The Tanaka–Thomas’s Vafa–Witten invariants via surface Deligne–Mumford stacks

Jiang, Yunfeng; Kundu, Promit

doi:10.4310/pamq.2021.v17.n1.a13

Cited by 6 publications

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“…The group H 2 (S, μ r ) classifies μ r -gerbes. In [20], the first-named author developed Vafa-Witten theory of μ r -gerbes (see also [21][22][23]25]). In this paper, which is inspired by [20], we take a different approach:…”

Section: Su(r)/z R Partition Functionmentioning

confidence: 99%

“…Although one could now painstakingly try to develop the analog of [43,44] for moduli of stable Y -sheaves (similar to [20][21][22][23]25] in the language of μ r -gerbes), we here take a short-cut when r is prime. Using the cone construction of Jiang-Thomas [24], we will see that N H Y ,ξ/r (r , D, n) has a natural symmetric perfect obstruction theory.…”

Section: Cone Constructionmentioning

confidence: 99%

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

Jiang

Kool

2021

Math. Ann.

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The SU(r ) Vafa-Witten partition function, which virtually counts Higgs pairs on a projective surface S, was mathematically defined by Tanaka-Thomas. On the Langlands dual side, the first-named author recently introduced virtual counts of Higgs pairs on μ r -gerbes. In this paper, we instead use Yoshioka's moduli spaces of twisted sheaves. Using Chern character twisted by rational B-field, we give a new mathematical definition of the SU(r )/Z r Vafa-Witten partition function when r is prime. Our definition uses the period-index theorem of de Jong. S-duality, a concept from physics, predicts that the SU(r ) and SU(r )/Z r partition functions are related by a modular transformation. We turn this into a mathematical conjecture, which we prove for all K 3 surfaces and prime numbers r .Communicated by Jean-Yves Welschinger.

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Section: Su(r)/z R Partition Functionmentioning

confidence: 99%

Section: Cone Constructionmentioning

confidence: 99%

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

Jiang

Kool

2021

Math. Ann.

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“…In order to get interesting information on the DM stack X, the sheaf Ξ is necessary. This can be seen for the µ r -gerbe p : X Ñ X, since the pushforward p ˚L of line bundle L is zero if the degree of L is not divisible by r. For another example, in [48, §7], [23] the modified Hilbert polynomial on a root stack X will corresponds to the parabolic Hilbert polynomial on the pair (X, D) with D Ă X a smooth divisor.…”

Section: Stability Condition Using Generating Sheavesmentioning

confidence: 99%

Counting twisted sheaves and S-duality

Jiang¹

2019

Preprint

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We provide a definition of Tanaka-Thomas's Vafa-Witten invariants for étale gerbes over smooth projective surfaces using the moduli spaces of µ r -gerbe twisted sheaves and Higgs sheaves. Twisted sheaves and their moduli are naturally used to study the period-index theorem for the corresponding µ r -gerbe in the Brauer group of the surfaces. Deformation and obstruction theory of the twisted sheaves and Higgs sheaves behave like general sheaves and Higgs sheaves. We define virtual fundamental classes on the moduli spaces and define the twisted Vafa-Witten invariants using virtual localization and the Behrend function on the moduli spaces. As applications for the Langlands dual group SU(r)/Z r of SU(r), we define the SU(r)/Z r -Vafa-Witten invariants using the twisted invariants for étale gerbes, and prove the S-duality conjecture of Vafa-Witten for the projective plane in rank two and for K3 surfaces in prime ranks. We also conjecture for other surfaces.CONTENTS 2.1. Algebraic stacks and Deligne-Mumford stacks 2.2. Gerbes and essentially trivial gerbes 2.3. Brauer group and optimal gerbes 3. Moduli space of twisted sheaves and twisted Higgs sheaves on surfaces 3.1. Étale gerbe twisted sheaves 3.2. Twisted stability and the moduli stack 3.3. Moduli of X-twisted sheaves for optimal gerbes 3.4. Twisted Higgs sheaves and the moduli stack 4. Perfect obstruction theory, virtual classes and Vafa-Witten invariants 4.1. Perfect obstruction theory for moduli of twisted stable sheaves 4.2. Deformation and obstruction theory for twisted Higgs sheaves 4.3. SU(rk)-twisted Vafa-Witten invariants 4.4. The G m -fixed loci 4.5. Invariants defined by the Behrend function 5. Joyce-Song twisted stable pairs 5.1. Background to count semistable objects 5.2. Hall algebras and the integration map 5.3. Application to semistable S-twisted Higgs sheaves 5.4. Generalized twisted SU(rk)-Vafa-Witten invariants VW tw 5.5. The SU(r)/Z r -Vafa-Witten invariants and proof in rank two for P 2 6. S-duality conjecture for K3 surfaces 1 2 YUNFENG JIANG 6.1. K3 surfaces-Picard numbers, Mukai Lattice and Brauer group 43 6.2. Tanaka-Thomas's Vafa-Witten invariants for K3 surfaces 44 6.3. Twisted Vafa-Witten invariants-essentially trivial gerbes 44 6.4. Twisted Vafa-Witten invariants-optimal gerbes 46 6.5. Proof of S-duality formula in rank two 49 6.6. Vafa-Witten's S-duality conjecture for K3 surfaces 51 6.7. Discussion on higher rank case 54 56 References 56

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On the Bogomolov–Gieseker inequality for tame Deligne–Mumford surfaces

Jiang,

Kundu

2024

Math. Z.

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The Tanaka–Thomas’s Vafa–Witten invariants via surface Deligne–Mumford stacks

Cited by 6 publications

References 0 publications

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

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

Counting twisted sheaves and S-duality

On the Bogomolov–Gieseker inequality for tame Deligne–Mumford surfaces

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