Degeneracy loci, virtual cycles and nested Hilbert schemes, I

Gholampour, Amin; Thomas, Robert Paul

doi:10.2140/tunis.2020.2.633

Cited by 25 publications

(58 citation statements)

References 30 publications

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“…Let r, c 1 be chosen such that r is prime and there exist no rank r strictly Gieseker H-semistable Higgs pairs on S with first Chern class c 1 . Then A. Gholampour and Thomas [GT1,GT2] express the monopole contribution to the Vafa-Witten invariants in terms of (virtual) intersection numbers on nested Hilbert schemes of points and curves on S (see also [GSY, Tho]). Based on this result, Laarakker expresses Z mono S,H,r,c 1 (q, y) in terms of Seiberg-Witten invariants and universal power series, which can be written in terms of intersection numbers on S [n 1 ] × · · · × S [nr] .…”

Section: (3)mentioning

confidence: 99%

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Refined $\mathrm{SU}(3)$ Vafa–Witten invariants and modularity

Kool

Göttsche

2018

Pure and Applied Mathematics Quarterly

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We conjecture a formula for the refined SU(3) Vafa-Witten invariants of any smooth surface S satisfying H 1 (S, Z) = 0 and p g (S) > 0. The unrefined formula corrects a proposal by Labastida-Lozano and involves unexpected algebraic expressions in modular functions. We prove that our formula satisfies a refined S-duality modularity transformation.We provide evidence for our formula by calculating virtual χ y -genera of moduli spaces of rank 3 stable sheaves on S in examples using Mochizuki's formula. Further evidence is based on the recent definition of refined SU(r) Vafa-Witten invariants by Maulik-Thomas and subsequent calculations on nested Hilbert schemes by Thomas (rank 2) and Laarakker (rank 3).

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Section: (3)mentioning

confidence: 99%

“…S,H,r,c 1 (q, y). In [GT1,GT2], Gholampour and Thomas express this contribution in terms of virtual cycles on nested Hilbert schemes of curves and points on S (see also [GSY]). This leads to an expression in terms of Seiberg-Witten invariants of S and intersection numbers on S [n 1 ] × · · · × S [nr] .…”

Section: Monopole Branchmentioning

confidence: 99%

Refined $\mathrm{SU}(3)$ Vafa–Witten invariants and modularity

Kool

Göttsche

2018

Pure and Applied Mathematics Quarterly

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“…The action A K M is defined via the correspondences Z 1 and Z • 2 (Definitions 4.11 and 4.16, respectively), which yield resolutions of singularities of the cohomological constructions of Nakajima [20] and Grojnowski [14] in rank 1, and Baranovsky [2] in rank r. In general, the correspondences Z 1 and Z • 2 are defined as dg schemes, but we show in Propositions 4.19 and 4.20 that they are smooth schemes whenever the moduli spaces M are smooth (more precisely, under Assumption S of (4.27)). The proof of the smoothness of Z 1 and Z • 2 relies on presenting their tangent spaces in terms of cones of Ext groups, which is an idea that has appeared repeatedly in the literature (see [13] for a development in the context of virtual degeneration loci).…”

Section: Introductionmentioning

confidence: 99%

Shuffle algebras associated to surfaces

Neguţ

2019

Sel. Math. New Ser.

123

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We define an integral form of the deformed W -algebra of type gl r , and construct its action on the K-theory groups of moduli spaces of rank r stable sheaves on a smooth projective surface S, under certain assumptions. Our construction generalizes the action studied by Nakajima, Grojnowski and Baranovsky in cohomology, although the appearance of deformed W -algebras by generators and relations is a new feature. Physically, this action encodes the AGT correspondence for 5d supersymmetric gauge theory on S×circle.

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“…Nested Hilbert schemes. There are components of the monopole branch which are nested Hilbert schemes of S. In [GT1,GT2] it was shown how to view these as degeneracy loci in smooth products of Hilbert schemes of S. This induces a virtual cycle which agrees with the one from Vafa-Witten theory. Its pushforward is described by the Thom-Porteous formula.…”

Section: K3 Surfacesmentioning

confidence: 96%

“…When p g (S) > 0, for any charge α with prime rank, Laarakker [La2] shows that the conjecture holds for the contribution of the monopole locus. He uses the vanishing Theorem 5.23 to remove many components, and [GT1,GT2] to calculate with the rest.…”

Section: Semistable Casementioning

confidence: 99%

Equivariant K-Theory and Refined Vafa–Witten Invariants

Thomas

2020

Commun. Math. Phys.

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In Maulik and Thomas (in preparation) the Vafa-Witten theory of complex projective surfaces is lifted to oriented C *-equivariant cohomology theories. Here we study the K-theoretic refinement. It gives rational functions in t 1/2 invariant under t 1/2 ↔ t −1/2 which specialise to numerical Vafa-Witten invariants at t = 1. On the "instanton branch" the invariants give the virtual χ −t-genus refinement of Göttsche-Kool, extended to allow for strictly semistable sheaves. Applying modularity to their calculations gives predictions for the contribution of the "monopole branch". We calculate some cases and find perfect agreement. We also do calculations on K3 surfaces, finding Jacobi forms refining the usual modular forms, proving a conjecture of Göttsche-Kool. We determine the K-theoretic virtual classes of degeneracy loci using Eagon-Northcott complexes, and show they calculate refined Vafa-Witten invariants. Using this Laarakker (Monopole contributions to refined Vafa-Witten invariants. arXiv:1810.00385) proves universality results for the invariants. Contents L ⊗ O C ] = [Lι * O C ] ∈ K 0 (M). The result is slightly different-differing by a Todd class, by virtual Riemann-Rochand therefore interesting! (Especially when we work equivariantly with respect to the T action.) To get the Nekrasov-Okounkov-twisted version of this used in our paper we instead take the intersection in KO-theory. This replaces the K-theoretic "fundamental classes" O Z of submanifolds by their twists by K 1/2 Z (this is the Atiyah-Bott-Shapiro complex orientation, and is well defined over Z only for spin manifolds). The universal case is complex cobordism theory; see [Sh,GK2] for instance. From this one can pass to all other oriented cohomology theories, such as "topological modular forms". 2 In Vafa-Witten theory, the three T-equivariant cohomology theories homology, KO-theory, tmf give rise to virtual versions of M 1, A(M), Witten genus (M) of the Vafa-Witten moduli space M respectively. On the "instanton locus" these produce the Euler characteristic, Hirzebruch χ y-genus, elliptic genus of the moduli space of instantons (or Gieseker stable sheaves) on the surface S. (This apparent paradox is because the Vafa-Witten obstruction theory on the instanton moduli space differs from its usual obstruction theory.) Calculations give generating series which seem to be (modular forms, Jacobi forms, Borcherds lifts of Jacobi forms) respectively; in particular see [GK2] for the instanton locus contributions in rank 2.

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Degeneracy loci, virtual cycles and nested Hilbert schemes, I

Cited by 25 publications

References 30 publications

Refined $\mathrm{SU}(3)$ Vafa–Witten invariants and modularity

Refined $\mathrm{SU}(3)$ Vafa–Witten invariants and modularity

Shuffle algebras associated to surfaces

Equivariant K-Theory and Refined Vafa–Witten Invariants

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