Two applications of instanton numbers

Instanton bundles on the Segre threefold with Picard number three

We study instanton bundles $E$ on $\mathbb{P}^1\times \mathbb{P}^1 \times \mathbb{P}^1$. We construct two different monads which are the analog of the monads for instanton bundles on $\mathbb P^3$ and on the flag threefold $F(0,1,2)$. We characterize the Gieseker semistable cases and we prove the existence of $μ$-stable instanton bundles generically trivial on the lines for any possible $c_2(E)$. We also study the locus of jumping lines.

Euler number of Instanton Moduli space and Seiberg-Witten invariants

We show that a partition function of topological twisted N=4 Yang-Mills theory is given by Seiberg-Witten invariants on a Riemannian four manifolds under the condition that the sum of Euler number and signature of the four manifolds vanish. The partition function is the sum of Euler number of instanton moduli space when it is possible to apply the vanishing theorem. And we get a relation of Euler number labeled by the instanton number $k$ with Seiberg-Witten invariants, too. All calculation in this paper is done without assuming duality.

Integrality of instanton numbers and p-adic B-model

We study integrality of instanton numbers (genus zero Gopakumar - Vafa invariants) for quintic and other Calabi-Yau manifolds. We start with the analysis of the case when the moduli space of complex structures is one-dimensional; later we show that our methods can be used to prove integrality in general case. We give an expression of instanton numbers in terms of Frobenius map on $p$-adic cohomology ; the proof of integrality is based on this expression.

在具有 Picard 数为二的三维光滑 toric 变分上研究 H-瞬子丛

H-Instanton Bundles on Three-Dimensional Smooth Toric Varieties with Picard Number Two

本文在具有 Picard 数为二的三维光滑 toric 变分上，研究 $H$-瞬子丛的构造与存在性，并推广了经典单调方法。

Frobenius transformation, mirror map and instanton numbers

We show that one can express Frobenius transformation on middle-dimensional p-adic cohomology of Calabi-Yau threefold in terms of mirror map and instanton numbers. We express the mirror map in terms of Frobenius transformation on p-adic cohomology . We discuss a $p$-adic interpretation of the conjecture about integrality of Gopakumar-Vafa invariants.

Even and odd instanton bundles on Fano threefolds of Picard number 1

We consider an analogue of the notion of instanton bundle on the projective 3-space, consisting of a class of rank-2 vector bundles defined on smooth Fano threefolds X of Picard number one, having even or odd determinant according to the parity of K_X. We first construct a well-behaved irreducible component of their moduli spaces. Then, when the intermediate Jacobian of X is trivial, we look at the associated monads, hyperdeterminants and nets of quadrics. We also study one case where the intermediate Jacobian of X is non-trivial, namely when X is the intersection of two quadrics in P5, relati…

Multi-Instanton Measure from Recursion Relations in N=2 Supersymmetric Yang-Mills Theory

By using the recursion relations found in the framework of N=2 Super Yang-Mills theory with gauge group SU(2), we reconstruct the structure of the instanton moduli space and its volume form for all winding numbers. The construction is reminiscent of the Deligne-Knudsen-Mumford compactification and uses an analogue of the Wolpert restriction phenomenon which arises in the case of moduli spaces of Riemann surfaces.

$O(-2)$ Blow-up Formula via Instanton Calculus on $\hat{C^2/Z_2}$ and Weil Conjecture

We calculate Betti numbers of the framed moduli space of instantons on $\hat{{\bf C}^2/{\bf Z}_2}$, under the assumption that the corresponding torsion free sheaves $E$ have vanishing properties ($Hom(E,E(-l_\infty))=Ext^2(E,E(-l_\infty))=0$). Moreover we derive the generating function of Betti numbers and obtain closed formulas. On the other hand, we derive a universal relation between the generating function of Betti numbers of the moduli spaces of stable sheaves on $X$ with an $A_1$-singularity and that on $\hat{X}$ blow-uped at the singularity, by using Weil conjecture. We call this the $O…

法诺三维流形上的偶与奇瞬子丛

Even and odd instanton bundles on Fano threefolds

该研究将瞬子丛概念推广到法诺三维流形上的非寻常瞬子丛，确定其量子数下界并证明存在性，研究相关模空间分量与直线限制，还给出特殊空间上的单子描述与跳跃线轨迹分析。

Instanton bundles on two Fano threefolds of index $1$

We deal with instanton bundles on the product ${\mathbb P}^1\times{\mathbb P}^2$ and the blow up of ${\mathbb P}^3$ along a line. We give an explicit construction leading to instanton bundles. Moreover, we also show that they correspond to smooth points of a unique irreducible component of their moduli space.

On the Instanton Complex of Holomorphic Morse Theory

Consider a holomorphic torus action on vector bundles over a complex manifold which lifts to a holomorphic vector bundle. When the connected components of the fixed-point set are partially ordered, we construct, using sheaf-theoretical techniques, two spectral sequences that converges to the twisted Dolbeault cohomology groups and those with compact support, respectively. These spectral sequences are the holomorphic counterparts of the instanton complex in standard Morse theory. The results proved imply holomorphic Morse inequalities and fixed-point formulas on a possibly non-compact manifold.…

Instanton bundles on the flag variety F(0,1,2)

Instanton bundles on $\mathbb{P}^3$ have been at the core of the research in Algebraic Geometry during the last thirty years. Motivated by the recent extension of their definition to other Fano threefolds of Picard number one, we develop the theory of instanton bundles on the complete flag variety $F:=F(0,1,2)$ of point-lines on $\mathbb{P}^2$. After giving for them two different monadic presentations, we use it to show that the moduli space $MI_F(k)$ of instanton bundles of charge $k$ is a geometric GIT quotient and the open subspace $MI^s_F(k)\subset MI_F(k)$ of stable instanton bundles has …

Mathematical Instantons In Characteristic Two

On P3, we show that mathematical instantons in characteristic two are unobstructed. We produce upper bounds for the dimension of the moduli space of stable rank two bundles on P3 in characteristic two. In cases where there is a phenomenon of good reduction modulo two, these give similar results in characteristic zero. We also give an example of a non-reduced component of the moduli space in characteristic two.

Scheme dependence of instanton counting in ALE spaces

There have been two distinct schemes studied in the literature for instanton counting in A_{p-1} asymptotically locally Euclidean (ALE) spaces. We point out that the two schemes---namely the counting of orbifolded instantons and instanton counting in the resolved space---lead in general to different results for partition functions. We illustrate this observation in the case of N=2 U(N) gauge theory with 2N flavors on the A_{p-1} ALE space. We propose simple relations between the instanton partition functions given by the two schemes and test them by explicit calculations.