Massless modes on cosmic strings

Physica B 178 (1992) 47-55 North-Holland PHYSICAI

Massless modes on cosmic strings

M a r k H i n d m a r s h 1 Department of Physics, The University, Newcastle-upon-Tyne NE1 7RU, UK

This constitutes a brief review of the physics of excitations in the core of a cosmic string, plus some generalisations of simple Abelian models to their non-Abelian counterparts. Similarities and differences to condensed matter systems are observed.

1. Introduction

Like many discoveries in particle physics, that of massless excitations on cosmic strings [1] was prefigured by condensed matter physicists [2]. In 1964 Caroli et al. pointed out that there exist very low energy excitations in type II superconductors, much lower than the mass gap, when there are Abrikosov vortices [3] present. These excitations are essentially quasiparticles confined to the vortex by the interaction with the Cooper pair field. It was some time later before particle physicists noted similar effects with elementary fermions in the background provided by a vortex in a relativistic field theory [4]. Working in two spatial dimensions, in was shown by Jackiw and Rossi in 1981 that for fermions that are massless in the symmetric phase, the Dirac operator has In] zero energy eigenvalues in the presence of a vortex of winding number n. The existence of such zero eigenvalues is intimately connected with the topology of the background fields [5].

It was not until 1985 that it was realised that for a vortex in three spatial dimensions, a cosmic string [6], that each zero mode of the transverse Dirac operator produces a massless fermion liv- ing on the string [1]. If some of these fermions are charged then the string becomes a perfect conductor, for in the idealised world of the

Present address: DAMTP, Silver St., Cambridge CB3 9EW, UK.

cosmic string there are no impurities to scatter the fermions and cause resistance. Although there is no mass gap at the Fermi surface, which is at zero momentum, the strings were termed superconducting.

Witten [1] also showed that there were strings with a better claim to being called superconducting, for it is quite easy to construct reasonable theories with cosmic strings which have elec- tromagnetically charged scalar field with a non zero expectation value at the core. The Ginz- bu rg -Landau theory of superconductivity now tells us that the string behaves exactly like a thin superconducting wire. By thin, we mean very thin; if the strings are formed when the universe cools through a Grand Unified phase transition, they will be about 1 0 - 2 9 m wide, with a critical current as high as 102o A. With a transition temperature of about 102s K, these would be the highest of high-T c superconductors.

It may seem like an unnecessary elaboration to propose not only that there exist cosmic strings, but also that they are superconducting. In fact, there are good reasons to suppose that strings in Grand Unified models are generically superconducting, for the order parameter in the lowest energy string solution need not vanish in the string core [7] (much as for vortices in 3He). Since many components of the order parameter in a Grand Unified theory are generally charged, that a string should be superconducting is not too surprising. If such strings exist, it has been sug- gest that they could produced a variety of highly

0921-4526/92/$05.00 © 1992- Elsevier Science Publishers B.V. All rights reserved

48 M. Hindmarsh / Massless modes on cosmic strings

energetic astrophysical phenomena [8], for an oscillating loop of string of size L carrying a current I is a radiating magnetic dipole with power ~ I 2 at frequency L -1 (here, as every- where, natural units are used in which h = c = k B = 1). Furthermore, the back reaction of the current on the motion of the string might signifi- cantly alter the usual string scenario [9].

In this talk paper, the essential physics behind SCS are outlined and, the connections to condensed matter physics explained. One of the surprises is that when there is simultaneous fermionic and bosonic superconductivity, a system looking like a relativistic Fr6hlich superconductor emerges [10, 11]. In order to compare and contrast with vortices in 3He, (global) non- Abelian vortices are also discussed, which also have the property of supporting superflows in the core.

2. Simple superconducting strings

The relativistic version of the Ginzburg-Land- au model of superconductivity is called the Abelian Higgs model* [13], which consists of a complex scalar field q~ and a real vector field fi~u (where ~ C {0, 1, 2, 3}). The Langrangian is

! F w.v 2 (1)

where D = 0 , - i Y f i ~ , is the covariant derivative. The potential (or bulk free energy density) is quartic, as renormalisability demands [13], and has the form

V(q') = ½A(I~] 2 - 71-12)2 (2)

at zero temperature. The ground state is therefore Iq~l = 7 , which breaks the U(1) symmetry @--> q~ e ia°~), fi,~ + 00. At finite temperature the parameter 7 is modified to 7 ( T ) = 7 ( 1 - T2/ T~) 1/2 [14], where T c = 7, and so there is a phase transition at T = T c which is usually second

