The Noncommutative Ward Metric

We analyze the moduli-space metric in the static non-Abelian charge-two sector of the Moyal-deformed CP^1 sigma model in 1+2 dimensions. After carefully reviewing the commutative results of Ward and Ruback, the noncommutative K"ahler potential is expanded in powers of dimensionless moduli. In two special cases we sum the perturbative series to analytic expressions. For any nonzero value of the noncommutativity parameter, the logarithmic singularity of the commutative metric is expelled from the origin of the moduli space and possibly altogether.


Introduction and summary
The CP 1 sigma model in 1 + 2 dimensions is a paradigm for soliton studies [1,2]. In particular, it provides the simplest example for a nontrivial dynamics of slowly-moving lumps, following the adiabatic approximation scheme of Manton [3]. In a slice of the charge-two sector, the moduli-space metric was worked out and the geodesic motion was analyzed by Ward [4]. The corresponding Kähler potential was then given by Ruback [5] (see also [6]).
In the case just mentioned, the (restricted) moduli space of static charge-two solutions is complex two-dimensional and contains ring-like as well as two-lump configurations. On the complex line where the lump size shrinks to zero, the metric develops a logarithmic singularity. Such divergencies can often be regulated by subjecting the system to a noncommutative deformation, which introduces a dimensionful deformation parameter θ. To explore this possibility, we analyze the Moyal-deformed CP 1 model [7] in this paper.
In fact, the (restricted) moduli-space metric for the charge-two sector of this noncommutative model was already investigated in [8]. There, the authors show that the metric in question is flat for θ → ∞ (corresponding to vanishing values of the dimensionless moduli) and possesses a smooth θ → 0 limit (which is attained for infinite values of the dimensionless moduli). However, these findings do not establish the removal of the logarithmic singularity for finite values of θ or amount to an explicit computation of the Kähler potential.
In this paper, we review the commutative results and present a power-series expansion of the deformed Kähler potential in the 'ring' regime of the moduli space. For the first time, this is achieved for arbitrary values of θ. We verify the commutative limit and sum up the perturbation series on the would-be singular line in the 'two-lump' domain via the Gel'fand-Yaglom method. There is a curious connection with the eigenvalues of the spheroidal wave equation. Around the origin of the moduli space, the Kähler potential is shown to be analytic, which substantiates the claim of [8]. Perturbative expressions for the moduli-space metric follow via differentiation, and the two-lump scattering behavior may be quantified.

The CP 1 model and its solitons
The CP 1 or, equivalently, the O(3) sigma model describes the dynamics of maps from R 1,2 with a metric (η µν ) = diag(−1, +1, +1) into CP 1 ≃ SU (2) U(1) ≃ S 2 . There are various ways to parametrize the target space, for instance by hermitian rank-one projectors P in C 2 , or else by vectors T ∈ C 2 modulo complex scale, so that the field degree of freedom is a single function u taking values in the extended complex planeĊ ≃ CP 1 . Introducing coordinates on R 1,2 , we can formulate the action as whereū is the complex conjugate of u, and only the last equality uses the commutativity of the functions. For later convenience, we also define the kinetic and potential energy density, respectively, so that yields the total energy of the configuration, which is conserved in time. Clearly, action and energy are form-invariant under translations and rotations of the domain R 2 (at fixed t), as well as under global SO(3) rotations of the target, Classically, one is interested in the extrema of S whose energy is finite. Among the static configurations,u = 0 (hence T = 0), those are all well known: with u being a rational function (of z orz) to ensure finite energy. Each rational analytic (or anti-analytic) function u = p q is a soliton (or anti-soliton) with a topological charge given by its degree n (or −n) and with energy E = 8π|n|. Hence, the soliton moduli space M n for charge n has complex dimension 2n + 1. Some of these moduli, however, correspond to isometries of the domain or the target.
In this paper, we shall investigate only charge-one and charge-two solitons. Let us characterize their static moduli spaces. By employing the SO(3) target rotations, in the numerator p we remove the highest monomial and restrict the coefficient of the second-highest one to be real and non-negative. This is also true for the third-highest one by means of a domain rotation. Furthermore, by a common rescaling of p and q we set the coefficient of the highest monomial in the denominator q to unity. Finally, the domain translation isometry allows us to remove the second-highest monomial of q, which corresponds to picking a center-of-mass frame for our configuration. These choices fix all isometries except possibly for special values of the remaining moduli. Of course, the full moduli space is recovered by acting with all isometries. In the chargeone case, we thus get with β ∈ R ≥0 and z = βz ′ . This is a single lump of height |β| −2 and width of order |β|.
For charge two, one finds with β, γ ∈ R ≥0 and ǫ ∈ C. In the last expression, we have introduced dimensionless quantities by the rescaling z = βz ′ , γ = β 2 γ ′ and ǫ = β 2 ǫ ′ , effectively putting β = 1. A different situation arises for the special value β = 0. Here, one can also rotate away the phase of ǫ and should rather use z = √ γz ′ to arrive at One may check that V integrates to 16π in both cases. This energy density can take a variety of shapes, depending on the values of the moduli. Two well-separated lumps appear for |ǫ| > |β| 2 and |ǫ| > |γ|, while ring-like structures emerge in the regime |γ| > |β| 2 and |γ| > |ǫ|.

