Recently, Minahan, Nemeschansky, Vafa and Warner computed partition functions for N=4 topological Yang-Mills theory on rational elliptic surfaces. In particular they computed generating functions of Euler characteristics of SU(2)-instanton moduli spaces. In mathematics, they are expected to coincide with those of Gieseker compactifications. In this paper, we compute Euler characteristics of these spaces and show that our results coincide with theirs. We also check the modular property of Z_{SU(2)} and Z_{SO(3)} conjectured by Vafa and Witten.
We study moduli spaces of framed perverse instantons on P^3. As an open subset it contains the (set-theoretical) moduli space of framed instantons studied by I. Frenkel and M. Jardim. We also construct a few counterexamples to earlier conjectures and results concerning these moduli spaces.
We use instanton numbers to: (i) stratify moduli of vector bundles, (ii) calculate relative homology of moduli spaces and (iii) distinguish curve singularities.
We use the recursive structure of the compactification of the instanton moduli space of N=2 Super Yang-Mills theory with gauge group SU(2), to construct, by inductive limit, a universal moduli space which includes all the multi-instanton moduli spaces. Furthermore, with the aim of understanding the field theoretic structure of the strong coupling expansion, we perform the Borel sum which acts on the parameter defining such a universal moduli space.
We give the general presciption for calculating the moduli of irreducible, stable SU(n) holomorphic vector bundles with positive spectral covers over elliptically fibered Calabi-Yau threefolds. Explicit results are presented for Hirzebruch base surfaces B=F_r. The transition moduli that are produced by chirality changing small instanton phase transitions are defined and specifically enumerated. The origin of these moduli, as the deformations of the spectral cover restricted to the ``lift'' of the horizontal curve of the M5-brane, is discussed. We present an alternative description of the trans…
We construct explicit deformation quantizations of the noncompact complex surfaces $Z_k := \operatorname{Tot} (\mathcal O_{\mathbb P^1} (-k))$ and describe their effect on moduli spaces of vector bundles and instanton moduli spaces. We introduce the concept of rebel instantons, as being those which react badly to some quantizations, misbehaving by shooting off extra families of noncommutative instantons. We then show that the quantum instanton moduli space can be viewed as the étale space of a constructible sheaf over the classical instanton moduli space with support on rebel instantons.
We provide a description of the moduli space of framed autodual instanton bundles on projective space, focusing on the particular cases of symplectic and orthogonal instantons. Our description will use the generalized ADHM equations which define framed instanton sheaves.
We show that the moduli space $M$ of holomorphic vector bundles on $CP^3$ that are trivial along a line is isomorphic (as a complex manifold) to a subvariety in the moduli of rational curves of the twistor space of the moduli space of framed instantons on $\R^4$, called the space of twistor sections. We then use this characterization to prove that $M$ is equipped with a torsion-free affine connection with holonomy in $Sp(2n,\C)$.
Let $M^K_n$ be the moduli space of framed $K$-instantons with instanton number $n$ when $K$ is a compact simple Lie group of classical type. Due to Donaldson's theorem, its scheme structure is given by the regular locus of a GIT quotient of $μ^{-1}(0)$ where $μ$ is the moment map on the associated symplectic vector space of ADHM data. A main theorem of this paper asserts that $μ$ is flat for $K=SO(3,\mathbb{R})$ and any $n\ge0$. Hence we complete the interpretation of the K-theoretic Nekrasov partition function for the classical groups in Nekrasov-Shadchin's work in term of Hilbert series of t…
We study two instanton correction problems of Hitchin's moduli spaces along with their wall crossing formulas. The hyperkahler metric of a Hitchin's moduli space can be put into an instanton-corrected form according to physicists Gaiotto, Moore and Neitzke. The problem boils down to the construction of a set of special coordinates which can be constructed as Fock-Goncharov coordinates associated with foliations of quadratic differentials on a Riemann surface. A wall crossing formula of Kontsevich and Soibelman arises both as a crucial consistency condition and an effective computational tool. …
I will explain my joint paper `Instantons moduli spaces and W-algebras' with A.Braverman, M.Finkelberg, arXiv:1406.2381. I will concentrate on the geometric part, that is a study of perverse sheaves on instanton moduli spaces. I place a particular emphasize on the hyperbolic restriction functor and stable envelop, which are our key tools, and appear also in other situations in geometric representation theory. These are lecture notes for a course in 2015 PCMI.
