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Fiber bundle from the Skyrme to the Yang-Mills model

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Claude Pierre Massé

Laboratoire de Physique Fondamentale, FRANCE

5 December 1995

Through the construction of a principal fiber bundle, a generalization of the Skyrme model equivalent to the Yang-Mills one is presented. In addition, a term in the lagrangian is shown to give mass to the gauge bosons by the Higgs mechanism.

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The Skyrme model [1] describes baryons through the meson-field by identifying them with topological solitons. What is of utmost interest is fermions emerge from a purely bosonic theory [2]. The Lagrangian density read $$\DeclareMathOperator{\Tr}{Tr}{\scr L}={f^2_\pi\over 4}\Tr[L^\mu L_\mu] -{1\over32g^2}\Tr([L_\mu,L_\nu][L^\mu,L^\nu])\tag1,$$ with $$L_\mu=-iU^\dagger \partial_\mu U=\exp\{i\pi_i(x,t)\tau^i\},\tag2$$ where $\tau^i$ are the Pauli matrices, generator of $SU(2),$ and $U$ represents the pion-field. The finiteness of energy imposes the boundary condition $U\rightarrow Cste$ as the distance from the origin of space tends to infinity. The ordinary space then has the topology of $S^3,$ the three dimensional sphere, like the group manifold of $SU(2),$ to which $U$ belongs. Therefore, the continuous mapping $U$ is from $S^3$ to $S^3,$ and falls into homotopy classes, the set of which is $\pi_3(S^3) = \Bbb Z.$ The baryon number is identified with the conserved winding number associated with the class, and the current is expressed as $$B_\mu={1\over3!}\varepsilon_{\mu\alpha\gamma\beta}L^\alpha L^\beta L^\gamma,\tag3$$ with $\varepsilon$ the totally antisymmetric tensor.

The strong likeness of the second term of (1) to the Yang-Mills Lagrangian has not been unobserved [3]. In fact, $L_\mu$ has the form of a pure gauge potential. Then a generalisation is necessary in order to have an equivalence. The one that will be exposed here is not entirely new [4], but go further and is applicable to the first term, thus allowing to reach new conclusions.

Note first that if $$W_\mu=\alpha^2 L_\mu,\tag4$$ Given that ($L_\mu$ acts on the right) $$[L_\mu,L_\nu]=-i(\partial_\mu L_\nu-\partial_\nu L_\mu),\tag5$$ this leads to $$\alpha^2(1-\alpha^2)[L_\mu,L_\nu]=[\partial_\mu-iW_\mu,\partial_\nu-iW_\nu].\tag6$$ Therefore, $\alpha^2L_\mu$ is no longer pure gauge, of course because of the non-abelian character, and then the non-linearity of the Yang-Mills pure field equation. This modification can be realised by taking $$V=\pmatrix{\alpha U\\\beta},\quad\alpha^2+\beta^2=1,\tag7$$ as field variable instead of $U,$ and by writing $$W_\mu=-iV^\dagger\partial_\mu V.\tag8$$ A gauge transformation $S\in SU(2)$ is represented by $$V\mapsto VS,\quad W_\mu\mapsto S^\dagger W_\mu S-iS^\dagger\partial_\mu S,\tag9$$ where now $V=\pmatrix{\alpha Q\\\beta R},$ and that’s all, the generalisation is complete. But it will be described again in a more formal way by using the fibre bundle theory [5] and the Hopf bundle [6] with its projector-value representation [7].

