$\mathrm{SU}(r)$ Vafa-Witten invariants, Ramanujan's continued fractions, and cosmic strings
L. Göttsche, M. Kool, T. Laarakker
TL;DR
This work proposes a comprehensive structure for the SU$(r)$ Vafa–Witten partition function on surfaces with holomorphic $2$-forms, combining Tanaka–Thomas theory, Gholampour–Thomas degeneracy loci, and $S$-duality to separate vertical and horizontal contributions. It introduces a universal framework with theta-function data, Seiberg–Witten invariants, and Ramanujan/Hauptmodul modular structures that encode the vertical fixed loci, and derives horizontal predictions via $S$-duality, yielding conjectural formulas for ranks $2$–$5$ (and partial results for higher ranks). The vertical computations are substantiated in low virtual dimensions through nested Hilbert-scheme calculations, while horizontal predictions connect to virtual Euler characteristics of Gieseker–Maruyama moduli spaces and display Galois-invariant structures for the corresponding invariants. The paper also extends to $K$-theoretic refinements and Jacobi-form enhancements, and develops blow-up relations that mirror known topological and Donaldson-theoretic results, signaling a rich, modular bridge between algebraic geometry and number theory in Vafa–Witten theory.
Abstract
We conjecture a structure formula for the $\mathrm{SU}(r)$ Vafa-Witten partition function for surfaces with holomorphic 2-form. The conjecture is based on $S$-duality and a structure formula for the vertical contribution previously derived by the third-named author using Gholampour-Thomas's theory of virtual degeneracy loci. For ranks $r=2,3$, conjectural expressions for the partition function in terms of the theta functions of $A_{r-1}, A_{r-1}^{\vee}$ and Seiberg-Witten invariants were known. We conjecture new expressions for $r=4,5$ in terms of the theta functions of $A_{r-1}, A_{r-1}^{\vee}$, Seiberg-Witten invariants, and continued fractions studied by Ramanujan. The vertical part of our conjectures is proved for low virtual dimensions by calculations on nested Hilbert schemes. The horizontal part of our conjectures gives predictions for virtual Euler characteristics of Gieseker-Maruyama moduli spaces of stable sheaves. In this case, our formulae are sums of universal functions with coefficients in Galois extensions of $\mathbb{Q}$. The universal functions, corresponding to different quantum vacua, are permuted under the action of the Galois group. For $r=6, 7$ we also find relations with Hauptmoduln of $Γ_0(r)$. We present $K$-theoretic refinements for $r=2,3,4$ involving weak Jacobi forms.