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On the quantum topology of strings
Volume 189, number 1,2
PHYSICS LETTERS B
30 April 1987
O N T H E Q U A N T U M T O P O L O G Y OF STRINGS B.V. IVANOV Institute for Nuclear Research and Nuclear Energy, BoulevardLenin 72, Sofia 1784, Bulgaria Received 14 January 1987
The role of the projective connection and the fundamental.differential of the seond kind in the determinants of the string integrand is emphasized. A new integrableterms is found.
At present, the supersymmetric string in ten dimensions is the most serious candidate for a unified theory of nature, including quantum gravity [ 1 ]. The string measure in the covariant path-integral approach was analysed in ref. [2]. While an application of strings to hadron physics was sought, the accent was laid on the dependence of the determinants on the conformal factor of the metric. The idea was that the Liouville mode could lead to a consistent quantization in four dimensions without tachyons in the bosonic string. Nowadays strings are studied once again in their critical dimension, where they are conformal invariant, and the descent to four dimensions is achieved by compactification. The accent has been transferred to the Teichmiiller parameters and thus from quantum geometry to quantum topology. Recently, the importance of the complex structure of the moduli space was emphasized [ 3 ]. Although the determinants cannot be found explicitly by Pauli-Villars [ 3 ], or heat kernel [ 4 ] regularization, their second variation vanishes, which means that the vacuum amplitude of the bosonic string is given by the squared modulus of a (unknown) holomorphic function:
Zs = 5rJdyi ^ dyi IF(y) [ 2(det Im z) -13
(1)
Here the y~ are the 3 g - 3 complex moduli for genus g, while r is the period matrix. It is possible to express F ( y ) in tel-ms o f R i e m a n n theta functions, holomorphic differentials and Green functions [ 5 ]. Green functions are inevitable anyway when external legs are present, but their appearance in the measure seems redundant. The Green function is a more complex object than the determinant, in the point splitting regularization it defines the latter, while the opposite is impossible. After all, the determinant is a product of the eigenvalues, while the Green function contains also the eigenvectors at two different points. This redundancy is reflected in ref. [ 5 ] by the independence of the formulas on the arguments of the Green function. Furthermore, the study of the conformal factor dependence has teached us that although the relevant operators do possess zero modes, their determinants are divided by the matrix determinants of the zero modes, so that finally no such modes appear in the Liouville effective action. Thus we expect a measure expressed solely in theta functions, which are simpler objects. Since the z o can be taken as the moduli, the arguments of the theta functions are also redundant and one should stop at last at theta constants (the z o are more than necessary for g > 3, but this is a problem of the Siegel upper half-plane, not of the determinants). This is exactly the result of modular form analyses [ 3,6 ]. The necessary parabolic modular forms of weight 12 f o r g = 1 and 1 2 - g f o r g > 1 are unique f o r g = 1, 2, 3 and are given by the product of all even theta constants. However, their properties for g > 3 are poorly understood. Modular analyses of the superstring leads to the vanishing of the cosmological constant for all g [ 7 ]. 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
39
Volume 189, number 1,2
PHYSICSLETTERSB
30 April 1987
It is clear that some direct method for computation of determinants is necessary. In this paper we shall use the point-splitting method since the Green functions are known explicitly [ 8,9]. It was applied in ref. [ 8 ] to find the spin-structure dependence (first found in ref. [ 4 ]) and in ref. [ 9 ] to prove the bosonization formula of ref. [ 10]. We obtain another integrable term, which is present in all determinants and for every genus. In the following we will not pay any attention to the conformal factor, since it finally cancels. The relevant operators are P, = ~x and the corresponding Green functions G~,, _, +~)(x, y) satisfy the equation
P,,G¢,,, _,,+ i)(x, y) = ~ t _,) (x, y) - H o _,) (x, y ) ,
(2)
where H o _,)(x, y) is the projection operator onto the space of zero modes of weight (1 - n , 0). For n = 1/2 we have the fermion Green funtion G[e](x, y) which depends on some even characteristics [e]. There are no zero modes for the even sector, while the odd sector does not contribute to amplitudes with less than five legs. The variation of P 1/2 is - q ( x ) Ox with r/(x ) the Beltrami differential. Thu s: 8 In det P~/2 = T r ~Pl/2G[e](x, y) ly=x •
(3)
The Green function possesses a pole which serves as a regulator. We must saturate this pole by a Taylor series expansion of the other factors. The mathematics used below can be found in ref. [ 11 ]. Let us introduce the prime form
E(x,y) =
O[o](y-x) [H[o](x)H[o](y)]~/2,
H[o](x) =
(O)Wi(x).
