Mocking the u-plane integral

The u-plane integral is the contribution of the Coulomb branch to correlation functions of N=2 gauge theory on a compact four-manifold. We consider the u-plane integral for correlators of point and surface observables of topologically twisted theories with gauge group SU(2), for an arbitrary four-manifold with (b1,b2+)=(0,1). The u-plane contribution equals the full correlator in the absence of Seiberg-Witten contributions at strong coupling, and coincides with the mathematically defined Donaldson invariants in such cases. We demonstrate that the u-plane correlators are efficiently determined using mock modular forms for point observables, and Appell-Lerch sums for surface observables. We use these results to discuss the asymptotic behavior of correlators as function of the number of observables. Our findings suggest that the vev of exponentiated point and surface observables is an entire function of the fugacities.


Introduction
A powerful approach to understand the dynamics of supersymmetric field theories is to consider such theories on a compact four-manifold without boundary [1,2,3,4,5,6,7]. We consider in this paper the topologically twisted counterpart of N = 2 supersymmetric Yang-Mills theory with gauge group SU (2) and in the presence of arbitrary 't Hooft flux [8]. The gauge group is broken to U(1) on the Coulomb branch B, which is parametrized by the vacuum expectation value u = 1 16π 2 Tr[φ 2 ] R 4 , where the subscript indicates that this is a vev in a vacuum state of the theory on flat R 4 . The Coulomb branch, also known as the "u-plane" can be considered as a three punctured sphere, where the punctures correspond to the weak coupling limit, u → ∞, and the two strong coupling singularities for u = ±Λ 2 .
The This paper considers the u-plane contribution Φ [O] for compact four-manifolds with b + 2 = 1 known as the u-plane integral [3]. 1 The integrand of Φ[O] for b + 2 = 1 does not receive perturbative corrections, such that the path integral reduces to a finite dimensional integral over the zero modes of the fields. After including the nonperturbative corrections to the integrand using the Seiberg-Witten solution [9], the u-plane integral has been evaluated for some four-manifolds with b 2 = 1 or 2, namely for four-manifolds which are rational or ruled complex surfaces [3,10,11,12,13,14]. The final expressions appeared to be in terms of mock modular forms [15,16], which could be traced to simplifying features, such as a vanishing chamber, wall-crossing, or birational transformations. For generic four-manifolds with b + 2 = 1, these simplifying features are not available. Nevertheless, we will demonstrate that u-plane integrals of arbitrary four-manifolds with b + 2 = 1 can be readily evaluated by integration by parts leading to expressions in terms of mock modular forms and Appell-Lerch sums. For simplicity, we will restrict to four-manifolds with (b 1 , b + 2 ) = (0, 1), but not necessarily simply-connected.
To achieve the evaluation of these u-plane integrals, we change variables from u to the effective coupling constant τ , such that Φ becomes an integral over the modular fundamental domain H/Γ 0 (4), where Γ 0 (4) is the duality group of the theory. We are able to express the integrand as a total derivative dτ ∧ dτ ∂τ ( du dτ H O ), for some H O which depends on the observable O. Reversing the change of variables, this demonstrates that the integrand takes the form du ∧ dū ∂ūH O , and the integral is thus reduced to integrals over the boundaries ∂ j B, j = 1, 2, 3 in the vicinity of each singularity {−1, +1, ∞} by Stokes' theorem. See Figure 1. More explicitly, we have (1.2) In order for this expression to be useful, it is necessary that H O (u,ū), when expressed in terms of τ,τ , has good modular properties allowing one to make the required duality transformation near strong coupling singularities. We will find, for a special choice of metric, that H O (τ,τ ) can be expressed in terms of mock modular forms. Then, given the expression for the wall-crossing formula using indefinite theta functions [17,18] the same result follows for general metric.  The expression for the u-plane integral as a modular integral over H/Γ 0 (4) paves the way for its evaluation. Earlier work has demonstrated that such modular integrals evaluate to the constant term of a q-series, or more specifically, the q 0 term of a mock modular form [19,20]. We thus establish a close connection between u-plane correlation functions and mock modular forms. Said more mathematically, we have established a connection between Donaldson invariants for general manifolds with b + 2 = 1 and mock modular forms. The explicit expressions are (5.44) for manifolds with odd intersection form and just point observables inserted, (5.65) for manifolds with odd intersection form and just surface observables inserted, and (5.84) for manifolds with even intersection form and just surface observables inserted. These expressions hold for a particularly nice choice of metric. The metric dependence only enters through the choice of period point, i.e. the unique self-dual degree two cohomology class in the forward light-cone in H 2 (M ; R). Using the expression for the wall-crossing formula in terms of indefinite theta functions [17,18] one can produce analogous mock modular forms relevant to other chambers. Expressions (5.44), (5.65) and (5.84) (or close cousins thereof) have appeared before in [10]. The derivations in [10] relied on the existence of a vanishing chamber and applied wall-crossing formulae. By contrast, in this paper we evaluate the u-plane integral directly, and do not rely on the existence of a vanishing chamber. Consequently, our formulae are justified for a larger class of manifolds.
Using the expression for Φ [O] in terms of mock modular forms (see for example Equation (5.44)), we can address analytic properties of the correlators for b + 2 = 1, analogously to the structural results for manifolds with b + 2 > 1 [21]. We study the asymptotic behavior of Φ[u ] for large , and find experimental evidence that Φ[u ] ∼ 1/( log( )) for any four-manifold with (b 1 , b + 2 ) = (0, 1). Remarkably, the asymptotic behavior of Φ [u ] suggests that Φ[e 2p u ] = ≥0 (2p) Φ[u ]/ ! is an entire function of p rather than a formal expansion. We find similar experimental evidence that the u-plane contribution to the exponentiated surface observable Φ[e I − (x) ] is an entire function of x ∈ H 2 (M, C). We leave a more rigorous analysis of these aspects for future work. The questions we address here would seem to be related to the analysis of correlation functions of large charge that have recently been studied in [22] and again we leave the investigation of this potential connection for future work.
One can change variables from q to the complex electric mass a in Φ [O], and express the u-plane integral as a residue of a around ∞ and 0. One may in this way connect to other techniques for the evaluation of Donaldson invariants, for examples those using toric localization [23,24]. Our results may also be useful for the evaluation of Coulomb branch integrals of different theories, such as those including matter and superconformal theories [18], and for four-manifolds with b 1 = 0.
The outline of the paper is as follows. Section 2 reviews Seiberg-Witten theory and its topological twist. Section 3 gives a lightning overview of compact four-manifolds with b + 2 = 1. Section 4 continues with introducing the path integral and correlation functions of the theory on these manifolds, which are evaluated in Section 5. We close in Section 6 with an analysis on the asymptotic behavior of correlation functions with a large number of fields inserted.

Seiberg-Witten theory and Donaldson-Witten theory
We give a brief review of pure Seiberg-Witten theory [9,25], and its topologically twisted counterpart aka Donaldson-Witten theory [8]. See [26,27] for a detailed introduction to both of these theories.

