The story of spontaneous symmetry breaking (SSB) and its effective field theory (EFT) description does not end here. There are several further facets of SSB that would have deserved place in the table of contents of the book. However, giving them proper credit would either increase the volume of the book beyond reasonable limits, or require a substantial amount of additional background. In this chapter, I will give a brief primer on some of these exciting advanced aspects of SSB. I will be able to go to detail where it is feasible based on the material covered elsewhere in the book. For topics that depart substantially from the main text, I will resort to a few basic comments augmented with references for further reading.

1 Effects of Nonzero Temperature

The great majority of Parts III and IV the book is restricted to the leading order of the derivative expansion of the EFT for Nambu–Goldstone (NG) bosons. This is just the classical approximation to the EFT, based on the leading-order effective Lagrangian and using only tree-level Feynman diagrams. At a few exceptional spots, such as the proof of the Goldstone theorem in Sect. 6.2 or of the Adler zero property in Sect. 10.1, the presented argument is clearly valid beyond the classical approximation. However, I have not discussed explicitly perturbative loop corrections except for an outline of their role in the derivative expansion; cf. Sects. 9.1.1 and 9.2.1.

The effects of nonzero temperature have thus fallen through the cracks together with other loop corrections. Namely, in the so-called imaginary time formalism, the temperature of a system in thermodynamic equilibrium enters through discrete sums over Matsubara frequencies in loop diagrams. This is a standard part of field theory and I will therefore not dwell on details. A reader interested in the application of thermal perturbation theory to EFT for NG bosons will find more information for instance in [1] (chiral perturbation theory of mesons) or [2] (EFT for ferromagnetic magnons). One generic aspect of SSB worth mentioning explicitly is that the order parameter tends to be reduced by thermal fluctuations. As a rule, the phase where a symmetry is spontaneously broken will persist only up to certain critical temperature, above which the symmetry is “restored.” This is of course mere jargon. The symmetry of a system constrains its dynamics at all temperatures. However, above the critical temperature, the order parameter for SSB vanishes and excitations in the spectrum are organized in multiplets of the full symmetry group. See Chap. 7 of [3] for a discussion of symmetry restoration within thermal field theory.

What are the thermal effects on the spectrum of NG bosons? First of all, the Goldstone theorem remains valid at nonzero temperature as long as the symmetry is spontaneously broken. See Chap. 26 of [4] for a proof of the existence of a stable gapless quasiparticle, the NG boson, at nonzero temperature. Likewise, the distinction between type-A and type-B NG bosons survives at nonzero temperature. A detailed investigation of the thermal spectrum of NG bosons was carried out in [5]. At nonzero temperature, quasiparticles manifest themselves by poles in the propagator in the lower half-plane of complex energy. The two types of NG modes differ in the way their thermal width scales with momentum in the long-wavelength limit. As a rule, the complex energy of a type-A NG boson is thus schematically \(E(\boldsymbol p)\propto \left \lvert {\boldsymbol p}\right \rvert -\mathrm{i} \boldsymbol p^2\), whereas that of a type-B NG boson is \(E(\boldsymbol p)\propto \boldsymbol p^2-\mathrm{i} \boldsymbol p^4\). In both cases, the ratio of thermal width and the real part of the energy tends to zero in the limit \(\boldsymbol p\to \mathbf 0\). This ensures the stability of the NG modes.

The imaginary time formalism is only suitable for describing matter in thermodynamic equilibrium. In the past couple of decades, much progress has been made in the development of quantum-field-theoretic methods for thermodynamic systems out of equilibrium. The extension of these techniques to EFTs based on nonlinear realization of symmetry is fairly recent. The reader will find further details in [6,7,8].

2 No-Go Theorems for Spontaneous Symmetry Breaking

The entire book is based on the assumption that the symmetry of a given system is spontaneously broken. It is however equally interesting to try to understand under what circumstances SSB may in fact occur. The most striking results in this regard form a collection of “no-go” theorems, forbidding SSB in a specific class of theories. What follows below is, to the best of my knowledge, a representative list. The results included differ somewhat in the level of rigor at which they have been proven, and in the mathematical techniques required to establish them. Having just discussed the effect of loop corrections, it is natural to start with several related statements where the fluctuations of the order parameter play a key role.

