Spontaneous symmetry breaking and topological charge

Volume 102B, number 4 PHYSICS LETTERS 18 June 1981

SPONTANEOUS SYMMETRY BREAKING AND TOPOLOGICAL CHARGE

C.J. ISHAM The Blackett Laboratory, Imperial College, London SW7 2BZ, UK

Received 9 March 1981

Connections between spontaneous symmetry breaking and topological charges in a gauge theory are studied in a general spacetime M. The topological properties of M may enhance or inhibit the breaking and this effect is shown to manifest it- serf in relations between the initial and final charges. 'It is also shown that, in the inhibited mode, the Higgs field solution = 0 can be stable for topological reasons and thus lead to a complete restoration Of symmetry.

The physical importance of spontaneously broken gauge theories hardly needs emphasising. Equally, it is generally appreciated that the existence of topological sectors and charges may reflect deep non-perturbative aspects of the quantized gauge theory. The purpose of this work is to illustrate some important connections between these phenomena. To ensure as broad a framework as possible the fields are d+fined on an ar- bitrary compact riemannian manifold. Topologically complex spaces appear in much recent work [1,2] on quantum gravity and they may play an important role at the Planck length and in the early stages of the evo- lution of the universe. Such manifolds also appear in magnetic monopole theory if the region containing the monopole is excised or in conventional instanton theory if unusual boundary conditions are imposed. Many of the results presented below have an interpre- tation within these frameworks.

My principal contention is that in a gauge theory with topological charge, topological properties of the spacetime M may inhibit or enhance spontaneous symmetry breakdown when compared with the usual case in which M is the S4-compactification of euclidean four-space. In addition homotopically inequivalent Higgs-Kibble fields may induce breaking into inequivalent gauge sectors. These phenomena involve a deli- cate interplay between the topologies of M and of the initial and final symmetry groups G and H.

A standard lagrangian is employed with a gauge field ~ and a G-multiplet of scalar * 1 Higgs fields

propagating in a background (unquantized) spacetime metric guy:

Live= + + (I)

where

D (~ = ~ 0 + tou" TdP.

The potential V(¢) is assumed to have an appropriate form for inducing symmetry breaking from G to H (denoted G ~ H). Conventional scaling arguments [3] are not directly applicable in a curved spacetime but nevertheless the usual ground state equations

D ~ t = O : a V / a ~ i, V(~)=0 , (2)

can be satisfied provided that there are no topological obstructions to the ensuing spontaneous symmetry breaking configuration in which ~b lies everywhere in the same G/H orbit. In many cases such obstructions will occur for certain initial G-sectors and it is evident- ly important to understand when this happens and the existence and properties of stable solutions to the full field equations in this situation where the residual symmetry group H is forced away from the normal type on certain spacetime regions. Such solutions can only satisfy (2) in the complement of these sets and in this respect they will resemble conventional monopole functions except that they are regular everywhere

, I This is for simplicity only. Similar effects arise in more complicated models involving fermions.

0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company 251


Of course genuine monopole solutions may also be sought i f H h a s a suitable U(1) factor [4].

Someinsight into the symmetry breaking problem may be obtained by considering the inverse situation in which an H-gauge theory is viewed as a G-theory by embedding H in G. There will be certain relations between the topological invariants of the initial and final theories and these will be necessary conditions for G

H. For example, if M has a non-vanishing second Betti number, U(1) topological sectors arise with an integral charge C 1(S) = (4~r)- 1 fs fur d°UV where S is any closed two-dimensional submanifold of M. Examples of such spaces are the basic building blocks S 2 × S 2, CP 2 and K 3 of spacetime foam [5]. If U(1) is em- bedded in SU(2), the integrand of the ensuing "instanton" charge C2(M) is related to C 1 by [6] ,2

(1/16~r 2) tr F ^ F = (1/167r2)fAf, (3)

where F =- T'FuvdxU ^ dx v and f - - fuvdXU A dx v. If no two-form f c a n be found such that (3) is satisfied for a given choice of C2(M ) (for example when M = $4), then the symmetry breaking from SU(2) to U(I) is inhibited in this sector. Conversely there may be many two-forms f satisfying (3) (up to an irrelevant total divergence) leading to breaking into different U(1) sectors [written SU(2):2; U(1)]. The algebraic topological techniques employed below show that the existence of a two-form fur such that (3) is satisfied is also a sufficient condition for SU(2) ~ U(1) and this two-form then becomes the U(1) charge density. This is by no means obvious and would indeed be false if dim M ~> 5.