* The relation between the two theories has been investigated by Davis [12].

order, but can be first order for some choices of the dimensionless coupling constant A [14]. The order parameter for this phase transition is just ( @ ) . It is of course well known [3, 15] that the equations of motion following from (1) have vortex solutions which can be written in the form

= f ( p ) e i'~, where p and q~ are cylindrical polar coordinates. The boundary conditions on f are f (0) = 0 and f(~) = 7 (T) , and in the vortex solution f approaches its value at infinity exponentially fast. The gauge field A , has a component in the azimuthal direction a(p) /Yp , with a(p )

2 p near the origin and a ( ~ ) = 1. It should be pointed out that the U(1) symmetry is not electromagnetic when we are discussing cosmic strings, for otherwise the whole universe would be a superconductor.

Now let us add a Dirac spinor which is massless in the symmetric phase but picks up a mass g7 when the symmetry is broken. The interesting effects occur because the left- and right-handed components of the spinor must have different charges ql and qr if the U(1) gauge symmetry is to be respected. The fermionic part of the Lag- rangian is therefore

~LP I = ~pi T • D q J - g@(OP+O- g@*~bP q,, (3)

where P+ = ½ ( 1 - + ys) are the projectors onto the right- and left-handed spinor subspaces [13]. Here , the covariant derivative is D~, = 0 , - i(c~rP + + ~lP )A~. Gauge invariance at the classical level** requires q r - ql = ~" W e will take c~r = - ~ = Y/2. In the vortex background, the Dirac equation then becomes

[iTA(0A -- i½YYSAa(P)) -- g f ( p ) ei~%]~0

= iy ~ 0,~b, (4)

where x A are coordinates perpendicular to the string (A E {1, 2)) and a takes the values 0 or 3, labeling coordinates which are tangent to the world sheet of the string. Jackiw and Rossi [4] showed that these equations have solutions with

** Note that this theory as it stands loses its gauge invariance unless 4= 0, i.e. the theory is anomalous, but we will correct the problem later.

M. Hindmarsh / Massless modes on cosmic strings 49

the left side equal to zero: these are the "zero modes" of the transverse Dirac operator in the square brackets. Note that this operator commutes with 3/03/3, so the solution can be generally written as a linear combination of the four eigen- states of this operator, with eigenvalues E = -+1. Note also that the covariant derivative term can be rewritten in a useful way:

1 2 3/AD A = 3/'(D 1 + 3/ 3 / D2) = 3/I(D 1 + i3/°3/33/5D2)

1 ei,?,( 1 ) = 3/ Op +ie3/5 P (09 - i l a ( p ) ) .

(5)

If we examine (1.4) in this light, we see that we i 5

can take out a factor of e ~ ~ if E = - 1 , and the equation simplifies to

1

= i3/~ OaO.

+ la)3/5)(iyyl) - gf ]t)

(6)

There is not too much extra work in showing that there exists a solution tp(p) which solves this equation, with the left side vanishing, and also that there is only one such solution (or In I if the background is an n quantum vortex). The modulus of gt tends to zero as exp( -gBp) away from the vortex, and so it is normalisable, while at the origin it tends to a constant. Up to now we have ignored any possible t and z dependence. In fact, it is clear that if ~P(p) is a transverse zero mode with ~ = - 1 then so is a(t + z)gt(p), since (0 0 - O3)a = 0. Thus in three spatial dimensions, the transverse zero mode produces a fermionic excitation with wave velocity 1 (the speed of light) which is confined to the string.

As mentioned above, this theory cannot be quantised as it stands, because it is afflicted by the famous axial anomaly [13] in the current ~ysy" O. This happened because the left- and right-handed components of the spinor were given different charges. The cure for this is simple: we add another fermion field X which couples to 05* instead of 05, so that its components have opposite charges to those of 0. This field will also have a zero mode, but because the

phase factor from the vortex appears with opposite sign in the Dirac equation, the zero mode has eigenvalue +1 under 7°73, and so it travels in the opposite direction. Let us write the zero mode solutions / 3 ( t - z )X(p ) . If we form a 2- component 2-dimensional spinor 7/= (/3, a) r then the equations of motion for a and/3 follow from the 2-dimensional action

