Abstract:We consider partition functions with insertions of surface operators of topologically twisted N = 2, SU(2) supersymmetric Yang-Mills theory, or Donaldson-Witten theory for short, on a four-manifold. If the metric of the compact four-manifold has positive scalar curvature, Moore and Witten have shown that the partition function is completely determined by the integral over the Coulomb branch parameter a, while more generally the Coulomb branch integral captures the wall-crossing behavior of both Donaldson polynomials and Seiberg-Witten invariants. We show that after addition of aQ-exact surface operator to the Moore-Witten integrand, the integrand can be written as a total derivative to the anti-holomorphic coordinateā using Zwegers' indefinite theta functions. In this way, we reproduce Göttsche's expressions for Donaldson invariants of rational surfaces in terms of indefinite theta functions for any choice of metric.

The presence of a BRST symmetry in topologically twisted gauge theories makes a precise analysis of these theories feasible. While the global BRST symmetry suggests that correlation functions of BRST exact observables vanish, this decoupling might be obstructed due to a contribution from the boundary of field space. Motivated by divergent BRST exact observables on the Coulomb branch of Donaldson-Witten theory, we put forward a new prescription for the renormalization of correlation functions on the Coulomb branch. This renormalization is based on the relation between Coulomb branch integrals and integrals over a modular fundamental domain, and establishes that BRST exact observables indeed decouple in Donaldson-Witten theory. arXiv:1901.03540v2 [hep-th] 9 Aug 2019 5. Evaluation of correlation functions of Q-exact observables 30 6. Discussion and conclusion 31 A. Modular forms and theta functions 33 B. Siegel-Narain theta function 34 C. The self-dual twisted operator 35 -1 -and whose restriction to H 2 (M, Z)×H 2 (M, Z) is an integral bilinear form with signature (1, b 2 − 1). The bilinear form provides the quadratic form Q(k) := B(k, k) ≡ k 2 , which can be brought to a simple standard form [15, Section 1.1.3]. We denote the period point by J, i.e. the harmonic two-form, satisfying * J = J ∈ H 2 (M, R), J 2 = 1, (3.2)

The u-plane integral is the contribution of the Coulomb branch to correlation functions of $${\mathcal {N}}=2$$ N = 2 gauge theory on a compact four-manifold. We consider the u-plane integral for correlators of point and surface observables of topologically twisted theories with gauge group $$\mathrm{SU}(2)$$ SU ( 2 ) , for an arbitrary four-manifold with $$(b_1,b_2^+)=(0,1)$$ ( b 1 , b 2 + ) = ( 0 , 1 ) . The u-plane contribution equals the full correlator in the absence of Seiberg–Witten contributions at strong coupling, and coincides with the mathematically defined Donaldson invariants in such cases. We demonstrate that the u-plane correlators are efficiently determined using mock modular forms for point observables, and Appell–Lerch sums for surface observables. We use these results to discuss the asymptotic behavior of correlators as function of the number of observables. Our findings suggest that the vev of exponentiated point and surface observables is an entire function of the fugacities.

We revisit the low-energy effective U(1) action of topologically twisted $${\mathcal {N}}=2$$ N = 2 SYM theory with gauge group of rank one on a generic oriented smooth four-manifold X with nontrivial fundamental group. After including a specific new set of $$\mathcal Q$$ Q -exact operators to the known action, we express the integrand of the path integral of the low-energy U(1) theory as an anti-holomorphic derivative. This allows us to use the theory of mock modular forms and indefinite theta functions for the explicit evaluation of correlation functions of the theory, thus facilitating the computations compared to previously used methods. As an explicit check of our results, we compute the path integral for the product ruled surfaces $$X=\Sigma _g \times \mathbb {CP}^1$$ X = Σ g × CP 1 for the reduction on either factor and compare the results with existing literature. In the case of reduction on the Riemann surface $$\Sigma _g$$ Σ g , via an equivalent topological A-model on $$\mathbb {CP}^1$$ CP 1 , we will be able to express the generating function of genus zero Gromov–Witten invariants of the moduli space of flat rank one connections over $$\Sigma _g$$ Σ g in terms of an indefinite theta function, whence we would be able to make concrete numerical predictions of these enumerative invariants in terms of modular data, thereby allowing us to derive results in enumerative geometry from number theory.

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