V01ume 1408, num6er 5,6 PHY51C5 LE77ER5 14 June 1984

0 N 7HE EQU1VALENCE 8E7WEEN 7HE WE55-2UM1N0 AC710N

AND 7HE FREE FERM1 7HE0RY 1N 7W0 D1MEN510N5

P. D1 VECCH1A and P. R0551 CERN, 6eneva, 5w1t2er1and

Rece1ved 24 Fe6ruary 1984

U51n9 the funct10na1 techn14ue we pr0ve the 6050n12at10n ru1e5 0f W1tten f0r the current51n a n0n-a6e11an tw0-d1men- 510na1 the0ry w1th a part1cu1ar re9u1ar12at10n 0f the Ferm1 the0ry.

Recent1y, W1tten [ 1 ] ha5 der1ved 6050n12at10n ru1e5 f0r n0n-a6e11an the0r1e5 1n tw0 d1men510n5 6y u51n9 ar9u- ment5 6a5ed 0n the un14uene55 0f the repre5entat10n 0f certa1n Kac-M00dy a19e6ra5 and 5h0w1n9 the e4u1va1ence 6etween a free Ferm1 the0ry and a 6050n1c ch1ra1 the0ry de5cr16ed 6y the We55-2um1n0 1a9ran91an 1n tw0 d1men- 510n5 [2].

7he 6050n12at10n ru1e5 f0r the current5 have 6een a150 der1ved [3] 6y u51n9 d1rect1y the funct10na1 1nte9ra1 techn14ue, 6ut 50me 11m1tat10n5 1n the app11cat10n 0fthe5e ru1e5 have 6een n0t1ced.

1n th15 paper, a 5tudy 0f the pr0pert1e5 0f the 6050n1c ch1ra1 m0de1 15 pre5ented and 6y u51n9 1t51nvar1ance under the Kac-M00dy a19e6ra a certa1n 0perat0r 1dent1ty 15 5h0wn. 7h15 1mp11e5 that th15 the0ry 15 e4u1va1ent t0 a free Ferm1 the0ry pr0v1ded that 1t 15 re9u1ar12ed 1n a very 5pec1f1c way 1n wh1ch the vect0r current re1at1ve t0 the 5U(N) part 0 f U(N) 15 n0t c0n5erved. 7he 6050n12at10n ru1e5 f0r the current5 pr0p05ed 1n ref. [1] f0110w then tr1v1a11y.

Let u5 c0n51der the tw0-d1men510na1 We55-2um1n0 act10n 1n a tw0-d1men510na1 M1nk0w5k1 5pace-t1me [ 1,2]:

1 fd3} d] k 7r [U-1a1U. U - 1 ~ / U ~ U - 1 ~ k U ] , (1) 5(t0 = fd2x 7 r 0 • U a . U - 1 ) + ] - ~ Q

where Q 15 a three-d1men510na1 hem15phere w1th c0mpact1f1ed tw0-d1men510na1 5pace a5 60undary. 7he f1e1d U1] 15 taken t0 6e an e1ement 0 f the U(N) 9r0up, 5uch that U + = U -1 . Later, we 5ha11 c0n51der the exten510n 0 f0u r re- 5u1t t0 0 (N) 9r0up5.

We def1ne the f0110w1n9 current5:

J/~ = (1/8rt) ( [U -1 0t2 U + U0/~ U -1 ] - e~(. [U -1 0 v U - U0 v U -1 ] } , J+ = (1/41r)U-1~+U, J• = ( 1 / 4 ~ ) U ~ U -1 ,

(2) w1th the n0tat10n5

J-+ =J0 -+J1 • ~ = ~0 + 01 " (3)

7he act10n (1) p055e55e5 a 10Ca1 U(N) X U(N) 5ymmetry C0rre5p0nd1n9 t0 1t5 1nVar1anCe Under the tran5f0rma- t10n

U --> a ( x - ) U6 - 1 (x+) , (4)

where a ( x - ) and 6 (x +) are ar61trary un1tary matr1ce5 depend1n9 0n1y 0n 0ne 0f the tw0 119ht-c0ne c00rd1nate5. A 5pec1a1 ca5e 0fth155ymmetry 15 the 9106a1 U(N)× U(N) 1nvar1ance wh05e 9enerat0r5 J• are, h0wever, n0t c0var1ant under the tran5f0rmat10n5 (4):

344 0.370-2693/84/$ 03.00 • E15ev1er 5c1ence Pu6115her5 8.V. (N0rth-H011and Phy51c5 Pu6115h1n9 D1v1510n)