(4,5)
Here [o] is any odd characteristics, the wi(x) are the abelian differentials and the argument of the theta function is y - x = f~ w, = z~. The prime form is antisymmetric and has a first-order zero for x = y. The fermion Green function is nothing but the Szego kernel:
1 O[e](y-x) G[e](x, y) = - ~ O[e](O)E(x, y) "
(6)
The following expansions are valid in the vicinity of some point p:
E(x, y)x//--~ d y = ( y - x ) - -~ ( y - x ) 3 S ( p ) + h.o.t.,
(7)
G[e] (x, y)v/--~ d y = - ( 1 / n ) { 1 / ( y - x ) + ( y - x ) [ - ~ S ( p ) + ½0" [ e ] ( 0 ) / 0 t e ] ( 0 ) ] } + h.o.t,
(8)
where the prime means differentiation with respect to y. S(p) is the projective connection. Inserting of (8) into (3) gives In det ell2
iI
~-~fr/(X)[ l / ( y - - x ) 2 --0" [e](y--x)/O[e](O) -- I S ( x ) ] ly=~.
(9)
The first term is divergent and should be dropped. The second term can be written as 8 In 0[e](0) after the use of the heat equation and the important identity (wi = v,dx in some local coordinates): - 2 i j dZx q(x)vi(x)vAx) = 8 % .
(10)
This term was found in refs. [4,8] and gives the dependence on the spin structure. Our main concern is the third term in (9). S(p) is defined as ( /Pi
s(.) = t J H.
h
,} +
]2(.)_ :[ H3
where the first brackets mean a schwarzian derivative, while 40
1(.).
Volume 189, number 1,2
PHYSICSLETTERSB
OnG Hn (f) = - (f) wi, (x)...wi, (x) , Ozi, ...Ozi,
30 April 1987
(12 )
a n d f i s a non-singular point belonging to the theta divisor. In order to integrate S(p) we must find a form for it of the type
~l(x)S(x)= ~l(x)wt(x)wj(x) o ~° In L(x, r ) .
(13)
Formula (11 ) contains abelian differentials in the denominators and it is hard to see how it can be integrated. On the torus, however, w(x) = d z ( x ) where z spans the universal cover of the torus, and
s(p)
=
- 2 O l n 0 ' [ l~](0)(dz) 2 ,
(14)
which when inserted in (9) and after using the Jacobi triple product identity 0[°](0)0[°/2](0)0[U](0)
= - ( 1 / ~ ) 0 ' [ I¢~1(0)
(15)
yields the well-known one-loop result
det P,/2~O[e](O)(~O[b](O) ) -'/3
(16)
The product goes over the three even theta functions. For higher genus one can consider another way, which makes use of the fundamental differential of the second kind: d2
d2
w(x, y)= -~-~yln E(x, y) d x d y = ~ y y l n O ( y - x - f ) dx dy .
(17)
It is independent of the non-singular point f a n d has a double pole:
w(x, y) = [ 1~(x-y) 2 + ~S(p) +h.o.t.] dx dy.
(18)
Before proceeding further, we shall show that the projective connection, or equivalently the fundamental differential, .are present in the variation of all determinants of the string integrand. The scalar Green function may be represented as [ 8,11 ] Y
G(x, y)= 2 ln[E(x, y)] + 1 f ( w - ~ ) ~ ( I m x
Y
~')/7 1
f(W--~}'j+Q(X)+~2(y),
(19,
x
where the last two factors are irrelevant as we shall see. The second factor is due to the zero modes. The Green function of P~ is given by the derivative of G(x, y) and yet another derivative comes from (3). In order to resolve the ambiguity which arguments exactly the derivatives act upon, we note that the point-splitting method is completely equivalent to the operator approach developed in ref. [ 12 ] and eq. (3) with i/deleted gives the effective energy-momentum tensor. For scalars it reads [ Tu~) ~ (OxXuOyX~) meaning that the derivatives are with respect to both arguments in (19). Then the last two terms vanish, the second term gives det Im z, while the first one becomes w(x, y). So the projective connection is present in the variation of the scalar determinant. The ghost Green function contains the x-derivative of the first term in (19) [ 8,12 ], and a different zero mode part. Since (Tgh) ~ (cXtSybyy+2O~c~byy) and the series expansion o r E ' (x, y)/E(x, y) shows that x- and yderivatives differ by a sign, we obtain once again the projective connection in the variation of the ghost determinant. The same conclusion is true for the superghost determinant since the expression for it given in ref. [ 8 ] 41
