Instanton Moduli Space Research Articles

Coulomb branch and the moduli space of instantons

The moduli space of instantons on C^2 for any simple gauge group is studied using the Coulomb branch of N=4 gauge theories in three dimensions. For a given simple group G, the Hilbert series of such an instanton moduli space is computed from the Coulomb branch of the quiver given by the over-extended Dynkin diagram of G. The computation includes the cases of non-simply-laced gauge groups G, complementing the ADHM constructions which are not available for exceptional gauge groups. Even though the Lagrangian description for non-simply laced Dynkin diagrams is not currently known, the prescription for computing the Coulomb branch Hilbert series of such diagrams is very simple. For instanton numbers one and two, the results are in agreement with previous works. New results and general features for the moduli spaces of three and higher instanton numbers are reported and discussed in detail.

Anti-Selfdual Connections on the Quantum Projective Plane: Instantons

We study one-instantons, that is anti-selfdual connections with instanton number 1, on the quantum projective plane with orientation which is reversed with respect to the usual one. The orientation is fixed by a suitable choice of a basis element for the rank 1 free bimodule of top forms. The noncommutative family of solutions is foliated, each non-singular leaves being isomorphic to the quantum projective plane itself.

Induced three-forms on instanton moduli spaces

In this note we study a correspondence between the space of three-forms on a four-manifold and the space of three-forms on the moduli space of instantons. We then specialize to the case where the base manifold is the four-sphere.

Generalized Kähler Geometry of Instanton Moduli Spaces

We prove that Hitchin’s generalized Kähler structure on the moduli space of instantons over a compact, even generalized Kähler four-manifold may be obtained by generalized Kähler reduction, in analogy with the usual Kähler case. The underlying reduction of Courant algebroids is a realization of Donaldson’s μ-map in degree three.

A vanishing result for a Casson-type instanton invariant

Casson-type invariants emerging from Donaldson theory over certain negative-definite four-manifolds were recently suggested by Teleman. These are defined by an algebraic count of points in a zero-dimensional moduli space of flat instantons. Motivated by the cobordism programme of proving Witten's conjecture, we use a moduli space of PU ( 2 ) Seiberg–Witten monopoles to exhibit an oriented one-dimensional cobordism of the instanton moduli space to the empty space. The Casson-type invariant must therefore vanish.

Generalised Chern-Simons Theory and G2-Instantons over Associative Fibrations

Adjusting conventional Chern-Simons theory to ${\rm G}_2$-manifolds, one describes ${\rm G}_2$-instantons on bundles over a certain class of $7$-dimensional flat tori which fiber non-trivially over $T^4$, by a pullback argument. Moreover, if $c_2\neq0$, any (generic) deformation of the ${\rm G}_2$-structure away from such a fibred structure causes all instantons to vanish. A brief investigation in the general context of (conformally compatible) associative fibrations $f:Y^7\to X^4$ relates ${\rm G}_2$-instantons on pullback bundles $f^*E\to Y$ and self-dual connections on the bundle $E\to X$ over the base, a fact which may be of independent interest.

Superconformal quantum mechanics and the exterior algebra

We extend the differential form representation of N = (n,n) supersymmetric quantum mechanics to the superconformal case. We identify the superalgebras occurring for n = 1,2,4, give necessary and sufficient conditions for their existence, and give explicit geometric constructions of their generators and commutation relations. Quantum mechanics on the moduli space of instantons is considered as an example.

Equivariant, string and leading order characteristic classes associated to fibrations

Infinite rank vector bundles often appear as pushdowns of finite rank bundles from the total space of a fibration to the base space. The infinite rank bundles have string and leading order characteristic classes related to the characteristic classes of the finite rank bundles. We rewrite the S1-index theorem as a statement about equivariant leading order classes on loop spaces, interpret certain Gromov–Witten invariants in terms of leading order and string classes, show that the generators of the cohomology of a loop group are Chern–Simons string classes, and relate Donaldson invariants to leading order currents.

Towards the Proof of AGT Relations with the Help of the Generalized Jack Polynomials

Original proofs of the AGT relations with the help of the Hubbard-Stratanovich duality of the modified Dotsenko-Fateev matrix model did not work for beta different from one, because Nekrasov functions were not properly reproduced by Selberg-Kadell integrals of Jack polynomials. We demonstrate that if the generalized Jack polynomials, depending on the N-ples of Young diagrams from the very beginning, are used instead of the N-linear combinations of ordinary Jacks, this resolves the problem. Such polynomials naturally arise as special elements in the equivariant cohomologies of the GL(N)-instanton moduli spaces, and this also establishes connection to alternative ABBFLT approach to the AGT relations, studying the action of chiral algebras on the instanton moduli spaces. In this paper we describe a complete proof of AGT in the simple case of GL(2) (N=2) Yang-Mills theory, i.e. the 4-point spherical conformal block of the Virasoro algebra.

