V01ume 248, num6er 1,2 PHY51C5 LE77ER5 8 27 5eptem6er 1990

70p0109y, char9ed part1c1e5 and an0ma11e51n 4uantum mechan1c5

6e0r9e Papad0p0u105 1,2

Mathemat1c5 Department, K1n9•5 C011e9e L0nd0n, 5trand, L0nd0n WC2R 2L5, UK

Rece1ved 28 Fe6ruary 1990; rev15ed manu5cr1pt rece1ved 27 June 1990

7he 4uant15at10n 0fa part1c1e c0up1ed t0 a U( 1 ) 9au9ed f1e1d and pr0pa9atm9 0n a 9ener1c man1f01d M w1th a 9r0up act10n f9 0f a 9r0up 6 15 5tud1ed. 1t 15 5h0wn that th15 m0de15uffer5 fr0m 10ca1 and 9106a1 an0ma11e5 a550c1ated w1th the 5ymmetr1e5 that the 9r0up act10nf9 9enerate5. 7he5e an0ma11e5 have a t0p01091ca11nterpretat10n and appear a5 065truct10n5 t0 c0n5truct1n9 un1tary repre5entat10n5 0f the 9r0up 6 0n the H116ert 5pace 0f the 4uantum mechan1ca15y5tem. M0re0ver, the re1at10n 6etween the 9106a1 an0ma11e5 and the t0r510n part 1n the c0h0m0109y 0fM and 6 15 exam1ned.

7he 4Uant15at10n 0f char9ed part1C1e5 C0up1ed t0 a U ( 1 ) 9aU9ed f1e1d ha5 attracted mUCh attent10n 51nCe D1raC 065erved the 4Uant15at10n C0nd1t10n 0f the C0up11n9 C0n5tant 0f the 1nteraCt10n term [ 1 ].

An0ther 1mp0rtant pr061em 15 the 4uantum treat- ment 0f the c1a551ca1 5ymmetr1e5 (the 5ymmetr1e5 0f the e4uat10n5 0f m0t10n) 0f the char9ed part1c1e pr0Pa9at1n9 0n an ar61trary curved 5pace M. 7h15 15 6ecau5e the 1mp1ementat10n 0f the c1a551ca1 5ymme- tr1e5 0f the char9ed part1c1e at the 4uantum 1eve1 15 a 9enera112at10n 0f the 5tandard 4uantum treatment 0f the tran51at10n and r0tat10n 5ymmetr1e5 0f a free par- t1c1e pr0Pa9at1n9 0n an euc11dean 5pace. 7he e4ua- t10n5 0f m0t10n 0f the char9ed part1c1e pr0Pa9at1n9 0n a curved 5pace M are 9106a11y def1ned 0n M 6ut 1t5 c1a551ca1 act10n 15 10ca11y def1ned 0n M. 7he c1a5- 51ca1 5ymmetr1e5 0f th15 m0de1 1eave 1t5 act10n 1nvar- 1ant up t0 (n0t nece55ar11y 9106a11y def1ned 0n M) 5urface term5. 7he 1atter 15 the ma1n 50urce 0f d1ff1- cu1t1e5 1n dea11n9 4uantum mechan1ca11y w1th the 5ymmetr1e5 0f the char9ed part1c1e and may 1ead t0 an0ma11e5. 1ndeed 1t wa5 065erved 1n ref. [2] 6y Mant0n that a U ( 1 ) 5ymmetry 0f the char9ed part1- c1e pr0Pa9at1n9 0n the f1at t0ru5 15 an0ma10u5. 7h15 wa5 extended 6y Ward [ 3 ] t0 the U ( 1 ) 5ymmetr1e5

E-ma11 addre55: udah130•0ak.cc.kc1.ac.uk Addre55 after 5eptem6er 1990:Phy51c5 Department, Queen Mary • We5tf1e1d C011e9e, M11e End R0ad, L0nd0n E 1 4N5, UK.

0f the char9ed part1c1e pr0pa9at1n9 0n a 9ener1c man- 1f01d and 1t wa5 5u99e5ted that the an0ma11e5 are due t0 the fact that after 4uant15at10n the U ( 1 ) char9e5 [4] 9enerated 6y the U( 1 ) 5ymmetr1e5 are n0t nec- e55ar11y 9106a11y def1ned 0perat0r5 0n M, 1n the 5ame reference 1t wa5 5tated that the cance11at10n 0f the5e an0ma11e5 nece551tate5 that the char9e5 mu5t 6e 9106- a11y def1ned 0perat0r5 0n M. 7he pre5ence 0f an0m- a11e51n th15 m0de1 wa5 a150 065erved 1n ref. [ 5 ] u51n9 path 1nte9ra1 meth0d5. An0ther 065ervat10n wa5 made 1n ref. [6 ] 6y Jack1w that the c1a551ca1 a19e6ra 0f char9e5 [ 4 ] 0f the char9ed part1c1e w1th 5ymmetr1e5 9enerated 6y a 9ener1c 9r0up 6 may deve10p a centra1 exten510n. 7h15 centra1 exten510n per515t5 after the 4uant15at10n 0f th15 m0de1 and 1t can 5p011 the 5ym- metr1e5 0f the the0ry at the 4uantum 1eve1. 7h15 pr0perty 15 51m11ar t0 the exten510n5 0f the a19e6ra 0f char9e5 wh1ch 0ccur51n f1e1d the0ry due t0 an0ma11e5 [7].

