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On the vanishing of the vacuum energy for superstrings
Volume 183, number 3,4
PHYSICS LETTERS B
15 Januar)' 1987
ON THE VANISHING OF THE VACUUM ENERGY FOR SUPERSTRINGS A. MOROZOV and A. PERELOMOV Institute of Theoretical and Experimental Physics, 117 259 Moscow, USSR Received 20 October 1986
A hypothesis concerning the structure o f formulae for vacuum diagrams in the f'ttst-quantized superstring theory is proposed. The analytical measure in the integration over moduli space is proportional to the sum X e ee0 [e] 4 (ee = ±1) over spin structures on Riemann surfaces and vanishes because o f the Riemarm identities for 0-constants.
1. Recently, on the basis of the analytical anomaly cancellation theorem due to Belavin and Knizhnik, the structure of many-loop contributions to amplitudes of the closed oriented bosonic string theory has been understood in terms of the analytical geometry of moduli space (see refs. [1-3] for details). In the supersymmetric case the formulae for amplitudes are yet unknown. Considering stat-sums (p-loop vacuum diagrams), they may be represented as integrals over Mv - the space of moduli of Riemann surfaces of genus p - with analytical measures d/as~S) (y) and d/a~s)(y) (y stands for holomorphic coordinates on Mp): f [d/a~'s)(y)12 Mt' (det Im 7) 5
for the superstring ,
/'j d/z~Ps)(y)d / ~ (y) Mp (det Im T)5
for the heterotic string. (I)
with a certain measure. The supermoduli space may be obtained, for example, as a factor over the modular group Sp(p, Z) of a direct product of 2P-1 (2P + 1) copies of Teichmiiller space Mp times (2p - 2)-dimensional flat space of odd moduli * ~ . Teichmiiller space Mp is the space of moduli of Riemann surfaces with marked systems of basic cuts. Mp is a factor of Mp over the modular group. The 2P- 1 (2P + 1) copies of Mp differ by even spinor structures. The measure on supermoduli space is defined by the ratio of superdeterrninants of the operators ~, acting on the superflelds ~(g) and ~(~) (ten-dimensional coordinates of string + fermions and superdiada). Expressions like (1) arise after the integration over odd moduli is performed. Though the supermoduff space itself is connected (we consider only even characteristics), such an integration gives rise to a discrete sum over 2P -1 (2P + 1) copies of ordinary moduli space Mp : d/~Ps)(y) ~ ~ c e (y) (dete~l/2)5. (y). e det'e ~3/2
(2)
Provided these theories really possess ten-dimensional supersymmetry on the quantum level, their predictions for vacuum loops should Vanish; more exactly, d#~PS)(y) -- 0. Further on a hypothesis about the structure of d/a~vb)(y) is proposed, which guarantees the validity of this identity.
We present only contributions, depending on the 0characteristic e-boundary conditions for half-integer differentials. The weights CeO' ) in eq. (2), appearing after integration over odd moduff, are some functions
2. The statistical sum for the superstring may be represented as an integral over supermoduli space
,1 We arc indebted to A.S. Schwarz for ascertaining this formulation.
