A conformally invariant σ-model in 4 dimensions

Volume 94B, number 2

PHYSICS LETTERS

28 July 1980

A CONFORMALLY INVARIANT o-MODEL IN 4 DIMENSIONS B. F E L S A G E R Niels Bohr Institute, DK-2100 Copenhagen O, Denmark and J.M. LEINAAS

Nordita, DK-2100 Copenhagen O, Denmark Received 25 April 1980

The two-dimensional O (3)-nonlinear o-model is generalized to a four-dimensional model. The generalization preserves conformal invariance and several other properties of the O (3)-nonlinear o-model. The instantons of the model are examined and shown to correspond to conformal mappings from R4 into S4. The close relationship with SU(2) Yang-Mills theory is pointed out.

As pointed out by Belavin and Polyakov a few years ago [1 ] there exist some remarkable similarities between the O (3) non-linear a-model in 2 dimensions and the SU(2) Yang-Mills theory in four dimensions. In particular they have the following four properties in common: (i) The action is conformally invariant. (ii) euclidean field configurations with finite action are characterized by an integer - the winding number of the configuration. (iii) The instanton configurations correspond to the solutions of a first order differential equation - the self-duality equation. (iv) The self-duality equation can be solved exactly, and for the 0 ( 3 ) non-linear o-model the instantons correspond to conformal mappings from the .euclidean plane into the two-dimensional sphere, R 2 ~ S 2. In this note we shall discuss a generalization of the 0 ( 3 ) non-linear o-model to a four-dimensional model, which has the same four properties. This generalization is based upon a geometrical representation of Yang-Mills fields, similar to the one which was used in a previous work to study the geometry of magnetic fields in three-dimensional space [2]. The model in fact has a geometrical and topological structure which makes it directly related to the SU(2) Yang-Mills theory. This geometric interpretation as well as a detailed discussion of other properties of the 192

model will be published elsewhere [3]. Here we will mainly focus attention on the properties listed above and in particular examine the self-duality equation o f the model. The model is characterized by an order parameter, which is a unit vector in 5 dimensions: ni(x), i = 1, .... 5; ni(x) ni(x) --- 1. The order parameter thus defines a map from four-dimensional s p a c e - t i m e into the four dimensional unit sphere, R 4 -+ S4. Introducing the skew-symmetric matrix field /~]v(x) = auni~vnJ-- avniaun/ ,

(1)

we define the lagrangian by ~=

lf~ij

^ ""

(2) = 1 {(~u niO¢ahi)2 -- ~, niov niO, nJOvnJ} " This lagrangian has the same form as the quartic part of a lagrangian originally introduced by Skyrme [4], but with n i here being a five-dimensional unit vector rather than a four-dimensional vector as in Skyrme's model. The action corresponding to this Lagrangian is conformally invariant in four dimensions. This follows from the fact that it is quartic in the derivatives. Note, however, that due to the skew symmetry in-

Volume 94B, number 2

PHYSICS LETTERS

volved the lagrangian includes only terms up to second order in the time derivatives. The .requirement of finite action forces the matrix field Fffv (x) to vanish at infinity, ile.

x2(OuniOvnJ-- ~ v n i ~ u n / ) x - ~

O.

(3)

This does not imply that the order parameter ni(x) necessarily approaches a constant at infinity, but only that the four vectors 31ni , ..., 04nl tangent to the four-sphere become parallel. This means that the asymptotic 3-sphere in R 4 is mapped into a one-dimensional curve in S4 . As a consequence of this the full space R 4 is mapped into a closed hypersurface in S4, and the mapping can therefore be characterized by a winding number, which is given by

•m =

1

.feOkl m euvtwnmO~niOvnfOonkOonl d4x

647r:~ - R4

(4)

Using the definition (1) of the matrix fields p~' (x) we can re-arrange the action in the following way

28 July 1980

FiJ pij = + 16rr2(SuoSv ° _ 8 u o S v o ) m ( x ) ~v- Oo -

(10)

This can be interpreted as an orthogonality relation between the matrices/~/~. Thus, if we interpret these matrices as vectors in the six-dimensional vector space consisting of skew symmetric matrices T i/, constrained by the condition TUn ~ = 0, then eq. (10) shows that at each point x the six vectors/?~ . . . . . /+~ are mutually orthogonal and all have the same length. This is consequently a necessary condition to saturate the lower bound ofeq. (6). However, this is also a sufficient condition. In order to show this we introduce the following twelve matrix fields AiJ

G+-Iav = 6 i j k l m

n,nkkl +

(11)

lay -

These matrices are also vectors in the six-dimensional vector space considered above. If we assume the vectors/~/2, ..., F ~ to be orthogonal, and all of the same length, then it follows from eq. (9) that all the twelve " 6 v are orthogonal. Therefore six of these vectors G+_u vectors have to vanish. The length of the vectors is furhter given by (no summation over/~v),

1 f nm~klT_euvoj~;i~)2dex+_16rc2m. S =~ (ei/klrn ,v (5)

Off

It is consequently bounded below by

When re(x) > 0 this shows that dPi,,,0P2,,,:, o and consequently the matrices G z'/'uvhave to vanish. The field configuration n Z(x) is then self-dual. When m (x) < 0 the matrices G/Ju~ similarly have to vanish and the configuration is then anti-self dual. We have now shown that the self duality equation (7) is equivalent to the orthogonality relation (10) between the matrix fields/?/~. As the last step in our proof we express this relation in terms of the tangent vectors ~u hi'

S~> 167r2 I m [ .