Seiberg-Witten theory
Seiberg-Witten theory is the low energy effective theory of N = 2 supersymmetric Yang-Mills theory with gauge group G = SU(2) or SO (3) and Lie algebra su (2). The building blocks of the theory contain a N = 2 vector multiplet which consists of a gauge field A, a pair of (chiral, anti-chiral) spinors ψ andψ, a complex scalar Higgs field φ (valued in su(2) ⊗ C), and an auxiliary scalar field D ij (symmetric in SU(2) R indices i and j, which run from 1 to 2). N = 2 hypermultiplets can be included in general. Here we will consider pure Seiberg-Witten theory with gauge group as above, so we assume no hypermultiplets. The gauge group is spontaneously broken to U(1) on the Coulomb branch B. The pair (a, a D ) ∈ C 2 are the central charges for a unit electric and magnetic charge. The parameters a and a D are expressed in terms of the holomorphic prepotential F of the theory a D = ∂F(a) ∂a . (2.1) Its second derivative equals the effective coupling constant where θ is the instanton angle with periodicity 4π, g is the Yang-Mills coupling and H is the complex upper half-plane. The Coulomb branch B is parametrized by the order parameter, where the trace is in the 2-dimensional representation of SU (2). The renormalization group flow relates the Coulomb branch parameter u and the effective coupling constant τ . Using the Seiberg-Witten geometry [9], the order parameter u can be exactly expressed as a function of τ in terms of modular forms, where Λ is a dynamically generated scale, q = e 2πiτ , and ϑ i (τ ) are the Jacobi theta functions, which are explicitly given in Appendix A. The function u(τ ) is invariant under transformations τ → aτ +b cτ +d given by elements of the congruence subgroup Γ 0 (4) ⊂ SL(2, Z). 2 See Equation (A.2) in Appendix A for the definition of this group. A change of variables from u to τ maps the u-plane to a fundamental domain of Γ 0 (4) in the upper-half plane H. We choose the fundamental domain as the union of the images of the familiar key-hole fundamental domain of SL(2, Z) under τ → τ + 1, τ + 2, τ + 3, −1/τ and 2 − 1/τ , which is displayed in Figure 2. Let u D be the vector-multiplet scalar for the dual photon vector multiplet with coupling constant τ D = −1/τ . Then At the cusp τ → 0 (respectively τ → 2) a monopole (respectively a dyon) becomes massless, and the effective theory breaks down since new additional degrees of freedom need to be taken into account. Another quantity which we will frequently encounter is the derivative da du . It is expressed as function of τ as (2.6) and transforms under a standard pair of generators of Γ 0 (4) as (2.7) Let us also give the expression of the dual of this quantity ( da

Donaldson-Witten theory
Donaldson-Witten theory is the topologically twisted version of Seiberg-Witten theory with gauge group SU(2) or SO(3), and contains a class of observables in its Qcohomology, which famously provide a physical realization of the mathematically defined Donaldson invariants [28,29]. Topological twisting preserves a scalar fermionic symmetry Q of N = 2 Yang-Mills on an arbitrary four-manifold 3 [8]. The twisting involves a choice of an isomorphism of an associated bundle to the SU(2) R-symmetry bundle with an associated bundle to the frame bundle. Namely, we choose an isomorphism of the adjoint bundle of the SU(2) R R-symmetry bundle with the bundle of anti-self-dual 2-forms, and we choose a connection on the R-symmetry bundle, which under this isomorphism becomes the Levi-Civita connection on the bundle of anti-self-dual 2-forms. In practice, this allows us to replace the quantum numbers of fields under the SU(2) − × SU(2) + factor of the N = 2 supergroup by the quantum numbers of a diagonally embedded SU (2) group.
The field content of the topologically twisted theory is a one-form gauge potential A, a complex scalar a, together with anti-commuting (Grassmann valued) self-dual two-form χ, one-form ψ and zero-form η. The auxiliary fields of the non-twisted theory combine to a self-dual two-form D. The action of the BRST operator Q on these fields is given by (2.10) The low energy Lagrangian of the Donaldson-Witten theory is given by [3] where y = Im(τ ) > 0.

A survey of four-manifolds with
We aim to evaluate and analyze the u-plane integral for compact four-manifolds with (b 1 , b + 2 ) = (0, 1) (and without boundary 4 ). This is a large class of manifolds which includes among others complex rational surfaces and examples of symplectic manifolds. The u-plane integral is well-defined and can be evaluated for all these fourmanifolds. This section gives a brief review of the standard geometric aspects of these four-manifolds.

Four-manifolds and lattices
Let M be a compact four-manifold, and let b j = dim(H j (M, R)) be the Betti numbers of M . For simplicity we restrict to manifolds with b 1 = 0, we do not require them to be simply connected. The torsion subgroups of H 1 (M, Z) and H 2 (M, Z) are naturally dual by Poincaré duality. They will not play an important role here, since they simply lead to an overall factor (the order) from the addition of flat connections.
We denote by L the image of the Abelian group H 2 (M, Z) ∈ H 2 (M, R), which effectively mods out the torsion in H 2 (M, Z). As a result, L is a lattice in a real vector space, and we can divide elements of L without ambiguity. If the context allows, we will occasionally use H 2 (M, Z) and L interchangeably. The intersection form on H 2 (M, Z) provides a natural non-degenerate bilinear form B : (L ⊗ R) × (L ⊗ R) → R that pairs degree two co-cycles, and whose restriction to L × L is an integral bilinear form. The bilinear form provides the quadratic form Q(k) := B(k, k) ≡ k 2 , which is uni-modular and possibly indefinite. For later use, recall that a characteristic element of L is an element c ∈ L, such that We let furthermore H 2 (M, R) ± be the positive definite and negative definite subspaces of H 2 (M, R), and set b ± 2 = dim(H 2 (M, R) ± ). Van der Blij's Lemma states that a characteristic element c of a lattice L satisfies Q(c) = σ L mod 8, where The second Stiefel-Whitney class w 2 (T M ) is a class in H 2 (M, Z 2 ), which distinguishes spinnable from non-spinnable manifolds. A smooth, spinnable manifold has w 2 (T M ) = 0, while w 2 (T M ) = 0 for non-spinnable manifolds. The class w 2 (T M ) has implications for the intersection form of the lattice L. In four (but not in higher) dimensions the Stiefel-Whitney class always has an integral lift. Any integral lift of the Stiefel-Whitney class defines a characteristic vector in L. Therefore, w 2 (T M ) = 0 implies that L is an even lattice. The converse is however only true if M is simply connected due to the possibility that w 2 (T M ) is represented by a torsion class in H 2 (M, Z). An even stronger statement for the intersection form of smooth, simply connected, spinnable four-manifolds is Rokhlin's theorem, which states that the signature of such manifolds satisfies σ L = 0 mod 16. Note that the Enriques surface is smooth while it has intersection form I 1,1 ⊕ L E 8 , where I 1,1 is the two-dimensional lattice with quadratic form ( 0 1 1 0 ), and L E 8 is minus the E 8 root lattice. This does not contradict Rokhlin's theorem since the Enriques surface is not simply connected. It is also worth noting that for complex manifolds the canonical class K is an integral lift of the Stiefel-Whitney class and therefore any other integral lift differs by twice a lattice vector in L.
Any closed, orientable four-manifold admits a Spin C structure. To a Spin C structure one attaches a first Chern class of a certain line bundle, which we refer to as the first Chern class of the Spin C structure. The first Chern class of a Spin C structure is an integral lift of w 2 (T M ) and is therefore a characteristic vector c ∈ L. Interestingly, the existence of an almost complex structure for a smooth four-manifold M is related to the existence of a characteristic vector c with fixed norm. Note that an almost complex structure ensures that the tangent bundle T M is complex, such that its Chern class c 1 (T M ) ∈ H 2 (M, Z) and canonical class K = −c 1 (T M ) are well-defined. The Riemann-Roch theorem for four-manifolds with an almost complex structure demonstrates that its canonical class K is a characteristic element of L. Moreover: • The modulo 2 reduction of K satisfies • By the Hirzebruch signature theorem In fact the converse holds as well: any characteristic vector c ∈ L, which satisfies (3.3) and (3.4), gives rise to an almost complex structure [31,32]. Combination of this statement with Van der Blij's Lemma demonstrates that if M admits an almost complex structure, then b + 2 + b 1 must be odd.
3.2 Four-manifolds with b + 2 = 1 We will specialize in the following to b + 2 = 1. In this case, the quadratic form Q can be brought to a simple standard form [29, Section 1.1.3], which will be instrumental to evaluate the u-plane integral in Section 5. The standard form depends on whether the lattice is even or odd: • If Q is odd, an integral change of basis can bring the quadratic form to the diagonal form with m = b 2 − 1. This has an important consequence for characteristic elements of such lattices. If K is a characteristic element, k 2 + B(K, k) ∈ 2Z for any k ∈ L. In the diagonal basis (3.5) this equivalent to b 2 j=1 k 2 j + K j k j ∈ 2Z with K = (K 1 , K 2 , . . . , K b 2 ). This can only be true for all k ∈ L if K j is odd for all j = 1, . . . , b 2 .
• If Q is even, the quadratic form Q can be brought to the form where I 1,1 and L E 8 as defined above and n = (b 2 − 2)/8. The components K j , j = 1, 2 must therefore be even in this basis.
Another important aspect of M is its period point J ∈ H 2 (M, R), which is the generator of H 2 (M, R) + , normalized such that Q(J) = 1. The period point depends on the metric due to the self-duality condition. In fact, the metric dependence in the expressions below only enters through a choice of J. Using J, we can project k ∈ L to the positive and negative definite subspaces H 2 (M, R) ± : k + = B(k, J) J is the projection of k to H 2 (M, R) + , and k − = k − k + is the projection to H 2 (M, R) − . Note that these projections are also the self-dual and anti-self-dual parts of k with respect to the Hodge * -operation.
Complex four-manifolds with b + 2 = 1 Complex four-manifolds with b + 2 = 1 are well-studied and classified by the Enriques-Kodaira classification. This classification starts with the notion of a minimal complex surface. This is a non-singular surface which can not be obtained from another nonsingular surface by blowing up a point. This is equivalent to the statement that the surface does not contain rational curves with self-intersection −1 (or (−1)-curves). The Enriques-Kodaira classification classifies minimal surfaces using the so-called Kodaira dimension.
The relevant surfaces for us are those with (b 1 , b + 2 ) = (0, 1), whose Kodaira dimension is either −∞, 0, 1 or 2: • Surfaces with Kodaira dimension −∞ are surfaces whose canonical bundle does not admit holomorphic sections. These surfaces are birational to more than one minimal surface. The simply connected surfaces with b + 2 = 1 in this family are the rational surfaces, i.e. the complex projective plane P 2 , Hirzebruch surfaces and blow-ups of these surfaces. A special property of these surfaces are vanishing chambers where the moduli spaces of instantons are empty. This has been useful for the explicit determination of partition functions on these geometries, including the u-plane integral [3,10,33,34].
• Surfaces with Kodaira dimension 0 are surfaces for which the canonical class K satisfies Q(K) = 0 and B(K, C) = 0 for any curve C. If they satisfy in addition (b 1 , b + 2 ) = (0, 1), they are known as Enriques surfaces. Their intersection form is I 1,1 ⊕L E 8 . Note that this four-manifold is not simply-connected, and that w 2 (T M ) is represented by a torsion class in H 2 (M, Z).
• Surfaces with Kodaira dimension 1 are surfaces for which the canonical class K satisfies Q(K) = 0, and B(K, C) > 0 for any curve C. Such surfaces are elliptic (but the converse is not always true). The Dolgachev surfaces are a family of simply-connected surfaces with Kodaira dimension 1.
• Surfaces with Kodaira dimension 2 are surfaces of general type. If a surface in this class is simply connected with b + 2 = 1, its holomorphic Euler character χ h equals 1. Their Euler numbers lie between 3 and 11, and there are examples for each integer in this set such as the Godeaux and Barlow surfaces which both have Euler number 11. See for example [35] for a more comprehensive list and details.