Volume 189,number 1,2
PHYSICSLETTERSB
30 April 1987
is built upon the fermion Green function. The presence of S(p) in all variations is analogous to the fact that the conformal factor integrates to the Liouville action for all determinants, the only difference being in the numerical coefficient. The integrand in (1) is given by the scalar and ghost determinants. Our analyses show that the zero modes supply the (det Im z) - 13part, while the projective connection should yield for g = 1, 2, 3 the product of all even theta functions, the latter being the unique modular form with the necessary properties. The fermion determinant reflects the features of the whole string integrand, but we need an expression for S(p) in terms of even theta functions in order to get close to the modular form results. The following equation: 0 N ( z [ z) =exp[ inaizoa j + 27tiai( zi-t-bi) ] O(z d- za+ bl z)
(20)
shows that in (17 ) we can pass from 0(z) to an arbitrary odd theta f u n c t i o n - for odd characteristics - f = za + b belongs to the theta divisor, while the exponential factor vanishes upon differentiation. The bridge to the even characteristics is provided by the identity
w(x,y)+ -~+-~wi(x)wjgy) ~ l n
0[b](z)
4g+2g z=0 --
~G[b](x,y) 2
(21)
Each side is a symmetric modular invariant differential. The second term on the LHS is integrable (being of the form (l 3)). Using (9) and ( 18 ) we obtain for the fermion determinant det Pl/2[e] = 0 [ e ] ( 0 )
0[b](0)
K.
(22)
The variation of the unknown modular invariant K is 1
5 lnK=-
/"
[
2~z 2
~ Jq(x)~-~
1
b~ G [ b ] ( x , y ) 2
~
)
,=x "
(23)
For the toms K = const, as seen from (16). For an arbitrary genus the modular properties of the determinant can be determined, in spite of the unknown factor, with the help of the formula for the change of w(x, y) after a change in the homology basis W(X, y) = w(x, y) -- ½[ Wi(X)Wj(y) "l- Wi(y ) Wj(X)]~
In det(cz + d ) , ~'= (az + b)/(cz +d) .
(24)
Eq. (24) means that the fermion determinant is multiplied by det (cz +d) - ~in addition to the changes in the theta function. For the bosonic string integrand (24) means that the part coming from w(x, y) is a modular form of fixed weight, independent of the genus. However the modular analyses ofref. [ 3 ] show that the product of the even theta functions must have weight 12 - g for g = 2, 3. In order to solve this paradox additional powers of the product should arise either from K or from the zero mode part. In any case (21 ) shows that the product of all even theta constants appears naturally both in the fermion determinant and in the string integrand for a surface of arbitrary genus. The further study of K can not be done from (23) since S(p) is present in the expansion of the RHS. For this purpose we use the special case of the addition theorem:
O[b](2(y-x)) •
~
O[b](y-x)4.~ 0[b](0)
4
~
/]E(x,
y)4
1 04 In 0[b] OZiOZjO,7,kOZl
(O)wi(x)wj(X)Wk(y)wl(y)
(25)
--2
Let us develop both sides of this identity in a Laurent series with the notation _ 0 ' = _dO[bl (0), dy
42
C . - d" In 0 [ b ] ( 0 ) . dy"
(26)
Volume 189, number 1,2
PHYSICS LETTERS B
30 April 1987
The expression in brackets in (25) is denoted as A (zi). Let us discuss the one-loop case first. It is easy to show that the series coefficients vanish up to and including A " ( 0 ) , so the pole term on the LHS of (24) cancels. The quadratic term gives a formula for S(p): (27)
S ( p ) = - [AVl(O)/AIV(O)]dz(p) 2 .
