Chamber structure and wallcrossing in the ADHM theory of curves II

This is the second part of a project concerning variation of stability and chamber structure for ADHM invariants of curves. Wallcrossing formulas for such invariants are derived using the theory of stack function Ringel-Hall algebras constructed by Joyce and the theory of generalized Donaldson-Thomas invariants of Joyce and Song. Some applications are presented, including strong rationality for local stable pair invariants of higher genus curves, and comparison with wallcrossing formulas of Kontsevich and Soibelman, and the halo formula of Denef and Moore.     

Equivariant Volumes of Non-Compact Quotients and Instanton Counting

Motivated by Nekrasov’s instanton counting, we discuss a method for calculating equivariant volumes of non-compact quotients in symplectic and hyper-Kähler geometry by means of the Jeffrey-Kirwan residue formula of non-abelian localization. In order to overcome the non-compactness, we use varying symplectic cuts to reduce the problem to a compact setting, and study what happens in the limit that recovers the original problem. We implement this method for the ADHM construction of the moduli spaces of framed Yang-Mills instantons on \({\mathbb{R}^{4}}\) and rederive the formulas for the equivariant volumes obtained earlier by Nekrasov-Shadchin, expressing these volumes as iterated residues of a single rational function.     

Symmetric instantons and the ADHM construction

We construct allSU(2) Yang-Mills instantons onS ⁴ that admit a certain symmetry (“quadrupole symmetry”). This is accomplished by an equivariant version of the “ADHM monad” classification of instantons. This work is part of an attempt to better understand the structure of non-self-dual Yang-Mills connections with the same symmetry. J.S. was supported by NSF Grants DMS-9106807 and DMS-9404468 Part of this work was done at the 1991 Regional Geometry Institute in Park City, Utah     

The Refined BPS Index from Stable Pair Invariants

A refinement of the stable pair invariants of Pandharipande and Thomas for non-compact Calabi–Yau spaces is introduced based on a virtual Bialynicki-Birula decomposition with respect to a ${\mathbb{C}^{*}}$ action on the stable pair moduli space, or alternatively the equivariant index of Nekrasov and Okounkov. This effectively calculates the refined index for M-theory reduced on these Calabi–Yau geometries. Based on physical expectations we propose a product formula for the refined invariants extending the motivic product formula of Morrison, Mozgovoy, Nagao, and Szendroi for local ${\mathbb{P}^1}$ . We explicitly compute refined invariants in low degree for local ${\mathbb{P}^2}$ and local ${\mathbb{P}^1\,\times\,\mathbb{P}^1}$ and check that they agree with the predictions of the direct integration of the generalized holomorphic anomaly and with the product formula. The modularity of the expressions obtained in the direct integration approach allows us to relate the generating function of refined PT invariants on appropriate geometries to Nekrasov’s partition function and a refinement of Chern–Simons theory on a lens space. We also relate our product formula to wall crossing.     

Symmetry Reduction of Ordinary Differential Equations Using Moving Frames

The symmetry reduction algorithm for ordinary differential equations due to Sophus Lie is revisited using the method of equivariant moving frames. Using the recurrence formulas provided by the theory of equivariant moving frames, computations are performed symbolically without relying on the coordinate expressions for the canonical variables and the differential invariants occurring in Lie’s original procedure.     

Observables in topological Yang-Mills theorieswith extended shift supersymmetry

We present a complete classification, at the classical level, of the observables of topological Yang-Mills theories with an extended shift supersymmetry of N generators, in any space-time dimension. The observables are defined as the Yang-Mills BRST cohomology classes of shift supersymmetry invariants. These cohomology classes turn out to be solutions of an N-extension of Witten's equivariant cohomology. This work generalizes results known in the case of shift supersymmetry with a single generator. Received: 8 March 2005, Published online: 21 October 2005 Supported in part by the Conselho Nacional de Desenvolvimento Científico e Tecnológico CNPq, Brazil     

Metrics on the moduli spaces of instantons over Euclidean 4-space

We prove that the natural hyper-Kähler metrics on the moduli space of chargek instantons over Euclidean four-space and on the space of ADHM matrices coincide. We use this to deduce formulae relating expressions in the curvature of a connection to invariant polynomials in the ADHM matrices corresponding to this connection. These arise from consideration of the group of symmetries acting on the moduli spaces.     

The complete system of algebraic invariants for the sixteen-vertex model: II. Derivation by means of the theory of algebraic invariants

The construction of a complete system of basic invariants for the sixteen-vertex model on an M x N lattice as described in part I is repeated by means of an alternative method based on the theory of algebraic invariants. We use a generalization of a theorem by Cayley and Sylvester to determine the characteristics of the covariants belonging to the basic system. In this way we arrive at the same set of 21 invariants that was found in part I. The present method offers the possibility of a generalization to the three-dimensional 64-vertex model and the vertex model on a triangular lattice.     

Polynomial invariants for SU(2) monopoles

We present an explicit expression for the topological invariants associated to SU(2) monopoles in the fundamental representation on spin four-manifolds. The computation of these invariants is based on the analysis of their corresponding topological quantum field theory, and it turns out that they can be expressed in terms of Seiberg-Witten invariants. In this analysis we use recent exact results on the moduli space of vacua of the untwisted N = 1 and N = 2 supersymmetric counterparts of the topological quantum field theory under consideration, as well as on electric-magnetic duality for N = 2 supersymmetric gauge theories.     

