ANNALS OF PHYSICS 119, 305-325 (1979)

On Finite Action Solutions of the Nonlinear +Model

WOLF-DJETERGARBER," SIMONN. M. RUIJSENAARS,'ANDERHARDSEILER'

Joseph Henry Laboratories of Physics, Princeton University, Princeton, New Jersey 08540

AND

DAN BURNT

Department of Mathematics, Princeton University, Princeton, New Jersey 08540

Received June 16. 1978

The instanton and anti-instanton solutions of the two-dimensional O(3) o-model are special examples of harmonic maps, which have been studied extensively in the mathe- matical literature. We give an elementary and self-contained proof that these solutions are the only continuous maps for which the action is fmite and stationary under variations, without assuming any additional boundary conditions at infinity. An element of the proof is the vanishing of the stress tensor for a finite action solution, which actually holds true for the general O(N) o-model. For the two-dimensional 0(2/t- 1) o-model we exhibit explicit finite action solutions that do not lie in any lower dimensional sphere; the existence of such solutions has been pointed out in the mathematical literature. We also present a rigorous proof, based on Derrick’s scaling argument, that there are no nonconstant finite action solutions in more than two dimensions.

1. INTRODUCTION

In this paper we consider finite action solutions for the Euclidean O(N) non-linear u-model in A dimensions, described by the Lagrangean

(1.1)

where ai stands for a/ax, , and where n satisfies the constraint

I n(x)1 = 1 Vx E Rd. (1.2)

* On leave from Institut fur TheoretischePhysik, Universitat Gottingen, Gottingen, F. R. Germany; work supported by a DAAD grant.

+ Supported in part by NSF Grant MPS 7422844. * Supported in part by NSF Grant MCS 7682196.

305 0003-4916/79/060305-21$05.00/O

Copyright 0 1979 by AcademicPress, Inc.

All rights of reproduction in any form reserved.

306 CiARBER ET AL.

For the case a’ = 2, N = 3 finite action solutions, i.e. solutions to the corresponding Euler-Lagrange equations satisfying

were recently discovered by Belavin and Polyakof [I]. In fact, these solutions were first obtained, in a quite different context, by Eells and Sampson in 1963 [2, p. 1301. They considered the problem of finding extrema for the “energy” E(f) associated to C” mapsffrom a compact Riemannian manifold M into a (not necessarily compact) Riemannian manifold M’, where E(f) is defined by

(1.4)

Here, the f” are local coordinates of the point f(x), g’ and g are the metrics of M’

and M, and dV is the volume element of M. In case M = S2, M’ = S”-l with their usual metrics one can obtain a map n: R2 + S-l C RN from any such map f by stereographic projection of S2 onto R2. The action of n, defined in (1.3), is then twice the energy off. In this way the harmonic mapsf: S2 ---f S”-l, i.e. the maps for which E(f) is extremal, give rise to finite action solutions of the two-dimensional O(N) o-model (note that E(f) is always finite since M is assumed to be compact and f is assumed to be C”“).

There is a large literature on the existence and other properties of harmonic maps. A recent survey with extensive references can be found in [3]. In particular, it is known [3, (3.5)] that any Co mapf: S2 + S2 that makes E(f) finite and stationary is actually C” (the derivatives occuring in E(f) are in the distributional sense and are assumed to be functions). Moreover, it is known that any such map can be described by a function that is rational in z = x1 + ix, (see Ref. [3, (11.5)] and below). This shows that the solutions found in [1] and their complex conjugates are the only continuous maps from S2 to S2 for which the action is finite and stationary under variations.

In a recent paper G. Woo (apparently unaware of the relevant mathematical literature) supplied a proof of a weaker assertion, viz. that the Belavin-Polyakov solutions (henceforth BP solutions) are the only smooth maps from S2 to S2 that satisfy the Euler-Lagrange equations associated with (1.1-2) [4]. In our opinion his proof is not fully satisfactory for reasons that will be pointed out below. It is the main pur- pose of this paper to provide an elementary, more or less self-contained proof of a stronger assertion, viz. that the only continuous maps from R2 to S2 for which the action is finite and stationary are the solutions known from Refs. [2, 1,4]. That this is a stronger result may not be immediately clear to the reader. Indeed, it has been repeatedly asserted in the physics literature that any smooth finite action map n: R2 + S+l is necessarily continuous at ccj, and as such gives rise to a continuous map from S2 to SNpl. However, this is false; for example, a C” map from R2 to S1 that behaves like (cos(Znr)“, sin(/nrp) at co (with r2 = xl2 + x2?, 0 < in < i-) has finite action, but is obviously discontinuous at co.l

FINITE ACTION SOLUTIONS OF A a-MODEL 307

Our assertion, which we prove in Section 3. implies that such maps fail to make the action stationary.

