String monopoles, string walls, vortex-skyrmions and nexus objects in polar distorted B-phase of $^3$He

The composite cosmological objects -- Kibble-Lazarides-Shafi (KLS) walls bounded by strings and cosmic strings terminated by Nambu monopoles -- could be produced during the phase transitions in the early Universe. Recent experiments in superfluid $^3$He reproduced the formation of the KLS domain walls, which opened the new arena for the detailed study of those objects in human controlled system with different characteristic lengths. These composite defects are formed by two successive symmetry breaking phase transitions. In the first transition the strings are formed, then in the second transition the string becomes the termination line of the KLS wall. In the same manner, in the first transition monopoles are formed, and then in the second transition these monopoles become the termination points of strings. Here we show that in the vicinity of the second transition the composite defects can be described by relative homotopy groups. This is because there are two well separated length scales involved, which give rise to two different classes of the degenerate vacuum states, $R_1$ and $R_2$, and the composite objects correspond to the nontrivial elements of the group $\pi_n(R_1,R_2)$. We discuss this on example of the so-called polar distorted B phase, which is formed in the two-step phase transition in liquid $^3$He distorted by aerogel. In this system the string monopoles terminate spin vortices with even winding number, while KLS string walls terminate on half quantum vortices. In the presence of magnetic field, vortex-skyrmions are formed, and the string monopole transforms to the nexus. We also discuss the integer-valued topological invariants of those objects. Our consideration can be applied to the composite defects in other condensed matter and cosmological systems.

short: only of few coherence length size. The similar wall between by HQVs appears in the polar distorted B-phase in nafen after transition from the polar phase. But due to strong vortex pinning in nafen, the KLS wall bounded by pinned vortices does not shrink and keeps its macroscopic size. This allows experimental detection and identification of such composite objects. 8 Strictly speaking, the wall bounded by strings is not a topological object, since after the second transition the topological charge of the string does not exist anymore. In the same way the string terminated by monopole is not topological. Here we show that under certain conditions these combined objects become topological, being described by relative homotopy groups. Originally the classification in terms of the relative homotopy groups has been used if there is the hierarchy of the energy scale or length scales in physical system, [31][32][33] when each energy scale has its own well defined vacuum manifold R i -the space of the degenerate states. Here are some examples: (i) The spin-orbit interaction in superfluid 3 He is small, and we have the order parameter vacuum manifold R 1 at short distances where the spin-orbit interaction can be neglected, and the submanifold R 2 ⊂ R 1 at large distances, where the space of the order parameter is restricted by spin-orbit interaction. 31 The relative homotopy groups π n (R 1 , R 2 ) give different types of the topologically stable combined objects. The π 1 (R 1 , R 2 ) describes the planar topological solitons terminated by strings. Examples are combined spin -mass vortices with soliton tail observed in superfluid 3 He-B, 34 and solitons terminated by half-quantum vortices observed in spinor Bose condensate. 35,36 The π 2 (R 1 , R 2 ) describes linear topological solitons and skyrmions terminated by monopoles.
(ii) Another example of the two-manifold system is when the boundary conditions restrict the order parameter on the boundary, with R 1 being the space in bulk and R 2 ⊂ R 1 is subspace on the boundary restricted by the boundary conditions. This gives the topological classification of the topological objects on the surface of an ordered system, 37 such as boojum. 38 (iii) The two-scale system also emerges when there is the hidden symmetry, which may soften the cores of topological defects. 39 In our case of two successive transitions, two energy scales arise in the vicinity of the second transition. There, the coherence length related to the first symmetry breaking G → H 1 is much smaller than the coherence length related to the second symmetry breaking H 1 → H 2 . This gives rise to two well defined vacuum manifolds, R 1 ∼ = G/H 2 and R 2 ∼ = H 1 /H 2 , and allows us to apply the relative homotopy groups π n (R 1 , R 2 ) for classification of the combined objects: monopole-string objects (analogs of Nambu monopoles); string-wall objects (analogs of KLS wall); nexus, 40 etc. That is because the order parameter fields are mapped into different degenerate vacuum manifolds at different spatial regions, thus the homotopy classes of order parameters constitute π n (R 1 , R 2 ) 41 , see details in supplementary material Sec. VIII. In superfluid 3 He, these topological objects live in the vicinity of transition between the polar phase and the polar distorted B-phase (PdB). This paper is organized as follows sequence. In Sec. II we consider the conventional scheme of the symmetry breaking and the vacuum manifolds of different superfluid phases appeared in the successive transitions. The topological defects in these phases emerge due to symmetry breaking and are described in terms of the conventional homotopy groups of vacuum manifolds are considered in Sec. III. In Sec.IV we discuss combined topological objects in the vicinity of the second transition, where the order parameter could be mapped into two different vacuum manifolds with different coherent lengths. We use the relative homotopy group and corresponding exact sequence of homomorphisms to describe the classes of combined objects, which are topologically stable in the vicinity of the transition. Based on the exact sequence of the homotopy groups, we find the topological stability of the string monopole (string terminated on monopole) and of the KLS string wall (KLS wall bounded by string). The latter has been observed in recent experiments. 8 In Sec. V we discuss vortex-skyrmions emerging in the presence of magnetic field, and the nexus object. In Sec. VI we summarize our results and discuss the possible disordering effect due to the appearance of those objects. All these objects contribute to the numerous supefluid glass states, which may exist in aerogel. [42][43][44]