296
Volume 183,number3,4
PHYSICSLETTERSB
on Mp. They may be determined from the dependence of the superdeterminant on odd moduli, or, differently, from modular properties of d/l~vS)(y). We shall use a somewhat different reasoning for their specification in section 4. Note, that the weights ce (y), entering the expressions for amplitudes, depend on the amplitude under consideration. 3. The dete ~n+ I/2 dependence on a 0-characteristic e is essentially the same as that of the corresponding 0-constant 0 [e] : dete ~n+ 1/2 ~ 0 [e] 0,)(0 [e] depends on the period matrix T, which is an analytical function of y: see refs. [1-4] about this proportionality and refs. [1-3,5] for the def'mition and properties of 0-constants.) However, this exhausts the dependence on e only forn = 0: dete ~1/2 "~ 0 [e] (y)
(3)
(more exactly, det ~01/2dete ~l/2(Y) = 0 [el (y) [4]). Iff there are holomorphic ~-differentials with appropriate boundary conditions on a Riemann surface, 0 [el(y) vanishes and the number of differentials equals the order of 0 [e] (y). Concerning ~-differentials, dete ff3/2(y) ~ {0 Iel/¢e )(Y)
If on a Riemann surface (i.e. for some y) 0 [e] Cv) = 0, . . . . . 1 . this implies the exastence of holomorphic ~-differenrials with boundary conditions, specified by e, ¢~1 --- g/k, their number k being the same as the order of zero of 0 [e] (y) (see eq. (3)). Then among holo. morphic ~-differentials there are ~k1v2 -.- g'k v2, and each of them has double zeroes at all the points Pu, so that det' [~'a(Pz) ~'~(Pu)] has a kth order zero. Thus the RI-15 ot eq. (4) never vanishes. Conversely, if the determinant has a kth order zero, then some k holomorphic ~-differentials ~'c~ have double zeroes at all the points Pu" From these k~"s k holomorphic . ~-differentaals 1 . . ~0 -_ ~"v_ 2 can be constructed. In turn, their existence according to eq. (3) implies the kth order zero of 0 [e](y). Recently Knizhnik [6] proposed a proof of identities like (3), (4), using conformal invariance and its anomalies. These identities are cited in eqs. (3), (4) in brackets, note, that in order to reproduce anomalies the contributions which are independent of characteristics should be also included. For instance, an anomaly in the product dett/2ff0det0ff3/2, calculated with the special metric Ip41 = O, iO,/o~i~/, is exactly reproduced by the dependence of the RHS of eq. (4) on the Pu. (the zeroes of the metric).
(4)
(more exactly, det 71/2 dete J3/2 (Y) = (0 [e]/¢e }(y). [4]). On the left-hand side det' ~ 0 for a n y y , so that function ee (Y) should cancel all zeroes of 0 [e] (y), and should be non-vanishing when 0 [e] (y) ~ 0. On the other hand, ~e (Y) should be a single-valued func. tion of boundary conditions, contrary to 0 [e], which acquire phase factors under modular transformations, leaving e invariant. For ¢e 0') the following expression may be substituted: ee = det [~a(P1) .-. ~Jo~(Pp-1)~(P1)... ~ ( P p - 1)] • Here ~1 -'- ~2p-2 are holomorphic ~-differentials on the Riemann surface with boundary conditions, corresponding to characteristic e;P 1 ...Pp_l axe positions of double zeroes of the holomorphic Prime differential 122 =O,icOi (the coi are canonically normalized holomorphic one-differentials: fat~ j = 8ij , fb t o~j = T//; the 0, i are derivatives of some odd 0function). In local coordinates ~ in the vicinity ofP u ~"= [~'(Pta) + ~"(P/J)~ + O(~2)] d~3/2 •
15 January 1987
4. From eqs. (2)-(4) it follows that
d4/s) (Y) "" ~ ce(y) ¢)e(y) 0 [e]4(y). e
Our hypothesis is that
Ce(Y) ~ ed¢~ 0,),
(s)
ee = -+i being some phase factors (see below). Of course, in eq. (5),just as in eq. (1), there are factors, depending on y, but not on the 0-characteristic e. If this hypothesis is valid, the vanishing of d/~(~) (y) is simply due to the Riemann identity
~eeOlel4(y) = 0 ,
(6)
e
and does not depend on anything else, in particular, on the choice of the metric on the Riemann surface. Otherwise an identical relation between 0-constants and values of holomorphic differentials at points Pu would exist, which seems extremely doubtful. It is worth mentioning that our reasoning is based exclusively on properties of systems of zeroes (divi297
Volume 183, number 3,4
PHYSICS LETTERS B
sors) of 0-constants and this is more simple than a possible approach to the definition of c e 0') from modular properties of d4~) 0'). Note that the modular properties of values of holomorphic differentials are rather complicated. Even in the bosonic case, only formulae for p ~<4 are yet known from modular properties [2], while those following from study of divisors are derived for any p [3] (however, they need some, probably, extra information about the parametrization of Riemann surfaces). 5. Let us prove now the formula (6). The thing is that for p >~ 2 there are many independent Riemann identities interrelating the fourth powers of even 0constants. (The total number of these relations is (4P - 1)/3, so that only (2P + 1)(2P- 1 + 1)/3 of the 2P -1 (2P + 1) 0 [e] 4 are linearly independent.) Identity (6), required for the vanishing of d p ~ ) , should certainly include all the even 0-constants on an equal footing, i.e. all the coefficients ee should be purely phase factors: [ee I = 1. Unfortunately, there are no a priori reasons for such an identity to exist, and it should be specially derived. In fact, if the 0characteristic e is parametrized as follows: [e] = [61...%1 el...e p j, then
(__)61 "l'e, 0 [e]40, ') ~ 0 .