It is obvious that the known finite action O(3) solutions and their transforms under O(N) give rise to finite action solutions for the O(N) a-model with N > 3. There is a wide-spread belief among physicists that apart from these trivial solutions, lying in a three-dimensional subspace of R .\, there are no other finite action solutions. This however is not true: in Section 4 we cite explicit examples from the mathematical literature [3, (Kl)] of finite action solutions for the two-dimensional O(21 t I) u-model (with I an integer greater than one) that do not lie in any lower-dimensional hyperplane of R 27i l We comment on work by Calabi [5] and Barbosa [6] that seems . to be relevant in this connection and mention some open problems.

Tn Section 5 we prove that the ti-dimensional o-model with d I- 2 has no regular finite action solutions.

Section 6 ends the paper with some concluding remarks.

2. PRELIMINARIES

In this section we summarize in a formal fashion some material on the non-linear a-model (cf. [l, 4, 81). It is included for reference purposes and to fix the notation.

The Euler-Lagrange (EL) equations associated with the Lagrangean (1 .l) can be found by using a Lagrange multiplier to enforce the constraint (1.2). They read

Lb’, $ ,U(n) 111 = 0 I == I,..., N. (2. I )

Here, A is the Laplacean in d dimensions. It is clear from (2.1) that from any solution it in d = do dimensions with N = N, one can obtain solutions 5 in d 3 d,, dimensions with N 3 N, by setting

(2.2)

where 0 is an arbitrary element of O(N). It is convenient to stereographically project P-l onto P-1, i.e. to choose new

coordinates Iii

,,‘i z .

1 + nN i = I , . . . , N - 1. (2.3)

More precisely, this is the stereographic projection from the south pole. The inverse transformation is

2wi l- W’W I’i = 1 + w . w “N ~ m- i=l ,..., N - I. (2.4)

1 We thank B. Simon for pointing out this example to us.

308 GARBER ET AL.

The Lagrangean expressed in terms of M’~ reads

aiw . 6iW 9 = 4 (1 + w . w)2

and the corresponding EL equations are

(1 + w . w) dw, - 4(w . Z,w) %& + 2(3iW . a,w> Mjc = 0. (2.6)

Here, as in the sequel, the summation convention is used. It is straightforward to verify that if n and w are related by (2.3)then n satisfies (2.1) if and only if w satisfies (2.6) (as expected of course).

Instead of the w’~ one can also introduce coordinates w; corresponding to projection from the north pole. Clearly, they satisfy (2.3-4), but with n,” replaced by -izN , and they are related to the wi by

Wi )1’! = - .

I W’W

(2.7)

Moreover, (2.5) holds true for w’ too. For the case d = 2, N = 3 it is convenient to set

z = xl + ix, , a, = $(a, - ia,), Z2 = ':;(Z1 $- ia,) (2.8)

and

W = w1 + iwz , W’ = w; + iw; .

We then have

w' = E-1 7

where the bar denotes complex conjugation, and

y _ * (I “‘Z I2 + I w.5 I”, (1 + wiq2

where e.g., w, stands for a,w. It is useful to introduce the charge density

and the charge

Q = j- dx, dx, 9(x, , x2)

(2.9)

(2.10)

(2.11)

(2.12)

(2.13)

FINITE ACTION SOLUTIONS OF A o-MODEL 309

(cf. Section 3). In terms of n the charge can be written

Q = & j dxl dx2 ~abcEiina(ain~)(ajnc~. (2.14)

Note that

Clearly, (2.11) also holds true if one replaces MT by w’, while the analogue of (2.12) for MY’ reads

9 = 1 (I 6 I2 - I 4 I”) 5-r (1 + M”W’)2

(2.16)

The BP solutions are the maps n: R2 + S2 C R3 such that the corresponding w or the corresponding w’ is a rational function of z. By using the Cauchy-Riemann equations one easily verifies that (2.6) is satisfied for such functions. From (2.11) and (2.12) and their analogues for w’ it is obvious that they are the only smooth maps satisfying

S(n) = 8~ I Q<n)l. (2.17)

3. UNIQUENESS OF THE BP SOLUTIONS

In this section we show that the BP solutions are the only continuous maps from R2 to S2 that have finite action and are stationary under a class of variations to be presently specified. To this purpose we introduce the set A? of all continuous maps n: R2 + S2 C R3 whose first order distributional derivatives are L2-functions (note that &’ is not a linear space). Thus, &’ consists of all triples of continuous functions

h q n 2 , n,) mapping R2 into [ - 1, l] and satisfying the constraints

and

Vx E R2 (3.1)

S(n) = j dx, dx, i i (ain,)2 (x1, XJ < CO i=l kzl

where the derivatives are in the weak sense. Let n E 2. We define a variation of n to be a function N from R2 x (--E, E) to

R3 having the following properties:

(i) N(*, t) E X Vt E c--E, 6);

(ii) N(x, 0) = n(x) Vx E R2;

310 GARBER ET AL.

(iii) N(x, t) is jointly continuous in x and t;

(iv) For any t E (-E, E): N(x, r) = n(x) in the complement of a compact subset K of R2; K may depend on N, but not on t.

Setting

4) = WV., t>h (3.3)

we define a map n E X to be stationary if for any variation of y1 for which CY. is differen- tiable at zero we have

a’(0) = 0. (3.4)

Our main result is the following theorem.