V01ume 1408, num6er 5,6 PHY51C5 LE77ER5 14 June 1984

J+ -+ 6 (X+)J•6-(X +) + (1/47r)6(X+)~+6-1(X+), J• ~ a(X-)J•a-1(X -) + (1/47r)a(X-)~a-1(X-). (5)

7he 10Ca1 1nVar1anCe [e4. (4)] 9enerate5 the man1f01d 0f the C1a551Ca1 501Ut10n5 (10Ca1 m1n1ma 0f the C1a551Ca1 act10n)

Uc1(X ) = a(X-) 6-1(X +) (6)

5UCh that 5(Uc1 ) = 0 and the C1a551Ca1 C0n5ervat10n e4Uat10n5

0+jc2 = ~ j c 1 = 0 , (7)

are 5at15f1ed. We are 1ntere5ted 1n the 9enerat1n9 funct10n f0r the 6reen funct10n5 0f the current5 J•+. F0r th15 purp05e, we

c0up1e them t0 externa1 50urce5 A•+, that we de1~me t0 6e herm1tean matr1ce5 6e10n91n9 t0 the a19e6ra 0f U(N) and we parametr12e 1n term5 0f un1tary matr1ce5 A , 8 , 6y mean5 0f

1A+ = A - 1 a + A , 1A~=8-1~8. (8)

E45. (8) have a tr1v1a1 reparametr12at10n 1nvar1ance

A ~ a(x-)A, 8 -~ (3(x+)8. (9)

We n0w can def•me

exp[1W(A+,A~)]-<exp(1fd2x7r[A.j~+A~J+])~

= [fexp (15(U)+ 1 fd2x 7r [A+J•+A•J+] )DU] (fexp [15(U)]DU) -1 . (10)

We 5ha11 c0mpute exact1yW(A+,A•). F0r th15 purp05e, we mu5t reca11 the f0110w1n9 re1at10n [3] that 15 a funda- menta1 pr0perty 0f the We55-2um1n0 act10n 5(U):

5(AU8 -1) = 5(U) + 5(A8 -1) + f d 2 x 7r [A+J• +A J+] + ~---~f d2x 7r [ A + A - A+UA•U-1]. (1 1)

7h15 e4uat10n a110w5 u5 t0 rewr1te (10) 1n the f0rm:

exp [1W(A+,A•)] = [ f D U e x p (15(AU8-1) + ~--~ f d2x7r(A+UA~U-1)) exp {-1[5(A)+5(8-1)]} ]

We can n0w make a chan9e 0f var1a61e 1n the funct10na11nte9ra1 6y exp101t1n9 the 10ca1 U(N) X U(80 1nvar1ance 0f the 11aar mea5ure and we 9et

exp[1W(A+,A~)]=(exp(-~5d2x7r[(AA+A-1)U(8A~8-1)U+]) > exp[--15(A)--15(8-1)]. (13)

1n 0rder t0 c0mpute the expectat10n va1ue 1nd1cated 1n (13), we can make u5e 0fthe 1nvar1ance under the tran5- f0rmat10n (4) and wr1te

(exp(~5d2x7r[A+UA~U-1])> = ( e x p ( - ~ f d 2 x 7 r [ ( a - 1 A + a ) U ( 6 - 1 A ~ 6 ) U + ] ) >

= (fda(x -1) fd6(x+)exp (-~-f62x 7r [(a-1A+a)U(6-1A••6)U +])>, (14)

345

V01ume 1408, num6er 5,6 PHY51C5 LE77ER5 14 June 1984

where we have def1ned f0r 51mp11c1ty 0f n0tat10n5

A + = A A + A -1 A• =8A 8 -1 (15)

and the 1nte9rat10n 0ver a, 6 15 1ntended t0 6e n0rma112ed

f da(x-) = f d6(x +) = 1. (16)

1n the der1vat10n 0f e4. (14), we have a55umed that the phy51ca1 vacuum 15 1nvar1ant under the 5ymmetry de- 5cr16ed 6y e4. (4). 7h15 5ymmetry 9enerate5 any c1a551ca1 501ut10n 5tart1n9 fr0m the tr1v1a1 U = 1.7heref0re, 0ur a55umpt10n 1mp11e5 that the 1nvar1ant vacuum t0 vacuum amp11tude5 can 6e c0mputed tak1n9 any c1a551ca1 501u- t10n a5 60undary c0nd1t10n at t = + ~ 0r e4u1va1ent1y avera91n9 0ver a11 501ut10n5.7he 1a5t 5tep 0f e4. (1 4) can 6e c0n515tent1y 1nterpreted 1n th15 way.