The two applications are: 1. sometimes instanton numbers stratify moduli of bundles better than Chern numbers. 2. sometimes instanton numbers distinguish singularities better than the classical numerical invariants.
We study the scheme of multi-jumping lines of an $n$-instanton bundle mainly for $n\leq 5$. We apply it to prove the irreducibility and smoothness of the moduli space of 5-instanton. Some particular situations with higher $c_2$ are also studied.
We prove the rationality and irreducibility of the moduli space of---what we call---the endomorphism-general instanton vector bundles of arbitrary rank on the projective space. In particular, we deduce the rationality of the moduli spaces of rank-two mathematical instantons. This problem was first studied by Hartshorne, Hirschowitz-Narasimhan in the late 1970s, and it has been reiterated within the framework of the ICM 2018.
We construct a representation of the affine W-algebra of gl_r on the equivariant homology space of the moduli space of U_r-instantons on A^2, and identify the corresponding module. As a corollary we give a proof of a version of the AGT conjecture concerning pure N=2 gauge theory for the group SU(r). Another proof has been announced by Maulik and Okounkov. Our approach uses a suitable deformation of the universal enveloping algebra of the Witt algebra W_{1+\infty}, which is shown to act on the above homology spaces (for any r) and which specializes to all W(gl_r). This deformation is in turn co…
This paper has been withdraw. A fully revised version with two new co-authors has been posted: "ADHM construction of perverse instanton sheaves", arXiv:1201.5657.
We study the irreducible components of the moduli space of instanton sheaves on $\mathbb{P}^3$, that is rank 2 torsion free sheaves $E$ with $c_1(E)=c_3(E)=0$ satisfying $h^1(E(-2))=h^2(E(-2))=0$. In particular, we classify all instanton sheaves with $c_2(E)\le4$, describing all the irreducible components of their moduli space. A key ingredient for our argument is the study of the moduli space ${\mathcal T}(d)$ of stable sheaves on $\mathbb{P}^3$ with Hilbert polynomial $P(t)=d\cdot t$, which contains, as an open subset, the moduli space of rank 0 instanton sheaves of multiplicity $d$; we desc…
We describe the (equivariant) intersection cohomology of certain moduli spaces ("framed Uhlenbeck spaces") together with some structures on them (such as e.g.\ the Poincaré pairing) in terms of representation theory of some vertex operator algebras ("$\mathscr W$-algebras").
We study semistable sheaves of rank $2$ with Chern classes $c_1=0$, $c_2=2$ and $c_3=0$ on the Fano 3-fold $V_5$ of Picard number $1$, degree $5$ and index $2$. We show that the moduli space of such sheaves has a component that is isomorphic to $\mathbb{P}^5$ by identifying it with the moduli space of semistable quiver representations. This provides a natural smooth compactification of the moduli space of minimal instantons, as well as Ulrich bundles of rank $2$ on $V_5$.
Gieseker-Nakajima moduli spaces $M_{k}(n)$ parametrize the charge $k$ noncommutative $U(n)$ instantons on ${\bf R}^{4}$ and framed rank $n$ torsion free sheaves $\mathcal{E}$ on ${\bf C\bf P}^{2}$ with ${\rm ch}_{2}({\mathcal{E}}) = k$. They also serve as local models of the moduli spaces of instantons on general four-manifolds. We study the generalization of gauge theory in which the four dimensional spacetime is a stratified space $X$ immersed into a Calabi-Yau fourfold $Z$. The local model ${\bf M}_{k}({\vec n})$ of the corresponding instanton moduli space is the moduli space of charge $k$ …
Symplectic instanton vector bundles on the projective space $\mathbb{P}^3$ constitute a natural generalization of mathematical instantons of rank 2. We study the moduli space $I_{n,r}$ of rank-$2r$ symplectic instanton vector bundles on $\mathbb{P}^3$ with $r\ge2$ and second Chern class $n\ge r,\ n\equiv r({\rm mod}2)$. We give an explicit construction of an irreducible component $I^*_{n,r}$ of this space for each such value of $n$ and show that $I^*_{n,r}$ has the expected dimension $4n(r+1)-r(2r+1)$.
