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Charge Violation and Alice Behavior in Global and Textured Strings

Spontaneous breaking of global symmetries can produce ``Alice'' strings: line defects which make unbroken symmetries multivalued, induce apparent charge violation via Aharonov-Bohm interactions, and form point defects when twisted into loops. We demonstrate this behavior for both divergent and textured global Alice strings. Both adiabatically scatter charged particles via effective Wilson lines. For textured Alice strings, such Wilson lines occur at all radii, and are multivalued only inside the string. This produces measurable effects, including path-dependent charge violation.


I. INTRODUCTION
Among the defects created in symmetry-breaking phase transitions are Alice strings. [1][2][3] Like monopoles [4], Alice strings obstruct the global extension of unbroken symmetries, making them multivalued when parallel transported around the string. This algebraic obstruction has two prominent physical consequences. First, it produces nonconservation of the associated charges under Aharonov-Bohm scattering around the string. Second, it induces topologically stable point defects, which are themselves twisted loops of Alice string.
These Alice features were first described for gauged strings, where a charged condensate winds asymptotically due to gauge flux on the string. The gauge flux acts asymptotically on all charged particles in the same way, through the Wilson line U(ϕ). Thus the condensate's winding, which makes Alice symmetries multivalued, is communicated to all charged particles circumnavigating the string. This communication induces both results above: Aharonov-Bohm scattering around the string changes both charge and monopole number. Loops of string, which leave charge and monopole number well-defined asymptotically, thus support deposited charge ("Cheshire charge") and deposited monopole number. [1,2] We have recently established, through topological arguments, that Alice loops can always carry such deposited monopole number by twisting. [5] More recent work confirms that Alice phenomena also occur for global strings. [6][7][8] Such strings arise when global symmetries -describing flavor, chiral, or condensed matter degeneracies -break down. Two types of global strings can form: divergent strings, which wind nontrivially far from the string and thus have logarithmically divergent tension; and textured strings, which approach constant field values asymptotically, but carry topological winding in their finite tension cores.
We note here that divergent global strings inherit their Alice characteristics from the gauged and textured cases. Their topological structure coincides with the gauged case: therefore the same topological arguments identify twisted Alice loops with global monopoles. [5] Their couplings to fundamental charged particles are the same as those of their textured cousins. Our careful Aharonov-Bohm analysis of these interactions shows that divergent strings induce an effective Wilson line for point charges, usually equivalent to the Wilson line for the gauged case. Thus divergent global Alice strings possess the entire constellation of Alice features: charge-altering Aharonov-Bohm scattering (first noted in [8], [7]), as well as twisted loops which are point defects.
Recently Benson, Manohar and Saadi found textured global strings with the same features. [6] At finite radius, their strings alter charges of particles that Aharonov-Bohm scatter around them. They also form twisted loops which are skyrmions, just as gauged Alice strings form twisted loops which are monopoles.
We show here that these Alice properties emerge generically for textured Alice strings.
The result is striking, as it stems from very different physics than that of the the gauged case. The topology differs: gauged strings have nontrivial first homotopy group π 1 , and form twisted loops with nontrivial π 2 ; while textured strings have nontrivial π 2 , and form twisted loops with nontrivial π 3 . Gauged strings vary nontrivially at infinite radius, yielding long-range flux-mediated interactions; while textured strings are asymptotically constant, yielding short-range direct interactions. Yet both are line defects with the Alice constellation of features: they induce apparent charge violation in Aharonov-Bohm scattering of charged particles (at least at finite scattering distances), and they form twisted loops that are topological point defects.
We demonstrate these results below as follows. The divergent global Alice string has the simplest structure, and thus appears in section II. Here we construct the divergent Alice string, assuming that the string couples naturally to fundamental charges via effective mass terms. We then show how global Aharonov-Bohm scattering converts such couplings into an effective charge-violating Wilson line related to that of the gauged case. This analysis generalizes earlier work of McGraw [7], and lets us develop tools for studying textured string scattering in a simpler context.
We then consider the textured Alice string. We construct it topologically in section III, again assuming representations with effective mass couplings to the fundamental charges.
Section IV explores the scattering of point charges in the textured string background. This scattering -which generically mixes degenerate states through a nonabelian Aharonov-Bohm effect -has a rich structure. It induces an effective Wilson line U(r, ϕ) which is radially dependent: while trivial at infinite radius, it becomes a multivalued group rotation at finite radii. This induces multiple effects. At finite radius, eigenstates of locally unbroken charge alter their charge, in an r-dependent way, in circumnavigating the string. Even asymptotic unbroken charges can interact non-trivially with the string, by approaching it, circumnavigating it at finite radius r o , and then returning with altered charge.
Finally, in section V, we discuss the connection -inherited from the Hopf fibrationbetween twisted textured strings and skyrmions (or textures). We note that the manner in which the string twists in internal space along the string loop is different from the gauged case: while the gauged case involves a string whose plane of rotation rotates along the loop, the textured string rotates always in the same plane, but with an offset angle increasing along the loop.
We review our conclusions in section VI. Intuition-building examples are developed throughout the text, including the original Alice symmetry-breaking pattern proposed by Schwarz [1], patterns realizable when a larger SO(6) symmetry breaks down, and the Benson-Manohar-Saadi textured string involving Majorana interactions with point charges. Of these, the first is most testable, as the global symmetry-breaking transition responsible for the nematic phase in liquid crystals; the others are motivated by particle physics GUT phase transitions.

