PPreprint typeset in JHEP style - HYPER VERSION
Mocking the u -plane integral Georgios Korpas , , Jan Manschot , , Gregory W. Moore and Iurii Nidaiev School of Mathematics, Trinity College, Dublin 2, Ireland Hamilton Mathematical Institute, Trinity College, Dublin 2, Ireland NHETC and Department of Physics and Astronomy, Rutgers University, 126Frelinghuysen Rd., Piscataway NJ 08855, USA
Abstract:
The u -plane integral is the contribution of the Coulomb branch to corre-lation 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 twistedtheories with gauge group SU(2), for an arbitrary four-manifold with ( b , b +2 ) = (0 , u -plane contribution equals the full correlator in the absence of Seiberg-Witten con-tributions at strong coupling, and coincides with the mathematically defined Donaldsoninvariants in such cases. We demonstrate that the u -plane correlators are efficientlydetermined using mock modular forms for point observables, and Appell-Lerch sumsfor surface observables. We use these results to discuss the asymptotic behavior ofcorrelators as function of the number of observables. Our findings suggest that the vevof exponentiated point and surface observables is an entire function of the fugacities. a r X i v : . [ h e p - t h ] O c t ontents
1. Introduction 22. Seiberg-Witten theory and Donaldson-Witten theory 5
3. A survey of four-manifolds with b +2 = 1 b +2 = 1 103.3 Donaldson invariants 12
4. Path integral and correlation functions 13
5. Evaluation of u -plane integrals 21 H / SL(2 , Z ) 215.2 General strategy 225.3 Factoring Ψ J µ u -plane integrands and mock modular forms 275.5 Evaluation of surface observables 31
6. Asymptology of the u -plane integral 39 A. Modular forms and theta functions 45B. Siegel-Narain theta function 48C. Indefinite theta functions for uni-modular lattices of signature (1 , n − – 1 – . Introduction A powerful approach to understand the dynamics of supersymmetric field theories isto 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 = 2supersymmetric Yang-Mills theory with gauge group SU(2) and in the presence ofarbitrary ’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 = π (cid:104) Tr[ φ ] (cid:105) R , wherethe subscript indicates that this is a vev in a vacuum state of the theory on flat R . TheCoulomb branch, also known as the “ u -plane” can be considered as a three puncturedsphere, where the punctures correspond to the weak coupling limit, u → ∞ , and thetwo strong coupling singularities for u = ± Λ .The contribution of the u -plane to a correlation function (cid:104)O O . . . (cid:105) is non-vanishingif the four-manifold M satisfies the topological condition b +2 ( M ) ≤
1, where b +2 is thenumber of positive definite eigenvalues of the intersection form of two-cycles of M . Foran observable O = O O . . . , the vev (cid:104)O(cid:105) can be expressed as a sum of two contri-butions: the Seiberg-Witten contribution (cid:104)O(cid:105) SW from the strong coupling singularities u = ± Λ , and the contribution from the u -plane Φ[ O ], (cid:104)O(cid:105) = (cid:104)O(cid:105) SW + Φ[ O ] , (1.1)This paper considers the u -plane contribution Φ[ O ] for compact four-manifoldswith b +2 = 1 known as the u -plane integral [3]. The integrand of Φ[ O ] for b +2 = 1does not receive perturbative corrections, such that the path integral reduces to afinite dimensional integral over the zero modes of the fields. After including the non-perturbative corrections to the integrand using the Seiberg-Witten solution [9], the u -plane integral has been evaluated for some four-manifolds with b = 1 or 2, namelyfor 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], whichcould be traced to simplifying features, such as a vanishing chamber, wall-crossing, orbirational transformations. For generic four-manifolds with b +2 = 1, these simplifyingfeatures are not available. Nevertheless, we will demonstrate that u -plane integrals ofarbitrary four-manifolds with b +2 = 1 can be readily evaluated by integration by partsleading to expressions in terms of mock modular forms and Appell-Lerch sums. Forsimplicity, we will restrict to four-manifolds with ( b , b +2 ) = (0 , The u -plane integral also contributes for manifolds with b +2 = 0. The integrand is one-loop exactin this case [3], but the one-loop determinants have never been worked out with great care. – 2 –o achieve the evaluation of these u -plane integrals, we change variables from u tothe effective coupling constant τ , such that Φ becomes an integral over the modularfundamental domain H / Γ (4), where Γ (4) is the duality group of the theory. We areable to express the integrand as a total derivative dτ ∧ d ¯ τ ∂ ¯ τ ( dudτ H O ), for some H O whichdepends on the observable O . Reversing the change of variables, this demonstrates thatthe integrand takes the form du ∧ d ¯ u ∂ ¯ u H O , and the integral is thus reduced to integralsover the boundaries ∂ j B , j = 1 , , {− , +1 , ∞} byStokes’ theorem. See Figure 1. More explicitly, we haveΦ[ O ] = (cid:90) B du ∧ d ¯ u ∂ ¯ u H O ( u, ¯ u ) = (cid:88) j =1 (cid:73) ∂ j B du H O ( u, ¯ u ) . (1.2)In order for this expression to be useful, it is necessary that H O ( u, ¯ u ), when expressedin terms of τ, ¯ τ , has good modular properties allowing one to make the required dualitytransformation near strong coupling singularities. We will find, for a special choice ofmetric, that H O ( τ, ¯ τ ) can be expressed in terms of mock modular forms. Then, giventhe expression for the wall-crossing formula using indefinite theta functions [17, 18] thesame result follows for general metric. •1 • • +1 u -plane1 Figure 1: Schematic representation of the u -plane, with the singularities {∞ , − , +1 } .The black circles indicate the boundaries ∂ j B of the u -plane after removing neighbor-hoods of the singularities, while the dashed circle denotes the wall of marginal stabilitybetween the strong and weak coupling regions of the u -plane.The expression for the u -plane integral as a modular integral over H / Γ (4) pavesthe way for its evaluation. Earlier work has demonstrated that such modular integrals– 3 –valuate to the constant term of a q -series, or more specifically, the q term of a mockmodular form [19, 20]. We thus establish a close connection between u -plane correlationfunctions and mock modular forms. Said more mathematically, we have established aconnection between Donaldson invariants for general manifolds with b +2 = 1 and mockmodular forms. The explicit expressions are (5.44) for manifolds with odd intersectionform and just point observables inserted, (5.65) for manifolds with odd intersection formand just surface observables inserted, and (5.84) for manifolds with even intersectionform and just surface observables inserted. These expressions hold for a particularlynice choice of metric. The metric dependence only enters through the choice of periodpoint , i.e. the unique self-dual degree two cohomology class in the forward light-conein H ( M ; R ). Using the expression for the wall-crossing formula in terms of indefinitetheta functions [17, 18] one can produce analogous mock modular forms relevant toother chambers. Expressions (5.44), (5.65) and (5.84) (or close cousins thereof) haveappeared before in [10]. The derivations in [10] relied on the existence of a vanishingchamber and applied wall-crossing formulae. By contrast, in this paper we evaluatethe 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 exampleEquation (5.44)), we can address analytic properties of the correlators for b +2 = 1,analogously to the structural results for manifolds with b +2 > u (cid:96) ] for large (cid:96) , and find experimental evidence that Φ[ u (cid:96) ] ∼ / ( (cid:96) log( (cid:96) )) for any four-manifold with ( b , b +2 ) = (0 , u (cid:96) ] suggests that Φ[ e p u ] = (cid:80) (cid:96) ≥ (2 p ) (cid:96) Φ[ u (cid:96) ] /(cid:96) ! is an entire function of p rather than a formal expansion. We find similar experimental evidence that the u -planecontribution to the exponentiated surface observable Φ[ e I − ( x ) ] is an entire function of x ∈ H ( 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 correlationfunctions of large charge that have recently been studied in [22] and again we leave theinvestigation of this potential connection for future work.One can change variables from q to the complex electric mass a in Φ[ O ], andexpress the u -plane integral as a residue of a around ∞ and 0. One may in this wayconnect to other techniques for the evaluation of Donaldson invariants, for examplesthose using toric localization [23, 24]. Our results may also be useful for the evaluationof Coulomb branch integrals of different theories, such as those including matter andsuperconformal theories [18], and for four-manifolds with b (cid:54) = 0.The outline of the paper is as follows. Section 2 reviews Seiberg-Witten theory andits topological twist. Section 3 gives a lightning overview of compact four-manifoldswith b +2 = 1. Section 4 continues with introducing the path integral and correlation– 4 –unctions of the theory on these manifolds, which are evaluated in Section 5. We closein Section 6 with an analysis on the asymptotic behavior of correlation functions witha large number of fields inserted.
2. Seiberg-Witten theory and Donaldson-Witten theory
We give a brief review of pure Seiberg-Witten theory [9, 25], and its topologicallytwisted counterpart aka Donaldson-Witten theory [8]. See [26, 27] for a detailed intro-duction to both of these theories.
Seiberg-Witten theory is the low energy effective theory of N = 2 supersymmetricYang-Mills theory with gauge group G = SU(2) or SO(3) and Lie algebra su (2). Thebuilding blocks of the theory contain a N = 2 vector multiplet which consists of agauge field A , a pair of (chiral, anti-chiral) spinors ψ and ¯ ψ , a complex scalar Higgsfield 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 ingeneral. 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 are the central charges for a unitelectric and magnetic charge. The parameters a and a D are expressed in terms of theholomorphic prepotential F of the theory a D = ∂ F ( a ) ∂a . (2.1)Its second derivative equals the effective coupling constant τ = ∂ F ( a ) ∂a = θπ + 8 πig ∈ H , (2.2)where θ is the instanton angle with periodicity 4 π , g is the Yang-Mills coupling and H is the complex upper half-plane. The Coulomb branch B is parametrized by the orderparameter, u = 116 π (cid:10) Tr [ φ ] (cid:11) R , (2.3)where the trace is in the 2-dimensional representation of SU(2). The renormalizationgroup flow relates the Coulomb branch parameter u and the effective coupling con-stant τ . Using the Seiberg-Witten geometry [9], the order parameter u can be exactlyexpressed as a function of τ in terms of modular forms, u ( τ )Λ = ϑ + ϑ ϑ ϑ = 18 q − + 52 q − q + O ( q ) , (2.4)– 5 –here Λ is a dynamically generated scale, q = e πiτ , and ϑ i ( τ ) are the Jacobi thetafunctions, which are explicitly given in Appendix A. The function u ( τ ) is invariantunder transformations τ (cid:55)→ aτ + bcτ + d given by elements of the congruence subgroup Γ (4) ⊂ SL(2 , Z ). See Equation (A.2) in Appendix A for the definition of this group. A changeof variables from u to τ maps the u -plane to a fundamental domain of Γ (4) in theupper-half plane H . We choose the fundamental domain as the union of the images ofthe familiar key-hole fundamental domain of SL(2 , Z ) under τ (cid:55)→ τ + 1, τ + 2, τ + 3, − /τ and 2 − /τ , which is displayed in Figure 2. Let u D be the vector-multiplet scalarfor the dual photon vector multiplet with coupling constant τ D = − /τ . Then u D ( τ D )Λ = u ( − /τ D )Λ = ϑ + ϑ ϑ ϑ = 1 + 32 q D + 256 q D + 1408 q D + O ( q D ) . (2.5) − − F ∞ S F ∞ T F ∞ T F ∞ T S F ∞ T F ∞ Re( τ )Im( τ ) Figure 2: Upper-half plane H with the area bounded by blue ( F ∞ ) a fundamentaldomain of H / SL(2 , Z ), and the shaded area a fundamental domain of H / Γ (4).At the cusp τ → τ →
2) a monopole (respectively a dyon) becomesmassless, and the effective theory breaks down since new additional degrees of freedom One way to understand this duality group is that the Seiberg-Witten family of curves is theuniversal family of family of elliptic curves with a distinguished order 4 point [9]. – 6 –eed to be taken into account. Another quantity which we will frequently encounter isthe derivative dadu . It is expressed as function of τ asΛ dadu ( τ ) = 12 ϑ ( τ ) ϑ ( τ ) , (2.6)and transforms under a standard pair of generators of Γ (4) as dadu ( τ + 4) = − dadu ( τ ) ,dadu (cid:18) ττ + 1 (cid:19) = ( τ + 1) dadu ( τ ) . (2.7)Let us also give the expression of the dual of this quantity ( dadu ) D Λ (cid:18) dadu (cid:19) D ( τ D ) = τ − D dadu ( − /τ D ) = − i ϑ ( τ D ) ϑ ( τ D ) , (2.8) Donaldson-Witten theory is the topologically twisted version of Seiberg-Witten the-ory with gauge group SU(2) or SO(3), and contains a class of observables in its Q -cohomology, which famously provide a physical realization of the mathematically de-fined Donaldson invariants [28, 29].Topological twisting preserves a scalar fermionic symmetry Q of N = 2 Yang-Millson an arbitrary four-manifold [8]. The twisting involves a choice of an isomorphismof an associated bundle to the SU(2) R -symmetry bundle with an associated bundleto the frame bundle. Namely, we choose an isomorphism of the adjoint bundle of theSU(2) R R -symmetry bundle with the bundle of anti-self-dual 2-forms, and we choosea connection on the R -symmetry bundle, which under this isomorphism becomes theLevi-Civita connection on the bundle of anti-self-dual 2-forms. In practice, this allowsus 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 original supersymmetry generators transform as the ( , , ) ⊕ ( , , ) repre-sentation of SU(2) + × SU(2) − × SU(2) R group. Their representation under the twistedrotation group SU(2) (cid:48) + × SU(2) − × U(1) R is ( , ) +1 ⊕ ( , ) − ⊕ ( , ) +1 . The firstterm ( , ) +1 corresponds to the BRST-type operator Q , whose cohomology providesoperators in the topological field theory. The second term ( , ) − corresponds to theone-form operator K , which provides a canonical solution to the descent equations {Q , O ( i +1) } = d O ( i ) , i = 0 , . . . , , (2.9) Note that in [3, 17] this operator is denoted as Q . – 7 –y setting O ( i ) = K i O (0) [3, 4, 30]. Integration of the operators O ( i ) over i − cycles givestopological observables since {Q , K } = d .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-dualtwo-form χ , one-form ψ and zero-form η . The auxiliary fields of the non-twisted theorycombine to a self-dual two-form D . The action of the BRST operator Q on these fieldsis given by [ Q , A ] = ψ, [ Q , a ] = 0 , [ Q , ¯ a ] = √ iη, [ Q , D ] = ( d A ψ ) + , {Q , ψ } = 4 √ da, {Q , η } = 0 , {Q , χ } = i ( F + − D ) . (2.10)The low energy Lagrangian of the Donaldson-Witten theory is given by [3] L DW = i π (¯ τ F + ∧ F + + τ F − ∧ F − ) + y π da ∧ ∗ d ¯ a − y π D ∧ ∗ D − π τ ψ ∧ ∗ dη + 116 π ¯ τ η ∧ d ∗ ψ + 18 π τ ψ ∧ dχ − π ¯ τ χ ∧ dψ + i √ π d ¯ τd ¯ a ηχ ∧ ( F + + D ) − i √ π dτda ψ ∧ ψ ∧ ( F − + D )+ i · d τda ψ ∧ ψ ∧ ψ ∧ ψ − i √ · π (cid:8) Q , χ µν χ νλ χ µλ (cid:9) √ g d x, (2.11)where y = Im( τ ) >