15 January 1987
period matrices on Riemann surfaces of genus p, see refs. [1-3] .) To derive eq. (7) one should substitute four squares ofeq. (8) in it, with [61 6 ] = 10 6], 06 16 lli el.~ tOg [1 E ], [0 E ] and [1 e]" After some stmple algebra one obtains identical zero. Relation (8) itself follows directly from the definition
#[6e12(T)= m E~3Z p exp(irrt(m + 6 / 2 ) T ( m + (m
+6/2)
+a/2)t ] }
X B
exp{ilr[(n+6/2)T(n+6/2)
n~Z p
+(n +6/2) t 1 } , if one sums over new variables u i = } (rn i + ni) and vt = ~ (mi - n/), taking simultaneously integer or half-integer values. It is enough to apply the following summation rule:
U1
+U 1 , oIEZq'~-
) " "-
B
,
Up,upEZ
(7)
Up,
e
All other identities are modular transforms of this one (plus linear combinations). In practice, the transformations from factor group Sp(p, Z)/Sp(p, 2 Z) are enough, since Riemann identities are invariant under the transformations, specified by symplectic matrices with even elements. Identity (7) is a trivial consequence of a well known relation [5]
0I|l
m =
o
o
= (_)t6 :=pj (-)=¢8 ~] (27') 0 [=+06] (279,
(8)
which expresses0 with an arbitrary characteristic through thetas with characteristics of the form [60] only, but depending on the doubled parameter 2T. (T is a symmetric p X p matrix with positively deftnite imaginary part - the element of Siegel space. T's entering expressions for string amplitudes are 298
--
"2?, eta{0, 1 } v
u,v~zP+~
References
[1] A. Belavinand V. Knizhnik, Phys. Lett. B 168 (1986) 201; Landau Institute preprint No. 32/7 (1986). [2] A. Belavinet al., Pis'ma Zh. Eksp. Teor. Fiz. 43 (1986) 319;preprint ITEP No. 59 (1986). [3] Yu.I. Martin, Pis'ma Zh. Eksp. Teor. Fiz. 43 (1986) 161; A. Morozov,preprint ITEP No. 88 (1986); Analytical anomaly and heterotic string in the formalismof continual integration, Phys. Lett. B 184 (1987), to be published. [4 ] L. Alvazez-Gaum~, G. Moore and C. Vafa, Harvard preprint HUTP-96/A017 (1986). [5 ] D. Humford, Tara lectures on theta (Birkhauser, Basel, (1983); J.D. Fay, Theta functions on Riernarm surfaces, Lecture Notes in Mathematics, VoL 352 (Springer, Berlin, 1973); H. Clemens, A scrapbook of complex curve theory (Plenum, New York, 1980). [6] V. Knizlmik, Europhys. Lett. 4 (1986) 27.