Ama21n91y en0u9h, we are a61e t0 perf0rm exp11c1t1y th15 1nte9rat10n 6y mean5 0f a 1att1ce re9u1ar12at10n where f d2x -~ a2Ex •x-, where a 15 the 1att1ce 5pac1n9.7he ma1n 1n9red1ent 0f 0ur ca1cu1at10n 15 a re5u1t due t0 1t2yk50n and 2u6er [4], wh0 perf0rmed a U(N) 9r0up 1nte9rat10n dV f0r any 91ven c0up1e 0f herm1tean matr1ce5 M1, M 2 :

N-1 /N-1 fdVexpL67r(M1VM2V+)]={~n}(~1n1k~=0kv/ 2 0 (k+nk),)X{n}(M1)X{n}(M2) , (17)

N-1 where {n} 15 a 5et 0 f N n0n-ne9at1ve 1nte9er5 character121n9 a p01yn0m1a1 repre5entat10n 0f U(N), 1n1 = Ek= 0 n k and ×(n} 15 the c0rre5p0nd1n9 character 0f U(N) ana1yt1ca11y c0nt1nued t0 an ar61trary N × N matr1x.

After 11tt1e th0u9ht, 1t 15 ea5y t0 c0nv1nce 0ne5e1f that:

fx1-1+x - d a 0 c - ) d 6 ( x +) exp( 1a2 ~ 7r[a-1(x-)A•+a(x-)U6-1(x+)A••6(x+)U+]) •4rr x+x•

N - 1 N-1

- L(n} 42r] 1 20 k (k+nk)• X{n}(A+(x ,x-))X{n}(A••(x+,x-)) . (18)

1n the c0nt1nuum 11m1t, 0n1y the term pr0p0rt10na1 t0 a 2 1n the exp0nent 15 n0n-van15h1n9 and theref0re

f da(X-)d6(X+)exp(~J~d2X 7r[(a-1A+a)U(6-1A••6)U-1])

=exp 4--~fd x7r(A+)7r(A~)) (19) =exp(~---~fd2x17r[A+(x)]7r[A~(x)]) ( 1 2

N0t1ce that the dependence 0n Uha5 a1ready d15appeared at the 1eve1 0 fe4 . (18). 7heref0re, tak1n9 the expecta- t10n va1ue 1n (14) 15 a tr1v1a1 0perat10n and we reach the f0110w1n9 f1na1 re5u1t:

W(A+, A •) = -5(A) - 5(8-1) + 4 4 fd2x 7r(A+) 7r(A •) . (20)

We can a150 re1nterpret 0ur re5u1t a5 1nd1cat1n9 that the pr0per n0rma1 0rder1n9 pre5cr1pt10n needed t0 def1ne the 0perat0r U1/(x)U£-1(x), appear1n9 1n (13), w0u1d 1ead t0 the 0perat0r 1dent1ty:

:U1](x)Uh1(x): =161k61h = f dUU1]1~hh k . (21)

1t 15 1ntere5t1n9 t0 rewr1te 0ur ma1n re5u1t, e4. (20), 6y c0n51der1n9 the natura1 dec0mp051t10n 0f the U(N) 9r0up e1ement 1n term5 0f an a6e11an fact0r 6e10n91n9 t0 U(1) and an e1ement 0f the 5U(N) 9r0up ~r:

U1] = exp(1~0/X/N)~]1], det U = 1 . (22)

346

V01ume 1408, num6er 5,6 PHY51C5 LE77ER5 14 June 1984

Under th15 dec0mp051t10n, the act10n 5(U) 5p11t5 1nt0 5eparate fact0r5:

5(U 1U) = 5(U1) + 5 ( U ) , 5(U 1) = -~nfd2x 0u~ 0~. (23)

When we app17 the 5ame pr0cedure t0 the current5 and 50urce5, we f1nd that they dec0mp05e 1nt0 1rreduc161e ten- 50r5

J~ =Ju-11 + (etav0v~p/8nV/~)611; j/+/=~/{ -7- (1/47rW/N ") 0+~0611 , 7r (.1+)~ = 0 , (24)

and

A+ =A+ + ( 1 / ~ ) a + a , A~ =.4~ + (1/X/~-)~/3 ; 7r(A~+) = 0 , (25)

where we have def1ned A 1 = exp(14/V~ and 8 1 = exp(1~/~/N-). 1n5ert1n9 the5e expre5510n5 1n (20), we f1nd that 1t a150 dec0mp05e5 acc0rd1n9 t0