Coleman Theorem

In Lorentz-invariant systems, NG bosons necessarily have a relativistic dispersion relation, \(E(\boldsymbol p)=\left \lvert {\boldsymbol p}\right \rvert \). It is easy to see that the corresponding free propagator in coordinate space,

$$\displaystyle \begin{aligned} \int\frac{\mathrm{d}^D\!p}{(2\pi)^D}\frac{\mathrm{e}^{-\mathrm{i} p\cdot x}}{p^2}\;, {} \end{aligned} $$
(15.1)

is infrared-divergent for \(D=2\) spacetime dimensions. This translates into a divergence of the two-point correlation function of the order parameter at long distances. In other words, the fluctuations arising from the NG boson destroy the assumed order parameter. This is the essence of Coleman’s theorem [9], which forbids spontaneous breaking of a continuous symmetry in two-dimensional relativistic systems.

This argument lends itself to a broad generalization. Consider a generic effective Lagrangian whose bilinear part starts at order 2n in spatial derivatives. While the \(n=1\) option is most natural, we also saw in Sect. 13.3.2 an example of a system where \(n=2\), at least in some spatial directions. More generally, the lack of low-derivative contributions to the spatial part of the kinetic term can be naturally explained by the presence of a coordinate-dependent symmetry [10, 11]. As to the part of the effective Lagrangian carrying time derivatives, we expect either one or two derivatives respectively for type-A and type-B NG bosons. This leads to a refined classification of NG bosons as type-\(\text A_n\) or type-\(\text B_{2n}\). See the first three columns of Table 15.1 for an overview of the schematic forms of the corresponding free-particle propagators and dispersion relations. A simple modification of the argument leading to Coleman’s theorem now shows that at zero temperature, type-\(\text A_n\) NG bosons are forbidden in \(D\leq n+1\) spacetime dimensions. Intriguingly, no such a constraint exists for type-\(\text B_{2n}\) NG bosons. In order to have any spectrum of quasiparticles, at least one spatial dimension must of course be present. However, for any \(D\geq 2\), the propagator of a type-\(\text B_{2n}\) boson at zero temperature is infrared-finite. See the fourth column of Table 15.1 for a summary.

Table 15.1 Subdivision of NG bosons into classes based on the noninteracting part of the effective Lagrangian. The latter determines the form of the (inverse) propagator, which in turn gives the dispersion relation of the NG mode. The last two columns of the table indicate spacetime dimensions, allowed by the condition that the coordinate-space propagator does not diverge at long distances
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Hohenberg–Mermin–Wagner Theorem

At nonzero temperature, the situation changes dramatically. Here the Minkowski-spacetime integral (15.1) is replaced with an imaginary-time sum-integral of the schematic type

$$\displaystyle \begin{aligned} T\sum_{k=-\infty}^{+\infty}\int\frac{\mathrm{d}^d\!\boldsymbol p}{(2\pi)^d}\frac{\exp(-\mathrm{i}\omega_k t+\mathrm{i}\boldsymbol{p}\cdot\boldsymbol{x})}{(\mathrm{i}\omega_k\ \text{or}\ \omega_k^2)+\boldsymbol p^{2n}}\;, {} \end{aligned} $$
(15.2)

where \(\omega _k\equiv 2k\pi T\) is a bosonic Matsubara frequency and T the temperature. The type-\(\text A_n\) and type-\(\text B_{2n}\) cases correspond respectively to \(\mathrm{i} \omega _k\) and \(\omega _k^2\) in the denominator. Due to the discrete nature of the Matsubara sum, the contributions of all terms with \(k\neq 0\) are infrared-finite. An infrared divergence can only arise from the zero Matsubara mode, and will be absent provided \(d\geq 2n+1\) or \(D\geq 2n+2\) [12] (the last column of Table 15.1). While the critical dimension is the same for type-\(\text A_n\) and type-\(\text B_{2n}\) NG bosons, note that n here counts the number of spatial derivatives in the Lagrangian. For fixed power of momentum in the dispersion relation, the critical dimension for type-\(\text A_{2n}\) NG bosons is higher than that for type-\(\text B_{2n}\) ones.

The original results on the absence of SSB at nonzero temperature are due to Hohenberg [13] (for superfluids) and Mermin and Wagner [14] (for isotropic lattice models of ferro- and antiferromagnets). These covered implicitly the cases of type-\(\text A_1\) and type-\(\text B_2\) NG bosons. However, their approach was different from the above intuitive argument and did not rely on the propagator of the would-be NG mode. Rather, they used the so-called Bogoliubov inequality to place an upper bound on the order parameter. This bound shows that the order parameter goes to zero in the limit of infinite volume and vanishing symmetry-breaking perturbation. See [15] for a review and further references.

