Gravitational instantons

The path-integral approach to quantum field theory assigns special importance to finite action Euclidean solutions of classical field equations. In Yang-Mills gauge theories, the instanton solutions of classical field equations with self-dual field strength have given rise to a new, nonperturbative treatment of the quantum field theory and its vacuum state. Since gravitation is also a species of gauge theory, one might think that similar phenomena would occur in gravity. The authors recently sought and found a new self-dual solution to Euclidean gravity which plays a role parallel to that of the Yang-Mills instanton. Gravitational instantons now promise to yield new insights into the nature of quantum gravity.This essay received the second award from the Gravity Research Foundation for the year 1979-Ed.     

Gravitational instantons and Euclidean supersymmetry

Supersymmetric instanton solutions in four-dimensional Euclidean ungauged Einstein–Maxwell theory are analysed and classified according to the fraction of supersymmetry they preserve, using spinorial geometry techniques.     

Gravitational gauge mediation

It is often the case that the naive introduction of the messenger sector to supersymmetry breaking models causes restoration of supersymmetry. We discuss a possibility of stabilizing the supersymmetry breaking vacuum by gravitational interaction.     

Perturbation theory with instantons

We develop “perturbation theory” rules for calculating the effect of instantons in a pure Yang-Mills theory with no fermions, in the “dilute gas” approximation in which the N-instanton solution is assumed to be the sum of N widely separated one-instanton solutions. These rules are then used to compute the gluon propagator and proper vertex function including all orders of the instanton interaction but only to lowest order in the gluon coupling. It is to be expected that such an approximation is valid only for momenta q larger than the physical mass μ. The result is that in this regime instantons cause variations in the propagator and vertex of the form where b is the coefficient in the expansion of the β function: β = bg³ + 3. .     

The role of instantons in scale-invariant gauge theories

Instanton calculations in scale-invariant gauge theories, such as QCD, have long been plaqued by divergences at large distances where strong coupling effects are important. Furthermore, Witten has argued that quantum effects may cause the instanton gas to disappear and has displayed this phenomenon in the CP^(N?1) model at large N. It is argued here that instantons can play a role in calculations involving an inherent infrared cut-off, and this is demonstrated in the CP^(N?1) model for large N at a finite temperature. Some results on finite-temperature QED are also obtained in passing.     

Small instantons in string theory

A long-standing puzzle about the heterotic string has been what happens when an instanton shrinks to zero size. It is argued here that the answer at the quantum level is that an extra SU(2) gauge symmetry appears that is supported in the core of the instanton. Thus in particular the quantum heterotic string has vacua with higher rank than is possible in conformal field theory. When k instantons collapse at the same point, the enhanced gauge symmetry is Sp(k). These results, which can be tested by comparison to Dirichlet five-branes of Type I superstrings and to the ADHM construction of instantons, give the first example for the heterotic string of a non-perturbative phenomenon that cannot be turned off by making the coupling smaller. They have applications to several interesting puzzles about string duality.     

Poincaré gauge theory from self-coupling

Poincaré gauge theory is derived from a linear theory by the method suggested by Gupta for deriving Einstein’s general relativity from the linear theory of a spin-2 field. Non-linearity is introduced by requiring that a set of tensor fields be coupled to the Noether currents of the Poincaré group (energy-momentum and spin).     

Simplicial gauge theory and quantum gauge theory simulation

We propose a general formulation of simplicial lattice gauge theory inspired by the finite element method. Numerical tests of convergence towards continuum results are performed for several SU(2) gauge fields. Additionally, we perform simplicial Monte Carlo quantum gauge field simulations involving measurements of the action as well as differently sized Wilson loops as functions of β.     

Gravitational correction to running of gauge couplings

We calculate the contribution of graviton exchange to the running of gauge couplings at lowest non-trivial order in perturbation theory. Including this contribution in a theory that features coupling constant unification does not upset this unification, but rather shifts the unification scale. When extrapolated formally, the gravitational correction renders all gauge couplings asymptotically free.     

Supersymmetric instantons and topological quantum field theory

We present an action which generates the supersymmetric self-dual equations corresponding to euclidean super Yang-Mills theory in four dimensions. By adding additional constraint fields with new local symmetries, the classical equations of this system are the usual super self-dual equations when a gauge is chosen for the constraint fields. This construction is a supersymmetric generalization of the Labastida-Pernici action which corresponds to a gauge unfixed version of Witten's topological quantum field theory. We discuss some topological prospects for this model, and the role of supersymmetric instantons in Donaldson theory.     

Finite-temperature gauge theory from the transverse lattice

Numerical computations are performed and analytic bounds are obtained on the excited spectrum of glueballs in SU(inifinity) gauge theory, by transverse lattice Hamiltonian methods. We find an exponential growth of the density of states, implying a finite critical (Hagedorn) temperature. It is argued that the Nambu-Goto string model lies in a different universality class.     

Kaluza-Klein theory from the viewpoint of gauge theory of gravity

We consider Kaluza-Klein theory based on the fiber bundle. We obtain the modified Kaluza-Klein metric as an invariant line element of a bundle. Its reduced action includes a higher derivative action in gravitation as well as a term linear inR.     

Zwanziger-Landau gauge in regularized gauge theory

The Green functions of Zwanziger-gaugefixed and continuum-regularized gauge theory are finite and transverse to all orders as the Zwanziger parameter α goes to zero.     

A canonical approach to instantons and multiple vacua in Abelian gauge theories

We formulate the concepts of instantons, multiple vacua and vacuum tunnelling in the canonical framework. This formalism enables us to give explicit expressions for previously unconsidered relations between the physical and unphysical sectors of the indefinite metric Hilbert space. As concrete examples we discuss the Higgs, Schwinger and Nambu-Jona-Lasinio models.