## Abstract

The Skyrme model is a classical field theory modelling the strong interaction between atomic nuclei. It has to be quantized in order to compare it to nuclear physics. When the Skyrme model is semi-classically quantized it is important to take the Finkelstein–Rubinstein constraints into account. Recently, a simple formula has been derived to calculate the constraints for Skyrmions which are well approximated by rational maps. However, if a pion mass term is included in the model, Skyrmions of sufficiently large baryon number are no longer well approximated by the rational map ansatz. This paper addresses the question how to calculate Finkelstein–Rubinstein constraints for Skyrme configurations which are only known numerically.

## 1. Introduction

The Skyrme model is a classical model of the strong interaction between atomic nuclei (Skyrme 1961). In order to compare the Skyrme model with nuclear physics, we have to understand the classical solutions and then quantize the model. The classical solutions have a surprisingly rich structure. Configurations in the Skyrme model are labelled by a topological winding number which can be interpreted as the baryon number *B*. Static minimal energy configurations for a given *B* are known as Skyrmions. The *B*=1 Skyrmion has spherical symmetry; the *B*=2 Skyrmion has axial symmetry; and for *B*>2 Skyrmions have various discrete symmetries (see Battye & Sutcliffe 2002 and references therein). The Skyrme model depends on a parameter which corresponds to the pion mass . For , all the Skyrmions for were found to be shell-like configurations with discrete symmetries (Battye & Sutcliffe 2002). Such configurations are very well described by the rational map ansatz (Houghton *et al*. 1998). However, if the value of the pion mass is increased to its physical value or higher, then for high enough baryon number, shell-like solutions are no longer the minimal energy solutions (Battye & Sutcliffe 2005).

In Adkins *et al*. (1983) and Adkins & Nappi (1984), Adkins *et al*. quantized the translational and rotational zero modes of the *B*=1 Skyrmion for zero and non-zero pion mass, respectively, and obtained good agreement with experiment. A subtle point is that Skyrmions can be quantized as fermions as has been shown in Finkelstein & Rubinstein (1968). Solitons in scalar field theories can consistently be quantized as fermions, provided that the fundamental group of configuration space has a subgroup generated by a loop in which two identical solitons are exchanged. All loops in configuration space give rise to so-called Finkelstein–Rubinstein constraints which depend on whether a loop in configuration space is contractible or not. In particular, symmetries of classical configurations induce loops in configurations space. After quantization, these loops give rise to constraints on the wave function.

The *B*=2 Skyrmion with axial symmetry was quantized in Verbaarschot (1987), Braaten & Carson (1988) and Kopeliovich (1988) using the zero-mode quantization. Later, the approximation was improved by taking massive modes into account (Leese *et al*. 1995). The *B*=3 Skyrmion was first quantized in Carson (1991) and the *B*=4 Skyrmion in Walhout (1992). Irwin performed a zero-mode quantization for (Irwin 2000), using the monopole moduli space as an approximation for the Skyrmion moduli space. The physical predictions of the Skyrme model for various baryon numbers were also discussed in Kopeliovich (2001). Recently, the Finkelstein–Rubinstein constraints have been calculated for Skyrmions which are well approximated by the rational map ansatz (Krusch 2003), and which we shall call as rational map Skyrmions. In this case, the Finkelstein–Rubinstein constraints are given by a simple formula. This formula is also valid if the Skyrme configuration can be deformed into a rational map Skyrmion while preserving the relevant symmetries. However, this is not always possible. The aim of this paper is to show how to calculate Finkelstein–Rubinstein constraints for more general configurations.

This paper is organized as follows. In §2, we first describe the rational map ansatz. Then we discuss the Finkelstein–Rubinstein constraints. Finally, we derive some constraints on the symmetries which are compatible with a rational map Skyrmion. In §3, we first introduce a truncated rational map ansatz which describes well-separated rational map Skyrmions. Then we calculate the Finkelstein–Rubinstein constraints for this class of Skyrme configurations. In §4, we describe how to calculate the Finkelstein–Rubinstein constraints for a minimal energy configuration which is only known numerically. We also give an example. In §5, we derive constraints on possible symmetries in order to make predictions about ground states for Skyrmions with even baryon number. We end with a conclusion.

