Wall-Crossing Formulae for Algebraic Surfaces with Positive Irregularity

Muñoz, Vicente

doi:10.1112/s0024610799008327

Cited by 7 publications

(20 citation statements)

References 10 publications

(21 reference statements)

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“…Notice that this result confirms the conjecture in p. 18 of [17]. As a check of this expression, and also to fix an overall coefficient depending on b 1 , we will compute the wall-crossing for the correlator p r S d−2r on a ruled surface, where d is half the dimension of the moduli space, and compare it to the expressions in [17].…”

Section: Donaldson Wall-crossingsupporting

confidence: 72%

“…If we compare with [17], we find perfect agreement except for an overall factor 1/2 (a standard discrepancy between topological and quantum field theory normalizations), and a factor (−i) b 1 /2 2 −3b 1 /4 . The latter factor is due to the normalization of the fermions in the physical theory.…”

Section: Donaldson Wall-crossingmentioning

confidence: 78%

“…The remaining constants a 1 , a 3 will be fixed below by comparing to certain topological results [17]. In principle, the coefficients in (2.20) are fixed by the proposal of equation (2.14) of the first paper in [10].…”

Section: )mentioning

confidence: 99%

“…In this section we want to compute the wall-crossing formulae associated to the region at infinity of the u-plane. To compare to mathematical results, it is better to write the expression in terms of ψ, S. There are two standard mathematical facts for manifolds of b + 2 = 1 that will be very useful in writing the resulting wall-crossing formula [17]. First, for any β 1 , β 2 , β 3 and…”

Section: Donaldson Wall-crossingmentioning

confidence: 99%

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Donaldson Invariants for Non-Simply Connected Manifolds

Mariño¹,

Moore²

1999

Communications in Mathematical Physics

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We study Coulomb branch ("u-plane") integrals for N = 2 supersymmetric SU (2), SO (3) Yang-Mills theory on 4-manifolds X of b 1 (X) > 0, b + 2 (X) = 1. Using wall-crossing arguments we derive expressions for the Donaldson invariants for manifolds with b 1 (X) > 0, b + 2 (X) > 0. Explicit expressions for X = CP 1 × F g , where F g is a Riemann surface of genus g are obtained using Kronecker's double series identity. The result might be useful in future studies of quantum cohomology.

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Section: Donaldson Wall-crossingsupporting

confidence: 72%

Section: Donaldson Wall-crossingmentioning

confidence: 78%

Section: )mentioning

confidence: 99%

Section: Donaldson Wall-crossingmentioning

confidence: 99%

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Donaldson Invariants for Non-Simply Connected Manifolds

Mariño¹,

Moore²

1999

Communications in Mathematical Physics

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“…Suppose we are in the situation of a closed 4-manifold X = X 1 ∪ Y X 2 , obtained as the union of two 4-manifolds with boundary, where ∂X 1 = Y and ∂X 2 = Y . Let w ∈ H 2 (X; Z) with w| Y = w 2 ∈ H 2 (Y ; Z/2Z) as above (this implies in particular b + (X) > 0, so the Donaldson invariants of X are defined; in the case b + = 1 relative to chambers [15] [22]). We need another bit of terminology from [20]…”

