Analytic fields on Riemannian surfaces

V01ume 180, num6er 3

PHY51C5 LE77ER5 8

13 N0vem6er 1986

ANALY71C F1ELD5 0N R1EMANN1AN 5URFACE5 V.6. KN12HN1K

L.D. Landau 1n5t1tutef0r 7he0ret1ca1Phy51c5, 7he Academy 0f 5c1ence50f the U55R, M05c0w 117334, U55R Rece1ved 27 June 1986

7he c0n•f0rma1 f1e1d the0ry 0f ana1yt1c d1fferent1a15 0n a 9enera1 R1emann 5urface 15 f0rmu1ated. A11the determ1nant5 and c0rre1at10n funct10n5 axe expre55ed 1n term5 0f them-funct10n5.

1t 15 we11 kn0wn that 1eft- and r19ht-m0v1n9 exc1tat10n5 0f free tw0-d1men510na1 4uantum f1e1d5 d0 n0t 1nteract, a110w1n9 t0 d1v1de a f1e1d 1nt0 ch1ra1 ••ha1f5••. 7he5e ••ha1f5•• p1ay a r01e 0f fundamenta1 ••6r1ck5•• 0f wh1ch 5tr1n9 t h e 0r1e5 and 2D c0nf0rma1 f1e1d the0r1e5 are 6u11t. 1n th15 1etter we w111 1nve5t19ate the 5tructure 0f the5e ••6r1ck5••. F0r th15, 1et u51ntr0duce 0n a tw0-d1men510na1 5urface 5 w1th c00rd1nate5 ~1, ~2 and metr1c 9a6 50me ana1yt1c c00rd1nate5 2, ~ 1n wh1ch the metr1c take5 the c0nf0rma1 f0rm 9a6 d~ad~ 6 = p(2, 5) d2 d9, and a150 1et u5 1ntr0duce a 5et (¢(])) 0f 5p1n-] f1e1d5, 0r ]-d1fferent1a15, wh1ch under the chan9e 0f an ana1yt1c 2D c00rd1nate 2 0n a 5urface tran5f0rm a5

¢(])(2) = [df(2)/d2]1¢(/)(f) .

(1)

U51n9 a pa1r 0f ant1c0mmut1n9 f1e1d5 ¢4) and ¢(1-D 0ne can c0n5truct an act10n

5j = f¢(1-5)~¢0~)d2

A d~,

~ = a1a~-,

(2)

where, 1n 9enera1, 1nte9rat10n 90e5 0ver a 9enu5 p r1emann1an 5urface. F0rma11y, the free ener9y F• 0f 5uch 5y5tem e4ua15 t0 F1 = -109 det 31.

(3)

(where the 1ndex ] 0f ~ den0te5 that ~ act5 0n ]-d1fferent1a15) and 0ur ma1n purp05e w111 6e t0 f1nd an exp11c1t f0rmu1a f0r 1t. We w111 f0cu5 0ur attent10n 0n the m05t n0ntr1v1a1 ca5e, p ~> 2, wh1ch a150 15 0f part1cu1ar 1ntere5t f0r mu1t1100p ca1cu1at10n5 1n 5tr1n9 the0ry. 06v10u51y expre5510n5 11ke (3) enter the heter0t1c 5tr1n9 p-100p c0ntr16ut10n and even f0r the m0de1 0f c105ed 0r1ented 6050n1c 5tr1n95 1n cr1t1ca1 d1men510n D = 26 (E5VM) a5 wa5 pr0ved 6y 8e1av1n and the auth0r [ 1 ], the mea5ure take5 the f0rm 3p - 3

f r1 Mp 1=1

dy1AdY11F(y)12(0et1m7) -13 ,

F=0et~1.(det~0)

-13 ,

(4)

Where (Y1, Y1, 1 = 1 ..... 3p -- 3} are 50me C0mp1eX ana1yt1C C00rd1nate5 0n the m0dU11 5paCe Mp 0f 9enU5 p r1emann1an 5urface5; r 15 a per10d matr1x. Actua11y there 15 n0 def1n1t10n 0f det ~ / t h a t c0u1d 5erve a5 a 5tart1n9 p01nt f0r a ca1cu1at10n, thu5 the f0rmu1ae 115ted 6e10w are m0re def1n1t10n5 than e4ua11t1e5. Anyway, they 5eem5 t0 u5 tran5parent and can 6e u5ed t0 c0mpute 5uch 06ject5 a5 F 1n (4), pr0v1d1n9 u5 w1th the c0nnect10n 6etween f1e1d the0ry 0fj-d1fferent1a15 (1), (2) and the 9e0metr1ca1 appr0ach 6y Qu111en [2] and Fa1t1n95 [3]. We w111 a150 5ee that the f0rmu1ae 06ta1ned 6y Man1n and 8e111n50n [4] natura11y ar15e 1n 0ur c0ntext. 0 u r 5trate9y w1116e t0 w0rk 0ut a11 nece55ary c0nd1t10n5 that any 247