## 2. Skyrmions and rational maps

In this section, we first recall some basic facts about the Skyrme model. Then we describe the rational map ansatz. We then discuss how to quantize a Skyrmion as a fermion. Finally, we derive which symmetries are compatible with a rational map Skyrmion of a given baryon number.

### (a) The rational map ansatz

The Skyrme model is a classical field theory of pions. The basic field is the -valued field , where . The static solutions can be obtained by varying the following energy:(2.1)where is a right invariant -valued current and *m* is a parameter proportional to the pion mass (Battye & Sutcliffe 2005). In order to have finite energy, Skyrme fields have to take a constant value, , at infinity, and such maps are characterized by an integer-valued winding number. This topological charge is interpreted as the baryon number and is given by the following integral:(2.2)We will denote the configuration space of Skyrmions by *Q*. *Q* splits into connected components labelled by the topological charge. Furthermore, the energy of configurations in is bounded below by (Faddeev 1976).

The minimal energy solutions have been calculated in the massless case, *m*=0, for all (Battye & Sutcliffe 2002). The solutions are shell-like structures which are very well approximated by the rational map ansatz (Houghton *et al*. 1998), which we will now describe.

The main idea is to write Skyrme fields which can be thought of as maps from in terms of rational maps which are holomorphic maps from . In algebraic topology, such a construction is known as a suspension. First, we introduce polar coordinates and note that the angular coordinates can be related to the complex plane *z* by the stereographic projection . Then the Skyrme field can be written as(2.3)where the profile function is a real function satisfying the boundary conditions and . The map is the eponymous rational map. It can be written as the quotient of two polynomials and , which satisfy and and have no common factors. Here, denotes the polynomial degree. The ansatz (2.3) can be inserted into the energy (2.1) and we obtain(2.4)where(2.5)To minimize the energy (2.4), one first determines the rational map which minimizes and then calculates the shape function numerically by solving the corresponding Euler–Lagrange equation. The rational maps which minimize have been determined numerically in Battye & Sutcliffe (2002) and Battye *et al*. (2003) for all . Note that the restriction that is a holomorphic map can be lifted and a generalized rational map ansatz can be introduced (Houghton & Krusch 2001). This generalized ansatz has been shown to improve the energy significantly for , and it also captures the singularity structure of Skyrmions better. However, it is difficult to use for higher baryon number, and from the point of view of discussing symmetries the original rational map ansatz is sufficient.

The rational map ansatz gives a good approximation to the energy of a Skyrmion and also gives a very accurate prediction of its symmetry (Battye & Sutcliffe 2002). By symmetry we mean that a rotation in space followed by a rotation in target space leaves the Skyrmion invariant. Namely,(2.6)where *A* and are matrices and is the associated rotation. It is therefore important to understand how a rational map transforms under rotations in space and target space. It can be shown that equation (2.6) gives rise to the following equation for the corresponding rational map :(2.7)where *M* and are Möbius transformations related to *A* and *A*′, respectively (see Krusch 2003 for further details).

### (b) Finkelstein–Rubinstein constraints

In the following, we recall the ideas of Finkelstein & Rubinstein (1968) on how to quantize a scalar field theory and obtain fermions (for further details see Krusch 2003; Krusch & Speight in press). The main idea is to define a wave function on the covering space of configuration space. Recall that the configuration space *Q* of the Skyrme model splits into connected components labelled by the degree *B*, and will be denoted by . The fundamental group of each component of the configuration space *Q* is . Therefore, the covering space of each component is a double cover. In order to have fermionic quantization, we have to impose the condition that if two different points correspond to the same point , then the wave function has to satisfy(2.8)The points can be interpreted as two paths in configuration space. The condition that implies that and differ by a non-contractible loop. Every symmetry of a classical configuration gives rise to a loop in configuration space. In particular, we are interested in symmetries given by a rotation by *α* in space followed by a rotation by *β* in target space. This leads to the following constraint on the wave function *ψ*:(2.9)where is the direction of the rotation axis in space, * N* is the rotation axis in target space, and are the angular momentum operators in space and target space, respectively.1 Note that rotations in target space will also be called isorotations. The Finkelstein–Rubinstein phase enforces the condition (2.8) and satisfies(2.10)Here is a good place to summarize some important and well-known results. Giulini showed that a rotation of a Skyrmion gives rise to if and only if the baryon number

