— DRAFT — IASSNS-HEP-97/72

Lecture 8

V. Moduli Dependence of Determinants and Green Functions

Eric D’Hoker

In §III and §IV, we have reduced the calculation of string transition amplitudes to integrals of combinations involving certain determinants and Green functions over the finite-dimensional moduli spaces ℳ_{h}. In this section, we complete the analysis by producing explicit formulas for these determinants and Green functions. We shall begin by collecting results on surfaces with constant curvature metrics. Then, we proceed and exploit the complex structure of ℳ_{h} to obtain explicit expressions in terms of Jacobi theta functions. (Derivations of the results stated in this section will not be given here; they may be found in E. D’Hoker, D. H. Phong, Rev. Mod. Phys. Vol. 60 (1988), p. 917–1065, and references therein.)

A) Worldsheets with constant curvature metric (h ≥ 2)

Let Σ be a compact surface of genus h ≥ 2. By the uniformization theorem, the simply connected cover of Σ is the upper half plane H, and Σ = H/Γ, where Γ = π_{1}(Σ) is the fundamental group of Σ. For any metric g on Σ, there is a unique Weyl transformation σ such that g = e^{2σ}ĝ with ĝ of constant curvature Rĝ = −1. The metric ĝ lifts to the Poincaré metric on H and Γ is a discrete subgroup of SL(2, ℝ) acting on H, with compact quotient Σ = H/Γ. Moduli space may now be identified as follows:

ℳ_{h} = {ĝ, Rĝ = −1} /Diff(Σ) .  (5.1)

We now have the following results for determinants and for the measure on ℳ_{h} arising in string amplitudes:

•  Det Δ(n)^{+} = S(n + 1) ; Det′Δ_{(0)} = S′(1)  (5.2)
•  ∏_{j} dm_{j}dm̄_{j}| det(μ_{j}, ϕ_{k})|^{2} det(ϕ_{j}, ϕ_{k})^{−1} = d(Weil − Peterson) .  (5.3)

V.1 Here S(s) is the Selberg zeta function, defined by

S(s) ≡ ∏_{γ primitive} ∏_{k=1}^{∞}(1 − e^{−(s+k)ℓ_{γ}})  (5.4)

for ℓ_{γ} the lengths of the simple closed (i.e. primitive) geodesics γ on Σ = H/Γ.

B) Holomorphicity in Moduli

The most powerful results on Green functions and determinants are obtained by exploiting the fact that ℳ_{h} is a complex orbifold, and that Green functions and determinants primarily involve complex analytic dependence on ℳ_{h}.

To examine this complex dependence, we first work locally on ℳ_{h}, in a neighborhood of a point m^{0} ∈ ℳ_{h}, represented by a metric g^{0}. We now parametrize the neighborhood U_{α} of m^{0} and thus of g^{0} by Beltrami differentials μ = μz ; z̄, which provide complex coordinates on ℳ_{h}, as established in §III. We have a general metric g in the neighborhood of g^{0}:

g = 2g_{zz̄}|dz + μdz̄|^{2}  where  g^{0} = 2g_{zz̄}|dz|^{2} .  (5.6)

We retain the Weyl factor g_{zz̄} for now, since we shall also study the interplay of Weyl dependence and moduli dependence.

The dependence of the functional determinant combinations Z_{(n)}^{±}(g) on the complex coordinates of ℳ_{h} is simple, and may be evaluated explicitly. This result is the content of

V.2 the so-called Belavin-Knizhnik theorem; or holomorphic anomaly theorem:

δ_{μ}δμ̄ ln Z_{(n)}^{±}(g) = −\frac{c_{(n)}^{±}}{12π} ∫_{Σ} dμ_{g} (∇_{z}δμ∇z̄δμ̄ − R_{g}δμδμ̄)  (5.7)

where the coefficients c_{(n)}^{±} = 6n^{2} ±6n+1 are the same as those entering the Weyl anomaly.

It follows immediately from the Belavin-Knizhnik theorem that Weyl invariant combinations of Z_{(n)}^{±}(g)’s are also especially simple in their dependence on moduli. Indeed, the δ_{μ}δμ̄ variation of any Weyl invariant combination vanishes, so that such a combination is a sum of a holomorphic and an anti-holomorphic part. We have, e.g.

