Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M,...
Transcript of Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M,...
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Vafa-Witten invariants of projective surfaces
Joint with Yuuji Tanaka
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The Vafa-Witten equations
“A Strong Coupling Test of S-Duality ” (1994)
Cumrun Vafa Ed Witten
Riemannian 4-manifold M, SU(r) bundle E → M, connection A,fields B ∈ Ω+(su(E )), Γ ∈ Ω0(su(E )),
F+A + [B.B] + [B, Γ] = 0,
dAΓ + d∗AB = 0.
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The Vafa-Witten equations
“A Strong Coupling Test of S-Duality ” (1994)
Cumrun Vafa Ed Witten
Riemannian 4-manifold M, SU(r) bundle E → M, connection A,fields B ∈ Ω+(su(E )), Γ ∈ Ω0(su(E )),
F+A + [B.B] + [B, Γ] = 0,
dAΓ + d∗AB = 0.
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Their prediction
VW invariants
Vafa and Witten told us to “count” (in an appropriate sense)solutions of these VW equations.
For c2 = n, let VWn (∈ Z? ∈ Q?) denote the resultingVafa-Witten invariants of M.
Modular forms“S-duality” voodoo should imply their generating series
q−e(S)12
∞∑n=0
VWn(M) qn
is a modular form.
In particular, the infinite collection of numbers VWn(M) should bedetermined by only finitely many of them.
![Page 5: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/5.jpg)
Their prediction
VW invariants
Vafa and Witten told us to “count” (in an appropriate sense)solutions of these VW equations.
For c2 = n, let VWn (∈ Z? ∈ Q?) denote the resultingVafa-Witten invariants of M.
Modular forms“S-duality” voodoo should imply their generating series
q−e(S)12
∞∑n=0
VWn(M) qn
is a modular form.
In particular, the infinite collection of numbers VWn(M) should bedetermined by only finitely many of them.
![Page 6: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/6.jpg)
Their prediction
VW invariants
Vafa and Witten told us to “count” (in an appropriate sense)solutions of these VW equations.
For c2 = n, let VWn (∈ Z? ∈ Q?) denote the resultingVafa-Witten invariants of M.
Modular forms“S-duality” voodoo should imply their generating series
q−e(S)12
∞∑n=0
VWn(M) qn
is a modular form.
In particular, the infinite collection of numbers VWn(M) should bedetermined by only finitely many of them.
![Page 7: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/7.jpg)
Vanishing theoremsThey were able to check their conjecture in many cases when Mhas positive curvature, due to a vanishing theorem B = 0 = Γ.
The equations then reduce to the anti-self-dual equations. Thesehave a compact moduli space Masd
n .
When there are no reducible solutions, the obstruction bundle isT (∗)Masd
n so we should have
VWn = ±e(Masdn ).
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Vanishing theoremsThey were able to check their conjecture in many cases when Mhas positive curvature, due to a vanishing theorem B = 0 = Γ.
The equations then reduce to the anti-self-dual equations. Thesehave a compact moduli space Masd
n .
When there are no reducible solutions, the obstruction bundle isT (∗)Masd
n so we should have
VWn = ±e(Masdn ).
![Page 9: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/9.jpg)
Vanishing theoremsThey were able to check their conjecture in many cases when Mhas positive curvature, due to a vanishing theorem B = 0 = Γ.
The equations then reduce to the anti-self-dual equations. Thesehave a compact moduli space Masd
n .
When there are no reducible solutions, the obstruction bundle isT (∗)Masd
n so we should have
VWn = ±e(Masdn ).
![Page 10: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/10.jpg)
Kahler case
For general M no one can yet define Vafa-Witten invariants sincethe moduli space is inherently noncompact. (|B|, |Γ| can →∞.)
When M = S is a Kahler surface we can rewrite B, Γ in terms ofan End0 E -valued (2, 0)-form φ and an End0 E -valued multiple ofthe Kahler form ω, giving
F 0,2A = 0,
F 1,1A ∧ ω +
[φ, φ
]= 0,
∂Aφ = 0.
So ∂A makes E into a holomorphic bundle with a holomorphicHiggs field
φ ∈ H0(End0 E ⊗ KS)
satisfying a moment map equation F 1,1A ∧ ω +
[φ, φ
]= 0.