= f dt dz(~ iF . Or/), (7) S (2)

where F" are 2-D Dirac matrices. Now suppose that we give qx and X elec-

tromagnetic charges q and q', respectively (since 05 is assumed electrically neutral, both components of each spinor have the same charge). Thus in the effective 2-D theory, the zero modes supplied by ~0, which we shall call left movers as they always travel in the - z direction, have charge q while the right movers have charge q'. Consider what happens when we apply a constant electric field E parallel to the string for a time t. Suppose that initially the Fermi levels are filled up to zero energy for both components. Ignoring any possible scattering for the moment, the Fermi momenta of the left and right movers will move to - q E t and q'Et, respectively, as the electric field accelerates the filled states. If q, q' and E are all positive, qEt/2w holes of charge - q and q'Et/2~v particles of charge q' have been created per unit length, resulting in a current I of ( q 2 + q,2)Et/2,~. Thus J ~ E , which is charac- teristic of a superconductor or a perfect conductor. That there really is no scattering in this case is assured by conservation of U(1) charge. In order for a left moving state to scatter into a right moving state, particles with total charge 8 must be created, and momentum must be conserved in the z direction. In this model there is no way of doing this. The limiting current is determined by when the top of the Fermi sea gets high enough so that the unbound states are accessible, that is when k v > g~.

There is a constraint on the charges q and q', for besides creating current at a rate (q,2 + q2)/ 2-rr the electric field also creates charge at a rate (q2-q'2)/2~r. Thus charge conservation requires that q , 2 = q2. T h i s is actually one of the


conditions that the theory has to satisfy if it is to be anomaly-free upon quantisation [11. Hence when there is mysterious talk of the necessity of cancelling anomalies all that is meant is that charge must be conserved.

There is another way of making a string superconducting, which is to arrange for a charged scalar condensate to exist at the string core, resulting in an effectively two-dimensional Ginz- burg-Landau theory. To this end we add another scalar field or to the model with electric charge e, and demand that there is another gauged U(1) symmetry in the theory which shall be identified with electromagnetism. The most general quartic potential that respects the U(1) x U(1) symmetry that we wish to impose is

V ( ( / ) , o - ) ~ - 1 9 _ yA](Iq) I- 712) 2+1A21o-I 4

+ A , ( I ~ I z - .2)l~rl2. (8)

I f /z : ~< ~/2, and A1A2/A ~ > ].Z4/T~ 4, then the absolute minimum of the potential is I q,]=T/ and cr =0 . In this case we still have the broken U(1) symmetry while retaining electromagnetism as a long range interaction. As before, string solutions exist with q) = f ( p ) e i~. At the core of the string, ]q)] must vanish and the potential takes the form

V(0, 0 9 = ' ~ ~ iA2]o" ] -- A3/J,-]o" ] (9)

If ~3/d, 2 is negative, the total potential energy of the string solution can be reduced by allowing o - # 0 at the core, albeit at the expense of gradient energy from the ]0o-] 2 terms. It can be shown [1, 16] that for p,/r I sufficiently close to 1 that the static string solution does indeed have non-vanishing ~r(p) as p--~0, so on the string electromagnetism is spontaneously broken. But

i0 if o-o(p) is a solution then so is e %(p ) , and so there exist low energy excitations of the form e i°(' Z)o-0(p). The effective 2-D action for these modes can be obtained just by integrating the 4-D action over the transverse directions. Thus

S~2'=fdtdzfdxdylD.o-I 2

= K f dt dz(O~O - e A , ) 2 (10)

where • = 2w .f 1~012p do, and Aa is evaluated on the string with the assumption that it changes little over the cross-section. The electromagnetic current Ja arising from non-trivial 0 dependence is

8 s ~ 2) J~ - 8A" - - 2 e K ( O a O - e A a ) , (11)

which is conserved through the equation of motion for 0. In 2-D we can also construct another

ab covariant equation for Ja through the tensor •

ab • OaJ b = 2 K e 2 E (12)

where E is the electric field parallel to the string. This is just a covariant version of the London equation. The supercurrent of the bosonic string is limited by the back reaction of the current on the condensate, and turns out to be of order /~ [1, 16].