Z_{(n)}^{±}(g) Z_{(0)}(g)^{−c_{(n)}^{±}} = |𝒵_{(n)}^{±}(m_{j})|^{2} .  (5.8)

where 𝒵_{(n)}^{±}(m_{j}) is holomorphic inside ℳ_{h}. In particular, the combination entering the integrand of the bosonic string amplitudes is precisely of this form, for superscript − and n = −1:

Z_{(−1)}^{−}(g) Z_{(0)}(g)^{−13} = |𝒵_{(−1)}^{−}(m_{j})|^{2} .  (5.9)

We shall make use of this powerful result in string theory in §C.

The BK Theorem may be understood geometrically in the language of determinant line bundles over ℳ_{h}, where the right hand side of BK — the holomorphic anomaly — arises as a non-zero curvature of the determinant line bundles Det(∇_{(n)}^{z}) and Det(∇_{z}^{(n)}) with certain hermitian metrics, the so-called Quillen metrics. The vanishing of the BK anomaly 𝒵_{(n)}^{±} then indicates that the corresponding line bundles

Det∇_{(n)}^{z} ⊗ (Det∇_{(1)}^{z})^{−⊗c_{(n)}^{−}} ,  5.10

are flat. In string theory, the case is n = 2, and this case is related to the flatness of 𝒦 ⊗ λ^{−13}, where 𝒦 is the canonical bundle of ℳ_{h} and λ is the Hodge bundle.

We shall not prove the BK Theorem here (it can be proven using heat-kernel methods, just as in the derivation of the Weyl anomaly), but only indicate the origin of the holomorphic anomaly. The key observation is that the covariant derivatives ∇_{(n)}^{z} depend only

V.3 on δμ, but not on δμ̄ to first order in δμ and δμ̄; thus ∇_{(n)}^{z} is holomorphic in δμ to first order.

\{ \begin{matrix}δ_{μ}∇_{(n)}^{z} = δμ∇z̄ ; (n) + n(∇z̄δμ) \\ δμ̄∇_{(n)}^{z} = 0 . \end{matrix}  (5.11)

Functional determinants of Laplace operators would then naively be expected to be products of a holomorphic times anti-holomorphic factor:

DetΔ_{(n)}^{−} ∼^{?} Det∇_{z}^{(n−1)}Det∇_{(n)}^{z} ,  (5.12)

with Det∇_{(n)}^{z} holomorphic on ℳ_{h} and Det∇_{z}^{(n−1)} anti-holomorphic. Actually, determinants must be properly defined (e.g. using ζ-function techniques, as in §III.), in a fashion that preserves their Diff(Σ) invariance. When this analysis is carried out, anomalies, such as the Weyl anomaly, may appear. Here, the above holomorphic factorization of determinants is also beset by an anomaly. In fact, these anomalies are intimately related, as can be seen from the argument below. As was explained in §III, the variations of μ, μ̄ and Weyl rescalings are independent. Hence, using the Weyl anomaly of Z_{(n)}^{±}(g):

\begin{aligned}δ_{σ}δ_{μ}δμ̄ ln Z_{(n)}^{±}(g) & = δ_{μ}δμ̄δ_{σ} ln Z_{(n)}^{±}(g) \\ & = −\frac{c_{(n)}^{±}}{6π} δ_{μ}δμ̄ ∫_{Σ} dμ_{g} R_{g} . \end{aligned}  (5.13)

But, the variation of the Gaussian curvature is easily computed

δ_{μ}δμ̄(dμ_{g} R_{g}) = (∇_{z}δμ∇z̄δμ̄)dμ_{g} ≠ 0 .  (5.14)

Thus, clearly, the holomorphic anomaly δ_{μ}δμ̄ ln Z_{(n)}^{±}(g) cannot cancel unless also the Weyl anomaly cancels.

C) The Chiral Splitting Theorem

In critical string theory, we also have strong results on the holomorphicity properties of the correlation functions for vertex operators of physical on-shell states, given by the Chiral Splitting Theorem. We begin by assembling the data we need to formulate this theorem.

V.4

Let V_{i} = ∫_{Σ} d^{2}z_{i} W_{i}(z_{i}, z̄_{i}) be vertex operators of physical on-shell states whose polarization tensors are factorized in left and right movers. Let A_{I}, B_{I}, I = 1, …, h be a canonical basis of homology 1-cycles on Σ, with

#(A_{I}, A_{J}) = #(B_{I}, B_{J}) = 0  #(A_{I}, B_{J}) = δ_{IJ} .  (5.15)

We define internal loop momenta p_{I} ∈ ℝ^{26} associated with each A_{I} cycle. Because the A_{I} cycles have trivial interaction numbers, these internal loop momenta are independent.