The unrestricted topological properties of M make the fibre bundle approach to gauge theories especially appropriate [7]. The starting point is a non-trivial principal G-bundle P over M in which the Yang-Mills field is a connection. I-Iiggs fields ~b are regarded as cross sections of an associated vector bundle PXGW and the lo- cal field equations 0 V/O~ i = 0 = V(¢) assign the range of ~ to a fixed G/H orbit in the vector space W. Thus defines a cross section of an associated fibre bundle Px G G/H (with fibre G/H) and as such induces a reduc- tion of the structure group from G to H [7,8]. A com- prehensive account of this mathematical analogue of spontaneous symmetry breaking may be found in ref. [9]. ,2 In cohomological terms C2 = -CxuCI ; the minus sign arises

from a factor of ~ in the Chern-Weil theory.

Thus the essential mathematical problem is to study the existence of "Higgs-Kibble" cross sections 4~ of Px G G/H. A non vanishing G-topological charge corre- sponds to a nontrivial bundle P and consequently there may be topological obstructions to the construction of q~ and hence to G --> H. Conversely, homotopically inequivalent q~ may exist, leading to inequivalent H bundles and G z~ H. One standard technique for construct- ing cross sections is to triangulate M and build ~ induc- tively on the simplices of increasing dimension contained in regions of M over which Px G G/H is trivial [ 10,11 ]. The section q~ is first defined arbitrarily on the vertices and then, if G (and hence G/H) is con- nected, it may be extended at once to the lines con- necting them. The continuation over the two-dimensional triangles will, however, be halted if q~ maps the boundary of a particular triangle into a loop in G/H which cannot be contracted to a point. This non-vanishing element of 7r 1 (G/H)is assigned to the triangle * 3 and constitutes a primary obstruction to the existence of the cross section. If dim M = D, higher (or second- ary) obstruction to G -+ H may appear in an analogous way involving n 2 (G/H), n 3 (G/H)... 7r D_ 1 (G/H). Simi- larly, two cross sections o fPx G G/Hmay be barred from being homotopic by elements of rt 1 (G/H) ... zrD(G/H) ,4 and thus provide the possibility that Gz~H.

Certain results now follow at once. For example SU(n + 1)/SU(n) = S 2n+l and rri(S 2n+l) = 0 for 0 ~< i ~< 2n. Thus if dim M ~< 5 there are no obstructions to the symmetry-breaking chain

-+ SU(n)-+ S U ( n - 1)-+ ... --> SU(3)-+ SU(2), (4)

and if dim M ~< 4, SU(3) :~ SU(2) does not occur. Sim- ilarly SO(n + 1)/SO(n) = S n and hence, without obstruction,

-+ SO(n) ~ SO(n - 1) --> ... ~ SO(6) ~ SO(S) ~ SO(4). (5)

However, in principle we could have SO(5) :~ SO(4) and SO(4) ~ SO(3) and SO(3) ~ SO(2) both obstructed. The weakness of this simple approach is that, although satisfyingly pictorial, it gives no general infor- mation on whether or not a potential obstruction ac-

,3 More precisely it defines an element of the simplicial cohomology group H2 (M; ~r I (G/H)).

,4 More precisely by elements of Hi(M; ~ri(G/H)), i = 1 ... D.

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tually arises. For example, SU(n + 1)/U(n)= CP n and rr 1 (CP n) = 0, n2(CP n) = Z but the potential lr2(cpn ) obstruction never in fact occurs (see later).