Let $X$ be a closed $6-$dimensional manifold with a half-closed $SU(3)-$structure. On the product manifold $X\times S^{1}$, with respect to the product $G_{2}-$structure and on a pullback vector bundle from $X$, we show that any $G_{2}-$instanton is equivalent to a Hermitian Yang-Mills connection on $X$ via a "broken gauge". This result reveals the topological type of the moduli of $G_{2}-$instantons on $X\times S^{1}$. In dimension $8$, similar result holds for moduli of $Spin(7)-$instantons. A generalization and an example are given.
We prove that the space of mathematical instantons with second Chern class 5 over ${\mathbb P}_3$ is smooth and irreducible. Unified and simple proofs for the same statements in case of second Chern class $\leq 4$ are contained.
The problem of irreducibility of the moduli space I_n of rank-2 mathematical instanton vector bundles with arbitrary positive second Chern class n on the projective 3-space is considered. The irreducibility of I_n was known for small values of n: Barth 1977 (n=1), Hartshorne 1978 (n=2), Ellingsrud and Stromme 1981 (n=3), Barth 1981 (n=4), Coanda, Tikhomirov and Trautmann 2003 (n=5). In this paper we prove the irreducibility of I_n for an arbitrary odd n.
We describe semiuniversal deformation spaces for the noncompact surfaces $Z_k := \operatorname{Tot} (\mathcal O_{\mathbb P^1} (-k))$ and prove that any nontrivial deformation $Z_k (τ)$ of $Z_k$ is affine. It is known that the moduli spaces of instantons of charge $j$ on $Z_k$ are quasi-projective varieties of dimension $2j-k-2$. In contrast, our results imply that the moduli spaces of instantons on any nontrivial deformation $Z_k (τ)$ are empty.
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.
We study the moduli space of rank stable based instantons over a connected sum of q copies of CP^2. For c_2=1 we give the homotopy type of the moduli space. For c_2=2 we compute the cohomology of the moduli space.
We survey some of the AGT relations between N=2 gauge theories in four dimensions and geometric representations of symmetry algebras of two-dimensional conformal field theory on the equivariant cohomology of their instanton moduli spaces. We treat the cases of gauge theories on both flat space and ALE spaces in some detail, and with emphasis on the implications arising from embedding them into supersymmetric theories in six dimensions. Along the way we construct new toric noncommutative ALE spaces using the general theory of complex algebraic deformations of toric varieties, and indicate how t…
We study the large rank limit of the moduli spaces of framed bundles on the projective plane and the blown-up projective plane. These moduli spaces are identified with various instanton moduli spaces on the 4-sphere and $\cpbar $, the projective plane with the reverse orientation. We show that in the direct limit topology, these moduli spaces are homotopic to classifying spaces. For example, the moduli space of $Sp(\infty)$ or $SO(\infty)$ instantons on $\cpbar $ has the homotopy type of $BU(k)$ where $k$ is the charge of the instantons. We use our results along with Taubes' result concerning …
We study semi-stable sheaves of rank $2$ with Chern class $c_1=0$, $c_2=2$ and $c_3=0$ on the Fano 3-folds $V_4$ of Picard number $1$, degree $4$ and index $2$. We show the moduli space of such sheaves is isomorphic to the moduli space of semi-stable rank $2$ even degree vector bundles on a genus $2$ curve. This provides a natural smooth compatification of the moduli space of Ulrich bundles of rank $2$ on $V_4$.
