Non-trivial Gauge Field Research Articles

Heterotic phase transitions and singularities of the gauge dyonic string

Heterotic strings on R 6 × K3 generically appear to undergo some interesting new phase transition at that value of the string coupling for which one of the six-dimensional gauge field kinetic energies changes sign. An exception is the E 8 × E 8 string with equal instanton numbers in the two E 8's, which admits a heterotic/heterotic self-duality. In this paper, we generalize the dyonic string solution of the six-dimensional heterotic string to include non-trivial gauge field configurations corresponding to self-dual Yang-Mills instantons in the four transverse dimensions. We find that vacua which undergo a phase transition always admit a string solution exhibiting a naked singularity, whereas for vacua admitting a self-duality the solution is always regular. When there is a phase transition, there exists a choice of instanton numbers for which the dyonic string is tensionless and quasi-anti-self-dual at that critical value of the coupling. For an infinite subset of the other choices of instanton number, the string will also be tensionless, but all at larger values of the coupling.

Ground-state projection multigrid for propagators in four-dimensional SU(2) gauge fields

The ground-state projection multigrid method is studied for computations of slowly decaying bosonic propagators in four-dimensional SU(2) lattice gauge theory. The defining eigenvalue equation for the restriction operator C is solved exactly. With the Galerkin choice A = C ∗ for the interpolation operator, the critical exponent z is not reduced in nontrivial gauge fields. Nevertheless, a considerable speedup is obtained compared to conventional relaxation, and the conjugate gradient algorithm is outperformed. Simulations with an “optimal” interpolation kernel A eliminate critical slowing down for any value of the gauge coupling. This proves that ground-state projection is a good choice of C (i.e., of the blockspin).

Holonomy and lattice simulation of topologically non-trivial gauge fields

The holonomy of a gauge configuration is calculated from the transition function and the path-ordered exponential of the gauge potential for the case of general base manifolds. A general rule for the lattice simulation of gauge fields is proposed by explicitly giving the link variables. It is applied to a cold-start Monte Carlo study of the U(1) gauge field on a torus and the average energy of a definite topological configuration is calculated.

A simplified model for orbifold compactification

The problem of vacuum stability in orbifold compactification is addressed. The spectrum of fermions and the effective potential in a simple model of compactification on the T2/Z2 orbifold with a topologically nontrivial gauge field background is calculated.

Construction of a Family of Non-Self-Dual Gauge Fields

Using the generalization of vector bundles by reflexive sheaves recently introduced by R. Hartshorne in [2] we construct a $15$-dimensional family of nontrivial complex gauge fields $(U,E,\nabla )$ which are not self-dual nor anti-self-dual. ($U$ is an affine neighborhood in ${Q_4} = \operatorname {Gr} (2,{{\mathbf {C}}^4})$ of the stereographic compactification ${S^4}$ of ${\mathbb {R}^4}$, $E$ is a vector bundle on $U$ and $\nabla$ is a connection on it whose curvature $\phi$ satisfies the inequalities ${}^{\ast }\phi \ne \phi$ and ${}^{\ast }\phi \ne - \phi$.)

Construction of a family of non-self-dual gauge fields

Using the generalization of vector bundles by reflexive sheaves recently introduced by R. Hartshorne in [2] we construct a 15 15 -dimensional family of nontrivial complex gauge fields ( U , E , ∇ ) (U,E,\nabla ) which are not self-dual nor anti-self-dual. ( U U is an affine neighborhood in Q 4 = Gr ⁡ ( 2 , C 4 ) {Q_4} = \operatorname {Gr} (2,{{\mathbf {C}}^4}) of the stereographic compactification S 4 {S^4} of R 4 {\mathbb {R}^4} , E E is a vector bundle on U U and ∇ \nabla is a connection on it whose curvature ϕ \phi satisfies the inequalities ∗ ϕ ≠ ϕ {}^{\ast }\phi \ne \phi and ∗ ϕ ≠ − ϕ {}^{\ast }\phi \ne - \phi .)

Stability of higher-dimensional Einstein-Yang-Mills theories on symmetric spaces

We study (4 + d)-dimensional Einstein-Yang-Mills theories with arbitrary gauge groups, G YM. The theory is compactified on a d-dimensional symmetric coset space G H with a symmetric, topologically non-trivial classical gauge field, embedded in an H-subgroup of the Yang-Mills group. These theories are known to be classically stabilized by gravity if G YM = H, G H is a sphere and d ≠ 3. We study classical instabilities caused by embedding H in a larger gauge group. The small fluctuation spectrum is completely calculable, and leads to a stability condition. For two-dimensional spheres this condition is precisely the Brandt-Neri stability condition for non-abelian monopole fields. For four-spheres we find stability for SU(2) instantons embedded in arbitrary gauge groups and we reproduce the fluctuation spectrum around instantons. For higher-dimensional spheres the stable solutions of this type are completely classified, and occur only for d = 5, 6, 8, 9, 10, 12 and 16. The results show a remarkable agreement with expected topological stability. We also give a few examples with other symmetric spaces, such as CP n , where the stability criterion appears less restrictive.

Solutions to Yang-Mills field equations in eight dimensions and the last Hopf map

We will show that the Hopf map\(S^{15} \xrightarrow{{S^7 }}S^8\) admits a sourceless, topologically non-trivial gauge field. This result will be cast in the form of a solution to eight dimensional Euclidean Yang-Mills field equations with topological chargeQ=1. This solution is Spin (9) symmetric and leads to a new generalized duality conditionF∧F=±(F∧F)*.

Vacuum energy in the presence of dielectric and conducting surfaces: P. Candelas. Center for Theoretical Physics, The University of Texas at Austin, Austin, Texas 78712

We study (4 + d)-dimensional Einstein-Yang-Mills theories with arbitrary gauge groups, GYM. The theory is compactified on a d-dimensional symmetric coset space GH with a symmetric, topologically non-trivial classical gauge field, embedded in an H-subgroup of the Yang-Mills group. These theories are known to be classically stabilized by gravity if GYM = H, GH is a sphere and d ≠ 3. We study classical instabilities caused by embedding H in a larger gauge group. The small fluctuation spectrum is completely calculable, and leads to a stability condition. For two-dimensional spheres this condition is precisely the Brandt-Neri stability condition for non-abelian monopole fields. For four-spheres we find stability for SU(2) instantons embedded in arbitrary gauge groups and we reproduce the fluctuation spectrum around instantons. For higher-dimensional spheres the stable solutions of this type are completely classified, and occur only for d = 5, 6, 8, 9, 10, 12 and 16. The results show a remarkable agreement with expected topological stability. We also give a few examples with other symmetric spaces, such as CPn, where the stability criterion appears less restrictive.