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Kahler case
For general M no one can yet define Vafa-Witten invariants sincethe moduli space is inherently noncompact. (|B|, |Γ| can →∞.)
When M = S is a Kahler surface we can rewrite B, Γ in terms ofan End0 E -valued (2, 0)-form φ and an End0 E -valued multiple ofthe Kahler form ω, giving
F 0,2A = 0,
F 1,1A ∧ ω +
[φ, φ
]= 0,
∂Aφ = 0.
So ∂A makes E into a holomorphic bundle with a holomorphicHiggs field
φ ∈ H0(End0 E ⊗ KS)
satisfying a moment map equation F 1,1A ∧ ω +
[φ, φ
]= 0.
![Page 12: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/12.jpg)
Kahler case
For general M no one can yet define Vafa-Witten invariants sincethe moduli space is inherently noncompact. (|B|, |Γ| can →∞.)
When M = S is a Kahler surface we can rewrite B, Γ in terms ofan End0 E -valued (2, 0)-form φ and an End0 E -valued multiple ofthe Kahler form ω, giving
F 0,2A = 0,
F 1,1A ∧ ω +
[φ, φ
]= 0,
∂Aφ = 0.
So ∂A makes E into a holomorphic bundle with a holomorphicHiggs field
φ ∈ H0(End0 E ⊗ KS)
satisfying a moment map equation F 1,1A ∧ ω +
[φ, φ
]= 0.
![Page 13: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/13.jpg)
Hitchin-Kobayashi correspondence
At least when S is projective, Alvarez-Consul–Garcıa-Prada andTanaka have proved an infinite dimensional Kempf-Ness theorem.
Solutions (modulo unitary gauge transformations) correspond topolystable Higgs pairs (E , φ) (modulo complex linear gauge).
Linearises the problem, and allows us to partially compactify byallowing E to be a (torsion-free) coherent sheaf.
When KS < 0 stability forces φ = 0.Similarly when KS ≤ 0 and stability = semistability.
Then we get the moduli space of (semi)stable sheaves E withdetE = OS on S , and VWn is some kind of Euler characteristicthereof.
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Hitchin-Kobayashi correspondence
At least when S is projective, Alvarez-Consul–Garcıa-Prada andTanaka have proved an infinite dimensional Kempf-Ness theorem.
Solutions (modulo unitary gauge transformations) correspond topolystable Higgs pairs (E , φ) (modulo complex linear gauge).
Linearises the problem, and allows us to partially compactify byallowing E to be a (torsion-free) coherent sheaf.
When KS < 0 stability forces φ = 0.Similarly when KS ≤ 0 and stability = semistability.
Then we get the moduli space of (semi)stable sheaves E withdetE = OS on S , and VWn is some kind of Euler characteristicthereof.
![Page 15: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/15.jpg)
Hitchin-Kobayashi correspondence
At least when S is projective, Alvarez-Consul–Garcıa-Prada andTanaka have proved an infinite dimensional Kempf-Ness theorem.
Solutions (modulo unitary gauge transformations) correspond topolystable Higgs pairs (E , φ) (modulo complex linear gauge).
Linearises the problem, and allows us to partially compactify byallowing E to be a (torsion-free) coherent sheaf.
When KS < 0 stability forces φ = 0.Similarly when KS ≤ 0 and stability = semistability.
Then we get the moduli space of (semi)stable sheaves E withdetE = OS on S , and VWn is some kind of Euler characteristicthereof.
![Page 16: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/16.jpg)
Spectral construction
Put eigenspaces of φ : E → E ⊗ KS over the correspondingeigenvalues in KS . Defines a torsion sheaf Eφ on X = KS .
spectral Page 1
Over a point of S we have a vector space V and a endomorphismφ. This makes V into a finite-dimensional C[x ]-module (and so atorsion sheaf) by letting x act through φ.
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Spectral construction
Put eigenspaces of φ : E → E ⊗ KS over the correspondingeigenvalues in KS . Defines a torsion sheaf Eφ on X = KS .
spectral Page 1
Over a point of S we have a vector space V and a endomorphismφ. This makes V into a finite-dimensional C[x ]-module (and so atorsion sheaf) by letting x act through φ.