The set of quaternions $\Bbb H$ will be used for convenience. The fibre bundle is constructed as follow. Let $V$ be a unit vector of $\Bbb H^2$: $$V=\pmatrix{q\\r},\quad V^\dagger V=q^\dagger q+r^\dagger r=1.\tag{10}$$ The set of all $V$ is $S^7,$ and is taken as total space. If $q$ and $r$ are written $$q=q_0+iq_i\tau^i,\quad r=r_0+ir_i\tau^i,\tag{11}$$ where $q_\mu$ and $r_\mu$ are real numbers, each vector can be represented by $$V=\pmatrix{\alpha Q\\\beta R},\quad \alpha^2+\beta^2=1.\tag{12}$$ Here $Q$ and $R$ are elements of $SU(2).$ The projection is taken as the one of the quaternionic projective space $\Bbb HP^1$ which projects the vector $V$ onto $p$ defined by $r=qp,$ the set of which, the base space, has the topology of $S^4$ since it is $\Bbb H$ completed by a point at infinity corresponding to $q=0.$ The fibre over $p$ is the intersection of the quaternionic line containing $V$ and the set of unit vectors. All its elements can then be parametrised by $VS,$ where $S$ is a unit quaternion: $S^\dagger S=1,~S\in SU(2).$ The base space will be more conveniently represented by the set of the projectors $P$ of $\Bbb H^2$ onto the quaternionic line through $V,$ so $PV=V.$ Consequently $$P=V V^\dagger=\pmatrix{\alpha^2& \alpha\beta U\\\alpha\beta U^\dagger& \beta^2},\quad U=QR^\dagger.\tag{13}$$ The set of all $P$ really has the topology of $S^4$ since $U^\dagger U=1.$ This defines the (second) Hopf bundle with total space $S^7$ and typical fibre $S^3$ on which $SU(2),$ the set of all $S,$ acts effectively. Then it is a principal bundle with projection $$p:~S^7\rightarrow S^4,\quad V\mapsto P=V V^\dagger.\tag{14}$$

Now, $P$ is taken as the field variable on the compactified space-time $S^4,$ or a representative $V,$ which amount to the choice of gauge owing to (9). The $P$-field induces a fiber bundle with structure group $SU(2)$ over space-time, whose equivalence class is the conserved quantity playing the role of the homotopy class, as shown by this theorem:

There is a homomorphism between the set of equivalence classes of the fibre bundles of base $B,$ and the homotopy group of the mappings from $B$ to $B’$ inducing them.

In case of the Hopf bundle, it is an isomorphism. The set of equivalence classes is then isomorphic to $\pi_4(S^4)=\Bbb Z,$ each element of which is characterised by the winding number of the $P$-field.

A covariant derivative of the fiber bundle can be defined as $${\frak D}_\mu=\partial_\mu-iW_\mu=\partial_\mu-V^\dagger\partial_\mu V,\tag{15}$$ which is just the one of the natural connection form associated with the section $V$ [5]. The Yang-Mills model is then constructed as usual, yielding for the Lagrangian: $${\scr L}_2=\frac12\Tr([\partial_\mu P,\partial_\nu P]P[\partial^\mu P,\partial^\nu P]).\tag{16}$$

On the other side, the Skyrme model can be recovered in the following way: for a particular solution, a representative $V$ is chosen such that $$\matrix{V_+=\pmatrix{\cos(\theta/2)U\\\sin(\theta/2)},\quad 0\leqslant\theta\leqslant\pi/2-\varepsilon\\& \varepsilon\gt0.\\ V_-=\pmatrix{\cos(\theta/2)\\\sin(\theta /2)U^\dagger},\quad \pi/2-\varepsilon\leqslant\theta\leqslant\pi}\tag{17}$$ For $\theta=\pi/2,$ $V_+= V_-U,$ $U$ representing the transition function between the two neighbourhoods, which is the interpretation of the Skyrme field. If the $P$-space is parametrised so that the subspace of space-time for which $\theta=\pi/2$ is space-like, the restriction of the field on it, in fact the $U$-one, is a mapping to $SU(2).$ This other theorem will now be useful:

The set of the equivalence classes of the principal bundles of base space $S^n$ and of connected structure group $G$ is in one-to-one correspondence with the homotopy group $\pi_{n-1}(G)$ of the homotopy classes of the transition function.