II. CONVENTIONAL SYMMETRY BREAKING SCHEME AND VACUUM MANIFOLDS
The continuous phase transition is understood as spontaneous symmetry breaking by order parameters about a primary symmetry group G. In 3 He liquid at low temperature, the order parameters space consists of two 3-dimensional vector spaces and one phase space, thus the order parameter usually notated as complex-valued dyadic tensor A αi . 22 Rotating transformations of spin, orbit and phase in G act on one A αi of the order parameter space. Stabilizer of those actions gives out residual symmetry group H of superfluid phase of 3 He. In our case, the symmetry group G of normal liquid 3 He in the "nematiclly ordered" aerogel with the uniaxial anisotropy is different from that in the bulk 3 He. 22 The uniaxial anisotropy in the orientation of aerogel strands reduces the symmetry under the SO L (3) group of rotations in the orbital space to O L (2) group. 8 Neglecting the tiny spin-orbit interaction we have the following symmetry group of normal phase vacuum: where SO S (3) is the group of spin rotations; U (1) is the global gauge group, which is broken in superfluid states; T is time reversal symmetry; P is parity; In what follows, we ignore the time reversal symmetry, since it is not broken in the polar and in PdB phases, and also ignore the parity P which is reduced to P e iπ in all superfluid phases, where e iπ is the π-rotation in phase space. Also, because we focus on the topological objects related to the spin and U (1) gauge parts of the order parameter, the Z 2 symmetry coming from C L 2x could be neglected in the rest parts. Then the relevant starting group G of symmetry breaking scheme in this paper is Starting from this normal phase vacuum, we discuss three types of phase transition: (i) from the normal phase to the polar phase; (ii) from the polar phase to the PdB phase; and (iii) the possible direct transition from the normal phase to the PdB phase. In this Section we consider the topological objects related to these symmetry breaking scenarios, using the conventional homotopy group approach.
A. Transition from normal phase to polar phase The order parameter in the p-wave spin-triplet superfluids is the dyadic tensor A αi as mentioned, 22 which transforms as a vector under spin rotation (the first index) and as a vector under orbital rotations (the second index). In the polar phase it has the form: where Φ is the phase,d α andẑ i are unit vectors of spin and orbital uniaxial anisotropy respectively, and ∆ P is the gap amplitude. The residual symmetry group of this order parameter is Here where C S 2x is π-rotation of the vectord about perpendicular axis and e iπ is phase rotation, Φ → Φ + π. Then the vacuum manifold of the polar phase is given as The coherent length ξ = v F /∆ P in the polar phase is the smallest length scale in our problem, which determines the size of singular (hard core) topological defects in the polar phase.