V01ume 180, num6er 3

PHY51C5 LE77ER5 8

13 N0vem6er 1986

expre5510n f0r det 9/. ha5 t0 06ey. 7hen we 9ue55 the m05t 51mp1e f0rmu1a, 5at15fy1n9 them and check the re5u1t f0r 5pec1a1 ca5e5. Let u5 6e91n w1th the ca5e/f > 1 and try t0 def•me f0rmaUy

0fexp(f~fd2~d-~),

det ~j = f 0 ¢

(5)

f a n d ~06e1n9 j- and (1 -])-d1fferent1a15.0f c0ur5e (5) cann0t 6e c0rrect 6ecau5e there are n/= (2j - 1)(p - 1) 2er0 m0de5 0f ~-h010m0rph1c/f-d1fferent1a15 f1 (2)(d2) j ..... fn/(2) (d2)1. We have t0 5u65tract fr0m (5) the 2er0 m0de c0ntr16ut10n and wr1te

det ~/. =

(f(21).-. f(2n1)}, det11~(2j)11

(6)

where the matr1x 11J~(2j)11 1n the den0m1nat0r ha5 f1(2j) 0n the 1nter5ect10n 0f the 1th c01umn and the kth r0w, and (f(21) ... f(2nj )) =f

cDc~(-Df f(21) ... f(2nj ) exp (f4~

~f d 2 A d , ).

(7)

1n 0rder that (6) d0e5 n0t depend 0n a ch01ce 0f p01nt5 21, the c0rre1at10n funct10n (7) ha5 t0 6e ant15ymmetr1c 1n a11 21 and 1n each 21 t0 6e a h010m0rph1c/f-d1fferent1a1.8ut we kn0w that due t0 the 9rav1tat10na1 an0ma1y [5 ] 1t 15 1mp055161e t0 c0n5truct the c0rre1at10n funct10n (7) that d0e5 n0t depend 0n the ch01ce 0f c00rd1nate5 at 0ther p01nt5 0f a 5urface 0r 0n the ch01ce 0f the c0nf0rma1 metr1c 0. Under the 9enera1 c00rd1nate tran5f0rmat10n 2 ~ 2 + e(2, 2) the var1at10n 0fFj 15 [5]

8eF1=~.~fp-1~(pe) A(1090)1d2

Ad~-,

Cj=6/f2--6/f+

1.

(8)

7hu5 Fj depend5 0n a c0mp1ex 5tructure X 0f a 5urface, 0n a c0nf0rma1 metr1c 0 and a ch01ce 0f c0nf0rma1 c00rd1nate 2:/71• = F/(X, 0, 2). We w111n0t c0n51der here the 9enera1 ca5e 0f an ar61trary X, 0, 2, 6ut 1n5tead f1x 50me part1cu1ar metr1c p = 0 , and 5tudy 0n1y the dependence 0n X, 2 . 7 h e m05t u5efu1 ch01ce 15 0 , = 1v,(2)[ 4 ,

(9)

where v,(2) 15 50me h010m0rph1c 1/2-d1fferent1a1 (ferm10n 2er0 m0de) and the a5ter15k den0te51t5 60undary c0nd1t10n5 0r a character15t1c (def1ned 6e10w). F0r u51t w1116e 1mp0rtant that 9enera11y v, ha5 exact1y p - 1 f1r5t-0rder 2er05, wh1ch we den0te 6y R 1 ... Rp-1, v,(R1) = 0. F0r the metr1c (9) the var1at10n (8) 0fFj take5 the f0rm p-1 1=1

0~v,(2)

=

=

0

e(2, 2e)v•,(2) + ~(a2e ) v,(2)),

that 91ve5 det ~ 1 [ X , p , , 2 ]

p -1 ( df(2)~2Q/3 =det ~j[X,p.,f] × 1=1• 1-1 d2 ]

(10)