3. A survey of four-manifolds with b +2 = 1 We aim to evaluate and analyze the u -plane integral for compact four-manifolds with( b , b +2 ) = (0 ,
1) (and without boundary ). This is a large class of manifolds whichincludes among others complex rational surfaces and examples of symplectic mani-folds. The u -plane integral is well-defined and can be evaluated for all these four-manifolds. This section gives a brief review of the standard geometric aspects of thesefour-manifolds. Let M be a compact four-manifold, and let b j = dim( H j ( M, R )) be the Betti numbersof M . For simplicity we restrict to manifolds with b = 0, we do not require them tobe simply connected. The torsion subgroups of H ( M, Z ) and H ( M, Z ) are naturally We will only consider four-manifolds without boundary in this paper. – 8 –ual by Poincar´e duality. They will not play an important role here, since they simplylead to an overall factor (the order) from the addition of flat connections.We denote by L the image of the Abelian group H ( M, Z ) ∈ H ( M, R ), whicheffectively mods out the torsion in H ( M, Z ). As a result, L is a lattice in a real vectorspace, and we can divide elements of L without ambiguity. If the context allows, we willoccasionally use H ( M, Z ) and L interchangeably. The intersection form on H ( M, Z )provides a natural non-degenerate bilinear form B : ( L ⊗ R ) × ( L ⊗ R ) → R that pairsdegree two co-cycles, B ( k , k ) := (cid:90) M k ∧ k , (3.1)and whose restriction to L × L is an integral bilinear form. The bilinear form providesthe quadratic form Q ( k ) := B ( k , k ) ≡ k , which is uni-modular and possibly indefinite.For later use, recall that a characteristic element of L is an element c ∈ L , such that Q ( k ) + B ( c , k ) ∈ Z . (3.2)We let furthermore H ( M, R ) ± be the positive definite and negative definite subspacesof H ( M, R ), and set b ± = dim( H ( M, R ) ± ). Van der Blij’s Lemma states that acharacteristic element c of a lattice L satisfies Q ( c ) = σ L mod 8, where σ L = b +2 − b − is the signature of L .The second Stiefel-Whitney class w ( T M ) is a class in H ( M, Z ), which distin-guishes spinnable from non-spinnable manifolds. A smooth, spinnable manifold has w ( T M ) = 0, while w ( T M ) (cid:54) = 0 for non-spinnable manifolds. The class w ( T M ) hasimplications 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 ofthe Stiefel-Whitney class defines a characteristic vector in L . Therefore, w ( T M ) = 0implies that L is an even lattice. The converse is however only true if M is sim-ply connected due to the possibility that w ( T M ) is represented by a torsion class in H ( M, Z ). An even stronger statement for the intersection form of smooth, simplyconnected, spinnable four-manifolds is Rokhlin’s theorem, which states that the sig-nature of such manifolds satisfies σ L = 0 mod 16. Note that the Enriques surfaceis smooth while it has intersection form I , ⊕ L E , where I , is the two-dimensionallattice with quadratic form ( ), and L E is minus the E root lattice. This does notcontradict Rokhlin’s theorem since the Enriques surface is not simply connected. It isalso worth noting that for complex manifolds the canonical class K is an integral lift ofthe Stiefel-Whitney class and therefore any other integral lift differs by twice a latticevector in L .Any closed, orientable four-manifold admits a Spin C structure. To a Spin C structureone attaches a first Chern class of a certain line bundle, which we refer to as the first– 9 –hern class of the Spin C structure. The first Chern class of a Spin C structure is anintegral lift of w ( T M ) and is therefore a characteristic vector c ∈ L . Interestingly, theexistence of an almost complex structure for a smooth four-manifold M is related to theexistence of a characteristic vector c with fixed norm. Note that an almost complexstructure ensures that the tangent bundle T M is complex, such that its Chern class c ( T M ) ∈ H ( M, Z ) and canonical class K = − c ( T M ) are well-defined. The Riemann-Roch theorem for four-manifolds with an almost complex structure demonstrates thatits canonical class K is a characteristic element of L . Moreover: • The modulo 2 reduction of K satisfies w ( T M ) = K mod 2 . (3.3) • By the Hirzebruch signature theorem Q ( K ) = 2 χ + 3 σ, (3.4)where χ = 2 − b + b is the Euler number of M , and σ = b +2 − b − is the signatureof M .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 ofthis statement with Van der Blij’s Lemma demonstrates that if M admits an almostcomplex structure, then b +2 + b must be odd. b +2 = 1We will specialize in the following to b +2 = 1. In this case, the quadratic form Q canbe brought to a simple standard form [29, Section 1.1.3], which will be instrumental toevaluate the u -plane integral in Section 5. The standard form depends on whether thelattice is even or odd: • If Q is odd, an integral change of basis can bring the quadratic form to thediagonal form (cid:104) (cid:105) ⊕ m (cid:104)− (cid:105) , (3.5)with m = b −
1. This has an important consequence for characteristic elementsof such lattices. If K is a characteristic element, k + B ( K, k ) ∈ Z for any k ∈ L . In the diagonal basis (3.5) this equivalent to (cid:80) b j =1 k j + K j k j ∈ Z with K = ( K , K , . . . , K b ). This can only be true for all k ∈ L if K j is odd for all j = 1 , . . . , b . – 10 – If Q is even, the quadratic form Q can be brought to the form I , ⊕ n L E , (3.6)where I , and L E as defined above and n = ( b − /
8. The components K j , j = 1 , M is its period point J ∈ H ( M, R ), which is thegenerator of H ( M, R ) + , normalized such that Q ( J ) = 1. The period point dependson the metric due to the self-duality condition. In fact, the metric dependence in theexpressions below only enters through a choice of J . Using J , we can project k ∈ L to the positive and negative definite subspaces H ( M, R ) ± : k + = B ( k , J ) J is theprojection of k to H ( M, R ) + , and k − = k − k + is the projection to H ( M, R ) − . Notethat these projections are also the self-dual and anti-self-dual parts of k with respectto the Hodge ∗ -operation. Complex four-manifolds with b +2 = 1Complex 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 complexsurface. This is a non-singular surface which can not be obtained from another non-singular surface by blowing up a point. This is equivalent to the statement that thesurface does not contain rational curves with self-intersection − − b , b +2 ) = (0 , −∞ , 0, 1 or 2: • Surfaces with Kodaira dimension −∞ are surfaces whose canonical bundle doesnot admit holomorphic sections. These surfaces are birational to more than oneminimal surface. The simply connected surfaces with b +2 = 1 in this family arethe rational surfaces, i.e. the complex projective plane P , Hirzebruch surfacesand blow-ups of these surfaces. A special property of these surfaces are vanishingchambers where the moduli spaces of instantons are empty. This has been usefulfor the explicit determination of partition functions on these geometries, includingthe 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 , b +2 ) = (0 , I , ⊕ L E . Note that this four-manifold is not simply-connected, and that w ( T M )is represented by a torsion class in H ( M, Z ).– 11 – Surfaces with Kodaira dimension 1 are surfaces for which the canonical class K satisfies Q ( K ) = 0, and B ( K, C ) > C . Such surfaces are elliptic(but the converse is not always true). The Dolgachev surfaces are a family ofsimply-connected surfaces with Kodaira dimension 1. • Surfaces with Kodaira dimension 2 are surfaces of general type. If a surface inthis 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 foreach integer in this set such as the Godeaux and Barlow surfaces which both haveEuler 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, mostfour-manifolds are not complex and their classification is an important open problem. Adistinguished class of four-manifolds with b +2 = 1 are symplectic ones, which partiallyoverlap with the complex four-manifolds. For a four-manifold to be symplectic, itsperiod point J must provide a symplectic structure. Reference [36] provides a surveyof such manifolds. Examples of symplectic four-manifolds which are not complex arefour-manifolds denoted by E (1) N , that is four-manifolds which are homotopy equivalentto a rational elliptic surface and whose construction relies on a fibered knot N in S [37, 38]. The manifolds in this class have b = 0. For recent progress on symplectic,non-complex manifolds with Kodaira dimension 1 ( Q ( K ) = 0 and B ( K, C ) >
0) with b (cid:54) = 0, see [39]. Donaldson invariants have been of crucial importance for the classification of four-manifolds, 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 theDonaldson invariants in the formalism of topological field theory. Let M γ be themoduli space of solutions to the ASD equations for gauge group SU(2) or SO(3), where γ = ( c , k ) represents the topological numbers of the solution, that is to say c = i π Tr( F ) ∈ H ( M, Z ) and k = π (cid:82) M Tr[ F ]. The map µ γD : H i ( M, Q ) → H − i ( M γ , Q )maps an i -cycle on M to a (4 − i )-form on M γ .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 A symplectic structure of a four-manifold is given by a two-form ω , which satisfies dω = 0 and ω ∧ ω > M . In other words, ω is closed and non-degenerate. – 12 –opological theory F = F + ψ + φ [40]. The class µ D is defined in terms of the firstPontryagin class of the universal bundle, µ D = − p ( U ) = 18 π Tr[ F ] , (3.7)where the trace is in the two-dimensional representation of the gauge group, i.e. thefundamental 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 finiteset of points of M and p = [ r ] + [ r ] + · · · ∈ H ( M, Z ) the corresponding 0-cycle. Then µ D ( p ) evaluates to µ D ( p ) = 18 π (cid:88) j Tr[ φ ( r j ) ] , (3.8)which we interpret here as a four-form on M γ . Since the cohomology class of Tr[ φ ] isindependent of position, we can express µ D ( p ) equivalently as µ D ( p ) = 2 p ( p ) u, (3.9)with u as in (2.3) and p : H ( M, Z ) → R the unique linear map satisfying p ( e ) = 1,where e is a generator of H ( M, Z ). For x ∈ H ( M, Z ), µ D ( x ) provides similarly a two-form on M γ . See Equation (4.13) for the precise expression in terms of the physicalfields. Using the linearity of the map µ D , we extend the definition of µ D from H ∗ ( M, Z )to H ∗ ( M, C ).Using the map µ D , we can define the Donaldson invariant D γ(cid:96),s ( p , x ) ∈ Q as theintersection number D γ(cid:96),s ( p , x ) = (cid:90) M γ µ D ( p ) (cid:96) ∧ µ D ( x ) s ∈ Q . (3.10)The number D γ(cid:96),s is only non-vanishing if 4 (cid:96) + 2 s = dim R ( M γ ). For smooth four-manifolds the virtual dimension of the moduli space isdim R ( M γ ) = − p −
32 ( χ + σ ) = 8 k −
32 ( χ + σ ) . (3.11)and in general this is in fact the dimension. For complex surfaces, we can write 2 (cid:96) + s =4 k − c − χ 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 partitionfunction in Subsection 4.1 and correlation functions in Subsection 4.2.– 13 – .1 Path integral We consider Donaldson-Witten theory on a four-manifold M with b +2 = 1 as discussedin the previous section. For the case of pure SYM with no hypermultiplets we are alwaysfree to consider the case where the principal SO (3) gauge bundle has a nontrivial ’tHooft flux w ( P ) ∈ H ( M ; Z ). We choose an integral lift w ( P ) (and we assume sucha lift exists) and embed it in H ( M ; R ), and we denote µ := w ( P ) ∈ L ⊗ R . Thedependence on the choice of lift will only enter through an overall sign. The pathintegral over the Coulomb branch of Donaldson-Witten theory, denoted by Φ J µ , is anintegral over the infinite dimensional field space, which reduces to a finite dimensionalintegral over the zero modes [3]. We restrict for simplicity to four-manifolds with b = 0,such that there are no zero modes for the one-form fields ψ . The path integral of theeffective theory on the Coulomb branch then becomesΦ J µ = Λ − (cid:88) U(1) fluxes (cid:90) B da ∧ d ¯ a ∧ dD ∧ dη ∧ dχ A ( u ) χ B ( u ) σ e − (cid:82) M L , (4.1)where L is the Lagrangian (2.11) specialized to the zero modes including the onesof the gauge field. The functions A ( u ) and B ( u ) are curvature couplings; they areholomorphic functions of u , given by [3, 41] A ( u ) = α (cid:18) duda (cid:19) ,B ( u ) = β ( u − Λ ) . (4.2)The coefficients α and β are numerical factors, which we choose to match with resultson Donaldson invariants from the mathematical literature. Note that A ( u ) χ B ( u ) σ hasdimension Λ since χ + σ = 4. Moreover, da ∧ d ¯ a ∧ dD ∧ dη ∧ dχ has dimension Λ, suchthat Φ J µ (4.1) is dimensionless. We denote the contribution of the Coulomb branch toa correlation function (cid:104)O O . . . (cid:105) J µ by Φ J µ [ O O . . . ]. This corresponds to an insertionof O O . . . 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 (cid:90) dD ∧ dη ∧ dχ e y π (cid:82) M D ∧∗ D − √ i π d ¯ τd ¯ a ηχ ∧ ( F + + D ) = − π √ y d ¯ τd ¯ a B ( k , J ) . (4.3) The dimensons of a , A µ , D µν , η , ψ µ and χ µν are respectively 1, 1, 2, , and in powers of Λ.The dimension of differential form fields is reduced by their form degree. For example, the dimensionof F = dA is 0. The dimensions of the differentials da , dD , dη and dχ are respectively 1, 0, − and . – 14 –here the vector k equals [ F ] / π and represents a class in L + µ with µ ∈ L/
2. Thefactor d ¯ τd ¯ a suggests that it is natural to change variables from a to the effective couplingconstant τ ∈ H / Γ (4) in (4.1). To this end, we define the holomorphic “measure factor”˜ ν ( τ ) := Λ − √ πi A χ B σ dadτ , (4.4)so that Equation (4.11) below will hold. Using Matone’s relation [42] dudτ = 4 πi ( u − Λ ) (cid:18) dadu (cid:19) , (4.5)and (2.4) and (2.6), we can express ˜ ν in terms of modular functions˜ ν ( τ ) = − i ϑ − b ( τ ) η ( τ ) , (4.6)where we fixed the constants α and β . The modular transformations of ˜ ν for the twogenerators ST − S : τ (cid:55)→ ττ +1 and T : τ (cid:55)→ τ + 4 of Γ (4) are:˜ ν (cid:18) ττ + 1 (cid:19) = ( τ + 1) − b / e − πiσ ˜ ν ( τ ) , ˜ ν ( τ + 4) = − ˜ ν ( τ ) . (4.7)The measure ˜ ν ( τ ) behaves near the weak coupling cusp τ → i ∞ as ∼ q − . Near themonopole cusp, we have ˜ ν ( − /τ D ) = ( − iτ D ) − b / ˜ ν D ( τ D ) with˜ ν D ( τ D ) = − Λ − i ( u D − Λ ) (cid:18) dadu (cid:19) D ϑ ( τ D ) σ , (4.8)whose q D -series starts at q σ D .The photon path integral takes the form of a Siegel-Narain theta function withkernel K Ψ J µ [ K ] ( τ, ¯ τ ) = (cid:88) k ∈ L + µ K ( k ) ( − B ( k ,K ) q − k − ¯ q k , (4.9)where K is a characteristic vector for L corresponding to an almost complex structureor Spin C structure. If one considers correlation functions rather than the partition The values of α and β are slightly different from those quoted in [17], since we have used a differentnormalization for the integral over D . Note that, compared to equation (3 .