Moduli space metric
So far, we have only considered static solutions to the sigma model. For dynamical issues, we must bring back the time dependence. Rather than attempting to solve the full equations of motion δS = 0 for u(t, z,z), we resort to the adiabatic approximation valid for slow motion [2], where u(z|α) denotes a static soliton depending holomorphically on moduli parameters α. For simplicity we suppress here the moduli labels but let α represent the holomorphic set {α}. 1 By allowing these moduli to vary with time, we approximate the true time-dependent solution by a sequence of snapshots of static solutions. In this way, the dynamics in the configuration space of maps, u : R → maps(C, CP 1 ) via t → u(t, ·), gets projected to the 'mechanics' of a particle moving in the finite-dimensional moduli space for a fixed topological charge, α : R → M n .
Since the potential energy of the soliton configurations is independent of α, the kinetic energy provides an action principle for α(t): the extrema of are just geodesics in M n endowed with the induced Kähler metric computes the Kähler potential from the static soliton configurations u = u(z|α). We remark that the freedom of rescaling T reappears in the ambiguity of K due to Kähler transformations, , and so we may also use the more divergent formal expression It turns out that gᾱ α diverges for the modulus β (and also for the removed z n coefficient in p). Hence, these particular moduli carry infinite inertia and do not participate in the dynamics, because changing their values requires an infinite amount of energy. Consequently, they get degraded to external parameters which are to be dialled by hand. In the charge-one case, no dynamics remains, which is consistent with the picture of a single lump sitting in its rest frame. Nevertheless, it is instructive to reinstate the translation moduli and verify the flat moduli space. With T = β z+δ we get This is formally independent of δ (by shifting z → z − δ) but it is logarithmically divergent, so we better compute its second derivatives as well as ∂δ∂ β K = −8πδ/β. Hence, we indeed get K = 8πδδ. Since the center-of-mass motion decouples from the remaining dynamics, we shall suppress it from now on.
For charge two, the Kähler potential K reads depending on whether β is chosen nonzero or not. In the first case, K is again divergent, and its derivatives are not elementary integrable. For the sake of simplicity, we therefore restrict ourselves to the second (special) case and put β = 0 from now on. The form of the relevant integral reveals that K is a function only of |γ| and |ǫ| which, up to an overall dimensional factor, depends merely on their ratio. The last integral can in fact be executed to yield 2 where E(m = k 2 ) denotes the complete elliptic integral of the second kind as a function of its parameter m (k is called the elliptic modulus), and we have parametrized 3 ǫ = re iω sin ϕ and γ = re iχ cos ϕ.
In the (|ǫ|, |γ|) plane, the Kähler potential grows linearly with the distance from the origin, with a slope varying between 8π 2 (for ǫ = 0) and 16π (for γ = 0). It is continuous but not smooth on the complex line γ = 0 (ϕ = π 2 ), which is the localization locus in the two-lump region because the lump width is of order |γ| √ |ǫ| at a lump separation of order 2 |ǫ|.

Moyal deformation
The task of this paper is the Moyal deformation of the Ward metric and the Kähler potential presented in the previous section. One way to describe such a noncommutative deformation of the zz plane is by giving the following 'quantization rule': where these operators may be realized as infinite matrices More generally, where 'sym' indicates a symmetric ordering of all monomials in (Z,Z). Naturally, derivatives turn into inner derivations, and the integral over the complex plane becomes a trace over the operator algebra, A highest-weight representation space F for the Heisenberg algebra, [a, a † ] = 1, is easily constructed from a vacuum |0 , where the basis states are the normalized eigenstates of the 'number operator' N = a † a, N |n = n |n and n|n = 1 for n = 0, 1, 2, . . . .
The Moyal-deformed CP 1 model is defined by copying most definitions of the previous section, but taking the entries of P and T to be operator-valued. Since, in this context, q may not have an inverse, we avoid using u as a variable and work with p and q instead. Because the deformation has traded functions on the xy plane with operators on F, densities such as T or V are less intuitive objects, but may still be visualized via the Moyal-Weyl map. The noncommutative solitons are found by taking T to be polynomial in a, i.e. both p and q are polynomial of degree n, and their moduli are identical to the commutative ones 4 . It is important to note that the deformation has introduced a new dimensionful parameter, θ. Therefore, we may relate all dimensional quantities to θ and pass to dimensionless parameters, As a consequence, K and gᾱ α will depend on all moduli individually and not only on their ratios. Of course, in the commutative limit θ → 0, the ratios will again dominate. As a warm-up, let us reconsider the charge-one soliton (with b frozen but including the translational moduli d), now given by Since by a unitary basis change in F we can shift a → a − d, this expression is again formally independent of d, but it is divergent: where the divergence is hidden in the ambiguous coefficients λ and µ, which may depend on b andb. To fix this ambiguity, we first take derivatives and then shift away the d dependence: while gb b is still infinite. Hence, λ remains arbitrary but µ =dd up to irrelevant terms. With b fixed, we therefore get K = 16πθdd = 8πδδ, the same flat metric as in the commutative case. Note that even though the modulus b has infinite inertia, it is needed to regulate the Kähler potential (4), which blows up at b = 0. 5