THEOREM 3.1. The BP solutions are the only maps in H that are stationary.

The proof of this theorem is divided into several lemmas. We first prove that if n E X is stationary, it is a BP solution (Lemmas I-5). We then show that the BP solutions are stationary (Lemmas 6-7). Accordingly, we first assume that some function n E Z is given that is stationary. We introduce its stress tensor Tij , defined by

Tij = -&(a,n) . (Q) + 2(8,n) . (a+). (3.5)

LEMMA 3.1. n has a vanishing stress tensor.

Proof. We first want to show that

aiTij = 0 j = 1,2 (3.6)

in the sense of distributions (note that Tij E L1(R2) and thus Tij E .9’(R2)). This is of course trivial for C2-solutions of the EL equations (2.1) but in our case n is a priori much less regular so that a little argument is needed (cf. also the remark after Lemma 2). Consider the family of maps ut : x H y “stretching” R2 in the following way:

Yl(L Xl 9 x2) = Xl + ml 1 x2)

Yz(t, Xl 2 x2) = x2

(3.7)

where fe C,,m(R2). According to the inverse function theorem ut is a diffeomorphism of R2 if t E (-6, E), where E < (sup 1 a,f I)-‘. We set

w-7 t> = 4@4) t E (-E, c). (3.8)

It is readily seen that N has properties (i)-(iv), i.e. N is a variation of 12. By making

FINITE ACTION SOLUTIONS OF A U-MODEL 311

a change of variables it follows that the function a(t) corresponding to N can be written

+ W2n a,n)(v)(a,f)(llll(L’)) + G2n . Gn)(Y)l. (3.9)

Again by the inverse function theorem the first coordinate of u,‘( J’) is a P-function of t, y1 and y, for f in the same interval (the second one equals ya of course). Hence, using the dominated convergence theorem, it follows that a(t) is differentiable at 0, and

However, by assumption n is stationary, and therefore, sincefwas arbitrary, it follows that ZiZiI = 0 in the sense of distributions. Of course, PiTi = 0 follows analogously, and thus (3.6) holds true.

Setting

T = T,, - iT,,( = -T,, - iT,,) (3.11)

it now follows from (3.6) that

ii,T = 0 (3.12)

(cf. (2.8)). However, it is well known that any distribution T(x, , x1) satisfying (3.12) is holomorphic in z = x1 + ix2 [9]. Therefore T is an entire function, which is also in L1(R2) by the finiteness of the action and the Schwartz inequality.

It is easy to see that this implies T = 0. (Indeed, by translation invariance it suffices to show that any entire function f in L1(R2) satisfies f(0) = 0. This follows from the inequality

< (R - 1)’ iR r dr [02n I f(re”“)l d+

< CR - I)-’ Ilflll (3.13)

by taking R -+ co.) Since T = 0, Tij must vanish too, as asserted. 1

We now want to show that n satisfies the EL equations (2.1) in the sense of distri- butions (since 5?(n) is in L1 and nk is continuous (2.1) is well defined in this sense). We first observe that

n 8p = 0 i = 1,2. (3.14)

312 GARBER ET AL.

Indeed, since nl;nk = 1 one has

0 = a,(n,n,) = bi I!$ (Nk *j,)(n, *.j<) = 2 l,1$ (r7k *j,)@,n,, *,j,) (3.15)

where the derivatives are in the weak sense and j, is an approximate identity [IO]. Since nk *j, -+ nL in the supremum norm, and ainlc *.j, + a,n,, in L2 (3.14) follows.

Consider now the variation of n, defined by

Nk(x, t) = @k(x) + gk(x)) 1 n(x) + tf(x)i-’ (3.16)

where fk E C,“(P) and t E (-6, E) for some E < (sup 1 f I)-“. By using (3.14) and dominated convergence it is straightforward to verify that the corresponding a(t) is differentiable at zero and satisfies

(3.17)

Since n is stationary and f is arbitrary, we conclude that

An, + (ain . a,n) rzk = 0 k = 1,2,3 (3.18)

in the weak sense. We have proven

LEMMA 3.2. The function n satisfies the EL equations (3.18) in the sense of distri- butions.

Remark. The last two lemmas and their proofs carry over to the general O(N) u-model.

It follows in particular from Lemma 2 that Ank is in L1. Since we did not make use of Lemma 1 to prove Lemma 2, the reader might ask at this point whether Lemma 2 could be used to give a simpler proof of Lemma 1. Indeed, it is easy to see that, e.g.,

aiTi, = 1:~ (&n * .j,) * (An * j,) (3.19)

where j, is an approximate identity and where the limit exists in the weak sense. Thus, if n would be C1 one would have

P,Ti, = a,n . An = --64(n) a,n . n = 0 (3.20)

where (3.18) and (3.14) have been used. However, so far we only know that aink is in L2. Since the product of an L2 function and an L1 function does not define a distribution in general it does not seem possible to use the above argument.