13) of [3] there is an overall phase difference. This phasedifference can be written as exp[ iπ (cid:0) k − B ( k , K ) (cid:1) ], where k is a lift of w ( P ) to H ( M, Z ). Because K is a characteristic vector this factor is a k -dependent sign. The choice of sign is related to a choiceof orientation of instanton moduli space. – 15 –unction, 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, thefactor (4.3) leads to Ψ J µ [ K ] with K ( k ) = i √ y B ( k , J ) , (4.10)where we have left out the factor d ¯ τd ¯ a , which provides the change of variables from theCoulomb branch parameters to a fundamental domain of Γ (4) in H . Combining allingredients, we arrive at the following expression for Φ J µ ,Φ J µ = (cid:90) H / Γ (4) dτ ∧ d ¯ τ ˜ ν ( τ )Ψ J µ [ K ]( τ, ¯ τ ) . (4.11)An important requirement for (4.11) is the modular invariance of the integrandunder Γ (4) transformations. We can easily determine the modular transformationsof Ψ J µ [ K ] =: Ψ J µ from those of Ψ J µ [1] (B.5). The effect of replacing 1 by K in Ψ J µ [1]is to increase the weight by ( , ). (The factor 1 / √ y contributes ( , ) and B ( k , J )contributes (0 ,
1) to the total weight.) We then arrive atΨ J µ (cid:18) ττ + 1 , ¯ τ ¯ τ + 1 (cid:19) = ( τ + 1) b (¯ τ + 1) e πi σ Ψ J µ ( τ, ¯ τ ) , Ψ J µ ( τ + 4 , ¯ τ + 4) = e πiB ( µ ,K ) Ψ J µ ( τ, ¯ τ ) , (4.12)where we used that Q ( K ) = σ mod 8. Combining (4.7) and (4.12), we deduce thatthe integrand of (4.11) is invariant under the τ (cid:55)→ ττ +1 transformation. Moreover, theintegrand is invariant under τ (cid:55)→ τ +4 if B ( µ , K ) = mod Z . However, if B ( µ , K ) = 0mod Z , the integrand is multiplied by − τ (cid:55)→ τ + 4. Since Ψ J µ vanishes identicallyin the latter case case, there is no violation of the duality.We conclude therefore that the Coulomb branch integral (4.11) is well defined sincethe measure dτ ∧ d ¯ τ transforms as a mixed modular form of weight ( − , −
2) while theproduct ˜ ν Ψ J µ is a mixed modular form of weight (2,2) for the group Γ (4) making theintegrand modular invariant. We close this subsection with Table 1 that collects theweights of the various modular forms that appear in the context of u -plane integrals.Evaluation of the integral is postponed to Section 5. Much more information about the theory is obtained if we include observables in thepath integral [3, 8], which contain integrals over positive degree homology cycles of the– 16 –ngredient Mixed weight dτ ∧ d ¯ τ ( − , − y ( − , − ∂ ¯ τ raises ( (cid:96),
0) to ( (cid:96), ν ( τ ) (2 − b / , J µ [ K ] ( b / , u -plane integral. Transforma-tions are in SL(2 , Z ) for the first three rows, while in Γ (4) for the last two rows.four-manifold M . Since we restrict to four-manifolds with b = b = 0, we will focusin this article on surface observables involving integrals over elements of H ( M, Q ) and H ( M, Q ).The Donaldson invariants are correlation functions of observables in Donaldson-Witten theory. The canonical UV surface observable of Donaldson-Witten theory isdefined using the descent operator K mentioned below Eq. (2.9), I − ( x ) = (cid:90) x K u = 14 π (cid:90) x Tr (cid:20) ψ ∧ ψ − √ φF (cid:21) , (4.13)with x ∈ H ( M, Q ). The Donaldson invariant D γ(cid:96),s (3.10) can be expressed as a corre-lation function of the twisted Yang-Mills theory, D γ(cid:96),s ( p , x ) = Λ − (cid:96) − s (cid:10) (2 p ( p ) u ) (cid:96) ( I − ( x )) s (cid:11) J µ (4.14)where on the rhs, γ = (2 µ , k ) with k ∈ Z − µ . The map p : H ( M, Z ) → R wasintroduced below (3.8).Note that D γ(cid:96),s ( p , x ) ∈ Z if p / ∈ H ( M, Z ) and x / ∈ H ( M, Z ), since thecoefficients of u ( τ ) are in Z / F ] / π ∈ H ( M, Z ). It is natural to form agenerating function of correlation functions by including exponentiated observables inthe path integral (cid:68) e p ( p ) u/ Λ + I − ( x ) / Λ (cid:69) J µ = (cid:88) k,(cid:96),s D γ(cid:96),s ( p , x ) (cid:96) ! s ! . (4.15)We will often suppress the argument of p , and consider it simply as a fugacity in whichwe can make a (formal) series expansion.– 17 –n the effective theory in the infrared, the operator I − ( x ) becomes˜ I − ( x ) = i √ π (cid:90) x (cid:32) d uda ψ ∧ ψ − √ duda ( F − + D ) (cid:33) . (4.16)Inclusion of this operator in the path integral gives rise to a contact term in the IR, e x T ( u ) [3, 4], with T ( u ) = − πi Λ (cid:18) duda (cid:19) ∂ τ log ϑ = q − q + O ( q ) , (4.17)where ϑ is the fourth classical Jacobi theta function. The dual contact term reads T D ( u D ) = − πi Λ (cid:18) duda (cid:19) D ∂ τ log ϑ = 12 + 8 q D + 48 q D + O ( q D ) . (4.18)We include moreover the Q -exact operator I + ( x ) [17], I + ( x ) = − π (cid:90) x {Q , Tr[ ¯ φχ ] } , (4.19)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 overthe u -plane are suitably defined. And more generally, if we add α I + ( x ), the integral isindependent of α . Nevertheless, the integrand depends in an interesting way on α . Wewill discuss this in more detail in Subsection 6.2. Here we will continue with α = 1. Inthe effective infrared theory, I + ( x ) becomes˜ I + ( x ) = − i √ π (cid:90) x (cid:32) d ¯ ud ¯ a η χ + √ d ¯ ud ¯ a ( F + − D ) (cid:33) . (4.20)With (4.16) and (4.20), we find that the contribution of the Coulomb branch to (cid:10) e I − ( x )+ I + ( x ) (cid:11) J µ readsΦ J µ (cid:2) e I − ( x ) / Λ+ I + ( x ) / Λ (cid:3) = Λ − (cid:88) U(1) fluxes (cid:90) B da ∧ d ¯ a ∧ dD ∧ dη ∧ dχ A ( u ) χ B ( u ) σ × e − (cid:82) M L +˜ I − ( x ) / Λ+˜ I + ( x ) / Λ+ x T ( u ) , (4.21)– 18 –s 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 gives2 πi (cid:114) y exp (cid:32) − πy b + i √ d ¯ τd ¯ a (cid:90) M b + ∧ ηχ (cid:33) . (4.22)where we have defined b ∈ L ⊗ R through ρ = x π Λ duda , b = Im( ρ ) y . (4.23)The variable ρ transforms with weight −
1. With this normalization, it will appear asa natural elliptic variable in the sum over fluxes. The dual variable is ρ D ( τ D ) = τ D ρ ( − /τ D ) = x π Λ (cid:18) duda (cid:19) D , (4.24)where (cid:0) duda (cid:1) D is given in (2.8).Substitution of (4.22) into the path integral and integration over the η and χ zeromodes modifies the sum over the U(1) fluxes to Ψ J µ [ K s ] (4.9) where the kernel K s givenby [3, 17] K s = exp (cid:0) − πy b − πiB ( k − , ρ ) − πiB ( k + , ¯ ρ ) (cid:1) ∂ ¯ τ (cid:16)(cid:112) y B ( k + b , J ) (cid:17) . (4.25)This gives the standard generalization of Ψ J µ ( τ, ¯ τ ) to a theta series with an ellipticvariable ρ . The holomorphic part couples to k − and the anti-holomorphic part to k + We will therefore also denote Ψ J µ [ K s ] asΨ J µ ( τ, ¯ τ , ρ , ¯ ρ ) = exp (cid:0) − πy b (cid:1) (cid:88) k ∈ L + µ ∂ ¯ τ (cid:16)(cid:112) y B ( k + b , J ) (cid:17) ( − B ( k ,K ) q − k − ¯ q k × exp (cid:16) − πiB ( k − , ρ ) − πiB ( k + , ¯ ρ ) (cid:17) , (4.26)Note that Ψ J µ ( τ, ¯ τ , ,
0) = Ψ J µ [ K ]( τ, ¯ τ ) (4.9). We postpone the remaining steps of theevaluation to Section 5.5.After describing the u -plane integrand, we can also give the Seiberg-Witten con-tribution of the strong coupling singularities u = ± Λ to (cid:10) e p u + I − ( x ) (cid:11) J µ . Setting Λ = 1,the contribution for u = 1 from a Spin C structure k is [3] (cid:10) e p u + I − ( x ) (cid:11) J SW , k , + = 2 SW( k ) + Res a D =0 (cid:20) da D a nD C ( u ) k / P ( u ) σ/ L ( u ) χ/ × exp (cid:18) p u + i duda B ( x , k ) + x T ( u ) (cid:19)(cid:21) , (4.27)– 19 –ith n = − (2 χ + 3 σ ) / k / C ( u ), P ( u ), L ( u ) given by C ( u ) = a D q D ,P ( u ) = − ϑ ( τ D ) ϑ ( τ D ) ϑ ( τ D ) a − D ,L ( u ) = 8 iϑ ( τ D ) ϑ ( τ D ) . (4.28)For four-manifolds of SW-simple type, the only k for which the (4.27) is non-vanishinghave n = 0. The expression then simplifies considerably. For the contribution from u = 1, (cid:10) e p u + I − ( x ) (cid:11) J SW , k , + = SW( k ) 2 K − χ h e p + x / iB ( x , k ) . (4.29)and for the contribution from u = − (cid:10) e p u + I − ( x ) (cid:11) J SW , k , − = SW( k ) 2 K − χ h e − p − x / B ( x , k ) . (4.30)The full correlation function for manifolds of simple type therefore reads (cid:10) e p u + I − ( x ) (cid:11) J µ = Φ J µ [ e p u + I − ( x ) ] + (cid:88) ± (cid:88) k ∈ L + 12 w TM ) k χ +3 σ ) / (cid:10) e p u + I − ( x ) (cid:11) J SW , k , ± . (4.31)Manifolds with b +2 = 1 are however rarely of SW-simple type [36]. These manifolds maygive rise to SW moduli spaces of arbitrarily high dimension. The SW contributions willthen be more involved, but are entire functions of p and x as is the case for (4.29) and(4.30). For compact four-manifolds with ( b , b +2 ) = (0 , u -plane tothe vev of an observable O , is given byΦ J µ [ O ] = (cid:90) H / Γ (4) ˜ ν ( τ ) Ψ J µ [ K O ]( τ, ¯ τ ) . (4.32)Besides the choice of O , it depends on the following data of the four-manifolds • the lattice L with signature (1 , b − • a period point J ∈ L ⊗ R , normalized to Q ( J ) = 1, • An integral lift K ∈ L of w ( T M ), • An integral lift w ( P ) of the ’t Hooft flux so that µ = w ( P ) ∈ H ( M, R ).– 20 – . 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 mod-ular fundamental domain [3, 19, 20]. Section 5.2 explains the strategy for arbitrarycorrelation functions. Subsection 5.3 factors the sum over fluxes into holomorphic andnon-holomorphic terms for a specific choice of J . We apply this result to the evaluationof the partition function and topological correlators in Subsections 5.4 and 5.5. 