B. From polar phase to PdB phase
Let us now consider the second symmetry breaking transition: from the polar phase vacuum to the PdB phase. In the vicinity of this transition the order parameter acquires the symmetry breaking term with amplitude q 1: Hereê 1 ,ê 2 andd form the triad of orthogonal vectors in spin space. The corresponding coherence length of the second transition ξ/q is large in the vicinity of this transition. This provides the hierarchy of the length scales, ξ and ξ/q ξ. The residual symmetry subgroup of the PdB phase is where SO J (2) represents the common rotations of spin and orbital spaces. The manifold of the vacuum states, which characterizes the second symmetry breaking is: Here SO L−S (2) is the broken symmetry with respect to relative rotations of spin and orbital spaces. Half-quantum vortex (HQV) described by the Z2 subgroup of π1(RP ) with core size ∼ ξ. This object also got the name "Alice string", because thed monopole transforms to the anti-monopole after going around the HQV, in the same manner as it happens for the charge going around Alice string. 46 The orange circle shows the path. Both defects loose topological stability after transition to the PdB phase. The HQV becomes the termination line of the KLS wall bounded by string, and the monopole-hedgehog becomes the termination point of spin vortices.

C. From normal phase to PdB phase
Here we consider the situation deep inside the PdB phase, where the parameter q is not necessarily small. In this general case there is only a single length scale which is relevant, and thus this situation becomes similar to that of the direct transition from the normal state to the PdB phase, G → H PdB . The order parameter Eq.(6) of the PdB phase could be written as where ∆ ⊥ ≤ ∆ . The corresponding residual symmetry subgroup is in Eq. (7), and the vacuum manifold of PdB phase in this scenario symmetry breaking is:

III. TOPOLOGICAL OBJECTS DUE TO SYMMETRY BREAKING TRANSITION FROM DIFFERENT VACUA
A. Defects in polar phase due to transition from the normal phase vacuum The polar phase vacuum manifold Eq.(5) has the homotopy groups In addition to the conventional vortices of group Z, the polar phase has the HQV (Alice string) of group Z 2 , and the hedgehogs (monopoles) in thed-field of group Z, see Fig. 1. The core size of these defects is on the order of the coherence length ξ. The topological classification of hedgehogs is modified by the phenomenon of influence of the homotopy group π 1 (R) on the group π 2 (R). 45 The monopole transforms to anti-monopole when circling around the Alice string (HQV), and thus in the presence of HQVs the hedgehogs (monopoles) could be described by the group Z 2 .
In the polar phase, the HQVs have been identified in NMR experiments. 16 By applying magnetic field tilted with respect to nafen strands, one creates the soliton attached to the HQVs, which produce the measured frequency shift in the NMR spectrum. Hedgehogs (monopoles) are still not identified in superfluid 3 He.
B. Defects in PdB phase due to transition from normal phase vacuum The vacuum manifold R 1 of the PdB phase in Eq.(10) has homotopy groups From all the defects of the polar phase with coherence length size ξ in Sec. III A, only the integer quantized vortices of group Z survive in the PdB phase. The other hard core defects (HQVs and hedgehogs) are not supported by topology any more. However, the new topological object appears -the Z 2 spin vortex, which becomes topologically stable in the PdB phase. This spin vortex is similar to that which has been observed in the bulk B-phase. 34

C. Defects in PdB phase due to transition from polar phase vacuum
The vacuum manifold R 2 of the PdB phase emerging at the transition from the polar phase in Eq.(8) has homotopy groups: These homotopy groups are responsible for the topological defects formed in the symmetry breaking transition from the fixed degenerate vacuum of the polar phase (withd = const and Φ = const) to the PdB phase. Let us consider them separately.