W(A +, A•) = --5(A ) -- 5(8 -1) • 5(A 181-1). (26)

7h15 f0rmu1at10n make5 1t ev1dent that there 15 an 1nvar1ance 0f the 9enerat1n9 funct10na1, c0rre5p0nd1n9 t0 the tran5f0rmat10n:

A-+ Aa1(x), 8 ~ 8a1(x); a# 1 =exp[14(x) /~]61j • (27)

7h15 10ca1 U(1) 1nvar1anCe 15 a ref1ect10n 0f the fact that the a6e11an current (1/87rV/~ eUv~v~611, 1ntr0duced 1n e4. (24), 15 aut0mat1ca11y c0n5erved and th15 feature mu5t 5urv1ve 1n the 4uant12ed ver510n 0f the m0de1.

We c0nc1ude the d15cu5510n 0f the 6050n1c the0ry de5cr16ed 6y (1) w1th an 065ervat10n C0nCern1n9 the 5truc- ture 0f the 4uantum effect1ve act10n 0f the m0de1.1n 0rder t0 def1ne th15 4uant1ty, we mu5t 1ntr0duce the ••c1a551- ca1•• current5

]+ = 6W/6A• = (1/4n)(8-14+8) + (1/4nN)7r(A+) 1 , j• = 6W/6A+ = (1/4rr)(A-10•A)+(1/4nN)7r(A ) 1 .

7he effect1ve act10n 15 def1ned t0 6e (28)

r4+,j•) = w(A+,A •) - fd2x 7r (A+]• + A •j+) = 5(A -1) + 5(8) + 5(81A-~1). (29)

N0t1ce that the re1at10n5h1p 6etween the ••c1a551ca1•• current57•+ and the1r re5pect1ve c0ntr16ut10n5 t0 the effect1ve act10n P 15 the 5ame a5 that 0 f the ••4uantum•• current5 J•+ and the act10n 5(U) (where U 15 th0u9ht a5 a funct10n 0f.1+~), 6ut the tw0 c0ntr16ut10n5 t0 1" dec0up1e, 1n c0ntra5t w1th what happen5 1n 5(U). 7he a6e11an fact0r 6e- have5 a5 a free f1e1d the0ry, a5 expected.

1n the 1a5t part 0f th15 w0rk, we want t0 5h0w that the 9enerat1n9 funct10na1 f0r the current5, 06ta1ned 1n the 6050n1c the0ry de5cr16ed 6y the act10n 5(U), 15 e4ua1 t0 the c0rre5p0nd1n9 0ne 1n the free Ferm1 the0ry pr0v1ded that we re9u1ar12e th15 the0ry 1n a very 5pec1f1c way.

7he 9enerat1n9 funct10na1 WF(Au) f0r the 6reen funct10n51nv01v1n9 current5 0f the free Ferm1 the0ry w1th D1rac ferm10n5 can 6e wr1tten a5

exp [1WF(A+,A•)] =f0(ff, ~k)exp( 1 f d2x ~1~J4J ) , (30)

where D u -- ~u + 1Au. 7he ferm10n determ1nant appear1n9 1n (30) ha5 6een exact1y c0mputed and the re5u1t 15

WF(A+,A•) = --5(A8-1) , (31)

where 5(A8 -1) 15 the 5ame expre5510n def1ned 1n (1) and the 50urce5 A+, A• have 6een parametr12ed a5 1n (8). 7he re5u1t (31) 15 6a5ed 0n the fact that 0ne re9u1ar12e5 the Ferm1 the0ry pre5erv1n9 the vect0r 5ymmetry.

347

V01ume 1408, num6er 5,6 PHY51C5 LE77ER5 14 June 1984

H0wever, 0ne c0u1d a150 re9u1ar12e 1t 1n 5uch a way that the 6reen funct10n5 1nv01v1n9 r19ht- and 1eft-handed current5 are fact0r12ed 1n a pr0duct 0f a 6reen funct10n w1th 0n1y the r19ht current5 and a 6reen funct10n w1th 0n1y the 1eft current5.1n th15 ca5e, 1n5tead 0f (31), 0ne w0u1d 9et the f0110w1n9 expre5510n f0r the determ1nant:

14]F(A +, A •) = --5(A) -- 5(8 -1 ) = - -5 (A8 - 1 ) -- ~-~fd2X 7r(A+A •) . (32)

F1na11y, 0ne C0U1d a150 re9U1ar12e 1t 1n 5UCh a Way that 0n1y the VeCt0r CUrrent, C0rre5p0nd1n9 t0 the U(1) fact0r 0f U(N), 15 C0n5erVed. 1n th15 Ca5e, 0ne W0U1d 9et * 1