An interesting side effect of having a current flowing in the string is to reduce its tension, and if the critical current is high enough the tension can vanish. In that case circular loops of string with large currents flowing in them would be prevented from collapse. Davis and Shellard have emphasised that giving the loop a charge reduces the backreaction on the condensate [91 and boosts the classical stability of the object. This is potentially disastrous in a cosmological context, for even if the probability of forming such objects were small, the matter in the universe would very early become dominated by these objects [9]. However, the current-carrying loops can only at best be metastable, for current can be lost through quantum tunnelling events at which one unit of flux is lost as ~ tunnels through ~r = 0 somewhere on the loop [1, 16]. The tunnelling rate depends exponentially on K, so by appropriate choice of parameters in the potential (8) it can be made very large or very small [16]. The fact that the string is no longer truly superconducting once quantum fluctuations are taken into account derives from a well-known property of the superconducting phase transition in 2-D [171.


3. The Fr6hl ich superconduct ing string

Suppose we combine the bosonic and fermionic models of the previous section, and allow couplings between fermions and the extra boson field o-, which must take the form

~w~,x = a(cr~hP+x + o-*xP_4,) (13)

if gauge invariance is to be respected. This requires that e = q - q'. Provided that the coupling to the core condensate is weak, the functional form of the fermionic zero modes will be little affected and to a first approximation aq t (p ) and /3X(p) will still solve the Dirac equation, although a and/3 are no longer functions of t +- z exclusively. In fact, by integrating (13) over the transverse coordinates we see that the coupling between left and right movers produces an effective mass term in the 2-D theory, whose action becomes

= f dt dz(~ i v . D r / - m~ eirS°r/), (14)

where F 5 is the 2-D chirality operator, and the effective mass m is given by

m = A f dx dy ~rt, l@X I . (15)

The existence of a mass gap for the modes on the string might seem to indicate that the string losses its superconductivity. However , this is not generally the case [11]. When e ~ 0 the cr condensate still exhibits London-type behaviour as in (12), although the coefficient relating the rate of current increase with the electric field is reduced. When e = 0 the physics that maintains the superconductivity is even more interesting: condensed matter physicists may recognise in (14) a relativistic version of the action for electrons in a 1-D lattice, for which it is known that (in the absence of impurities) a supercurrent can be carried by charge density waves [10].

The neatest way to elucidate the physics of this model is to use the technique of bosonisation [18]. Under this procedure operators bilinear in

Fermi fields are replaced by bosonic operators according to certain rules, listed in table 1.

The resulting bosonic theory can be shown to be completely equivalent to the fermionic one. If (14) is bosonised, and we include the kinetic term for the 0 field, we find

S(u2) = f dt dt (K(O aO)2 + ½(O aga) 2

1 .b M2 )) + ~ qe ~31 a (~b(~ - - COS(0 -t- g ~ ( ~ .

(16)

This theory is very like the sine-Gordon model, which is a bosonised version of a simple massive fermion in 2-D [18]. There, the sine-Gordon solitons are actually fermions, and the coefficient of the potential is fixed by the requirement that the mass of the soliton be equal to the mass of the fermion of the unbosonised theory. Thus for weak coupling between the fermions and o-, M m in (16).

The electromagnetic current J, has the expres- sion in terms of the bosonic fields

1 Ja = ~ qe'ab Obq b , (17)

which is clearly conserved, while the equations of motion are

2K 020 - M 2 sin(0 + V ~ ) = 0 ,

024) - V~w M 2 sin(0 + V ~ ) - 1 v ' ~ q E , (18)

where, as before, E is the electric field along the string. The sine potential can only give one combination of the fields a mass. The other, s c = 2 K 0 - ~b/X/-~, has the equation of motion

Table 1

Fermionic Bosonic

~iF. On ½ O o d)O"4)

1 - 2 nr Fort ~ 0o4,

1

~P ± 77 I* e ~vTg*

52 M. Hindmarsh Massless

qE (19) O2s~- 2ax '

and so in a uniform electric field, s ~ increases quadratically with time. One can solve the equations of motion exactly in the constant field case, and one finds that the rate of change of the current is given by

] _ 1 q2E (20) (1 + 8~rK)

Thus the effect of coupling the left and right movers through the condensate is not to render the conductivity finite but merely to reduce the rate of increase of the current.

As mentioned above, this mechanism is practi- cally identical to transport of current by charge density waves in certain effectively one-dimensional systems such as NbSe3 [10]. There, the Peierls instability produces a lattice distortion of wavevector 2k v, which opens up a mass gap at the Fermi surface. Charge neutrality is preserved by the electron density also acquiring a modulation of wave vector 2k v, the charge density wave (CDW). If the wavelength of the CDW is incom- mensurate with the lattice spacing then it is free to slide, and will continue to do so (in the absence of impurities) once the applied electric field in removed. In the above system, 0 corre- sponds to the phase of the lattice distortion and 4) to the phase of the CDW [10]. The analogy is not quite exact, for the Fermi surface was at zero momentum, so the modulation of the CDW effectively has infinite wavelength.