II. DIVERGENT GLOBAL ALICE STRINGS
Consider a global symmetry group G broken to a residual subgroup H by some condensate φ . Divergent strings form under the same topological conditions as gauged strings: whenever the vacuum manifold G/H of degenerate φ orientations has nontrivial loops π 1 (G/H). Such loops determine the string's topologically nontrivial boundary conditions: that is, the winding of φ asymptotically. We parametrize this winding via the angledependent G rotation U(ϕ) which generates it: Here U(ϕ) acts on φ o according to its group representation. Recall that, in the gauged case, U(ϕ) is the Wilson line due to the string's nonabelian flux. Here, it is the same group rotation, inducing the same condensate winding, but with divergent gradient energy cost per unit length.
By the exact sequence for π 1 (G/H), U(ϕ) has two topologically nontrivial forms: it can be a nontrivial loop in G, which cannot lie wholly in H; or it can be a path in G, ending on some element h o in a disconnected component of H.
To find an Alice string, which makes some H generators multivalued, we must adopt the second possibility. This is because the symmetries H leaving φ (ϕ) invariant, and their generators T h , are parallel transported around the string: Thus T h (ϕ) becomes multivalued only when a nontrivial U(2π) fails to commute with T h,o .
Thus divergent strings are Alice, with multivalued generators T h (ϕ) , whenever U(2π) is an element h o which 1) lies in a disconnected component of H o , and 2) doesn't commute with Alice generators T h,o . We note further that U (ϕ) rotates the condensate, and connects disconnected components of H, so that it must be generated, at least partially, by broken symmetry generators T b . To minimize string energy, we take U (ϕ) to be generated uniformly by a single broken generator T b : Given this construction of the divergent global Alice string, existence of global monopoles formed by twisted string loops is guaranteed by [5]. Charge violation due to Aharonov-Bohm scattering requires more careful analysis. Fundamental fermions ψ carry charges and interact with the condensate φ through effective mass terms of either Dirac or Majorana form: Here the condensate field φ is in a group representation transforming either as (II.6) The Dirac mass term is simpler and generalizes readily to charged bosons, so we focus on it here. The Majorana interaction arises in some symmetry breakdowns of SU(n) groups, as for the textured global string model of Benson, Manohar and Saadi. We return to its role in Aharonov-Bohm scattering in section IV.
For the Dirac-type interaction, the fermion ψ has a spatially varying effective mass matrix outside the global string. This determines the fermion's behavior in Aharonov-Bohm scattering around the string. For, under Aharonov-Bohm scattering, a fermion in mass eigenstate ψ a does three things: 1) it tracks the local mass eigenstate ψ a (ϕ); 2) it acquires the time evolution phase e −imat ; and 3) it acquires a "Berry's phase." This Berry's phase is generally nonabelian, as the residual symmetries H induce fermion mass degeneracies. Thus ψ a (ϕ) mixes with other degenerate eigenstates ψ b (ϕ), according to the nonabelian Wilson line [9] W (ϕ) = e − ϕ dϕ ′ A ϕ ′ (II. 8) with the (matrix) gauge field A ϕ ′ given by in the degenerate subspace. Altogether, then, a fermion ψ a (ϕ = 0) becomes, after circumnavigating the string, For the divergent Alice string, the spatially varying mass eigenstates are  In the third case, the rotated state T b | ψ a has components in the degenerate subspace.
By basis choice we diagonalize T b within the degenerate subspace, taking there. This produces an overall numeric phase in Berry's phase W (ϕ) , one that exactly cancels an overall numeric phase in ψ a (ϕ) = U(ϕ) ψ a . This shows that T b 's action in the degenerate subspace is spurious, causing a nonphysical phase variation in ψ a (ϕ) which is eliminated via the self-consistent Berry's phase approach. string generator T b fails to rotate the mass eigenstate entirely out of its degenerate subspace.
In that uncommon case, W (2π) cancels any nonphysical phase which creeps into the Wilson loop U(2π).
Behind our calculation of fermion scattering lies the physical question, "Is charge conservation violated in Aharonov-Bohm scattering around the divergent global Alice string?" Details are model-dependent, but the mass eigenstate scattering above lays the foundation for possible charge violation. This is because the effective Wilson loop, W (2π) U(2π) = W (2π) h o , makes only some mass eigenstates multivalued, while others remain single-valued.
Thus a charge eigenstate where the first sum ranges over single-valued mass eigenstates, the second over multivalued ones, becomes after circumnavigating the string. Typically, c ′ b differ from c b by a phase. Thus, so long as the charge eigenstate has both single-and multivalued mass eigenstate components, the scattered state is physically distinct, and carries distinct charge, from the initial |q . This means that charges alter, just as in the gauged case, when Aharonov Bohm scattered around the divergent global Alice string. In the usual case where W (2π) = 1l , it is precisely those charges associated with Alice generators T ho -which do not commute with the Wilson loop U(2π) -which may be altered. For in this case, the effective Wilson loop is just the gauge Wilson loop, U(2π), and the Alice charge eigenstates obey Thus, in this usual case, all Alice charges not annihilated by [T ho , U(2π)] are nonconserved in scattering around the divergent global Alice string.

