Instanton Moduli Space Research Articles

RG equations from Whitham hierarchy

The second derivatives of the prepotential with respect to Whitham time variables in the Seiberg-Witten theory are expressed in terms of Riemann theta-functions. These formulas give a direct transcendental generalization of the algebraic ones for the Kontsevich matrix model. In particular case they provide an explicit derivation of the renormalization group (RG) equation proposed recently in papers on Donaldson theory.

Wall-Crossing Formulas, the Bott Residue Formula and the Donaldson Invariants of Rational Surfaces

Wall-Crossing Formulas, the Bott Residue Formula and the Donaldson Invariants of Rational Surfaces

Symplectic topology in the nineties

This is a survey of some selected topics in symplectic topology. In particular, we discuss lowdimensional symplectic and contact topology, applications of generating functions, Donaldson's theory of approximately complex manifolds and some other recent developments in the field.

On matrix models of M5 branes

We compare the (0,2) theory of the single M5 brane decoupled from gravity in the lightcone with transverse R 4, and a matrix nmodel description in terms of quatum mechanics on instanton moduli space. We give some tests of the Matrix model in the case of multi-5-branes on R 4. We We extract constraints on the operator content of the field theory of the multi-5-brane system by analyzing the Matrix model. We also begin a study of compactifications of the (0,2) theory in this framework, arguing that, for large compactification scale, the (0,2) theory is described by super-quantum mechanics on appropriate instanton moduli spaces.

M theory on AdSp×S11−p and superconformal field theories

We study the large N limit of the interacting superconformal field theories associated with N M5 branes or M2 branes using the recently proposed relation between these theories and M theory on AdS spaces. We first analyze the spectrum of chiral operators of the 6d (0,2) theory associated with M5 branes in flat space, and find full agreement with earlier results obtained using its DLCQ description as quantum mechanics on a moduli space of instantons. We then perform a similar analysis for the D_N type 6d (0,2) theories associated with M5 branes at an R^5/Z_2 singularity, and for the 3d N=8 superconformal field theories associated with M2 branes in flat space and at an R^8/Z_2 singularity respectively. Little is known about these three theories, and our study yields for the first time their spectrum of chiral operators (in the large N limit).

Reducible connections in massless topological QCD and 4-manifolds

The role of reducible connections in non-Abelian Seiberg-Witten invariants is analyzed with massless topological QCD where the monopole is extended to the non-Abelian group version. By giving small external fields, we found that the vacuum expectation value can be separated into a part from Donaldson theory, a part from Abelian monopole theory and a part from non-Abelian monopole theory. As a by-product, we find identities of U(1) topological invariants. In our proof, the duality relation and the Higgs mechanism are not necessary.

Non-$c_{i}$-self-dual quaternionic Yang-Mills connections and $L_{2}$-gap theory

Non-$c_{i}$-self-dual quaternionic Yang-Mills connections and $L_{2}$-gap theory

Euler Characteristics of SU (2) Instanton Moduli Spaces on Rational Elliptic Surfaces

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.

Extended self-dual Yang-Mills from the N = 2 string

We show that the physical degrees of freedom of the critical open string with N = 2 superconformal symmetry on the worldsheet are described by a self-dual Yang-Mills field on a hyperspace parametrised by the coordinates of the target space R 2,2 together with a commuting chiral spinor. A prepotential for the self-dual connection in the hyperspace generates the infinite tower of physical fields corresponding to the inequivalent pictures or spinor ghost vacua of this string. An action is presented for this tower, which describes consistent interactions amongst fields of arbitrarily high spin. An interesting truncation to a theory of five fields is seen to have no graphs of two or more loops.

Jacobi forms and the structure of Donaldson invariants for 4-manifolds with b+ = 1

We prove structure theorems for the Donaldson invariants of 4-manifolds with b_+=1, similar to those of Kronheimer and Mrowka in the case b_+>1: We show that for a 4-manifold with b_+=1 and two different period points F, G on the boundary of the positive cone, the difference of the Donaldson invariants at F and G satisfies the k^{th}-order simple type condition for a number k, explicitely given in terms of F and G. We give a formula for this difference in terms of a set of basic classes, expressed in a universal formula involving modular forms. The leading terms coincide with the formulas of Kronheimer and Mrowka in the simple type case. The basic classes are related to the Seiberg Witten invariants. The proof of these results is based on the conjecture of Kotschick and Morgan on the structure of the wall-crossing terms. We relate the problem to new theta functions for indefinite lattices.

F-theory on K3 × K3 and instantons on 7-branes

We discuss N = 2 supersymmetric compactifications to four dimensions from the point of view of F-theory and heterotic theory. In a relatively simple setup, we illustrate the spectral theory for vector bundles on K3 × T 2 and discuss the heterotic-F-theory map. The moduli space of instantons on the 7-brane wrapped around K3 is discussed from the point of view of the Higgs mechanism in the effective four-dimensional N = 2 theory. This allows us to elaborate on the transitions between various branches of the moduli space of the heterotic-F-theory. We also describe the F-theory compactification on smooth K3 × K3 without the 3-branes in which the anomaly is canceled by the gauge fields.

Blow-up formulae for $SO(3)$ -Donaldson polynomials

We calculate first through seventh order terms of a blow-up formula for $SO(3)$ -Donaldson polynomials, using equivariant cohomology and techniques similar to those of [Oz]. These formulas are used to extend the results of [B2] to any rank one negative definite four manifold and to describe the extension of the equivalence proved in [M] of the Donaldson polynomial and O'Grady's algebro-geometric analogues to t he four dimensional classes.