Here, $n=4$ and $G= SU(2)\sim S^3.$ The relevant homotopy group is then $\pi_3(S^3),$ which is the one of the Skyrme field homotopy classes. Hence, its topological charge is shown to be identical to the class of the bundle. The three considered classes can thus be used indifferently, and there is even a natural isomorphism between $\pi_3(S^3)$ and $\pi_4(S^4).$

For $\theta=\pi/2$ and $V_+,$ it follows from a simple calculation the Yang-Mills Lagrangian (16) is equal to the second term of (1), apart from a multiplying constant. If $V_-$ is taken, $L_\mu$ is replaced by $R_\mu=-iU\partial_\mu U^\dagger,$ the right current, from which the same Lagrangian can be obtained in a symmetric way.

By this explicit example, what has been shown is skyrmions actually are instantons [8] located at time infinity. Then, they can be constructed from known parametrisations of instantons, as has already been done [9]. The important point is, as baryons can be described by meson-fields only, leptons could also emerge from gauge field theory as topological solitons. The leptonic density would have the expression similar to (3): $$\ell=\varepsilon_{\mu\nu\alpha\beta}{\frak D^\mu D^\nu D^\alpha D^\beta}=\varepsilon_{\mu\nu\alpha\beta}F^{\mu\nu}F^{\alpha\beta}.\tag{18}$$ The direct generalisation of (3), \(\varepsilon_{\mu\nu\alpha\beta}L^\mu L^\nu L^\alpha L^\beta\) cannot be used, since it is equal to zero.

The first term of (1) has now to be examined. A possible generalisation reads $${\scr L}_1\propto-\Tr[\partial_\mu P~\partial^\mu P],\tag{19}$$ since it reduces to $-\frac12Tr(\partial_\mu U^\dagger~\partial^\mu U)=\frac12Tr(L_\mu L^\mu)$ for $\theta=\pi/2.$ Explicitely: $${\scr L}_1\propto-\frac12\partial_\mu\theta~\partial^\mu\theta-\frac12\sin^2\theta~ \Tr[\partial_\mu U^\dagger~\partial^\mu U].\tag{20}$$ In the arbitrary gauge defined by (12), as $U=QR^\dagger$: $${\scr L}_1\propto-\frac12\partial_\mu\theta~\partial^\mu\theta-\frac12\sin^2\theta~\Tr[Q^\dagger\partial_\mu Q-R^\dagger\partial_\mu R]^2.\tag{21}$$ Using the expression of $W_\mu$ in this gauge: $$W_\mu=-i\cos^2{\theta\over2}~Q^\dagger\partial_\mu Q -i\sin^2{\theta\over2}~R^\dagger\partial_\mu R,\tag{22}$$ it follows $${\scr L}_1\propto-\frac12\partial_\mu\theta~\partial^\mu\theta +2\tan^2{\theta\over2}~\Tr[R^\dagger\partial_\mu R-iW_\mu]^2,\tag{23}$$ where a mass term of the $W$-field appears, although from (19) $\scr L_1$ is gauge invariant. It can actually be written $${\scr L}_1\propto-\frac12\partial_\mu\theta~\partial^\mu\theta -2\tan^2\frac\theta2~\Tr[(\partial_\mu R -iRW_\mu)^\dagger(\partial^\mu R-iRW^\mu)],\tag{24}$$ where it can be seen $R$ plays the rôle of the Higgs doublet in the Weinberg-Salam model. So, if the gauge $R=1$ is chosen, only the mass term remains and there is no component left since $R^\dagger R=1.$ In other words, the generalised first term of the Skyrme Lagrangian breaks spontaneously the gauge symmetry, the Higgs mechanism without the Higgs scalar! The same reasoning could be done with $Q,$ so the real degree of freedom giving rise to the mass is the gauge one.

Finally, the full Lagrangian of the generalised theory reads $${\scr L}=-{f^2\over2}\Tr[\partial_\mu P~\partial^\mu P] +{1\over2g^2}\Tr([\partial_\mu P,\partial_\nu P]P[\partial^\mu P,\partial^\nu P]).\tag{25}$$ It describes three massive $SU(2)$ gauge bosons with field strength $$F_{\mu\nu}=P[\partial_\mu P,\partial_\nu P]P,\tag{26}$$ while fermions appear as topological solitons.

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