A. The Canonical Divergent Global Alice String
We illustrate this general formalism with the simplest example, the Schwarz Alice string, whose scattering in the divergent global string case was considered by McGraw. [7] Here G is SO (3), and the Higgs φ transforms as a matrix and interacts with fundamental fermions through Dirac mass terms. φ develops the vev with an O(2) residual symmetry H containing z-rotations R z (α) and the discrete symmetry π 1 (G/H) = Z 2 so we have global Z 2 strings generated by This string is Alice, as U(2π) fails to commute with the unbroken symmetry generator T z ; in fact, on parallel transport around the string, We now show that this formal Alice property yields physical Alice behavior -with twisted Alice string loops forming monopoles, and charge-violating Aharonov Bohm scattering. As always for divergent global strings, twisted loops form monopoles due to the arguments of [5]. Charge violation occurs because some fermion mass eigenstates are doublevalued; in particular, we have spatially varying mass eigenstates (II.20) Note that the first two mass eigenstates are degenerate, while the last two are double-valued.
The nonabelian Berry's phase vanishes, as the string generator T x does not connect any degenerate mass eigenstates. Thus this example has the same effective Wilson loop U(2π) as in the gauged case, making two of three mass eigenstates double-valued. As expected, fermions reverse charge in circumnavigating the global string, for charge eigenstates scatter as |± → |∓ , |0 → −|0 in traversing the string. This can be seen directly from the action of the effective Wilson loop U(2π) = h o on the charge eigenstates, or from the underlying spatial variation of the constituent mass eigenstates.