The correlation function of unintegrated vertex operators W_{i}(z_{i}, z̄_{i}) at fixed internal loop momenta p_{I} are defined by

\begin{aligned}⟨W_{1} … W_{N}⟩_{g} (p_{I}) ≡ ∫_{Map(Σ,M)} & Dx W_{1}(z_{1}, z̄_{1}) … W_{N}(z_{N}, z̄_{N}) \\ & ∏_{I=1}^{h} δ (p_{I}^{μ} − ∫_{A_{I}} \frac{dz}{2πi} ∂_{z}x^{μ}) e^{−S[x,g]} \end{aligned}  (5.16)

The Chiral Splitting Theorem states that

⟨W_{1} … W_{N}⟩_{g} (p_{I}) = δ(k)Z_{0}(g)^{−13}ℱℱ̃  (5.17)

where ℱ(z_{i}, m_{j}; p_{I}, k_{i}) is a complex analytic function of z_{i} and m_{j}, and ℱ̃ is its complex conjugate

ℱ̃(z̄_{i}, m̄_{j}; p_{I}, k_{i}) = ℱ(z_{i}, m_{j}; p_{I}, k_{i})^{∗} .  (5.18)

Here, we have suppressed the dependence of W and ℱ on polarization tensors occurring in the construction of W. Also, the function ℱ depends upon the choice of curves representing

V.5 the A_{I} boundary cycles, and this dependence has been subsumed in the definition of the internal momenta p_{I}.

An outline of the proof of this Theorem will be given in §E. Let us simply remark here that the proper conformal field theory interpretation of the integration over loop momenta is the following. At fixed values of the internal loop momenta, p_{I} through all A_{I} cycles, only a single conformal family traverses each A_{I}-cycle. The contributions from left and right movers are factorized, so that the entire amplitude — at fixed momenta p_{I} — is factorized.

Mathematically, we have gained considerably through the Belavin-Knizhnik and Chiral Splitting Theorems, since we may now apply techniques of complex function theory, thetafunctions and algebraic geometry to study amplitudes.

Physically, we have uncovered a method for separating the contributions of left and right movers in the integrand of transition amplitudes, via the holomorphic versus antiholomorphic dependence on ℳ_{h} and Σ. (This will be crucial in the study of superstrings.)

An immediate result from the combination of the Belavin-Knizhnik and Chiral Splitting Theorems is that the full transition amplitude at genus h is an integral over a product of a holomorphic times anti-holomorphic function.

A_{h} = δ(k) ∫_{ℝ^{26h}} ∏_{I} d p_{I}^{26} ∫_{ℳ_{h}} ∏_{j} d m_{j}m̄_{j} ∫_{Σ} d^{2}z_{1} … ∫_{Σ} d^{2}z_{N} C C̃ ,  (5.19)

where

C(z_{i}, m_{j}; p_{I}, k_{i}) = det(μ_{j}, ϕ_{k})𝒵_{(−1)}^{−}(m_{j})ℱ(z_{i}, m_{j}; p, k)  (5.20)

and

C̃(z̄_{i}, m̄_{j}; p_{I}, k_{i}) = C(z_{i}, m_{j}; p_{I}, k_{i})^{∗} .  (5.21)

(Of course, for h ≥ 1, these expressions are only formal, since the integration ultimately diverges. The generalization to the case of the superstring on the other hand will be welldefined.)

V.6 D) Holomorphic and Meromorphic Differentials (a brief review of basics)

We begin by recalling some basic tools of complex fucntion theory on Riemann surfaces. The space of holomorphic line bundles of degree d is the Picard variety

Pic_{d}(Σ) = {holomorphic line bundles L on Σ , c_{1}(L) = d} .

The subspace of line bundles L of degree 0 is the Jacobian variety: J(Σ) = Pic_{0}(Σ), which may also be viewed as

J(Σ) = H^{1}(Σ, ℝ)/H^{1}(Σ, ℤ) .  (5.22)

Choosing a reference L_{0} ∈ Pic_{d}(Σ), any other L ∈ Pic_{d}(Σ) may be obtained as a tensor product L = L_{0} ⊗ ℓ, ℓ ∈ J(Σ).