Section: Relative Invariants Of 4-manifoldsmentioning

confidence: 99%

Fukaya-Floer homology of $\Sigma\times \mathbb{S}^1$ and applications

Muñoz¹

1999

J. Differential Geom.

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We determine the Fukaya-Floer (co)homology groups of the three-manifold Y = Σ×S 1 , where Σ is a Riemann surface of genus g ≥ 1. These are of two kinds. For the 1-cycle S 1 ⊂ Y , we compute the Fukaya-Floer cohomology HF F * (Y, S 1 ) and its ring structure, which is a sort of deformation of the Floer cohomology HF * (Y ). On the other hand, for 1-cycles δ ⊂ Σ ⊂ Y , we determine the Fukaya-Floer homology HF F * (Y, δ) and its HF * (Y )-module structure.We give the following applications:• We show that every four-manifold with b + > 1 is of finite type.• Four-manifolds which arise as connected sums along surfaces of four-manifolds with b 1 = 0 are of simple type and we give constraints on their basic classes.• We find the invariants of the product of two Riemann surfaces both of genus greater or equal than one. * Supported by a grant from Ministerio de Educación y Cultura of Spain A(X) with that of Kronheimer and Mrowka [16], as we do not consider 3-homology classes (this is done in this way since the techniques here contained are not well suited to deal with these classes).We say that X is of w-simple type when D w X ((x 2 − 4)z) = 0, for any z ∈ A(X). If X has b 1 = 0 and it is of w-simple type, then it is of w ′ -simple type, for any other w ′ and it is said to be of simple type for brevity. Analogously, we say that X is of w-finite type when there is some n ≥ 0 such that D w X ((x 2 − 4) n z) = 0, for any z ∈ A(X). The order is the minimum such n, so order 1 means simple type and order 0 means that the Donaldson invariants are identically zero. X is of finite type if it is of w-finite type for any w. For completeness we introduce the notion of X being of w-strong simple type when D w X ((x 2 − 4)z) = 0, for any z ∈ A(X) and D w X (γz) = 0, for any γ ∈ H 1 (X) and any z ∈ A(X). This condition is the right one for extending the concept of simple type in the case b 1 = 0 to the case b 1 > 0. It gives the same structure theorem for the invariants presented in [16]. Introducing 3-homology classes, we would have (potentially) different definitions, but we shall not deal here with that issue. Also the order of w-finite type of X does not depend on w (see [25]).Let Σ = Σ g be a Riemann surface of genus g ≥ 1 and consider the three-manifold Y = Σ × S 1 .In [23] we computed the ring structure of the (instanton) Floer (co)homology of Y , together with the SO(3)-bundle with w 2 = P.D.[S 1 ]. This gadget encondes all the relations R ∈ A(Σ) satisfied by all 4-manifolds X containing an embedded surface Σ, representing an odd homology class and with Σ 2 = 0. More accurately, for such X, D w X (Rz) = 0, for any z ∈ A(Σ ⊥ ) and w ∈ H 2 (X; Z) with w · Σ ≡ 1 (mod 2). This is so since we have a decomposition X = X 1 ∪ Y A, where A is a tubular neighbourhood of Σ, and we can consider R ∈ A(A) and z ∈ A(X 1 ). Then the (relative) Donaldson invariants for A corresponding to R are already vanishing.In order to drop the condition z ∈ A(Σ ⊥ ), the useful space to work in is no longer the Floer homology, but the extension developed by F...

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Donaldson Invariants of Product Ruled Surfaces¶and Two-Dimensional Gauge Theories

Lozano¹,

Mariño²

2001

Communications in Mathematical Physics

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Using the u-plane integral of Moore and Witten, we derive a simple expression for the Donaldson invariants of product ruled surfaces Σ g × S 2 , where Σ g is a Riemann surface of genus g. This expression generalizes a theorem of Morgan and Szabó for g = 1 to any genus g. We give two applications of our results: (1) We derive Thaddeus' formulae for the intersection pairings on the moduli space of rank two stable bundles over a Riemann surface.(2) We derive the eigenvalue spectrum of the Fukaya-Floer cohomology of Σ g ×S 1 .

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Wall-Crossing Formulae for Algebraic Surfaces with Positive Irregularity

Abstract: Abstract. We extend the ideas of Friedman and Qin [5] to find the wall-crossing formulae for the Donaldson invariants of algebraic surfaces with p g = 0, q > 0 and anticanonical divisor −K effective, for any wall ζ with l ζ = 1 4 (ζ 2 − p 1 ) being 0 or 1.

Cited by 7 publications

References 10 publications

Donaldson Invariants for Non-Simply Connected Manifolds

Donaldson Invariants for Non-Simply Connected Manifolds

Fukaya-Floer homology of $\Sigma\times \mathbb{S}^1$ and applications

Donaldson Invariants of Product Ruled Surfaces¶and Two-Dimensional Gauge Theories

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