III. BUILDING TEXTURED ALICE STRINGS
Again consider the breaking of a global symmetry G to residual subgroup H by Higgs condensate φ . Finite tension strings form only if they are textured; that is, if they approach a constant vev φ o asymptotically and carry topological winding in their cores.
Such winding occurs whenever the vacuum manifold has nontrivial homotopy π 2 (G/H).
This describes the condensate's nontrivial variation through degenerate vacuum states in the plane transverse to the string; because each point at spatial infinity approaches the same vacuum state φ o , the transverse plane becomes identified with a sphere S 2 , and has topologically nontrivial vacuum configurations classified by π 2 (G/H).
We display the condensate's spatial variation in Figure 1, noting that it is generated by some group rotation U(r, ϕ) acting on the single asymptotic vev φ o : (III.1) U(r, ϕ) acts on φ o according to its group representation, and varies over G in some singlevalued, nonsingular way. By the exact sequence for π 2 (G/H), U(r, ϕ) has two topologically   Construction could, of course, vary greatly by model; here, however, we suggest a useful ansatz. Say our nontrivial loop h(ϕ) is generated by unbroken generator T h , which anticommutes with some group element b o . (This happens whenever G contains SU(2) or SO (3) subgroups, but could occur for more exotic symmetry breaking patterns as well.) Taking b o to be generated by broken generator a nontrivial map U(r, ϕ) is given by

and approaches the nontrivial loop h(ϕ)
at infinite radius. The only caveat is that U(r, ϕ) must be single-valued. This could occur if h(π) commutes with T b , a possibility we exploit in the canonical Alice model. For our GUTinspired models, however, we insure single-valuedness by taking T b and T h to lay within a partially broken SU(2) subalgebra. We then choose Lie algebra bases so that the generators within the subspace where T b and T h are nontrivial. (We note that we can make these choices whenever G includes SU(N) factors, for N ≥ 2, or SO(N) factors, for N ≥ 4, as we show by construction below). Thus, for these models, Thus far, our ansatz constructs a nontrivial textured string. When is this string Alice; that is, when does an unbroken generator T h ′ (r) determine non-conserved charge eigenstates in circumnavigating the string? Surprisingly, the answer is nontrivial, and depends on the nonabelian Berry phase to resolve ambiguities. Formally, unbroken generators are parallel transported throughout the textured string: Naively, since U(r, 2π) = U(r, 0), all generators are single-valued; none would appear to be Alice. Closer analysis, however, in all our examples below and in section IV, reveals that any generator T h ′ which fails to commute with U (r, ϕ), at any angle ϕ, is Alice: its charges alter when Aharonov-Bohm scattered around the string at finite, nonzero radius.
We construct textured Alice strings in three models below, following this ansatz. In the It is in this subspace, of an SO(4) theory broken to SO(2) × SO (2), where our textured string will vary. We therefore reindex these rows from 1 to 4 in what follows.
A natural basis for so(4) in our subspace is the set { T ij } generating rotations in the six planes ij. However we exploit the basis which makes the isomorphism SO(4) = SU(2) × SU(2) clear: in the SO(4) subspace, with acts on φ ✷ to create a nontrivial textured string. This is because U(r, ϕ) is single-valued and nonsingular, giving b o = −iJ 2 at the origin and the nontrivial loop h(ϕ) asymptotically.
Because φ transforms in the adjoint, the textured string looks like We can also use the unbroken symmetry to rewrite this as We see here how ambiguities arise in establishing the string as Alice. For the first form (III.15) for φ ✷ encourages us to defines spatially varying mass eigenstates as Applying the transformation law (III.19), the textured string looks like Again, the existence of double-valued mass eigenstates in the (12) plane, makes all (13) and (23) unbroken generators appear to be Alice (since the mass eigenstate (0, 0, 1) is spatially invariant). But such omnipresent Alice behavior again cannot be physical -our solution is nonsingular at the origin and constant at spatial infinity, and thus cannot physically cause Alice scattering at those radii.
Again we resolve the conundrum in section IV, via careful nonabelian Aharonov-Bohm scattering analysis. At the origin and at spatial infinity, we show that no charges are Alice.
However, at finite intermediate r, all charges whose generators T h ′ fail to commute with U (r, ϕ) show r-dependent Alice behavior. Given this, our ansatz gives a lowest winding textured string, with Note that U is manifestly single-valued. It approaches the nontrivial loop R z (2ϕ) at infinite radius, and the value R x (π) = h o at the origin, since U determines the textured string