Integrating over Higgs Branches

We develop some useful techinques for integrating over Higgs branches in supersymmetric theories with 4 and 8 supercharges. In particular, we define a regularized volume for hyperkahler quotients. We evaluate this volume for certain ALE and ALF spaces in terms of the hyperkahler periods. We also reduce these volumes for a large class of hyperkahler quotients to simpler integrals. These quotients include complex coadjoint orbits, instanton moduli spaces on R^4 and ALE manifolds, Hitchin spaces, and moduli spaces of parabolic Higgs bundles on Riemann surfaces. In the case of Hitchin spaces the evaluation of the volume reduces to a summation over solutions of Bethe Ansatz equations for the non-linear Schroedinger system. We discuss some applications of our results.

F-theory, geometric engineering and N = 1 dualities

We consider geometric engineering of N = 1 supersymmetric QFTs with matter in terms of a local model for compactification of F-theory on Calabi-Yau fourfold. By bringing 3-branes near 7-branes we engineer N = 1 supersymmetric SU( N c ) gauge theory with N f flavors in the fundamental. We identify the Higgs branch of this system with the instanton moduli space on the compact four-dimensional space of the 7-brane world-volume. Moreover we show that the Euclidean 3-branes wrapped around the compact part of the 7-brane world-volume can generate superpotential for N f = N c − 1 as well as lead to quantum corrections to the moduli space for N f = N c . Finally we argue that Seiberg's duality for N = 1 supersymmetric QCD may be mapped to T-duality exchanging 7-branes with 3-branes.

The BRST operator of quantum symmetries: The quantum analogs of Donaldson invariants

We construct the BRST operator of quantum symmetries and show that the nilpotency of this operator can either be derived from the Hopf axiom structure of the quantum group symmetries or from the Jacobi identity of their quantum Lie algebra. We extend this BRST operator to the topological transformations and we investigate the properties of invariant polynomials of curvatures from which we derive the descent equations for Donaldson invariants.

Special Quantum Field Theories¶in Eight and Other Dimensions

We build nearly topological quantum field theories in various dimensions. We give special attention to the case of 8 dimensions for which we first consider theories depending only on Yang-Mills fields. Two classes of gauge functions exist which correspond to the choices of two different holonomy groups in SO(8), namely SU(4) and Spin(7). The choice of SU(4) gives a quantum field theory for a Calabi-Yau fourfold. The expectation values for the observables are formally holomorphic Donaldson invariants. The choice of Spin(7) defines another eight dimensional theory for a Joyce manifold which could be of relevance in M- and F-theories. Relations to the eight dimensional supersymmetric Yang-Mills theory are presented. Then, by dimensional reduction, we obtain other theories, in particular a four dimensional one whose gauge conditions are identical to the non-abelian Seiberg-Witten equations. The latter are thus related to pure Yang-Mills self-duality equations in 8 dimensions as well as to the N=1, D=10 super Yang-Mills theory. We also exhibit a theory that couples 3-form gauge fields to the second Chern class in eight dimensions, and interesting theories in other dimensions.

Correlation functions in topological Yang-Mills theory with two ferminionic charges

The solution of the Donaldson cohomology problem for the topological Yang-Mills theory with two fermionic symmetries needs besides the gauge field and its descendants additional fields, hereafter called ascendants of the gauge field. It is shown that the dependence of the ascendants disappears in the all the correlation functions. This property allows one for the usual interpretation of the Donaldson invariants as cocycles of the instanton moduli space.

Automorphic forms with singularities on Grassmannians

We construct some families of automorphic forms on Grassmannians which have singularities along smaller sub Grassmannians, using Harvey and Moore's extension of the Howe (or theta) correspondence to modular forms with poles at cusps. Some of the applications are as follows. We construct families of holomorphic automorphic forms which can be written as infinite products, which give many new examples of generalized Kac-Moody superalgebras. We extend the Shimura and Maass-Gritsenko correspondences to modular forms with singularities. We prove some congruences satisfied by the theta functions of positive definite lattices, and find a sufficient condition for a Lorentzian lattice to have a reflection group with a finite volume fundamental domain. We give some examples suggesting that these automorphic forms with singularities are related to Donaldson polynomials and to mirror symmetry for K3 surfaces.

Some computations of Donaldson's invariants via flat connections

The Donaldson invariants for the elliptic surfaces X with p g ( X) > 0 were first studied by Donaldson (1990), and Friedman and Morgan (1994) using stable bundles. Kronheimer (1991) gave a topological calculation of the degree d = 0 Donaldson invariant of the K3 surface. Here we generalize Kronheimer's approach to calculate a degree d = g + 1 invariant for each elliptic surface X with p g ( X) = g > 0. We shall construct a moduli space of flat connections M with virtual dimension 2 g + 2, derive that the degree g + 1 Donaldson invariant satisfies q d = aQk g − 1 + bk g + 1 , and compute the leading coefficient a and how it changes under logarithmic transformations. The result agrees with Friedman and Morgan's, but our proofs do not use algebraic geometry. We will also prove a relation over the anti-self-dual moduli space between the Pontryagin class of the base-point fibration and Donaldson's μ-class for certain smoothly embedded two-spheres with self-intersection −2.

Langevin approach for Abelian topological gauge theory

An Abelian topological action is constructed from the quantization of Seiberg–Witten monopole equations as “Langevin equations.” The starting point is an analogous action to the Labastida–Pernici’s non-supersymmetric action for Donaldson theory. As the local symmetry of the action is first stage reducible, the quantum action is obtained by using Batalin–Vilkovisky quantization procedure. We can also obtain off-shell quantum action and BRST transformation.