Edward Corrigan

The construction of self-dual solutions to SU(2) gauge theory

Ignoring the problem of sources and singularities, explicit expressions are constructed for the ansätze of Atiyah and Ward. These take an especially simple form in theR gauge of Yang. Some non-linear transformation properties of the self-duality equations in this gauge provide an inductive proof of the ansätze. There is a six-parameter family of these Bäcklund transformations. They take real SU(2) gauge fields into real SU(1, 1) gauge fields and vice versa.

Construction of instanton and monopole solutions and reciprocity

Abstract An elementary argument demonstrating the completeness of the Atiyah-Drinfeld-Hitchin-Manin construction of self-dual instanton solutions to Euclidean gauge theories is presented. The adaptation of this discussion to Nahms construction for SU(2) monopoles is outlined. These constructions are shown to establish a reciprocity or duality between self-dual theories in zero and four dimensions and in one and three dimensions, respectively.

A green function for the general self-dual gauge field

Abstract The recent general solution by Atiyah, Hitchin, Drinfeld and Manin of the self-duality equations for an arbitrary compact classical group is discussed. The Green function for a scalar field transforming as a vector under the group is shown to take an elegant form in the background field of the general self-dual solution. The massless solutions of the Dirac equation, for this representation of the group, are also explicitly exhibited.

Instanton green functions and tensor products

Abstract The structure of the Green function for a scalar field, transforming under the adjoint representation of a gauge group, in the background field of an arbitrary self-dual instanton field, is considered. It is shown that it can be written in the same elegant form as the Green function for the fundamental vector representation of the group, given the introduction of a certain conformally invariant matrix. The same method solves the more general problem of finding the Green function for a field transforming under the direct product of two groups. Simple expressions for the solutions of the corresponding massless Dirac equation are obtained. The results on the products of representations may be applied, iteratively, to construct the Green function and massless solutions of the Dirac equation for any higher representation of the group.

Zeta-function regularization and multi-instanton determinants☆

Abstract The determinant of the covariant laplacian is discussed using the ζ-function regularization procedure. For a self-dual solution to a general euclidean Yang-Mills gauge theory, an integral expression for its variation, with respect to the parameters of the solution, is obtained in terms of the general construction of Atiyah et al. The conformal properties are analyzed and an explicit expression is obtained for a part of the determinant containing all the non-trivial behavior. The remaining conformally invariant part has not been determined. The removal of the infrared divergences by considering the problem on a sphere is discussed in some detail.

A Functional Equation for Exponential Loop Integrals in Gauge Theories

Abstract We analyze the behavior of the operator ψ(Γ) = Tr P Γ exp (∲A· d x) under variation of the entire closed contour Γ. An equation for the second contour variation ofψ (Γ) is derived. It codes the dynamics of the gauge theory in a gauge-invariant and frame-independent way, without reference to the connection.

A note on the quark content of large color groups

Abstract We point out an ambiguity in the generalization of the quark content of SUN color groups for N > 3. We argue that the likely quarks in such groups appear in different representation of the antisymmetrized variety, allowing for a natural three-quark color singlet baryon in the large N expansion.

General dual operatorial vertices

Abstract The N -reggeon vertex in constructed for a large class of factorizable dual resonance models by representing the constituent Mobius transformations in different unitary irreducible representations of SU(1, 1). The conventional model and the model of Neveu and Schwarz emerge as special cases. The requirement of the existence of an infinite set of gauge operators restricts the number of representations that can be used.

Non-local charges for the supersymmetric σ-model☆

Abstract The classical non-local currents for the two dimensional supersymmetric non-linear σ-model are constructed recursively. They are polynomial in the canonical fields and generalize naturally the corresponding currents of the constituent Gross-Neveu and ordinary σ -models.

Static non-abelian forces and the permutation group☆

Abstract It is shown that the classical static interaction energy of two non-abelian point sources is ambiguous. The ambiguity is discrete and corresponds to the elements of the permutation group on n symbols, if the gauge group is U( n ).