4. N o n - A b e l i a n zero modes

So far the discussion has been limited to a simple model where the symmetries were all Abelian and also gauged. In this section we will discuss vortices in non-Abelian theories with global symmetries,* neglecting fermions for sire-

* A global symmetry is an invariance of the Hamiltonian under a t ransformation which is independent of space and time. A symmetry can be gauged, that is made space and time dependent , by the addition of a vector field.

modes" on cosmic strings"

plicity (fermionic excitations on vortices in superfluids and superconductors have been considered by Volovik [19], and zero modes on gauged vortices have been considered in ref. [20]). This should enable closer contact to be made with the vortices in 3He [21], although there are a number of important differences between 3He vortices and cosmic strings. The principal one is that the symmetries of the Hamiltonian of a field theory can be divided into internal and external ones. The latter symmetries form the Poincar6 group consisting of transla- tions plus Lorentz transformations, while the internal symmetries commute with the Poincar4 group and generally form some compact Lie group. The order parameter is invariably a scalar under the Lorentz group (which includes spatial rotations), while in superfluid 3He the order parameter transforms as a vector under spatial rotations. This means that two extra terms are permitted in the gradient free energy density, which have the effect of considerably complicat- ing the equations of motion.

Firstly, we establish some notation. The order parameter is a scalar field q) whose Hamiltonian density

H = 4, t4 , + a ,q , t o'4, + (21)

is invariant under a group of internal symmetries G. The potential V (bulk free energy density) may have a non-trivial minimum for certain values of its parameters. Suppose this minimum is at q )= q)0, which is invariant under a subgroup H C G. It is well known that static vortex solutions exist if rr I ( G / H ) is non-trivial [22]. We may write the general axisymmetric vortex solution as

¢ ) = g ( p , ¢ ) q ' 0 ( p ) (22)

where g(p, ¢) is a non-contractible loop in G/H. Although g is itself badly defined at the origin, provided g(0, ~p)q)0(0) = q)0(0) there is no phys- ical discontinuity there. This means that either q)o(0) vanishes, or that g(0, ¢) is entirely con- tained within the stability group of q)0(0) (the subgroup of G that leaves q)0(0) invariant). Let us call the generator of rotations of the string at


spatial infinity O. This can be determined from

i ~ a ~ O=-5(ev ace)To (23)

where {T,} forms a basis for G and q~ is evaluated at infinity. Note that g(oc, q~) = e x p ( i ~ ) ) need not commute with all the elements of H, so that in general the stability group at infinity is a function of position H(~) . In fact, H(cp) may not even be globally defined, by which we mean that g(o% 2v)hg ~(% 2-rr) ¢ h for some h E H. This is the case in uniaxial nematic liquid crystals, for example, where G = SO(3) and H = 0 ( 2 ) [22]. The generator of the unbroken symmetry group can be taken to be T 3, and Q can be chosen from T1/2 and 7"2/2. The factors of 2 are important , for they mean that T 3 changes sign on being taken around the vortex, or disclination.

Without too much loss of generality it is sim- pler to restrict the discussion to vortex solutions for which g is independent of p, and we may therefore write the solution

O(p, ~) = g(¢)(f(p)q)o(~) + O-o(p)) , (24)

change the solution at all. Let this subgroup of H~ be H0, the group of unbroken internal symmetries of the vortex. Another set of elements which leave (26) invariant are those which leave ~) invariant, i.e. those for which h ~)h = ~). Let this subgroup of H~ be ho. In some cases there may also be some special elements which fall into neither subgroup, but here we shall assume that H o kJ H o exhausts all the possibilities, as it seems to in solutions studied to date. Therefore, the t ransformation (25) acts non-trivially on the vortex solution without changing its energy when acted upon by those elements of H 0 that are not in H o. The elements of H o that are in H 0 form a subgroup of H O, say 14oo.* Thus the generators of non-trivial t ransformations in the vortex gen- erate a coset space Ho/HOo.

If we now promote (25) to a space and time dependent t ransformation with h = h(t, z), the action picks up an extra piece

6S=f dtdz2 T f p d p ( ~ ( p ) O~h l a O ho'o(p) ) .