*B*is odd (Giulini 1993). Finkelstein & Rubinstein (1968) showed that a rotation of a Skyrmion of degree

*B*is homotopic to an exchange of two Skyrmions of degree

*B*. This also implies that an exchange of two identical Skyrmions gives rise to if and only if their baryon number

*B*is odd. In Krusch (2003), it was shown that a isorotation of a Skyrmion also gives rise to if and only if the baryon number

*B*is odd. These results agree with the physical intuition since atomic nuclei can be modelled by interacting point-like fermionic particles.

### (c) Symmetries of rational maps

Shell-like Skyrmions are described very well using the rational map ansatz. If a rational map Skyrmion of degree *B* is symmetric under a rotation by *α* followed by an isorotation by *β*, then the Finkelstein–Rubinstein phase of this symmetry is given by(2.11)which has been proven in Krusch (2003). For *B*>2, all the known Skyrmions are invariant under discrete subgroups, so they contain cyclic groups as subgroups. Let be a cyclic group of order *n*, which is generated by a rotation by followed by an isorotation by , where . Equation (2.11) imposes a constraint on the values of *B* which are compatible with a given symmetry. Namely, *N* has to be an integer. A stronger constraint can be derived if we work directly with rational maps.

*A rational map of degree B can have a* *symmetry if and only if* *or* .

Without the loss of generality consider rational maps with boundary condition and assume that the symmetry corresponds to a rotation around the third axis in space followed by a rotation around the negative third axis in target space. With this choice of axes, the boundary conditions are preserved by the relevant rotation and also by the relevant isorotation, and the sign choice corresponds to the sign choice for (2.11) in Krusch (2003). The rational map can be written as(2.12)where the polynomials and have no common factors. The symmetry condition is given by(2.13)Note that a rotation can be interpreted as a rotation followed by a isorotation. Since for a isorotation, the Finkelstein–Rubinstein phase is simply given by , we can restrict our attention to .

First we show existence. Let , so . Then the rational map(2.14)where is a polynomial of degree *l* and is a polynomial of at most degree *l*. For *k*=0, the degree of has to be less than *l* in order to respect the boundary conditions . This rational map is invariant under . To make sure that it is a rational map of degree , the polynomials and are required not to have any common factors. Furthermore, for , we need to impose , since the polynomial has a zero at *z*=0. For *k*=0, we also impose the condition , and we will discuss the case in the next paragraph. The simplest example of such a rational map is(2.15)Hence, a rational map of degree *B* with symmetry exists for .

Similarly, let , so *B*=*nl*. Again, we only consider . Then the rational map,(2.16)is invariant under . Here is again a polynomial of degree *l* and is a polynomial of at most degree which has no common factors with . Furthermore, we also require . One example of such a rational map is(2.17)Hence a rational map of degree *B* with symmetry exists for . This completes the proof of existence. The classification of rational maps into types (2.14) and (2.16) will become useful in §5.

Now, we assume that the rational map (2.12) is invariant under . In homogeneous coordinates, the rational map is given by subject to the relation that for any complex number . Under the symmetry , the polynomials and in equation (2.12) transform as(2.18)for . Assuming the rational map is invariant under leads to the following constraints on the coefficients:(2.19)In order to have a rational map of degree *B*, and cannot have any common factors. This implies that and cannot both be zero. First, assume . Then equation (2.19) implies that(2.20)The coefficient of the highest power in the numerator is also not allowed to vanish, which implies(2.21)so that . Now consider that , which implies(2.22)Again, the coefficient of the highest power in the numerator is not allowed to vanish. Therefore,(2.23)so that , which completes the proof. ▪

The lemma is more restrictive than the condition that *N* is an integer. For example, suggests that a symmetry is possible for *B*=2. However, since , our lemma excludes such a symmetry.