(III.32)
This determines apparently single-valued mass eigenstates, However, degeneracy of the first two eigenstates again forces a careful nonabelian Aharonov-Bohm analysis, which again reveals nontrivial Alice behavior at finite radius.

IV. CHARGE SCATTERING BY TEXTURED ALICE STRINGS
We have identified three models with textured strings. Each string is generated by the action on a vev φ o of a single-valued group map We now consider Aharonov-Bohm scattering of fundamental fermions ψ by such a string.
Details of the scattering analysis differ, for Dirac and Majorana fermions; however, both give the same result: charge-violating scattering at finite nonzero radii only. We treat each in turn.

A. For Dirac Fermions
When ψ couples to φ through Dirac couplings (II.5), φ transforms under U by conjugation and produces the effective mass matrix within a subspace of degenerate mass eigenstates ψ a (r, ϕ). For our mass eigenstates (IV.3), this is determined by Here T b ′ is another broken generator (J 3 for the SO (6)  More specifically, consider the angle α(r) ≡ π [1 − cos (kF (r)) ]. h( α(r) ) truly acts as an r-dependent Wilson loop, for Aharonov-Bohm scattering around the textured Alice string. To see consequences of this Wilson loop, we consider two specific Aharonov-Bohm scattering experiments: in the first, we bring a charge radially inward from spatial infinity to r, circumnavigate the string, then return radially to spatial infinity to compare or interfere with untransported asymptotic charges. In the second, an observer within the texture defines charges, as related to the unbroken symmetries he observes locally, and observes parallel transport of those charges around the string. Either case gives the same criterion for apparent charge violation, and in each, this criterion is met generically for circumnavigation at finite nonzero r.

Reference Charges at Infinity
First consider comparison of charges defined at spatial infinity. Here we define charge eigenstates |q h ′ for all unbroken generators T h ′ . We note first that the nonabelian Aharonov Bohm gauge field A r (analogous to equation (IV.5)) vanishes for our models. Thus radial paths contribute no Aharonov-Bohm phase, and mass eigenstates simply track the r-dependent eigenstate (IV.3) during any radial transit. Thus, on the radial transit inward, the charge eigenstate |q h ′ of generator T h ′ evolves to taking ϕ = 0. After parallel transport around the string, the charge eigenstate acquires the effective Wilson loop h( α(r) ): Now this circumnavigated charge must be transported back to spatial infinity, for comparison with other asymptotic charges. This gives for the final transported charge Note we can see this by decomposing our circumnavigated charge (IV.10) at r into mass eigenstates e −iF (r) T b |e a at r: Each mass eigenstate e −iF (r) T b |e a then adiabatically tracks back to its value |e a at r = ∞, Thus, asymptotic symmetry generators T h ′ define charge eigenstates |q h ′ which acquire the effective Wilson loop e +iF (r) T b h( α(r) ) e −iF (r) T b in our thought experiment: when brought in to radius r, parallel transported about the string, then returned to spatial infinity.