Spin vortices
The homotopy group π 1 (R 2 ) ∼ = Z describes the spin vortices with 2πn 1 rotation of vectorsê 1 andê 2 about the fixedd vector of the polar phase. The winding number is: In the vicinity of the transition, these spin vortices have the soft core of size of the coherence length, which corresponds to the transition from the polar to the PdB phase. This is ξ/q ξ. As distinct from the topological defects in the polar phase, which have the "normal" core, the spin vortices in the PdB phase with R 2 have the "polar" core. Quotation marks mean that in multicomponent systems the order parameter is not necessarily equal to zero on the axis of the topological defects. Proliferation of spin vortices in PdB marks the transition to the polar phase.
As follows from Sec. III B, deep in the PdB phase only the Z 2 spin vortices survive. The other spin vortices loose the topological stability and thus can live only in the vicinity of the transition from the polar phase vacuum. Far from transition between polar to PdB phase, their topological stability can be restored by applying the magnetic field. In this case spin vortices have thed skyrmions in the core, if the winding number is even (say, doubly quantized spin vortices). The skyrmions in thed-field are described by the relative π 2 group and thus are the combined topological objects. All this is discussed in detail in the sections IV and V. vortices), as discussed in detail in the section IV. As a result thed-hedgehog becomes the analog of Nambu monopole, which terminates the electroweak cosmic string, 1 see Fig. 2(b). The analog of electroweak string in the PdB phase is served either by the doubly quantized spin vortex with n 1 = 2, or by pair of n 1 = 1 spin vortices. In the presence of magnetic field, the hedgehog (monopole) separates the string on one side of the monopole and the skyrmion on the other side of the monopole.
3. The fate of half-quantum vortices in the PdB phase and the KLS wall Similar situation takes place with the HQVs, which are not topologically stable in the PdB phase. They become the termination lines of the KLS cosmic walls, as discussed in section IV, see Fig. 2(a). In 3 He experiments, after transition from the polar phase the PdB in the presence of HQVs, the KLS walls appear between the neighbouring vortices, and. in spite of the tension of domain walls, the HQVs remain pinned by the nafen strands. 8 In general the KLS wall is not topologically stable, and can be stabilized only due to symmetry reasons. 47 However, in the vicinity of the transition to PdB phase from the polar phase vacuum, the KLS wall becomes topological. The topological domain wall of the thickness ξ/qis is described by the nonzero element of the homotopy group π 0 (R P→PdB ) ∼ = Z 2 . Example of such a wall is the domain wall between the domains with A αi = ∆ P Diag(1, q, q) and A αi = ∆ P Diag(1, q, −q).

IV. COMBINED OBJECTS AND CLASSIFICATION BY RELATIVE HOMOTOPY GROUPS
As mentioned before, in the vicinity of the second transition, the system has two different length scales, ξ and ξ/q ξ. This leads to the new classes of objects, which combine the topology of both vacuum spaces, R 1 and R 2 . Such combined objects are described by the relative homotopy groups 31,37,39,41 In particular, the group π 1 (R 1 , R 2 ) describes the string wall in Fig. 2(a) (the wall bounded by strings, such as KLS wall 2 ) and the group π 2 (R 1 , R 2 ) describes string monopoles in Fig. 2(b) (such as string terminated by Nambu monopole 1 ). This combined topology can be illustrated by the following example of the string wall. At small distances ξ r ξ/q from the core of HQV, the HQV is described by the homotopy group π 1 (R P ). However, at larger distances r ξ q , the HQV becomes the termination line of the wall, which is described by the π 0 (R 2 ) topology, see Fig.2(a). This figure demonstrates the degeneracy spaces R 1 and R 2 , which are involved in the topology of the combined object. The similar physics takes place for string monopoles. At small distances ξ r ξ/q from the core of the hedgehog, it is described by the homotopy group π 2 (R P ), while at larger distances r ξ q , the monopole becomes the termination point of spin vortices described by the π 1 (R 2 ) topology, see Fig.2(b). Such combinations of π n+1 and π n groups needed for the description of the object with two different length scales and two different dimensions are the relative homotopy groups.
In all the cases the relative homotopy groups as well as topological charges can be found by using the exact sequence of homomorphisms, see details in Supplementary material Sec. VIII. In our case of two successive transitions this gives The mapping diagram of exact sequence in Fig. 5 depicts the relation between different topological objects in R 1 and R 2 .