Landau–Peierls Instability

There is another related result that applies to systems where the order parameter is spatially modulated in one direction, thus spontaneously breaking translations. This is the case for instance for liquid crystals in the smectic-A phase, as we saw in Sect. 13.3.2. Let us briefly recall the line of reasoning therein to stress its generality. Suppose that the order parameter is given by a scalar field \(\phi \) that develops a nonzero gradient in the ground state. We can treat \(\langle {\boldsymbol \nabla \phi }\rangle \) as a secondary, vector order parameter. The assumption that translations are spontaneously broken only in one dimension amounts to the condition that \(\langle {\boldsymbol \nabla \phi }\rangle \) points in the same direction everywhere in space. Next, we expand \(\phi \) around its expectation value as \(\phi \equiv \langle {\phi }\rangle +\pi \) and focus on the part of the effective Lagrangian bilinear in the fluctuation \(\pi \). The assumed (and also spontaneously broken) rotation invariance dictates that there cannot be any term in the Lagrangian, bilinear in the part of \(\boldsymbol \nabla \pi \), perpendicular to \(\langle {\boldsymbol \nabla \phi }\rangle \). The gradient expansion of the Lagrangian starts with terms proportional to \((\boldsymbol \nabla _\parallel \pi )^2\) and \((\boldsymbol \nabla _\perp ^2\pi )^2\), where \(\parallel \) and \(\perp \) denote projections to subspaces parallel and perpendicular to \(\langle {\boldsymbol \nabla \phi }\rangle \). The denominator of the large fraction in (15.2) should then be replaced with \((\mathrm{i} \omega _k\ \text{or}\ \omega _k^2)+\boldsymbol p_\parallel ^2+\boldsymbol p_\perp ^4\). Integrating the \(k=0\) term over \(\boldsymbol p_\parallel \) shows that the order parameter is washed out by thermal fluctuations at any nonzero temperature whenever \(d\leq 3\). This is known as the Landau–Peierls instability; see Sect. 1.6 of [16] and [17] for a more detailed discussion within the condensed-matter and nuclear physics context, respectively.

Absence of Time Crystals

Speaking of spontaneously breaking translations in a single direction, it is mandatory to consider the possibility that this direction corresponds to time. The idea that time translations could be spontaneously broken was proposed in [18, 19]. Such systems have since been known as time crystals. The reason why this possibility had not been noticed until the twenty-first century is that time crystals are not easy to realize. It was shown soon after the original proposal that their existence is forbidden in thermodynamic equilibrium of any Hamiltonian with sufficiently short-range interactions [20, 21]. There is however a nonequilibrium route towards quantum time crystals; see [22] for a recent review of the subject.

Vafa–Witten Theorem

Finally, the list of no-go theorems would not be complete without the Vafa–Witten theorem [23]. This is of quite a different nature than all the other results above, and applies to Lorentz-invariant gauge theories coupled to fermionic matter in a vector-like manner. An important example of such a theory is the quantum chromodynamics (QCD). Suppose the theory possesses a vector-like symmetry, that is one that acts in the same way on left- and right-handed fermions. In QCD, this could be for instance the \(\mathrm {U}(1)_{\mathrm {B}}\) baryon number symmetry or, in the limit of equal quark masses, the \(\mathrm {SU}(2)_{\mathrm {V}}\) isospin symmetry. The theorem states that such vector-like symmetries cannot be spontaneously broken in the ground state. The proof is technical. See also [24] for a review of the relevant mathematical methods and their applications to QCD and hadron physics.

3 Topological Aspects of Spontaneous Symmetry Breaking

We already encountered various topological aspects of EFTs for NG bosons on several occasions. Below, I will generalize some of the observations made previously, and add new interesting physics.

Topological Defects and Solitons

The classification of topological defects was the first major application of topology to physics; see [25] for an introduction to the relevant mathematics. Recall that the coset space \(G/H\) can be viewed as the vacuum manifold of a theory with SSB, whose points indicate possible values of the order parameter in the ground state. A defect is a nonuniform configuration of the \(G/H\)-valued order parameter that is singular on some subdimensional domain in space. See Fig. 15.1 for the simple examples of a point and line defect. The presence of a defect can be established from the properties of the order parameter away from the singularity. Thus, a p-dimensional defect in \(\mathbb {R}^d\) can be enclosed by a \((d-p-1)\)-dimensional hypersurface \(\Sigma _{d-p-1}\) that is topologically a sphere, \(S^{d-p-1}\). Equivalence classes of defects can then be determined by studying maps \(S^{d-p-1}\to G/H\), which are classified by the homotopy group\(\pi _{d-p-1}(G/H)\). This assigns the defect a unique label: its topological charge. An exception are domain walls, which are of codimension one in space, dividing it into two halves. A domain wall is thus characterized by the values of the order parameter on both sides. See Table 15.2 for an overview of basic types of topological defects.