A completely explicit realization is obtained by making a choice of homology basis (A_{I}, B_{I}), I = 1, …, h, with canonial intersection matrix:

#(A_{I}, A_{J}) = #(B_{I}, B_{J}) = 0  #(A_{I}, B_{J}) = −#(B_{J}, A_{I}) = δ_{IJ} .  (5.23)

To this choice corresponds a unique basis of holomorphic Abelian differentials (holomorphic sections of K), (ω_{1}, …, ω_{h}) = ω⃗ or denoted simply ω by

∮_{A_{I}} ω_{J} = δ_{IJ} ,  I, J = 1, …, h .  (5.24)

The period matrix is then defined by

∮_{B_{I}} ω_{J} = Ω_{IJ}  (5.25)

and the Jacobian is a complex torus of dimension h:

J(Σ) = ℂ^{h}/(ℤ^{h} + Ω ℤ^{h}) .  (5.26)

V.7 The intersection from #( , ) is left invariant under the action of the modular group on the homology cycles

( \begin{matrix}B \\ A \end{matrix} )_{I} ⟶ ( \begin{matrix}B′ \\ A′ \end{matrix} )_{I} = \underbrace{( \begin{matrix}a & b \\ c & d \end{matrix} )_{IJ}}_{∈ Sp(2h; ℤ)} ( \begin{matrix}B \\ A \end{matrix} )_{J}  (5.27)

under which

Ω → Ω′ = (aΩ + b)(cΩ + d)^{−1} .  (5.28)

The Riemann bilinear relations guarantee that Ω lies in the Siegel upper half space: Ω ∈ ℋ_{h}, where

ℋ_{h} = {h × h complex Ω; Ω_{IJ} = Ω_{JI}; Im Ω > 0} .  (5.29)

In fact, Torelli’s theorem guarantees that to each Riemann surface in ℳ_{h} there corresponds a unique point in the fundamental domain of the modular group

𝒜_{h} ≡ ℋ_{h}/Sp(2h; ℤ) ,  (5.30)

(the converse holds for the torus, but not for h ≥ 2.)

ℋ_{h} carries a Sp(2h; ℤ)-invariant metric

ds^{2} = Tr ((Im Ω)^{−1}dΩ(Im Ω)^{−1}dΩ^{∗}) .  (5.31)

Since ω_{I} depends holomorphically on ℳ_{h}, so does Ω_{IJ}; in fact, we have an explicit formula for the variation of Ω_{IJ} with holomorphic Beltrami differentials μz ; z̄ associated with a Riemann surface deformation:

δΩ_{IJ} = −i ∫_{Σ} d^{2}z μz ; z̄ ω_{Iz} ω_{Jz}  (5.32)

The induced complex structure on ℳ_{h} is the one associated with the metric ds^{2} of (5.31).

The Jacobi ϑ-function, with characteristics δ = (δ′, δ′′) ∈ [0, 1]^{2h}, is defined as a function on ℂ^{h} × ℋ_{h}:

ϑ[δ](ζ, Ω) ≡ ∑_{n∈ℤ^{h}+δ′} e^{π in^{T} Ωn+2πi n^{T} (ζ+δ′′)} .  (5.33)

V.8 It obeys, for any M, N ∈ ℤ^{h}:

ϑ[δ](ζ + M + ΩN, Ω) = ϑ[δ](ζ, Ω)e^{−iπ N^{T} Ω N−2πi N^{T} (ζ+δ′′)+2πiδ′M} ,  (5.34)

and this property may be used to define ϑ as the unique (up to constant factor) holomorphic section of a holomorphic line bundle on J(Σ), with the above transition functions. One also often uses ϑ(ζ, Ω) ≡ ϑ[0](ζ, Ω).

Using the Jacobi ϑ-functions, and the Riemann vanishing theorem (which yields the zeros of ϑ as a function of ζ, but which we shall not discuss here), one may uniquely (up to constant factor) reconstruct a meromorphic function on Σ from its divisor D = z_{1} + ⋯ + z_{d} − w_{1} − ⋯ − w_{d}:

f_{D}(z) = ∏_{i=1}^{d} ϑ (ζ + ∫_{z_{i}}^{z} ω, Ω) ϑ (ζ + ∫_{w_{i}}^{z} ω, Ω)^{−1} .  (5.35)

Here ζ is such that ϑ(ζ, Ω) = 0, but so that the above ϑ functions do not vanish identically; f_{D}(z) is then independent of ζ.