1n th15 paper, the re5u1t5 0fref5. [2,3 ] are extended t0 1nc1ude the 4uantum mechan1ca1 treatment 0f the c1a551ca1 5ymmetr1e5 0f the char9ed part1c1e 9ener- ated 6y the act10n f9 (f9f9, =fm• andfe= 1dM, e, 9, 9• ~ 6 and e 15 the 1dent1ty 0f 6 ) 0f a 9ener1c c0mpact 9r0up 6 0n the man1f01d M. 7h1515 c0m61ned w1th the cen- tra1 exten510n 0f ref. [6 ] 0f the c1a551ca1 a19e6ra 0f char9e5 0f the char9ed part1c1e w1th 5ymmetr1e5 9en- erated 6y the 9r0up 6 and D1rac•5 4uant15at10n c0n- d1t10n t0 pr0v1de a c0mp1ete de5cr1pt10n 0f the 4uan- tum 6ehav10r 0f the5e 5ymmetr1e5. 1n part1cu1ar, the

0370-2693/90/$ 03.50 • 1990 - E15ev1er 5c1ence Pu6115her5 8.V. ( N0rth-H011and ) 1 13

V01ume 248, num6er 1,2 PHY51C5 LE77ER5 8 27 5eptem6er 1990

nece55ary and 5uff1c1ent c0nd1t10n5 t0 1mp1ement the c1a551ca1 5ymmetr1e5 0f the char9ed part1c1e 9ener- ated 6y a c0mpact 9r0up 6 w1th un1tary tran5f0rma- t10n5 0n the H116ert 5pace 0f the a550c1ated 4uantum mechan1ca1 5y5tem are pre5ented. 51nce 6 15 a110wed t0 6e d15c0nnected, the 1mp1ementat10n at the 4uan- tum 1eve1 0f d15crete c1a551ca1 5ymmetr1e5, 11ke 5pace 1nver510n5, can 6e 1nve5t19ated w1th1n th15 appr0ach. 7he prec15e nature 0f the an0ma11e5 that man1fe5t them5e1ve5 a5 065truct10n5 t01mp1ement the c1a551ca1 5ymmetr1e5 0f the char9ed part1c1e 6y un1tary tran5- f0rmat10n5 0n the H116ert 5pace 0f the a550c1ated 4uantum the0ry 15 111u5trated. 1t 15 5h0wn that 1f the 9r0up 6 15 c0mpact the5e an0ma11e5 have a t0p0109- 1ca1 0r191n and t0p01091ca1 meth0d5 can 6e u5ed t0 06ta1n a ref1nement 0f the re5u1t5 pre5ented 1n ref5. [2,3,6].

70 c0nt1nue, we 6e91n 6y rec0n51der1n9 the c1a551- ca1 the0ry 0f th15 m0de1. 7he pr0Pa9at10n 0f the char9ed part1c1e 15 de5cr16ed 0n a man1f01d M w1th a 9r0up act10n f~ 0f an ar61trary (c0mpact) L1e 9r0up 6 (M c0mpact, 0r1enta61e, c0nnected and w1th0ut 60undary). 1n the 4uantum the0ry, 1t 15 5h0wn that the an0ma11e5 due t0 the 10ca1 def1n1t10n 0fthe char9e5 0n M c0rre5p0nd1n9 t0 the 5ymmetr1e5 9enerated 6y f9, and the an0ma11e5 a550c1ated w1th the centra1 ex- ten510n 0f the a19e6ra 0f char9e5, appear a5 065truc- t10n5 t0 the ex15tence 0f a 11ft1n9f ~9 0f the 9r0up ac- t10nf9 t0 a pr1nc1pa1 U( 1 ) 6und1e P = P ( M , U( 1 ), n) 0ver M w1th pr0ject10n n. P natura11y ar15e51n the 4uantum the0ry 0f the5e m0de15 fr0m D1rac•5 4uan- t15at10n c0nd1t10n. U51n9f~ and the 5tandard meth- 0d5 0f ref5. [ 8,9 ], we c0n5truct un1tary repre5enta- t10n5 0f 6 1n the H116ert 5pace 0f the 4uantum the0ry. 1f the 9r0up 6 15 c0mpact and c0nnected, 1t wa5 pr0ved 1n ref. [ 10 ] that the 065truct10n5 t0 the ex15- tence 0 f f 9 ~ are e1ement5 0f the c0h0m0109y 9r0up5 H1(6 , H1(M, 2 ) ) and H2(M, 2) #1, 2 1nte9er5. 8e- cau5e 0f the re1at10n 6etween an0ma11e5 and t0p0109- 1ca1065truct10n5, t0p01091ca1 meth0d5 can 6e app11ed t0 5tudy the an0ma11e5 and extend the re5u1t5 pre- 5ented 1n ref5. [ 2,3,6 ]. 1n part1cu1ar 1t 15 5h0wn that even 1f the 065truct10n5 van15h 1n the t0r510n free part 0f the c0h0m0109y ( 6 c0nnected), there m19ht 5t111

#1 1fthe 9r0up act10nf~ 0fthe 9r0up 6 0n M d0e5 n0t 11ft t0 P, 1t m19ht 6e p055161e t0 11ft the 9r0up act10n 0f an exten510n 6• 0 f 6 .

ex15t an 065truct10n that 11e 1n the t0r510n part 0f H2(6, 2) and c0n5e4uent1y the the0ry rema1n5 an0ma10u5. F1na11y we a110w 6 t0 6e d15c0nnected, the 065truct10n5 are 6r1ef1y d15cu55ed and the ex15- tence 0f a 9106a1 an0ma1y that re5em61e the 9106a1 an0ma11e5 0f 9au9e the0r1e5 [ 11 ] 15 dem0n5trated. Under certa1n a55umpt10n5, a nece55ary c0nd1t10n f0r the ex15tence 0f the5e an0ma11e515 that H2(M, 2 ) ha5 a t0r510n term.