Fig. 15.1
figure 1

Examples of topological defects in \(d=3\) spatial dimensions together with the surfaces defining their topological charge. The left panel shows a point defect (monopole), the right panel a line defect (vortex)

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Table 15.2 Classification of basic topological defects by their codimension in space \( \mathbb {R}^d\)
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Example 15.1

We saw examples of domain walls in Sect. 13.4. The double-well potential (13.78) has a discrete \(G\simeq \mathbb {Z}_2\) symmetry under \(\phi \to -\phi \). This is broken in its minima to \(H\simeq \{e\}\), hence \(G/H\simeq \mathbb {Z}_2\) and likewise \(\pi _0(G/H)\simeq \mathbb {Z}_2\). The cosine potential (13.80), on the other hand, has the symmetry group \(G\simeq \mathbb {Z}_2\ltimes \mathbb {Z}\). The \(\mathbb {Z}_2\) factor is generated by the inversion \(\phi \to -\phi \) whereas the \(\mathbb {Z}\) factor by the shift \(\phi \to \phi +2\pi v\). In any of the minima of the potential, the symmetry is broken to a subgroup isomorphic to \(H\simeq \mathbb {Z}_2\). Therefore, in this case \(G/H\simeq \mathbb {Z}\) and accordingly \(\pi _0(G/H)\simeq \mathbb {Z}\). The elements of \(\pi _0(G/H)\times \pi _0(G/H)\) shown in Table 15.2 correspond to domain wall solutions interpolating between different pairs of minima.

The focus on singular field configurations may be surprising; the physics should not change if we “smooth down” the order parameter. This is however only possible at the cost of embedding the coset space \(G/H\) in a larger order parameter manifold \(\mathcal {M}\).

Example 15.2

Consider a superfluid in \(d=2\) dimensions. Here the order parameter \(\langle {\psi }\rangle \) takes values from \(\mathbb {C}\), while the vacuum manifold is \(G/H\simeq \mathrm {U}(1)/\{e\}\simeq S^1\) for any \(\langle {\psi }\rangle \neq 0\). A point defect (vortex) in the superfluid is described by a smooth complex field \(\psi (\boldsymbol x)\). The corresponding topological charge is the winding number, defining an element of \(\pi _1(G/H)\simeq \pi _1(S^1)\simeq \mathbb {Z}\); see also Example 12.4. Importantly, nonzero winding number implies by the argument principle (Chap. 7 of [26]) the existence of a point \(\boldsymbol x\in \mathbb {R}^2\) where \(\psi (\boldsymbol x)=0\): the core of the vortex. This is our “singularity” where the order parameter cannot lie on the vacuum manifold \(G/H\simeq S^1\).

For another example, consider a spin system in \(d=3\) dimensions with \(G/H\simeq S^2\). Suppose that on the sphere \(\Sigma _2\) as shown in Fig. 15.1, the order parameter represented by a unit vector \(\boldsymbol n\in S^2\) points radially outwards everywhere. This spin configuration is sometimes called hedgehog. If we now try to continue the field to the inside of the sphere, we will inevitably encounter a point singularity: a monopole. Possible values of its topological charge are classified by \(\pi _2(G/H)\simeq \pi _2(S^2)\simeq \mathbb {Z}\). The singularity can be avoided only at the cost of deforming the spin configuration out of the coset space \(G/H\). The natural way to do so is by embedding \(G/H\simeq S^2\) in \(\mathbb {R}^3\) and allowing \(\boldsymbol n\) to vary its magnitude. One can then extend the hedgehog field on \(\Sigma _2\) to the inside by scaling down its magnitude so that it vanishes at the location of the monopole.