We study the local holomorphic Euler characteristic $χ(x,\mathcal{F})$ of sheaves near a surface singularity obtained from contracting a line $\ell$ inside a smooth surface $Z$. We prove non-existence of sheaves with certain prescribed numerical invariants. Non-existence of instantons on $Z$ with certain charges follows, and we conclude that $\ell^2$ poses an obstruction to instanton decay. A Macaulay 2 algorithm to compute $χ$ is made available at http://www.maths.ed.ac.uk/~s0571100/Instanton/
We perform an exact computation of the gauged linear sigma model associated to a D1-D5 brane system on a resolved A_1 singularity. This is accomplished via supersymmetric localization on the blown-up two-sphere. We show that in the blow-down limit C^2/Z_2 the partition function reduces to the Nekrasov partition function evaluating the equivariant volume of the instanton moduli space. For finite radius we obtain a tower of world-sheet instanton corrections, that we identify with the equivariant Gromov-Witten invariants of the ADHM moduli space. We show that these corrections can be encoded in a…
In order to obtain existence criteria for orthogonal instanton bundles on $\mathbb{P}^n$, we provide a bijection between equivalence classes of orthogonal instanton bundles with no global sections and symmetric forms. Using such correspondence we are able to provide explicit examples of orthogonal instanton bundles with no global sections on $\mathbb{P}^n$ and prove that every orthogonal instanton bundle with no global sections on $\mathbb{P}^n$ and charge $c\geq 3$ has rank $r\leq (n-1)c$. We also prove that when the rank $r$ of the bundles reaches the upper bound, $\mathcal{M}_{\mathbb{P}^n}…
In the physicist's language, a brane in a hyperkahler manifold is a submanifold which is either complex or lagrangian with respect to three Kahler structures of the ambient manifold. By considering the fixed loci of certain involutions, we describe branes in Nakajima quiver varieties of all possible types. We then focus on the moduli space of framed torsion free sheaves on the projective plane, showing how the involutions considered act on sheaves, and proving the existence of branes in some cases.
Let ${\mathcal I}(n)$ denote the moduli space of rank $2$ instanton bundles of charge $n$ on ${\mathbb P}^3$. We know from several authors that ${\mathcal I}(n)$ is an irreducible, nonsingular and affine variety of dimension $8n-3$. Since every rank $2$ instanton bundle on ${\mathbb P}^3$ is stable, we may regard ${\mathcal I}(n)$ as an open subset of the projective Gieseker--Maruyama moduli scheme ${\mathcal M}(n)$ of rank $2$ semistable torsion free sheaves $F$ on ${\mathbb P}^3$ with Chern classes $c_1=c_3=0$ and $c_2=n$, and consider the closure $\overline{{\mathcal I}(n)}$ of ${\mathcal I…
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.
We study the instanton contributions of N=2 supersymmetric gauge theory and propose that the instanton moduli space is mapped to the moduli space of punctured spheres. Due to the recursive structure of the boundary in the Deligne-Knudsen-Mumford stable compactification, this leads to a new recursion relation for the instanton coefficients, which is bilinear. Instanton contributions are expressed as integrals on M_{0,n} in the framework of the Liouville F-models. This also suggests considering instanton contributions as a kind of Hurwitz numbers and also provides a prediction on the asymptotic …
Explicit construction of the basic SU(2) anti-instantons over the multi-Taub--NUT geometry via the classical conformal rescaling method is exhibited. These anti-instantons satisfiy the so-called weak holonomy condition at infinity with respect to the trivial flat connection and decay rapidly. The resulting unital energy anti-instantons have trivial holonomy at infinity. We also fully describe their unframed moduli space and find that it is a five dimensional space admitting a singular disk-fibration over R^3. On the way, we work out in detail the twistor space of the multi-Taub--NUT geometry t…