WF(A +, A •) = --5(A) -- 5(8 -1 ) + 4•fd2x 7r(A •) 7r(A•) = - -1 ) • 5(A 181-1), (33)

that 15 exact1y e4ua1 t0 the c0rre5p0nd1n9 expre5510n 1n the ca5e 0f the 6050n1c the0ry de5cr16ed 6y the act10n 1n

( 0 . 7h15 e4ua11ty 1mp11e5 the f0110w1n9 6050n12at10n ru1e5 f0r the current5 1n a n0n-a6e11an Ferm1 the0ry:

-~7+4J ~ (1/2rr)U -1 a + U , - ~ 7 - ~ ~ (1 /22 r )U~U -1 , (34)

that are the 5ame pr0p05ed 6y W1tten 1n ref. [1 ]. H0wever, 1t mu5t 6e 5tre55ed that the e4u1va1ence 15 exact1y va11d 0n1y 1f the ferm10n the0ry 15 re9u1ar12ed 1n the way 1mp11ed 6y (33). 1f we re9u1ar12e the Ferm1 the0ry a5 1n (31), the 6050n12at10n ru1e5 (34) are n0t exact1y va11d anym0re and 50me caut10n mu5t 6e u5ed 1n 6050n121n9 an 1nteract1n9 the0ry a5, f0r 1n5tance, the n0n-a6e11an 7h1rr1n9 m0de1. N0t1ce a150 that the re9u1ar12at10n 1mp11ed 6y (33) de5tr0y5 vect0r 9au9e 1nvar1ance and theref0re 1t 15 n0t a110wed 1n a 10ca11y 1nvar1ant the0ry a5 QCD 2 . 7h15 15 ref1ected 1n the fact that the 9enerat1n9 funct10na1 (10) 15 n0t vect0r 9au9e 1nvar1ant [3].

F1na11y, the e4u1va1ence 5h0wn 1n the ca5e 0f U(N) 15 a150 true f0r an 0 ( N ) 9r0up that wa5 c0n51dered 6y W1tten 1n re f. [ 1 ]. 7he 0n1y d1fference5 are that the act10n (1) mu5t have an add1t10na1 fact0r ~ t0 6e e4u1va1ent t0 a Fe rm1 the0ry w1th Maj0rana ferm10n5 and that f 0 r N > 2 there 15 n0 a6e11an fact0r 9r0up and theref0re the 1a5t term 1n (26) 15 a65ent. N0t1ce a150 that f 0 r N = 2, the the0ry 15 c0mp1ete1y e4u1va1ent t0 the U(1) ca5e.

We thank 8. Durhuu5 and J.L. Peter5en f0r the1r cr1t1c15m 0n 50me p01nt5 0 f the paper 6y K. Y05h1da, 6 .C. R0551 and M. 7e5ta f0r a u5efu1 d15cu5510n.

,1 N0t1ce that the three re5u1t5, that we f1nd f0r the ferm10n determ1nant, d1ffer 6y a 10ca1 p01yn0m1a1 4uadrat1c 1n the externa1 f1e1d A/~. 7h15 15 prec15e1y the ar61trar1ne55 due t0 the 1nf1n1t1e5 0f the the0ry, a5 d15cu55ed 6y 6a55er and Leutwy1er 1n Append1x A 0fref. [5].

Reference5

[ 1 ] E. W1tten, N0n-a6e11an 6050n12at10n 1n tw0 d1men510n5, Pr1ncet0n Un1ver51ty prepr1nt (1983). [2] A. D•Adda, A. Dav15 and P. D1 Vecch1a, Phy5. Lett ~. 1218 (1982) 335 ;

0. A1vare2, Ferm10n determ1nant5, ch1ra1 5ymmetry and the We55-2um1n0 an0ma1y, 8erke1ey prepr1nt (1983); A. P01yak0v and A.5. W1e9man, Phy5. Lett. 1318 (1983) 121.

[ 3 ] P. D1 Vecch1a, 8. Durhuu5, J.L. Peter5en, 7he We55-2um1n0 act10n 1n tw0 d1men510n5 and n0n-a6e11an 6050n12at10n, prepr1nt N81-HE-84-02 (1984).

[4] C. 1t2yk50n and J.8.2u6er, J. Math. Phy5.21 (1980) 411. [5] J. 6a55er and H. Leutwy1er, CERN prepr1nt 7H.3689 (1983).

348