In contrast to defects are topological solitons (Sect. IX of [25]). These are fields in \(\mathbb {R}^d\) that are smooth and take values from \(G/H\) everywhere. Nontrivial topology arises from imposing a specific boundary condition at spatial infinity. Thus, the field configuration of a p-dimensional soliton is required to converge to a constant far from a fixed p-dimensional hypersurface \(C_p\) in \(\mathbb {R}^d\). Now consider an open hypersurface \(\Sigma _{d-p}\) that is “transverse” to \(C_p\). The boundary condition effectively compactifies this hypersurface to a sphere, \(S^{d-p}\). Equivalence classes of p-dimensional solitons in \(\mathbb {R}^d\) are therefore classified by the homotopy group \(\pi _{d-p}(G/H)\).

Example 15.3

The simplest type of a topological soliton is that with \(p=0\). Such solitons can be viewed as quasiparticles localized in all directions in space, and typically carry a finite amount of energy. The transverse hypersurface \(\Sigma _d\) is in this case the entire space \(\mathbb {R}^d\). We already met two examples of such solitons in Chap. 9: skyrmions in QCD, corresponding to \(d=3\) and \(\pi _3(G/H)\simeq \pi _3(S^3)\simeq \mathbb {Z}\), and baby skyrmions in two-dimensional ferromagnets, for which \(\pi _2(G/H)\simeq \pi _2(S^2)\simeq \mathbb {Z}\). The latter are easily generalized to higher dimensions, simply by making the fields independent of whatever extra coordinates are present. Thus a skyrmion in a three-dimensional ferromagnet is localized to a line, \(C_1\). Its topological charge is determined by the properties of a unit vector field on a two-dimensional surface \(\Sigma _2\), transverse to \(C_1\).

Topological Terms in the Action

We saw in Sect. 8.1 that the part of the effective Lagrangian for NG bosons with a single time derivative may be topologically nontrivial. Similarly, it turned out in Sect. 9.1 that the part of the low-energy EFT of QCD, taking into account the chiral anomaly, is topological in nature. We shall now generalize these observations, thus uncovering a broad class of contributions to EFTs for NG bosons, arising from the topology of the coset space \(G/H\).

Suppose that \(G/H\) is compact and consider a closed p-form \(\omega ^{(p)}\) that belongs to a nontrivial cohomology class in the p-th de Rham cohomology group, \(H^p(G/H)\).Footnote 1 Upon pulling \(\omega ^{(p)}\) back to a spacetime M of dimension D by the map \(M\to G/H\) defining the NG fields, it becomes a likewise closed p-form on M. There are now three different possibilities depending on the relation between p and D. If \(p=D\), the spacetime p-form can be directly integrated and added to the action of the EFT. This is called a \(\theta \)-term. Due to its origin in a closed D-form, it is a mere surface term in the Lagrangian and thus does not affect the perturbative dynamics of the EFT. This is the reason why \(\theta \)-terms have not featured in the book at all. If present, they contribute a phase factor to the generating functional though, which may affect the nonperturbative physics of the theory.

Another relevant possibility is \(p=D+1\). In this case, there is a locally well-defined D-form \(\omega ^{(D)}\) on \(G/H\) such that \(\mathrm{d} \omega ^{(D)}=\omega ^{(p)}\). Upon pulling back to the spacetime, \(\omega ^{(D)}\) gives a contribution to the Lagrangian that is quasi-invariant under the action of G. This is a Wess–Zumino (WZ) term. The anomalous contribution to the low-energy effective Lagrangian of QCD is obviously of this kind; cf. Sect. 9.1.4. However, the single-time-derivative part of the effective Lagrangian, , constructed in Sect. 8.1.3, belongs to the same category. Here only the variation of the NG fields in time matters and thus effectively \(D=1\), which makes \(H^2(G/H)\) the relevant de Rham cohomology group. Following the line of reasoning of Sect. 9.2.4, we see moreover that such WZ terms generally give rise to a Berry phase when the ground state is driven by an external field [27]. See Table 15.3 for a summary. Further technical details on the construction of topological WZ and \(\theta \)-terms in the action can be found in the recent study [28]. For a somewhat more condensed-matter oriented perspective on topological terms, see Chap. 9 of [29].