To proceed further a more sophisticated approach is required. I will only sketch the method here and will publish the details elsewhere. There exists a space BG such that the set B G (M) of isomorphism classes o f principal G-bundles over M is equal to the set [M, BG] of homotopy classes of maps from M into BG [12]. Since H is a subgroup of G, any H-bundle (represented by h: M ~ BH) is necessarily a G-bundle and there is a map J : BH ~ BG (a fibre map with fibre G/H) such that this G-bundle is represented by J o h: M ~ BG. Conversely, given a bundle P represented by f : M

BG (i.e. a topological G-sector), then G ~ H if and only if there exists h: M ~ BH such that J o h = f. The relation of h with the Higgs field is contained in the diagram:

H )Q . ~ P

M-----~ P/H w ~BH

f

Here P/H = Px G G/H and co: P/H-~ BH is the classify- ing map of P viewed as a principal H-bundle over P/H.

Homotopically equivalent Higgs fields yield isomor- phic bundles Q (i.e. H-sectors) and every H-bundle obtained by "lifting" f to h has a Higgs field ~b associated with it. It is possible that in certain cases homotopically inequivalent Higgs fields could yield the same H- bundle in which case there would be an additional topological charge associated with ~. This is currently being investigated.

The fibration BH ~ BG is split into a Postnikov tower [13] , s

J4 J3 J2 J1 BH 4 ~ BH 3--> BH 2 ~ B H 1 ~ BG

O i_ 1 : BHi_ 1 -+ K(~r i(G/H), i + 1)

= "BK(ni(G/H), i )" (BH 0 = BG).

We now try to lift f t o h 1 : M -~ BH1, then lift h 1 to h2: M ~ BH 2 and so on. The process stops at BH4, since if dim M ~< 4, [M, BH4] = [M, BH]. Each hi(h 0 = f ) can be lifted to hi+ 1 iff 0 i o h i ~ , (i.e. is homotopically trivial) and if, for some i, this is not so then G ~ H is obstructed. Now we recall that [X, K(rr, n)] equals the cohom01ogy group Hn(x ; rr). A function 0: BG ~ KQr, n) represents a characteristic class c(O) E Hn(BG; rr) and if f : M ~ BG, 0 o f : M -~ K(~r, n) represents the characteristic class c ( f ) (and hence a topological charge) of the corresponding G-bundle * 7 Thus the maps 0 i oh i correspond to elements in Hi(M; ~i_x(G/H)) and in practice (if dim M ~< 4) can be identified with functions of characteristic classes whose vanishing gives the necessary and sufficient conditions for G ~ H. For example in SU(n + 1) -~ U(n) we note that 0 0 of ~* since H3(BSU(2); Z) = 0 thus explaining my earlier comment on the absence of the lr2(Cpn ) obstruction.

Calculations of this type may be performed for any pair of groups G and H provided that the first four homotopy groups of G/H are known and that the 0 maps can be identified with relations between characteristic classes * s. ~4(G/H ) only arises when studying the G z~ H effect but lack of space precludes any further discussion here. Note that by virtue of the Palais -Mostow theorem [15] any subgroup H of G appears as the little group of some linear representation of G but of course the polynomial V(¢) which leads to the appropriate orbit may not correspond to a renormal- izable interaction. I have collected together a sample set of useful results in table 1. Some of these are easily derived whereas others require the full Postnikov treat- ment.

The initial and final G and H bundles are labelled

in which each Ji is a fibration with fibre the Eilenberg -Maclane space * 6 K(ni(G/H), i) induced by a classi- fying map

,s For accounts aimed at a theoretical physics audience, see refs. [1,14].

,6 Recall that 7ri(K(n, n)) = 7r ffi = n and 0 otherwise.

¢7

=t=8

It cannot be overemphasised that it is cohomology groups and not homotopy groups, which are the natural mathematical objects for describing topological charge. This homotopic representation of cohomology and its connection with Postnikov towers is a particularly potent tool for describing the types of problem that arise in theoretical physics. These are obtained using the Serre exact cohomology s e- quences of the fibrations Ji.

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Table 1 Spontaneous symmetry breaking for dim M < 4.