Reference Charges at r
Now consider charges as defined by an observer within the texture, at radius r and angle ϕ = 0. Because φ assumes a nonasymptotic value at finite r, its residual symmetry group H is not the asymptotic one. In particular, for our φ -solutions (IV.2), the unbroken symmetry generators leaving φ invariant at r are again for ϕ = 0. Thus the charge eigenstates at r are given by related to the asymptotic charges by change of basis. Note that this is the same equation as equation (IV.9) for asymptotic charge eigenstates parallel transported to r; however, here the meaning is different, as these are the charge eigenstates as defined at r.
After parallel transport around the string, the charge eigenstate acquires the effective Wilson loop h( α(r) ): Thus, for an observer within the texture, symmetry generators e −iF (r) We thus see that our two experiments, involving asymptotic or r-defined charge, are related by change of basis, and give the same criterion for charge nonconservation. Recall that any charge q h ′ is nonconserved if 1) its associated generator fails to commute with the Wilson loop; and 2) the charge eigenstate is not annihilated by the nontrivial commutator, as discussed in equation (II.15). Here that means charge q h ′ will be nonconserved, in either thought experiment, if or equivalently, for finite nonzero r, Recalling the algebraic result (IV.6) for our models, our criterion for q h ′ charge violation becomes with k and T b ′ as given after equation (IV.6). That is, in each model, the textured string varies only over some SU(2) (or SO (3) Thus scattered charge remains unchanged at the origin and spatial infinity; elsewhere, it becomes a radially dependent admixture of variously charged states. Note that, although this textured Alice string is a Z 2 string topologically, its charge violation is generally not Z 2 : two traversals around the string are not trivial, at most radii. Only at the special "midpoint" radius, where α = π, kF = π/2, does a Z 2 scattering occur: |± → |∓ .
Similarly, for our canonical Alice/nematic liquid textured Alice string of section III C, |± charge eigenstates of T h = T z scatter to Again we find nontrivial, but non-Z 2 , mixing of the charge eigenstate into states of different charge. At the string "midpoint," α = π, F = π/2, the charge is actually conserved, as |± → |± . Elsewhere, however, for 0 < F < π (finite nonzero r), the charge eigenstates mix nontrivially.
We have thus shown that circumnavigation of the string at finite radius violates charge conservation, for charges defined either asymptotically or locally, in accordance with local unbroken symmetries. Those charges associated with unbroken symmetries are simply nonconserved, at finite radius.
In the gauged or divergent global Alice string case, charge violation occurs because local residual symmetries h cannot be consistently extended to singlevalued symmetries h (ϕ). Here that is not true -the fact that the group map U(r, ϕ) generating the string is single-valued induces perfectly single-valued symmetries, and symmetry generators T h ′ , when parallel transported around the string by (II.2). However, no set of associated conserved charges can be defined, at finite radius. One might hope some "hidden" set of charge eigenstates, murkily connected with the local unbroken symmetry, might emerge. However, the only candidates for such conserved charge eigenstates are those simultaneous charge eigenstates |q h , q h ′ , . . . , for all T h ′ commuting with T h . These, at least, are conserved in circumnavigating the string, acquiring only the r-dependent anyonic phase e −iq h α . However, any deviation from a circular path in traversing the string violates these eigenstate's charge, for the same reason that asymptotically defined charge becomes violated at finite radius.
That is, charge is nonconserved because radial transit causes the charge eigenstates to change and become misaligned with the Wilson loop h( α ).