7he an0ma11e5 d15cu55ed here are d1fferent fr0m the an0ma11e5 appear1n9 1n the 4uant15at10n 0f 5uper- 5ymmetr1c 519ma m0de15 1n 0ne d1men510n. 7he 1at- ter are due t0 the 065truct10n5 t0 the ex15tence 0f a 5p1n 5tructure 0n the 519ma m0de1 man1f01d M [ 12 ] 0r after an appr0pr1ate ref1nement 0f the the0ry t0 the ex15tence 0f a 5p1nc 5tructure 0n M [ 13 ].

7he e4uat10n5 0f m0t10n 0fthe char9ed part1c1e are

--Vt0t(Y +91J0)jk0t0k ~-0, 0t = 0/0t , (1)

Where ~ 15 a map fr0m the W0r1d11ne 1= [0,1 ] ( t e d t0 the 519ma m0de1 man1f01d M W1th metr1C 9, 1, j = 1 . . . . , d1m M, and a~ 15 a C105ed 6Ut n0t nece55ar11y eXaCt tW0-f0rm 0n M (d0~=0). Let {U5} 6e a 900d C0Ver 0f M. 0 n each U5 0f M, 0J5 = d65 Where 65 15 10Ca11y def1ned and 06ey5 the patch1n9 C0nd1t10n5

6p=65+dw~ (w5p=-w~) . (2)

w5, 15 a funct10n def1ned 0n the 1nter5ect10n 0f tw0 0pen 5et5 U5,= U5c~ U,. V/=0t~1V~ and V115 the Lev1- C1v1ta c0nnect10n 0f the metr1c 9. E4. ( 1 ) 15 1nvar- 1ant under reparametr15at10n5 0f M.

5upp05e that M adm1t5 a 9r0up act10n f9 0f the 9r0up 6 . f915 a 5ymmetry 0f the e4uat10n5 0f m0t10n 0f the char9ed part1c1e pr0v1ded that

f~9=9 (3)

and

f~09=09, V9e 6 , (4)

wheref~9(f~09) 15 the pu116ack metr1c (tw0-f0rm). E4. (3) 1mp11e5 that f5- V9e 6 , are 150metr1e5 0f the r1emann1an man1f01d M.

1t 15 a1way5 p055161e t0 f1nd a metr1c 91 that 501ve5 the c0nd1t10n (3) pr0v1ded that the 9r0up 6 15 c0m- pact. 1ndeed 5et

91 = f f~9 d/.t ( 9 ) , (5) 6

114

V01ume 248, num6er 1,2 PHY51C5 LE77ER5 8 27 5eptem6er 1990

where d/115 the n0rma112ed Haar mea5ure 0f 6 .51m- 11ar1y, 1t 15 a1way5 p055161e t0 c0n5truct a c105ed tw0- f0rm t0t 0n M that 06ey5 (4). H0wever, 1fthe 9r0up 6 15 d15c0nnected the c1a55 [ t0t ] 0f t01 1n the Rham c0h0m0109y can 6e d1fferent fr0m the c1a55 [t0] 0f the c105ed f0rm t0.

Let k~, a = 1 ..... d1m 6 , 6e vect0r f1e1d5 0n M 9en- erated 6y the 9r0up act10n f9- 1nf1n1te51ma11y, e45. (3) and (4) are 91ven 6y

La9=0 (6)

and

L~t0=0, (7)

where L~ 15 the L1e der1vat1ve w1th re5pect t0 the vec- t0r f1e1d ka. La=1ad+d1a, where 1a 15 the der1vat10n w1th re5pect t0 vect0r f1e1d ka that map5 4 f0rm5 t0 ( 4 - 1 ) f0rm5. 1ndeed

(1a2) 1,.....14•t = 4k1a2u,.....14•, . ( 8 )

7he 1a9ran91an 0f the the0ry

L0t "~- •91j0t 0 t ( ~1 0 t~ ) j + 61a 0 t ~ 1 (9)

d0e5 n0t patch a5 5ca1ar 0n M 6ecau5e 0f the 1nh0- m09ene0u5 patch1n9 c0nd1t10n5 (2) 0f 6. 1ndeed,

Lp=L~ + 0t0101wf10t . ( 1 0 )

7he m 0 m e n t a pa1=0LcJ00t01 0f the the0ry 06ey the patch1n9 c0nd1t10n5

p~ =p~ +dw~,~. ( 1 1 )

U51n9 e4. (2) and e4. (11), we Can pr0ve that the ham11t0n1an 0f the the0ry

1 1j Ha = ~9,~(P,,~ - 61,~ ) (Pja -6j,~ ) (12)

tran5f0rm5 a5 a 5ca1ar, 1.e. H~=Hp. M0re0ver, the 5ymp1ect1c f0rm 92~= dp~,, • dx~ 15 a150 a 9106a11y de- f1ned tw0-f0rm, 92,~ = 12p. H0wever, the act10n 0f the the0ry 1n 60th 1a9ran91an and ham11t0n1an f0rm 1510- ca11y def1ned. 1f the metr1c 9 and the c105ed tw0-f0rm 09 5at15fy e45. (6) and (7), the c0n5erved char9e5 c0rre5p0nd1n9 t0 the 5ymmetr1e5f9 are

- k a ~ ( p ~ - 6~) - m ~ , (13) Qa0t -- 1

where

1~t0a =dma~ . (14)

E4. (14) f0110w5 fr0m the e4Uat10n5 dt0 = 0 and (7).