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 theform I f = (cid:90) F ∞ dτ ∧ d ¯ τ y − s f ( τ, ¯ τ ) , (5.1)where f is a non-holomorphic modular form of weight (2 − s, − s ), and F ∞ is thestandard keyhole fundamental domain for the modular group, F ∞ = H / SL(2 , Z ). Theintegral is naturally independent of the choice of fundamental domain due to the mod-ular properties of f . We assume that f has a convergent Fourier series expansion f ( τ, ¯ τ ) = (cid:88) m,n (cid:29)−∞ c ( m, n ) q m ¯ q n , (5.2)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 beboth 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 beexpressed as a total anti-holomorphic derivative, ∂ ¯ τ (cid:98) h ( τ, ¯ τ ) = y − s f ( τ, ¯ τ ) , (5.3)where (cid:98) h transforms as a modular form of weight (2 , − d ( dτ (cid:98) h ). If only terms with n > (cid:98) h is a mock modular form and can be expressed as (cid:98) h ( τ, ¯ τ ) = h ( τ ) + 2 s (cid:90) i ∞− ¯ τ f ( τ, − v )( − i ( v + τ )) s dv, (5.4)where h is a (weakly) holomorphic function with Fourier expansion h ( τ ) = (cid:88) m (cid:29)−∞ m ∈ Z d ( m ) q m . (5.5)– 21 –ote 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 → − /τ implies for h ( τ ), h ( − /τ ) = τ (cid:18) h ( τ ) + 2 s (cid:90) i ∞ f ( τ, − v )( − i ( v + τ )) s dv (cid:19) . (5.6)Reference [19] gives a definition of the integral I r f such that the value turns out to be I r f = d (0) . (5.7)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 non-negative [3, 44] but is new if both n, m are negative. It is shown in [19] that, at leastfor 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 ambiguityin the anti-derivative h ( τ ). Different choices for h ( τ ) differ by a weakly holomorphicmodular form of weight 2. However, this ambiguity does not lead to an ambiguityin the final result, d (0), since the constant term of such weakly holomorphic modularforms vanishes. This can be understood from the cohomology of F ∞ . Since the firstcohomology of F ∞ is trivial, any closed one-form ξ is necessarily exact. Such a one-form ξ can be expressed as C ( τ ) dτ , with C ( τ ) a (weakly holomorphic) modular formof weight two. Since ξ is exact, the period (cid:82) Y +1 Y C ( τ ) dτ vanishes, which implies thatthe constant term of C ( τ ) vanishes. Indeed, a basis of weakly holomorphic modularforms of weight 2 is given by derivatives of powers of the modular invariant J -function, ∂ τ (cid:0) J ( τ ) (cid:96) (cid:1) , (cid:96) ∈ N , which all have vanishing constant terms. Recall that in Section 4 we analyzed the partition function of Donaldson-Witten theory,which led to an integrand of the form ˜ ν ( τ ) Ψ J µ [ K ]( τ, ¯ τ ), with a specific kernel K (4.10).For more general correlation functions, the integrand takes a similar form,Φ J µ [ O ] = (cid:90) H / Γ (4) dτ ∧ d ¯ τ ˜ ν ( τ ) Ψ J µ [ K O ] . (5.8)where the kernel K O depends on the insertion O = O O . . . . This can be expressed asan integral of the form (5.1), whose integrand could consist of several terms (cid:80) j y − s j f j .Moreover, one can express the integral over Γ (4) as the sum of six integrals over F ∞ – 22 –sing modular transformations. As explained in the previous subsection, an efficienttechnique to evaluate these integrals is to express the integrand as a total derivativewith respect to ¯ τ , 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)as dd ¯ τ (cid:98) H J µ [ O ]( τ, ¯ τ ) = ˜ ν ( τ ) Ψ J µ [ K O ]( τ, ¯ τ ) , (5.9)which by a change of variables is equivalent to an anti-holomorphic derivative in u asdiscussed in the Introduction. The inverse map u − : B → H / Γ (4) maps each of theboundaries ∂ j B to arcs in H / Γ (4) in the vicinity of the cusps { i ∞ , , } displayed inFigure 2.The function (cid:98) 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 usingmethods from analytic number theory, especially the theory of mock modular forms[15, 16]. To derive a suitable (cid:98) H J µ [ O ], we will choose a convenient period point J .Once (cid:98) H J µ [ O ] is known it is straightforward to apply the discussion of Section 5.1. Torelate the integral over H / Γ (4) to an integral over F ∞ , we use coset representativesof SL(2 , Z ) / Γ (4) to map the six different images of F ∞ within H / Γ (4), displayed inFigure 2, back to F ∞ . After this inverse mapping, we use the modular properties ofthe integrand to express each of the six integrands as a series in q and ¯ q , after whichthe 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 for ˜ ν ( τ ) follows from the standard relations for Jacobitheta functions.Since the maps τ (cid:55)→ τ − n , n = 1 , , J µ [ O ] evaluates toΦ J µ [ O ] = 4 (cid:104) (cid:98) H J µ [ O ]( τ, ¯ τ ) (cid:105) q + (cid:104) τ (cid:55)→ − τ (cid:105) q + (cid:104) τ (cid:55)→ τ − τ (cid:105) q , (5.10)where for the second and third brackets on the rhs, one makes the indicated modulartransformation for τ , S and T S , and then determines the q coefficient of the resultingFourier expansion.An important point is the possibility to add to (cid:98) H J µ [ O ] a holomorphic integration“constant” s O , which is required to be a weight 2 modular form for Γ (4). Of course,Φ J µ [ O ] should be independent of s O , since definite integrals do not depend on theintegration constant. To see the independence of Φ J µ [ O ] on s O , note that s O will bemapped to a weight 2 form for SL(2 , Z ) by the inverse mapping. As discussed in Section5.1, there are no holomorphic SL(2 , Z ) modular forms with weight 2, and the weaklyholomorphic ones have a vanishing constant term. There is therefore no ambiguityarising from the holomorphic integration constant.– 23 –n the other hand, the integration constant s O can modify the contribution fromeach cusp, since a non-vanishing holomorphic modular form of weight 2 for Γ (4) exists.It is explicitly given by ϑ ( τ ) + ϑ ( τ ) , 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 choiceof the integration constant by requiring that the exponential behavior of H J µ for τ → i ∞ matches the behavior of ˜ ν Ψ J µ in this limit.Once we have determined Φ J µ [ O ] for a specific period point J , one can change toan 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 . Ψ J µ To evaluate the partition function Φ J µ , we will choose a convenient period point J sothat Ψ J µ , as a function of τ , has a simple factorization as a holomorphic times an anti-holomorphic function. In this way, we can easily determine an anti-derivative usingthe 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 intersectionform. Odd intersection form
Let us first assume that the intersection lattice L is odd, such that its quadratic formcan be brought to the standard form in Equation (3.5). Since the wall-crossing formulafor Donaldson invariants is known [3], it suffices to determine Φ J µ for a convenient choiceof J . To this end, we choose the polarization J = (1 , ) , (5.11)where is the ( b − J , the orthogonaldecomposition of the lattice, L = L + ⊕ L − into a 1-dimensional positive definite lattice L + and ( b − L − , implies that the sum overthe U(1) fluxes Ψ J µ ( τ, ¯ τ ) factors. To see this explicitly, we let k = ( k , k − ) ∈ L , and k ∈ Z + µ , k − ∈ L − + µ − and µ = ( µ , µ − ). The Siegel-Narain theta functionΨ J µ = Ψ J µ [ K ] (4.9) now factors asΨ J µ ( τ, ¯ τ ) = − i ( − µ ( K − f µ ( τ, ¯ τ ) Θ L − , µ − ( τ ) , (5.12)with f µ ( τ, ¯ τ ) := − e πiµ √ y (cid:88) k ∈ Z + µ ( − k − µ k ¯ q k / , Θ L − , µ − ( τ ) = (cid:88) k − ∈ L − + µ − ( − B ( k − ,K − ) q − k − / , (5.13)– 24 –e used also that K is odd since K is a characteristic vector, as discussed below Eq.(3.5). Using (cid:88) k ∈ Z + ( − k − k q k / = η ( τ ) , we can express f µ in terms of the Dedekind eta function η , f µ ( τ, ¯ τ ) = , µ = 0 mod Z , − i √ y η ( τ ) , µ = mod Z . (5.14)We can similarly evaluate Θ L − , µ − . Since all K j are odd, Θ L − , µ − ( τ ) vanishes, except if µ − = mod Z b − . In that case, Θ L − , µ − is a power of the Jacobi theta function ϑ ,Θ L − , µ − ( τ ) = ϑ ( τ ) b − , µ − = mod Z b − , , else . (5.15)After substitution of ˜ ν (4.6), we find for the integrand˜ ν Ψ J µ = ( − ( K / f ( τ, ¯ τ ) ϑ ( τ ) η ( τ ) , µ = ( , ) mod Z b , , else . (5.16)Note that the dependence of the integrand on b has disappeared, and that the integranddiverges for τ → ∞ . Even intersection form
We continue with the even lattices, whose quadratic form can be brought to the formgiven in Equation (3.6), L = I , ⊕ n L E . We choose for the period point J = 1 √ , , ) , (5.17)where the first two components correspond to I , ⊂ L , and is now the ( b − k ∈ L , k = 12 ( k + k ) , k − = −
12 ( k − k ) + k n , (5.18)where k n ∈ nL E . Note k n ≤
0, since L E is the negative E lattice.The sum over fluxes Ψ J µ factors for this choice of J ,Ψ J µ ( τ, ¯ τ ) = Ψ I , ( µ + ,µ − ) ( τ, ¯ τ ) Θ nL E , µ n ( τ ) , (5.19)– 25 –here the subscript is µ = ( µ + , µ − , µ n ), and Ψ I , ( µ + ,µ − ) ( τ, ¯ τ ) is given byΨ I , ( µ + ,µ − ) ( τ, ¯ τ ) = i √ y (cid:88) k ∈ I , +( µ + ,µ − ) ( k + k ) ( − k K + k K q ( k − k ) / ¯ q ( k + k ) / . (5.20)Moreover, the theta series Θ nE , µ for the negative definite lattice equalsΘ nE , µ n ( τ ) = (cid:88) k n ∈ nL E + µ n q − k n / . (5.21)As before, the K j are components of the characteristic element K ∈ L , thistime in the basis (3.6). Recall K and K ∈ Z since they are components of acharacteristic vector of I , . Changing the sign of k and k in the summand givesΨ I , ( µ + ,µ − ) = − Ψ I , ( µ + ,µ − ) , hence Ψ I , ( µ + ,µ − ) vanishes identically. Nevertheless, it is in-structive to evaluate the integral using the approach of Section 5.2, to set up notationfor working with the closely analogous function in Equation (5.69), which is definitelynonzero.To express Ψ I , ( µ + ,µ − ) as an anti-holomorphic derivative, we split the lattice into apositive and negative definite one, by changing summation variables to n + = k + k , n − = k − k , (5.22)and similarly for the ’t Hooft flux and the canonical class, µ + = µ + µ , µ − = µ − µ ,K + = 12 ( K + K ) , K − = 12 ( K − K ) , (5.23)where µ j ∈ Z / µ ± , the summation over n ± runs over two sets, namely n ± ∈ Z + µ ± + j with j = 0 ,