For Majorana Fermions
Understanding the Aharonov Bohm scattering for Majorana fermions in the Benson-Manohar-Saadi model is slightly more complex. Here fermions ψ interact with the textured via the mass interaction −mψ φ ψ c + h.c. At first glance, this gives the mass eigenstates (III.27), with mass eigenvalues m e ∓2iF (r) , and with nonabelian Aharonov Bohm phase cancelling the apparent angular variation of the mass eigenstates. However, to get the masses and nonabelian Berry's phase correct, we must look more closely at the bispinor (ψ, ψ c ) and how it obtains its mass. We include a self-conjugate bare Majorana mass matrix M for clarity. The full mass matrix is acting on (ψ, ψ c ). It has mass eigenstates where ω = M + m e −2iF 1/2 , (IV.26) ψ ± are the mass eigenstates (III.27) in flavor space and χ is the self-conjugate field ψ + ψ c in spin space. These have associated, spatially varying mass eigenvalues with degeneracies between ψ 1+ and ψ 2+ , ψ 1− and ψ 2− .
We now calculate the nonabelian Aharonov Bohm phase (IV.5) due to these degeneracies.
(IV.28) Thus our nonabelian Aharonov Bohm gauge field A ϕ mixes only degenerate mass eigenstates, with matrix elements

V. SKYRMIONS AS TWISTED TEXTURED ALICE STRINGS
We now show that the textured Alice strings constructed here can be twisted to form topologically nontrivial skyrmions -that is, point defects with nontrivial π 3 (G/H). The twist we consider is the following: bend the string into a loop, with angle θ along the string core 2π-periodic. As θ increases along the loop, twist the textured string by starting its In all these cases, then, we can exploit the isomorphism SU(2)/SO(2) ∼ S 2 to identify the twisted loop's π 3 index with its Hopf number, the linking number of any two fibers of constant φ in physical space. [10] Our twisted textured strings (V.2) were constructed to be exactly 2π-periodic in the angle ϕ + θ at finite r, and angle-independent at r = 0. Thus we obtain one fiber by taking r = 0 along the core of the loop, and a second by taking finite r = r o , with θ + ϕ = 0. This second fiber, or constant value for φ , occurs at angle ϕ = −θ as we traverse the string loop taking θ from 0 to 2π. The two fibers are drawn below in Figure 2. Note that endpoints in θ and ϕ are identified. The linking number is exactly one, showing that the twisted textured Alice string is a fundamental skyrmion. Thus, as in the gauged case, twisted textured Alice strings form fundamental point defects. Unfortunately, here the argument is not generic, as in the gauge case; it relies specifically on the Hopf map for the vacuum submanifold SU(2)/U(1). θ ϕ fibers. r fiber crosses under then over r=0 fiber; ends are identified. ο ο φ ϕ= −θ Figure 2 : Linking of fibers for < > : shown are r=0 and r ,

VI. CONCLUSIONS
We have demonstrated Alice behavior for global strings of both types: divergent global strings with nontrivial π 1 (G/H), and finite tension textured strings with nontrivial π 2 (G/H).
In both cases, point defects emerge as twisted strings. Charge violation also emerges in both cases: in the first case, the global string inherits its Alice characteristics from its gauged counterpart; while in the second, charge violation is generic, at finite radius. Specifically, for the divergent global string, charge violation occurs for all generators that fail to commute with W (2π) U(2π), where U(2π) is the Wilson loop of the associated gauged string, and W (2π) is a rare correction due to the nonabelian Aharonov Bohm effect. In the textured case, charge violation occurs for all generators which interact with the string at all, as a direct consequence of the nonabelian Aharonov Bohm effect. These textured strings are associated with topologically nontrivial loops in H, h(ϕ). We showed that the nonabelian Aharonov Bohm effect determines an effective Wilson loop for the string at finite radius of h(α), where α ranges from 0 at infinite radius to 2π at r = 0. At finite radius, a mismatch between this Wilson loop, and locally defined unbroken symmetry generators, leads to generic and observable charge violation.

ACKNOWLEDGMENTS
This work was supported by NSF grant PHY-9631182, and by the University Research Committee of Emory University. I thank Tom Imbo for useful discussions, particularly on subtleties of the Hopf map.