## 3. A truncated rational map ansatz

In Krusch (2003), the Finkelstein–Rubinstein constraints were calculated for Skyrmions which are well approximated by the rational map ansatz. In this section, we calculate the Finkelstein–Rubinstein constraints for Skyrme configurations , which are given by a truncated rational map ansatz defined as follows. Let be a Skyrme configuration of degree which is given by (2.3) and the shape function is a smooth, decreasing function which satisfies and for . Then the Skyrme configuration is given by(3.1)From formula (2.2) it is obvious that the configuration has the degree . The parameters are the positions of the Skyrmions, and we assume that for . Such an ansatz provides reasonable initial conditions for numerical simulations (R. A. Battye & P. M. Sutcliffe 2005, personal communication). A related ansatz is the product ansatz. This ansatz produces Skyrme configurations which are closer in energy to the true solutions. Topologically, these two ansatze are equivalent. However, the product ansatz has the disadvantage that it is non-commutative, since in general for matrices and , so that it is slightly more difficult to discuss symmetries. Therefore, we restrict our attention to the truncated rational map ansatz.

Consider a configuration which is invariant under . The symmetry relates different Skyrmions with each other. For each individual Skyrmion , there are two possibilities. Either the centre of this Skyrmion lies on the symmetry axis or it is one constituent of a regular *n*-gon of Skyrmions, which transform into each other under the symmetry.

### (a) Skyrmions centred on the symmetry axis

Assume two Skyrmions with baryon numbers and have a common axis of symmetry, say the -axis, and are centred around the origin and the point for *c*>*L*. Then the symmetry loop is homotopic to a product of two loops, each acting only on one Skyrmion. This can be seen as follows. The configuration can be written as(3.2)Under a rotation, the configuration transforms as(3.3)Here, is a rotation in target space acting by conjugation and is a rotation around the -axis. Note, in particular, that the vacuum is invariant under isorotation. The map provides a homotopy, such that is a rotation of the whole configuration while corresponds to a loop which first rotates one Skyrmion and then the other,(3.4)where(3.5)and similarly(3.6)Therefore, the Finkelstein–Rubinstein phase can be calculated as a product of the two loops of the individual Skyrmions. Since we are assuming that the and the Skyrmion are both well described by the rational map ansatz, we can apply formula (2.11) and obtain , where(3.7)

Naive application of formula (2.11) gives an incorrect result, namely , where(3.8)This is no longer well defined. Let and be invariant under a rotation. Then *N*=0, but . Also, consider and with symmetry , then *N*=0, but .

In this context, it is worth mentioning another interesting ansatz for Skyrmions, namely, the multi-shell ansatz by Manton & Piette (2001). The main idea is to construct multiple concentric shells of Skyrmions, where each shell is given by the usual rational map ansatz. The multi-shell ansatz can then be written as(3.9)where is a monotonically decreasing function with boundary conditions and . Here, and are two rational maps of degree and , respectively. The degree of such a Skyrme configuration is . Note that for , the Skyrme field takes the value , which is invariant under rotations and isorotations. As before, we can split a symmetry loop into two loops, each of which only acting on the inner or the outer Skyrmion, similar to (3.4). Therefore, formula (3.7) is also valid in this case.