Beyond complex four-manifolds
Although many four-manifolds with b + 2 = 1 admit an almost complex structure, most four-manifolds are not complex and their classification is an important open problem. A distinguished class of four-manifolds with b + 2 = 1 are symplectic ones, which partially overlap with the complex four-manifolds. For a four-manifold to be symplectic, its period point J must provide a symplectic structure. 5 Reference [36] provides a survey of such manifolds. Examples of symplectic four-manifolds which are not complex are four-manifolds denoted by E(1) N , that is four-manifolds which are homotopy equivalent to a rational elliptic surface and whose construction relies on a fibered knot N in S 3 [37,38]. The manifolds in this class have b 1 = 0. For recent progress on symplectic, non-complex manifolds with Kodaira dimension 1 (Q(K) = 0 and B(K, C) > 0) with b 1 = 0, see [39].

Donaldson invariants
Donaldson invariants have been of crucial importance for the classification of fourmanifolds, since they can distinguish among smooth structures on four-manifolds [29,21]. These invariants are based on ASD equations and via the Donaldson-Uhlenbeck-Yau theorem to semi-stable vector bundles. We briefly recall the definition of the Donaldson invariants in the formalism of topological field theory. Let M γ be the moduli space of solutions to the ASD equations for gauge group SU(2) or SO(3), where γ = (c 1 , k) represents the topological numbers of the solution, that is to say This map is constructed using the universal curvature F of the universal bundle U over M × M γ if it exists, which can be expressed as a formal sum of the fields of the topological theory F = F + ψ + φ [40]. The class µ D is defined in terms of the first Pontryagin class of the universal bundle, where the trace is in the two-dimensional representation of the gauge group, i.e. the fundamental representation for SU(2) and the spinor representation for SO(3). We will only consider the image of µ D for 0-and 2-cycles of M . Let {r j } be a finite set of points of M and p = [ Using the map µ D , we can define the Donaldson invariant D γ ,s (p, x) ∈ Q as the intersection number The number D γ ,s is only non-vanishing if 4 + 2s = dim R (M γ ). For smooth fourmanifolds the virtual dimension of the moduli space is and in general this is in fact the dimension. For complex surfaces, we can write 2 +s = 4k − c 2 1 − 3χ h with k ∈ Z and χ h the holomorphic Euler characteristic, χ h = (χ + σ)/4.

Path integral and correlation functions
This section reviews general properties the u-plane integral. We will treat the partition function in Subsection 4.1 and correlation functions in Subsection 4.2.