1. We can now express the sum over fluxes asΨ I , µ ( τ, ¯ τ ) = − i ( − µ + K + − µ − K − (cid:88) j =0 , h µ + + j ( τ, ¯ τ ) t µ − + j ( τ ) , (5.24)where µ = ( µ + , µ − ) and h ν ( τ, ¯ τ ) = − √ y (cid:88) n ∈ Z + ν n ¯ q n / ,t ν ( τ ) = (cid:88) n ∈ Z + ν q n / , (5.25)with ν ∈ Z / Z . For the four conjugacy classes of ν , we find that h ν equals h ν ( τ, ¯ τ ) = , ν = 0 mod Z , i √ y e πiν η ( τ / , ν = mod Z . (5.26)– 26 –e have similarly for t ν t ν ( τ ) = ϑ (2 τ ) , ν = 0 mod 2 Z ,ϑ (2 τ ) , ν = 1 mod 2 Z , ϑ ( τ / , ν = mod Z . (5.27)Substitution of the expressions (5.26), (5.27) in (5.24) confirms the vanishing of Ψ I , µ . u -plane integrands and mock modular forms Our next aim is to express the integrand as an anti-holomorphic derivative of a non-holomorphic modular form. We will determine functions (cid:98) F µ (respectively (cid:98) H µ ), whichtransform as weight modular forms, and such that ∂ ¯ τ (cid:98) F µ = f µ , ∂ ¯ τ (cid:98) H ν = h ν , (5.28)for odd and even lattices respectively. The holomorphic parts of (cid:98) F µ and (cid:98) H ν are knownas 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 ∈ Z , we can take (cid:98) F µ = 0. We thus only need to be concerned withfinding an anti-derivative (cid:98) F of − i √ y η . Let us reduce notation by setting (cid:98) F = (cid:98) F ,then (cid:98) F takes the general form (cid:98) F ( τ, ¯ τ ) = F ( τ ) − i (cid:90) i ∞− ¯ τ η ( w ) (cid:112) − i ( w + τ ) dw, (5.29)and is required to transform as a Γ (4) modular form with (holomorphic) weight .The first term on the rhs is holomorphic and is a mock modular form [15, 16], while thesecond term on the right hand side is known as a period integral and transforms witha shift under transformations of SL(2 , Z ). The function η is known as the shadow ofthe mock modular form F . Similarly to the discussion above Eq. (5.6), we deduce thatthe 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 indefinitetheta functions provides a constructive approach to derive a suitable F ( τ ). AppendixC provides a brief introduction to these functions, and derives an explicit expressionfor F : F ( τ ) = − ϑ ( τ ) (cid:88) n ∈ Z ( − n q n − − q n − = 2 q (cid:16) q + 7 q + 14 q + . . . (cid:17) . (5.30)– 27 –o evaluate Φ J µ following (5.10), we need to determine the q − expansion of F atthe other cusps. We introduce to this end F D and (cid:98) F D , (cid:98) F D ( τ, ¯ τ ) := − ( − iτ ) − (cid:98) F ( − /τ, − / ¯ τ )=: F D ( τ ) − i (cid:90) i ∞− ¯ τ η ( w ) (cid:112) − i ( w + τ ) dw, (5.31)where τ is now the local coordinate which goes to i ∞ near the strong coupling cusp u → Λ . Appendix C discusses how to derive the q -expansion of F D using the trans-formations of the indefinite theta function (C.5). One finds F D ( τ ) = − ϑ ( τ ) (cid:88) n ∈ Z q ( n + ) − q n = 14 q − (cid:0) − − q + 7 q − q + 21 q + . . . (cid:1) , (5.32)For the cusp τ →
2, the q -expansion is − i F D ( τ ).We leave the precise evaluation for later in this Subsection, and continue with theeven lattices, which can be treated more briefly. We see from (5.26) that h µ ( τ, ¯ τ ) = f µ ( τ / , ¯ τ / f µ as in (5.14). We can thus easily determine a suitable anti-derivative for h µ , namely (cid:98) H µ ( τ ) = , µ ∈ Z , (cid:98) F ( τ / , µ ∈ Z + , (5.33)with (cid:98) F as in (5.29). We similarly define H ( τ /
2) = F ( τ /
2) with F as in (5.30). Remark
Malmendier and Ono have emphasized the connection between q -series appearing inthe context of Mathieu moonshine and the u -plane integral for the complex projectiveplane P [45]. See [46, 47, 48, 49, 50] for overviews of the moonshine phenomenon. Ourdiscussion above demonstrates that the appearance of these q -series is quite generic forfour-manifolds with b +2 = 1. In particular, the function F (5.30) equals 1 / H (4)1 A, , which appears in the context of umbral moonshine on page 107 of [51].Similarly, F D (5.32) equals 1 / H (2)2 A, on page 103 of [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 irreduciblerepresentations of the finite sporadic group M , and which appeared in the elliptic– 28 –enus of the K ,
4) supersymmetry. We have for F , F ( τ ) = 124 (cid:18) H (2) ( τ ) + 2 ϑ ( τ ) + ϑ ( τ ) η ( τ ) (cid:19) , (5.34)where [53], H (2) ( τ ) = 2 ϑ ( τ ) − ϑ ( τ ) η ( τ ) − ϑ ( τ ) ∞ (cid:88) n =1 q n − q n − = 2 q − ( − q + 231 q + 770 q + . . . ) . (5.35)Whereas F is a mock modular form for the subgroup Γ (4) ⊂ SL(2 , Z ), H (2) is a mockmodular form for the full SL(2 , Z ). The completion (cid:98) H (2) ( τ, ¯ τ ) = H (2) ( τ ) − i (cid:90) i ∞− ¯ τ η ( w ) (cid:112) − i ( w + τ ) dw, (5.36)transforms under the two generators of SL(2 , Z ) as (cid:98) H (2) ( − /τ, − / ¯ τ ) = −√− iτ (cid:98) H (2) ( τ ) , (cid:98) H (2) ( τ + 1 , ¯ τ + 1) = e − πi (cid:98) H (2) ( τ ) . (5.37)The holomorphic part H (2) therefore transforms as H (2) ( − /τ ) = −√− iτ (cid:32) H (2) ( τ ) − i (cid:90) i ∞ η ( w ) (cid:112) − i ( w + τ ) dw (cid:33) . (5.38)We can express F D in terms of H (2) as F D ( τ ) = 124 (cid:18) H (2) ( τ ) − ϑ ( τ ) + ϑ ( τ ) η ( τ ) (cid:19) . (5.39)As a last example of a mock modular form with shadow η , we mention the function Q + , which was introduced by Malmendier and Ono in the context of the u -plane integral[12, 45] Q + ( τ ) = 112 H (2) ( τ ) + 76 ϑ ( τ ) + ϑ ( τ ) η ( τ ) = q − (1 + 28 q + 39 q + 196 q + 161 q + . . . ) . (5.40)Since F , H (2) and Q + are all weight mock modular forms for Γ (4) and have shadowsproportional to η , 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 behavessimilarly to its derivative f in this limit.– 29 – valuation We continue by evaluating the u -plane integral for an arbitrary four-manifold with( b , b +2 ) = (0 , J µ is onlynon-vanishing for odd lattices, and with ’t Hooft flux µ with µ = in the standardbasis. We have then using (5.16) and (5.28),Φ J µ = ( − ( K +1) / (cid:90) H / Γ (4) dτ ∧ d ¯ τ ϑ ( τ ) η ( τ ) ∂ ¯ τ (cid:98) F ( τ, ¯ τ )= ( − ( K +1) / (cid:32) (cid:20) F ( τ ) ϑ ( τ ) η ( τ ) (cid:21) q + 2 (cid:20) F D ( τ ) ϑ ( τ ) η ( τ ) (cid:21) q (cid:33) , (5.41)where the first term in the straight brackets is due to the contribution at i ∞ and thesecond term due to the two strong coupling singularities, which contribute equally. Thestrong coupling singularities do not contribute to the q term. We finally arrive atΦ J µ = ( − ( K +1) / , µ = ( , ) mod Z b , , else . (5.42)This is in agreement with the results for P for which K = 3 [54].It is straightforward to include the exponentiated point observable e p u in the pathintegral. One then arrives atΦ J µ [ e p u ] = ( − ( K +1) / (cid:32) (cid:20) F ( τ ) ϑ ( τ ) η ( τ ) e p u ( τ ) (cid:21) q + (cid:20) F D ( τ ) ϑ ( τ ) η ( τ ) (cid:0) e p u D + e − p u D (cid:1)(cid:21) q (cid:33) , (5.43)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 -term for any power of p . The resultis therefore completely due to the weak coupling cusp,Φ J µ [ e p u ] = ( − ( K +1) / (cid:20) F ( τ ) ϑ ( τ ) η ( τ ) e p u ( τ ) (cid:21) q . (5.44)Only even powers of p contribute to the constant term, which is in agreement with theinterpretation of the point observable as a four-form on the moduli space of instantons.Except for the mild dependence of (5.43) on K , this equation demonstrates that thecontribution from the u -plane to (cid:104) e p u (cid:105) J µ is universal for any four-manifold with oddintersection form and period point J (5.11).– 30 – (cid:96) Φ J µ [ u (cid:96) ]0 12 194 6806 29 5578 1 414 696Table 2: List of non-vanishing Φ J µ [ u (cid:96) ] with 0 ≤ (cid:96) ≤ b , b +2 ) = (0 , K = 3 mod 4, period point J (5.11), and µ = mod Z . For K = 1 mod 4, they differ by a sign, while they vanish for any four-manifold with aneven intersection form.We list Φ J µ [ u (cid:96) ] for small (cid:96) in Table 2. See [54] for a more extensive list. Section 6will discuss these numbers and the convergence of Φ J µ [ e p u ] = (cid:80) (cid:96) ≥ Φ J µ [ u (cid:96) ] (2 p ) (cid:96) /(cid:96) ! inmore detail. This subsection continues with the evaluation of the contribution of the u -plane to vevsof surface observables. Odd intersection form
We proceed as in Section 5.3 choosing J = (1 , ) and set ρ + = B ( ρ , J ) and ρ − = ρ − ρ + J . Specializing (4.26) givesΨ J µ ( τ, ¯ τ , ρ , ¯ ρ ) = − i ( − µ ( K − f µ ( τ, ¯ τ , ρ + , ¯ ρ + ) Θ L − , µ − ( τ, ρ − ) . (5.45)with f µ ( τ, ¯ τ , ρ, ¯ ρ ) = i e πiµ exp( − πy b ) (cid:88) k ∈ Z + µ ∂ ¯ τ ( (cid:112) y ( k + b )) ( − k − µ ¯ q k / e − πi ¯ ρ k , (5.46)where b = Im( ρ ) /y , and we removed the subscript of µ as before in Section 5. More-over, Θ L − , µ − ( τ, ρ − ) is given byΘ L − , µ − ( τ, ρ − ) = (cid:88) k − ∈ L − + µ − ( − B ( k − ,K − ) q − k − / e − πiB ( ρ − , k − ) . (5.47)– 31 –he 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 and ϑ (A.8) depending on the precise value of µ − . We define the dual theta seriesΘ D,L − , µ − asΘ D,L − , µ − ( τ, ρ D, − ) = ( − iτ ) − ( b − / e − πiτ ρ D, − Θ L − , µ − ( − /τ, ρ D /τ )= (cid:88) k − ∈ L − + K − / ( − B ( k − , µ − ) q − k − / e − πiB ( ρ D, − , k − ) . (5.48)We aim to write this as a total anti-holomorphic derivative of a real-analyticfunction (cid:98) 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 \{ Z τ + Z } , which has appeared at many places in mathemat-ics 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 re-viewed in Appendix D. Equation (D.10) is the main property for us. It states that (cid:99) M ( τ, ¯ τ , u, ¯ u, v, ¯ v ) = M ( τ, u, v ) + i R ( τ, ¯ τ , u − v, ¯ u − ¯ v ) transforms as a multi-variableJacobi form of weight , where R ( τ, ¯ τ , u, ¯ u ) is real-analytic in both τ and u . To de-termine (cid:98) F µ ( τ, ¯ τ , ρ, ¯ ρ ), we express f µ ( τ, ¯ τ , ρ, ¯ ρ ) in terms of ∂ ¯ τ R ( τ, ¯ τ , u, ¯ u ) (D.8). With w = e πiρ , we find f µ ( τ, ¯ τ , ρ, ¯ ρ ) = e πiν q − ν / w − ν ∂ ¯ τ R ( τ, ¯ τ , ρ + ντ, ¯ ρ + ν ¯ τ ) , ν = µ − . (5.49)We can thus determine the anti-derivative of f µ in terms of the completion (cid:99) M (D.9)in Appendix D by choosing u and v such that u − v = ρ + ντ while avoiding the polesin u and v . We will find below that the choice u = ρ + µτ and v = τ is particularlyconvenient. From Appendix D, we find that a candidate for the completed function is (cid:98) F µ ( τ, ¯ τ , ρ, ¯ ρ ) = − i e πiν q − ν / w − ν (cid:0) M ( τ, ρ + µτ, τ ) + i R ( τ, ¯ τ , ρ + ντ, ¯ ρ + ν ¯ τ ) (cid:1) . (5.50)We will find that for this choice of u and v , the holomorphic part F ( τ, ρ ) reduces to F ( τ ) = F ( τ ) (5.30) for ρ = 0. Indeed, substitution of this choice in M ( τ, u, v ) givesfor F ( τ, ρ ) F ( τ, ρ ) = − w ϑ ( τ ) (cid:88) n ∈ Z ( − n q n / − − w q n − , (5.51)which satisfies F ( τ,
0) = F ( τ ). – 32 –et us move on to µ ∈ Z , or µ = 0 to be specific. The function (cid:98) F ( τ, ¯ τ , ρ, ¯ ρ ) (5.50)evaluates then to (cid:98) F ( τ, ¯ τ , ρ, ¯ ρ ) = − i wϑ ( τ ) (cid:88) n ∈ Z ( − n q n / n − w q n − i q − w R ( τ, ¯ τ , ρ − τ, ¯ ρ − ¯ τ ) . (5.52)where we used ϑ ( τ, τ ) = − i q − ϑ ( 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 mentionfirst that we have to be careful with singling out the holomorphic part of (5.52), i.e. thepart which does not vanish in the limit y → ∞ , b →