### (b) Configurations of *n* Skyrmions related by symmetry

*For n Skyrmions of degree B which are related by a* *symmetry, the Finkelstein–Rubinstein phase is* *, where*(3.10)

The relevant Skyrme configuration corresponds to a regular *n*-gon of Skyrmions. The loop *L* which is induced by the symmetry has two effects on this configuration. Each Skyrmion is rotated and isorotated by . Furthermore, the Skyrmions are exchanged via the permutation , keeping the orientation in space and target space fixed. Therefore, the loop *L* can be divided into *n* rotation and isorotation loops for each Skyrmion and a permutation loop, which in turn can be split up into exchanges of two Skyrmions. The individual Skyrmions are given by rational maps, so we can apply formula (2.11) for each rotation and isorotation. Furthermore, Finkelstein & Rubinstein have shown that an exchange of two Skyrmions of degree *B* is homotopic to a rotation of a Skyrmion of degree *B* and we can again apply formula (2.11). Note that *N* in formula (2.11) is only well-defined modulo 2. The Finkelstein–Rubinstein phase for the loop *L* is then given by(3.11)Using (2.11) this gives rise to(3.12)(3.13)which completes the proof. ▪

A more heuristic way of understanding the result of lemma 3.1 is the following. Consider a regular *n*-gon of Skyrmions of degree *B* which transform into each other under a symmetry. Intuitively, this configuration can be deformed into a torus of degree under a homotopy which preserves the symmetry. Then the Finkelstein–Rubinstein phase can be calculated with formula (2.11) and we obtain(3.14)as above.

### (c) General configurations

Given a general configuration in the truncated rational map ansatz, which is symmetric under , we split up the configuration into regular *n*-gons of Skyrmions which transform into each other and Skyrmions which are on the symmetry axes. Assume that there are *l* regular *n*-gons of Skyrmions with degree for and *m* Skyrmions of degree for , which are located on the symmetry axis. Then the Finkelstein–Rubinstein phase for this symmetry is given by(3.15)This formula follows by constructing a homotopy between the symmetry loop and a product of loops for the individual groups of Skyrmions as in §3*a*.

## 4. How to calculate Finkelstein–Rubinstein constraints from numerical configurations

In this section, we describe how to calculate the Finkelstein–Rubinstein constraints for a Skyrme configuration, which is only known numerically.

Calculate the minimal energy configuration and analyse its symmetry properties.

Confirm the symmetry by starting with a symmetric configuration as initial condition and relaxing to the same final configuration.2

For each generator of the symmetry group identify

*k*for .Approximate the Skyrmion by a truncated rational map ansatz with the right symmetries.

Calculate the Finkelstein–Rubinstein constraints using formula (3.15).

In the following, we calculate the Finkelstein–Rubinstein constraints for the *B*=32 cube, which is displayed in figure 1. This configuration is one of the first examples of a Skyrmion which cannot be described with the rational map ansatz (Battye & Sutcliffe 2005). The configuration has been calculated numerically in Battye & Sutcliffe (2005) (step 1). It can be approximated by a chunk of the Skyrmion crystal (Baskerville 1996) and has cubic symmetry. Starting with a chunk of the crystal as initial conditions imposes the symmetries and corresponds to step 2. Step 3 is comparatively easy in this example, since we have an analytic ansatz for the initial condition. The cubic symmetry is generated by a and a symmetry. The Skyrme field can be parametrized as , where are the Pauli matrices. With the choice of fields as in Battye & Sutcliffe (2005), the symmetries areSo, the cubic symmetry is generated by and . The *B*=32 Skyrmion can be thought of as eight *B*=4 cubes. Under the symmetry, the top four cubes transform into each other and the bottom four cubes transform into each other (step 4). So, we can use formula (3.15) with and *l*=2, *n*=4 and *k*=2 (step 5). There are no Skyrmions on the symmetry axis, ,Therefore, the Finkelstein–Rubinstein phase for this symmetry. Under the symmetry, two *B*=4 Skyrmions are on the rotation axes and there are two groups of three *B*=4 Skyrmions which transform into each other. Therefore, , *l*=2, *n*=3, *k*=1, and *m*=2. So, formula (3.15) givesTherefore, the Finkelstein–Rubinstein phase is again trivial. In the following section, we show that we can derive some results from general principles, so that we only have to carry through steps 1–5 for a very small subset of all the possible symmetries.