Natural operators of smooth mappings of manifoldswith metric fields

In this paper we introduce the concepts of both a natural bundle and a natural operator generalized for the case of the category Mf_m × Mf_m of cartesian products of two manifolds and products of local diffeomorphisms. It is shown that any r-th order natural bundle over M × N has a structure of an associated bundle (P^rM × P^rN)Z G_{m^r × Gm^r}]. We consider prolongations of such associated bundles and their reduction with respect to a chosen subgroup. The existence of a bijective correspondence between natural operators of order k and the equivariant mappings of the corresponding type fibers are proved. A basis of invariants of arbitrary order is constructed for natural operators of smooth mappings of manifolds endowed with metric fields or connections, with values in a natural bundle of order one.     

The ADHM variety and perverse coherent sheaves

We study the full set of solutions to the ADHM equation as an affine algebraic set, the ADHM variety. We determine a filtration of the ADHM variety into subvarieties according to the dimension of the stabilizing subspace. We compute dimension, and analyze singularity and reducibility of all of these varieties. We also establish a connection between arbitrary solutions of the ADHM equation and coherent perverse sheaves on P²${\mathbb{P}}^2$ in the sense of Kashiwara.     

Heterotic p-branes from massive sigma models

We explicitly construct massive (0,4) supersymmetric ADHM sigma models which have heterotic p-brane solitons as their conformal fixed points. These yield the familiar gauge 5-brane and a new 1-brane solution which preserve half and a quarter of the space-time supersymmetry, respectively. We also discuss an analogous construction for the type 11 NS-NS p-branes using (4,4) supersymmetric models.     

Explicit non-'t Hooft rational solutions of the Atiyah-Drinfeld-Hitchin-Manin instanton matrix equations for SU(2r)

The Atiyah-Drinfeld-Hitchin-Manin instanton matrix equations may be written as a pair, one quadratic and one linear. The explicit general solution of the quadratic equation is given in terms of rational functions of free parameters, thus reducing the ADHM equations to a single equation. For SU(2r), large families of explicit solutions, rational in free parameters, are constructed by this method. These new families are generally about twice as large as the previously known 't Hooft families, and contain non-'t Hooft solutions for every r ? 1.     

Curve counting,instantons and McKay correspondences

We survey some features of equivariant instanton partition functions of topological gauge theories on four and six dimensional toric Kähler varieties, and their geometric and algebraic counterparts in the enumerative problem of counting holomorphic curves. We discuss the relations of instanton counting to representations of affine Lie algebras in the four-dimensional case, and to Donaldson–Thomas theory for ideal sheaves on Calabi–Yau threefolds. For resolutions of toric singularities, an algebraic structure induced by a quiver determines the instanton moduli space through the McKay correspondence and its generalizations. The correspondence elucidates the realization of gauge theory partition functions as quasi-modular forms, and reformulates the computation of noncommutative Donaldson–Thomas invariants in terms of the enumeration of generalized instantons. New results include a general presentation of the partition functions on ALE spaces as affine characters, a rigorous treatment of equivariant partition functions on Hirzebruch surfaces, and a putative connection between the special McKay correspondence and instanton counting on Hirzebruch–Jung spaces.     

Moduli of ADHM sheaves and the local Donaldson-Thomas theory

The ADHM construction establishes a one-to-one correspondence between framed torsion free sheaves on the projective plane and stable framed representations of a quiver with relations in the category of complex vector spaces. This paper studies the geometry of moduli spaces of representations of the same quiver with relations in the abelian category of coherent sheaves on a smooth complex projective curve X. In particular it is proven that this moduli space is virtually smooth and related by relative Beilinson spectral sequence to the curve counting construction via stable pairs of Pandharipande and Thomas. This yields a new conjectural construction for the local Donaldson-Thomas theory of curves as well as a natural higher rank generalization.     

Left chiral “zero modes” in a super-self-dual Yang-Mills background

A super-self-duality constraint can be imposed on the curvature two-form in supersymmetric Yang-Mills theories formulated in euclidean superspace. If left covariantly chiral superfields are coupled to such a background, part of this space of superfields is annihilated by a superspace operator constructed from the quadratic part of the action. These are analogous to the zero modes of the Dirac equation which occur in the presence of a multi-instanton. Using the supersymmetric version of the ADHM formalism, this space of left covariantly chiral “zero mode” superfields is constructed in the fundamental representation of an SU(n) gauge group. It is shown to be spanned by a set of k superfields, where k is the instanton number of the bosonic component of the Yang-Mills background.     

The Hilbert series of the one instanton moduli space

The moduli space of k G-instantons on \( {\mathbb{R}^4} \) for a classical gauge group G is known to be given by the Higgs branch of a supersymmetric gauge theory that lives on Dp branes probing D(p + 4) branes in Type II theories. For p = 3, these (3 + 1) dimensional gauge theories have \( \mathcal{N} = 2 \) supersymmetry and can be represented by quiver diagrams. The F and D term equations coincide with the ADHM construction. The Hilbert series of the moduli spaces of one instanton for classical gauge groups is easy to compute and turns out to take a particularly simple form which is previously unknown. This allows for a G invariant character expansion and hence easily generalisable for exceptional gauge groups, where an ADHM construction is not known. The conjectures for exceptional groups are further checked using some new techniques like sewing relations in Hilbert Series. This is applied to Argyres-Seiberg dualities.     

Wall-crossing between stable and co-stable ADHM data

We prove formula between Nekrasov partition functions defined from stable and co-stable ADHM data for the plane following method by Nakajima and Yoshioka (Kyoto J Math 51(2):263–335, 2011) based on the theory of wall-crossing formula developed by Mochizuki (Donaldson type invariants for algebraic surfaces: transition of moduli stacks, Lecture notes in mathematics, vol 1972, Springer, Berlin, 2009). This formula is similar to conjectures by Ito et al. [J High Energy Phys 2013(5):045, 2013, (4.1), (4.2)] for \(A_{1}\) singularity.