Path integral
We consider Donaldson-Witten theory on a four-manifold M with b + 2 = 1 as discussed in the previous section. For the case of pure SYM with no hypermultiplets we are always free to consider the case where the principal SO(3) gauge bundle has a nontrivial 't Hooft flux w 2 (P ) ∈ H 2 (M ; Z 2 ). We choose an integral lift w 2 (P ) (and we assume such a lift exists) and embed it in H 2 (M ; R), and we denote µ := 1 2 w 2 (P ) ∈ L ⊗ R. The dependence on the choice of lift will only enter through an overall sign. The path integral over the Coulomb branch of Donaldson-Witten theory, denoted by Φ J µ , is an integral over the infinite dimensional field space, which reduces to a finite dimensional integral over the zero modes [3]. We restrict for simplicity to four-manifolds with b 1 = 0, such that there are no zero modes for the one-form fields ψ. The path integral of the effective theory on the Coulomb branch then becomes where L 0 is the Lagrangian (2.11) specialized to the zero modes including the ones of the gauge field. The functions A(u) and B(u) are curvature couplings; they are holomorphic functions of u, given by [3,41] A(u) = α du da 1 2 , The coefficients α and β are numerical factors, which we choose to match with results on Donaldson invariants from the mathematical literature. Note that A(u) χ B(u) σ has dimension Λ 2 since χ + σ = 4. Moreover, da ∧ dā ∧ dD ∧ dη ∧ dχ has dimension Λ, such that Φ J µ (4.1) is dimensionless. 6 We denote the contribution of the Coulomb branch to a correlation function . This corresponds to an insertion of O 1 O 2 . . . in the rhs of (4.1) plus possible contact terms depending on the O j .
We proceed by reviewing the evaluation of Φ J µ . Integration over D, and the fermions η and χ gives The dimensons of a, A µ , D µν , η, ψ µ and χ µν are respectively 1, 1, 2, 3 2 , 3 2 and 3 2 in powers of Λ. The dimension of differential form fields is reduced by their form degree. For example, the dimension of F = dA is 0. The dimensions of the differentials da, dD, dη and dχ are respectively 1, 0, − 3 2 and 1 2 .
where the vector k equals [F ]/4π and represents a class in L + µ with µ ∈ L/2. The factor dτ dā suggests that it is natural to change variables from a to the effective coupling constant τ ∈ H/Γ 0 (4) in (4.1). To this end, we define the holomorphic "measure factor" so that Equation (4.11) below will hold. Using Matone's relation [42] du and (2.4) and (2.6), we can expressν in terms of modular functions where we fixed the constants α and β. 7 The modular transformations ofν for the two generators ST −1 S : τ → τ τ +1 and T 4 : τ → τ + 4 of Γ 0 (4) are: The measureν(τ ) behaves near the weak coupling cusp τ → i∞ as ∼ q − 3 8 . Near the monopole cusp, we haveν The photon path integral takes the form of a Siegel-Narain theta function with kernel K where K is a characteristic vector for L corresponding to an almost complex structure or Spin C structure. 8 If one considers correlation functions rather than the partition 7 The values of α and β are slightly different from those quoted in [17], since we have used a different normalization for the integral over D. 8 Note that, compared to equation (3.13) of [3] there is an overall phase difference. This phase difference can be written as exp . Because K is a characteristic vector this factor is a k 0 -dependent sign. The choice of sign is related to a choice of orientation of instanton moduli space. function, the sum over U(1) fluxes can be expressed as Ψ J µ [K], with the kernel K dependent on the fields in the correlation function [19]. For the partition function, the factor (4. where we have left out the factor dτ dā , which provides the change of variables from the Coulomb branch parameters to a fundamental domain of Γ 0 (4) in H. Combining all ingredients, we arrive at the following expression for Φ J µ , An important requirement for (4.11) is the modular invariance of the integrand under Γ 0 (4) transformations. We can easily determine the modular transformations . (The factor 1/ √ y contributes ( 1 2 , 1 2 ) and B(k, J) contributes (0, 1) to the total weight.) We then arrive at where we used that Q(K) = σ mod 8. Combining (4.7) and (4.12), we deduce that the integrand of (4.11) is invariant under the τ → τ τ +1 transformation. Moreover, the integrand is invariant under τ → τ +4 if B(µ, K) = 1 2 mod Z. However, if B(µ, K) = 0 mod Z, the integrand is multiplied by −1 for τ → τ + 4. Since Ψ J µ vanishes identically in the latter case case, there is no violation of the duality.
We conclude therefore that the Coulomb branch integral (4.11) is well defined since the measure dτ ∧ dτ transforms as a mixed modular form of weight (−2, −2) while the productν Ψ J µ is a mixed modular form of weight (2,2) for the group Γ 0 (4) making the integrand modular invariant. We close this subsection with Table 1 that collects the weights of the various modular forms that appear in the context of u-plane integrals. Evaluation of the integral is postponed to Section 5.

Correlators of point and surface observables
Much more information about the theory is obtained if we include observables in the path integral [3,8], which contain integrals over positive degree homology cycles of the Ingredient Mixed weight Table 1: Modular weights of various ingredients for the u-plane integral. Transformations are in SL(2, Z) for the first three rows, while in Γ 0 (4) for the last two rows.
four-manifold M . Since we restrict to four-manifolds with b 1 = b 3 = 0, we will focus in this article on surface observables involving integrals over elements of H 0 (M, Q) and H 2 (M, Q). The Donaldson invariants are correlation functions of observables in Donaldson-Witten theory. The canonical UV surface observable of Donaldson-Witten theory is defined using the descent operator K mentioned below Eq. (2.9), 10) can be expressed as a correlation function of the twisted Yang-Mills theory, where on the rhs, γ = (2µ, k) with k ∈ Z − 2µ 2 . The map p : We will often suppress the argument of p, and consider it simply as a fugacity in which we can make a (formal) series expansion.
In the effective theory in the infrared, the operator I − (x) becomes Inclusion of this operator in the path integral gives rise to a contact term in the IR, e x 2 T (u) [3,4], with where ϑ 4 is the fourth classical Jacobi theta function. The dual contact term reads We include moreover the Q-exact operator I + (x) [17], which can aid the analysis in the context of mock modular forms. As explained in [19], addition of this observable to I − (x) does not change the answer, once the integrals over the u-plane are suitably defined. And more generally, if we add α I + (x), the integral is independent of α. Nevertheless, the integrand depends in an interesting way on α. We will discuss this in more detail in Subsection 6.2. Here we will continue with α = 1. In the effective infrared theory, I + (x) becomes With (4.16) and (4.20), we find that the contribution of the Coulomb branch to As a first step towards evaluating this integral, we carry out the integral over D. If we just consider the terms in (4.21) that depend on D, this gives The variable ρ transforms with weight −1. With this normalization, it will appear as a natural elliptic variable in the sum over fluxes. The dual variable is where du da D is given in (2.8). Substitution of (4.22) into the path integral and integration over the η and χ zero modes modifies the sum over the U(1) fluxes to Ψ J µ [K s ] (4.9) where the kernel K s given by [3,17] This gives the standard generalization of Ψ J µ (τ,τ ) to a theta series with an elliptic variable ρ. The holomorphic part couples to k − and the anti-holomorphic part to k + We will therefore also denote Ψ . We postpone the remaining steps of the evaluation to Section 5.5.
After describing the u-plane integrand, we can also give the Seiberg-Witten contribution of the strong coupling singularities u = ±Λ 2 to e 2p u+I − (x) J µ . Setting Λ = 1, the contribution for u = 1 from a Spin C structure k is [3] with n = −(2χ + 3σ)/8 + k 2 /2 and the functions C(u), P (u), L(u) given by (4.28) For four-manifolds of SW-simple type, the only k for which the (4.27) is non-vanishing have n = 0. The expression then simplifies considerably. For the contribution from and for the contribution from u = −1, The full correlation function for manifolds of simple type therefore reads Manifolds with b + 2 = 1 are however rarely of SW-simple type [36]. These manifolds may give rise to SW moduli spaces of arbitrarily high dimension. The SW contributions will then be more involved, but are entire functions of p and x as is the case for (4.29) and (4.30).

Summary
For compact four-manifolds with (b 1 , b + 2 ) = (0, 1), the contribution of the u-plane to the vev of an observable O, is given by Besides the choice of O, it depends on the following data of the four-manifolds • An integral lift w 2 (P ) of the 't Hooft flux so that µ = 1 2 w 2 (P ) ∈ H 2 (M, R).

Evaluation of u-plane integrals
This section discusses the evaluation of u-plane integrals using mock modular forms. Subsection 5.1 reviews the evalulation and renormalization of integrals over a modular fundamental domain [3,19,20]. Section 5.2 explains the strategy for arbitrary correlation functions. Subsection 5.3 factors the sum over fluxes into holomorphic and non-holomorphic terms for a specific choice of J. We apply this result to the evaluation of the partition function and topological correlators in Subsections 5.4 and 5.5.

