Mocking the u-plane integral

Journal Article · 24 June 2021 · Research in the Mathematical Sciences (Print)

DOI: https://doi.org/10.1007/s40687-021-00280-5 · OSTI ID:1798402

Korpas, Georgios; Manschot, Jan ; Moore, Gregory W.; Nidaiev, Iurii

Abstract The u -plane integral is the contribution of the Coulomb branch to correlation functions of $$$${\mathcal {N}}=2$$$$ $N=2$ gauge theory on a compact four-manifold. We consider the u -plane integral for correlators of point and surface observables of topologically twisted theories with gauge group $$$$\mathrm{SU}(2)$$$$ $\mathrm{SU}\left(2\right)$ , for an arbitrary four-manifold with $$$$(b_1,b_2^+)=(0,1)$$$$ $(b_1,b_2^{+})=(0,1)$ . 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.

• Sponsoring Organization: USDOE
• Grant/Contract Number: SC0010008
• OSTI ID: 1798402
• Journal Information: Research in the Mathematical Sciences (Print), Journal Name: Research in the Mathematical Sciences (Print) Journal Issue: 3 Vol. 8; ISSN 2522-0144
• Publisher: Springer Science + Business MediaCopyright Statement
• Country of Publication: Germany
• Language: English