Spin bundles on Σ belong to Pic_{h−1}(Σ), and may be described in terms of their spin structure, of which there are 2^{2h}. A choice of homology basis singles out a particular spin bundle S_{0}, and all other spin structures may be labeled by points in the Jacobian, as indicated earlier:

S = S_{0} ⊗ [ \begin{matrix}δ′ \\ δ′′ \end{matrix} ]  δ′′ + Ωδ′ ∈ J(Σ) ,  (5.36)

where δ′ and δ′′ are half-points, i.e. δ′_{I}, δ′′_{I} ∈ \{0, \frac{1}{2}\}. Spin structures are partitioned into even or odd spin structures, according to the parity of the ϑ-function

ϑ[δ](−ζ, Ω) = (−)^{4δ′·δ′′}ϑ[δ](ζ, Ω) .  (5.37)

Thus, there are 2^{h−1}(2^{h} ± 1) even/odd spin structure with 4δ′ · δ′′ even/odd, and the number of holomorphic sections is then even/odd as well, in fact generally 0/1. The generic holomorphic section h_{δ}(z) for an odd spin structure δ is obtained from the square root of an Abelian differential with exactly h − 1 double zeros, (this h_{δ}(z) is a Dirac zero mode)

h_{δ}(z) ≡ (∑_{I} ∂_{I}ϑ[δ](0, Ω) ω_{I}(z))^{1/2} .  (5.38)

V.9

Finally, we define one of the fundamental objects that enter Green functions and determinants in string theory. The prime form is a function on the universal cover of Σ×Σ and ℋ_{h}, given by

E(z, w) ≡ \frac{ϑ[δ] (∫_{w}^{z} ω, Ω)}{h_{δ}(z) h_{δ}(w)}  (5.39)

where δ is any odd half characteristic. E(z, w) is a holomorphic (−\frac{1}{2}, 0) differential form in both z and w, with a single zero at z = w. As such, E(z, w) generalizes the function z − w on ℂ to higher genus surfaces. On the torus, we have the well-known expression

E(z, w) = \frac{ϑ_{1}(z − w, Ω)}{ϑ′_{1}(0, Ω)} .  (5.40)

The monodromies of E(z, w) around A_{I} homology cycles are simple: E(z, w) picks up a − sign; the monodromy as z moves around B_{I} once is:

E(z, w) → − exp [−iπΩ_{II} − 2πi ∫_{z}^{w} ω_{I}] E(z, w) .  (5.41)

Under modular transformations (represented by their action on the homology cycles as in (5.27)), we have

E(z, w) → exp [πi ∫_{z}^{w} ω^{T} (cΩ + d)^{−1}c ∫_{z}^{w} ω] E(z, w) .  (5.42)

Meromorphic Abelian differentials (meromorphic sections of K) are readily obtained

\begin{aligned}ω_{xy}(z) & = ∂_{z} ln \frac{E(z, x)}{E(z, y)} dz \\ ω_{w}(z) & = ∂_{z}∂_{w} ln E(z, w)dz . \end{aligned}  (5.43)

Meromorphic \frac{1}{2} differentials (meromorphic sections of S_{δ}) are

δ even  S_{δ}(z, w) = \frac{1}{E(z, w)} \frac{ϑ[δ] (∫_{z}^{w} ω, Ω)}{ϑ[δ](0, Ω)}
δ odd  S_{δ}(z, w) = \frac{1}{E(z, w)} \frac{∂_{I}ϑ[δ] (∫_{z}^{w} ω, Ω) ω_{I}(y)}{∂_{I}ϑ[δ](0, Ω)ω_{I}(y)}

(5.44)

For δ even, S_{δ} is the Szegö kernel, while for δ odd, §_{δ} is not single-valued on Σ as defined.

V.10 E) Green Functions, Determinants and Chiral Splitting

The Green function of scalar fields on Σ is simply expressed in terms of the prime form and Abelian differentials

G(z, w) = − ln |E(z, w)|^{2} + 2π Im ∫_{w}^{z} ω_{I}(Im Ω)_{IJ}^{−1} Im ∫_{w}^{z} ω_{J}  (5.45)

The second term on the right hand side is required to render G(z, w) single-valued on Σ×Σ, as well as to render it modular invariant. This expression is readily seen to generalize the 1-loop result (and of course the 0-loop result as well.)