0, keeping ρ and τ fixed. Since theelliptic argument of R is shifted by − τ , we have lim y →∞ q − w R ( τ, ¯ τ , − τ, − ¯ τ ) = 1.The holomorphic part of (5.52) is thus F ( τ, ρ ) = i − iϑ ( τ ) (cid:88) n ∈ Z ( − n q n / − w q n . (5.53)To write the non-holomorphic part, we define for µ ∈ { , } mod Z , R µ ( τ, ¯ τ , ρ, ¯ ρ ) = − e πiν δ ν, Z + + e πiν q − ν / w − ν R ( τ, ¯ τ , ρ + ντ, ¯ ρ + ν ¯ τ )= − i e πiµ (cid:88) n ∈ Z + µ (cid:16) sgn( n ) − Erf(( n + b ) (cid:112) πy ) (cid:17) ( − n − µ e − πiρn q − n / . (5.54)Note that R µ vanishes in the limit y → ∞ with b = 0 for any µ . Using these expressions,we can write the completed functions (cid:98) F µ as (cid:98) F µ ( τ, ¯ τ , ρ, ¯ ρ ) = F µ ( τ, ρ ) + R µ ( τ, ¯ τ , ρ, ¯ ρ ) . (5.55)We mentioned in the previous subsection the connection to dimensions of repre-sentations of sporadic groups. Other arithmetic information that has appeared in thecontext of the u -plane integral are the Hurwitz class numbers [3], which count binaryintegral quadratic forms with fixed determinant. Using relations for the Appell-Lerchsum (D.3), we can make this connection more manifest for the F µ . To this end, let usconsider the functions g ( τ, z ) = 12 + q − e πiz ϑ (2 τ, z ) (cid:88) n ∈ Z q n + n e − πinz − e πiz q n − ,g ( τ, z ) = q − e πiz ϑ (2 τ, z ) (cid:88) n ∈ Z q n e − πinz − e πiz q n − . (5.56)– 33 –hese functions appear in the refined partition function of SU (2) and SO (3) Vafa-Witten theory on P [57, 58, 59]. They vanish for z →
0, while their first derivativegive generating functions of Hurwitz class numbers H ( n ) [59]:lim z → πi ∂g j ( τ, z ) ∂z = 3 (cid:88) n ≥ H (4 n − j ) q n − j . (5.57)Using (D.3), we can express the functions g j in terms of F µ as g ( τ / , z ) = − i F ( τ, − z + ) − i η ( τ ) ϑ ( τ, z ) ϑ ( τ, z ) ϑ ( τ, z ) ϑ ( τ ) ϑ ( τ, z ) ϑ ( τ, z ) ,g ( τ / , z ) = − i F ( τ, − z + ) − i η ( τ ) ϑ ( τ, z ) ϑ ( τ, z ) ϑ ( τ, z ) ϑ ( τ ) ϑ ( τ, z ) ϑ ( τ, z ) , (5.58)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-definedpower series in x after integration, since the integrand involves expressions with polesin x . This could be avoided by the addition of a meromorphic Jacobi form of weight2 and index 0, with a pole at ρ = 0 with opposite residue. The reason is that theaddition of such a meromorphic Jacobi form does not alter the value of the integral. Tosee this, note that a meromorphic Jacobi form φ of weight 2 and index 0 has a Laurentexpansion in x of the form φ ( τ, ρ ) = (cid:88) (cid:96) φ (cid:96) ( τ ) x (cid:96) , (5.59)where (cid:96) = ( (cid:96) , . . . , (cid:96) b ) ∈ N b and x (cid:96) = x (cid:96) · · · x (cid:96) b b . The φ (cid:96) are weakly holomorphicmodular forms for Γ (4) of weight 2, since x is invariant under Γ (4). Mapping the siximages of F ∞ in H / Γ (4) to F ∞ gives us a meromorphic Jacobi form (cid:101) φ for SL(2 , Z )with expansion (cid:101) φ ( τ, ρ ) = (cid:88) (cid:96) (cid:101) φ (cid:96) ( τ ) x (cid:96) , (5.60)where the (cid:101) φ are modular forms for SL(2 , Z ) of weight 2. These have a vanishing constantterm as discussed before, and thus do not contribute to Φ J µ .To illustrate this, we present an alternative for F ( τ, ρ ) (5.53), i − iϑ ( τ ) (cid:88) n ∈ Z ( − n q n / − w q n − iϑ ( τ, ρ ) ∂ ρ ln (cid:18) ϑ ( τ, ρ ) ϑ ( τ, ρ ) (cid:19) . (5.61)This series is analytic for ρ →
0, and can be expressed as1 η ( τ ) (cid:88) k ∈ Z k ∈ Z + 12 (sgn( k + k ) − sgn( k )) k ( − k + k e πiρk q ( k − k ) . (5.62)– 34 –his is the series in terms of which G¨ottsche expressed the Donaldson invariants of P [60, Theorem 3.5].Let us return to the evaluation of Φ J µ [ e I − ( x ) ]. To this end, we also need to determinethe magnetic dual versions F D,µ . We let w D = e πiρ D , and define (cid:98) F D,µ by (cid:98) F D,µ ( τ, ¯ τ , ρ D , ¯ ρ D ) = − ( − iτ ) − e πiρ Dτ (cid:98) F µ ( − /τ, − / ¯ τ , ρ D /τ, ¯ ρ D / ¯ τ ) . (5.63)We evaluate the rhs using the transformation of (cid:99) M (D.10). Subtracting the subleadingnon-holomorphic part gives for F D,µ ( τ, ρ D ) F D,µ ( τ, ρ D ) = − w D ϑ ( τ ) (cid:88) n ∈ Z q n ( n +1) / − ( − µ w D q n , (5.64)which indeed reduces for µ = and ρ D → F D (5.32).Having determined F D,µ ( τ, ρ D ), we can write down our final expression for Φ J µ [ e I − ( x ) ]for four-manifolds with an odd intersection form. Similarly to Section 5.4, we expressΦ J µ [ e I − ( x ) ], as a sum of three terms, one from each cusp,Φ J µ [ e I − ( x ) ] = − i ( − µ ( K −
1) 3 (cid:88) s =1 Φ Js, µ [ e I − ( x ) ] , (5.65)withΦ J , µ [ e I − ( x ) ] = 4 (cid:104) ˜ ν ( τ ) e x T ( u ) F µ ( τ, ρ ) Θ L − , µ − ( τ, ρ − ) (cid:105) q , Φ J , µ [ e I − ( x ) ] = (cid:104) ˜ ν D ( τ ) e x T D ( u D ) F D,µ ( τ, ρ D, ) Θ D,L − , µ − ( τ, ρ D, − ) (cid:105) q , Φ J , µ [ e I − ( x ) ] = i e − πi µ (cid:104) ˜ ν D ( τ ) e − x T D ( u D ) F D,µ ( τ, − iρ D, ) Θ D,L − , µ − ( τ, − i ρ D, − ) (cid:105) q . Note that for this choice of J , Φ J µ [ e I − ( x ) ] only depends on µ , K and b (assuming b = 0). We list Φ J µ [ I s − ( x )] for the first few non-vanishing s in Table 3. If we specializeto the four-manifold P , 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 discussionof Section 5.3, we choose for the period point J = √ (1 , , ) ∈ L ⊗ R . To factorthe sum over fluxes Ψ J µ in the presence of the surface observable, we introduce the– 35 – Φ J µ [ I s − ( x )]1 − s Φ J µ [ I s − ( x )]0 14 3 · − · −
12 69525 · −
16 6 231 285 · − Table 3: For a smooth four-manifold with ( b , 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 , ). No assumptionis made about the value of b . The left table is for µ ∈ Z , and the right table is for µ = + Z . For µ ∈ Z , I − ( x ) s is an integral class, while for µ ∈ + Z , 2 s I − ( x ) s is an integral class. The first entry at s = 1 is fractional, but (we believe) this arisesbecause the moduli space is a stack with nontrivial stabilizer group.vector C = √ (1 , − , ) ∈ L ⊗ R . The vectors J and C form an orthonormal basisof I , ⊗ R ⊂ L ⊗ R . We denote by ρ + and ρ − the projections of the elliptic variable ρ ∈ L ⊗ C to J and C , ρ + = √ B ( ρ , J ) , ρ − = √ B ( ρ , C ) . (5.66)With respect to the basis (3.6), ρ reads ρ = ( ρ + , ρ − , ρ n ) , (5.67)with ρ n ∈ nL E ⊗ C . As in the case of the partition function (5.19), the sum over fluxesΨ J µ ( τ, ¯ τ , ρ , ¯ ρ ) (4.26) factors,Ψ J µ ( τ, ¯ τ , ρ , ¯ ρ ) = Ψ I , ( µ + ,µ − ) ( τ, ¯ τ , ρ − , ρ + , ¯ ρ + ) Θ nE , µ n ( τ, ρ n ) , (5.68)withΨ I , ( µ + ,µ − ) ( τ, ¯ τ , ρ − , ρ + , ¯ ρ + ) = exp( − πy b ) (cid:88) k ∈ I , +( µ + ,µ − ) ∂ ¯ τ ( √ y ( k + k + b + )) × ( − k K + k K q ( k − k ) / ¯ q ( k + k ) / e πiρ − ( k − k ) − πi ¯ ρ + ( k + k ) , (5.69)where b + = Im( ρ + ) /y , andΘ nE , µ n ( τ, ρ n ) = (cid:88) k n ∈ nL E + µ n q − k n / e − πiB ( ρ n , k n ) . (5.70)– 36 –he modular transformations are easily determined if we express Θ nE , µ n ( τ, ρ n ) in termsof Jacobi theta series. We define the dual theta series Θ D,nE , µ n asΘ D,nE , µ n ( τ, ρ D,n ) = τ − n e − πiρ D,n /τ Θ nE , µ n ( − /τ, ρ D,n /τ )= (cid:88) k n ∈ nL E ( − B ( µ n , k n ) q − k n / e − πiB ( ρ D,n , k n ) . (5.71)Unlike equation (5.20), (5.69) is definitely nonzero. The series can be decomposedfurther as Ψ I , ( µ + ,µ − ) ( τ, ¯ τ , ρ − , ρ + , ¯ ρ + ) = − i ( − µ + K + − µ − K − (cid:88) j =0 , h µ + + j ( τ, ¯ τ , ρ + , ¯ ρ + ) t µ − + j ( τ, ρ − ) , (5.72)with h ν ( τ, ¯ τ , ρ, ¯ ρ ) = i exp (cid:0) − π y b (cid:1) (cid:88) n ∈ Z + ν ∂ ¯ τ ( √ y ( n + b )) ¯ q n / e − πi ¯ ρ n ,t ν ( τ, ρ ) = (cid:88) n ∈ Z + ν q n / e πi ρ n = e πiρ ν q ν / ϑ (2 τ, ρ + ντ ) . (5.73)where b = Im( ρ ) /y . These functions reduce to those in (5.25) in the limit ρ → I , ( µ + ,µ − ) is a modular form for Γ (4), the functions h ν and t ν are not. Tocontinue working with modular forms for Γ (4), we rewrite Ψ I , ( µ + ,µ − ) (5.72) asΨ I , ( µ + ,µ − ) ( τ, ¯ τ , ρ − , ρ + , ¯ ρ + ) = − i ( − µ + K + − µ − K − × (cid:0) g + µ + ( τ, ¯ τ , ρ + , ¯ ρ + ) θ + µ − ( τ, ρ − ) + g − µ + ( τ, ¯ τ , ρ + , ¯ ρ + ) θ − µ − ( τ, ρ − ) (cid:1) , (5.74)with g ± ν ( τ, ¯ τ , ρ, ¯ ρ ) = ( h ν ( τ, ¯ τ , ρ, ¯ ρ ) ± h ν +1 ( τ, ¯ τ , ρ, ¯ ρ )) ,θ ± ν ( τ, ρ ) = t ν ( τ, ρ ) ± t ν +1 ( τ, ρ ) . (5.75)These functions are modular forms for Γ (4), which becomes manifest when we expressthem in terms of functions we encountered before. We can express the g ± ν in terms of f ν , g + ν ( τ, ¯ τ , ρ, ¯ ρ ) = f ν ( τ / , ¯ τ / , ( ρ + 1) / , ( ¯ ρ + 1) / ,g − ν ( τ, ¯ τ , ρ, ¯ ρ ) = e πiν f ν ( τ / , ¯ τ / , ρ/ , ¯ ρ/ . (5.76)– 37 –he θ ± ν can be expressed in terms of the Jacobi theta functions ϑ j as θ + ν ( τ, ρ ) = ϑ ( τ / , ρ/ , ν = 0 mod Z ,ϑ ( τ / , ρ/ , ν = mod Z ,θ − ν ( τ, ρ ) = ( − ν ϑ ( τ / , ρ/ , ν = 0 mod Z , − e πiν ϑ ( τ / , ρ/ , ν = mod Z . (5.77)We define the dual functions as θ ± D,ν ( τ, ρ D ) = ( − iτ ) − e − πiρ D τ θ ± D,ν ( − /τ, ρ D /τ ) . (5.78)These are explicitly given by θ + D,ν ( τ, ρ D ) = ϑ (2 τ, ρ D ) , ν = 0 mod Z ,ϑ (2 τ, ρ D ) , ν = mod Z ,θ − D,ν ( τ, ρ D ) = ( − ν ϑ (2 τ, ρ D ) , ν = 0 mod Z ,i e πiν ϑ (2 τ, ρ D ) , ν = mod Z . (5.79)Since the g ± ν can be expressed in terms of the f ν , we can determine anti-derivatives (cid:98) G ± ν in terms of (cid:98) F µ . Namely, (cid:98) G + ν ( τ, ¯ τ , ρ, ¯ ρ ) = (cid:98) F ν ( τ / , ¯ τ / , ( ρ + 1) / , ( ¯ ρ + 1) / , (cid:98) G − ν ( τ, ¯ τ , ρ, ¯ ρ ) = e πiν (cid:98) F ν ( τ / , ¯ τ / , ρ/ , ¯ ρ/ . (5.80)The holomorphic parts of these completed functions are G + ν ( τ, ρ ) = F ν ( τ / , ( ρ + 1) / ,G − ν ( τ, ρ ) = e πiν F ν ( τ / , ρ/ , (5.81)with the F ν given by (5.51) and (5.53). We define the dual (cid:98) G ± D,ν as (cid:98) G ± D,ν ( τ, ¯ τ , ρ D , ¯ ρ D ) = − ( − iτ ) − e πiρ D τ (cid:98) G ± ν ( − /τ, − / ¯ τ , ρ D /τ, ¯ ρ D / ¯ τ ) . (5.82)These evaluate to G + D,ν ( τ, ρ D ) = −
12 + q ϑ (2 τ ) (cid:88) n ∈ Z q n ( n +1) − ( − ν w D q n +1 ,G − D,ν ( τ, ρ D ) = − e πiν w D ϑ (2 τ ) (cid:88) n ∈ Z q n ( n +1) − ( − ν w D q n . (5.83)– 38 –ith these expressions, we can present our final expression Φ J µ [ e I − ( x ) ] for four-manifolds with even intersection form,Φ J µ [ e I − ( x ) ] = − i ( − µ + K + − µ − K − (cid:88) s =1 Φ Js, µ [ e I − ( x ) ] , (5.84)withΦ J , µ [ e I − ( x ) ] = 4 (cid:34) ˜ ν ( τ ) e x T ( u ) Θ nE , µ n ( τ, ρ n ) (cid:88) ± G ± µ + ( τ, ρ + ) θ ± µ − ( τ, ρ − ) (cid:35) q , Φ J , µ [ e I − ( x ) ] = 2 (cid:104) ˜ ν D ( τ ) e x T D ( u D ) Θ D,nE , µ n ( τ, ρ D,n ) × (cid:88) ± G ± D,µ + ( τ, ρ D, + ) θ ± D,µ − ( τ, ρ D, − ) (cid:35) q , Φ J , µ [ e I − ( x ) ] = 2 i e − πi µ (cid:104) ˜ ν D ( τ ) e − x T D ( u D ) Θ D,nE , µ n ( τ, − i ρ D,n ) × (cid:88) ± G ± D,µ + ( τ, − iρ D, + ) θ ± D,µ − ( τ, − iρ D, − ) (cid:35) q . (5.85)The overall factor 2 for the strong coupling contributions is due to the factors of √ u -plane to Donaldson polynomials for smallinstanton number. The expressions confirm that I − ( x ) is an integral class for gaugegroup SU(2) µ = mod Z , and half-integral for µ (cid:54) = mod Z .