Integrating over H/SL(2, Z)
In the previous section we arrived at the general form (4.32) for the contribution of the u-plane to the correlators. Order by order in x we encounter modular integrals of the form where f is a non-holomorphic modular form of weight (2 − s, 2 − s), and F ∞ is the standard keyhole fundamental domain for the modular group, F ∞ = H/SL(2, Z). The integral is naturally independent of the choice of fundamental domain due to the modular properties of f . We assume that f has a convergent Fourier series expansion where the exponents m, n are bounded below. They may be real and negative, but m − n ∈ Z by the requirement that f is a modular form. Since m and n can be both negative, the integral I f is in general divergent and needs to be properly defined [19,20,43,44]. While the definition of the regularized and renormalized integral I r f is quite involved, the final result is quite elegant and compact, at least if f can be expressed as a total anti-holomorphic derivative, where h transforms as a modular form of weight (2, 0). In this case, the integrand of (5.1) is exact and equal to −d(dτ h). If only terms with n > 0 contribute to the sum in (5.2), h is a mock modular form and can be expressed as where h is a (weakly) holomorphic function with Fourier expansion Note that the two terms on the rhs of (5.4) are separately invariant under τ → τ + 1, while the transformation of the integral under τ → −1/τ implies for h(τ ), Reference [19] gives a definition of the integral I r f such that the value turns out to be As a result the only contribution to the integral arises from the constant term of h(τ ). The definition in [19] reduces to the older definition for I f if either m or n is nonnegative [3,44] but is new if both n, m are negative. It is shown in [19] that, at least for Donaldson-Witten theory, the new definition is physically sensible in the sense that Q-exact operators decouple. Note that the absence of holomorphic modular forms of weight two for SL(2, Z) implies that h(τ ) is uniquely determined by the polar coefficients, that is to say those d(m) with m < 0. The ambiguity in polar coefficients gives thus rise to an ambiguity in the anti-derivative h(τ ). Different choices for h(τ ) differ by a weakly holomorphic modular form of weight 2. However, this ambiguity does not lead to an ambiguity in the final result, d(0), since the constant term of such weakly holomorphic modular forms vanishes. This can be understood from the cohomology of F ∞ . Since the first cohomology of F ∞ is trivial, any closed one-form ξ is necessarily exact. Such a oneform ξ can be expressed as C(τ ) dτ , with C(τ ) a (weakly holomorphic) modular form of weight two. Since ξ is exact, the period Y +1 Y C(τ ) dτ vanishes, which implies that the constant term of C(τ ) vanishes. Indeed, a basis of weakly holomorphic modular forms of weight 2 is given by derivatives of powers of the modular invariant J-function, ∂ τ J(τ ) , ∈ N, which all have vanishing constant terms.

General strategy
Recall that in Section 4 we analyzed the partition function of Donaldson-Witten theory, which led to an integrand of the formν(τ ) Ψ J µ [K 0 ](τ,τ ), with a specific kernel K 0 (4.10). For more general correlation functions, the integrand takes a similar form, . . . This can be expressed as an integral of the form (5.1), whose integrand could consist of several terms j y −s j f j . Moreover, one can express the integral over Γ 0 (4) as the sum of six integrals over F ∞ using modular transformations. As explained in the previous subsection, an efficient technique to evaluate these integrals is to express the integrand as a total derivative with respect toτ , which has indeed been used in a few special cases to evaluate the u-plane integral [3,12,13,14]. We express the integrand of the generic integral (5.8) which by a change of variables is equivalent to an anti-holomorphic derivative in u as discussed in the Introduction. The inverse map u −1 : B → H/Γ 0 (4) maps each of the boundaries ∂ j B to arcs in H/Γ 0 (4) in the vicinity of the cusps {i∞, 0, 2} displayed in Figure 2.
The function H J µ [O](τ,τ ) is required to transform as a modular form of weight (2, 0) with trivial multiplier system, which one may hope to determine explicitly using methods from analytic number theory, especially the theory of mock modular forms [15,16]. To derive a suitable H J µ [O], we will choose a convenient period point J.
is known it is straightforward to apply the discussion of Section 5.1. To relate the integral over H/Γ 0 (4) to an integral over F ∞ , we use coset representatives of SL(2, Z)/Γ 0 (4) to map the six different images of F ∞ within H/Γ 0 (4), displayed in Figure 2, back to F ∞ . After this inverse mapping, we use the modular properties of the integrand to express each of the six integrands as a series in q andq, after which the techniques of Section 5.1 can be applied. To this end, one can use the relations (B.4) for Ψ J µ , while the q-series forν(τ ) follows from the standard relations for Jacobi theta functions.
Since the maps τ → τ − n, n = 1, 2, 3 do not change the constant part of the integrand, we find that Φ where for the second and third brackets on the rhs, one makes the indicated modular transformation for τ , S and T 2 S, and then determines the q 0 coefficient of the resulting Fourier expansion. An important point is the possibility to add to H J µ [O] a holomorphic integration "constant" s O , which is required to be a weight 2 modular form for Γ 0 (4). Of course, should be independent of s O , since definite integrals do not depend on the integration constant. To see the independence of Φ J µ [O] on s O , note that s O will be mapped to a weight 2 form for SL(2, Z) by the inverse mapping. As discussed in Section 5.1, there are no holomorphic SL(2, Z) modular forms with weight 2, and the weakly holomorphic ones have a vanishing constant term. There is therefore no ambiguity arising from the holomorphic integration constant.
On the other hand, the integration constant s O can modify the contribution from each cusp, since a non-vanishing holomorphic modular form of weight 2 for Γ 0 (4) exists. It is explicitly given by ϑ 2 (τ ) 4 +ϑ 3 (τ ) 4 , and while it contributes 4 at the cusp at infinity, the contributions of the three cusps together add up to 0. We can make a natural choice of the integration constant by requiring that the exponential behavior of H J µ for τ → i∞ matches the behavior ofν Ψ J µ in this limit.
for a specific period point J, one can change to an arbitrary J quite easily using indefinite theta functions as discussed in [10,17,18]. The integrand can thus be expressed as a total derivative (5.8) for any J.

Factoring Ψ J µ
To evaluate the partition function Φ J µ , we will choose a convenient period point J so that Ψ J µ , as a function of τ , has a simple factorization as a holomorphic times an antiholomorphic function. In this way, we can easily determine an anti-derivative using the theory of mock modular forms. Using the classification of the uni-modular lattices, Equations (3.5) and (3.6), a convenient factorisation is possible for any intersection form.

Odd intersection form
Let us first assume that the intersection lattice L is odd, such that its quadratic form can be brought to the standard form in Equation (3.5). Since the wall-crossing formula for Donaldson invariants is known [3], it suffices to determine Φ J µ for a convenient choice of J. To this end, we choose the polarization J = (1, 0), (5.11) where 0 is the (b 2 − 1)-dimensional 0-vector. For this choice of J, the orthogonal decomposition of the lattice, L = L + ⊕ L − into a 1-dimensional positive definite lattice L + and (b 2 − 1)-dimensional negative definite sublattice L − , implies that the sum over the U(1) fluxes Ψ J µ (τ,τ ) factors. To see this explicitly, we let k = (k 1 , k − ) ∈ L, and k 1 ∈ Z + µ 1 , k − ∈ L − + µ − and µ = (µ 1 , µ − ). The Siegel-Narain theta function We used also that K 1 is odd since K is a characteristic vector, as discussed below Eq. (3.5). Using we can express f µ in terms of the Dedekind eta function η, (5.14) We can similarly evaluate Θ L − ,µ − . Since all K j are odd, In that case, Θ L − ,µ − is a power of the Jacobi theta function ϑ 4 , After substitution ofν (4.6), we find for the integrand Note that the dependence of the integrand on b 2 has disappeared, and that the integrand diverges for τ → ∞.