We will now consider the functions MJ and MI’ determined by n via stereographic

FINITE ACTION SOLUTIONS OF A U-MODEL 313

projection (and determining n, cf. Section 2). It is readily seen that w is a finite-valued continuous function in the open set

R,,={.xER21ng(X) # -I} (3.21)

having weak first-order derivatives in L&,(R,). We set MI = co on E,, = R2\R,,, . The sets R,, and E,, are defined analogously. Clearly, R,,: u R,, = R.

One easily verifies that

,&p T(X) = 16 [ (1 + ’ ),.qj(‘))2 3

(a~) Vx E R,,(,, C R2, (3.22)

where u’(‘) stands for either MJ or ~1’. Thus T is the same as Woo’s “first integral” of the EL equations (2.6) [4]. It seems he did not realize that the relation &T = 0 holds not only in R,. but also in R ,L,’ and hence in R2; he therefore supplied a formal argument to show that T cannot have poles. He also concluded from the relation

(WZW,)(X) = 0 Vx E R,, (3.23)

(a consequence of the vanishing of T) that either M’, or W3 (= z) vanishes identically. This does not follow without an argument; a counterexample is the function w that is equal to z in the upper half-plane and to Z in the lower half-plane. However, as a result of the next two lemmas such functions cannot be stationary.

LEMMA 3.3. w and w’ are harmonic functions in R,. resp. R,, .

Proof. We will show this for ~1. The proof for w’ is similar. We first observe that A wi is in L:,JR,). Indeed, this easily follows from (2.3) by making use of an approxi- mate identity and by using the fact that o?gxl, is in L2(R2) and dnl, in L1(R2). Using (3.18) it is now a routine calculation to verify that the EL equations (see (2.6))

(1 + w . w) dw, - 4(w . a,w)(a,w,) + 2(8,w . l&w) Wk = 0 k = 1, 2 (3.24)

hold true in R, in the weak sense. However, in view of (3.23) we have on R,,,

a,w . a,w = 0

alw . a,w = a,w . a,w = 12. (3.25)

Using this we can write (3.24) as

i(l + w . w) dw = (w . &w) aiw - Pw. (3.26)

Since (3.25) implies that 8,~ and a,w are orthogonal vectors of equal length they form a basis for R2 wherever they are non-zero. Hence the right-hand side of (3.26) vanishes

314 GARBER ET AL.

on R, . Since the Laplacean is hypoelliptic [9] it follows that JZ?~ and therefore also IV is a harmonic function in R,,: . 1

LEMMA 3.4. w or +t” is meromorphic in C N R”.

Proof. First assume E, = R2\R, is a discrete set. If w, vanishes on R, then ~1 is antiholomorphic in R,, . Since ~1’ = l/W is continuous on the open set R,,:, 3 E,, it follows that w is meromorphic in C. Hence w’ is meromorphic in C.

On the other hand, if M’,(z,J f 0 for some Z” E R,, then by continuity IV, ;/- 0 on an open set U C R,,. containing z,, . Thus by (3.23) uk vanishes on U. Since ui is har- monic by Lemma 3, it vanishes on R,, . Hence M’ is meromorphic in C.

We now assume E,. is not discrete. Let z1 be a limit point of E, , and let UC R,,, be an open connected set containing z1 Reasoning as above it follows that IV’ is holomorphic or antiholomorphic in U. But :I is a limit point of zeros of w’, and so u” = 0 on U. Since IV’ is harmonic in R,,, it then vanishes on the (open) maximal connected component K of R,u!, that contains U. However, if K # C then K has a boundary point Z$ E E,,,, , which would contradict the continuity of n. Hence, ~3’ y- 0 on C. 1

In view of Lemma 4 and (2.11) it follows that

where MJ’) equals IV or 1~’ depending on whether w or IV’ is meromorphic. It is known that for a function that is meromorphic in C this integral is finite if and only if it is a rational function. This clearly implies

LEMMA 3.5. n is a BP solution, i.e. the corresponding w or the corresponding w’ is a rational function of z.

Although the above-mentioned fact can be found-by looking very hard-in the mathematical literature [IL] we include a proof of Lemma 5 both for reasons of com- pleteness and because it shows in an elementary way why the action is quantized.

Proof. As noted above, we only need to show that finiteness of the integral (3.27) (with IV(‘) = ~1, say) implies u’ is rational. This will follow if IV assumes any value in C u {cc} only a finite number of times, since Picard’s theorem then implies M’ has no essential singularity at cc. As S(n) is invariant under the rotations MJ - (w - b)/@w -t I) it is sufficient to show +V has only finitely many poles. To this end we define

and

F(R) = 1”” d$ log(1 + I ,z(Rei$)12) 0

(3.28)

(3.29)

FINITE ACTION SOLUTIONS OF A G-MODEL 315

Denoting by P(R) the number of poles (counting their order) in ( z 1 < R we claim that

A(R) = %rP(R) + 2RF’(R) (3.30)

for any R such that there are no poles on 1 z 1 = R. To see that (3.30) holds, note that

A log(l + M-W) = 4 II’, I2

(I + II+”

and apply Gauss’ theorem for the domain 1 z 1 < R where small discs around the poles have been removed; (3.30) follows by letting the radii of the discs shrink to zero.