6. Asymptology of the u -plane integral Up to this point we have treated the u -plane integral Φ J µ [ e pu + I − ( x ) ] as a formal gener-ating series in the homology elements p and x . However, one might ask if the integralactually expresses a well-defined function on the homology of the four-manifold M . Inother words, one might ask if the formal series is in fact convergent. The contributionof the Seiberg-Witten invariants is a finite sum and hence in fact defines an entirefunction on H ∗ ( M, C ). Therefore the Donaldson-Witten partition function is a well-defined 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 analyticstructure of the resulting partition function, which, in turn, might signal interesting– 39 – + s P µ ( x , x ) for µ = (0 , − x − x x − x x + x x + x x − x x + x − x + 24 x x − x x + 4 x x + 4 x x − x x + 24 x x − x s + s P µ ( x , x ) for µ = ( , − x + 2 x x − x x + x x − x − x x + x x − x x + x x − x x + x x − x x + 3 x x − x s + s P µ ( x , x ) for µ = ( , )3 x − x x − x x + x − x − x x + x x − x x − x x + x x − x x − x Table 4: Let M be a smooth four-manifold with ( b , b +2 , b − ) = (0 , , K , = 2 mod 4 and period point J (5.17). Examples of such manifolds are S × S , and the Hirzebruch surfaces F n with n even. Let x = ( x ,
0) and x = (0 , x ) ∈ L ⊗ R in the basis (3.6). The tables list the non-vanishing polynomials P µ ( x , x ) = (cid:80) s ,s Φ J µ [ I − ( x ) s I − ( x ) s ] with s + s ≤
9, and ’t Hooft flux µ = ( µ , µ ) = (0 , ,
0) and ( , ) mod Z . The polynomials for ( µ , µ ) = (0 , ) follow from those for( µ , µ ) = ( ,
0) by the exchange x ↔ x .physical effects. In this Section we will explore that question, starting with the pointobservable in Subsection 6.1. We will find strong evidence that in fact the u -planeintegral is indeed an entire function of p .The situation for x is less clear, since the numerical results are more limited. Wediscuss in Subsection 6.1 that the results do suggest that Φ J µ [ e pu + I − ( x ) ] is also an entirefunction x . As a step towards understanding the analytic structure in x , we willconsider 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 bestwhen α = 1. – 40 – .1 Asymptotic growth of point observables We will analyze the dependence of the contribution from the u -plane Φ J µ [ e p u ] to thecorrelation function (cid:104) e p u (cid:105) J µ . Due to the exponential divergence of u (2.4) for τ → i ∞ , the divergence of e p u is doubly exponential. The u -plane integral thus formallydiverges. The discussion of Section 5.1 does not provide an immediate definition ofsuch divergent expressions. On the other hand, the vev of the exponentiated pointobservable e p u should be understood as a generating function of correlation functions,and we can define Φ J µ [ e p u ] asΦ J µ [ e p u ] = ∞ (cid:88) (cid:96) =0 (2 p ) (cid:96) (cid:96) ! Φ J µ [ u (cid:96) ] . (6.1)As discussed in Section 5.4, there is no problem evaluating Φ J µ [ u (cid:96) ] using the definitionof Section 5.1.We consider the case of odd lattices, and the period point J (5.11). Modifying(5.43), we express Φ J µ [ u (cid:96) ] asΦ J µ [ u (cid:96) ] = 1192 (cid:90) dτ H (2) ( τ ) η ( τ ) ϑ ( τ ) u ( τ ) (cid:96) − (cid:90) dτ H (2) ( τ ) η ( τ ) ϑ ( τ ) u D ( τ ) (cid:96) , (6.2)We have replaced here F by H (2) , since its completion is an equally good choice ofanti-derivative. It is straightforward to determine Φ J µ [ u (cid:96) ] using this expression. We listin Table 5 values of Φ J µ [ u (cid:96) ] for various large values of (cid:96) .Before giving evidence that the Φ J µ [ e p u ] is an entire function of p , let us discuss theintegrand in more detail. We first write Φ J µ [ u (cid:96) ] as an integral from 0 to 1:Φ J µ [ u (cid:96) ] = 1192 (cid:90) dτ H (2) ( τ ) η ( τ ) (cid:0) (1 + ( − (cid:96) ) u ( τ ) (cid:96) ϑ ( τ ) − (1 + ( − (cid:96) ) u ( τ − (cid:96) ϑ ( τ ) + 2 u D ( τ ) (cid:96) ϑ ( τ ) (cid:1) , (6.3)We can express the integrand in a SL(2 , Z ) invariant form. To this end, note u ( τ ) = − u ( τ −
2) = ϑ ( ϑ + ϑ )8 η u ( τ −
1) = i ϑ ( − ϑ + ϑ )8 η u D ( τ ) = ϑ ( ϑ + ϑ )8 η (6.4)– 41 – Φ J µ [ u (cid:96) ] (cid:96) Φ J µ [ u (cid:96) ] log( (cid:96) ) (cid:96) Φ J µ [ u (cid:96) ]0 1 0 -100 5 . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − J µ [ u (cid:96) ] for large (cid:96) .For (cid:96) even, we find thus that Φ J µ can be expressed asΦ J µ [ u (cid:96) ] = 196 (cid:20) H (2) ( τ )8 (cid:96) η ( τ ) (cid:96) Q (cid:96) ( τ ) (cid:21) q , (6.5)where Q (cid:96) is the weight 6 + 3 (cid:96) modular form defined by Q (cid:96) ( τ ) = ϑ (cid:96) ( ϑ + ϑ ) (cid:96) − ( − (cid:96)/ ϑ (cid:96) ( ϑ − ϑ ) (cid:96) + ϑ (cid:96) ( ϑ + ϑ ) (cid:96) . (6.6)The first few terms are Q (cid:96) ( τ ) = (cid:96) − q + . . . , (cid:96) = 0 mod 4 , − (cid:96) + 400 (cid:96) ) q + . . . , (cid:96) = 2 mod 4 . (6.7)– 42 –he most straightforward way of trying to establish the large (cid:96) asymptotics is bya saddle point analysis. Expressing u (cid:96) as u D ( − /τ ) (cid:96) ∼ e (cid:96) e − πiτ , we find that, tofirst approximation, the saddle point is at τ ∗ = πi log( − (cid:96) ) for large (cid:96) . We find that thecontribution of this saddle point to (cid:12)(cid:12) Φ J µ [ u (cid:96) ] (cid:12)(cid:12) behaves as C/(cid:96) for some constant C . Weleave an investigation into the difference between the saddle point contribution andTable 5 for another occasion.Let us explore the consequences of the large (cid:96) asymptotics for Φ J µ [ e pu ]. We deducefrom the Table 5 that Φ J µ [ u (cid:96) ] also decreases faster than C ( (cid:96) + 1) − . This estimatestrongly suggests that radius of convergence for (cid:80) (cid:96) ≥ p (cid:96) Φ J µ [ u (cid:96) ] /(cid:96) ! is infinite and thusthat Φ J µ [ e pu ] is an entire function of p . Moreover, we can easily bound | Φ J µ [ e pu ] | forreal p by (cid:12)(cid:12) Φ J µ [ e pu ] (cid:12)(cid:12) < C sinh(2 p )2 p . (6.8)The exponentials in sinh(2 p ) resemble the SW contribution at u = ± Λ . Comparingwith the SW-simple type expression (4.29), we see that the SW contribution to (cid:10) u (cid:96) (cid:11) is O (1), while the u -plane contribution is subleading.While we have focused in this Subsection on the point observables, the behaviorof Φ J µ [ I s − ( x )] for large s is equally if not more interesting. We list in Table 6 somenumerical data for log(Φ[ I − ( x ) s ]) /s . These data, while admittedly limited, do give theimpression that the asymptotic growth of Φ J µ [ I − ( x ) s ] is bounded by e αs log( s ) for somepositive constant α , and that α <
1. Assuming that this is the correct behavior forlarge s , the radius of convergence for x of Φ J x [ e I − ( x ) ] = (cid:80) s ≥ Φ J µ [ I − ( x ) s ] /s ! is infinite,implying that Φ J x [ e I − ( x ) ] is entire in x . We hope to get back to the asymptotics of thesecorrelators in future work. As a first step towards understanding the asymptotic behavior of the series in x weinvestigate here the growth of the integrand of the u -plane integral in the weak couplingregion. In order to do this it is useful to recall that one can add to the surface observablethe 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 integralis subtle and formally divergent this statement requires careful justification, but it turnsout 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 worthexploring in more detail, in particular the behavior in the weak coupling limit. In this– 43 – log(Φ J µ [ I − ( x ) s ]) /s log(Φ J µ [ I − ( x ) s ]) / ( s log( s ))17 0.4612 0.162837 0.9396 0.260257 1.2079 0.298777 1.3925 0.320697 1.5326 0.3350 s log(Φ J µ [ I − ( x ) s ]) /s log(Φ J µ [ I − ( x ) s ]) / ( s log( s ))20 0.5541 0.184940 0.9880 0.267860 1.2395 0.302780 1.4157 0.3231100 1.5509 0.3368Table 6: For a four-manifold with odd intersection form, these tables list data for non-vanishing Φ J µ [ I s − ( x )] for s ≤
100 in steps of 20, and with x = (1 , ). The top table isfor µ ∈ Z and the bottom table for µ ∈ + Z .limit, duda diverges as q − = e − i θ + πy . As a result, the elliptic variable ρ = x π duda diverges.This subsection studies this divergence as function of α .The u -plane integral with observable e I − + α I + results in a modified sum over fluxesΨ J µ ,α . This sum is defined as in (4.26), but with ¯ ρ replaced by α ¯ ρ , and readsΨ J µ ,α ( τ, ¯ τ , ρ , ¯ ρ ) = e π ( ρ + − α ¯ ρ +)22 y (cid:88) k ∈ L + µ ∂ ¯ τ ( (cid:112) yB ( k + ρ − α ¯ ρ iy , J ) ( − B ( k ,K ) × q − k − / ¯ q k / e − πiB ( ρ , k − ) − πiαB (¯ ρ , k + ) . (6.9)By completing the squares in the exponent, we can write this asΨ J µ ,α ( τ, ¯ τ , ρ , ¯ ρ ) = e π ( ρ + − α ¯ ρ +)22 y + πiτ b − + πi ¯ τα b (cid:88) k ∈ L + µ ∂ ¯ τ ( (cid:112) yB ( k + ρ − α ¯ ρ iy , J ) ( − B ( k ,K ) × q − ( k + b ) − / ¯ q ( k + α b ) / e − πiB ( a, k − ) − πiB ( αa, k + ) , (6.10)– 44 –here b = Im( ρ ) /y as before. The sum over k ∈ L + µ is dominated by the k forwhich − ( k + b ) − + ( k + b ) is minimized. For a generic choice of period point J , thereis only one k ∈ L + µ which minimizes this quantity. The leading asymptotic behavioris given by the exponential multiplying the sum. This evaluates to (cid:12)(cid:12) Ψ J µ ,α (cid:12)(cid:12) ∼ e − πy b + π (1 − α )22 y | ρ | . (6.11)Thus we see that α = 1 is special, since for this choice the exponent is negativedefinite for x >
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 ofthe 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), (cid:12)(cid:12) Φ J µ [ e I − ( x )+ I + ( x ) ] (cid:12)(cid:12) ∼ (cid:90) dy ∧ dϕ e πy − π x y e πy sin( π ϕ ) = (cid:90) dy e πy − π x y e πy (cid:90) − dϕ e π x y e πy cos( π ϕ ) , (6.12)where we only consider terms which are non-vanishing in the limit for y → ∞ . Theintegral over ϕ is a Bessel function I ( z ) with z = π x y e πy , which behaves for large z as e z / √ πz . This leads to a single exponential divergence, (cid:82) dy e πy , which can be treatedas 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.
Acknowledgments
We thank Johannes Aspman, Aliakbar Daemi, Elias Furrer, Lothar G¨ottsche, JeffHarvey, John Morgan, Tom Mrowka and Hiraku Nakajima for discussions. GK wouldlike to thank the Institute Of Pure and Applied Mathematics of UCLA and the PhysicsDepartment of the National and Kapodistrian University of Athens for hospitality. JMwould like to thank the NHETC, Rutgers University for hospitality. JM is supportedby the Laureate Award 15175 “Modularity in Quantum Field Theory and Gravity”of the Irish Research Council. GM and IN are supported by the US Department ofEnergy under grant DE-SC0010008.
A. Modular forms and theta functions
In this appendix we collect a few essential aspects of the theory of modular forms,Siegel-Narain theta functions and indefinite theta functions. For more comprehensivetreatments we refer the reader to the available literature. See for example [61, 62, 63].– 45 – odular groups
The modular group SL(2 , Z ) is the group of integer matrices with unit determinantSL(2 , Z ) = a bc d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a, b, c, d ∈ Z ; ad − bc = 1 . (A.1)We introduce moreover the congruence subgroup Γ ( n )Γ ( n ) = a bc d ∈ SL(2 , Z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b = 0 mod n . (A.2) Eisenstein series
We let τ ∈ H and define q = e πiτ . Then the Eisenstein series E k : H → C for even k ≥ q -series E k ( τ ) = 1 − kB k ∞ (cid:88) n =1 σ k − ( n ) q n , (A.3)with σ k ( n ) = (cid:80) d | n d k the divisor sum. For k ≥ E k is a modular form of SL(2 , Z ) ofweight k . In other words, it transforms under SL(2 , Z ) as E k (cid:18) aτ + bcτ + d (cid:19) = ( cτ + d ) k E k ( τ ) . (A.4)Note that the space of modular forms of SL(2 , Z ) forms a ring that is generated preciselyby E ( τ ) and E ( τ ). On the other hand E is a quasi-modular form, which means thatthe SL(2 , Z ) transformation of E includes a shift in addition to the weight, E (cid:18) aτ + bcτ + d (cid:19) = ( cτ + d ) E ( τ ) − iπ c ( cτ + d ) . (A.5) Dedekind eta function
The Dedekind eta function η : H → C is defined as η ( τ ) = q ∞ (cid:89) n =1 (1 − q n ) . (A.6)It is a modular form of weight under SL(2 , Z ) with a non-trivial multiplier system.It transforms under the generators of SL(2 , Z ) as η ( − /τ ) = √− iτ η ( τ ) ,η ( τ + 1) = e πi η ( τ ) . (A.7) For an unambiguous value of the square root, we define the phase of z ∈ C ∗ by log z := log | z | + i arg( z ) with − π < arg( z ) ≤ π . – 46 – acobi theta functions The four Jacobi theta functions ϑ j : H × C → C , j = 1 , . . . ,
