arXiv subjects

Paul M. N. Feehan

Publications and source records attributed to Paul M. N. Feehan.

47 records · Page 3Linked to original sources

arXiv2001

PU(2) monopoles. II: Top-level Seiberg-Witten moduli spaces and Witten's conjecture in low degrees

In this article we complete the proof---for a broad class of four-manifolds---of Witten's conjecture that the Donaldson and Seiberg-Witten series coincide, at least through terms of degree less than or equal to c-2, where c is a linear combination of the Euler characteristic and signature of the four-manifold. This article is a revision of sections 4--7 of an earlier version, while a revision of sections 1--3 of that earlier version now appear in a separate companion article (math.DG/0007190). Here, we use our computations of Chern classes for the virtual normal bundles for the Seiberg-Witten strata from the companion article (math.DG/0007190), a comparison of all the orientations, and the PU(2) monopole cobordism to compute pairings with the links of level-zero Seiberg-Witten moduli subspaces of the moduli space of PU(2) monopoles. These calculations then allow us to compute low-degree Donaldson invariants in terms of Seiberg-Witten invariants and provide a partial verification of Witten's conjecture.

dg-ga↗

arXiv2001

PU(2) monopoles and links of top-level Seiberg-Witten moduli spaces

This is the first of two articles in which we give a proof - for a broad class of four-manifolds - of Witten's conjecture that the Donaldson and Seiberg-Witten series coincide, at least through terms of degree less than or equal to c-2, where c is a linear combination of the Euler characteristic and signature of the four-manifold. This article is a revision of sections 1-3 of an earlier version of the article dg-ga/9712005, now split into two parts, while a revision of sections 4-7 of that earlier version appears in a recently updated dg-ga/9712005. In the present article, we construct virtual normal bundles for the Seiberg-Witten strata of the moduli space of PU(2) monopoles and compute their Chern classes.

math.DG↗

arXiv2000

A Kato-Yau inequality for harmonic spinors and decay estimate for eigenspinors

We show that harmonic spinors obey a strengthened version of the well-known pointwise Kato inequality for sections of a vector bundle with a connection. We then prove a decay estimate for eigenspinors using this Kato-Yau estimate and resulting differential inequality. We briefly describe some applications to gauge theory---specifically to integral estimates which are used when gluing and ungluing PU(2) monopoles (math.DG/9907107).

math.DG↗

arXiv2000

Generic metrics, irreducible rank-one PU(2) monopoles, and transversality

We prove that the moduli space of solutions to the PU(2) monopole equations is a smooth manifold of the expected dimension for simple, generic parameters such as (and including) the Riemannian metric on the given four-manifold. In a previous article, dg-ga/9710032, we proved transversality using an extension of the holonomy-perturbation methods of Donaldson, Floer, and Taubes, together with the existence of an Uhlenbeck compactification for the perturbed moduli space. However, it remained an important and interesting question to see whether there were simpler, more intrinsic alternatives to the holonomy perturbations and this is the issue we settle here. The idea that PU(2) monopoles might lead to a proof of Witten's conjecture (hep-th/9411102, hep-th/9709193) concerning the relation between the two types of four-manifold invariants was first proposed by Pidstrigach and Tyurin in 1994 (dg-ga/9507004): the space of PU(2) monopoles contains the moduli space of anti-self-dual connections together with copies of the various Seiberg-Witten moduli spaces, these forming singular loci in the higher-dimensional space of PU(2) monopoles. Results in this direction, due to the author and Leness, are surveyed in dg-ga/9709022, with a detailed account appearing in dg-ga/9712005. Our transversality theorem ensures that the anti-self-dual and Seiberg-Witten loci are the only singularities and that the PU(2) monopole moduli space forms a smooth - though non-compact, because of bubbling - cobordism between the links of the singularities.

math.DG↗

arXiv1999

PU(2) monopoles. III: Existence of gluing and obstruction maps

This is the third installment in our series of articles (dg-ga/9712005, dg-ga/9710032) on the application of the PU(2) monopole equations to prove Witten's conjecture (hep-th/9411102) concerning the relation between the Donaldson and Seiberg-Witten invariants of smooth four-manifolds. The moduli space of solutions to the PU(2) monopole equations provides a noncompact cobordism between links of compact moduli spaces of U(1) monopoles of Seiberg-Witten type and the moduli space of anti-self-dual SO(3) connections, which appear as singularities in this larger moduli space. In this paper we prove the first part of a general gluing theorem for PU(2) monopoles, with the proof of the second half to appear in a companion article. The ultimate purpose of the gluing theorem is to provide topological models for neighborhoods of ideal Seiberg-Witten moduli spaces appearing in lower levels of the Uhlenbeck compactification of the moduli space of PU(2) monopoles and thus permit calculations of their contributions to Donaldson invariants using the PU(2)-monopole cobordism.

math.DG↗

arXiv1999

PU(2) monopoles and a conjecture of Marino, Moore, and Peradze

In this article we show that some of the recent results of Marino, Moore, and Peradze (math.DG/9812042, hep-th/9812055) -- in particular their conjecture that all closed, smooth four-manifolds with b_2^+ > 1 (and Seiberg-Witten simple type) are of `superconformal simple type' -- can be understood using a simple mathematical argument via the PU(2)-monopole cobordism of Pidstrigach and Tyurin (dg-ga/9507004) and results of the first and third authors (dg-ga/9712005, dg-ga/9709022).