7he char9e Qa tran5f0rm5 a5 a 5ca1ar pr0v1ded that ma~ tran5f0rm5 a5 a 5ca1ar 0n M, 1.e. rn0~ = ma8.

7he a19e6ra 0f char9e5 under P01550n 6racket5 15

(Qa, Q6}=-f~a6Qc+ca6, (15)

where f ~ are the 5tructure c0n5tant5 0f the L1e a19e- 6ra (L1e 6 ) 0f the 9r0up 6 and ca6 15 the centra1 ex- ten510n 91ven 6y

Ca6 =1a16t0(X) - - fc6mc. (16)

065erve that Ca615 a C0n5tant. NeXt def1ne,

1tar= f 1at0 (17) Cr

and

2,6=(1a16t0(x) d u ( x ) , (18)

I

M

where Cr are 1-cyc1e5 0 fM and du 15 the v01ume f0rm 0f M a550c1ated w1th a metr1c 9. 80th nar and 7a6 are L1e a19e6ra c0h0m0109y c0cyc1e5 [ 14]. 1ndeed, u51n9 (17), (18) and (19), we can 5h0w that 7reH ~ (L1e 6 , H ~ (M, ~) ) (~ rea1 num6er5) and the c1a55 0f 715 1n H2(L1e 6 ) . L1e 6 act5 tr1v1a11y 0n H2(M, ~) . 1f ~rar van15he5 (2tar = 0), 1t 1mp11e5 that m~ 15 a 5ca1ar 0n M and c0n5e4uent1y the char9e Qa 15 a 5ca1ar a5 we11. 7hen ca615 c0h0m01090u5 t0 2a6, and 1f 2a6 repre5ent5 the tr1v1a1 c1a55 1n H 2 (L1e 6 ) after a 5u1ta61e redef1- n1t10n 0f the char9e5 the centra1 exten510n c 1n (15) van15he5.

5ummar121n9 the re5u1t5 0f the c1a551ca1 ana1y515 0f the the0ry, we 065erve that the e4uat10n5 0f m0t10n rema1n 1nvar1ant under the 9r0up act10n f9 pr0v1ded that f9, V9~ 6, are 150metr1e5 (3) and 1eave the c105ed tw0-f0rm t0 1nvar1ant (4). H0wever, the char9e5 Qa c0rre5p0nd1n9 t0 the5e 5ymmetr1e5 are 9enera11y 10- ca11y def1ned and the1r a19e6ra w1th re5pect t0 P015- 50n 6racket5 may have a centra1 exten510n c. 7he c0n- d1t10n f0r Qa t0 6e 5ca1ar 15 that (17) van15he5. F1na11y, the centra1 exten510n 15 tr1v1a1 pr0v1ded that (18 ) 15 the tr1v1a1 c0cyc1e 1n H2(L1e 6 ) . 7he5e tw0 c0nd1t10n5 a1way5 h01d 1f L1e 6 15 a 5em151mp1e L1e a19e6ra.

70 4uant15e the the0ry, we u5e the can0n1ca1 c0m- mutat10n re1at10n510ca11y 0n M. 7he5e pr0duce a 10- ca11y def1ned ham11t0n1an 0perat0r/4,~ f0r each U,~,

115

V01ume 248, num6er 1,2 PHY51C5 LE77ER5 8 27 5eptem6er 1990

/-]r = • 199( V,, -161~) (Vj,~ -16j~) , (19)

where 71 15 the Lev1-C1v1ta c0nnect10n 0f the metr1c 9. 7he 5et {H~) 0f 10ca11y def1ned 0perat0r5 can 6e patched t09ether t0 91ve a 9106a11y def1ned 0perat0r /1 0n M. 1ndeed, 1et {~/,~} 6e the 5et 0f 10ca11y def1ned funct10n5 ~u, 0n M 0n wh1ch {/~,,) act5. 5upp05e that ~u,~ 06ey5 the patch1n9 c0nd1t10n5

~up=h~u~. (20)

A55um1n9 that H89p= h t ~ r ~ t ~ and u51n9 e4. (2), we can pr0ve that

h~ 8 = exp (1w~) (21)

0n the 1nter5ect10n5 U~p. 7e5t1n9 the c0n515tency 0f the patch1n9 c0nd1t10n5 0n tr1p1e 0ver1ap5 U~p~, we 9et

h,~php~h~,~ = 1 . (22)

E4. (22) 1mp11e5 that {h~} are the tran51t10n func- t10n5 0f a c0mp1ex 11ne 6und1e L 0ver M. 7heref0re {6~} 15 a c0nnect10n 0 f L and F,~=d6,~ 15 the c0rre- 5p0nd1n9 curvature. 7h15 15 D1rac•5 4uant15at10n c0nd1t10n.

7he H116ert 5pace 0f the the0ry 15 the 5et 0f 54uare 1nte9ra61e 5ect10n5 F(L) 0fthe 11ne 6und1e L. 7he 1n- ner pr0duct 1n F(L) 15 def1ned 6y

(~, , 9t2)=~ (9t1, ~2 ) (x)dv(x) (23) M

where ( , ) 15 the 1nner pr0duct 0n the f16re5 0fL, d v the v01ume f0rm 0fM and ~1, ~u2eF(L). N0te that 1f M 15 a tw0-d1men510na1 man1f01d w1th a 5p1n 5truc- ture, 1t 15 p055161e t0 arran9e 5uch that L 15 the 5p1n 6und1e and theref0re the wave funct10n ~u 15 a 5p1n0r.