It can be shown next that A w (0)
S(p)
= - (6c2
+c6/2c4)
=
c4/2 and 6A vI (0)
= 6 c 2 c 4 -{-
½C6 SO that (28)
dz(p) 2 ,
where c2 = 0 " / 0 ,
c4 = 0 I V / 0 - 3 0 " 2 / 0 2 ,
c6 = 0 v l / 0 - 150"0IV/02 +300"2/03 •
(29)
When inserted in (9) the first term of S(p) is of the form (13) and integrates to 0[b](0). Exploiting the fact that theta functions with even characteristics are even functions, we get for the second term: c6/c4= (In c4)" = (0/0T)ln c4, thus it is also integrable and finally In det .el/2[e] = 0 [ e ] ( 0 ) 0[b](0)-lc~-~/~2,
(30)
where [e] is the given, while [b] is an arbitrary even characteristic. Eq. (30) may be written also as (bl~I
~t-- I,3//
0[b](0))
lndetP,/z[e]=O[e](O)
~-- 1/36
~bC4)
(31)
,
and the comparison with (16) shows that I-[bc4= const., an identity we were unable to prove directly. In the case of an arbitrary genus several complications occur. Now the derivatives of the abelian differentials do not vanish and the RHS of (25) has a quadratic piece. For the same reason 0 " [b] (0) # 0. This does not alter c2 and c, but adds the term - 100 " 2/02 to c6. Furthermore 0~")(2z) ~ 2"O~n)(z). With all these modifications taken into account, the expression for the projective connection becomes
S(p)
= -6c2 -
(C2/20)[0(27")IV -- 1601v] - 6 X TM- 160 TM]
½e6 - 0 " 2 / 0 2 "~- (ll5!O)[O(2z)Vl -- 6 4 0 v I ] --
c4 + ( 1 / 1 2 0 ) [ 0 ( 2 z )
z=o
,
(32)
where 0 4 In 0
d 2
x=~1 OZiOT"JOgkO2!(O)~y2Wi(X) Wj(X) Wk(y ) wl(y) y-x _.
(33)
The integrable c2 term is still present, but it is hard to guess whether the second term is integrable. If we set w }")(x) = 0 it transforms into the second term in the one-loop case (28). Since the even characteristics in (32) is arbitrary, we can take the product over all of them. Then the c2 term integrates to the product in eq. (22), while the other term in (32) provides an explicit expression for 8 In K in terms of the even theta functions. Thus we have shown once again that the product of all even theta functions is present in all determinants of the string integrand for any genus. In order to reproduce the results of the modular analyses, the second term in (32) should supply the absent powers of 1-Ib0[b](0) (perhaps when combined with the zero mode contributions) for g = 2 or 3. This means that an identity, similar to the one on the torus, should exist. References [ 1 ] M. Green and J. Schwarz, Phys. Lett. B 149 (1984) 117; B 151 (1985) 21; D. Gross, J. Harvey, E. Martinec and R. Rohm, Nucl. Phys. B 256 (1985) 253; B 267 (1986) 75. [2] A. Polyakov, Phys. Lett. B 103 (1981) 207, 211;
43
Volume 189, number 1,2
PHYSICS LETTERS B
O. Alvarez, Nucl. Phys. B 216 (1983) 125; E. D'Hoker and D. Phong, Nucl. Phys. B 269 (1986) 205; B 278 (1986) 225; P. Nelson and G. Moore, Nucl. Phys. B 274 (1986) 509; G. Moore, P. Nelson and J. Polchinski, Phys. Lett. B 169 (1986) 47. [3] A. Belavin and V. Knizhnik, Phys. Lett. B 168 (1986) 201; Zh. Eksp. Teor. Fiz. 91 (1986) 364. [4] L. Alvarez-Gaum6, G, Moore and C. Vafa, Commun. Math. Phys. 106 (1986) 40. [5] Y. Manin, Pis'ma Zh. Eksp. Teor. Fiz. 43 (1986) 161; A. Beilinson and Y. Manin, Commun. Math. Phys. 107 (1986) 359; V. Knizhnik, Phys. Lett. B 180 (1986) 247; L. Alvarez-Gaum6, J.B. Bost, G. Moore, P. Nelson and C. Vafa, Phys. Lett. B 178 (1986) 41. [6] A. Belavin, V. Knizhnik, A. Morozov and A. Perelomov, Phys. Lett. B 177 (1986) 324; G. Moore, Phys. Lett. B 176 (1986) 369; A. Kato, Y. Matsuo and S. Odake, Phys. Lett. B 179 (1986) 241. [7] G. Moore, J. Harris, P. Nelson and I. Singer, Phys. Lett. B 178 (1986) 167; M. Chang and Z. Ran, preprint (1986). [8] M. Namazie and M. Sarmadi, Phys. Lett. B 177 (1986) 329. [9] H. Sonoda, Phys. Lett. B 178 (1986) 390. [ 10 ] J. Bost and P. Nelson, Harvard preprint HUTP-86/A044 (1986). [ 11 ] J. Fay, Theta functions on Riemann surfaces, Lecture Notes in Mathematics, Vol. 352 (Springer, Berlin, 1973). [ 12 ] D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B 271 (1986) 93; S. Giddings and E. Martinec, Nucl. Phys. B 278 (1986) 91; E. Martinec, Princeton University preprint (1986).