math.DG↗

arXiv1999

Donaldson invariants and wall-crossing formulas. I: Continuity of gluing maps

The present article is the first in a series whose ultimate goal is to prove the Kotschick-Morgan conjecture concerning the wall-crossing formula for the Donaldson invariants of a four-manifold with b^+ = 1. The conjecture asserts that the wall-crossing terms due to changes in the metric depend at most on the homotopy type of the four-manifold and the degree of the invariant. Our principal interest in this conjecture is due to the fact that its proof is expected to resemble that of an important intermediate step towards a proof of Witten's conjecture concerning the relation between Donaldson and Seiberg-Witten invariants (hep-th/9411102, hep-th/9709193), using PU(2) monopoles as described in (dg-ga/9709022, dg-ga/9712005). Moreover, it affords us another venue in which to address some of the technical difficulties arising in our work on Witten's conjecture. The additional difficulties in the case of PU(2) monopoles are due to the more complicated gluing theory, the presence of obstructions to deformation and gluing, and the need to consider links of positive-dimensional families of `reducibles' even in the presence of `simple type' assumptions. Witten's conjecture should then follow from a final step analogous to Goettsche's computation of the wall-crossing terms, assuming that the Kotschick-Morgan conjecture holds (alg-geom/9506018).

math.DG↗

arXiv1999

Critical-exponent Sobolev norms and the slice theorem for the quotient space of connections

The use of certain critical-exponent Sobolev norms is an important feature of methods employed by Taubes to solve the anti-self-dual and similar non-linear elliptic partial differential equations. Indeed, the estimates one can obtain using these critical-exponent norms appear to be the best possible when one needs to bound the norm of a Green's operator for a Laplacian, depending on a connection varying in a non-compact family, in terms of minimal data such as the first positive eigenvalue of the Laplacian or the L^2 norm of the curvature of the connection. Following Taubes, we describe a collection of critical-exponent Sobolev norms and general Green's operator estimates depending only on first positive eigenvalues or the L^2 norm of the connection's curvature. Such estimates are particularly useful in the gluing construction of solutions to non-linear partial differential equations depending on a degenerating parameter, such as the approximate, reference solution in the anti-self-dual or PU(2) monopole equations. We apply them here to prove an optimal slice theorem for the quotient space of connections. The result is optimal in the sense that if a point [A] in the quotient space is known to be just L^2_1-close enough to a reference point [A_0], then the connection A can be placed in Coulomb gauge relative to the connection A_0, with all constants depending at most on the first positive eigenvalue of the covariant Laplacian defined by A_0 and the L^2 norm of the curvature of A_0. In this paper we shall for simplicity only consider connections over four-dimensional manifolds, but the methods and results can adapted to manifolds of arbitrary dimension to prove slice theorems which apply when the reference connection is allowed to degenerate.

dg-ga↗

arXiv1997

PU(2) Monopoles, I: Regularity, Uhlenbeck Compactness, and Transversality

We prove the existence of perturbations for the PU(2) monopole equations, yielding transversality on the complement of the anti-self-dual or reducible solutions, and the existence of an Uhlenbeck compactification for the moduli space of solutions to these perturbed PU(2) monopole equations. In December 1994, V. Pidstrigach and A. Tyurin and then others proposed a method to prove Witten's conjecture concerning the relation between the Donaldson and Seiberg-Witten invariants of smooth four-manifolds. Their proposal uses a moduli space of solutions to the PU(2) monopole equations, which are a natural generalization of the U(1) monopole equations of Seiberg and Witten and the equation for anti-self-dual SO(3) connections, to construct a cobordism between links of compact moduli spaces of U(1) monopoles of Seiberg-Witten type and the moduli space of anti-self-dual connections, which appear as singularities in this larger moduli space. A basic requirement of this cobordism technique is the existence of an Uhlenbeck compactification for the moduli space of PU(2) monopoles and of generic-parameter transversality results for all the moduli spaces of PU(2) monopoles which appear in this compactification, on the complement of the anti-self-dual and U(1) solutions.

dg-ga↗

arXiv1997

Uhlenbeck compactness and transversality for the moduli space of PU(2) monopoles

This research announcement gives a brief report of the main results in our paper "PU(2) monopoles, I: Regularity, Uhlenbeck compactness, and transversality" (Journal of Differential Geometry, to appear). We describe the existence of perturbations for the PU(2) monopole equations, yielding both useful transversality properties and an Uhlenbeck compactification for this perturbed moduli space.

dg-ga↗

arXiv1997

PU(2) monopoles and relations between four-manifold invariants

Using quantum field-theoretic arguments, Witten has established a relation between the Donaldson and Seiberg-Witten invariants of smooth four-manifolds. In this survey article, we describe the program to prove this relation using a moduli space of PU(2) = SO(3) monopoles as a cobordism between the Donaldson moduli space of anti-self-dual SO(3) connections and moduli spaces of U(1) monopoles. We provide an overview of some of our transversality and Uhlenbeck compactness results for PU(2) monopoles, along with some of our calculations of Donaldson invariants in terms of Seiberg-Witten invariants. We give a brief outline of issues concerning the gluing theory, focussing on some of the analytical difficulties that are particular to PU(2) monopoles, and its application to the PU(2) monopole program to prove the relation between Donaldson and Seiberg-Witten invariants.

dg-ga↗