Next a55ume that the 9r0up act10n f9 0f 6 0n M that 1eave5 the e4uat10n5 0f m0t10n 0f the char9ed part1c1e ( 1 ) 1nvar1ant. 7he 5ymmetr1e5 9enerated 6y f9 are 1mp1emented 4uantum mechan1ca11y pr0v1ded that there 15 a un1tary (n0t nece55ar11y 1rreduc161e) repre5entat10n D 0f 6 0n F(L) a n d / 1 15 1nvar1ant under f9. 1f th15 15 n0t p055161e, the 5ymmetr1e5 9en- erated 6yf9 are an0ma10u5 at the 4uantum 1eve1.7he 0perat0r ~ rema1n5 1nvar1ant underf9 pr0v1ded that there are 9au9e tran5f0rmat10n515uch that

f~6,~ = 19 1 (9, x)d1,~(9, x ) . (24)

E4. (24) ha5 501ut10n5 pr0v1ded that the 9r0up ac- t10nf911ft5 t0 an act10nf r9 0f 6 0n the pr1nc1pa1 U ( 1 ) 6und1e P(M, U( 1 )) w1th tran51t10n funct10n5 the 5ame a5 the tran51t10n funct10n5 0 fL and f16re U( 1 ) [ 15 ]. P (M, U ( 1 ) ) 15 the a550c1ated pr1nc1pa16und1e 0fL.

N0w a55ume that there 15 a 11ft1n9f~ 0f the 9r0up act10n f9 0n the pr1nc1pa1 6und1e P. 7hen the 9r0up act10n 11ft5 t0 any a550c1ated vect0r 6und1e L = P × uV (L={(p , v ) 5 P × V 5.t. (p, v)~(pk, U(k-1)V), Vke U ( 1 ) } ) where V (C) 15 a vect0r 5pace that U ( 1 ) 15 repre5ented w1th a un1tary (1rreduc161e) repre5en- tat10n u. 7he 11ft1n9 f~9,u 0ff9 0n L 15 def1ned 6y f ~9,, ( [p, v] ):= [f~ (p), v1.61ven the 11ft1n9f*9,,, we can c0n5truct a un1tary repre5entat10n D 0 f 6 ( 6 n0t nece55ar11y c0mpact) 0n F (L) pr0v1ded that there 15 a 4ua51-1nvar1ant mea5ure d/2 0f 6 1n M, 1.e. the tran5f0rmat10n 0f the mea5ure d/t under the 9r0up act10n f915

dU (f9(x)) =p(9, x ) d # ( x ) , (25)

where p > 0 15 the Rad0n-N1k0dym der1vat1ve 0f d# that 5at15f1e5 the c0nd1t10n

p(9•9, x)=p(9•,f9(x))p(9, x), 9•,9e6. (26)

7hen, the repre5entat10n D 15 91ven 6y

(D(9) ~,) (x) =p(9-1, x)1/2f~9.,~(f 91(x) ),

~t~A ( L ) . (27)

1n the ca5e 0f char9ed part1c1e de5cr16ed 6y the ham11t0n1an H, we 5et d/~=dv. 1fthe 1nvar1ant met- r1c9 (e4. (3) ) 15 u5ed t0 c0n5truct dv, p= 1 and D 15 un1tary w1th re5pect t0 the 1nner pr0duct 1n F (L) 91ven 6y e4. (23). M0re0ver the ham11t0n1an 0pera- t 0 r /~ (19) 15 herm1t1an w1th re5pect t0 th15 1nner pr0duct.

E45. (23) and (27) 1mp1y that there are an0ma11e5 whenever there 15 an 065truct10n t0 11ft1n9 the 9r0up act10nf9 t0 the pr1nc1pa1 U( 1 ) 6und1e P.

Next, we exam1ne the c0nd1t10n5 under wh1ch f9 11ft5. F1r5t a55ume that 6 15 c0mpact and c0nnected. 7hen, 1et f9 6e a 9r0up act10n 0f 6 0n M and P (M, U( 1 ), 9) 6e a pr1nc1pa1 U( 1 ) 6und1e 0ver M w1th pr0ject10n 7t and f1r5t Chern c1a55 c1 (P)eH2(M, 71). 1t can 6e 5h0wn [ 10] that the 9r0up act10nf911ft5 t0 P 1f and 0n1y 1f

f*(c1(P) )=1×c~(P)~H2(6×M, 2) (28)

116

v01ume 248, num6er 1,2 PHY51C5 LE77ER5 8 27 5eptem6er 1990

where f. 6 X M - M 15 def1ned 6y f(9, X).•=f9(X). 1 X Ct (P) 15 the pu116aCk 0f the C1a55 Ct (P) w1th re- 5pect t0 the pr0ject10n 0f 6 × M 0nt0 M.