PHYSICS LETTERS B
15 Januar)' 1987
ON THE VANISHING OF THE VACUUM ENERGY FOR SUPERSTRINGS A. MOROZOV and A. PERELOMOV Institute of Theoretical and Experimental Physics, 117 259 Moscow, USSR Received 20 October 1986
A hypothesis concerning the structure o f formulae for vacuum diagrams in the f'ttst-quantized superstring theory is proposed. The analytical measure in the integration over moduli space is proportional to the sum X e ee0 [e] 4 (ee = ±1) over spin structures on Riemann surfaces and vanishes because o f the Riemarm identities for 0-constants.
1. Recently, on the basis of the analytical anomaly cancellation theorem due to Belavin and Knizhnik, the structure of many-loop contributions to amplitudes of the closed oriented bosonic string theory has been understood in terms of the analytical geometry of moduli space (see refs. [1-3] for details). In the supersymmetric case the formulae for amplitudes are yet unknown. Considering stat-sums (p-loop vacuum diagrams), they may be represented as integrals over Mv - the space of moduli of Riemann surfaces of genus p - with analytical measures d/as~S) (y) and d/a~s)(y) (y stands for holomorphic coordinates on Mp): f [d/a~'s)(y)12 Mt' (det Im 7) 5
for the superstring ,
/'j d/z~Ps)(y)d / ~ (y) Mp (det Im T)5
for the heterotic string. (I)
with a certain measure. The supermoduli space may be obtained, for example, as a factor over the modular group Sp(p, Z) of a direct product of 2P-1 (2P + 1) copies of Teichmiiller space Mp times (2p - 2)-dimensional flat space of odd moduli * ~ . Teichmiiller space Mp is the space of moduli of Riemann surfaces with marked systems of basic cuts. Mp is a factor of Mp over the modular group. The 2P- 1 (2P + 1) copies of Mp differ by even spinor structures. The measure on supermoduli space is defined by the ratio of superdeterrninants of the operators ~, acting on the superflelds ~(g) and ~(~) (ten-dimensional coordinates of string + fermions and superdiada). Expressions like (1) arise after the integration over odd moduli is performed. Though the supermoduff space itself is connected (we consider only even characteristics), such an integration gives rise to a discrete sum over 2P -1 (2P + 1) copies of ordinary moduli space Mp : d/~Ps)(y) ~ ~ c e (y) (dete~l/2)5. (y). e det'e ~3/2
(2)
Provided these theories really possess ten-dimensional supersymmetry on the quantum level, their predictions for vacuum loops should Vanish; more exactly, d#~PS)(y) -- 0. Further on a hypothesis about the structure of d/a~vb)(y) is proposed, which guarantees the validity of this identity.
We present only contributions, depending on the 0characteristic e-boundary conditions for half-integer differentials. The weights CeO' ) in eq. (2), appearing after integration over odd moduff, are some functions
2. The statistical sum for the superstring may be represented as an integral over supermoduli space
,1 We arc indebted to A.S. Schwarz for ascertaining this formulation.