44
30 April 1987
PHYSICS LETTERS B
30 April 1987
O N T H E Q U A N T U M T O P O L O G Y OF STRINGS B.V. IVANOV Institute for Nuclear Research and Nuclear Energy, BoulevardLenin 72, Sofia 1784, Bulgaria Received 14 January 1987
The role of the projective connection and the fundamental.differential of the seond kind in the determinants of the string integrand is emphasized. A new integrableterms is found.
At present, the supersymmetric string in ten dimensions is the most serious candidate for a unified theory of nature, including quantum gravity [ 1 ]. The string measure in the covariant path-integral approach was analysed in ref. [2]. While an application of strings to hadron physics was sought, the accent was laid on the dependence of the determinants on the conformal factor of the metric. The idea was that the Liouville mode could lead to a consistent quantization in four dimensions without tachyons in the bosonic string. Nowadays strings are studied once again in their critical dimension, where they are conformal invariant, and the descent to four dimensions is achieved by compactification. The accent has been transferred to the Teichmiiller parameters and thus from quantum geometry to quantum topology. Recently, the importance of the complex structure of the moduli space was emphasized [ 3 ]. Although the determinants cannot be found explicitly by Pauli-Villars [ 3 ], or heat kernel [ 4 ] regularization, their second variation vanishes, which means that the vacuum amplitude of the bosonic string is given by the squared modulus of a (unknown) holomorphic function:
Zs = 5rJdyi ^ dyi IF(y) [ 2(det Im z) -13
(1)
Here the y~ are the 3 g - 3 complex moduli for genus g, while r is the period matrix. It is possible to express F ( y ) in tel-ms o f R i e m a n n theta functions, holomorphic differentials and Green functions [ 5 ]. Green functions are inevitable anyway when external legs are present, but their appearance in the measure seems redundant. The Green function is a more complex object than the determinant, in the point splitting regularization it defines the latter, while the opposite is impossible. After all, the determinant is a product of the eigenvalues, while the Green function contains also the eigenvectors at two different points. This redundancy is reflected in ref. [ 5 ] by the independence of the formulas on the arguments of the Green function. Furthermore, the study of the conformal factor dependence has teached us that although the relevant operators do possess zero modes, their determinants are divided by the matrix determinants of the zero modes, so that finally no such modes appear in the Liouville effective action. Thus we expect a measure expressed solely in theta functions, which are simpler objects. Since the z o can be taken as the moduli, the arguments of the theta functions are also redundant and one should stop at last at theta constants (the z o are more than necessary for g > 3, but this is a problem of the Siegel upper half-plane, not of the determinants). This is exactly the result of modular form analyses [ 3,6 ]. The necessary parabolic modular forms of weight 12 f o r g = 1 and 1 2 - g f o r g > 1 are unique f o r g = 1, 2, 3 and are given by the product of all even theta constants. However, their properties for g > 3 are poorly understood. Modular analyses of the superstring leads to the vanishing of the cosmological constant for all g [ 7 ]. 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
39
Volume 189, number 1,2
PHYSICSLETTERSB
30 April 1987
It is clear that some direct method for computation of determinants is necessary. In this paper we shall use the point-splitting method since the Green functions are known explicitly [ 8,9]. It was applied in ref. [ 8 ] to find the spin-structure dependence (first found in ref. [ 4 ]) and in ref. [ 9 ] to prove the bosonization formula of ref. [ 10]. We obtain another integrable term, which is present in all determinants and for every genus. In the following we will not pay any attention to the conformal factor, since it finally cancels. The relevant operators are P, = ~x and the corresponding Green functions G~,, _, +~)(x, y) satisfy the equation
P,,G¢,,, _,,+ i)(x, y) = ~ t _,) (x, y) - H o _,) (x, y ) ,
(2)
where H o _,)(x, y) is the projection operator onto the space of zero modes of weight (1 - n , 0). For n = 1/2 we have the fermion Green funtion G[e](x, y) which depends on some even characteristics [e]. There are no zero modes for the even sector, while the odd sector does not contribute to amplitudes with less than five legs. The variation of P 1/2 is - q ( x ) Ox with r/(x ) the Beltrami differential. Thu s: 8 In det P~/2 = T r ~Pl/2G[e](x, y) ly=x •
(3)
The Green function possesses a pole which serves as a regulator. We must saturate this pole by a Taylor series expansion of the other factors. The mathematics used below can be found in ref. [ 11 ]. Let us introduce the prime form
E(x,y) =
O[o](y-x) [H[o](x)H[o](y)]~/2,
H[o](x) =
(O)Wi(x).
