Instantons with fractional topological charge

Volume

72B, number

3

PHYSICS

INSTANTONS

WITH

LETTERS

FRACTIONAL

2 January

TOPOLOGICAL

CHARGE

1978

*

Lay-Nam CHANGl Sranford Lmear Accelerator Center, Stanford Umverslty, Stanford, CA 94305, USA

Ngee-Pong CHANG Physzcs Department, City College of the Czty, Umversrty of New York, New York, NY 10031, USA Received

10 July

1977

We mtroduce and discuss the propertles of multivalent gauges for Yang-Mills of mstantons with fractlonal topological charges m such gauges

The study of Yang-Mills gauge fields has so far been carried out m univalent gauges Under a rotation m physical space by 277,the field strengths return to their orlgmal value There exist, however, multivalent gauges for non-abehan groups where the field strengths do not return to their orlgmal value but are, of course, gauge equivalent to their orlgmal value, I e , for the field strength matruc, &,,

where no(~) E G, the gauge group, but 1s not the IdentIty For an abehan group, such multivalent gauges are physically meaningless, since the ffiV are gauge-mvariant and directly measurable For non-abehan groups since the fpvare only gauge-covanant, such multivalence is, m prmclple, admissible The action density remains angle-valued for these gauges In view of the existence of these new gauges, we must now dlstmgulsh between mvarlance under single and multiple-values gauge transformations. The action, energy, and topological charge densltles are all generally gauge-invariant However, the closed loop mtegral

1s only invariant under single-valued gauge transformations In eq (2), an ordermg of the matruc b, 1s lmWork supported m part by the Energy Research and Development Admmlstratlon and m part by a grant from the National Science Foundation ’ On leave of absence from the University of Pennsylvania, Philadelphia, Pennsylvama, USA

l

fields, and demonstrdte

the existence

plied. In a quantum theory, beyond the tree level, the allowed gauge transformations which leave the S matruc mvarlant are the smgle-valued gauge transformations Presumably, however, m the functional mtegral formalism, the mtegral over b, must mclude conflguratlons of different homotopy classes m both kinds of gauges The structure of the vacuum could then be more complicated In this paper we present a remarkable feature of such multivalent gauges They give rise to mstantons of fractional topological charge We shall exhlblt this for the SU(2) group Consider the generatmg matruc, S *I S = cos x + i sm x cos 8 u3 + I sin x sin e(o, cos n$~ + a2 sin n@),

(3)

where, m Euclidean space, x4 = R cos x, x3 = R sin x cos e,x - R sm x sm 0 sm $J,x1= x2 cot @ For n fl2 a >3’> S generates a multi-valued gauge bp = 2f Q!? - St

(4)

Suppose that f 1s a function of R only, with f+ 1 as R + 00, and f+ 0 as R + 0 If f has no smgularltles m between these hmlts, the Chern-Pontryagm number 4,~ given by q = --!_Jd4x 128 rr2

Tr (fpyfAp)?‘hp

=n

*’ This parametrlzatlon of SU(2) has been consldered prevlously m ref [I] However, no attempt was made there to muumlze the Yang-Mills actlon density

341

Volume

72B, number

PHYSICS

3

For n equal to a fraction, S generates a fractional topological charge. The result 1s independent of whether eq (4) satisfied the Yang-Mills field equation *2 The field strengths associated with eq (4) are not mvarlant under a rotation 1n $ by 2 71,but satisfies eq (1) with S20(~)=cosnn-1a3s1nnn,

nfl,2,3,

(6)

It 1s convenient for our calculation to perform a multlple-valued gauge transformation to a univalent gauge Of course thus does not change the topological charge nor the Yang-Mills field equations The field strengths obtained from eq (4) are not self-dual with respect to a flat Euclidean metnc. They are instead anti-self-dual with respect to the metric QR=l’

2 January

LETTERS

with the other three components fP” = -f

$vhpfhP

simply given by (10)

77,“^, = fiEpvAp

Since the field strengths are anti-self-dual, the gauge 1s sourceless The space has negative curvature with the Rlemann curvature tensor given by RX

ki@

=-

n-l n

sln2xj

RXQxe = n(n - 1) s1n2x s1n28,

RB bed = n(n - 1) s1n2x s1n2e ,

and R

80

gse = (R2/n) m2x,

gxx = R2fn,

2

1978

2(n - 1)

R xx=---g

R2

xx’ (11)

(7)

= nR2 sm2x sin26 ,

G0

1ff 1s given by f=RY/(hY+R^3,

y=2&,

(8)

and h 1s a scale parameter The field strengths 1n the univalent by the tensorlal components fxe = -41~3

s1n2x sin 0 f(1 -f)

+ 41 s1n2x cos 6 f(1 -f)

-41sm~c0s~f(l~f)[u~ fxG = t41u3s1nxcosx -41nfll-f)s1nxcosxs1n8 - 4inf(l feG = 41nf(i + 4inf(l

gauge are given

-f) -f) -f)

One of us (NPC) wishes to thank Professor ChenN1ng Yang for an encouraging and stlmulatlng conversation The other (LNC) thanks Professor I Singer for a discussion

[aI cos @+ u2 sin $1 sin@-u2

cos$]

,

an28 *rzf(l -f) cos8(ol

cos@+u2s1n@)

s1n2x sin B (aI s1n @- u2 c0S 4))

References

SIIY~Xsin e cos e a3 s1n2x s1n2@(ul cos @+ u2 sin

$) ,

(9)

Non-mtegral values of q can also be obtamed by modlfymg the angular dependence of S on 8 and x. but the resultmg field strengths are neither self-dual nor ant]-self-dual with respect to any non-smgular metric

342

This space 1s not conformally flat It also has a vanish1ng Euler characterlstlc, and zero topological index. The fractional Chern-Pontryag1n number may come as a surprise However, the mapping defined by S 1s not from S3 to S3, but to a Rlemannlan manifold and the Chern number which IS not necessarily integral [2] Our solution fin the complex C, space of Yang [3] 1s not everywhere holomorphic

111L J Swank, Umverslty of Maryland TechnIcal Report 77I21 S S Chern and J Slmons, Proc Nat Acad Scl 68 (1971)

[31

791, R Bott and A Haefhger, Bull Am Math Sot 78 (1972) 1039, R Bott, Lectures on algebraic and dlfferentlal topology (Sprmger, New York, 1972) C N Yang, Phys Rev Lett 38 (1977) 1377