A. Strings terminated by monopole -String Monopole
The objects described by the π 2 (R 1 , R 2 ) are monopoles (hedgehogs) ofd field, which terminate the spin vortices. The string monopole has two topological charges, which are connected: where S 2 is the surface encircling monopole and n 1 is the winding number of spin vortices in Eq. (14). This means that the total winding number of spin vortices, which are terminated by monopoles, is even. This situation is similar (a) KLS string wall. In general the KLS wall is non-topological, but it acquires the nontrivial topology in the vicinity of the second phase transition. In this limit case there are two well separated length scales: the coherence length ξ of the first transition, which determines the size of the hard core of string (the black dot), and the much larger coherence length ξ/q ξ of the second transition, which determines the soft core size of the wall (the pink region). The hierarchy of scales gives rise to two types of the degenerate vacua in the PdB phase, R1 and R2. The R1 vacua include all the degenerate vacua of the PdB phase, while the R2 vacua are those, which are obtained from the fixed order parameter of the polar phase, i.e. at fixedd and Φ in Eq.(3). This is the region, where the asymptotic condition |δd| 1 is achieved. The blue line shows the characteristic border between the regions of two classes of vacuum spaces. The topology of the string wall is determined by relative homotopy group π1(R1, R2), in which the green loop is mapped to the space R1, with the ends of the loop mapped to R2. (b) The string monopole is described by relative homotopy group π2(R1, R2). In this case the black dot shows the core of the hedgehog in thed-field and the pink region is the core of 4π spin vortex, which is terminated by the hedgehog. The green 2-loop is mapped to the space R1, with its 1-loop edge mapped to R2. to the monopole in the chiral A-phase, 51-53 which either terminates a single vortex with n 1 = 2, or forms the nexus with two singly quantized vortices with n 1 = 1 + 1 = 2, or with four HQVs with n 1 = 1/2 + 1/2 + 1/2 + 1/2 = 2. Those vortices, which connect with monopoles (n 2 > 0) or antimonopoles (n 2 < 0) allow the existences of complex monopole-antimonopole networks. 4,54-56 . Fig.3 illustrates the configuration of the string monopole with n 2 = 1 and with two strings each with n 1 = 1. The spin vortices have a soft core with size ξ/q.

B. Wall bounded by string -KLS string wall
The relative homotopy group π 1 (R 1 , R 2 ) = Z 2 × Z in Eq.(16) describes the following objects. The trivial element of Z 2 means the absence of the string wall, and one has the integer valued phase vortices of group Z, which are topologically protected. The nontrivial element of the Z 2 subgroup is responsible for the existence of the wall bounded by string, i.e. the KLS string wall. Fig.(4) illustrates the configuration of this nontrivial case. This kind of topologically protected KLS string wall induces the cosmological catastrophe in the axion solution of strong CP problem. 6,7 This configurations can be accompanied by the integer valued phase vortices of group Z, which can coexist with the string wall. However the Z 2 spin vortices discussed in Sec. III C 3 cannot coexist with the string wall, since they can be absorbed by this object. So, in the presence of the string wall, the Z 2 spin vortices are topological unstable and become "invisible".