(4,5)
Here [o] is any odd characteristics, the wi(x) are the abelian differentials and the argument of the theta function is y - x = f~ w, = z~. The prime form is antisymmetric and has a first-order zero for x = y. The fermion Green function is nothing but the Szego kernel:
1 O[e](y-x) G[e](x, y) = - ~ O[e](O)E(x, y) "
(6)
The following expansions are valid in the vicinity of some point p:
E(x, y)x//--~ d y = ( y - x ) - -~ ( y - x ) 3 S ( p ) + h.o.t.,
(7)
G[e] (x, y)v/--~ d y = - ( 1 / n ) { 1 / ( y - x ) + ( y - x ) [ - ~ S ( p ) + ½0" [ e ] ( 0 ) / 0 t e ] ( 0 ) ] } + h.o.t,
(8)
where the prime means differentiation with respect to y. S(p) is the projective connection. Inserting of (8) into (3) gives In det ell2
iI
~-~fr/(X)[ l / ( y - - x ) 2 --0" [e](y--x)/O[e](O) -- I S ( x ) ] ly=~.
(9)
The first term is divergent and should be dropped. The second term can be written as 8 In 0[e](0) after the use of the heat equation and the important identity (wi = v,dx in some local coordinates): - 2 i j dZx q(x)vi(x)vAx) = 8 % .
(10)
This term was found in refs. [4,8] and gives the dependence on the spin structure. Our main concern is the third term in (9). S(p) is defined as ( /Pi
s(.) = t J H.
h
,} +
]2(.)_ :[ H3
where the first brackets mean a schwarzian derivative, while 40
1(.).
Volume 189, number 1,2
PHYSICSLETTERSB
OnG Hn (f) = - (f) wi, (x)...wi, (x) , Ozi, ...Ozi,
30 April 1987
(12 )
a n d f i s a non-singular point belonging to the theta divisor. In order to integrate S(p) we must find a form for it of the type
~l(x)S(x)= ~l(x)wt(x)wj(x) o ~° In L(x, r ) .
(13)
Formula (11 ) contains abelian differentials in the denominators and it is hard to see how it can be integrated. On the torus, however, w(x) = d z ( x ) where z spans the universal cover of the torus, and
s(p)
=
- 2 O l n 0 ' [ l~](0)(dz) 2 ,
(14)
which when inserted in (9) and after using the Jacobi triple product identity 0[°](0)0[°/2](0)0[U](0)
= - ( 1 / ~ ) 0 ' [ I¢~1(0)
(15)
yields the well-known one-loop result
det P,/2~O[e](O)(~O[b](O) ) -'/3
(16)
The product goes over the three even theta functions. For higher genus one can consider another way, which makes use of the fundamental differential of the second kind: d2
d2
w(x, y)= -~-~yln E(x, y) d x d y = ~ y y l n O ( y - x - f ) dx dy .
(17)
It is independent of the non-singular point f a n d has a double pole:
w(x, y) = [ 1~(x-y) 2 + ~S(p) +h.o.t.] dx dy.
(18)
Before proceeding further, we shall show that the projective connection, or equivalently the fundamental differential, .are present in the variation of all determinants of the string integrand. The scalar Green function may be represented as [ 8,11 ] Y
G(x, y)= 2 ln[E(x, y)] + 1 f ( w - ~ ) ~ ( I m x
Y
~')/7 1
f(W--~}'j+Q(X)+~2(y),
(19,
x
where the last two factors are irrelevant as we shall see. The second factor is due to the zero modes. The Green function of P~ is given by the derivative of G(x, y) and yet another derivative comes from (3). In order to resolve the ambiguity which arguments exactly the derivatives act upon, we note that the point-splitting method is completely equivalent to the operator approach developed in ref. [ 12 ] and eq. (3) with i/deleted gives the effective energy-momentum tensor. For scalars it reads [ Tu~) ~ (OxXuOyX~) meaning that the derivatives are with respect to both arguments in (19). Then the last two terms vanish, the second term gives det Im z, while the first one becomes w(x, y). So the projective connection is present in the variation of the scalar determinant. The ghost Green function contains the x-derivative of the first term in (19) [ 8,12 ], and a different zero mode part. Since (Tgh) ~ (cXtSybyy+2O~c~byy) and the series expansion o r E ' (x, y)/E(x, y) shows that x- and yderivatives differ by a sign, we obtain once again the projective connection in the variation of the ghost determinant. The same conclusion is true for the superghost determinant since the expression for it given in ref. [ 8 ] 41