Let $\mathcal{M}$ be a moduli space of polystable rank 2-bundles bundles with fixed determinant (a moduli space of $\mathrm{PU}(2)$-instantons) on a Gauduchon surface with $p_g=0$ and $b_1=1$. We study the holomorphic structure of $\mathcal{M}$ around a circle $\mathcal{T}$ of regular reductions. Our model space is a "blowup flip passage", which is a manifold with boundary whose boundary is a projective fibration, and whose interior comes with a natural complex structure. We prove that a neighborhood of the boundary of the blowup $\hat{\mathcal{M}}_{\mathcal{T}}$ of $\mathcal{M}$ at $\mathcal{…
This thesis is an investigation of the moduli spaces of instanton bundles on the Fano threefold $Y_5$ (a linear section of $\mathbb{G}r(2,5)$). It contains new proofs of classical facts about lines, conics and cubics on $Y_5$, and about linear sections of $Y_5$. The main original results are a Grauert-Mülich theorem for the splitting type of instantons on conics, a bound to the splitting type of instantons on lines and an $SL_2$-equivariant description of the moduli space in charge 2 and 3. Using these results we prove the existence of a unique $SL_2$-equivariant instanton of minimal charge an…
We study the moduli space $I_{n,r}$ of rank-$2r$ symplectic instanton vector bundles on $\mathbb{P}^3$ with $r\ge2$ and second Chern class $n\ge r+1,\ n-r\equiv 1(\mathrm{mod}2)$. We introduce the notion of tame symplectic instantons by excluding a kind of pathological monads and show that the locus $I^*_{n,r}$ of tame symplectic instantons is irreducible and has the expected dimension, equal to $4n(r+1)-r(2r+1)$. The proof is inherently based on a relation between the spaces $I^*_{n,r}$ and the moduli spaces of 't Hooft instantons
We discuss two-dimensional sigma models on moduli spaces of instantons on K3 surfaces. These N=(4,4) superconformal field theories describe the near-horizon dynamics of the D1-D5-brane system and are dual to string theory on AdS_3. We derive a precise map relating the moduli of the K3 type IIB string compactification to the moduli of these conformal field theories and the corresponding classical hyperkaehler geometry. We conclude that, in the absense of background gauge fields, the metric on the instanton moduli spaces degenerates exactly to the orbifold symmetric product of K3. Turning on a s…
We model A_k and D_k asymptotically locally flat gravitational instantons on the moduli spaces of solutions of U(2) Bogomolny equations with prescribed singularities. We study these moduli spaces using Ward correspondence and find their twistor description. This enables us to write down the Kähler potential for A_k and D_k gravitational instantons in a relatively explicit form.
In this Part II, D(10.2), of D(10), we take D(10.1) (arXiv:1302.2054 [math.AG]) as the foundation to define the notion of $Z$-semistable morphisms from general Azumaya nodal curves, of genus $\ge 2$, with a fundamental module to a projective Calabi-Yau 3-fold and show that the moduli stack of such $Z$-semistable morphisms of a fixed type is compact. This gives us a counter moduli stack to D-strings as the moduli stack of stable maps in Gromov-Witten theory to the fundamental string. It serves and prepares for us the basis toward a new invariant of Calabi-Yau 3-fold that captures soft-D-string …
Yang-Mills instantons on ALE gravitational instantons were constructed by Kronheimer and Nakajima in terms of matrices satisfying algebraic equations. These were conveniently organized into a quiver. We construct generic Yang-Mills instantons on ALF gravitational instantons. Our data are formulated in terms of matrix-valued functions of a single variable, that are in turn organized into a bow. We introduce the general notion of a bow, its representation, its associated data and moduli space of solutions. For a judiciously chosen bow the Nahm transform maps any bow solution to an instanton on a…
Effective field theories in type I and II superstring theories for D-branes located at points in the orbifold C^2/Z_n are supersymmetric gauge theories whose field content is conveniently summarized by a `quiver diagram,' and whose Lagrangian includes non-metric couplings to the orbifold moduli: in particular, twisted sector moduli couple as Fayet-Iliopoulos terms in the gauge theory. These theories describe D-branes on resolved ALE spaces. Their spaces of vacua are moduli spaces of smooth ALE metrics and Yang-Mills instantons, whose metrics are explicitly computable. For U(N) instantons, the …
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.