7he 065truct10n5 can 6e under5t000 a5 f0110w5: 1n 9enera11t 15 expected fr0m the K11nneth f0rmu1a that f * (c1 (P ) ) ~ H 2 ( 6 X M , 7/) dec0mp05e5 a5

f*(c1(P))=1Xc~(P)+a+f~(c~(P))×1, (29)

where a e H ~ ( 6 , H~(M, 7/)) a n d f ~ ( c~ (P ) )~H2(6 , 2). f0: 6 ~ M 15 def1ned 6y f 0 ( 9 ) = f ( 9 , x0) where x0 15 a 91ven p01nt 1n M. f * ( c ~ ( P ) ) 5pec1f1e5 the pu11- 6ack 6und1ef*P = ( (p, 9, x) ~ P X ( 6 X M ) 5.t. 7t (p) = f(9, x)}. 7he c1a55 a c0rre5p0nd5 t0 the m1xed tw15t5 0 f f * P 0n 6 and M, a n d f ~ ((c~P)) c0rre5p0nd5 t0 the tw15t5 0 f f * P 0n 6 . f ~ (c1 ( P ) ) X 1 15 the pu116ack 0f the c1a55f~ (c~ (P ) ) w1th re5pect t0 the pr0ject10n 0f 6 × M 0nt0 6 . 7 h e an0ma11e5 are repre5ented 6y the c1a55e5 a a n d f ~ c 1 ( P ) . 1t f0110w5 that 60th 06- 5truct10n5 van15h 1f 6 15 a c0mpact, c0nnected, 51m- p1y c0nnected and 5em151mp1e L1e 9r0up. F0r exam- p1e, the act10n 0f 5U(2) 0n 5 2 = 5 U ( 2 ) / U ( 1 ) a1- way511ft5 t0 a11 U ( 1 ) pr1nc1pa16und1e5 0ver the tw0- 5phere 52.

Next c0n51der 6 = U ( 1 ) , the 0n1y an0ma1y that may appear 15 repre5ented 6y the 065truct10n a and 1t wa5 5tud1ed 1n ref. [ 3 ]. H0wever, 1f 6 = U ( 1 ) ~, n > 1 1nte9er, 60th c0n5truct10n5 a a n d f ~ c~ (P) may 0ccur and c0ntr16ute t0 the an0ma1y.

51nce we are dea11n9 w1th c0h0m0109y w1th 1nte9er c0eff1c1ent5 the t0r510n part 0f H 2 ( 6 , 2) 151mp0rtant (H~(6 , H~(M, 2 ) ) d0e5 n0t have a t0r510n part). A1th0u9h H 1 (6 , H ~ (M, 2) ) and the t0r510n free part 0f H2(6 , 2) can 6e de5cr16ed u51n9 L1e a19e6ra c0- h0m0109y and the c0rre5p0nd1n9 a and f ~ (c~ (P) ) 065truct10n5 van15h pr0v1ded that the c1a55e5 (17) and (18) are tr1v1a1, the t0r510n part 0 fH2(6 , 2) may 5t111 pr0v1de an 065truct10n t0 the 11ft1n9 0fthe 9r0up act10n. F0r examp1e, c0n51der the 11ft1n9 0f the 5tan- dard 5 0 (3) 9r0up act10n 0n 52 t0 the pr1nc1pa1 U ( 1 ) 6und1e P 0ver 52 w1th f1r5t Chern num6er e4ua1 t0 1 (the 5p1n 6und1e) 0r any 0ther pr1nc1pa1 U( 1 ) 6un- d1e P 0ver 52 w1th 0dd Chern num6er. 7he L1e a19e- 6ra c0h0m0109y 0f 5 0 (3) 15 2er0 at d1men510n5 0ne and tw0 51nce L1e 5 0 ( 3 ) 15 5em151mp1e, theref0re 60th c1a55e5 (17) and (18) van15h. H0wever, H 2 ( 5 0 (3) , 2) = 71 2 and 1t can 6e 5h0wn that the 9r0up act10n 0f 5 0 (3) cann0t 6e 11fted t0 P whenever the f1r5t Chern num6er 0f P 15 an 0dd 1nte9er. 51nce

5 0 (3) 15 c0nnected, th15 an0ma1y 15 n0t 9106a11n the 5en5e that the 4uantum the0ry fa115 t0 6e 1nvar1ant under tran5f0rmat10n5 n0t c0nnected t0 the 1dent1ty. 7he an0ma1y 15 due t0 the t0r510n 1n the c0h0m0109y 0f 5 0 (3) that cann0t 6e repre5ented 1n term5 0f d1f- ferent1a1 f0rm5. F1na11y the 1mp1ementat10n 0f the 5 0 (3) 5ymmetry 1n the a60ve 4uantum mechan1ca1 m0de1 nece551tate5 the 4uant15at10n 0f the 1nterac- t10n term 1n term5 0f even 1nte9er5. 7h15 15 a ref1ne- ment 0f the D1rac•5 4uant15at10n c0nd1t10n.

70 de5cr16e 9106a1 an0ma11e5 51m11ar t0 th05e 0f 9au9e the0r1e5, 1et 6 6e a d15c0nnected c0mpact L1e 9r0up. 1n th15 ca5e pr0v1ded t h a t f ~ P ~ P, V9e 6 , the 065truct10n5 t0 the ex15tence 0ff*9 are e1ement5 0fthe c0h0m0109y 9r0up5 H 2 ( 8 6 , H 1 (M, 2) ) and H3(86 , 2) [16] where f~P~.={(p,x)~PXM 5uch that f9(x) =7r(p)} and 8 6 [ 17] 15 the un1ver5a1 c1a551fy- 1n9 5pace 0f the 9r0up 6 ~2. 70 5tudy a type 0f 9106a1 an0ma1y, 1et u5 a55ume that the c0nnected part 60 0f 6 that c0nta1n5 the 1dent1ty e1ement 0f 6 11ft5 t0 the pr1nc1pa1 U( 1 ) 6und1e P 0ver M and that the 06- 5truct10n5 that 11e 1n H3(86 , 1) and H 2 ( 8 6 , H 1 (M, 2) ) van15h (a m0re deta11ed exp051t10n 15 91ven 1n ref. [ 18] ). 1fH2(M, 7/) 15 t0r510n free, the c0nd1t10n (4) f f ~ F = F , V9~6) der1ved fr0m the c1a551ca1 1n- var1ance 0fthe the0ry 1mp11e5 thatf~ P -~ P, V9e 6 , and the wh01e 9r0up 6 11ft5. H0wever 1f H2(M, 2) ha5 t0r510n, f ~ F = F may n0t 1mp1yf~P~P, V9~6, and the 9r0up 6 d0e5 n0t nece55ar11y 11ft.