## 5. Symmetries and Finkelstein–Rubinstein constraints for even *B*

In this section, we collect a set of general results about symmetries and Finkelstein–Rubinstein constraints. We start with the following lemma.

*Negative Finkelstein–Rubinstein constraints cannot occur for* *, for* *if B is even*.

Applying a symmetry *n* times corresponds to a rotation in space followed by a rotation in target space. If *B* is even, then a rotation (and also a isorotation) is homotopic to the trivial loop in the Skyrme configuration space. In this case, the Finkelstein–Rubinstein constraints correspond to one-dimensional and hence irreducible representations of which are obtained by mapping the generator of to . This representation can be thought of as a homomorphism from and therefore can only be non-trivial if *n* is even. ▪

In the following, we address the question of which symmetries can lead to negative Finkelstein–Rubinstein phases. Lemma 5.1 greatly simplifies the discussion, so we only consider even baryon number *B*. Furthermore, we restrict our attention to the symmetries which have been found empirically (Battye & Sutcliffe 2002). These are cyclic symmetry *C*_{2}, dihedral symmetry for , the tetrahedral group *T*, the octahedral group *O* and the icosahedral group *Y*. Therefore, we first discuss the cyclic subgroups for . For even baryon number *B*, the following picture emerges for Skyrmions which are well approximated by rational maps. Using formula (2.11) and lemma 5.1, the cyclic groups can be grouped into three groups:

The following symmetries never lead to negative , if : .

The following symmetries give rise to negative if : .

The following symmetries give rise to negative if : .

In order to understand symmetry, we need to examine under which conditions can there be an additional symmetry for a given realization of a symmetry? Since isorotations are always a symmetry, we can restrict our attention to *j*=0 and 1. The two different realizations of a symmetry can be characterized by their zeros and poles at zero and infinity. A rational map of type (2.14) has a zero of multiplicity at *z*=0 and a pole of multiplicity at . A rational map of type (2.16) has a pole of multiplicity at *z*=0, and a pole of order at .

For our purpose, it is sufficient to discuss the case that the rotation axis is orthogonal to the rotation axis. This generates the group . For a symmetry, it is important whether the isorotation axis is parallel to the isorotation or orthogonal to it, and we will introduce the notation and . For , such a distinction is not necessary. A symmetry around an axis orthogonal to the symmetry axis maps *z*=0 to ∞. In target space, *R*=0 is mapped to if the axis is orthogonal to the axis in target space, namely for , but *R*=0 and ∞ are invariant for and for . The numbers of zeros and poles imply that a rational map of type (2.14) cannot have an additional symmetry of type or . However, a symmetry is possible for all values of *k*. A rational map of type (2.16) can only have an additional or symmetry if . An additional symmetry is never allowed.

Now, we can apply the above result to the case . The above list shows that a negative Finkelstein–Rubinstein phase is only possible for and symmetry. A symmetry empirically only occurs as a subgroup of a symmetry which itself might be a subgroup of the cubic group *O*. For , the corresponding rational map can be of type (2.14). Then an additional symmetry is only possible for , which excludes *k*=1 and 3. The rational map can also be of type (2.16), provided that . Therefore, symmetry is not compatible with and , so that no negative Finkelstein–Rubinstein constraints can occur for .

Now, we discuss the case . Negative can occur for , and . can occur as subgroup of , *T*, *O* and *Y*. From the point of view of Finkelstein–Rubinstein constraints, only the subgroup of *T* and *Y* contributes and, similarly, only the subgroup of *O* contributes. Rational maps of type (2.14) always allow a symmetry which leads to . For rational maps of type (2.16), the symmetry is either or , and only the latter leads to negative . Note, however, that a symmetry of type (2.16) is only possible if , so that a symmetry always leads to negative . In Krusch (2003), the symmetries and Finkelstein–Rubinstein phases have been discussed for . The results for are displayed in table 1. Note that type (2.16) can only occur if for the maximal subgroup. All symmetries are either or are of type (2.14), so that for only symmetries with negative Finkelstein–Rubinstein phases have been observed. However, from symmetry arguments alone, it is not possible to exclude that the symmetries act in such a way that all the phases are positive.