4, are defined as ϑ ( τ, v ) = i (cid:88) r ∈ Z + ( − r − q r / e πirv ,ϑ ( τ, v ) = (cid:88) r ∈ Z + q r / e πirv ,ϑ ( τ, v ) = (cid:88) n ∈ Z q n / e πinv ,ϑ ( τ, v ) = (cid:88) n ∈ Z ( − n q n / e πinv . (A.8)We let ϑ j ( τ,
0) = ϑ j ( τ ) for j = 2 , ,
4. These have the following transformationsfor modular inversion ϑ ( − /τ ) = √− iτ ϑ ( τ ) ,ϑ ( − /τ ) = √− iτ ϑ ( τ ) ,ϑ ( − /τ ) = √− iτ ϑ ( τ ) , (A.9)and for the shift ϑ ( τ + 1) = e πi/ ϑ ( τ ) ,ϑ ( τ + 1) = ϑ ( τ ) ,ϑ ( τ + 1) = ϑ ( τ ) . (A.10)Their transformations under the generators of Γ (4) are ϑ ( τ + 4) = − ϑ ( τ ) , ϑ (cid:18) ττ + 1 (cid:19) = √ τ + 1 ϑ ( τ ) ,ϑ ( τ + 4) = ϑ ( τ ) , ϑ (cid:18) ττ + 1 (cid:19) = √ τ + 1 ϑ ( τ ) ,ϑ ( τ + 4) = ϑ ( τ ) , ϑ (cid:18) ττ + 1 (cid:19) = e − πi √ τ + 1 ϑ ( τ ) . (A.11)Two useful identities are ϑ ϑ ϑ = 2 η ,ϑ + ϑ = ϑ . (A.12)– 47 – . Siegel-Narain theta function Siegel-Narain theta functions form a large class of theta functions of which the Jacobitheta functions are a special case. For our applications in the main text, it is sufficientto consider Siegel-Narain theta functions for which the associated lattice L is a uni-modular lattice with signature (1 , n −
1) (or a Lorentzian lattice). We denote thebilinear form by B ( x , y ) and the quadratic form by B ( x , x ) ≡ Q ( x ) ≡ x . Let K bea characteristic vector of L , such that Q ( k ) + B ( k , K ) ∈ Z 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 k + = B ( k , J ) J, k − = k − k + . (B.1)Given this notation, we can introduce the Siegel-Narain theta function of our in-terest Ψ J µ [ K ] : H → C , asΨ J µ [ K ]( τ, ¯ τ ) = (cid:88) k ∈ L + µ K ( k ) ( − B ( k ,K ) q − k − / ¯ q k / (B.2)where µ ∈ L/ K : L ⊗ R → C is a summation kernel. Let us be slightlymore generic and include the elliptic variables which are relevant for the Donaldsonobservables. We defineΨ J µ [ K ]( τ, ¯ τ , z , ¯ z ) = e − πy b (cid:88) k ∈ L + µ K ( k ) ( − B ( k ,K ) q − k − / ¯ q k / × exp ( − πiB ( z , k − ) − πiB (¯ z , k + )) , (B.3)with b = Im( z ) /y .The modular properties of Ψ J µ [ K ] depend on the kernel K . The modular transfor-mations under the SL(2 , Z ) generators for Ψ J µ [1] areΨ J µ + K/ [1]( τ + 1 , ¯ τ + 1 , z , ¯ z ) = e πi ( µ − K / Ψ µ + K/ [1]( τ, ¯ τ , z + µ , ¯ z + µ ) , Ψ J µ + K/ [1] ( − /τ, − / ¯ τ , z /τ, ¯ z / ¯ τ ) = ( − iτ ) n − ( i ¯ τ ) exp( − πi z /τ + πiK / × ( − B ( µ ,K ) Ψ JK/ [1]( τ, ¯ τ , z − µ , ¯ z − µ ) (B.4)For the case of the partition function, we set the elliptic variables z , ¯ z to zero. Usingthe 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 (4). The transformations under the– 48 –enerators of this group readΨ J µ [1] (cid:18) ττ + 1 , ¯ τ ¯ τ + 1 (cid:19) = ( τ + 1) n − (¯ τ + 1) exp (cid:0) πi K (cid:1) Ψ J µ [1]( τ, ¯ τ ) , Ψ J µ [1]( τ + 4 , ¯ τ + 4) = e πiB ( µ ,K ) Ψ µ [1]( τ, ¯ τ ) , (B.5)where we have set z = ¯ z = 0. Transformations for other kernels appearing in the maintext are easily determined from these expressions. C. Indefinite theta functions for uni-modular lattices of signa-ture (1 , n − In this appendix we discuss various aspects of the theory of indefinite theta functionsand their modular completion. We assume that the corresponding lattice L is unimod-ular and of signature (1 , n − JJ (cid:48) µ , we let µ ∈ L/ J ∈ L ⊗ R and a vector J (cid:48) ∈ L , such that(i) J is positive definite, Q ( J ) = 1,(ii) J (cid:48) is a null-vector, Q ( J (cid:48) ) = 0,(iii) B ( J, J (cid:48) ) > B ( k , J (cid:48) ) (cid:54) = 0 for all k ∈ L + µ .The indefinite theta function Θ JJ (cid:48) µ ( τ, z ) is then defined asΘ JJ (cid:48) µ ( τ, z ) = (cid:88) k ∈ L + µ (cid:104) sgn( B ( k , J )) − sgn( B ( k , J (cid:48) )) (cid:105) ( − B ( k ,K ) q − k / e − πiB ( k , z ) . (C.1)The kernel within the straight brackets ensures that the sum over the indefinite latticeis convergent, since it vanishes on positive definite vectors [15]. This is also the caseif both J and J (cid:48) are positive definite, without the need to impose condition (iv). Onemay start from this situation and obtain the conditions above by taking the limit that J (cid:48) approaches a null vector. The indefinite theta series Θ JJ (cid:48) µ can also be defined, for µ which do not satisfy requirement (iv) above, but this requires more care.We can express Θ JJ (cid:48) µ also as Ψ J µ [ K ] (B.2) with the kernel K ( k ) = (cid:104) sgn( B ( k , J )) − sgn( B ( k , J (cid:48) )) (cid:105) e πy k +4 πyB ( k , b ) , (C.2)– 49 –here b = Im( z ) /y .While the sum over L is convergent, Θ JJ (cid:48) µ only transforms as a modular form afteraddition of certain non-holomorphic terms. References [15, 16] explain that the modularcompletion (cid:98) Θ JJ (cid:48) µ of Θ JJ (cid:48) µ is obtained by substituting (rescaled) error function for thesgn-function in (C.1). We let E ( u ) : R → ( − ,
1) be defined as E ( u ) = 2 (cid:90) u e − πt dt = Erf( √ πu ) . (C.3)The completion (cid:98) Θ JJ (cid:48) µ then transforms as a modular form of weight n/
2, and is explicitlygiven by (cid:98) Θ JJ (cid:48) µ ( τ, ¯ τ , z , ¯ z ) = (cid:88) k ∈ L + µ (cid:16) E ( (cid:112) y B ( k + b , J )) − sgn( B ( k , J (cid:48) )) (cid:17) × ( − B ( k ,K ) q − k / e − πiB ( k , z ) . (C.4)Note that in the limit y → ∞ , E ( √ y u ) approaches the sgn-function of (C.1),lim y →∞ E ( (cid:112) y u ) = sgn( u ) . The transformation properties under SL(2 , Z ) follow from Chapter 2 of Zwegers’thesis [15] or Vign´eras [64]. One finds (cid:98) Θ JJ (cid:48) µ + K/ ( τ + 1 , ¯ τ + 1 , z , ¯ z ) = e πi ( µ − K / (cid:98) Θ JJ (cid:48) µ + K/ ( τ, ¯ τ , z + µ , ¯ z + µ ) , (cid:98) Θ JJ (cid:48) µ + K/ ( − /τ, − / ¯ τ , z /τ, ¯ z / ¯ τ ) = i ( − iτ ) n/ exp (cid:0) − πi z /τ + πiK / (cid:1) × ( − B ( µ ,K ) (cid:98) Θ JJ (cid:48) K/ ( τ, ¯ τ , z − µ , ¯ z − µ ) . (C.5)The argument of E in (C.4) depends only on the imaginary part of z in and is valued in R . Reference [18] demonstrates that if one formally sets ¯ z = 0 such that the argumentof E is complex-valued, the modular properties of (cid:98) Θ JJ (cid:48) µ remain unchanged.When z = 0, we set (cid:98) Θ JJ (cid:48) µ ( τ, ¯ τ , ,
0) = (cid:98) Θ JJ (cid:48) µ + K/ ( τ, ¯ τ ). One finds for the action of thegenerators on (cid:98) Θ JJ (cid:48) µ ( τ, ¯ τ ) (cid:98) Θ JJ (cid:48) µ (cid:18) ττ + 1 , ¯ τ ¯ τ + 1 (cid:19) = ( τ + 1) n/ exp (cid:0) πi K (cid:1) (cid:98) Θ JJ (cid:48) µ ( τ, ¯ τ ) . (cid:98) Θ JJ (cid:48) µ ( τ + 4 , ¯ τ + 4) = e πiB ( µ ,K ) (cid:98) Θ JJ (cid:48) µ ( τ, ¯ τ ) . (C.6)For our application, the ¯ τ -derivative of (cid:98) Θ JJ (cid:48) µ is of particular interest. This gives the“shadow” of Θ JJ (cid:48) µ , whose modular properties are easier to determine than those of Indefinite theta functions can often be expressed as product of a mock modular form and modularform, in other words they are a mixed mock modular form. We therefore use the notion of “shadow”slightly differently from its definition for mock modular forms [16]. – 50 – JJ (cid:48) µ . We obtain here ∂ ¯ τ (cid:98) Θ JJ (cid:48) µ ( τ, ¯ τ ) =Ψ J µ [ K ]( τ, ¯ τ ) , (C.7)with Ψ J µ (B.2) the same function discussed in Appendix B and K , K = i √ y B ( k , J ) . (C.8) An example
Let us now specialize to an example which is useful in Section 5.4. We consider a two-dimensional lattice L ∼ = Z , with quadratic form − (cid:0) (cid:1) . The positive and negativedefinite cones of this lattices are illustrated in Figure 3. The upper-right componentof the negative cone for this lattice is generated by the vectors (0 ,
1) and (2 , − J and J (cid:48) as follows: J = ( − ,
1) and J (cid:48) = (0 , k = ( n, (cid:96) ), the kernel in (C.1) becomes (sgn( (cid:96) ) + sgn( n )). Then the only elements of L which contribute to the theta series are those in the two yellow areas in Figure 3.For the characteristic vector we choose K = (0 , − µ = (1 , JJ (cid:48) µ becomes the following q -series,Θ JJ (cid:48) µ ( τ ) = (cid:88) n,(cid:96) ∈ Z + (sgn( (cid:96) ) + sgn( n )) ( − n q n + n(cid:96) = − i (cid:88) n ∈ Z ( − n q n − − q n − , (C.9)where we performed the geometric sum over (cid:96) on the second line. The first partof this Appendix discussed that Θ JJ (cid:48) µ can be completed by replacing sgn( (cid:96) ) in (C.1)by E ( √ y (cid:96) ), where E is the rescaled error function defined in (C.3). We can write E ( √ y (cid:96) ) as sgn( (cid:96) ) plus a non-holomorphic period integral, E ( (cid:112) y (cid:96) ) = sgn( (cid:96) ) + i(cid:96) q (cid:96) (cid:90) i ∞− ¯ τ e πi(cid:96) w (cid:112) − i ( w + τ ) dw. (C.10)The completion can then be written as (cid:98) Θ JJ (cid:48) µ ( τ ) = Θ JJ (cid:48) µ ( τ ) + ϑ ( τ ) (cid:90) i ∞− ¯ τ η ( w ) (cid:112) − i ( w + τ ) dw, (C.11)and transforms as a non-holomorphic modular form for Γ (4) as discussed in AppendixC. We thus find that F = − i Θ JJ (cid:48) µ /ϑ is the holomorphic part of (cid:98) F in Section 5.4.– 51 – J + + + + Figure 3: The lattice L ∼ = Z , with quadratic form − ( ). The negative definite set ofthis lattice is the union of the yellow and purple regions. For the given choices of J and J (cid:48) , only the lattice vectors in the yellow area contribute to the sum in the indefinitetheta function.We conclude this appendix by deriving F D , which is the holomorphic part of (cid:98) F D ( τ, ¯ τ ) = − ( − iτ ) − (cid:98) F ( − /τ, − / ¯ τ ). We are instructed by (C.5) to determine (cid:98) Θ JJ (cid:48) K/ ( τ, − µ + K/
2) with µ = (1 , JJ (cid:48) K/ ( τ, − µ + K/
2) = − i (cid:88) n ∈ Z (cid:96) ∈ Z + 12 (sgn( (cid:96) ) + sgn( n )) ( − (cid:96) − q n + n(cid:96) = − i (cid:88) n ∈ Z q ( n + ) − q n . (C.12)This gives for F D F D ( τ ) = − ϑ ( τ ) (cid:88) n ∈ Z q ( n + ) − q n . (C.13)– 52 – . The Appell-Lerch sum We recall the definition and properties of the Appell-Lerch sum. We will denote thisfunction by M ( τ, u, v ) rather then the more common µ ( τ, u, v ) to avoid a class ofnotation with the ’t Hooft fluxes. We will mostly follow the exposition of Zwegers [15].For fixed τ , the Appell-Lerch function is a function of two complex variables M :( C \{ Z τ + Z } ) → C , defined as M ( τ, u, v ) := M ( u, v ) = e πiu ϑ ( τ, v ) (cid:88) n ∈ Z ( − n q n ( n +1) / e πinv − e πiu q n . (D.1)It has single order poles at Z τ + Z for both u and v .We list a number of useful properties, whose proofs can be found in [15]:1. Periodicity of u and v : M ( u + 1 , v ) = M ( u, v + 1) = − M ( u, v ) . (D.2)2. Quasi-periodicity of M under simultaneous translations of u and v . For u, v, u + z, v + z (cid:54) = Z τ + Z , M satisfies M ( u + z, v + z ) − M ( u, v ) = i η ϑ ( u + v + z ) ϑ ( z ) ϑ ( u ) ϑ ( v ) ϑ ( u + z ) ϑ ( v + z ) . (D.3)This relation can be demonstrated by showing that the periodicity, zeroes andpoles of the variable z are identical on the left and right hand side.3. Inversion of the elliptic arguments leaves µ invariant: M ( − u, − v ) = M ( u, v ) (D.4)4. M is symmetric under exchange of u and v : M ( v, u ) = M ( u, v ) . (D.5)Note that this relation follows from (D.3) and (D.4) using z = − u − v .A further property of M is that M transforms as a Jacobi form after the additionof a suitable non-holomorphic function R , which is analytic in its arguments. It isdefined explicitly as R ( τ, ¯ τ , u, ¯ u ) := R ( u ) = (cid:88) n ∈ Z + (cid:16) sgn( n ) − Erf (cid:16) ( n + a ) (cid:112) πy (cid:17)(cid:17) ( − n − e − πiun q − n / , (D.6)– 53 –here a = Im( u ) /y , and Erf( t ) is the error functionErf( t ) = 2 √ π (cid:90) t e − u du. (D.7)The anti-holomorphic derivative of R ( τ, ¯ τ , u, ¯ u ) is ∂ ¯ τ R ( τ, ¯ τ , u, ¯ u ) = − e − π y a (cid:88) n ∈ Z + ∂ ¯ τ (cid:16)(cid:112) y ( n + a ) (cid:17) ( − n − e − πi ¯ un ¯ q n / . (D.8)Addition of this function to M provides a function (cid:99) M , which transforms as a weight Jacobi form. This non-holomorphic completion (cid:99) M of M is explicitly given by (cid:99) M ( τ, ¯ τ , u, ¯ u, v, ¯ v ) = M ( τ, u, v ) + i R ( τ, ¯ τ , u − v, ¯ u − ¯ v ) . (D.9)This function transforms under SL(2 , Z ) as (cid:99) M (cid:18) aτ + bcτ + d , a ¯ τ + bc ¯ τ + d , ucτ + d , ¯ uc ¯ τ + d , vcτ + d , ¯ vc ¯ τ + d (cid:19) = ε ( γ ) − ( cτ + d ) e − πic ( u − v ) / ( cτ + d ) (cid:99) M ( τ, ¯ τ , u, ¯ u, v, ¯ v ) , (D.10)where ε ( γ ) is the multiplier system of the Dedekind η function. The anti-holomorphicderivative of (cid:99) M is given by ∂ ¯ τ (cid:99) M ( τ, ¯ τ , u, ¯ u, v, ¯ v ) = − i (cid:16) ∂ ¯ τ (cid:112) y (cid:17) e − π ( a − b ) (cid:88) n ∈ Z + ( n + a − b )( − n − ¯ q n / e − πi (¯ u − ¯ v ) n , (D.11)where a = Im( u ) /y and b = Im( v ) /y , and we hope there is no confusion with the a, b, c, d used in Equation (D.10). References [1] E. Witten,
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