0 (27)

with f = 1 and % = 0 at p = w. We shall denote the unbroken symmetry group at p = 0% q~ = 0 by H~. Then, just as for the U(1) x U(1) bosonic superconducting string, we can ask the result of making global transformations of the form

h(¢) = g(¢)hg ' (~) , (25)

Provided o- 0 vanishes fast enough (at least as fast as l / p , a logarithmic divergence is not serious in a real situation where there are many vortices, where their separation provides an upper cut-off on the integration), then we may define a matrix K = 2w f p dpo-o(P)O'o(p) and arrive at an effective action for the non-Abelian zero modes

with h @ H~, which leave the solution invariant at infinity, but may affect the fields at the vortex core. Under this t ransformation, the free energy per unit length of the vortex becomes

P

+ v(4,o(p))). (26)

This is unchanged if [Oh~0(p) [ = IO 0(p)l for all p. Clearly, group elements h which leave ~0 invariant for all p satisfy this condition, but they are trivial t ransformations because they do not

f S~2) = J dt dz tr(Oah-' OahK). (28)

The matrix K is the generalisation of the quanti- ty K in (10). This is the action for a kind of non-linear sigma model [23] on the coset space HQ/HQo, which has dim H 0 - dim Hoo massless modes.

So far, the discussion has unfortunately been somewhat abstract. Let us make it concrete by discussing the implications for some of the pos-

* It is easy to see that they do form a supergroup , for if h~ and h 2 are bo th in H 0 and both leave Q invariant, then hlh 2 also has the same propert ies .


sible vortices in superfluid 3He. As mentioned above, the fact that the order parameter transforms as a vector under spatial rotations means that the analysis is not as clean as for the cosmic string. The gradient free energy density is con- ventionally written [21] in terms of a complex 3 x 3 matrix A,~ i"

possibility of a zero mode in the hard core of the vortex.

We now turn to the B phase, in the region where the minimal energy vortices are axisymmetric. There the order parameter is a complex orthogonal matrix R,~ e i°, and at infinity the axisymmetric vortex solution may be written [21]

fg = ]/l OiA,~] OiA * i + ] /20 iA~i OjA *j

+ Z~ OiA~ OjA*i, (29)

where a labels spin indices and i, j label spatial and orbital angular momentum indices. In theories of spontaneous symmetry breaking in the context of relativistic field theories, the order parameter is always a scalar under the group of spatial rotations, and so the last two terms in (29) do not appear. These are the terms that complicate the issue, because they couple the derivatives of components with different angular momentum quantum numbers.

Nonetheless, we can at least show that there is no possibility for bosonic zero modes in the hard core of the vortices in 3He-A. In this region we can neglect the dipole interaction between the spin and orbital angular momentum degrees of f reedom, and treat the orbital part separately. The relevant part of the order parameter q) is then a complex vector 4h + i~b2, with ~bl • ~b 2 = 0 and I~bl] = I~b2[. The continuous part of the symmetry group of the Hamiltonian is G = SO(3)L X U(1), which is broken in the superfluid phase to a U(1) which is generated by a linear combination of one of the generators of the SO(3)L and the generator of phase rotations of q~, which we shall call I. The minimal energy vortex solution at infinity may then be written [21]

@ ( ~ , q~) = ei~L3AA(e2 + i e 3 ) , (30)

where L 3 generates rotations around the z axis, and e 2 and e 3 are unit vectors in the y and z directions, respectively. The continuous part of the unbroken symmetry group H~ is therefore generated by L I - I . This does not commute with L3, and so HO_ is trivial. There is thus no

I~(O0 @) = AB(~c~i e '~ . (31)

The continuous part of the unbroken subgroup is just the SO(3) of total angular momentum, which commutes with the generator of rotations at infinity I, and so H 0 is also equal to this SO(3). But it is well known [21] that the core of the 3He-B vortex has a rather complicated struc- ture, with several other components of the order parameter appearing, and this SO(3) symmetry is completely broken. Thus Hoo is just the identi- ty, and the effective 2-D theory in the core has at least the possibility of three massless modes, corresponding to the three generators of SO(3). Whether or not these modes can propagate along the string is an open question.

Acknowledgements

I am very grateful to Martti Salomaa for the invitation to the K6rber symposium, and I thank G. Volovik for enlightenment on the subject of vortices in superfluid 3He.

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Massless modes on cosmic strings