$\mathrm{SU}(r)$ Vafa-Witten invariants, Ramanujan's continued fractions, and cosmic strings

Abstract

We conjecture a structure formula for the SUprq 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 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 , 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 give 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 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 prq. We present K-theoretic refinements for r " 2, 3, 4 involving weak Jacobi forms. c vd pT vir M q " p´1q vd e vir pMq.Keeping S, H, r, c 1 fixed and varying c 2 (still assuming there are no strictly H-semistable objects), we define 3 the SUprq Vafa-Witten partition function by H,c 1 pqq " r ´1Z prq S,H,c 1 pqq `r´1 Z p1 r q S,H,c 1 pqq.3 The factor in front of the sum is required for modularity later.

Twisted sheaves and $$\mathrm {SU}(r) / {\mathbb {Z}}_{r}$$ Vafa–Witten theory

“…This conjectural formula has been verified, up to some order in q, in many cases [10,30]. For the SU(3) and SU(5) partition function, conjectural closed expressions are also known and tested [11,12,30]. Each of these partition functions is invariant under replacing c 1 by c 1 + r γ for any algebraic class γ ∈ H 2 (S, Z).…”

“…In particular, there are no rank r twisted strictly semistable Y -sheaves, so M is projective and ( 13) is an equality. By [24], we obtain diagram (12).…”

“…In [11,12], the authors took the following approach to the S-duality transformation (1). For SU(2), SU(3), and SU(5), the conjectural closed formulae for the right hand side of ( 6) in [4,11,12,48] are explicit formulae involving modular forms and χ(O S ), K 2 S , a i a j , K S ξ, a i ξ, ξ 2 , where a i runs over the Seiberg-Witten basic classes of S. These formulae satisfy Conjecture 1.1 and make sense for non-algebraic classes ξ , thereby providing an ad hoc definition of Z…”

Virtual Segre and Verlinde numbers of projective surfaces

“…□ Remark 3.4. We verified the formulae for 𝑄 𝑠 , 𝑅 𝑠 , 𝑇 𝑠 of Conjecture 1.4 for 𝜌 = 1 modulo 𝑧 11 . For this, we choose a collection of triples (𝑆, 𝛼, 𝐿) such that the corresponding vectors (𝑐 2 (𝛼), 𝑐 1 (𝛼) 2 , 𝜒, 𝑐 1 (𝛼)𝐾, 𝐾 2 , 𝐿 2 , 𝑐 1 (𝛼)𝐿, 𝐿𝐾) are ℚ-linearly independent.…”

The Blowup Formula for the Instanton Part of Vafa-Witten Invariants on Projective Surfaces

“…The virtual invariants have the benefit of being invariant under deformations of the surface that preserve the polarization, and -strikingly -they are in many cases predicted to depend only on finitely many basic invariants of the underlying surface. For example, explicit formulas up to rank 5 have been conjectured for the virtual Euler characteristics in [GKL21] (see also [GK18], [GK20], [GK19a], [GKW21]; and [GK19b] for an excellent survey).…”