(6)

This bound is saturated for the field configurations satisfying the following self-duality equation

etlt~t??Z ..... n ' F ,UP kl = +~ e la vp c~F'i!o P '

(7)

which therefore characterize the instantons of the model. In terms o f n i ( x ) i t corresponds to the following non-linear first-order equation 6ijklm

n m OvnkOvnl = +_euvpeOpniOon] .

(8)

We will now prove that the solutions of this equation correspond to conformal maps R 4 -~ S4. The proof is based upon the following identity, which follows from the definition (1) of t ?i]

~ij ±,v--+uv = 8 ~" /' u F^~' -+ 128~2m(x) •

~u ni~v n J~p ni~an j - ~ ni~ v n J3 o ni~p nJ (13) = _+87r2(SuoSvo -- 6 # e S v a ) m ( x )

=

%va~F~f3 ~ ij

647r2(8~pSvo -- 8uaSvp)m(x ) ,

where m ( x ) is the winding number density defined by the integrand of eq. (3). When F~/v satisfies the selfduality equation (7) it follows from eq. (9) that we have

•

One can readily show that this implies the following equation

~uniOvnirn(x) = 0 , eiJ k l m n m ~ k l vv

(12)

12 "/=v .

(14)

Thus, either m ( x ) = 0, which according to eq. (13) means that one has the vacuum configuration Fu/~(x) = 0, or otherwise, the vectors 3un i are all orthogonal. With the length of the vectors determined by eq. (13) one can write the orthogonality relation in the form

193

Volume 94B, number 2

PHYSICS LETTERS

OuniOun i = (8rr 2 Im(x)l)l/26 ,v .

(15)

This shows that the tangent vectors alni .... , ~4 ni are orthogonal and have the same length at any point x. Consequently the mapping R 4 ~ S4 defined by ni(x) is a conformal mapping. This concludes our proof. We have shown that selfduality of the field configuration means that either the vacuum equation/~]u = 0 is satisfied, or the mapping ni(x) is a conformal mapping. This is analogous to what is the case for the 0 ( 3 ) non-linear o-model in two dimensions. One should however note that the conformal mappings are much more restrictive in four dimensions than they are in two dimensions. They include the stereographic projection from R 4 to S4

ni(x) -

2xi

i = 1,...,4

n 5 ( X ) - x 2x 2 -+ 11 , (16)

l+x2, which defines a one-instanton configuration. However, all other configurations corresponding to conforreal maps R 4 -+ S4 are obtained from this and the corresponding anti-instanton configuration by applying the transformations from the continuous 15-parameter group o f conformal mappings in four dimensions. As a consequence of this there are no stable multiinstanton configurations included in the model. In addition to the one-instanton and anti-instanton solutions, it is trivial to verify that the model includes meron solutions. A one-meron solution is given by the spherically symmetric configuration

ni(x)=xi/x 2 ,

i = 1 . . . . . 4,

n5(x)=0.

(17)

Two-meron solutions are obtained by applying conformal transformations to this one-meron solution. Finally, we will make some comments on the connection between the present four-dimensional model and SU(2) Yang-Mills theory. For a more detailed discussion of this point we refer to ref. [3]. One can associate with each configuration o f the order parameter ni(x) an 0 ( 5 ) gauge potential defined by

~l~'(x) = ni(x)3un/(x ) - n/(x)3uni(x).

(18)

The corresponding 0 ( 5 ) gauge field/?~/~ (x) is precisely the one defined by eq. (1). With this choice of gauge potential the order parameter is covariantly constant,

bun i +ft~'nJ = 0 .

194

(19)

28 July 1980

As a consequence of this the 0 ( 5 ) gauge field can be reduced to an 0 ( 4 ) gauge field corresponding to rotations around the vector ni(x). Since the group 0 ( 4 ) is locally the product of two SU(2) groups this shows that one can associate with the field configurations ni(x) o f the model two SU(2) gauge fields. As shown in ref. [3] these two fields are strongly correlated. Their lagrangian density is, apart from an irrelevant constant, the same as in the O(5)-model and their Pontryagin densities coincide, up to a sign, with the winding number density in the O(5)-model. Further, the self-duality equation of the O(5)-model reduces to the conventional self-duality equation for the SU(2) gauge fields, and the instantons in the 0 ( 5 ) model correspond to the BPST-instantons [5]. One should however note that the present model is not equivalent to the SU(2) Yang-Mflls theory. This is in particular demonstrated by the fact that the model does not include stable multi-instantons. As a final remark we will point out that it is possible to give the O (5) model considered here a quarter. nionic form. In this formulation the model is closely related to some recently discussed HP n-models [6]. In fact, as shown in ref. [3], the 0 ( 5 ) model is equivalent to a HP 1-model, and the results reported in this note therefore apply equally well to this quarternionic model.

References [1 ] A.A. Belavin and A.M. Polyakov, JETP Lett. 22 (1975) 245. [2] B. Felsager and J.M. Leinaas, Nucl. Phys. B166 (1980) 162. [3] B. Felsager and J.M. Leinaas, A generalization of the nonlinear o-model to four dimensions, Nordita preprint 80/8 (1980). [4] T.H.R. Skyrme, Proc. Roy. Soc. A260 (1961) 127. [5] A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Y.S. Tyupkin, Phys. Lett. 59B (1975) 85. [6] J. Lukierski, Four dimensional quaternionic o-models and SU(2)gauge fields, preprint TH 2678 CERN (1979); F. Gtirsey and H.C. Tze, Complex and quaternionic analyticity in chiral and gauge theories, Ann. Phys. (N.Y.), to be published; D. Maison, Some facts about non-linear o-models, preprint Max-Planck-Institut fiir Physik and Astrophysik (Mtinchen, November 1979).