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HiggsKS(S)←→Cohc(X )
spectral Page 1
Globally over S , we make E into a π∗OS =⊕
i K−iS -module by
E ⊗ K−iS
φi−−→ E .
Thus we get a sheaf Eφ over X := KS .
Conversely, from E over X we recover E := π∗E and then φ fromthe action of η · id, where η is the tautological section of π∗KS .
detE = OS , tr φ = 0 ⇐⇒E has centre of mass 0 on each fibre, and detπ∗E = OS .
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HiggsKS(S)←→Cohc(X )
spectral Page 1
Globally over S , we make E into a π∗OS =⊕
i K−iS -module by
E ⊗ K−iS
φi−−→ E .
Thus we get a sheaf Eφ over X := KS .
Conversely, from E over X we recover E := π∗E and then φ fromthe action of η · id, where η is the tautological section of π∗KS .
detE = OS , tr φ = 0 ⇐⇒E has centre of mass 0 on each fibre, and detπ∗E = OS .
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HiggsKS(S)←→Cohc(X )
spectral Page 1
Globally over S , we make E into a π∗OS =⊕
i K−iS -module by
E ⊗ K−iS
φi−−→ E .
Thus we get a sheaf Eφ over X := KS .
Conversely, from E over X we recover E := π∗E and then φ fromthe action of η · id, where η is the tautological section of π∗KS .
detE = OS , tr φ = 0 ⇐⇒E has centre of mass 0 on each fibre, and detπ∗E = OS .
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Virtual cycle
When stability = semistability, deformation-obstruction theory ofsheaves E on Calabi-Yau 3-fold X is perfect, symmetric:
Deformations Ext1X (E , E)
Obstructions Ext2X (E , E) ∼= Ext1X (E , E)∗
Higher obstructions Ext≥3X (E , E)0 = 0
Therefore inherits a virtual cycle of virtual dimension 0.
Noncompact, but moduli space admits C∗ action scaling KS fibresof X
π−−→ S (equivalently, scaling Higgs field φ).
C∗-fixed locus compact, so can define an invariant by virtualC∗-localisation. Local DT invariant.
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Virtual cycle
When stability = semistability, deformation-obstruction theory ofsheaves E on Calabi-Yau 3-fold X is perfect, symmetric:
Deformations Ext1X (E , E)
Obstructions Ext2X (E , E) ∼= Ext1X (E , E)∗
Higher obstructions Ext≥3X (E , E)0 = 0
Therefore inherits a virtual cycle of virtual dimension 0.
Noncompact, but moduli space admits C∗ action scaling KS fibresof X
π−−→ S (equivalently, scaling Higgs field φ).
C∗-fixed locus compact, so can define an invariant by virtualC∗-localisation. Local DT invariant.
![Page 23: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/23.jpg)
Virtual cycle
When stability = semistability, deformation-obstruction theory ofsheaves E on Calabi-Yau 3-fold X is perfect, symmetric:
Deformations Ext1X (E , E)
Obstructions Ext2X (E , E) ∼= Ext1X (E , E)∗
Higher obstructions Ext≥3X (E , E)0 = 0
Therefore inherits a virtual cycle of virtual dimension 0.
Noncompact, but moduli space admits C∗ action scaling KS fibresof X
π−−→ S (equivalently, scaling Higgs field φ).
C∗-fixed locus compact, so can define an invariant by virtualC∗-localisation. Local DT invariant.
![Page 24: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/24.jpg)
Virtual cycle
When stability = semistability, deformation-obstruction theory ofsheaves E on Calabi-Yau 3-fold X is perfect, symmetric:
Deformations Ext1X (E , E)
Obstructions Ext2X (E , E) ∼= Ext1X (E , E)∗
Higher obstructions Ext≥3X (E , E)0 = 0
Therefore inherits a virtual cycle of virtual dimension 0.
Noncompact, but moduli space admits C∗ action scaling KS fibresof X
π−−→ S (equivalently, scaling Higgs field φ).
C∗-fixed locus compact, so can define an invariant by virtualC∗-localisation. Local DT invariant.
![Page 25: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/25.jpg)
Better virtual cycle
But this U(r) VW invariant zero if H0,1(S) 6= 0 or H0,2(S) 6= 0.