V. SKYRMIONS AND NEXUS IN THE PRESENCE OF MAGNETIC FIELD
In the presence of magnetic field H, a new length scale appears in the PdB phase -the magnetic length ξ H ∝ |H| −1 . The magnetic length ξ H is the longest length scale if we neglect the spin-orbit coupling. In this case, the system is the two scale system of type (i) in Introduction Section. In the region with length scale larger than ξ H , the magnetic anisotropy locks the directions ofd vector in the plane perpendicular to H to minimize the magnetic energy, which is proportional to |H ·d| 2 . The degenerate space of the order parameter is reduced from R 1 ∼ = SO L−S (3) × U (1) in Eq.(10) to R H 1 = S 1 × S 1 × U (1) in the regions which are larger than ξ H . The first S 1 is the manifold of in planê d vector, while the second S 1 is the manifold of rotations ofê 1 andê 2 about thed-axis. Then the second relative . This demonstrates that the elements of πn(R1, R2) group have two sources: from the kernel of the mapping πn−1(R2) → πn−1(R1) and from the factor group of πn(R1) over the image of the mapping πn(R2) → πn(R1). These mapping diagram prescribes the relation between the elements of the composite topological defects. In particular, it shows that the relative homotopy group π2(R1, R2) is determined by the kernel of the mapping π1(R2) → π1(R1) ∼ = Z/Z2. It demonstrates that the nontrivial monopoles are termination points of spin vortices with the total winding number being even. As we will see in next section, this mapping relations is identical with that describing vortiex-skyrmions. On the other hand, the relative homotopy group π2(R1, R2) ∼ = Z2 × Z is determined by both sources: kernel of the mapping π0(R2) → π0(R1) ∼ = Z2 and by the factor group related to image of homomorphisms π1(R1)/π1(R2) → π1(R1) ∼ = Z. As a result, there are two different kinds of string defects, that terminate and do not terminate the KLS wall. These are correspondingly the vortices with half of odd integer circulation numbers and the vortices with integer circulation quanta.
homotopy group of combined objects with length scale ξ H is π 2 (R 1 , R H 1 ) ∼ = Z × Z. However, the gradient energy of thed-textures is much larger than that of the textures inê 1 andê 2 fields. 22 That is why we consider only the S 1 manifold ofê 1 andê 2 , and neglect the S 1 manifold ofd. Then the relative second homotopy group which we need in this case is This results for the relative homotopy group has been confirmed by calculations using the exact sequence, see details in supplementary materials Sec. VIII. The mapping diagram of exact sequence is shown in Fig. 6. The relative homotopy group π 2 (R 1 , S 1 × U (1)) describes the composite object in the PdB phase in the presence of magnetic field. This object is the spin vortex with even winding number, which has the soft core of size ξ H represented by skyrmion, see Fig. 7. The topological charge of skyrmion is where D 2 is the cross-section of skyrmion and n 1 is the winding number of spin vortices in Eq. (14). The Eq. (19) is the analog of the Mermin-Ho relation in 3 He-A. 59 Eq.(19) is identical with Eq.(17) because of π 2 (R 1 , S 1 × U (1)) ∼ = π 2 (R 1 , R 2 ). Due to this relation the vortex-skyrmion can be connected to Z spin vortices with core size ξ/q via the string monopole. Such composite objects, where the monopole connects several linear objects is called nexus. It demonstrates the interplay between π 1 and π 2 topologies. Originally vortex-skyrmions have been suggested in 3 He-A by Anderson and Toulouse 60 and by Chechetkin. 61 The lattice of vortex-skyrmions in rotating 3 He-A has been discussed in Ref. 62 . These object have been identified in different experiments made under rotation. 63,64 The dynamics of the vortex-skyrmions provides an effective electromagnetic fields, which induces the observed effect of chiral anomaly experienced by fermionic excitations living in the soft core of a vortex-skyrmion. 65 FIG. 6: Mapping diagram of exact sequence between R1 and S 1 × U (1). This diagram shows that the skyrmions soften the core of Z spin vortices with size ξ/q to size ξH in the presence of magnetic field. In regions larger than ξ H , vortex-skyrmions can be connected with spin vortices via string monopole, if their total topological charge is even according to Eq. (19). This is because π2(R1, R2) ∼ = π2(R1, S 1 × U (1)). There are also phase vortices described by Z, but here we ignore them because they do not influence the connection between the spin vortices and skyrmions.

