On blowup formulae for the S -duality conjecture of Vafa and Witten

Li, Wei Ping; Qin, Zhenbo

doi:10.1007/s002220050316

Cited by 43 publications

(60 citation statements)

References 26 publications

(16 reference statements)

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“…This formula is proved by Li and Qin [8]. Of course, their results are inspired by the work of Yoshioka [16].…”

Section: O(−2) Curve Blow-upmentioning

confidence: 84%

“…In this section, we derive the partition function of the untwisted sector of T 4 /Z 2 [14], [11], [18], [8], [12].…”

Section: Untwisted Sectormentioning

confidence: 99%

“…Following [8], we first introduce virtual Hodge number e s,t (X) and virtual Hodge polynomial e(X; x, y) of X. The virtual Hodge number is labeled by a pair of integers (s, t) and is given by…”

Section: Göttsche Formulamentioning

confidence: 99%

“…In this section, we introduce O(−2) blow-up formula and reconstruct K3 partition function [16], [8], [14].…”

Section: Twisted Sectormentioning

confidence: 99%

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N=4 SUPERSYMMETRIC YANG–MILLS THEORY ON ORBIFOLD-T⁴/Z₂

Jinzenji

Sasaki

2001

Mod. Phys. Lett. A

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We derive the partition function of N = 4 supersymmetric Yang-Mills theory on orbifold-T 4 /Z 2 . In classical geometry, K3 surface is constructed from the orbifold-T 4 /Z 2 . Along the same way as the orbifold construction, we construct the partition function of K3 surface from orbifold-T 4 /Z 2 . The partition function is given by the product of the contribution of the untwisted sector of T 4 /Z 2 , and that of the twisted sector of T 4 /Z 2 i.e., O(−2) curve blow-up formula.

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“…This formula is proved by Li and Qin [8]. Of course, their results are inspired by the work of Yoshioka [16].…”

Section: O(−2) Curve Blow-upmentioning

confidence: 84%

“…In this section, we derive the partition function of the untwisted sector of T 4 /Z 2 [14], [11], [18], [8], [12].…”

Section: Untwisted Sectormentioning

confidence: 99%

Section: Göttsche Formulamentioning

confidence: 99%

“…In this section, we introduce O(−2) blow-up formula and reconstruct K3 partition function [16], [8], [14].…”

Section: Twisted Sectormentioning

confidence: 99%

See 2 more Smart Citations

N=4 SUPERSYMMETRIC YANG–MILLS THEORY ON ORBIFOLD-T⁴/Z₂

Jinzenji

Sasaki

2001

Mod. Phys. Lett. A

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“…Similar results are shown in [58] for abelian surfaces. Other motivating examples for the S-duality conjecture were the case of P 2 [56] and the blowup formula relating the generating function for the Euler numbers of the moduli spaces of rank 2 sheaves on a surface S to that on the blowup of S in a point, which has since been established ( [40], [41], see also [27]). …”

Section: Moduli Of Vector Bundlesmentioning

confidence: 99%

On the Motive of the Hilbert scheme of points on a surface

Göttsche

2001

Mathematical Research Letters

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The Hilbert scheme S[n] of points on an algebraic surface S is a simple example of a moduli space and also a nice (crepant) resolution of singularities of the symmetric power S (n) . For many phenomena expected for moduli spaces and nice resolutions of singular varieties it is a model case. Hilbert schemes of points have connections to several fields of mathematics, including moduli spaces of sheaves, Donaldson invariants, enumerative geometry of curves, infinite dimensional Lie algebras and vertex algebras and also to theoretical physics. This talk will try to give an overview over these connections. IntroductionThe Hilbert scheme S[n] of points on a complex projective algebraic surface S is a a parameter variety for finite subschemes of length n on S. It is a nice (crepant) resolution of singularities of the n-fold symmetric power S (n) of S. If S is a K3 surface or an abelian surface, then S[n] is a compact, holomorphic symplectic (thus hyperkähler) manifold. Thus S [n] is at the same time a basic example of a moduli space and an example of a nice resolution of singularities of a singular variety. There are a number of conjectures and general phenomena, many of which originating from theoretical physics, both about moduli spaces for objects on surfaces and about nice resolutions of singularities. In all of these the Hilbert scheme of points can be viewed as a model case and sometimes as the main motivating example. Hilbert schemes of points on a surface have connections to many topics in mathematics, including moduli spaces of sheaves and vector bundles, Donaldson invariants, Gromov-Witten invariants and enumerative geometry of curves, infinite dimensional Lie algebras and vertex algebras, noncommutative geometry and also theoretical physics.