Even intersection form
We continue with the even lattices, whose quadratic form can be brought to the form given in Equation (3.6), L = I 1,1 ⊕ n L E 8 . We choose for the period point where the first two components correspond to I 1,1 ⊂ L, and 0 is now the (b 2 − 2)dimensional 0-vector. We have then for the positive and negative definite components of k ∈ L, where k n ∈ nL E 8 . Note k 2 n ≤ 0, since L E 8 is the negative E 8 lattice. The sum over fluxes Ψ J µ factors for this choice of J, where the subscript is µ = (µ + , µ − , µ n ), and Ψ I,(µ + ,µ − ) (τ,τ ) is given by Moreover, the theta series Θ nE 8 ,µ for the negative definite lattice equals As before, the K j are components of the characteristic element K ∈ L, this time in the basis (3.6). Recall K 1 and K 2 ∈ 2Z since they are components of a characteristic vector of I 1,1 . Changing the sign of k 1 and k 2 in the summand gives Ψ I,(µ + ,µ − ) = −Ψ I,(µ + ,µ − ) , hence Ψ I,(µ + ,µ − ) vanishes identically. Nevertheless, it is instructive to evaluate the integral using the approach of Section 5.2, to set up notation for working with the closely analogous function in Equation (5.69), which is definitely nonzero.
To express Ψ I,(µ + ,µ − ) as an anti-holomorphic derivative, we split the lattice into a positive and negative definite one, by changing summation variables to and similarly for the 't Hooft flux and the canonical class, where µ j ∈ Z/2 as before. Given µ ± , the summation over n ± runs over two sets, namely n ± ∈ 2Z + µ ± + j with j = 0, 1. We can now express the sum over fluxes as where µ = (µ + , µ − ) and  We have similarly for t ν

u-plane integrands and mock modular forms
Our next aim is to express the integrand as an anti-holomorphic derivative of a nonholomorphic modular form. We will determine functions F µ (respectively H µ ), which transform as weight 1 2 modular forms, and such that for odd and even lattices respectively. The holomorphic parts of F µ and H ν are known as mock modular forms and contain interesting arithmetic information [15,16]. We consider first the case that the lattice L is odd. We deduce from Equation (5.16) that for µ 1 ∈ Z, we can take F µ 1 = 0. We thus only need to be concerned with finding an anti-derivative F 1 2 of − i 2 √ 2y η 3 . Let us reduce notation by setting F = F 1 2 , then F takes the general form and is required to transform as a Γ 0 (4) modular form with (holomorphic) weight 1 2 . The first term on the rhs is holomorphic and is a mock modular form [15,16], while the second term on the right hand side is known as a period integral and transforms with a shift under transformations of SL(2, Z). The function η 3 is known as the shadow of the mock modular form F . Similarly to the discussion above Eq. (5.6), we deduce that the holomorphic part F (τ ) must be non-vanishing to cancel the shift.
The derivation of such a function is in general non-trivial. The theory of indefinite theta functions provides a constructive approach to derive a suitable F (τ ). Appendix C provides a brief introduction to these functions, and derives an explicit expression for F :  To evaluate Φ J µ following (5.10), we need to determine the q−expansion of F at the other cusps. We introduce to this end F D and F D , where τ is now the local coordinate which goes to i∞ near the strong coupling cusp u → Λ 2 . Appendix C discusses how to derive the q-expansion of F D using the transformations of the indefinite theta function (C.5). One finds For the cusp τ → 2, the q-expansion is −i F D (τ ).

Remark
Malmendier and Ono have emphasized the connection between q-series appearing in the context of Mathieu moonshine and the u-plane integral for the complex projective plane P 2 [45]. See [46,47,48,49,50] for overviews of the moonshine phenomenon. Our discussion above demonstrates that the appearance of these q-series is quite generic for four-manifolds with b + 2 = 1. In particular, the function F (5.30) equals 1/8 times the function H   [51]. Moreover, F and F D can be expressed in terms of the famous q-series H (2) (τ ) of Mathieu moonshine [52], whose coefficients are sums of dimensions of irreducible representations of the finite sporadic group M 24 , and which appeared in the elliptic genus of the K3 sigma model with (4, 4) supersymmetry. We have for F , where [53], Whereas F is a mock modular form for the subgroup Γ 0 (4) ⊂ SL(2, Z), H (2) is a mock modular form for the full SL(2, Z). The completion transforms under the two generators of SL(2, Z) as The holomorphic part H (2) therefore transforms as We can express F D in terms of H (2) as As a last example of a mock modular form with shadow η 3 , we mention the function Q + , which was introduced by Malmendier and Ono in the context of the u-plane integral [12,45] Q + (τ ) = 1 12 Since F , H (2) and Q + are all weight 1 2 mock modular forms for Γ 0 (4) and have shadows proportional to η 3 , each of them can be used for the evaluation of the u-plane integral. Note that among these functions, only F vanishes in the limit τ → i∞. Thus it behaves similarly to its derivative f 1

Evaluation
We continue by evaluating the u-plane integral for an arbitrary four-manifold with (b 1 , b + 2 ) = (0, 1). As mentioned in Subsection 5.3, the partition function Φ J µ is only non-vanishing for odd lattices, and with 't Hooft flux µ with µ 1 = 1 2 in the standard basis. We have then using (5.16) and (5.28), where the first term in the straight brackets is due to the contribution at i∞ and the second term due to the two strong coupling singularities, which contribute equally. The strong coupling singularities do not contribute to the q 0 term. We finally arrive at This is in agreement with the results for P 2 for which K 1 = 3 [54]. It is straightforward to include the exponentiated point observable e 2p u in the path integral. One then arrives at with u D given in (2.5). We deduce from the expansion of F D (5.32) and u D (τ ) = 1+O(q) that the monopole cusps do not contribute to the q 0 -term for any power of p. The result is therefore completely due to the weak coupling cusp, Only even powers of p contribute to the constant term, which is in agreement with the interpretation of the point observable as a four-form on the moduli space of instantons. Except for the mild dependence of (5.43) on K 1 , this equation demonstrates that the contribution from the u-plane to e 2p u J µ is universal for any four-manifold with odd intersection form and period point J (5.11).  2 ) = (0, 1). These numbers are universal for all four-manifolds with odd intersection form, K 1 = 3 mod 4, period point J (5.11), and µ 1 = 1 2 mod Z. For K 1 = 1 mod 4, they differ by a sign, while they vanish for any four-manifold with an even intersection form.
We list Φ J µ [u ] for small in Table 2. See [54] for a more extensive list. Section 6 will discuss these numbers and the convergence of

Evaluation of surface observables
This subsection continues with the evaluation of the contribution of the u-plane to vevs of surface observables.