F0r examp1e c0n51der M = 52 × RP (3) × RP (3) where RP(3) 15 the 3-rea1 pr0ject1ve 5pace and 6 = 5 U ( 2 ) X22. 7he e1ement5 0f 5U(2) r0tate the p01nt5 0f 52 and the n0n-tr1v1a1 e1ement x 0f 7/2 1nter- chan9e the tw0 pr0ject1ve 5u65pace5 0f M. Fr0m the K11nneth f0rmu1a the 5ec0nd c0h0m0109y 0f M 15 H2(M, 2) = 2 ~ 7/2 ~ 7 ~ 2 and 1t 15 9enerated 6y the e1e- ment5 a, 0t andf1, a 15 the 9enerat0r 0fthe t0r510n free part 0f H2(M, 7/) wh1ch 15 1nduced fr0m H2(52, 2) =2 , and 0t, f1 are the 9enerat0r5 0fthe t0r510n part 0f H2(M, 2) wh1ch are 1nduced fr0m the c0h0m01- 09y H 2 (RP ( 3 ), 7/) = 7/2 0f the pr0ject1ve 5pace5. 7he n0n-tr1v1a1 e1ement x 0f 22 1n 6 act5 0n H 2 (M, 2 ) a5 f0110W5:

#2 1f6 15 c0nnected 1t can 6e 5h0wn thatf~P-~P, V9~6. M0re- 0ver the 065truct10n5 that 11e 1n the c0h0m0109Y 9r0up5 H 2 ( 86, H 1 (M, 2 ) ) and H 3 ( 86, 2 ) can 6e re1ated t0 the 065truct10n5 a andf~ (c1 (P)) 0fe4. (29).

117

V01ume 248, num6er 1,2 PHY51C5 LE77ER5 8 27 5eptem6er 1990

f~1,~)(0t)=f1, f~/,~)(f1)=01, f ~ 1 , , 0 ( a ) = a . (30)

7he c0nnected c0mp0nent 60 (60 = 5U ( 2 ) ) 0f 6 a1- way511ft5 t0 a11 pr1nc1pa1 U ( 1 ) 6und1e5 0ver M, 51nce 60 15 c0mpact, c0nnected, 51mp1y c0nnected and 5em151mp1e. M0re0ver, u51n9 the K11nneth f0rmu1a we can 5h0w that H 3 ( 8 6 , 2 ) = H 2 ( 8 6 , H • (M, 2 ) ) = 0.

Let F 6e the curvature tw0-f0rm 0f a U ( 1 )-pr1nc1- pa1 6und1e 1nvar1ant under the wh01e 9r0up 6 , and [F] the t0r510n free repre5entat1ve 0 f F 1 n H2(M, 7/). H0wever, F d0e5 n0t c0mp1ete1y c1a551fy the pr1nc1- pa1 6und1e P 51nce 60th pr1nc1pa1 6und1e5 w1th f1r5t Chern c1a55e5 [F] and [F] + t~ have the 5ame curva- ture f0rm (0t and f1 c0rre5p0nd t0 f1at 6und1e5). 7heref0re, 1f c 1 ( P ) = [ F ] + 0 t , f ~ , ~ ) ( c 1 ( P ) ) = [F] +f1~ [F] +0t, and the act10n d0e5 n0t 11ft t0 P even 1f 1t5 curvature tw0-f0rm F 151nvar1ant. F1na11y, 065erve that the 9r0up 6 11ft5 t0 the pr1nc1pa1 U ( 1 ) 6und1e5 0ver M w1th f1r5t Chern c1a55e5, na and na + 0t + f1, n 1nte9er.

1n c0nc1u510n, we have d15cu55ed h0w the tran5f0r- mat10n5 that 1eave the c1a551ca1 e4uat10n5 0f m0t10n 0f a char9ed part1c1e c0up1ed t0 a U ( 1 ) 9au9ed f1e1d 1nvar1ant may have an0ma11e5 at the 4uantum 1eve1. 7he5e an0ma11e5 are 60th 0f10ca1 and 9106a1 type and they man1fe5t them5e1ve5 a5 065truct10n5 t0 the 11ft- 1n9 0f the 9r0up act10n that 9enerate5 the tran5f0r- mat10n5 t0 a pr1nc1pa1 U ( 1 ) 6und1e P that ar15e5 nat- ura11y 1n the 4uantum the0ry 0f the5e m0de15 fr0m D1rac•5 4uant15at10n c0nd1t10n. 7he pre5ence 0fthe5e an0ma11e5 d0e5 n0t affect the un1tar1ty 0f the the0ry 6ut they d0 n0t a110w the 4uantum mechan1ca1 1m- p1ementat10n 0f the c1a551ca15ymmetr1e5.