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Hilbert Schemes of Points on Surfaces

Göttsche¹

2005

Frobenius Splitting Methods in Geometry and Representation Theory

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n distinct points of X, if d ≥ 2 (Lemma 7.1.6). On the other hand, every X (n) is nonsingular if d = 1 (Exercise 7.1.E.5).Section 7.2 presents some general results on Hilbert schemes of points on quasiprojective schemes: their existence (Theorem 7.2.3) and the description of their Zariski tangent spaces (Lemma 7.2.5). Also, the punctual Hilbert scheme X [n] x (parametrizing length-n subschemes supported at a given point x) is introduced, and it is shown that X [n] x is projective and connected (Proposition 7.2.9). The Hilbert-Chow morphism is introduced and studied in Section 7.3, again in the setting of quasi-projective schemes. The existence of a projective morphism of schemes γ : X [n] −→ X (n) which yields the cycle map on closed points is deduced from work of Iversen (Theorem 7.3.1). The fibers of γ are products of punctual Hilbert schemes; their connectedness implies that X [n] is connected if X is (Corollary 7.3.4). Then, the loci X [n] * * , resp. X [n] * , consisting of subschemes supported at n distinct points, resp. at least n − 1 distinct points, are considered. In particular, it is shown that X [n] * is a nonsingular variety of dimension nd, and the complement X [n] * \ X [n] * * is a nonsingular 208 Chapter 7. Hilbert Schemes of Points prime divisor, if X is a nonsingular variety of dimension d (Lemma 7.3.5). Section 7.4 is devoted to Hilbert schemes of points on a nonsingular surface X. Each X [n] is shown to be a nonsingular variety of dimension 2n (Theorem 7.4.1) and the Hilbert-Chow morphism is shown to be birational, with exceptional set being a prime divisor (Proposition 7.4.5). Finally, the Hilbert-Chow morphism is shown to be crepant (Theorem 7.4.6), a result due to Beauville in characteristic 0 and to Kumar-Thomsen in characteristic p ≥ 3.Section 7.5 begins with the observation that any symmetric product of a split quasiprojective scheme is split as well (Lemma 7.5.1). Together with the crepantness of the Hilbert-Chow morphism, this implies the splitting of X [n] , where X is a nonsingular split surface (Theorem 7.5.2). In turn, this yields the vanishing of higher cohomology groups of any ample invertible sheaf on X [n] , if X is split and proper over an affine variety (Corollary 7.5.4). This applies, in particular, to the nonsingular projective split surfaces and also to the nonsingular affine surfaces (since these are split by Proposition 1.1.6). Further, we obtain a relative vanishing result for the Hilbert-Chow morphism (Corollary 7.5.5).Notation. Throughout this chapter, X denotes a quasi-projective scheme over an algebraically closed field k of characteristic p ≥ 0, and n denotes a positive integer. By schemes, as earlier in the book, we mean Noetherian separated schemes over k; their closed points will just be called points.

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On blowup formulae for the S -duality conjecture of Vafa and Witten

Cited by 43 publications

References 26 publications

N=4 SUPERSYMMETRIC YANG–MILLS THEORY ON ORBIFOLD-T⁴/Z₂

N=4 SUPERSYMMETRIC YANG–MILLS THEORY ON ORBIFOLD-T⁴/Z₂

On the Motive of the Hilbert scheme of points on a surface

Hilbert Schemes of Points on Surfaces

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On blowup formulae for the S -duality conjecture of Vafa and Witten

Cited by 43 publications

References 26 publications

N=4 SUPERSYMMETRIC YANG–MILLS THEORY ON ORBIFOLD-T4/Z2

N=4 SUPERSYMMETRIC YANG–MILLS THEORY ON ORBIFOLD-T4/Z2

On the Motive of the Hilbert scheme of points on a surface

Hilbert Schemes of Points on Surfaces

Contact Info

Product

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About

N=4 SUPERSYMMETRIC YANG–MILLS THEORY ON ORBIFOLD-T⁴/Z₂

N=4 SUPERSYMMETRIC YANG–MILLS THEORY ON ORBIFOLD-T⁴/Z₂