Odd intersection form
We proceed as in Section 5.3 choosing J = (1, 0) and set ρ + = B(ρ, J) and ρ − = ρ − ρ + J. Specializing (4.26) gives where b = Im(ρ)/y, and we removed the subscript of µ 1 as before in Section 5. Moreover, Θ L − ,µ − (τ, ρ − ) is given by The functions f µ (τ,τ , ρ + ,ρ + ) and Θ L − ,µ − (τ, ρ − ) specialize to f µ (τ,τ ) and Θ L − ,µ − (τ ) (5.13) for ρ = 0. The theta series Θ L − ,µ − (τ, ρ − ) can be expressed as a product of ϑ 1 and ϑ 4 (A.8) depending on the precise value of µ − . We define the dual theta series Θ D,L − ,µ − as We aim to write this as a total anti-holomorphic derivative of a real-analytic function F µ (τ,τ , ρ,ρ) of τ and ρ. We can achieve this using the Appell-Lerch sum M (τ, u, v), with u, v ∈ C\{Zτ + Z}, which has appeared at many places in mathematics and mathematical physics. See for examples [53,55,56]. The function M (τ, u, v) is meromorphic in u and v, and weakly holomorphic in τ . More properties are reviewed in Appendix D. Equation (D.10) is the main property for us. It states that transforms as a multi-variable Jacobi form of weight 1 2 , where R(τ,τ , u,ū) is real-analytic in both τ and u. To determine F µ (τ,τ , ρ,ρ), we express f µ (τ,τ , ρ,ρ) in terms of ∂τ R(τ,τ , u,ū) (D.8). With w = e 2πiρ , we find f µ (τ,τ , ρ,ρ) = 1 2 e πiν q −ν 2 /2 w −ν ∂τ R(τ,τ , ρ + ντ,ρ + ντ ), We can thus determine the anti-derivative of f µ in terms of the completion M (D.9) in Appendix D by choosing u and v such that u − v = ρ + ντ while avoiding the poles in u and v. We will find below that the choice u = ρ + µτ and v = 1 2 τ is particularly convenient. From Appendix D, we find that a candidate for the completed function is We will find that for this choice of u and v, the holomorphic part F 1 2 (τ, ρ) reduces to Let us move on to µ ∈ Z, or µ = 0 to be specific. The function F 0 (τ,τ , ρ,ρ) (5.50) evaluates then to where we used ϑ 1 (τ, 1 2 τ ) = −i q − 1 8 ϑ 4 (τ ). Note that (5.52) contains a pole for ρ = 0, since n = 0 is included in the sum. We will discuss this in more detail later. Let us mention first that we have to be careful with singling out the holomorphic part of (5.52), i.e. the part which does not vanish in the limit y → ∞, b → 0, keeping ρ and τ fixed. Since the elliptic argument of R is shifted by − 1 2 τ , we have lim y→∞ q − 1 8 w 1 2 R(τ,τ , − 1 2 τ, − 1 2τ ) = 1. The holomorphic part of (5.52) is thus To write the non-holomorphic part, we define for µ ∈ {0, 1 2 } mod Z, Note that R µ vanishes in the limit y → ∞ with b = 0 for any µ. Using these expressions, we can write the completed functions F µ as We mentioned in the previous subsection the connection to dimensions of representations of sporadic groups. Other arithmetic information that has appeared in the context of the u-plane integral are the Hurwitz class numbers [3], which count binary integral quadratic forms with fixed determinant. Using relations for the Appell-Lerch sum (D.3), we can make this connection more manifest for the F µ . To this end, let us consider the functions 1 − e 8πiz q 2n−1 .

(5.56)
These functions appear in the refined partition function of SU (2) and SO(3) Vafa-Witten theory on P 2 [57,58,59]. They vanish for z → 0, while their first derivative give generating functions of Hurwitz class numbers H(n) [59]: Using (D.3), we can express the functions g j in terms of F µ as , which demonstrates the connection of the integrand to the class numbers. It might come as a surprise that the expressions we have defined give well-defined power series in x after integration, since the integrand involves expressions with poles in x. This could be avoided by the addition of a meromorphic Jacobi form of weight 2 and index 0, with a pole at ρ = 0 with opposite residue. The reason is that the addition of such a meromorphic Jacobi form does not alter the value of the integral. To see this, note that a meromorphic Jacobi form φ of weight 2 and index 0 has a Laurent expansion in x of the form The φ are weakly holomorphic modular forms for Γ 0 (4) of weight 2, since x is invariant under Γ 0 (4). Mapping the six images of F ∞ in H/Γ 0 (4) to F ∞ gives us a meromorphic Jacobi form φ for SL ( where the φ are modular forms for SL(2, Z) of weight 2. These have a vanishing constant term as discussed before, and thus do not contribute to Φ J µ . To illustrate this, we present an alternative for F 0 (τ, ρ) (5.53), This series is analytic for ρ → 0, and can be expressed as This is the series in terms of which Göttsche expressed the Donaldson invariants of P 2 [60, Theorem 3.5].
Let us return to the evaluation of Φ J µ [e I − (x) ]. To this end, we also need to determine the magnetic dual versions F D,µ . We let w D = e 2πiρ D , and define F D,µ by We evaluate the rhs using the transformation of M (D.10). Subtracting the subleading non-holomorphic part gives for F D,µ (τ, ρ D ) Note that for this choice of J, Φ J µ [e I − (x) ] only depends on µ, K and b 2 (assuming b 1 = 0). We list Φ J µ [I s − (x)] for the first few non-vanishing s in Table 3. If we specialize to the four-manifold P 2 , these results are in agreement with the results in Reference [54, Theorem 4.2 and Theorem 4.4].

Even intersection form
We proceed similarly for the case that the lattice L is even. As in the discussion of Section 5.3, we choose for the period point J = 1  Table 3: For a smooth four-manifold with (b 1 , b + 2 ) = (0, 1) and odd intersection form, these tables list non-vanishing Φ J µ [I s − (x)] for 0 ≤ s ≤ 17 and x = (1, 0). No assumption is made about the value of b 2 . The left table is for µ 1 ∈ Z, and the right table is for µ 1 = 1 2 + Z. For µ 1 ∈ Z, I − (x) s is an integral class, while for µ 1 ∈ 1 2 + Z, 2 s I − (x) s is an integral class. The first entry at s = 1 is fractional, but (we believe) this arises because the moduli space is a stack with nontrivial stabilizer group.

(5.80)
The holomorphic parts of these completed functions are with the F ν given by (5.51) and (5.53). We define the dual G ± D,ν as These evaluate to With these expressions, we can present our final expression Φ J µ [e I − (x) ] for fourmanifolds with even intersection form, The overall factor 2 for the strong coupling contributions is due to the factors of √ 2 in (5.78) and (5.82). Table 4 lists the contribution from the u-plane to Donaldson polynomials for small instanton number. The expressions confirm that I − (x) is an integral class for gauge group SU(2) µ = 0 mod Z, and half-integral for µ = 0 mod Z.

Asymptology of the u-plane integral
Up to this point we have treated the u-plane integral Φ J µ [e 2pu+I − (x) ] as a formal generating series in the homology elements p and x. However, one might ask if the integral actually expresses a well-defined function on the homology of the four-manifold M . In other words, one might ask if the formal series is in fact convergent. The contribution of the Seiberg-Witten invariants is a finite sum and hence in fact defines an entire function on H * (M, C). Therefore the Donaldson-Witten partition function is a welldefined function on the homology if and only if the u-plane is a well-defined function. If that were the case then one could explore interesting questions such as the analytic structure of the resulting partition function, which, in turn, might signal interesting  physical effects. In this Section we will explore that question, starting with the point observable in Subsection 6.1. We will find strong evidence that in fact the u-plane integral is indeed an entire function of p.
The situation for x is less clear, since the numerical results are more limited. We discuss in Subsection 6.1 that the results do suggest that Φ J µ [e 2pu+I − (x) ] is also an entire function x. As a step towards understanding the analytic structure in x, we will consider in Subsection 6.2 the magnitude of the integrand in the weak-coupling limit. Although the integral is independent of α we will see that the integrand behaves best when α = 1.