7he 10ca1 an0ma11e5 part1a11y appear even at the c1a551ca11eve1 a5 065truct10n5 t0 def1n1n9 the c1a551ca1 char9e5 that 9enerate the 5ymmetr1e5 0f the the0ry 9106a11y 0n the tar9et man1f01d and a5 a centra1 exten- 510n 1n the a19e6ra 0f the5e char9e5 under P01550n 6racket5. 7he5e 065truct10n5 are e1ement5 0f the L1e a19e6ra c0h0m0109y 0f 6 .

H0wever, the 065truct10n5 t0 11ft1n9 the 9r0up ac- t10n5 t0 pr1nc1pa1 U ( 1 ) 6und1e5 P 11e 1n c0h0m0109y 9r0up5 w1th 1nte9er c0eff1c1ent5 and the t0r510n part5 0f the5e c0h0m0109y 9r0up5 may a110w the appear- ance 0f an0ma11e5 that cann0t 6e detected fr0m the c1a551ca1 ana1y515 0f the the0ry. 7w0 type5 0f the5e an0ma11e5 were dem0n5trated. 7he f1r5t type may ap- pear f0r 6 c0nnected, pr0v1ded that H 2 ( 6 , 2) ha5 a t0r510n term. 7he 5ec0nd type 15 a 9106a1 an0ma1y

wh1ch ex15t5 pr0v1ded that 6 15 d15c0nnected and f ~ P 15 n0t 150m0rph1c t0 P, V 9 ~ 6 . F0r the 1atter ca5e 1n 0rder th1515 the 0n1y 50urce 0 f the 9106a1 an0ma1y, 1t wa5 a55umed that the 065truct10n5 that 11e 1n H3 ( 8 6 , 2 ) and H 2 ( 8 6 , H• (M, 2) ) van15h (f0r m0re deta115 5ee ref. [18] ). 7he t0r510n part 1n H2(M, 7/) 15 nece55ary, 6ecau5e 0n1y 1n th15 ca5e the c1a551ca1 5ymmetry c0nd1t10n ( f ~ F = F , V 9 ~ 6 ) 0n the curva- ture F 0 f the pr1nc1pa1 U( 1 ) 6und1e P may n0t 1mp1y t h a t f ~ P 15 150m0rph1c t0 P f0r every e1ement 0 f the 9r0up 6 .

1 w0u1d 11ke t0 thank P.5. H0we and L. H0d9k1n f0r u5efu1 adv1ce, and 8 .5pence f0r d15cu5510n5.7h15 w0rk wa5 5upp0rted 6y 5ERC.

Reference5

[ 1 ] P.A.M. D1rac, Pr0c. R. 50c. A 133 ( 1931 ) 60. [2] N.5. Mant0n, Ann. Phy5. (NY) 159 (1985) 220. [3] R.5. Ward, Phy5. Rev. D 36 (1987) 640. [4] R. Jack1w and N. Mant0n, Ann. Phy5. (NY) 127 (1980)

257. [ 5 ] 6. Papad0p0u105, C1a55. Quantum 6rav. 7 (1990) 41. [ 6 ] R. Jack1w, 70p01091ca1 1nve5t19at10n5 0f 4uant15ed 9au9e

the0r1e5, 1n: Current a19e6ra and an0ma11e5 (W0r1d 5c1ent1f1c, 51n9ap0re 1985 ).

[7] L.D. Faddeev, Phy5. Lett. 8 145 (1984) 81; 8.2um1n0, Nuc1. Phy5. 8 253 (1985) 477.

[8] C.J. 15ham, 70p01091ca1 and 9106a1 a5pect5 0f 4uantum the0r1e5, 1n: Re1at1v1ty, 9r0up5 and t0p0109y 11, ed5. 8.5. DeW1tt and R. 5t0ra (N0rth-H011and, Am5terdam 1984).

[9] 6. Mackey, Un1tary 9r0up repre5entat10n5 1n phy51c5, pr06a6111ty, and num6er the0ry (Add150n-We51ey, Read1n9, MA, 1989); N.R. Wa11ach, Harm0n1c ana1y515 0n h0m09ene0u5 5pace5 (Marce1 Dekker, New Y0rk, 1973 ).

[ 10 ] D.H. 60tt11e6, L1ft1n9 act10n5 0n f16rat10n5, Lecture N0te5 1n Mathemat1c5, V01. 757 (5pr1n9er, 8er11n, 1978).

[ 11 ] E. W1tten, Phy5. Lett. 8 117 (1982), 324. [12]D. Fr1edan and P. W1ndey, Nuc1. Phy5. 8 235 [F511]

(1984) 395; E. W1tten, 6106a1 an0ma11e51n 5tr1n9 the0ry, 1n: 5ymp. 0n 6e0metry, an0ma11e5 and t0p0109y, ed5. W. 8ardeen and A. Wh1te (W0r1d 5c1ent1f1c, 51n9aP0re, 1985 ).

[ 13 ] P.5. H0we and P.K. 70wn5end, Chern-51m0n5 4uantum mechan1c5, CERN prepr1nt CERN/7H-5307/89.

[ 14 ] 6. H0ch5ch11d and J.-P. 5erre, Ann. Math. 57 ( 1953 ) 591. [ 15 ] J. Harnad, 5. 5hn1der and L. V1net, J. Math. Phy5. 21 (1980)

2719. [ 16 ] A. Hatt0r1 and 7. Y05h1da, Japan J. Math. 2 ( 1976 ) 13. [ 17 ] J. M11n0r, Ann. Math. 63 (1956) 272, 430. [ 18 ] 6. Papad0p0u105, 1n preparat10n.

118