Volume 189,number 1,2
PHYSICSLETTERSB
30 April 1987
is built upon the fermion Green function. The presence of S(p) in all variations is analogous to the fact that the conformal factor integrates to the Liouville action for all determinants, the only difference being in the numerical coefficient. The integrand in (1) is given by the scalar and ghost determinants. Our analyses show that the zero modes supply the (det Im z) - 13part, while the projective connection should yield for g = 1, 2, 3 the product of all even theta functions, the latter being the unique modular form with the necessary properties. The fermion determinant reflects the features of the whole string integrand, but we need an expression for S(p) in terms of even theta functions in order to get close to the modular form results. The following equation: 0 N ( z [ z) =exp[ inaizoa j + 27tiai( zi-t-bi) ] O(z d- za+ bl z)
(20)
shows that in (17 ) we can pass from 0(z) to an arbitrary odd theta f u n c t i o n - for odd characteristics - f = za + b belongs to the theta divisor, while the exponential factor vanishes upon differentiation. The bridge to the even characteristics is provided by the identity
w(x,y)+ -~+-~wi(x)wjgy) ~ l n
0[b](z)
4g+2g z=0 --
~G[b](x,y) 2
(21)
Each side is a symmetric modular invariant differential. The second term on the LHS is integrable (being of the form (l 3)). Using (9) and ( 18 ) we obtain for the fermion determinant det Pl/2[e] = 0 [ e ] ( 0 )
0[b](0)
K.
(22)
The variation of the unknown modular invariant K is 1
5 lnK=-
/"
[
2~z 2
~ Jq(x)~-~
1
b~ G [ b ] ( x , y ) 2
~
)
,=x "
(23)
For the toms K = const, as seen from (16). For an arbitrary genus the modular properties of the determinant can be determined, in spite of the unknown factor, with the help of the formula for the change of w(x, y) after a change in the homology basis W(X, y) = w(x, y) -- ½[ Wi(X)Wj(y) "l- Wi(y ) Wj(X)]~
In det(cz + d ) , ~'= (az + b)/(cz +d) .
(24)
Eq. (24) means that the fermion determinant is multiplied by det (cz +d) - ~in addition to the changes in the theta function. For the bosonic string integrand (24) means that the part coming from w(x, y) is a modular form of fixed weight, independent of the genus. However the modular analyses ofref. [ 3 ] show that the product of the even theta functions must have weight 12 - g for g = 2, 3. In order to solve this paradox additional powers of the product should arise either from K or from the zero mode part. In any case (21 ) shows that the product of all even theta constants appears naturally both in the fermion determinant and in the string integrand for a surface of arbitrary genus. The further study of K can not be done from (23) since S(p) is present in the expansion of the RHS. For this purpose we use the special case of the addition theorem:
O[b](2(y-x)) •
~
O[b](y-x)4.~ 0[b](0)
4
~
/]E(x,
y)4
1 04 In 0[b] OZiOZjO,7,kOZl
(O)wi(x)wj(X)Wk(y)wl(y)
(25)
--2
Let us develop both sides of this identity in a Laurent series with the notation _ 0 ' = _dO[bl (0), dy
42
C . - d" In 0 [ b ] ( 0 ) . dy"
(26)
Volume 189, number 1,2
PHYSICS LETTERS B
30 April 1987
The expression in brackets in (25) is denoted as A (zi). Let us discuss the one-loop case first. It is easy to show that the series coefficients vanish up to and including A " ( 0 ) , so the pole term on the LHS of (24) cancels. The quadratic term gives a formula for S(p): (27)
S ( p ) = - [AVl(O)/AIV(O)]dz(p) 2 .