So define SU(r) VW invariant by restricting to E with centre ofmass 0 on each fibre, and detπ∗E = OS .
Has a perfect, symmetric deformation theory governed by last termof splitting
Ext∗X (E , E)0∼= H∗(OS)⊕ H∗−1(KS)⊕ Ext∗X (E , E)⊥.
C∗-localisation then defines an invariant
VWn ∈ Q
when stability = semistability.
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Better virtual cycle
But this U(r) VW invariant zero if H0,1(S) 6= 0 or H0,2(S) 6= 0.
So define SU(r) VW invariant by restricting to E with centre ofmass 0 on each fibre, and detπ∗E = OS .
Has a perfect, symmetric deformation theory governed by last termof splitting
Ext∗X (E , E)0∼= H∗(OS)⊕ H∗−1(KS)⊕ Ext∗X (E , E)⊥.
C∗-localisation then defines an invariant
VWn ∈ Q
when stability = semistability.
![Page 27: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/27.jpg)
Better virtual cycle
But this U(r) VW invariant zero if H0,1(S) 6= 0 or H0,2(S) 6= 0.
So define SU(r) VW invariant by restricting to E with centre ofmass 0 on each fibre, and detπ∗E = OS .
Has a perfect, symmetric deformation theory governed by last termof splitting
Ext∗X (E , E)0∼= H∗(OS)⊕ H∗−1(KS)⊕ Ext∗X (E , E)⊥.
C∗-localisation then defines an invariant
VWn ∈ Q
when stability = semistability.
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Better virtual cycle
But this U(r) VW invariant zero if H0,1(S) 6= 0 or H0,2(S) 6= 0.
So define SU(r) VW invariant by restricting to E with centre ofmass 0 on each fibre, and detπ∗E = OS .
Has a perfect, symmetric deformation theory governed by last termof splitting
Ext∗X (E , E)0∼= H∗(OS)⊕ H∗−1(KS)⊕ Ext∗X (E , E)⊥.
C∗-localisation then defines an invariant
VWn ∈ Q
when stability = semistability.
![Page 29: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/29.jpg)
C∗-fixed loci
Invariant computed from two types of C∗-fixed locus:
1. φ = 0. We get the moduli space Masd of stable sheaves on S .
2. φ 6= 0 nilpotent. We call this moduli space M2.
The first are supported on S , and contribute the virtual signedEuler characteristic
(−1)vdevir(Masd) ∈ Z.
When KS ≤ 0 they give all stable terms.
Heavily studied, giving modular forms [. . . , Gottsche-Kool].
The second are supported on a scheme-theoretic thickening ofS ⊂ X . When they have rank 1 on their support, they can bedescribed in terms of nested Hilbert schemes of S .
Unstudied. Our (limited) computations give more modular formspredicted by Vafa-Witten 100 years ago by “cosmic strings”.
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C∗-fixed loci
Invariant computed from two types of C∗-fixed locus:
1. φ = 0. We get the moduli space Masd of stable sheaves on S .
2. φ 6= 0 nilpotent. We call this moduli space M2.
The first are supported on S , and contribute the virtual signedEuler characteristic
(−1)vdevir(Masd) ∈ Z.
When KS ≤ 0 they give all stable terms.
Heavily studied, giving modular forms [. . . , Gottsche-Kool].
The second are supported on a scheme-theoretic thickening ofS ⊂ X . When they have rank 1 on their support, they can bedescribed in terms of nested Hilbert schemes of S .
Unstudied. Our (limited) computations give more modular formspredicted by Vafa-Witten 100 years ago by “cosmic strings”.
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C∗-fixed loci
Invariant computed from two types of C∗-fixed locus:
1. φ = 0. We get the moduli space Masd of stable sheaves on S .
2. φ 6= 0 nilpotent. We call this moduli space M2.
The first are supported on S , and contribute the virtual signedEuler characteristic
(−1)vdevir(Masd) ∈ Z.
When KS ≤ 0 they give all stable terms.
Heavily studied, giving modular forms [. . . , Gottsche-Kool].
The second are supported on a scheme-theoretic thickening ofS ⊂ X . When they have rank 1 on their support, they can bedescribed in terms of nested Hilbert schemes of S .