We now assert that F is absolutely continuous. Since F is Cm except for values R, for which there are poles on / z / = R, this will follow if lim,,,O F(R) = F(R,) < co. However, this follows from the dominated convergence theorem by writing w as the sum of its principal parts at the poles and a regular function and then estimating the logarithm of the former function in the obvious way.

We are now in a position to derive a contradiction from the assumption that P( co) = 00. Indeed, if this would be true, one would have P(R) > S(n)/477 for R > i?. But since F is absolutely continuous we can integrate (3.30) and obtain the inequality

S(n) Ja” $ > kR dr q = 8~ lRR 4 P(r) + 2F(R) - 2F(E)

> 2S(n) sRR $ - 2F(R). (3.32)

Because S(n) < a3 this gives the desired contradiction by taking R big enough. Thus, P(a) < 00, as claimed. i

From this proof one can draw an instructive conclusion. Since w is rational it has a pole of order n >, 0 at co. Now a straightforward computation shows that

iz RF’(R) = 4ntl. (3.33)

Combining this with (3.30) it follows that

S(n) = 87-s, (3.34)

where p is the total number of poles of w (counting their order) in C u ( co]. By using the invariance of S(n) under rotations it follows that S(n)/87~ is also equal to the number of times M’ takes on any finite value in C u {cc> (counting its multiplicity).

To prove the second half of the theorem, i.e. stationarity of the BP solutions, we will make use of a result from differential topology, viz. that for any smooth map n from S2 to S2 C R3 the topological degree of n (characterizing its homotopy class) equals Q(n) as defined in (2.14) (see e.g. [12]; note that Q(n) equals 1/4~ times the integral over S2 of the area form x1 dx, A dx, $- x2 dx, A dx, + x3 dx, A dx, on

316 GARBER ET AL.

S2 C R3, pulled back to S2 with n). We define { as the set of elements of Z that are continuous at co, i.e. those elements that correspond to the continuous finite action maps from S2 to S2 C R3. It is a natural question to ask whether the topological degree d(n) of such an (in general quite rough) map is also equal to Q(n). Since we are not aware of a discussion of this question in the literature, we include the next lemma, which gives an affirmative answer.

LEMMA 3.6. For any n E [ Q(n) equals the topological degree d(n) of n.

ProoJ: As mentioned above this is known for smooth maps. We shall make use of this to prove the lemma.

We first consider n E 5 such that n(x) is a constant vector for r greater than some R > 0. We set

hr = nk *jc, (3.35)

where j, is an approximate identity, and

4c = IpJ. (3.36)

Clearly,

n, = p&h (3.37)

is a smooth map for E small enough, and

I$ n, = n, (3.38)

uniformly on R2. Since n, and n are homotopic we have d(n) = d(n,) = Q(nJ. Thus i;rmffices to show Q(n) = lim,,, Q(n,). I n view of (2.14) and (3.38) this will follow

E-O bk,E = ainle in L2(R2). However, this can be seen by writing

aink, - ai& = q;l(aipk.E - aink) + &“k($ - I)

and estimating the four terms on the right-hand side in a straightforward way. Therefore our assertion holds for this class of elements of [.

Setting for such an element n of 5

i&(x) = &(-r-2x) r2 = xl2 + x22, (3.40)

one readily verifies that Q(Z) = -Q(n). Since fi equals n 0 a, where a is the antipodal map of S2, one has d(E) = -d( n and so Q($) = d(E). This proves our claim for ), any n E 5 such that n(x) is a constant vector for r smaller than some E > 0.

FINlTE ACTlON SOLUTIONS OF A o-MODEL 317

Let n now be any element of 5 that is Cr in a neighborhood of the origin. We set

(6 > 0)

44 r>S

n,(x) = n(2.x - r-l 6x) 6 z < r < 6 (3.41)

n(O) 6

r <-. 2

It is easy to see that n, is in < (note that this may be false if n is not C’ near the origin) and that d&J = d(n). Moreover, it is straightforward to verify that Q(nJ -+ Q(n) as 6 + 0. Thus,

Q(n) = 1~~ Q(ns) = 1~~ d(nJ = d(n). (3.42)

Finally, let n be a general element of 1. Defining n, by (3.35-37) it follows as before that Q(n) = lim,,, Q(nJ. However, n, is in { and Cl in a neighborhood of the origin. Consequently Q(n) = d(n), which proves the lemma. m

We are now in a position to show the stationarity of the BP solutions in the sense defined in the beginning of this section, which concludes the proof of the theorem.

LEMMA 3.7. The BP solutions are stationary.

Proof. Let N(x, t) be a variation of a BP solution n(x) such that 01’(o) exists (cf. (3.3)). We must show a’(O) = 0. This will follow if

4) 3 43 vt E (-c, E), (3.43)

since then the derivative at 0 from the right (resp. left) is 30 (resp. GO). But AJ(*, t) is in 5 and belongs to the same homotopy class as n, SO, using Lemma 6,

Thus,

Q(N(., t)) = d(N(., t)) = d(n) = Q(n). (3.44)

a(t) = S(N(*, t)) b 87r I Q(N(., t))l = 8~ I Q(n)] = S(n) = e(O), (3.45)

where we used (2.15) and (2.17). This proves (3.43) and hence the lemma and the theorem. 1

We finally should like to point out that Lemma 6 and the proof of Lemma 7 imply that the BP solutions are in fact stationary under any homotopy in 5, and that their action is not only stationary, but even minimal under these more general variations.