7hu5 det ~1 tran5f0rm5 a5 a pr0duct 0f ( - ~Cj)-d1fferent1a15at p01nt5 R1. 7he 1a5t c0nd1t10n 0f det ~j 15 that 1t ha5 n0t t0 depend 0n the c00rd1nate5 Y1 0n the m0du11 5pace Mp. 7he 1mp0rtant a5pect 0f (6) 15 that 1t depend5 0n the ch01ce 0f a 6a515 {f~} 0f h010m0rph1c j-d1fferent1a15 1n acc0rdance w1th the def1n1t10n 6y Qu111en 0f det ~j a5 a 5ect10n 0f the determ1nant 6und1e 0nMp [2]. 1n 9enera1, there 15 n0 part1cu1ar ch01ce 0f the 6a515 {f/} except f0r the ca5ej = 1, when the n0rma112ed 6a515 {c0 6 1 = 1, ..., p} 248

V01ume 180, num6er 3

PHY51C5 LE77ER5 8

13 N0vem6er 1986

ex15t5.70 def1ne 1t we f1x a 5ymp1ect1c 6a515 0f h0m01091ca11y n0ntr1v1a1 cyc1e5 a1, 61, 1 = 1, ..., p 0n a 9enu5 p 5urface:

a10aj=61061=0

1:/=f,

a1061=81j

(11)

(where 0 den0te5 the a19e6ra1c num6er 0f 1nter5ect10n5) and put

f W](2) d2 = 61]. a1

(12)

1t 15 C0nven1ent t0 U5e th15 part1CU1ar 6a515 t0 def1ne det 90: det 3 0 - f ¢0(21) ...

~(2p) ~(2) exp(f

~0 ~0 d2 • d9)

(13)

det 11<-01(2/)11 where p 1-d1fferent1a15 c0 (21)and a 5ca1ar ~0(2) appeared due t0 p 2er0 m0de5 0f w and 1 2er0 m0de 0f ~0: ~0(~) = c0n5t., (13) 6e1n9 5upp05ed t0 6e 1ndependent 0n 2. F0r a 91ven 9enu5 p the 5urface 6a515 ~a1,61), and c0n5e4uent1y (w1) are def1ned up t0 11near tran5f0rmat10n5,~re5erv1n9 (1 1). 5uch tran5f0rmat10n5 f0rm the de9ree p m0du1ar 9r0up 5p (p, 2) --- Pp, n0ntr1v1a11y act1n9 0n d e t - 11~1(2/)11, the 1atter 6e1n9 ca11ed the we19ht-0ne m0du1ar f0rm w1th re5pect t0 Fp. 5urpr151n91y, the m05t 51mp1e f0rmu1ae can 6e 06ta1ned n0t f0r det ~/1t5e1f 6ut f0r the c0m61nat10n ~7~= det ~/(det ~0) 1/2 .

(14)

Fr0m the prev10u5 d15cu5510n 1t f0110w5 that X/ha5 t0 6e a -(21 - 1)2-d1fferent1a1 w1th re5pect t0 c0nf0rma1 tran5f0rmat10n5 at p01nt5 R 1and the we19ht-0ne-ha1f m0du1ar f0rm w1th re5pect t0 1•p ,1. F0r ha1f-1nte9er j there are a150 2 2p type5 0f 60undary c0nd1t10n5 and k/• ha5 t0 depend 0n them. We n0w exp1a1n h0w the5e 60undary c0nd1t10n5 are parametr12ed and 1ntr0duce the 1a5t c0ncept5 and n0tat10n5 needed t0 c0n5truct the f0rmu1a f0r ~./-. Let u5 c0n51der f0r a 91ven j C 2 + 1/2 a mer0m0rph1c f-d1fferent1a1 f and den0te 6y (f) the f0rma1 5um (f)=

~"

~k mkPk

Where ~Q1) (~Pk)) 15 a 5et 0f 2er05 (p01e5) 0 f f 0 f 0rder5 the map

n1 (mk). We a150 f1x a p01nt Q 0n a 5UrfaCe 5 and

def1ne

P

e-+P=--f 10d2,

~=(a91,...,60p) 7, P E 5

(15)

Q 0f a 5urfaCe 5 1nt0 the c0mp1ex t0ru5,1 =

r1k =,~r~°kd2, 61

7-7 = 7-.

