Mocking the \(u\)-plane integral (Q2042085)

The authors consider the topologically twisted counterpart of \(\mathcal{N} = 2\) supersymmetric Yang-Mills theory with gauge group SU(2) in the presence of arbitrary 't~Hooft flux [\textit{E. Witten}, Commun. Math. Phys. 117, No. 3, 353--386 (1988; Zbl 0656.53078)]. The gauge group is broken to U(1) on the Coulomb branch. The Coulomb branch, also known as the \(u\)-plane, can be considered as a three-punctured sphere, where the punctures correspond to the weak coupling limit, and the two strong coupling singularities. The authors consider compact four-manifolds \(M\) with \((b_1, b^+_2 ) = (0, 1)\) and without boundary. Here \(b_j\) denotes the Betti numbers of \(M\), and \(b^+_2\) is the number of positive definite eigenvalues of the intersection form of two-cycles of \(M\). Complex four-manifolds with \(b^+_2= 1\) are well-studied and classified by the Enriques-Kodaira classification reviewed in section 3.2 of this paper. The authors evaluate and analyze the \(u\)-plane contribution to the correlation functions known as the \(u\)-plane integral and show that they can be evaluated by integration by parts leading to expressions in terms of mock modular forms for point observables, and Appell-Lerch sums for surface observables. In the absence of hypermultiplets, we are always free to consider the case where the principal SO(3) gauge bundle has a nontrivial 't~Hooft flux \(w_2\in H^2(M;\mathbb{Z}_2)\). The authors choose an integral lift \(\overline{w_2}\) (which is supposed to exist) such that \(\mu:=\frac{1}{2}\overline{w_2}\in H^2(M;\mathbb{R})\). The path integral over the Coulomb branch of Donaldson-Witten theory, denoted by \(\Phi^J_\mu\) is an integral over the infinite dimensional field space, which reduces to a finite dimensional integral over the zero modes [\textit{G. Moore} and \textit{E. Witten}, Adv. Theor. Math. Phys. 1, No. 2, 298--387 (1997; Zbl 0899.57021)]. \(J\in H^2(M,\mathbb{R})\) is the period point which depends on the metric due to the self-duality condition. The explicit expressions for the \(u\)-plane integrals are given in equation (5.44) for manifolds with odd intersection form and just point observables inserted, equation (5.65) for manifolds with odd intersection form and just surface observables inserted, and equation (5.84) for manifolds with even intersection form and just surface observables inserted. These expressions hold for a special choice of the metric, though the metric dependence only enters through the choice of period point \(J\). Using the expression for the wall-crossing formula in terms of indefinite theta functions, analogous mock modular forms relevant to other chambers can be obtained [\textit{L. Göttsche} and \textit{D. Zagier}, Sel. Math., New Ser. 4, No. 1, 69--115 (1998; Zbl 0924.57025)]. Using the expression for \(\Phi^J_\mu[\mathcal{O}]\) in terms of mock modular forms, one can address analytic properties of the correlators for \(b^+_2= 1\), analogously to the structural results for manifolds with \(b^+_2 > 1\) [\textit{P. B. Kronheimer} and \textit{T. S. Mrowka}, J. Differ. Geom. 41, No. 3, 573--734 (1995; Zbl 0842.57022)]. The authors study the asymptotic behavior of \(\Phi[u^\ell]\) for large \(\ell\) and find experimental evidence that \(\Phi^J_\mu[u^\ell]\sim 1/(\ell\log(\ell))\) for any four-manifold with \((b_1, b^+_2 ) = (0, 1)\). The asymptotic behavior of \(\Phi^J_\mu[u^\ell]\) suggests that \[ \Phi^J_\mu[e^{2pu}]=\sum_{\ell\ge0}(2p)^\ell\Phi^J_\mu[u^\ell]/\ell! \] is an entire function of \(p\) rather than a formal expansion. The authors find similar experimental evidence for the \(u\)-plane contribution to the exponentiated surface observable.