Let $\M_{k}^{n}$ be the moduli space of based (anti-self-dual) instantons on $\cpbar$ of charge $k$ and rank $n$. There is a natural inclusion of rank $n$ instantons into rank $n+1$. We show that the direct limit space is homotopy equivalent to $BU(k)\times BU(k)$. The moduli spaces also have the following algebro-geometric interpretation: Let $\linf$ be a line in the complex projective plane and consider the blow-up at a point away from $\linf$. $\M _{k}^{n}$ can be described as the moduli space of rank $n$ holomorphic bundles on the blownup projective plane with $c_{1}=0$ and $c_{2}=k$ and w…
We describe explicitly the moduli spaces $M^{pst}_g(S,E)$ of polystable holomorphic structures $E$ with $\det E\cong K$ on a rank 2 vector bundle $E$ with $c_1(E)=c_1(K)$ and $c_2(E)=0$ for all minimal class VII surfaces $S$ with $b_2(S)=1$ and with respect to all possible Gauduchon metrics $g$. These surfaces $S$ are non-elliptic and non-Kaehler complex surfaces and have recently been completely classified. When $S$ is a half or parabolic Inoue surface, $M^{pst}_g(S,E)$ is always a compact one-dimensional complex disc. When $S$ is an Enoki surface, one obtains a complex disc with finitely man…
Let $MI_{Simp,P^{2n+1}}(k)$ be the moduli space of stable symplectic instanton bundles on $P^{2n+1}$ with second Chern class $c_2=k$ (it is a closed subscheme of the moduli space $MI_{P^{2n+1}}(k)$), We prove that the dimension of its Zariski tangent space at a special (symplectic) instanton bundle is $2k(5n-1)+4n^2-10n+3, k\geq 2$. It follows that special symplectic instanton bundles are smooth points for $ k \leq 3 $
Let $\mathcal{M}^{\mathrm{st}}$ ($\mathcal{M}^{\mathrm{pst}}$) be a moduli space of stable (polystable) bundles with fixed determinant on a complex surface with $b_1=1$, $p_g=0$, and let $Z\subset \mathcal{M}^{\mathrm{st}}$ be a pure $k$-dimensional analytic set. We prove a general formula for the homological boundary $δ[Z]^{BM}\in H_{2k-1}^{BM}(\partial\hat{\mathcal M}^{\mathrm{pst}},\mathbb{Z})$ of the Borel-Moore fundamental class of $Z$ in the boundary of the blow up moduli space $\hat {\mathcal M}^{\mathrm{pst}}$. The proof is based on the holomorphic model theorem (proved in a previous a…
Denote by MI(k) the moduli space of k-instanton bundles E of rank 2 on $\PP^3=\PP(V)$ and by $Z_k(E)$ the scheme of k-jumping lines. We prove that $[E]\in MI(k)$ is not stable for the action of SL(V) if $Z_k(E)\neq\emptyset$. Moreover $\dim Sym(E)\ge 1$ if $length Z_k(E)\ge 2$. We prove also that E is special if and only if $Z_k(E)$ is a smooth conic. The action of SL(V) on the moduli of special instanton bundles is studied in detail.
We calculate the Euler characteristic of associated vector bundles over the moduli spaces of stable parabolic bundles on smooth curves. Our method is based on a wall-crossing technique from Geometric Invariant Theory, certain iterated residue calculus and the tautological Hecke correspondence. Our work was motivated by the results of Teleman and Woodward on the index of K-theory classes on moduli stacks.
Combining the MPS degeneration formula for the Poincaré polynomial of moduli spaces of stable quiver representations and localization theory, it turns that the determination of the Euler characteristic of these moduli spaces reduces to a combinatorial problem of counting certain trees. We use this fact in order to obtain an upper bound for the Euler characteristic in the case of the Kronecker quiver. We also derive a formula for the Euler characteristic of some of the moduli spaces appearing in the MPS degeneration formula.