CP/7-2p ~ 2 p, where

a per10d matr1X 7- 15 def1ned a5 (16)

7he fundamenta1 fact (A6e1•5 the0rem) 15 that f0r any mer0m0rph1C ]-d1fferent1a1 f the 1ma9e 0f (f) Under (16)

(f)=-- ~

-- ~k mkPk

(17)

*1 5tr1Ct1y5peak1n9We eXpeCt that h1 W1116e We19ht-0ne f0rm 0n1y W1thre5pect t0 the 5U69r0Up0f rp, that pre5erve5 the 60Undary C0nd1t10n5 0n V.(2) 1n (9). 249

V01ume 180, num6er 3

PHY51C5 L E 7 7 E R 5 8

d0e5 n0t depend 0n f a n d e4ua15 2er0 f0r/' 2(50 = 2 ] ( ~ )

= 0.

13 N0vem6er 1986

Fr0m th15 f0110w5 that

(1n 3),

(18)

where w 15 any mer0m0rph1c 1-d1fferent1a1, and thu5 (f)=j(~)+~(m

1

t)7 7 +

1 t• , ~m

(19)

where the c0mp0nent5 0fp-vect0r5 m•, m•• e4ua1 0 0r 1 . 7 h e 5et 0 f 2 2p d1fferent pa1r5 (m•, m••) - m, ca11ed character15t1c5 0 f f , parametr12e a11 p055161e 60undary c0nd1t10n5 0 n f 1 n an 1nvar1ant way. 7he num6er e(m) = (m•) 7 • m•• (m0d 2) 15 ca11ed the par1ty 0f m. 6enera11y, the R1emann-R0ch the0rem 5tate5 that the num6er 0f h010m0rph1c /-d1fferent1a15 f0r/' > 1 d0e5 n0t depend 0n m and e4ua15 n / = (2j - 1) (p - 1), 6ut f0r j = 1/2 the 51tuat10n 15 m0re 5u6t1e. F0r th15 ca5e 1t 15 kn0wn (fr0m the R1emann 51n9u1ar1ty the0rem [6]) that the par1ty 0f the num6er 0f h010m0rph1c 1/2-d1fferent1a15 c01nc1de5 w1th the par1ty 0f the character15t1c; f0r a 9enera1 5urface n1/2 = e(m). F0r an 0dd character15t1c m the h010m0rph1c 1/2-d1fferent1a1 vm (2) can 6e f0und exp11c1t1y:

p2 (2) = 0m,1~1(2) ,

(20)

where the R1emann 0-funct10n 15 def1ned a5

0m(2) =

~ e x p [ r r 1 ( n + ~1 m )~ 7 ff~7~.P

1 t r(n+~m)+22r1(n+~m~)7(2+~m~)],

(21)

and

0m,1 =-~0m (2)/~2112=0"

(22)

N0w everyth1n9 15 ready t0 c0n5truct X/. F 0 r / E 2 + 1/2 we pr0p05e the f0110w1n9 expre5510n: 2]•-1 ~]=u. (21) .

p2j-1(2n])(V1(21) .

. . . . V1(2n])V1-2](R1)~V1-2j(RP-1))*0m(~121--(2]-1)~R~). det 11~"(2j)11 (v,(R 1) ..• v~,(Rp•1)) (2/- 1)2

Here * den0te5 an ar61trary 0dd character15t1c, (v,) = R 1 + --• + Rp•1,

m 15 the

(23)

character15t1c 0ff1 and (24)

U51n9 kn0wn ana1yt1c pr0pert1e5 0f 0-funct10n5 [6,7], 1t 15 n0t hard t0 ver1fy that (23) 5at15f1e5 a11 c0nd1t10n5 115ted a60ve. F0r ] E 2 the expre5510n 15 the 5ame except f0r that m ha5 t0 6e e4ua1 t0 *. 7he 9enera112at10n 0f (23) t0 the ca5e 0f the c0rre1at10n funct10n 1n 5tra19htf0rward: t0 each add1t10na1 pa1r f(~), ¢ ( ~ ) 1n (7) c0rre5p0nd add1t10na1 0perat0r5 U21-1(~) V1(~), U1-21(~~) V~1(~ ) 1n (23) and 0ne a150 ha5 t0 add - - ~ t0 the ar9Ument 0f the 0-fUnCt10n. 1t 15 W0rth n0t1n9 that (24) 15 a Ch1ra1 part 0f the C0rre1at10n fUnCt10n 0f the exp0nent5 0f the free 5Ca1ar f1e1d ~00n a 5UrfaCe, a5 f0110W5 fr0m the C0rre5p0nd1n9 f0rmU1a 1n ref. [1 ]:

~1 V41(21)~*12exp(-2rr1~<]414]1m(91-

9J)7(1m f ) - 1 1 m ( 9 1 - 2 ] ) ) = ( k[-1

exp[14k~°(2k~9k)]) ~

~. 41 = 0 . N0w, 1et u5 d15cu55 50me part1cu1ar ca5e5. F1r5t 0f a11 we c0n51der j = 0.1n th15 ca5e (12), (23) take the f0rm

250

(25)

v01ume 180, num6er 3

PHY51c5 LE77ER5 8

13 N0vem6er 1986

V,1(2) V.(21) ... V.(2p)(V1(2) V-1(21) ... V-1(2p) V1(R1) ... V1(Rp-1)). det 11~1(2j)11" v•,(R 1) .-. v•,(Rp•1)

=

X 0,(2 -- 21 -- ... - 2p + R 1 + ... +Rp•1) •

(26)

N0w, we u5e an 1ndependence 0f (26) 0n 2, 21 and put 2p = 2; 21 = R1, 1 = 1, ..., p - 1. After th15 we 06ta1n

0*, 1~°1(2)

(27)

(det ~0) 3/2 = det11~ (2) ~ (R 1)... a) (Rp • 1)11 7h15 expre5510n d0e5 n0t depend 0n 2 and 100k5 11ke a 5tra19htf0rward 9enera112at10n 0f 0] f0r the ca5e p = 1. A5 wa5 exp1a1ned a60ve a) (R1) 1n (27) appeared due t0 the 9rav1tat10na1 an0ma1y wh1ch d0e5 n0t man1fe5t 1t5e1f f0r p = 1 6ecau5e acc0rd1n9 t0 (9), p = c0n5t, and the RH5 0f (8) turn5 t0 2er0. C0m61n1n9 (23) and (27) 0ne can f1nd f0rmu1ae f0r det 8j 1t5e1f. An0ther 1ntere5t1n9 and 1mp0rtant ca5e 15j = 1/2. F0r even character15t1c5 we 9et det m 81/2(det ~0) 1/2 =

0m

(0),

e(m) = 0 ,

(28)

and f0r 0dd m detm 31/2(det 30)1/2 = (V1(21)V(21)V-1(22))V(22)0m(21 -- 22) =

0m,1~1(2) V2(2) ,

e(m) = 1 ,

(29)

Where tW0 2er0 m0de5 1n (29) appeared 6eCaU5e the act10n 51/2 = f ~ 8ff d2 • d~ C0nta1n5 tw0 d1fferent 5p1n-1/2 f1e1d5 ff and t~ and there 15 1 2er0 m0de Vm (2) f0r each. We a150 06ta1n the f0110W1n9 expre5510n f0r the C0rre1at10n fUnCt10n 0f ferm10n5 ~0, ~, f0r any m 1t e4Ua15: N

(1~=1~(21)~(~1))m~det1/2-~0 N

N

= 1=11-1[0,,1601(21) 0,,kc0k(~1)] 1/2 1
6,7109X1/2=2~1(fr17d2~d9)+17r((1m~c)-16r~r),

(31)

where 7 =- 722 15 the 1eft c0mp0nent 0f the e n e r 9 y - m 0 m e n t u m ten50r 0f f1e1d5 4, ~ and ~0: (7) = ( 7 (1/2)) - (7(0)).

(32)

251

V01ume 180, num6er 3

PHY51C5 LE77ER5 8

13 N0vem6er 1986

1 7he f1r5t term 7 (1/2) = ~(3ff ¢- -- 0ff~ •k) 15 the ener9y--m0mentUm ten50r 0f C0mp1eX ferm10n5 ¢, ~k and the 5ec0nd term 7 (0) = --(~2 ~0)2 15 that 0f the 5ca1ar f1e1d ~0.0ne can f1nd the1r vacuum expectat10n5 fr0m the 0perat0r pr0duct expan510n5 5u65t1tuted 1n tw0-p01nt c0rre1at10n funct10n5:

• = 1/(2 - 2•) + (2 - 2•)<7(1/2)(2•)> + 0((2 - 2•) 2) (~ (2) ¢~ (2)>m 2~2~ = { [0., 1601 (2) 0 . , k 60k(2•)] 1[2/0 * (2 -- 2•)} 0 m (2 -- 2•)/0 m ( 0 ) ,

(33)

2-~2~ 12 -- 2•1-2 [1 + 7(0)(2•)(2 -- 2•) 2 + 0((2 -- 2•)3)] ~-=%~

(~5) 1[0*•1W1(2) 0*•kWk(2•)]

1/2/0*(2 -- 2•)12 eXp [27r 1m(2 -- 2•)(1m 7")-1 1m(2 -- 2•)] .