296
Volume 183,number3,4
PHYSICSLETTERSB
on Mp. They may be determined from the dependence of the superdeterminant on odd moduli, or, differently, from modular properties of d/l~vS)(y). We shall use a somewhat different reasoning for their specification in section 4. Note, that the weights ce (y), entering the expressions for amplitudes, depend on the amplitude under consideration. 3. The dete ~n+ I/2 dependence on a 0-characteristic e is essentially the same as that of the corresponding 0-constant 0 [e] : dete ~n+ 1/2 ~ 0 [e] 0,)(0 [e] depends on the period matrix T, which is an analytical function of y: see refs. [1-4] about this proportionality and refs. [1-3,5] for the def'mition and properties of 0-constants.) However, this exhausts the dependence on e only forn = 0: dete ~1/2 "~ 0 [e] (y)
(3)
(more exactly, det ~01/2dete ~l/2(Y) = 0 [el (y) [4]). Iff there are holomorphic ~-differentials with appropriate boundary conditions on a Riemann surface, 0 [el(y) vanishes and the number of differentials equals the order of 0 [e] (y). Concerning ~-differentials, dete ff3/2(y) ~ {0 Iel/¢e )(Y)
If on a Riemann surface (i.e. for some y) 0 [e] Cv) = 0, . . . . . 1 . this implies the exastence of holomorphic ~-differenrials with boundary conditions, specified by e, ¢~1 --- g/k, their number k being the same as the order of zero of 0 [e] (y) (see eq. (3)). Then among holo. morphic ~-differentials there are ~k1v2 -.- g'k v2, and each of them has double zeroes at all the points Pu, so that det' [~'a(Pz) ~'~(Pu)] has a kth order zero. Thus the RI-15 ot eq. (4) never vanishes. Conversely, if the determinant has a kth order zero, then some k holomorphic ~-differentials ~'c~ have double zeroes at all the points Pu" From these k~"s k holomorphic . ~-differentaals 1 . . ~0 -_ ~"v_ 2 can be constructed. In turn, their existence according to eq. (3) implies the kth order zero of 0 [e](y). Recently Knizhnik [6] proposed a proof of identities like (3), (4), using conformal invariance and its anomalies. These identities are cited in eqs. (3), (4) in brackets, note, that in order to reproduce anomalies the contributions which are independent of characteristics should be also included. For instance, an anomaly in the product dett/2ff0det0ff3/2, calculated with the special metric Ip41 = O, iO,/o~i~/, is exactly reproduced by the dependence of the RHS of eq. (4) on the Pu. (the zeroes of the metric).
(4)
(more exactly, det 71/2 dete J3/2 (Y) = (0 [e]/¢e }(y). [4]). On the left-hand side det' ~ 0 for a n y y , so that function ee (Y) should cancel all zeroes of 0 [e] (y), and should be non-vanishing when 0 [e] (y) ~ 0. On the other hand, ~e (Y) should be a single-valued func. tion of boundary conditions, contrary to 0 [e], which acquire phase factors under modular transformations, leaving e invariant. For ¢e 0') the following expression may be substituted: ee = det [~a(P1) .-. ~Jo~(Pp-1)~(P1)... ~ ( P p - 1)] • Here ~1 -'- ~2p-2 are holomorphic ~-differentials on the Riemann surface with boundary conditions, corresponding to characteristic e;P 1 ...Pp_l axe positions of double zeroes of the holomorphic Prime differential 122 =O,icOi (the coi are canonically normalized holomorphic one-differentials: fat~ j = 8ij , fb t o~j = T//; the 0, i are derivatives of some odd 0function). In local coordinates ~ in the vicinity ofP u ~"= [~'(Pta) + ~"(P/J)~ + O(~2)] d~3/2 •
15 January 1987
4. From eqs. (2)-(4) it follows that
d4/s) (Y) "" ~ ce(y) ¢)e(y) 0 [e]4(y). e
Our hypothesis is that
Ce(Y) ~ ed¢~ 0,),
(s)
ee = -+i being some phase factors (see below). Of course, in eq. (5),just as in eq. (1), there are factors, depending on y, but not on the 0-characteristic e. If this hypothesis is valid, the vanishing of d/~(~) (y) is simply due to the Riemann identity
~eeOlel4(y) = 0 ,
(6)
e
and does not depend on anything else, in particular, on the choice of the metric on the Riemann surface. Otherwise an identical relation between 0-constants and values of holomorphic differentials at points Pu would exist, which seems extremely doubtful. It is worth mentioning that our reasoning is based exclusively on properties of systems of zeroes (divi297
Volume 183, number 3,4
PHYSICS LETTERS B
sors) of 0-constants and this is more simple than a possible approach to the definition of c e 0') from modular properties of d4~) 0'). Note that the modular properties of values of holomorphic differentials are rather complicated. Even in the bosonic case, only formulae for p ~<4 are yet known from modular properties [2], while those following from study of divisors are derived for any p [3] (however, they need some, probably, extra information about the parametrization of Riemann surfaces). 5. Let us prove now the formula (6). The thing is that for p >~ 2 there are many independent Riemann identities interrelating the fourth powers of even 0constants. (The total number of these relations is (4P - 1)/3, so that only (2P + 1)(2P- 1 + 1)/3 of the 2P -1 (2P + 1) 0 [e] 4 are linearly independent.) Identity (6), required for the vanishing of d p ~ ) , should certainly include all the even 0-constants on an equal footing, i.e. all the coefficients ee should be purely phase factors: [ee I = 1. Unfortunately, there are no a priori reasons for such an identity to exist, and it should be specially derived. In fact, if the 0characteristic e is parametrized as follows: [e] = [61...%1 el...e p j, then
(__)61 "l'e, 0 [e]40, ') ~ 0 .