We use this result to prove the Chiral Splitting theorem for the case of vertex operators for tachyon states. (Higher mass states and their vertex operators may be obtained by factorization or equivalently by using the OPE for exponential operators.) To restrict the loop momenta and bring out an integration over them, we can insert the identity (as we did in §III for one loop amplitudes)

1 = ∏_{I} ∫_{ℝ^{26}} dp_{I} δ (p_{I}^{μ} − \frac{1}{2πi} ∫_{A_{I}} dz ∂_{z}x^{μ})  (5.46)

into the correlation function ⟨W_{1}(z_{1}, z̄_{1}) … W_{N}(z_{N}, z̄_{N})⟩_{g}. In fact, here, we may use a short-cut, yielding the same result, carried out as follows. If W_{i}(z_{i}, z̄_{i}) = exp ik_{i} · x(z_{i}, z̄_{i}) then we have

\begin{aligned}⟨∏_{i=1}^{N} e^{ik_{i}·x(z_{i},z̄_{i})}⟩_{g} & = δ(k) (\frac{Det′Δ_{(0)}}{∫_{Σ} dμ_{g}} )^{−13} ∏_{i<j} e^{−k_{i}·k_{j}G(z_{i},z_{j})} \\ & = δ(k)Z_{(0)}(g)^{−13} ∏_{i<j} |E(z_{i}, z_{j})|^{2k_{i}·k_{j}}X \end{aligned}  (5.47)

where

X = det(ω_{I}, ω_{J})^{−13} ∏_{i<j} e^{−2πk_{i}·k_{j} Im ∫_{w}^{z} ω (Im Ω)^{−1} Im ∫_{w}^{z} ω} .  (5.48)

Using the Riemann bilinear relations, we have

det(ω_{I}, ω_{J}) = det(2Im Ω_{IJ})  (5.49)

V.11 and we may now represent X by a single Gaussian integral over real parameters p_{I}^{μ}, which are just the internal loop momenta:

X = ∏_{I} ∫_{ℝ^{26}} d^{26}p_{I} |e^{iπp_{I}^{μ}Ω_{IJ} p_{J}^{μ}+2πip_{I}^{μ}k_{i}^{μ} ∫_{P}^{z_{i}} ω_{I}} |^{2} .  (5.50)

It is now straightforward to read off the function ℱ that enters the Chiral Splitting Theorem:

ℱ(z_{i}, m_{j}; p_{I}, k_{i}) = ∏_{i<j} E(z_{i}, z_{j})^{k_{i}·k_{j}} · exp [iπp_{I}^{μ}Ω_{IJ}p_{j}^{μ} + 2πip_{I}^{μ}k_{i}^{μ} ∫_{P}^{z_{i}} ω_{I}]  (5.51)

The point P is arbitrary in view of the momentum conservation δ(k)-function factor. ℱ above is determined only up to phases. In particular, ℱ is multiple-valued, with monodromies that may be easily computed from the monodromies of E(z, w), as given in (5.41). ℱ also has branch cuts along the homology cycles: the effect of letting vertex insertion point z_{i} cross a cycle A_{I} is to add the external momentum k_{i} to the loop momentum p_{I}. Modular transformation properties of ℱ are also non-trivial, and may be read off from (5.42) and (5.28).

It remains to express the functional determinant combinations 𝒵_{(n)}^{−}(m_{j}) in terms of ϑ-functions and differentials. This may be achieved by a variety of techniques, inducing bosonization and integration over ℳ_{h} of the stress tensor. We shall not develop these techniques here but limit ourselves to stating the result for 𝒵_{(−1)}^{−}:

𝒵_{(−1)}^{−}(m_{j}) = \frac{ϑ(3Δ_{w}|Ω)}{det ∂_{w}^{i}ϕ_{j}(w)} (\frac{∂_{w}^{h}ϑ(Δ_{w}|Ω)}{det ∂_{w}^{I}ω_{J}(w)})^{−9} .

Here, w is an arbitrary point on Σ, ∂_{w}^{α} stands for the derivative of order α in w and Δ_{w} is the Riemann vector

Δ_{w}^{I} = \frac{1}{2} − \frac{1}{2}Ω_{II} + ∑_{J≠I} ∮_{A_{J}} ω_{J}(z) ∫_{w}^{z} ω_{I} .

V.12