. 31 (1990), 928. [5] J. A. Smoller and A. G. Wassermann: Commun. Math. Phys. 151 (1993) 303. J. A. Smoller, A. G. Wassermann and S. T. Yau: Commun. Math. Phys. 154 (1993) 377. P. Breitenlohner, P. Forgacs and D. Maison: Commun. Math. Phys. 163 (1994), 141. J. A. Smoller, A. G. Wassermann, S. T. Yau and J. B. McLeod: Commun. Math. Phys. 143 (1992). 115. [6] N. Straumann and Z.-H. Zhou: Phys. Lett. B 237 (1990) 353. N. Straumann and Z.-H. Zhou: Phys. Lett. B 243 (1990) 33. Z.-H. Zhou and N. Straumann: Nucl. Phys. B 360 (1991) 180. 0. Brodbeck and N. Straumann: Phys. Lett. B 324 (1994) 309. P. Boschung, 0. Brodbeck, F. Moser, N. Straumann and M. S. Volkov: Phys. Rev. D 50 (1994) 3842. [7] K.-Y. Lee, V. P. Nair and E. Weinberg: Phys. Rev. Lett. 68 (1992) 1100. K.-Y. Lee, V. P. Nair and E. Weinberg: Phys. Rev. D 45 (1992), 2751. P. Breitenlohner, P. Forgacs and D. Maison: Nucl. Phys. 383 (1992), 357. P. C. Aichelburg and P. Bizon: Phys. Rev. D 48 (1993) 607. B. R. Greene, S. D. Mathur and C. M. O’Neill: Phys. Rev. D 47 (1993), 2242.
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