Symmetric calorons

Calorons (periodic instantons) interpolate between monopoles and instantons, and their holonomy gives approximate Skyrmion conﬁgurations. We show that, for each caloron charge N (cid:1) 4, there exists a one-parameter family of calorons which are symmetric under subgroups of the three-dimensional rotation group. In each family, the corresponding symmetric monopoles and symmetric instantons occur as limiting cases. Symmetric calorons therefore provide a connection between symmetric monopoles, symmetric instantons and Skyrmions.

generates caloron solutions [14], possibly all of them (see [15] for a recent analysis). In the last few years, this construction has been used to investigate and interpret caloron solutions, especially those for which the holonomy Ω is non-trivial at spatial infinity [16][17][18]; but this recent work was not concerned with symmetric solutions as such. In this Letter, we shall see how symmetric calorons arise from the ADHMN construction.
A special case of this is where A µ is independent of x 4 = t; this is a monopole, where we make the usual interpretation of A t as a Higgs field Φ. The holonomy (or Wilson loop) in the t-direction takes values in the gauge group; under a periodic gauge transformation, it transforms as The quantity Ω(x j ) is, in general, non-trivial at spatial infinity [11]; but for the examples below, Ω(x j ) tends to a constant group element (in fact the identity) as r → ∞. Such a field may be viewed as an approximate Skyrmion configuration; the Skyrmion number is the degree of Ω, and the normalized Skyrme energy is Provided Ω is asymptotically trivial, the topological charge (caloron number) is an integer, and is equal to the Skyrmion number of Ω [11]. In the t-independent (monopole) case, it is also the monopole number, provided we take β to be related to the asymptotic norm of the Higgs field by A large number of caloron solutions can be generated [10] by the Corrigan-Fairlie-'t Hooft [19] or Jackiw-Nohl-Rebbi [20] ansatz. These express the gauge potential in terms of a solution φ (periodic, in the caloron case) of the four-dimensional Laplace equation. For example, the component A t is given by where σ j are the Pauli matrices. For the JNR solutions one has φ → 0 as r → ∞, whereas for the CF'tH solutions one has φ → 1 as r → ∞. In the case of instantons on R 4 , one regards the CF'tH solutions as being limiting cases of the JNR solutions, but for calorons it is the other way round: to produce an N -caloron in JNR form, one uses a φ with N poles (not N + 1 as for instantons), and this is a limiting case of the CF'tH form with N poles.
To illustrate this, let us review the N = 1 case. The 1-caloron (with trivial holonomy at infinity) is generated [10] by the 1-pole function where µ = 2π/β, and W > 0 is a constant. This caloron is spherically-symmetric; it depends on the period β and on the parameter W . The gauge field is not affected by an overall scale factor in φ, so the W → ∞ limit of (7) gives, in effect, the JNR-type solution with this corresponds to a 1-caloron which is in fact gauge-equivalent to the 1-monopole [21]. Another way of viewing things is to use the dimensionless combination θ = β/W 2 : for θ = 0 (or W → ∞) we get the 1-monopole, while for θ → ∞ (or β → ∞) we get the 1-instanton on R 4 . In other words, we have a one-parameter family of sphericallysymmetric calorons, with the 1-monopole at one end and the 1-instanton at the other end. The holonomy Ω(x j ) can be computed exactly in this case [22,23]; if one restricts to spherically-symmetric gauges, then Ω is actually gauge-invariant. The Skyrme energy (3) of this configuration Ω attains a minimum for θ ≈ 7; this minimum is only slightly less [22] than the value obtained from 1-instanton holonomy. It is straightforward to produce spherically-symmetric calorons of higher charge in this way: for example, the function generates a spherically-symmetric 2-caloron, for any t 0 ∈ (0, β) and W, W > 0. The holonomy of this is a spherically-symmetric (hedgehog) 2-Skyrmion configuration (cf. [1,4]). The limits β → ∞ and W, W → ∞ are both regular; the former is a 2-instanton, but the latter is not a 2-monopole (since, unlike in the N = 1 case, the t-dependence cannot be gauged away). It seems very unlikely that the CF'tH ansatz can yield any examples (other than for N = 1) of symmetric calorons having symmetric monopoles as a limiting case-for that, one needs more general solutions. A way of generating such solutions is described in the next section.