G H G ~ H Obstructions

SU(n) (1~- C2(~) = 0 SO(n) {1) P(~) = '~2(~) = 0 SO(5) SO(4) No

Implies

SO(4) SO(3) e(~) = 0

SO(3) SO(2) P(O = -e(n) A e(n), toa(~) = e(~/) rood 2

U(n) SU(n) C 1 (~) = 0 U(n + m) UR(n) X UL(m) No

SU(2) U(1) C2 (O = -Ca (n) ̂ C~ (n) SU(n + 1) , n/> 2 U(n) No SU(2) L × U(1) U(1)e.m. C a L(~) = -Ca (/~) ̂ C1 (~) U(2) U(1) Ca(~) = 0 SO(2n) SU(n) co2(~) = 0 SU(5) SU(2) L X U(1) X SU(3) c No

SU(2) L x U(1) × SU(3) c Z6

U(3) C2L(~) = 0

Z6

P(~) = p(n), w2(~) = wa(n) w4(~) = e(n) mod 2. p(~) = p(n), w2(~) = w2(~)

G _ ~ H

No

No Yes

No

Yes

C2(~) = C2(n) No C1(~) = C1L(~) + C1R(r/) Yes Ca(~) = C2L(n ) + C2R(n)

+ CxL(n) ̂ C1R(n) Yes

C2(~) = Ca(n) - C x ( n ) A C l ( n ) Yes CI(~) = Ca(n) No C1 (~) = Ca (r~) No P(~) = Ca (77) No C2(~) = C2L(n) + C2c(n) Yes

- Ca(n) ^Ca(n) C2c(~) = Ca(n) No Cl(~) = Ca(T/)

by ~ and 77 whilst {1) is the trivial group. An entry in the G -+ H column specifies charge relationships that m u s t necessarily be satisfied if symmetry breaking occurs. Conversely they are s u f f i c i e n t conditions and al- low G -+ H if the ~ (or ~ and r/) classes are chosen to satisfy them ,9 . The number of possible non-trivial choices depends on the detailed topology of M.

The "implies" column lists additional ~- r /charge relationships implied by the existence of G ~ H but which can always be satisfied for any ~ and M (if dim M ~< 4). A "Yes" under G Z~Hmeans that the rl classes can take on different values and still satisfy the G ~ H and "implies" conditions. The actual characteristic classes are the Chern classes C 1 E H2(M; Z), C 2 E H4(M; Z); the Pontryagin class p and Euler class e [if G or H = SO(4)] in H4(M; Z) and the Euler class for SO(2): e E H2(M; Z). The St ief fe l -Whitney classes

When dim M < 4 any elements of the appropriate cohomology groups for the cases considered in table 1 can be characteristic classes for some bundle 2. The only excep- tion is that, ff G = SO(n), the identity p rood 2 ~- co 2 u w a must be satisfied and e mod 2 ~- w 4 if n = 4.

are 602 EH2( M: Z2) and 6o 4 EH4(M; Z2). I have wr i t - ten the cup products as ^ rather than U, to remind the reader of the corresponding differential form relations between the real classes.

As in monopole theory [16], the precise global forms of G and H are crucial. Thus if the S a l a m - Weinberg group was globally SU(2)L × U(1), the breaking to U(1)e m would be obstructed unless the "weak instanton" charg e and U(1) charge were related by

C2L(~ ) = - C I ( ~ ) ^ CI(~ ). In fact the usual ( e - ) L , (~)L etc, assignments are such that the global group is SU(2) L × U(1)/Z 2 = U(2) and the requirement is sim- ply vanishing weak charge: C2L(~ ) = 0 * 1 o. Note also how the ~ topological charge can "f low" into different r/channels. For example in the GUT breaking SU(5)

[SU(2)L × U(1) × SU(3)c ] / Z 6 (this is the correct global form of the subgroup [17] ) t h e ~ charge can be shared amongst the weak and coloured " instanton" and the U(1) charges. If this 1015 GeV breaking ends in a sector with C 2 L ( T I ) ~ 0 then the lower energy breaking

, lo See ref. [23] for a discussion of this effect in the context of primordial magnetic monopoles.