318 GARBER ET AL.

4. FINITE ACTION SOLUTIONS FOR THE O(N) U-MODEL WITH N> 3

It has been known for some time that there exist smooth harmonic maps from S2 to Szz C R2’+~’ (with I > I) such that the image of S2 does not lie in any lower- dimensional subspace of R 21+1 [5, 6, 13, 14, 151. As explained in the introduction such maps give rise to smooth finite action solutions to the EL equations (2.1). It is the purpose of this section to briefly describe these solutions, to comment on related work and to point out some problems that seem to be open.

From the viewpoint of the u-model a natural way to obtain the solutions mentioned above is by the following argument. Consider instead of (2.1) the seemingly more general equation

An, + f(n, x) nt = 0 k = I,..., N (4.1)

under the usual constraint 1 II ( = 1. For this to be consistent one must clearly have f = -n . a,+~ - n . a,a,n = a,n . a,n + a,n . a,n = Z(n) which follows by differ- entiating n . n = 1. Thus we are led back to (2.1). One easily concludes from this that any solution of the equation

t(l + r2)2 dn, = cml, CXER (4.2)

(where r2 = x12 i x2”) satisfying the constraint / n 1 = 1 is actually a finite action solution to (2.1) with S(n) = -4 z-01. However, the operator on the left-hand side is just the Laplace-Beltrami operator d, on S2 (i.e. minus the ‘total angular momentum’ operator) in the coordinate chart obtained by stereographic projection. (Note that the metric induced on R2 by stereographic projection is

and remember the well-known formula

A, = g-l/2ai gijgl/2aj (4.4)

where g = det gij .) As every physicist knows, the eigenvalues of A, are --1(/ + I), I = 0, 1, 2,... and any eigenfunction for this eigenvalue is a linear combination of the well-known spherical harmonics YLm, transformed to x1 , x2 variables instead of the customary polar angles 0 and r$. Once this observation is made it is easy to find solutions of the type mentioned above: one simply sets

ndx, , x2) = (&) & D;i,m+zi-lYzmn(xl, x2) k = L..., 2-t 1, (4.5)

where U is a unitary (2Z+ 1) x (2Z+ 1) matrix such that nJc is real-valued. Then by the addition theorem for the spherical harmonics one has 1 n(x)/ = 1 for any

FINITE ACTION SOLUTIONS OF A U-MODEL 319

x in RZ, and therefore IZ is a solution with action 47~l(l+ l), not lying in a lower- dimensional hyperplane of RzL+' (since this would contradict the linear independence of the spherical harmonics). As a simple example we mention the solution

n(xl , x2) = (1 + r”)-“(2(3)‘12(x12 - x,~), 1 - 4r2 + r4, 4(3)‘/” x1x2 ,

2(3)'/" x2(1 - r2), 2(3)‘/” x,(1 - F)). (4.6)

Clearly, several questions arise at this point. Are these polynomial maps (and their transforms under conformal maps of S2) the only harmonic maps from S2 to S2z (with I > l)? Do they correspond to local minima of the action? Are there any finite action C2 solutions of the EL equations (2.1) that are discontinuous at co (and therefore do not correspond to harmonic maps from S2 to S”) ? It seems that the answer to these questions is not known. Regarding the last question it may be useful to point out that any C2 solution of (2.1) satisfies (3.6). Thus, by the same argument as in Lemma 3.1 T<j must vanish if the solution has finite action, i.e. one must have

a,n . a,n = 0

a,n - a,n = a,n . a,n. (4.7)

In geometrical terms this means that the map II: R* -+ S” has to be conformal. (As a consequence one obtains the known, but weaker result that a harmonic map from S2 to S” must be conformal.)

In connection with the first question we should like to mention work by Calabi [5] and Barbosa [6] on minimal immersions of S2 into S”. Instead of considering extrema of the action functional they study extrema of the closely related functional

A(n) = J dx, dx, [(aIn . a,n)(a,n * a,n) - (&n . a2n)2]1/2 (4.8)

where n is a smooth map from S2 to P. (Geometrically this functional is the total unoriented area of the image of the map.) Using the inequality (det M)‘j2 < 4 Tr M for M a positive 2 x 2 matrix it easily follows that

A(n) < G(n) (4.9)

with equality sign if and only if it is conformal. Hence, if 12 is a conformal map mini- mizing the area it is a harmonic map. (The converse is also true, but less trivial [16].) Now it is obvious that A(n) = A(n o d) for any diffeomorphism d of S2. As a result one can obtain from the area minimizing maps n studied in [5] and [6] finite action solutions n’ by setting ~1’ = n . d whenever a diffeomorphism d exists such that n 0 d is a conformal map. It seems not to be known whether such a diffeomorphism exists if the map fails to be an immersion at some isolated points, as is the case here. Thus it is not clear whether Barbosa’s maps lead to harmonic maps that are not polynomial in the sense described above.

sgs/I w/2-6

320 GARBER ET AL.

5. ABSENCE OF FINITE ACTION SOLUTIONS IN d > 2 DIMENSIONS

The result of this section, viz. that all C2 solutions of the EL equations (2.1) in d > 2 dimensions have infinite action (apart from the constant maps of course), will not come as a surprise to many readers. Indeed, it is well known that the action is not invariant under the scale transformation n(x) + n(tx); in the physics literature (e.g. [17, 181) it is argued that this implies absence of finite action solutions, since the action of a solution to the EL equations should be stationary under variations.