Unstudied. Our (limited) computations give more modular formspredicted by Vafa-Witten 100 years ago by “cosmic strings”.
![Page 32: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/32.jpg)
C∗-fixed loci
Invariant computed from two types of C∗-fixed locus:
1. φ = 0. We get the moduli space Masd of stable sheaves on S .
2. φ 6= 0 nilpotent. We call this moduli space M2.
The first are supported on S , and contribute the virtual signedEuler characteristic
(−1)vdevir(Masd) ∈ Z.
When KS ≤ 0 they give all stable terms.
Heavily studied, giving modular forms [. . . , Gottsche-Kool].
The second are supported on a scheme-theoretic thickening ofS ⊂ X . When they have rank 1 on their support, they can bedescribed in terms of nested Hilbert schemes of S .
Unstudied. Our (limited) computations give more modular formspredicted by Vafa-Witten 100 years ago by “cosmic strings”.
![Page 33: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/33.jpg)
C∗-fixed loci
Invariant computed from two types of C∗-fixed locus:
1. φ = 0. We get the moduli space Masd of stable sheaves on S .
2. φ 6= 0 nilpotent. We call this moduli space M2.
The first are supported on S , and contribute the virtual signedEuler characteristic
(−1)vdevir(Masd) ∈ Z.
When KS ≤ 0 they give all stable terms.
Heavily studied, giving modular forms [. . . , Gottsche-Kool].
The second are supported on a scheme-theoretic thickening ofS ⊂ X . When they have rank 1 on their support, they can bedescribed in terms of nested Hilbert schemes of S .
Unstudied. Our (limited) computations give more modular formspredicted by Vafa-Witten 100 years ago by “cosmic strings”.
![Page 34: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/34.jpg)
The Vafa-Witten prediction
For general type surfaces with a smooth connected canonicaldivisor, [VW] predicts
modular_form Page 1
In particular, this convinces us that our virtual localisationdefinition is the right one. An alternative definition viaBehrend-weighted Euler characteristic has various advantages(integers in stable case, natural generalisation to semistable case,natural refinement and categorification) and also gives modularforms, but the wrong ones.
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The Vafa-Witten prediction
For general type surfaces with a smooth connected canonicaldivisor, [VW] predicts
modular_form Page 1
In particular, this convinces us that our virtual localisationdefinition is the right one.
An alternative definition viaBehrend-weighted Euler characteristic has various advantages(integers in stable case, natural generalisation to semistable case,natural refinement and categorification) and also gives modularforms, but the wrong ones.
![Page 36: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/36.jpg)
The Vafa-Witten prediction
For general type surfaces with a smooth connected canonicaldivisor, [VW] predicts
modular_form Page 1
In particular, this convinces us that our virtual localisationdefinition is the right one. An alternative definition viaBehrend-weighted Euler characteristic has various advantages(integers in stable case, natural generalisation to semistable case,natural refinement and categorification) and also gives modularforms, but the wrong ones.
![Page 37: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/37.jpg)
Semistable case
Motivated by Mochizuki and Joyce-Song, we consider pairs
(E , s)
of a torsion sheaf E on X and a section s ∈ H0(E(n)), n 0,(E has centre of mass 0 on the fibres of X → S , and detπ∗E ∼= OS .)(Equivalent to consider triples (E , φ, s) on S with detE ∼= OS , tr φ = 0
and s ∈ H0(E (n)).)
which are stable
I E is semistable,
I if F ( E has the same Giesker slope then s does not factorthrough F .
These also admit a perfect symmetric obstruction theory governedby
Ext∗X (I •, I •)⊥ where I • =OX (−N)→ E
.
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Semistable case
Motivated by Mochizuki and Joyce-Song, we consider pairs
(E , s)
of a torsion sheaf E on X and a section s ∈ H0(E(n)), n 0,(E has centre of mass 0 on the fibres of X → S , and detπ∗E ∼= OS .)(Equivalent to consider triples (E , φ, s) on S with detE ∼= OS , tr φ = 0
and s ∈ H0(E (n)).)
which are stable
I E is semistable,
I if F ( E has the same Giesker slope then s does not factorthrough F .
These also admit a perfect symmetric obstruction theory governedby
Ext∗X (I •, I •)⊥ where I • =OX (−N)→ E
.