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to U(3) can be obstructed, even though a single irreduc- ible Higgs-Kibble field (if one exists) always allows the direct breaking SU(5) ~ U(3).

Finally I would like to consider the nature of the ground states when symmetry breaking is inhibited. As an example consider SU(2)-+ U(1) with an isotrip- let ~ and the usual V(q~) = X/4 (~'q~-a2) 2 . Then, not- withstanding the shape of V(~b), q~ = 0 can be stable for topological reasons. If (say) Fur = Fu~,* then writing in the usual way S = S 1 (6o) + $2(6o, q~) we have S 1 (co + 66o) t> S 1 (co) and

8S 2 - $2(6o + 86o, ~b + h) - $2(6o, ~b)

=fM[-½h'VDUh-('ha2/2)h'h]v~d4x (6)

+ O(h 3, 6o3).

Now the leading mode h which leads to the instability of ¢ = 0 in the conventional theory is the zero eigen- function V~DUh = 0. Then 3u(h .h/2) = h ,D~h and

A(h "hi2) =- VuSU(h "hi2) (7)

=Duh.DUh +h'V DUh =D h'DUh.

Integrating both sides of (7) over the compact manifold M gives Duh .DUh = 0 and hence D~h = 0. In particular ~u(h "h) = 0 which implies that h "h is a constant. However, this cannot be so since h would then be a cross section of the SU(2)/U(1) = S 2 bundle and in this inhibited sector there are none. Thus for this topological reason this zero eigenmode cannot exist and the smallest magnitude eigenvalue is p = -p/R 2 where O is a strictly positive real number and R is some quantity in the metric guy with units of length [e.g. R = (GA) -1/2 where A and G are, respectively, the cosmological constant and Newton's constant]. It follows from (6) that if Xa 2 ~< 1/21, then q~ = 0 is stable and there is complete symmetry restoration.

The existence of this phenomenon was first demon- strated in ref. [18] for the much simpler pure scalar field case in which G = Z 2 , H = (1} and M was chosen to be a multitorus. In that example a phase transition occurred at the critical value Xa 2 = [/~[ and a new stable solution appeared just as q~ = 0 became unstable. It seems unlikely, however, that this is a generic feature. Indeed, if we imagine that the new solution ~ vanishes on a D-dimensional surface A in M (hence H = G there)

and rises approximately to the I-Iiggs vacuum value ~b.~b = a 2 within a normal distance e, then stability under small deformations of e and A requires that D = 3 and that A is a minimal area hypersurface. Such surfaces may or may not arise metrically but will certainly be present for topological reasons [19], if the third (and hence by duality, the first) Betti number of M is non zero as in the toroidal example in ref. [18] . Similar comments apply to many other examples of G and H.

An example of a manifold with a topology which can cause the obstruction of SU(2) -+ U(1) is the ex- treme Schwarzchild-de Sitter space with GA = (3GM) -2 [20] which has topology S 2 × S 2 (see earlier). In this example the regime of symmetry restoration is Xa2G2M 2 <~ I/9 and the possibility arises of a bifurcating property at the critical mass. This effect is purely classical but the possibility of phase transitions in the presence of a thermally radiating black hole [21] has been recently discussed by Hawking in a quantum-mechanical context [22]. It would be most interesting to combine these two results.

Finally note that the action of the q~ = 0 solution is suppressed with respect to the G ~ H value by a factor exp[-Vol(M)Xa4/4] which will be of relevance when the initial and]or final topological sectors are summed in the functional integral. Eventually of course it will be necessary to quantize the gravitational field itself. This will not change the essential features of the arguments presented above but will lead to new effects as the eigenvalue g of VuDU (which is metric dependent) sweeps through the critical value.

References

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[5] S.W. Hawking, Nucl. Phys. B144 (1978) 349. [6] M.J. Duff and J. Madore, Phys. Rev. D18 (1979) 2788. [7] M.E. Mayer, The fibre bundle approach to gauge theories,

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Spontaneous symmetry breaking and topological charge