VI. CONCLUSION AND DISCUSSION
Here we discussed the topology, which emerges in two-step phase transition in the vicinity of the second transition. An example is provided by the second order phase transition from the normal 3 He to the polar phase followed by the second order phase transition from the polar phase to the PdB phase experimentally observed in superfluid 3 He in nafen. 8 Here the composite object -the analog of the KLS wall bounded by cosmic string has been observed. We demonstrated that in the vicinity of the second transition, such composite object is described by the relative homotopy groups. The reason for that is the existence of the two well separated length scales. The smaller length scale determines the core size of the half-quantum vortex (the analog of Alice cosmic string). It is the coherence length ξ related to the symmetry breaking phase transition form the normal liquid to the polar phase. The larger length scale ξ/q ξ determines the soft core size of the KLS wall terminated by this string. It is the coherence length related to the second symmetry breaking phase transition -the transition form the polar phase to the PdB phase.
The two-scale composite defects are described by relative homotopy groups π n (R 1 , R 2 ). Here R 1 is the vacuum manifold of the PdB phase, while R 2 is also the vacuum manifold of the PdB phase, but at a fixed value of the order parameter of the polar phase before the transition to the PdB phase. The observed KLS wall terminated by the half-quantum vortex is determined by the nontrivial element of π 1 (R 1 , R 2 ). The other composite object, which is still waiting for its observation, is the monopole (hedgehog), which terminates the string (the spin vortex). Its topology is determined by the nontrivial element of π 2 (R 1 , R 2 ). The core of the monopole is of coherence length size ξ, while the spin vortices have the soft core of size ξ/q ξ. The relative homotopy groups π n (R 1 , R 2 ) are calculated using the exact sequence of the group homomorphisms.
The topology of these combined objects demonstrates new application of the relative homotopy groups. Earlier the relative homotopy groups have been applied for classification of topological defects on the surface of the ordered system, 37 and for classification of topological solitons terminated by point or linear defects. 31 Here we also considered the nexus objects, which combines the monopole, the string terminated by monopole, and also skyrmion (topological soliton) terminated by the same monopole. Such object in the PdB phase arises in the presence of magnetic field, which provides another length scale. The situation becomes even richer, when the spin-orbit interaction is included, which provides the fourth length scale and extends the multi-scale topology. The vortex-skyrmion binding objects was recently considered for superconductor-ferromagnet heterostructures, in which the existence of Majorana bound states were suggested. 66,67 In principle, the KLS walls discussed here can also emerge as the ground state of the system. Earlier it was suggested that the suppression of the B-phase on the boundary of superfluid 3 He may lead to formation of stripe phase in superfluid 3 He-B under nanoscale confinement in a slab geometry. 68 On a microscopic level, this inhomogeneous phase is thought as the periodic array of the KLS domain walls between the degenerate states of the B-phase, see Refs. 29,47 . The possible observation of such spatially modulated phase has been reported 69,70 . Similar situation may take place in another kind of confined geometry, in nafen. The strands of nafen could play the same role as the boundaries in the slab confinement. The suppression of the order parameter near the strands may result in the spontaneous proliferation of the composite defects leading to the stripe phases or stripe glasses.

VII. ACKNOWLEDGMENTS
We thank Jaakko Nissinen and Ilya M. Eremin for discussions. This work has been supported by the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (Grant Agreement No. 694248).

VIII. SUPPLEMENTARY MATERIAL
A. Exact Sequences: with and without magnetic field

No Magnetic Field
The exact sequence of (relative) homotopy groups means that the image of any homomorphism x * : A → B (the sets of the elements of the group B into which the elements of A are mapped) is the kernel of the next homomorphism B → C (the sets of the elements of B which are mapped to the zero element of C) i.e. Imgx k+1 * ∼ = Kerx k * , with k ∈ Z 48 . The k + 1th relative homotopy classes of π k+1 (R 1 , R 2 ) are mapped to the kth homotopy classes of π k (R 2 ) by mapping the k-dimension subset of k + 1 sphere, which surrounds the defects, into R 2 . This mapping between two homotopy classes with different dimensions is called boundary homomorphism ∂ * 48 . Boundary homormophism shows how topological objects with different dimensions connect to each other. In the PdB phase the exact sequence of homomorphisms is where the ∂ k * and ∂ p * are boundary homomorphisms. This gives the following relative homotopy groups: The ∂ k * maps the homotopy classes of string monopoles to the homotopy classes of spin vortices. The ∂ p * maps the homotopy classes of KLS string wall to homotopy classes of domain wall. The kernels and images of every relative homotopy group are analyzed in the section VIII B of supplementary material.