Table 15.3 Physical interpretation of generators of the de Rham cohomology group \(H^p(G/H)\), depending on the dimension D of spacetime they are pulled back to. For \(p<D\), the pull-back leads to a closed p-form on the spacetime, Hodge-dual to an identically conserved tensor current of rank \(D-p\). In the table, the rank of currents \(J^{\mu \nu \dotsb }\) is indicated by the number of indices
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4 Generalized Symmetries

It remains to clarify the content of the lower left corner of Table 15.3. For \(p<D\), the spacetime p-form arising from pulling back \(\omega ^{(p)}\) can be expressed as the Hodge dual of a \((D-p)\)-form, \( \operatorname {{\star }} J^{(D-p)}\). The form \(J^{(D-p)}\) has a vanishing codifferential; cf. Appendix A.6.5. In a flat spacetime, this can be expressed component-wise as

$$\displaystyle \begin{aligned} \partial_\mu J^{\mu\nu\dotsb}=0\;, {} \end{aligned} $$
(15.3)

where \(J^{\mu \nu \dotsb }\) is fully antisymmetric in all its indices. Equation (15.3) is nothing but a local conservation law with a tensor current of rank \(D-p\). It encodes, on the classical level, a generalized symmetry. This (and other) kind of generalization of conservation laws has attracted much attention in the last decade. An interested reader will find a comprehensive introduction for instance in [30,31,32].

In the context of EFT for NG bosons, the identically conserved currents \(J^{\mu \nu \dotsb }\) give us a tool for calculation of topological charges of defects and solitons, classified in Sect. 15.3. Mathematically, this is based on the correspondence between the homotopy and de Rham cohomology groups of \(G/H\) via the Hurewicz theorem (see Appendix A.8.2). Physically, the conservation law (15.3) suggests the existence of a conserved charge, obtained by integrating the temporal component, \(J^{0rs\dotsb }\), over a spatial hypersurface. For a static defect or soliton, it is easiest to pull \(\omega ^{(p)}\) back directly to the space \(\mathbb {R}^d\) and express it therein as \( \operatorname {{\star }}\mathcal {J}^{(d-p)}\). The \((d-p)\)-form \(\mathcal {J}^{(d-p)}\) is the density of our topological charge. The topological charge itself is obtained by integration over an effectively closed p-dimensional surface \(\Sigma _p\) in \(\mathbb {R}^d\),

$$\displaystyle \begin{aligned} Q(\Sigma_p)\equiv\int_{\Sigma_p}\operatorname{{\star}}\mathcal{J}^{(d-p)}\;. {} \end{aligned} $$
(15.4)

The fact that \(\Sigma _p\) is closed guarantees that the value of \(Q(\Sigma _p)\) does not change under smooth deformations of the surface. Depending on circumstances, (15.4) represents the topological charge of a \((d-p-1)\)-dimensional defect enclosed by \(\Sigma _p\), or that of a \((d-p)\)-dimensional soliton intersecting \(\Sigma _p\).

Example 15.4

The coset space \(G/H\simeq \mathrm {U}(1)\simeq S^1\) of superfluids has nontrivial first de Rham cohomology group, \(H^1(G/H)\simeq \mathbb {R}\). Pulling its generator back to space and integrating over a closed loop \(\Sigma _1\) defines the winding number of a \((d-2)\)-dimensional defect: the vortex. The coset space \(G/H\simeq \mathrm {SU}(2)/\mathrm {U}(1)\simeq S^2\) of ferro- and antiferromagnets has nontrivial second de Rham cohomology group, \(H^2(G/H)\simeq \mathbb {R}\). Pulling its generator back to \(\mathbb {R}^2\) and integrating over the entire plane, \(\Sigma _2=\mathbb {R}^2\), defines the charge of a 0-dimensional soliton: the baby skyrmion.

All the conservation laws of the type (15.3) discussed until now were emergent. Namely, their existence follows from the topology of the coset space \(G/H\), which itself parameterizes the breaking of the symmetry we started with. There are however also other, important examples of generalized conservation laws. For instance, the Maxwell equations in absence of matter take the form of a rank-two conservation law, \(\partial _\mu F^{\mu \nu }=0\). In this case, the charge (15.4) obtained by integration over a closed spatial surface is nothing but the electric flux through that surface. The “defect” is a localized electric charge enclosed by the surface. Intriguingly, the spontaneous breakdown of the generalized symmetry of electrodynamics is the ultimate reason why the photon is massless. This brings us back to the beginning of the book: we are constantly surrounded by a sea of NG bosons, namely sound and light. A reader wondering whether the classification of NG bosons into type-A and type-B also applies to generalized symmetries will be glad to hear that this is indeed the case. See [33] for a discussion of the classification and counting of NG bosons of generalized symmetries, and [34] for a nontrivial example of a system where the photon coupled to a particular type of matter behaves as a type-B NG mode.