(34)

7he 1a5t expre5510n 1n (33) 15 the Un14Ue (1/2, 1/2)-d1fferent1a11n (2, 2•) W1th the CharaCter15t1C m and a f1r5t-0rder p01e W1th a Un1t re51due at 2 = 2• Wh1Ch 15 ant15ymmetr1C 1n 2 *+ 2• and h010m0rph1C 1n 2 1f2 V~2•. 5tr1Ct1y 5peak1n9 (33), (34) are C0rreCt 0n1y When a metr1C 0 1n a V1C1n1ty C0nta1n1n9 2, 2• 15 Ch05en t0 6e a C0n5tant, 6Ut the C0m61nat10n (32) that We Want t0 06ta1n, d0e5 n0t depend 0n P due t0 the CanCe11at10n 0f a11 an0ma11e5. C0mpa1r1n9 (32), (33) We 1mmed1ate1y 9et

7(2) = 10m,11601(2 ) ~](2)/0 m (0) + ~7r(1m ~)~1~1(2) ~ k ( 2 ) = 2rr1[a 109 0 m (0)/0r111 601(2 ) 60/(2) + ~r(1m r)~k1~1(2) ~ k ( 2 ) .

(35)

5U65t1tUt1n9 th15 1nt0 (31) and U51n9 the f0rmU1a f0r the Var1at10n 0f r

6,~r1/=fn,~1~/d2A d%,

(36)

we 06ta1n 5,7109 k1/2 = 6 n 109 0 m ( 0 ) , fr0m wh1ch (28) f0110w5. 1t 15 1ntere5t1n9 that the expre5510n f0r 7 (0) 06ta1ned fr0m (34) c0nta1n5 the 50-ca11ed pr0ject1ve c0nnect10n F: 1

,,

3

,

2

1" = 1.-~[0.,1¢01/0,,1(.01 • ~(0.,1601/0.,1¢01) ] • -~0.,1]kC,91¢0jt.0k/0.,k09k, 1 "

7 (0) = F -- ~r(1m r ) ~ 1 ~ 1 ~ k

.

(37)

1n C0ntra5t W1th 7 1n (35), 7 (0) 15 n0t a 4Uadrat1C d1fferent1a1 and a5 f0110W5 fr0m (37) 1t tran5f0rm5 Under 2 -+ f(2) thr0U9h the 5ChWar2 der1Vat1Ve:

~(0)(f(2))[df(2)/d212 + ~ [ f , . / f , - ~3 ( f ,,/f ,) 2 ] = 7 ( 0 ) ( 2 ) ,

(38)

acc0rd1n9 t0 the 9enera1 ru1e5 0f c0nf0rma1 f1e1d the0ry [9]. 7he expre5510n (37) turn5 1nt0 a 2-d1fferent1a1 1fwe re5t0re the dependence 0n the c0nf0rma1 metr1c ~0 = 109 0: 2 . 7(0)(~0) = P -- ~7rC07(1m r ) - 1 m + ~ [1~(a2~0) 2 -- ~2~0] (39) An a1ternat1ve way t0 06ta1n (37) 15 t0 u5e the 0perat0r expan510n 0f the pr0duct 0f current5 w and ~6:

a2,6~(2, 2•) = <<~0(2) 02,¢(2•)>> 2-~2~ 1/(2 - 2•) 2 -- .r(0) ~ ,.) + ... ~ h01.(2

~f ~2~ 252

<$a, ~(21) d21 ... $ap w(2p) d2p ¢(~)>

(40)

V01ume 180, num6er 3

PHY51C5 L E 7 7 E R 5 8

13 N0vem6er 1986

Fr0m (40) 1t f0110w5 that 6~(2, 2•) 15 a (1,0)-d1fferent1a11n (2, 2•) w1th p01e5 at 2 = 2•, ~, 5at15fy1n9

f6~(2.2~)d2=0,

f, 6~(2,2~)d2=-f6~(2.2~)d2=2.1.

a1

2

7he5e C0nd1t10n5 def1ne 6~(2, 2•) un14ue1y:

6~(2, 2•) = 0 2 109 [0,(2 - 2•)/0,(2 - { ) 1 , and we 9et h°) = F 01.