15 January 1987
period matrices on Riemann surfaces of genus p, see refs. [1-3] .) To derive eq. (7) one should substitute four squares ofeq. (8) in it, with [61 6 ] = 10 6], 06 16 lli el.~ tOg [1 E ], [0 E ] and [1 e]" After some stmple algebra one obtains identical zero. Relation (8) itself follows directly from the definition
#[6e12(T)= m E~3Z p exp(irrt(m + 6 / 2 ) T ( m + (m
+6/2)
+a/2)t ] }
X B
exp{ilr[(n+6/2)T(n+6/2)
n~Z p
+(n +6/2) t 1 } , if one sums over new variables u i = } (rn i + ni) and vt = ~ (mi - n/), taking simultaneously integer or half-integer values. It is enough to apply the following summation rule:
U1
+U 1 , oIEZq'~-
) " "-
B
,
Up,upEZ
(7)
Up,
e
All other identities are modular transforms of this one (plus linear combinations). In practice, the transformations from factor group Sp(p, Z)/Sp(p, 2 Z) are enough, since Riemann identities are invariant under the transformations, specified by symplectic matrices with even elements. Identity (7) is a trivial consequence of a well known relation [5]
0I|l
m =
o
o
= (_)t6 :=pj (-)=¢8 ~] (27') 0 [=+06] (279,
(8)
which expresses0 with an arbitrary characteristic through thetas with characteristics of the form [60] only, but depending on the doubled parameter 2T. (T is a symmetric p X p matrix with positively deftnite imaginary part - the element of Siegel space. T's entering expressions for string amplitudes are 298
--
"2?, eta{0, 1 } v
u,v~zP+~
References
[1] A. Belavinand V. Knizhnik, Phys. Lett. B 168 (1986) 201; Landau Institute preprint No. 32/7 (1986). [2] A. Belavinet al., Pis'ma Zh. Eksp. Teor. Fiz. 43 (1986) 319;preprint ITEP No. 59 (1986). [3] Yu.I. Martin, Pis'ma Zh. Eksp. Teor. Fiz. 43 (1986) 161; A. Morozov,preprint ITEP No. 88 (1986); Analytical anomaly and heterotic string in the formalismof continual integration, Phys. Lett. B 184 (1987), to be published. [4 ] L. Alvazez-Gaum~, G. Moore and C. Vafa, Harvard preprint HUTP-96/A017 (1986). [5 ] D. Humford, Tara lectures on theta (Birkhauser, Basel, (1983); J.D. Fay, Theta functions on Riernarm surfaces, Lecture Notes in Mathematics, VoL 352 (Springer, Berlin, 1973); H. Clemens, A scrapbook of complex curve theory (Plenum, New York, 1980). [6] V. Knizlmik, Europhys. Lett. 4 (1986) 27.