It can be shown next that A w (0)
S(p)
= - (6c2
+c6/2c4)
=
c4/2 and 6A vI (0)
= 6 c 2 c 4 -{-
½C6 SO that (28)
dz(p) 2 ,
where c2 = 0 " / 0 ,
c4 = 0 I V / 0 - 3 0 " 2 / 0 2 ,
c6 = 0 v l / 0 - 150"0IV/02 +300"2/03 •
(29)
When inserted in (9) the first term of S(p) is of the form (13) and integrates to 0[b](0). Exploiting the fact that theta functions with even characteristics are even functions, we get for the second term: c6/c4= (In c4)" = (0/0T)ln c4, thus it is also integrable and finally In det .el/2[e] = 0 [ e ] ( 0 ) 0[b](0)-lc~-~/~2,
(30)
where [e] is the given, while [b] is an arbitrary even characteristic. Eq. (30) may be written also as (bl~I
~t-- I,3//
0[b](0))
lndetP,/z[e]=O[e](O)
~-- 1/36
~bC4)
(31)
,
and the comparison with (16) shows that I-[bc4= const., an identity we were unable to prove directly. In the case of an arbitrary genus several complications occur. Now the derivatives of the abelian differentials do not vanish and the RHS of (25) has a quadratic piece. For the same reason 0 " [b] (0) # 0. This does not alter c2 and c, but adds the term - 100 " 2/02 to c6. Furthermore 0~")(2z) ~ 2"O~n)(z). With all these modifications taken into account, the expression for the projective connection becomes
S(p)
= -6c2 -
(C2/20)[0(27")IV -- 1601v] - 6 X TM- 160 TM]
½e6 - 0 " 2 / 0 2 "~- (ll5!O)[O(2z)Vl -- 6 4 0 v I ] --
c4 + ( 1 / 1 2 0 ) [ 0 ( 2 z )
z=o
,
(32)
where 0 4 In 0
d 2
x=~1 OZiOT"JOgkO2!(O)~y2Wi(X) Wj(X) Wk(y ) wl(y) y-x _.
(33)
The integrable c2 term is still present, but it is hard to guess whether the second term is integrable. If we set w }")(x) = 0 it transforms into the second term in the one-loop case (28). Since the even characteristics in (32) is arbitrary, we can take the product over all of them. Then the c2 term integrates to the product in eq. (22), while the other term in (32) provides an explicit expression for 8 In K in terms of the even theta functions. Thus we have shown once again that the product of all even theta functions is present in all determinants of the string integrand for any genus. In order to reproduce the results of the modular analyses, the second term in (32) should supply the absent powers of 1-Ib0[b](0) (perhaps when combined with the zero mode contributions) for g = 2 or 3. This means that an identity, similar to the one on the torus, should exist. References [ 1 ] M. Green and J. Schwarz, Phys. Lett. B 149 (1984) 117; B 151 (1985) 21; D. Gross, J. Harvey, E. Martinec and R. Rohm, Nucl. Phys. B 256 (1985) 253; B 267 (1986) 75. [2] A. Polyakov, Phys. Lett. B 103 (1981) 207, 211;
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Volume 189, number 1,2
PHYSICS LETTERS B
O. Alvarez, Nucl. Phys. B 216 (1983) 125; E. D'Hoker and D. Phong, Nucl. Phys. B 269 (1986) 205; B 278 (1986) 225; P. Nelson and G. Moore, Nucl. Phys. B 274 (1986) 509; G. Moore, P. Nelson and J. Polchinski, Phys. Lett. B 169 (1986) 47. [3] A. Belavin and V. Knizhnik, Phys. Lett. B 168 (1986) 201; Zh. Eksp. Teor. Fiz. 91 (1986) 364. [4] L. Alvarez-Gaum6, G, Moore and C. Vafa, Commun. Math. Phys. 106 (1986) 40. [5] Y. Manin, Pis'ma Zh. Eksp. Teor. Fiz. 43 (1986) 161; A. Beilinson and Y. Manin, Commun. Math. Phys. 107 (1986) 359; V. Knizhnik, Phys. Lett. B 180 (1986) 247; L. Alvarez-Gaum6, J.B. Bost, G. Moore, P. Nelson and C. Vafa, Phys. Lett. B 178 (1986) 41. [6] A. Belavin, V. Knizhnik, A. Morozov and A. Perelomov, Phys. Lett. B 177 (1986) 324; G. Moore, Phys. Lett. B 176 (1986) 369; A. Kato, Y. Matsuo and S. Odake, Phys. Lett. B 179 (1986) 241. [7] G. Moore, J. Harris, P. Nelson and I. Singer, Phys. Lett. B 178 (1986) 167; M. Chang and Z. Ran, preprint (1986). [8] M. Namazie and M. Sarmadi, Phys. Lett. B 177 (1986) 329. [9] H. Sonoda, Phys. Lett. B 178 (1986) 390. [ 10 ] J. Bost and P. Nelson, Harvard preprint HUTP-86/A044 (1986). [ 11 ] J. Fay, Theta functions on Riemann surfaces, Lecture Notes in Mathematics, Vol. 352 (Springer, Berlin, 1973). [ 12 ] D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B 271 (1986) 93; S. Giddings and E. Martinec, Nucl. Phys. B 278 (1986) 91; E. Martinec, Princeton University preprint (1986).
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30 April 1987