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Invariants in the semistable case
Again we use virtual C∗-localisation to define pairs invariantsP⊥α (n), where α = (rank(E ), c1(E ), c2(E )).
From these we define VW invariants by the conjectural formula
P⊥α (n) =∑
`≥1, (αi=δiα)`i=1:∑`
i=1 δi=1
(−1)`
`!
∏i=1
(−1)χ(αi (n))χ(αi (n))VWαi (S)
when H0,1(S) = 0 = H0,2(S). If either is 6= 0 we instead use onlythe first term
P⊥α (n) = (−1)χ(α(n))χ(α(n))VWα(S).
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Invariants in the semistable case
Again we use virtual C∗-localisation to define pairs invariantsP⊥α (n), where α = (rank(E ), c1(E ), c2(E )).
From these we define VW invariants by the conjectural formula
P⊥α (n) =∑
`≥1, (αi=δiα)`i=1:∑`
i=1 δi=1
(−1)`
`!
∏i=1
(−1)χ(αi (n))χ(αi (n))VWαi (S)
when H0,1(S) = 0 = H0,2(S). If either is 6= 0 we instead use onlythe first term
P⊥α (n) = (−1)χ(α(n))χ(α(n))VWα(S).
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Results
I When stability = semistability for the sheaves E , then our pairsconjecture holds and the invariants equal the invariants wedefined directly earlier.
I When degKS < 0 the same is true. (Here we prove the pairsmoduli space is smooth, and the invariants equal thosedefined by (Behrend-weighted) Euler characteristic. To thesewe can apply Joyce-Song’s work.)
I When S is a K3 surface the same is true. (Joint work withDavesh Maulik. We work on compact S × E , whereBehrend-weighted Euler characteristic invariants equal virtualinvariants. We then degenerate E to a nodal rational curve toaccess S × C. This introduces an exponential, which accountsfor the difference between our simplified pairs formula andJoyce-Song’s.)
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Results
I When stability = semistability for the sheaves E , then our pairsconjecture holds and the invariants equal the invariants wedefined directly earlier.
I When degKS < 0 the same is true. (Here we prove the pairsmoduli space is smooth, and the invariants equal thosedefined by (Behrend-weighted) Euler characteristic. To thesewe can apply Joyce-Song’s work.)
I When S is a K3 surface the same is true. (Joint work withDavesh Maulik. We work on compact S × E , whereBehrend-weighted Euler characteristic invariants equal virtualinvariants. We then degenerate E to a nodal rational curve toaccess S × C. This introduces an exponential, which accountsfor the difference between our simplified pairs formula andJoyce-Song’s.)
![Page 43: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/43.jpg)
Results
I When stability = semistability for the sheaves E , then our pairsconjecture holds and the invariants equal the invariants wedefined directly earlier.
I When degKS < 0 the same is true. (Here we prove the pairsmoduli space is smooth, and the invariants equal thosedefined by (Behrend-weighted) Euler characteristic. To thesewe can apply Joyce-Song’s work.)
I When S is a K3 surface the same is true. (Joint work withDavesh Maulik. We work on compact S × E , whereBehrend-weighted Euler characteristic invariants equal virtualinvariants. We then degenerate E to a nodal rational curve toaccess S × C. This introduces an exponential, which accountsfor the difference between our simplified pairs formula andJoyce-Song’s.)
![Page 44: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/44.jpg)
Refinement
(Behrend-weighted) Euler characteristics e have naturalrefinements H∗ (of the perverse sheaf of vanishing cycles).
For invariants defined by virtual localisation we need to dosomething different. Maulik proposes taking the virtual K-theoreticinvariant (in the stable = semistable case)
χ((
K virMVW
) 12
)defined by K-theoretic localisation, and refining it to a polynomialin t1/2 by using the C∗ action.
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Refinement
(Behrend-weighted) Euler characteristics e have naturalrefinements H∗ (of the perverse sheaf of vanishing cycles).
For invariants defined by virtual localisation we need to dosomething different. Maulik proposes taking the virtual K-theoreticinvariant (in the stable = semistable case)
χ((
K virMVW
) 12
)defined by K-theoretic localisation, and refining it to a polynomialin t1/2 by using the C∗ action.