Donaldson Invariants of Symplectic Manifolds

We prove that symplectic 4-manifolds with $b_1 = 0$ and $b^+ > 1$ have nonvanishing Donaldson invariants, and that the canonical class is always a basic class. We also characterize in many situations the basic classes of a Lefschetz fibration over the sphere which evaluate maximally on a generic fiber.

Hilbert series for moduli spaces of instantons on ℂ2/ℤ n

We study chiral gauge-invariant operators on moduli spaces of G instantons for any classical group G on A-type ALE spaces using Hilbert Series (HS). Moduli spaces of instantons on an ALE space can be realized as Higgs branches of certain quiver gauge theories which appear as world-volume theories on Dp branes in a Dp-D(p + 4) system with the D(p + 4) branes (with or without O(p + 4) planes) wrapping the ALE space. We study in detail a list of quiver gauge theories which are related to G-instantons of arbitrary ranks and instanton numbers on a generic A n−1 ALE space and discuss the corresponding brane configurations. For a large class of theories, we explicitly compute the Higgs branch HS which reveals various algebraic/geometric aspects of the moduli space such as the dimension of the space, generators of the moduli space and relations connecting them. In a large number of examples involving lower rank instantons, we demonstrate that HS for equivalent instantons of isomorphic gauge groups but very different quiver descriptions do indeed agree, as expected.

Coset conformal field theory and instanton counting on ℂ2/ℤ p

Abstract We study conformal field theory with the symmetry algebra $ \mathcal{A}\left( {2,\ p} \right)={{{\widehat{\mathfrak{gl}}{(n)_2}}} \left/ {{\widehat{\mathfrak{gl}}{{{\left( {n-p} \right)}}_2}}} \right.} $ . In order to support the conjecture that this algebra acts on the moduli space of instantons on ℂ2/ℤ p , we calculate the characters of its representations and check their coincidence with the generating functions of the fixed points of the moduli space of instantons. We show that the algebra $ \mathcal{A}\left( {2,\ p} \right) $ can be realized in two ways. The first realization is connected with the cross-product of p Virasoro and p Heisenberg algebras: $ {{\mathcal{H}}^p} $ × Vir p . The second realization is connected with: $ {{\mathcal{H}}^p} \times \widehat{\mathfrak{sl}}{(p)_2}\times \left( {\widehat{\mathfrak{sl}}{(2)_p}\times {{{\widehat{\mathfrak{sl}}{(2)_{n-p }}}} \left/ {{\widehat{\mathfrak{sl}}{(2)_n}}} \right.}} \right) $ . The equivalence of these two realizations provides the non-trivial identity for the characters of $ \mathcal{A}\left( {2,\ p} \right) $ . The moduli space of instantons on ℂ2/ℤ p admits two different compactifications. This leads to two different bases for the representations of $ \mathcal{A}\left( {2,\ p} \right) $ . We use this fact to explain the existence of two forms of the instanton pure partition functions.

Poincaré invariants are Seiberg–Witten invariants

Recently there has been a renewed interest in Donaldson invariants and Seiberg–Witten invariants due to the influx of virtual intersection theory. See Mochizuki [15], Gottsche, Nakajima and Yoshioka [6], and [4], for instance. The purpose of this paper is to prove a conjecture (Theorem 1.1 below) of Durr, Kabanov and Okonek in [4], which provides a natural algebro-geometric theory of Seiberg–Witten invariants. Our main technique is the cosection localization principle in Kiem and Li [8] that effectively localizes the virtual cycle when there is a cosection of the obstruction sheaf.