Deformed rings
We now turn to the nontrivial charge-two case with the choice of β = 0, defined by Of course, K is divergent, but the singularity is removable, and ∂ḡ g K and ∂ē∂ e K already converge. Like the modulus b in the previous section, here g plays the role of a regulator, but this time its inertia is finite. This expression is not amenable to exact analytic computation, but we can attempt to establish power series expansions inēe or inḡg. In this section, we investigate the 'ring' regime |e| ≪ |g|.
It is easy to set up an expansion around e = 0, since T † T | e=0 is already diagonal in our basis (3). Note that no zero-mode issue arises sinceḡg > 0. Writing and E = ea †2 +ēa 2 +ēe =: տ + ւ + ←, the Taylor series of ln(1 + x) unfolds to displaying all terms to order (G −1 E) 6 and (ēe) 3 . One sees that for a given power k of G −1 E, there is a sum over all cyclic paths of length k, where each step is either տ or ւ or ←, separated by a factor of 1 G . All terms containing ← can be resummed into the shift operator exp(ēe ∂ḡ g ), which shortens the above to = exp ēe ∂ḡ g tr ln G −ēe տւ −(ēe) 2 1 2 տւտւ + տտււ − (ēe) 3 Using a †2 |n = (n + 1)(n + 2)|n + 2 , a 2 |n = n(n − 1)|n − 2 and 1 G |n = 1 gg + n(n − 1) |n , the above traces convert into infinite sums of rational functions of n. After repeated partial fraction decomposition these sums can be evaluated to K 16πθ = lnḡg + ln cos W +ēeπ 2ḡ g 4ḡg + 3 tan W W with the definition The leading term was determined via ∂ḡ g tr ln G = tr G −1 = n≥0 1 gg + n(n − 1) and a constant as well as the lnḡg term in K may be omitted.
To each order inēe, the expression (5) is exact inḡg and, hence, valid for arbitrary values of θ. For strong noncommutativity, when g → 0 but | e g | ≪ 1 fixed, potential poles due to tan W ∼ sec W ∼ (πḡg) −1 are always compensated by suitable powers ofḡg. To check our computation, let us take the opposite, commutative limit, For g → ∞, the expansion (5)

Deformed lumps
More interesting however is the |γ| ≪ |ǫ| 'two-lump' domain, which at θ = 0 featured a weak logarithmic singularity for γ → 0, where the two lumps are localized infinitely sharply. To analyze this situation, we need to expand K around g = 0, in powers and perhaps also logarithms ofḡg, generalizing (2) to finite values of θ. To this end, we are interested in the eigenvalues of Representing the noncommutative coordinates on which via Fourier transformation and change of variables is equivalent to This equation matches with the one defining spheroidal (scalar) wave functions [13,14], with m ∈ Z and n = 0, 1, 2, . . . (in our convention) counting the discrete spheroidal eigenvalues λ mn (γ). Clearly, we have m = 1 and γ 2 = −ēe (the oblate case), hence λ n (e) = λ 1n (i|e|). For small values ofēe one finds the expansion [13] λ n (e) = n(n − 1) 1 + 2 (2n − 3)(2n + 1)ē e where the two zero modes of F , namely λ 0 = λ 1 = 0, are explicit. Therefore, we may write The role of g as a regulator is obvious; the first term carries the F zero modes. After expanding the logarithm under the last sum, one can perform the sums and nicely reproduces all terms in (5).
All these traces converge and should be finite in the entire γǫ plane. The coefficient gǭ ǫ may be read off (5) by replacing (ēe) k with k 2 (ēe) k−1 in the series. For the other two, one has to work out the derivatives.

Conclusions
We have investigated the charge-two moduli-space metric in the noncommutative CP 1 sigma model in 2+1 dimensions. After decoupling the center of mass and a convenient dialling of frozen moduli, we find that the Kähler potential depends only on the combinationsḡg andēe of the dynamical complex-valued dimensionless moduli g and e. The noncommutativity strength √ θ sets the single scale of the system. In the limit |e| ≪ |g| → ∞, where the solitonic energy density has a ring-like profile, our power series inēe matches with the known commutative Kähler potential, which depends only on the ratio |e| |g| . In the complementary regime |g| ≪ |e|, where the configuration splits into two lumps, we observe that the logarithmic singularity of the commutative Kähler potential is smoothed out by the deformation, which pushes it to the θ = 0 boundary of the moduli space. The (|γ|, |ǫ|) = 2θ(|g|, |e|) plane is depicted in Fig. 1.
We have expanded the Kähler potential to order (ēe) 4 and to any order inḡg, but an analytic expression remains a challenge, which amounts to computing the spectrum of the spheroidal wave equation for m = 1 but any e. However, at g = 0 we only needed the lowest (regularized) eigenvalue, and theēe series could be summed to an analytic function via the Gel'fand-Yaglom trick.