### (a) Physical interpretation of the symmetry calculations

The physical interpretation of this result is as follows. In nuclear physics, we are interested in the quantum ground states for a given number of nucleons. In our semi-classical approximations, the ground state is given by the lowest values of the angular momentum quantum numbers *J* and *I* which are compatible with the symmetries of the classical configurations. In particular, the wave function *ψ* has to satisfy the symmetry condition (2.9) for all classical symmetries of the given Skyrmion. We decompose the wave function *ψ* in angular momentum eigenfunction and write for a wave function with angular momentum quantum numbers *J* and *I*.

Particularly interesting is the even–even situation when there is an even number of protons and an even number of neutrons. For small nuclei, the number of protons and neutrons are equal, so that even–even nuclei have . In this case, experiment shows that the ground state is generally given by . This is only possible if the Finkelstein–Rubinstein constraints are trivial. The above discussion showed that the ground state is allowed for , in agreement with experiment. In this calculation, we assumed that the relevant Skyrmions are well approximated by rational maps, and that symmetry only occurs as a subgroup of a symmetry (which might itself be a subgroup of an octahedral symmetry).

For higher pion mass, the configurations deviate significantly from the rational map ansatz (R. A. Battye & P. M. Sutcliffe 2005, personal communication), which raises the question whether it is possible to classify symmetries allowing for the more general truncated rational map ansatz. The Skyrmions would be allowed to split into groups and the Finkelstein–Rubinstein constraints have to be calculated with formula (3.15). Note that if Skyrmions can be thought of as being composed only of *B*=4 Skyrmions (*α* particles), then formula (3.15) and the above symmetry discussion also imply that there are no negative Finkelstein–Rubinstein phases, so that the ground state for such Skyrmions is .

For odd–odd nuclei, which implies for equal number of protons and neutrons, experiment shows that the ground state is usually not given by . This is consistent with our observation that negative Finkelstein–Rubinstein phases occur. However, symmetry arguments alone are not sufficient to prove the occurrence of negative Finkelstein–Rubinstein phases. For odd baryon number, lemma 5.1 cannot be applied and there is little hope of finding simple rules.

## 6. Conclusion

In this paper, we discussed how to calculate Finkelstein–Rubinstein constraints for configurations which are only known numerically. This is an important problem, since recent calculations show that for large pion mass Skyrmions are not very well described by rational maps (R. A. Battye & P. M. Sutcliffe 2005, personal communication). Moreover, there is mounting evidence that the pion mass in the Skyrme model should be interpreted as an effective mass with a value at least twice the physical value (Battye *et al*. 2005). The calculation of the Finkelstein–Rubinstein constraints can be incorporated into the algorithm for finding the minimal energy configurations with only minor modifications.

Note that for a wide range of values of the pion mass, the rational map ansatz works well for small enough baryon number *B*. Therefore, the truncated rational map ansatz in its current form has a good chance of capturing all the physically relevant Skyrmions. An obvious but slightly tedious generalization would be to allow for the constituent Skyrmions to be themselves approximated by a truncated rational map ansatz.

The paper also discussed which symmetries occur for rational map Skyrmions and when to expect negative Finkelstein–Rubinstein constraints for even baryon numbers. In particular, the Skyrme model calculations suggest the correct phenomenological trend, namely that the ground state of even–even nuclei have the ground state .

## Acknowledgments

The author would like to thank N. S. Manton, J. M. Speight and P. M. Sutcliffe for many fruitful discussions. The author also acknowledges an EPSRC Research fellowship GR/S29478/01.

## Footnotes

↵For a more detailed discussion on body fixed and space fixed angular momentum operators in this context, see Krusch & Quantization (2005).

↵This provides a homotopy from the initial condition to the final configuration which is invariant under the symmetry.

- Received November 21, 2005.
- Accepted January 6, 2006.

- © 2006 The Royal Society