A lot can be learned by simply focussing on the leading singularities in the moduli space of the effective theory. However, for the sake of performing really non-trivial quantitative tests of the heterotic/F-theory duality, we should try harder in order to reproduce the exact functional form of the couplings ∆_{eff}(T) from K3 geometry. The hope is, of course, to learn something new about how to do exact non-perturbative computations in D-brane physics.

More specifically, the issue is to eventually determine the extra contributions to the geometric Green’s functions. Having *a priori* no good clue from first principles how to do this, the results of the previous section, together with experience with four dimensional compactifications with N = 2 supersymmetry, suggest that somehow mirror symmetry should be a useful tool.

The starting point is the observation that threshold couplings of similar structure appear also in four dimensional, N = 2 supersymmetric compactifications of type II strings on Calabi-Yau threefolds. More precisely, these coupling functions multiply operators of the form TrF_{G}^{2} (in contrast to quartic operators in d = 8), and can be written in the form

∆^{(4d)}_{eff} ∼ ln[λ^{α1}(1-λ)^{α2}(λ’)^{3}] + γ(λ) —– (1)

which is similar to Green’s function

∆_{eff} (λ) = ∆^{∑N-1}_{prime form} (λ) + δ(λ)

It is to be noted that a Green’s function is in general ambiguous up to the addition of a finite piece, and it is this ambiguous piece to which we can formally attribute those extra non-singular, non-perturbative corrections.

The term δ(λ) contributes to dilation flat coordinate. The dilation S is a period associated with the CY threefold moduli space, and like all period integrals, it satisfies a system of linear differential equations. This differential equation may then be translated back into geometry, and this then would hopefully give us a clue about what the relevant quantum geometry is that underlies those quartic gauge couplings in eight dimensions.

The starting point is the families of singular K3 surfaces associated with which are the period integrals that evaluate to the hypergeometric functions. Generally, period integrals satisfy the Picard-Fuchs linear differential equations.

The four-dimensional theories are obtained by compactifying the type II strings on CY threefolds of special type, namely they are fibrations of the K3 surfaces over P^{l}. The size of the P^{1} yields then an additional modulus, whose associated fiat coordinate is precisely the dilaton S (in the dual, heterotic language; from the type II point of view, it is simply another geometric modulus). The K3-fibered threefolds are then associated with enlarged PF systems of the form:

L_{N}(z, y) = θ_{z}(θ_{z} – 2θ_{y}) – z(θ_{z} + 1/2N)(θ_{z} + 1/2 – 1/2N)

L_{2}(y) = θ_{y}^{2} – 2y(2θ_{y} +1)θ_{y} —– (2)

For perturbative, one-loop contributions on the heterotic side (which capture the full story in d = 8, in contrast to d = 4), we need to consider only the weak coupling limit, which corresponds to the limit of large base space: y ∼ e^{-S} → 0. Though we might now be tempted to drop all the θ_{y }≡ y∂y terms in the PF system, we better note that the θ_{y} term in L_{N}(z, y) can a non-vanishing contribution, namely in particular when it hits the logarithmic piece of the dilaton period, S = -In[y] + γ. As a result one finds that the piece , that we want to compute satisfies in the limit y → 0 the following inhomogenous differential equation

L_{N} . (γϖ_{0})(z) = ϖ_{0}(z) —– (3)

We now apply the inverse of this strategy to our eight dimensional problem. Since we know from the perturbative heterotic calculation what the exact answer for δ must be, we can work backwards and see what inhomogenous differential equation the extra contribution δ(λ) obeys. It satisfies

L_{N}^{⊗2} . (δϖ_{0})(z) = ϖ_{0}^{2}(z) —– (4)

whose homogenous operator

L_{N}^{⊗2}(z) = θ_{z}^{3} – z(θ_{z} + 1 – 1/N)(θ_{z} + 1/2)(θ_{z} + 1/N) —– (5)

is the symmetric square of the K3 Picard-Fuchs operator. This means that its solution space is given by the symmetric square of the solution space of L_{N}(z), i.e.,

L_{N}^{⊗2 }. (ϖ_{0}^{2}, ϖ_{0}ϖ_{1}, ϖ_{1}^{2}) = 0 —– (6)

Even though the inhomogenous PF equation (4) concisely captures the extra corrections in the eight-dimensional threshold terms, the considerations leading to this equation have been rather formal and it would be clearly desirable to get a better understanding of what it mathematically and physically means.

Note that in the four dimensional situation, the PF operator L_{N}(z), which figures as homogenous piece in (3), is by construction associated with the K3 fiber of the threefold. By analogy, the homogenous piece of equation (4) should then tell us something about the geometry that is relevant in the eight dimensional situation. Observing that the solution space (6) is given by products of the K3 periods, it is clear what the natural geometrical object is: it must be the symmetric square Sym^{2}(K3) = (K3 x K3)/Ζ_{2}. Being a hyperkähler manifold, its periods (not subject to world-sheet instanton corrections) indeed enjoy the factorization property exhibited by (6).

*Formal similarity of the four and eight-dimensional string compactifications: the underlying quantum geometry that underlies the quadratic or quartic gauge couplings appears to be given by three- or five-folds, which are fibrations of K3 or its symmetric square, respectively. The perturbative computations on the heterotic side are supposdly reproduced by the mirror maps on these manifolds in the limit where the base P ^{l}‘s are large.*

The occurrence of such symmetric products is familiar in D-brane physics. The geometrical structure that is relevant to us is however not just the symmetric square of K3, but rather a fibration of it, in the limit of large base space – this is precisely what the content of the inhomogenous PF equation (4) is. It is however not at all obvious to us why this particular structure of a hyperkähler-fibered five-fold would underlie the non-perturbative quantum geometry of the quartic gauge couplings in eight dimensions.