The ADHMN construction for calorons
There is a construction which produces caloron solutions [14]; for gauge group SU (2), and for calorons which have trivial holonomy at infinity, it is as follows. As before, N is a positive integer which will turn out to be the caloron charge, and β is a positive number which will turn out to be the caloron period. It is convenient to use quaternion notation, with a quaternion q being represented by the 2 × 2 matrix q 4 + iq j σ j ; in particular, x µ corresponds to the quaternion x = t + ix j σ j . The unit quaternion (q 4 = 1, q j = 0) is denoted 1.
The Nahm data consists of four Hermitian N × N matrix functions T µ (s), and an N -row-vector W of quaternions, such that T µ (s) is periodic in the real variable s with period 2π/β, and the Nahm equation is satisfied. The trace is over quaternions, so the right-hand side is an N × N Hermitian matrix (as is the left-hand side). Given such data, we construct a caloron as follows. Let U(s, x) be an N -column-vector of quaternions, and V (x) a single quaternion, such that Note that both T j and U are periodic in s, and have jump discontinuities at one value of s, which we have taken to be s = π/β. The discontinuities could equally well be located anywhere else; the choice in (10) and (11) is for later convenience. Note also that the overall quaternionic phase of the N -vector W = [W 1 . . . W N ] is irrelevant; so we may, without loss of generality, take W 1 to be real. The pair (U, V ) determines the caloron gauge potential according to where Λ is a quaternion satisfying Λ † Λ = 1; this corresponds exactly to the gauge freedom in A µ .
By contrast, the usual formulation of the ADHMN construction for monopoles involves three matrices T j (s), satisfying In this case, the T j (s) are not periodic in s, but rather are smooth on the open interval |s| < 1, with poles at the endpoints s = ±1. (The length of this interval sets the scale of the monopole.) In addition, the T j satisfy The idea here is that given a solution of the monopole Nahm equation (13), one may re-interpret it as a solution of the caloron Nahm equation (10), with T 4 = 0 and with a suitable choice of W , namely such that We need to take β > π, so that the T j are bounded for |s| π/β. The symmetric part of T j can, because of (14), be regarded as a continuous periodic function on [−π/β, π/β]; while the antisymmetric part of T j has a jump discontinuity as in (15). The limit β → π is the original monopole, while the limit β → ∞ gives an instanton on R 4 . This instanton limit works as follows. For β π , we are solving (11) on the small interval |s| π/β, so we may approximate the solution as U(s) = U 0 + U 1 s. Eq. (15) then gives where T j = T j (0), and where U 0 and V satisfy the constraint then this is exactly the ADHM construction [24] for instantons, with the ADHM matrix ∆ being given by This ∆ is an (n + 1) × n matrix of quaternions, satisfying the condition that ∆ † ∆ is an n × n real matrix.
Let us now consider calorons which are symmetric under subgroups of the three-dimensional rotation group acting on x j . For any rotation R, let R 2 ∈ SU(2) denote the image of R in the 2-dimensional irreducible representation of SO (3); in other words, R acts on the quaternion x according to For the corresponding caloron to be G-invariant, we need an additional condition on W , and this is easily seen (from (10) and (11)) to be where τ R , for each R ∈ G, is some quaternionic phase (namely a quaternion with τ † R τ R = 1). So given a symmetric monopole, there is a family of symmetric calorons parametrized by the solutions W (if there are any) of (15) and (20). In the N = 1 case, for example, we have G = SO(3) (spherical symmetry) and T j = 0; and W is an arbitrary positive constant, which is precisely the parameter appearing in the expression (7). In the next section, we shall see that analogous one-parameter families of symmetric calorons exist for N = 2, 3 and 4.

Symmetric examples for N = 2, 3, 4
We begin with the N = 2 case, taking G = SO(2) (corresponding to rotations about the x 2 -axis). The solution of (13) which generates the axially-symmetric N = 2 monopole is T j (s) = f j (s)σ j (not summed over j ), where Then (15) and (20) have a solution W which is unique (given that W 1 is real), namely So we get a family of N = 2 axially-symmetric caloron solutions, depending on the parameter β > π. It is possible to solve (11) analytically, and hence obtain exact expressions for the caloron (cf. [25] for the monopole case), although the expressions are rather complicated. The limit β → π is the 2-monopole, and β → ∞ is a 2-instanton on R 4 , generated by the ADHM matrix This axially-symmetric 2-instanton can be obtained in the JNR form, and its holonomy was used to approximate the minimum-energy 2-Skyrmion [4,26]. The holonomy Ω of the caloron gives a one-parameter family of axiallysymmetric 2-Skyrmion configurations; as in the N = 1 case, this gives an approximation to the true Skyrmion which is better than the instanton one, but only marginally so.
Let us now consider the N = 3 case. There is a 3-monopole with tetrahedral symmetry [8,27], corresponding to the following Nahm data. (Note that the T j in [8,27] have to be multiplied by a factor of −i to agree with the conventions used here.) Define Here ℘ is the Weierstrass p-function satisfying ℘ (u) 2 = 4℘ (u) 3 − 4. The unique solution of (15), with W 1 > 0, is Explicit calculation then verifies that (20) is satisfied for each of the elements of the tetrahedral group. So we have a one-parameter family of tetrahedrally-symmetric 3-calorons, interpolating between the tetrahedral 3-monopole and a tetrahedrally-symmetric 3-instanton. The latter is generated by the ADHM matrix A tetrahedrally-symmetric 3-instanton can also be obtained in JNR form, and its holonomy was used to approximate the minimum-energy 3-Skyrmion [5].
In conclusion, we have seen that, at least for charge N 4, there is an intimate connection between symmetric monopoles, symmetric calorons, symmetric instantons, and (via holonomy) Skyrmions. Many open questions remain, of which the following are a few.
• Several more symmetric monopoles (of higher charge) are known-do all of these arise as limiting cases of calorons with the same symmetry? More generally, is it true that any symmetric monopole has to be a special case of a symmetric caloron? • Similarly, does every symmetric instanton [6] extend to a family of symmetric calorons? Note that such families are much more general, in that there may not be a symmetric monopole at the 'other end'; • What is the role of harmonic maps, which are known to be related to symmetric monopoles and Skyrmions [9]?
Does this involve the interpretation of calorons as monopoles with a loop group as their gauge group [28,29]?