Asymptotic growth of point observables
We will analyze the dependence of the contribution from the u-plane Φ J µ [e 2p u ] to the correlation function e 2p u J µ . Due to the exponential divergence of u (2.4) for τ → i∞, the divergence of e 2p u is doubly exponential. The u-plane integral thus formally diverges. The discussion of Section 5.1 does not provide an immediate definition of such divergent expressions. On the other hand, the vev of the exponentiated point observable e 2p u should be understood as a generating function of correlation functions, and we can define As discussed in Section 5.4, there is no problem evaluating Φ J µ [u ] using the definition of Section 5.1.
We consider the case of odd lattices, and the period point J (5.11). Modifying We have replaced here F by 1 24 H (2) , since its completion is an equally good choice of anti-derivative. It is straightforward to determine Φ J µ [u ] using this expression. We list in Table 5 values of Φ J µ [u ] for various large values of .
Before giving evidence that the Φ J µ [e 2p u ] is an entire function of p, let us discuss the integrand in more detail. We first write Φ J µ [u ] as an integral from 0 to 1: We can express the integrand in a SL(2, Z) invariant form. To this end, note where Q is the weight 6 + 3 modular form defined by  The most straightforward way of trying to establish the large asymptotics is by a saddle point analysis. Expressing u as u D (−1/τ ) ∼ e 32 e − 2πi τ , we find that, to first approximation, the saddle point is at τ * = 2πi log(−32 ) for large . We find that the contribution of this saddle point to Φ J µ [u ] behaves as C/ for some constant C. We leave an investigation into the difference between the saddle point contribution and Table 5 for another occasion.
Let us explore the consequences of the large asymptotics for Φ J µ [e 2pu ]. We deduce from the Table 5 Table 6 some numerical data for log(Φ[I − (x) s ])/s. These data, while admittedly limited, do give the impression that the asymptotic growth of Φ J µ [I − (x) s ] is bounded by e αs log(s) for some positive constant α, and that α < 1. Assuming that this is the correct behavior for large s, the radius of convergence for We hope to get back to the asymptotics of these correlators in future work.

Weak coupling limit of the integrand
As a first step towards understanding the asymptotic behavior of the series in x we investigate here the growth of the integrand of the u-plane integral in the weak coupling region. In order to do this it is useful to recall that one can add to the surface observable the operator I + (x) discussed in section 4.2 with an arbitrary coefficient α. Since I + (x) is Q exact such an addition does not modify the resulting integral. (Because the integral is subtle and formally divergent this statement requires careful justification, but it turns out to be correct [19].) In this way, we can interpolate between α = 0 [3] and α = 1 [17]. While the result is independent of α, the dependence of the integrand is worth exploring in more detail, in particular the behavior in the weak coupling limit. In this  The u-plane integral with observable e I − +α I + results in a modified sum over fluxes Ψ J µ,α . This sum is defined as in (4.26), but withρ replaced by αρ, and reads By completing the squares in the exponent, we can write this as where b = Im(ρ)/y as before. The sum over k ∈ L + µ is dominated by the k for which −(k + b) 2 − + (k + b) 2 + is minimized. For a generic choice of period point J, there is only one k ∈ L + µ which minimizes this quantity. The leading asymptotic behavior is given by the exponential multiplying the sum. This evaluates to Thus we see that α = 1 is special, since for this choice the exponent is negative definite for x 2 > 0. Moreover, for large y, Ψ J µ,α will only remain finite in the domain ϕ = Re(τ ) ∈ [0, 4] for this choice of parameters. The double exponential divergence of the exponentiated surface observable is therefore mitigated at α = 1.
Let us make a rough estimate for the magnitude of the u-plane integral using (6.11), where we only consider terms which are non-vanishing in the limit for y → ∞. The integral over ϕ is a Bessel function I 0 (z) with z = πx 2 2y e 1 2 πy , which behaves for large z as e z / √ 2πz. This leads to a single exponential divergence, dy e 1 4 πy , which can be treated as discussed before. We leave a more detailed analysis including the dependence on µ and possible cancellations in the integral along the interval ϕ ∈ [0, 4] for future work.

Modular groups
The modular group SL(2, Z) is the group of integer matrices with unit determinant (A.1) We introduce moreover the congruence subgroup Γ 0 (n)

Eisenstein series
We let τ ∈ H and define q = e 2πiτ . Then the Eisenstein series E k : H → C for even k ≥ 2 are defined as the q-series with σ k (n) = d|n d k the divisor sum. For k ≥ 4, E k is a modular form of SL(2, Z) of weight k. In other words, it transforms under SL(2, Z) as Note that the space of modular forms of SL(2, Z) forms a ring that is generated precisely by E 4 (τ ) and E 6 (τ ). On the other hand E 2 is a quasi-modular form, which means that the SL(2, Z) transformation of E 2 includes a shift in addition to the weight,

B. Siegel-Narain theta function
Siegel-Narain theta functions form a large class of theta functions of which the Jacobi theta functions are a special case. For our applications in the main text, it is sufficient to consider Siegel-Narain theta functions for which the associated lattice L is a unimodular lattice with signature (1, n − 1) (or a Lorentzian lattice). We denote the bilinear form by B(x, y) and the quadratic form by B(x, x) ≡ Q(x) ≡ x 2 . Let K be a characteristic vector of L, such that Q(k) + B(k, K) ∈ 2Z for each k ∈ L.
Given an element J ∈ L ⊗ R with Q(J) = 1, we may decompose the space L ⊗ R in a positive definite subspace L + spanned by J, and a negative definite subspace L − , orthogonal to L + . The projections of a vector k ∈ L to L + and L − are then given by Given this notation, we can introduce the Siegel-Narain theta function of our in- where µ ∈ L/2 and K : L ⊗ R → C is a summation kernel. Let us be slightly more generic and include the elliptic variables which are relevant for the Donaldson observables. We define For the case of the partition function, we set the elliptic variables z,z to zero. Using the above SL(2, Z) transformations and Poisson resummation one may verify that Ψ J µ [1] is a modular form for the congruence subgroup Γ 0 (4 where we have set z =z = 0. Transformations for other kernels appearing in the main text are easily determined from these expressions. C. Indefinite theta functions for uni-modular lattices of signature (1, n − 1) In this appendix we discuss various aspects of the theory of indefinite theta functions and their modular completion. We assume that the corresponding lattice L is unimodular and of signature (1, n − 1), and use the notation introduced in Appendix B. To define the indefinite theta function Θ JJ µ , we let µ ∈ L/2 and choose a vector J ∈ L ⊗ R and a vector J ∈ L, such that The kernel within the straight brackets ensures that the sum over the indefinite lattice is convergent, since it vanishes on positive definite vectors [15]. This is also the case if both J and J are positive definite, without the need to impose condition (iv). One may start from this situation and obtain the conditions above by taking the limit that J approaches a null vector. The indefinite theta series Θ JJ µ can also be defined, for µ which do not satisfy requirement (iv) above, but this requires more care.
We can express Θ JJ µ also as Ψ J where b = Im(z)/y. While the sum over L is convergent, Θ JJ µ only transforms as a modular form after addition of certain non-holomorphic terms. References [15,16] explain that the modular completion Θ JJ µ of Θ JJ µ is obtained by substituting (rescaled) error function for the sgn-function in (C.1). We let E(u) : R → (−1, 1) be defined as The completion Θ JJ µ then transforms as a modular form of weight n/2, and is explicitly given by Θ JJ µ (τ,τ , z,z) = (C.4) Note that in the limit y → ∞, E( √ 2y u) approaches the sgn-function of (C.1), lim y→∞ E( 2y u) = sgn(u).

An example
Let us now specialize to an example which is useful in Section 5.4. We consider a twodimensional lattice L ∼ = Z 1,1 with quadratic form − 1 1 1 0 . The positive and negative definite cones of this lattices are illustrated in Figure 3. The upper-right component of the negative cone for this lattice is generated by the vectors (0, 1) and (2, −1). Any linear combination of these vectors with positive definite coefficients will be negative definite. We choose the vectors J and J as follows: J = (−1, 1) and J = (0, 1). For k = (n, ), the kernel in (C.1) becomes (sgn( ) + sgn(n)). Then the only elements of L which contribute to the theta series are those in the two yellow areas in Figure 3.