1t 15 amu51n9 t0 ver1fy var10u5 c1a551ca1 1dent1t1e5 6etween 1-d1fferent1a15 u51n9 the1r repre5entat10n 1n term5 0f Ward 1dent1t1e5 0f the c0nf0rma1 f1e1d the0ry 0f c0, ¢. F0r examp1e we can der1ve (36) 5tart1n9 fr0m the 1dent1ty

r1x = Y~1

6k

(a~) d2•

.

7w0 1a5t examp1e5 we w0u1d 11ke t0 d15cu55 are the ca5e5 j = 3/2 a n d / = 2. A9a1n 0ur 5trate9y w0u1d 6e t0 put a11 p01nt5 21 1n (23) t0 R 1. After that we 06ta1n 3.3/2(m ) = 0m(0 ) det-111~(R1)~(R 1) ... ~(Rp~1)~(Rp~1)11 , 1#

X3/2(m ) = 0m,1~1(R1) det

--

1

t

11~ (R1)~

tt#

e(m) = 0 ,

(R1)~(R2)9~(R2) ... ~(Rp~1)~(Rp~1)11,

(41) (42)

where 915 a c01umn 0f 2 p - 2 h010m0rph1c 3/2-d1fferent1a15 w1th character15t1c m. X2 = 0., 1w~(R 1) d e t - 111f(R 1) f••(R 1) f••••(R

1) f(R 2) f•(R 2) [••(R 2) f(R 3)••" f••(Rp • 1) 11,

(43)

where f15 a c01umn 0f h010m0rph1c 2-d1fferent1a15. E5pec1a11y 51mp1e 15 (41) wh1ch 1n part1cu1ar de5cr16e5 the dependence 0f the 9h05t determ1nant 1n the heter0t1c 5tr1n9 0n the 60undary c0nd1t10n 0n w0r1d-5heet ferm10n5 and a110w5 t0 5upp05e that 5uper5tr1n9 n0nren0rma112at10n the0rem5 are the c0n5e4uence 0f the R1emann 1deht1ty [6]. 7he f0rmu1ae f0r det ~] can 6e u5ed t0 c0mpute the c0nf0rma11y 1nvar1ant pr0duct5 0f var10u5 Lap1ace 0perat0r5 A5. = -0/-180-]~, act1n9 0n/-d1fferent1a15. Acc0rd1n9 t0 [1 ] we expect

6e<, ;;1=

1" detN--~-~N1-j~

1-1/(det~]) n]

1,

~C]n]=0/,

(N/)1k=ff14)(2)f0~)(2)p1-/d2 A d5.

(44)

1t f0110w5 fr0m (9), (23) that the pr0duct 11](det ~])n~ d0e5 n0t depend 0n the ch01ce 0f c00rd1nate5 at p01nt5 R 1 due t0 E/C/n/= 0.7h15 dem0n5trate5 u5 a c105e c0nnect10n 6etween the cance11at10n 0f ana1yt1c an0ma1y [1 ] 1n (44) and the cance11at10n 0f 9rav1tat10na1 an0ma1y 1n 11] det 3/. N0w, 1et xt5 exp1a1n the c0nnect10n w1th the f0rmu1a f0r the 6050n1c 5tr1n9 mea5ure 06ta1ned 6y 8e111n50n and Man1n [4]. 7he mea5ure 1n E5VM 15 the part1cu1ar ca5e 0f (44) and fr0m (4), (27), (43) we 9et X2 •

(Rp • 1) 1[ f(R 2) .•.1 ,t(Rp-1) 11•

det 911 c0 (R 1) ~ ( R 1) c0 (R 2)--• ~

F= --X9- (0*~1~1(R1)) 8. , .det [[f(R1) . f••(R1) . f••••(R1)

(45)

where the h010m0rph1c 4uadrat1c d1fferent1a15 f1(2) 1n the c01umn f = (]•1 .... , f3p•3) 7 def1ne the c00rd1nate5 Y1 0n Mp, u5ed 1n (4) 1n a 5tandard way, de5cr16ed 1n ref. [1]. 0ne can check that (45) d0e5 n0t depend 0n the ch01ce 0f c0nf0rma1 c00rd1nate5 at p01nt5 R 1 and we f1x 1t 6y the c0nd1t10n5 t

v.(R1) = 1 ,

1 = 1 ..... p -

1.