In the presence of magnetic field
In magnetic field the corresponding exact sequence is i.e. π 2 (R 1 , S 1 × U (1)) ∼ = Z/Z 2 ∼ = Z. We found Ker∂ k * ∼ = 0 and Img∂ k * ∼ = Z/Z 2 ∼ = Z + . That means that only those objects are topological protected, which have an even total winding number of spin rotation. These objects are thê d-vector skyrmions. Since π 2 (R 1 , S 1 × U (1)) ∼ = π 2 (R 2 , R 1 ), thesed-skyrmions can terminate on thed-monopole, which in turn is the end point of spin vortices with the total even number of spin rotation. As a result one obtains the composite effect -the nexus in Fig. 7.
B. relative homotopy groups 1. π2(R1, R2) The objects described by the π 2 (R 1 , R 2 ) are monopoles ofd-vector, because the manifold of the degenerate states of thed-vector is S 2 ∼ = SO S (3)/SO S (2), and we have the mapping from S 2 in real space to S 2 manifold ofd-vectors.
The boundary homomorphism ∂ k * maps S 1 ⊂ S 2 to R 2 . Then the Img∂ k * describes all classes of string defects terminated by the monopoles. We found Img∂ k * ∼ = Z/Z 2 ∼ = Z + which is the set of even numbers. This means that only the spin vortices with the total even winding number can form the string monopole. This situation is similar to the monopole connected with four half-quantum vortices in the A-phase, where the total winding number is 2. 51 The topologically trivial monopole cannot connect with the string defects because of Ker∂ k * ∼ = 0. Actually this trivial class is identical to π 2 (R 1 ) because Kerj * ∼ = 0.

π1(R1, R2)
The relative homotopy group is π 1 (R 1 , R 2 ) ∼ = Z 2 × Z. From Kerq * ∼ = Img∂ p * ∼ = π 0 (R 2 ) ∼ = Z 2 , we know there are domain walls bounded by string defects. The Z 2 group of π 1 (R 1 , R 2 ) represents the two topologically different classes of string defects, which can or cannot be connected with the domain wall respectively. The trivial element of Z 2 describes the situation in which only the phase vortices with integer winding number are topologically protected. This is because Ker∂ p * ∼ = Z. On the contrary, the nontrivial element of the Z 2 describes the situation in which the wall is bounded by string. This is because the nontrivial img∂ p * means the domain wall terminated by string defects. This geometry corresponds to the string wall -the KLS wall terminated by HQV.
In general, the string parts of the string wall are the either the HQVs or any phase vortex with half of odd integer winding number, N = n + 1/2 with n ∈ Z. There are also the Z 2 spin vortices of π 1 (R 1 ). However, they can be absorbed by the HQVs, and thus become invisible in the presence of string wall. 3. π1(R1, S 1 × U (1)) From exact sequence in Eq. (22) it follows that π 2 (R 1 , S 1 ×U (1)) ∼ = Z/Z 2 . This group describes the linear skyrmions in thed-vector, which are the linear analogs of the original point-like skyrmion. 57,58 The spin texture inside the crosssection D 2 of the skyrmion corresponds to continuous mapping to SO S (3), which is implemented by choosing first a direction ofd outside the skyrmion and then making SO S (2) rotation ofê 1 andê 2 around this direction. This skyrmion also represents the spin vortex with even winding number, because of img∂ k * ∼ = Z/Z 2 ∼ = Z + and ker∂ k * ∼ = 0.