We should like to point out that this argument is incomplete. A smooth functionf is a solution to the EL equations if and only if its action in the ball 1 x / < R is stationary (for any R > 0) under any smooth variation that differs from fin a fixed compact subset of this ball (cf. the variation (3.16) and Lemma 3.2). A priori it is by no means clear that this implies the total action of a solution is stationary under a scale transformation, which has a global character. The Prasad-Sommerfield monopole is a case in point [19,20].” Nevertheless, in the case under consideration this heuristic argument can be made rigorous, as will be seen in the proof of the following theorem. (A similar argument for Higgs models has been used in [2].)

THEOREM 5.1. The only C2 jinite action solutions of the EL equations (2.1) in d > 2 dimensions are the constant maps.

Proof. Let (n, ,..., nN) be a C2 finite action solution of (2.1) with S(n) > 0. It

is sufficient to show S(n) = 00. To this end we set

N(x, t) = n(tx) tER (5.1)

and introduce the functions

and

D(Ry t, = s,,,=, dW9 WW, t)) (5.2)

A(R, t) = I” dr D(r, t), 0

(5.3)

where dS(R) is the usual surface measure on SN-l(R). Evidently, we have the relations

and

4R, 1) T S(n) for RA co (5.4)

A(R, t) = t2-d 1 ,<tR dx, ... dxx, S?(n(x)). z\

(5.5)

2 We thank R. Weder for pointing out this example to us.

FINITE ACTlON SOLUTIONS OF A U-MODEL 321

Since D(r, 1) is continuous in r it follows from this that

ii(R, 1) = (2 - d)A(R, 1) 1 RD(R, I). (5.6)

On the other hand we also have

A(R, t) = f dY(P,N,J(.Y, f)(8,N,)(S, t) * irj<R

(5.7)

whence by dominated convergence

(5.8)

(5.9)

Ai(R, 1) == 2 j;,,,, cix-(ai~?,)(x)(aii~~)(-Y, 1).

In view of Green’s theorem and the fact that

(dn,)(x) Nk(X, 1) =.= -s?(n(x)) n,,(x) xL(2LnJ(x) = 0

we then have

(5.10)

From this one easily infers, using (5.4) and (5.6), that there is an R, > 0 such that

RD(R, 1) > E > 0 VR > R,, (5.11)

Consequently,

(5.12)

as claimed. 1

6. CONCLUDING REMARKS

(1) The BP solutions can be naturally associated with finite energy solutions

for the Minkowski 2-dimensional O(3) a-model. Indeed, for the BP solutions w is a rational function of x1 + ixz or x1 - ix2 and thus it makes sense to set

u(x, r) = w(x, it). (6.1)

Evidently, the real and imaginary parts u1 and v2 are rational functions in s - t or x + t and therefore the corresponding n is also a rational function of s - t

322 GARBER E.-I AL.

(resp. x + t). However, any function of x - f or x + t trivially satisfies the Minkowski EL equations

an, + (&n . &n - a,n . L&n) nk = 0 k = 1, 2, 3. (6.2)

Thus, n is indeed a solution. To see that its energy,

e(n) = + j dx (a tn . &n + 0 . %9)(x, j), (6.3)

is finite observe that

c(n) = s dx(8,n . a,n)(x, 0)

= 4 s ( dx azv . accv ) (x, 0) (I + v - v)” < 4 j- Wl + x2) ( (la; ;:tj2) (x, 0). (6.4)

However, the last expression is up to a constant just the energy (1.4) associated with n(x, 0), considered as a smooth map from S1 to S2, where S1 is mapped onto R by stereographic projection (n defines a smooth map since it is a rational function of x). It follows that c(n) is finite, as claimed.

Conversely, any unit vector function n(x, t) whose three components are rational functions in x - t or x + t has finite energy and gives rise to a BP solution through (6.1). Thus the BP solutions naturally correspond to a special class of “massless lumps” (cf. [22]). Note that this class contains O(2) solutions like

n(x, t) = [l + (x * t)““]-‘(2(x If t)“, 0, 1 - (x * t)2% (6.5)

corresponding to w = Z~ (resp. w = 5”).