Loop quantum gravity without the Hamiltonian constraint

We show that under certain technical assumptions, including the existence of a constant mean curvature (CMC) slice and strict positivity of the scalar field, general relativity conformally coupled to a scalar field can be quantized on a partially reduced phase space, meaning reduced only with respect to the Hamiltonian constraint and a proper gauge fixing. More precisely, we introduce, in close analogy to shape dynamics, the generator of a local conformal transformation acting on both, the metric and the scalar field, which coincides with the CMC gauge condition. A new metric, which is invariant under this transformation, is constructed and used to define connection variables which can be quantized by standard loop quantum gravity methods. Since this connection is invariant under the local conformal transformation, the generator of which is shown to be a good gauge fixing for the Hamiltonian constraint, the Dirac bracket associated with implementing these constraints coincides with the Poisson bracket for the connection. Thus, the well developed kinematical quantization techniques for loop quantum gravity are available, while the Hamiltonian constraint has been solved (more precisely, gauge fixed) classically. The physical interpretation of this system is that of general relativity on a fixed spatial CMC slice, the associated ‘time’ of which is given by the CMC. While it is hard to address dynamical problems in this framework (due to the complicated ‘time’ function), it seems, due to good accessibility properties of the CMC gauge, to be well suited for problems such as the computation of black hole entropy, where actual physical states can be counted and the dynamics is only of indirect importance. The corresponding calculation yields the surprising result that the usual prescription of fixing the Barbero–Immirzi parameter β to a constant value in order to obtain the well-known formula S = a(Φ)A/(4G) does not work for the black holes under consideration, while a recently proposed prescription involving an analytic continuation of β to the case of a self-dual space-time connection yields the correct result. Also, the interpretation of the geometric operators gets an interesting twist, which exemplifies the deep relationship between observables and the choice of a time function and has consequences for loop quantum cosmology.

Exact construction of noncommutative instantons

We discuss the Atiyah-Drinfeld-Hitchin-Manin (ADHM) construction of U(N) instantons in noncommutative (NC) space and give some exact instanton solutions for various noncommutative settings. We also present a new formula which is crucial to show an origin of the instanton number for U(1) and to prove the one-to-one correspondence between moduli spaces of the noncommutative instantons and the ADHM data.

Noncommutative Instantons Revisited

We discuss the Atiyah-Drinfeld-Hitchin-Manin (ADHM) construction of U(N) instantons in noncommutative (NC) space and prove the one-to-one correspondence between moduli spaces of the noncommutative instantons and the ADHM data, together with an origin of the instanton number for U(1). We also give a derivation of the ADHM construction from the viewpoint of the Nahm transformation of instantons on four-torus.

Moduli stacks of maps for supermanifolds

We consider the moduli problem of stable maps from a Riemann surface into a supermanifold; in twistor-string theory, this is the instanton moduli space. By developing the algebraic geometry of supermanifolds to include a treatment of super-stacks we prove that such moduli problems, under suitable conditions, give rise to Deligne-Mumford superstacks (where all of these objects have natural definitions in terms of super-geometry). We make some observations about the properties of these moduli super-stacks, as well as some remarks about their application in physics and their associated Gromov-Witten theory.

Floer homology for 2-torsion instanton invariants

We construct a variant of Floer homology groups and prove a gluing formula for a variant of Donaldson invariants. As a corollary, the variant of Donaldson invariants is non-trivial for connected sums of 4-manifolds which satisfy a condition for Donaldson invariants. We also show a non-existence result of compact, spin 4-manifolds with boundary some homology 3-spheres and with certain intersection forms.

SO(3)–Donaldson invariants of ℂP2and mock theta functions

We compute the Moore-Witten regularized u-plane integral on CP^2, and we confirm their conjecture that it is the generating function for the SO(3)-Donaldson invariants of CP^2. We prove this conjecture using the theory of mock theta functions and harmonic Maass forms. We also derive further such generating functions for the SO(3)-Donaldson invariants with 2N_f massless monopoles using the geometry of certain rational elliptic surfaces (N_f=0,2,3,4). We show that the partition function for N_f=4 is nearly modular. When combined with one of Ramanujan's mock theta functions, we obtain a weight 1/2 modular form. This fact is central to the proof of the conjecture.

Instantons, concordance, and Whitehead doubling

We use moduli spaces of instantons and Chern-Simons invariants of flat connections to prove that the Whitehead doubles of (2;2 n − 1) torus knots are independent in the smooth knot concordance group; that is, they freely generate a subgroup of infinite rank. Call two knots in the 3-sphere concordant if they arise as the boundary of a smooth and properly embedded cylinder in the 3-sphere times an interval. Concordance is clearly an equivalence relation. Modulo this relation, the set of knots forms an abelian group C, with the role of addition played by connected sum and inverse given by considering the mirror image, with reversed orientation. This concordance group is a much studied object, well motivated by its role as a gateway into the mysterious world of 4-dimensional topology. Indeed, even in this relative situation of studying 3-dimensional manifolds in relation to the 4-dimensional manifolds they bound, one can observe the distinction between the smooth and topological categories in dimension four. Moreover, the group structure afforded by passing to concordance paves a clearer path through the often intractable field of knot theory. Our results focus on a particular satellite operation, (positive, untwisted) Whitehead doubling, and its effect on the concordance group, see Figure 1 and Section 3.1 for a definition. Our motivation comes from the following conjecture. To state it recall that a knot is slice if it is concordant to the unknot or, equivalently, if it bounds a smooth and properly embedded disk in the 4-ball.