![Page 46: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/46.jpg)
Refinement
(Behrend-weighted) Euler characteristics e have naturalrefinements H∗ (of the perverse sheaf of vanishing cycles).
For invariants defined by virtual localisation we need to dosomething different. Maulik proposes taking the virtual K-theoreticinvariant (in the stable = semistable case)
χ((
K virMVW
) 12
)defined by K-theoretic localisation, and refining it to a polynomialin t1/2 by using the C∗ action.
![Page 47: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/47.jpg)
RefinementBy (virtual) Riemann-Roch this amounts to replacing the virtuallocalisation definition
VW =
∫[MC∗
VW
]vir 1
e(Nvir)
by ∫[MC∗
VW ]
ch((K virMVW
)12
)Td(T virMVW
)e(Nvir)
.
Both are integrals in equivariant cohomology taking values in
H∗(BC∗) ∼= Z[t] (localised and extended to Q[t±12 ]). The first is a
constant, whereas the second can be a more general Laurentpolynomial in t1/2.
On the component Masd this recovers the χ−y refinement of
(−1)vdevir studied by Gottsche-Kool.
On M2 computations are work in progress, trying K-theoreticcosection localisation, and refined Carlsson-Okounkov operators.
![Page 48: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/48.jpg)
RefinementBy (virtual) Riemann-Roch this amounts to replacing the virtuallocalisation definition
VW =
∫[MC∗
VW
]vir 1
e(Nvir)
by ∫[MC∗
VW ]
ch((K virMVW
)12
)Td(T virMVW
)e(Nvir)
.
Both are integrals in equivariant cohomology taking values in
H∗(BC∗) ∼= Z[t] (localised and extended to Q[t±12 ]). The first is a
constant, whereas the second can be a more general Laurentpolynomial in t1/2.
On the component Masd this recovers the χ−y refinement of
(−1)vdevir studied by Gottsche-Kool.
On M2 computations are work in progress, trying K-theoreticcosection localisation, and refined Carlsson-Okounkov operators.
![Page 49: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/49.jpg)
RefinementBy (virtual) Riemann-Roch this amounts to replacing the virtuallocalisation definition
VW =
∫[MC∗
VW
]vir 1
e(Nvir)
by ∫[MC∗
VW ]
ch((K virMVW
)12
)Td(T virMVW
)e(Nvir)
.
Both are integrals in equivariant cohomology taking values in
H∗(BC∗) ∼= Z[t] (localised and extended to Q[t±12 ]). The first is a
constant, whereas the second can be a more general Laurentpolynomial in t1/2.
On the component Masd this recovers the χ−y refinement of
(−1)vdevir studied by Gottsche-Kool.
On M2 computations are work in progress, trying K-theoreticcosection localisation, and refined Carlsson-Okounkov operators.
![Page 50: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/50.jpg)
RefinementBy (virtual) Riemann-Roch this amounts to replacing the virtuallocalisation definition
VW =
∫[MC∗
VW
]vir 1
e(Nvir)
by ∫[MC∗
VW ]
ch((K virMVW
)12
)Td(T virMVW
)e(Nvir)
.
Both are integrals in equivariant cohomology taking values in
H∗(BC∗) ∼= Z[t] (localised and extended to Q[t±12 ]). The first is a
constant, whereas the second can be a more general Laurentpolynomial in t1/2.
On the component Masd this recovers the χ−y refinement of
(−1)vdevir studied by Gottsche-Kool.
On M2 computations are work in progress, trying K-theoreticcosection localisation, and refined Carlsson-Okounkov operators.
![Page 51: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/51.jpg)
Nested Hilbert schemes
The simplest nontrivial C∗-fixed component in M2 is whenrank(E ) = 2.
Then we get sheaves supported on the doubling 2S of S ⊂ X .
Up to ⊗L they are ideal sheaves of C∗-fixed subschemes of 2S .
Back on S they can be interpreted as nested subschemes Z1 ⊇ Z2
in S . The simplest case is when both are 0-dimensional.
We get the nested Hilbert scheme
S [n1,n2] =I1 ⊆ I2 ⊂ OS : length(OS/Ii ) = ni
embedded in S [n1] × S [n2] as the locus of ideals (I1, I2) where
HomS(I1, I2) 6= 0.