(2) In a recent paper D. Gross has studied meron configurations in the 2-dimen- sional O(3) u-model [8]. The most general singular 2N meron solution he obtains (satisfying w + 1 for z --+ co) can be written

. i-6, )I 112

z-bi ’ U( ) bi E c. (6.6)

This clearly is a smooth solution of the EL equations except at the points ai , bi , where it is discontinuous. Since 1 w 1 = 1 the corresponding n maps S2 onto the equator of S2 C R3 (cf. (2.4)). Since the charge of a subset U of R2 (i.e. the integral over U of the charge density) equals 1/4~ times the oriented area of n(U) it follows that the charge density of (6.6) vanishes outside the points ai , bi . (Alternatively, this directly follows from (2.12).)

FINITE ACTION SOLUTIONS OF A U-MODEL 323

Considering the singular 2 meron solution

U’ = (z/Z)‘l”, (6.7)

Gross points out it may be regarded as a limit of the continuous map

’ z/r1

I

1 z I < rl “‘QJ, = (z/2)1/2 rl < 1 z i < r, (6.8)

4r2 r2 < 1 z I

by taking rl + 0, r, ---f co. This map is clearly homotopic to w = z; the charges inside and outside the ring equal $. This leads him to associate a charge density

Q(x) = $6(x) + 4 8(x - co) (6.9)

with the map (6.7). It may be of interest to point out that the general solution (6.6) can be regarded

as a similar limit of continuous maps with degree -N < d < N. Indeed, one can leave out small circles around cli , bi , map a, and bi to either the north pole or the south pole, and interpolate continuously in the obvious way. The charge inside the circle is then &+ depending on whether the center is mapped to the north or south pole. (We tacitly assume here that all a, , bi are different. If, say, a, = a2 = ... = ale = a

then the charge inside the circle around a will be &tk/2.) For the map (6.7) one could e.g. take rJ.7 in (6.8) instead of z/r1 , corresponding to a map of degree 0. (Gross calls the latter choice a meron-anti-meron configuration.) Thus, the 2N meron solution (6.6) can be regarded as a “boundary point” of the homotopy classes with degree -N<d<N.

(3) Since the map n corresponding to (6.6) maps S2 onto the equator of s” C R3

n is also a singular solution for the O(2) o-model. For such maps n: s” + S’ C R3 one clearly has u’ = exp(i$), corresponding to n = (cos 4, sin 4,O). It is easy to see n is a smooth O(2) solution if and only if on R2

04 =O, (6.10)

i.e. if and only if 4 is a harmonic function in R2. The action of an O(2) map IZ (infinite for smooth non-constant solutions) equals the Dirichlet integral (electrostatic energy) of&

S(n) = j- d.y, dx, a,~$ ai+. (6.11)

Singular solutions can be obtained by considering Poisson’s equation A+ = p

where p is a given distribution. If p is a finite sum of delta functions (corresponding to charged point particles) it is well known that the electrostatic energy (6.11) is infinite and that the potential r$ goes to co at the location of the particles.

324 GARBER ET AL.

If we look at the simplest example, 4 = log r, corresponding to a point charge at the origin, we see that it produces the same energy density as the meron solution (6.7) (corresponding to 4 = tan-‘(x.JxJ which is the conjugate harmonic function to log r). The corresponding map can also be obtained as a limit of continuous maps (in any homotopy class in this case) by a similar interpolation as in (6.8).

We may also consider line charges and thereby obtain a continuous 4 and a finite action (6.11). An example is the potential

where

f(z) Ez [(Z‘J - a”)(z” - by-v. (6.13)

Here, c > b > a > 0, and the integration contour in (6.12) is to be taken in the upper (lower) half plane for x2 > 0 (x2 < 0). The corresponding map II is a smooth finite action solution of the EL equations except on the intervals [--b, -a], [a, b], where it is only continuous and its x2 derivative has a jump discontinuity. (Note that S(n) can be written

on account of (2.1 l).) Gross’ meron solutions (6.6) do not correspond to ordinary electrostatics solutions:

A map like (6.7) for instance is from the point of view of electrostatics not only singular at the origin but also on a ray extending from the origin to infinity. In the O(2) model these rays disappear since 4 only has to be continuous modulo 2~r.

It would be of interest to investigate the relevance of these singular O(2) solutions for the quantum field theory version of the O(2) u-model.

(4) Let us close with one more slightly speculative remark: It has been noticed that there is a rather close, if somewhat mysterious, analogy between the nonlinear u-model in two and the Yang-Mills theory in four dimensions. In the case at hand, it was the vanishing of the stress tensor that reduced the second order field equations to the first order (anti-) Cauchy-Riemann equations. In the instanton problem for the four-dimensional SU(2) Yang-Mills theory it is widely believed that a similar reduction to the first order (anti-) selfdual’ity equations is possible. There is an inter- esting paper [23] showing that vanishing of the stress tensor actually does imply at least pointwise (anti-) selfduality. It would therefore be most interesting to find out whether in this case the stress tensor also has to vanish for finite action solutions.

Note added in proof. After completion of this manuscript we were informed about a paper by P. Butera and M. Enriotti [7] which also recognizes that the vanishing of the stress tensor (Lemma 3.1) is important for the determination of all finite-action solutions.

FINITE ACTION SOLUTIONS OF A U-MODEL 325

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