(46) 253

V01ume 180, num6er 3

PHY51C5 LE77ER5 8

13 N0vem6er 1986

Next we w111 try t0 51mp11fy the determ1nant5 1n (45) mak1n9 an appr0pr1ate chan9e 0f 6a515 { c0/}, { ft•)- 7h15 purp05e 15 ach1eved thr0u9h the f0110w1n9 ch01ce [4] : ~(R1)

= 6~1,

~k = 6 0 k P . , f k = V*~k

a = 1 ..... p -- 1 , k = 1 ..... p ,

~p = v 2 , ~p+1(R1+1)=611

k = 1 ..... 2p -- 2 ,

f2p-2+1(R1) = 611

1 = 1. . . . , p - - 2 ,

1 = 1 ..... p -- 1 .

(47)

7he n0rma112at10n C0nd1t10n51n (47) are wr1tten 1n C00rd1nate5 (46) and def1ne Wa, ~k .... Un14ue1y. 5U65t1tut1n9 (47) 1nt0 (45) we 9et the 8e111n50n--Man1n f0rmu1a: F = [ 0 , , 1 W ~ ( R 1 ) 1 - 8 = [V2(2)/0.,1W1(2)18 ,

(48)

Where We U5ed that the fa5t rat10 d0e5 n0t depend 0n 2.1f We add1t10na11y f1x V. a5 1n (20) We 06ta1n F = 1, 1n acc0rdance w1th the the0rem 6y Mumf0rd [9] a60Ut the tr1v1a11ty 0f the c0rre5p0nd1n9 6und1e 0n Mp. 7hu5 we have c0n5tructed c0nf0rma1 f1e1d the0r1e5 0f ana1yt1c ]-d1fferent1a15 0n a 9enu5 p r1emann1an 5urface. A 5pec1a1 metr1c p = [v. 14 wa5 u5ed and f0r th15 ch01ce the f0110w1n9 ferm10n12at10n ru1e5 take p1ace (c0mpare (23), (24) and (30)): #)

+~ V2/-1t~ ,

45(1-]) ~ V1,-2] ~ ,

p-1 1VaCuUm) +~ [•1 [~(R1) ~ ( R 1 ) . . . ~(21-1)(R1)/[V~,(R1)](21-1)2 ] d e t - 1 / 2 ~ 0 , 1=1

(49)

where 4, ~ are 1/2-d1fferent1a15 5u6jeCted t0 60Undary C0nd1t10n * f0r f E 2 and t0 that 0 f f 4) f 0 r / E 2 + 1/2. HenCe we 5ee that the 6050n12at10n f0r h19her 9enera 15 50mewhat n0ntr1V1a1. 1t w0U1d 6e 1ntere5t1n9 t0 tran5f0rm the f0rmU1ae 06ta1ned a60ve t0 an ar61trary metr1c 0. Unf0rtUnate1y, 0Ur appr0ach wa5 rather 9ue55ed. 7he m05t 1mp0rtant pr061em 0f the C0rreCt def1n1t10n 0f det ~j rema1n5 t0 0Ur 0p1n10n un501Ved, 6Ut We h0pe that the re5U1t5 de5Cr16ed 1n th15 1etter w111 he1p t0 f1nd 1t. 7he auth0r 15 9ratefU1 t0 A. 8e1aV1n and A. 8e111n50n f0r he1pfU1 C0nver5at10n5. Reference5

[1] [2] [3] [4] [5] [6] [7] [8] [9]

254

A.A. 8e1av1n and V.6. Kn12hn1k, Phy5. Lett. 8 168 (1986) 201; Landau 1n5t1tute prepr1nt - 9 (1986). D. Qu1Uen, Funct. Ana1. App1. 19 (1986) 31. 6. Fa1t1n95, Ann. Math. 119 (1984) 387. A. 8e111n50n and Yu.1. Man1n, C0mmun. Math. Phy5., t0 6e pu6115hed. L. A1vare2-6aum6 and E. w1tten, NucL Phy5. 8 234 (1983) 269. 6. C1emen5, A 5crap600k 0f c0mp1ex curve the0ry (P1enum, New Y0rk, 1980). J. Fay, Lecture N0te51n Mathemat1c5 (5pr1n9er, 8er11n, 1973) p. 352. L. A1vare2-6aum6, 6. M00re and C. Vafa, Harvard prepr1nt HU7P-86/A017. D. Mumf0rd, L•En5. Math. 23 (1977) 39.