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Nested Hilbert schemes
The simplest nontrivial C∗-fixed component in M2 is whenrank(E ) = 2.
Then we get sheaves supported on the doubling 2S of S ⊂ X .
Up to ⊗L they are ideal sheaves of C∗-fixed subschemes of 2S .
Back on S they can be interpreted as nested subschemes Z1 ⊇ Z2
in S . The simplest case is when both are 0-dimensional.
We get the nested Hilbert scheme
S [n1,n2] =I1 ⊆ I2 ⊂ OS : length(OS/Ii ) = ni
embedded in S [n1] × S [n2] as the locus of ideals (I1, I2) where
HomS(I1, I2) 6= 0.
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Nested Hilbert schemes
The simplest nontrivial C∗-fixed component in M2 is whenrank(E ) = 2.
Then we get sheaves supported on the doubling 2S of S ⊂ X .
Up to ⊗L they are ideal sheaves of C∗-fixed subschemes of 2S .
Back on S they can be interpreted as nested subschemes Z1 ⊇ Z2
in S . The simplest case is when both are 0-dimensional.
We get the nested Hilbert scheme
S [n1,n2] =I1 ⊆ I2 ⊂ OS : length(OS/Ii ) = ni
embedded in S [n1] × S [n2] as the locus of ideals (I1, I2) where
HomS(I1, I2) 6= 0.
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Carlsson-Okounkov operators
In this way we can see S [n1,n2]ι
−→ S [n1] × S [n2] as the degeneracylocus of the complex of vector bundles RHomπ(I1, I2).
(We have to modify this complex in a clever way whenH0,1(S) 6= 0 or H0,2(S) 6= 0.)
With Gholampour and Sheshmani we show such degeneracy locicarry natural perfect obstruction theories, and that in this case itreproduces the one from VW theory.
By the Thom-Porteous formula for the degeneracy locus, this gives
ι∗[S [n1,n2]
]vir= cn1+n2
(RHomπ(I1, I2)[1]
)on S [n1] × S [n2]. The latter has been computed by Carlsson-Okounkov in terms of Grojnowski-Nakajima operators on H∗(S [∗]).
![Page 55: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/55.jpg)
Carlsson-Okounkov operators
In this way we can see S [n1,n2]ι
−→ S [n1] × S [n2] as the degeneracylocus of the complex of vector bundles RHomπ(I1, I2).
(We have to modify this complex in a clever way whenH0,1(S) 6= 0 or H0,2(S) 6= 0.)
With Gholampour and Sheshmani we show such degeneracy locicarry natural perfect obstruction theories, and that in this case itreproduces the one from VW theory.
By the Thom-Porteous formula for the degeneracy locus, this gives
ι∗[S [n1,n2]
]vir= cn1+n2
(RHomπ(I1, I2)[1]
)on S [n1] × S [n2]. The latter has been computed by Carlsson-Okounkov in terms of Grojnowski-Nakajima operators on H∗(S [∗]).
![Page 56: Vafa-Witten invariants of projective surfaces · Cumrun Vafa Ed Witten Riemannian 4-manifold M, SU(r) bundle E !M, connection A, elds B 2 +(su(E)); 2 0(su(E)), F+ A + [B:B] + [B;]](https://reader035.fdocuments.net/reader035/viewer/2022071212/611a45e8ffc9dd3cfa5ceb98/html5/thumbnails/56.jpg)
Carlsson-Okounkov operators
In this way we can see S [n1,n2]ι
−→ S [n1] × S [n2] as the degeneracylocus of the complex of vector bundles RHomπ(I1, I2).
(We have to modify this complex in a clever way whenH0,1(S) 6= 0 or H0,2(S) 6= 0.)
With Gholampour and Sheshmani we show such degeneracy locicarry natural perfect obstruction theories, and that in this case itreproduces the one from VW theory.
By the Thom-Porteous formula for the degeneracy locus, this gives
ι∗[S [n1,n2]
]vir= cn1+n2
(RHomπ(I1, I2)[1]
)on S [n1] × S [n2]. The latter has been computed by Carlsson-Okounkov in terms of Grojnowski-Nakajima operators on H∗(S [∗]).