Abstract
We consider Yang-Mills theory with a compact structure group G on four-dimensional de Sitter space dS4. Using conformal invariance, we transform the theory from dS4 to the finite cylinder mathcal{I} × S3, where mathcal{I} = (−π/2, π/2) and S3 is the round three-sphere. By considering only bundles P → mathcal{I} × S3 which are framed over the temporal boundary ∂ mathcal{I} × S3, we introduce additional degrees of freedom which restrict gauge transformations to be identity on ∂ mathcal{I} × S3. We study the consequences of the framing on the variation of the action, and on the Yang-Mills equations. This allows for an infinite-dimensional moduli space of Yang-Mills vacua on dS4. We show that, in the low-energy limit, when momentum along mathcal{I} is much smaller than along S3, the Yang-Mills dynamics in dS4 is approximated by geodesic motion in the infinite-dimensional space mathcal{M} vac of gauge-inequivalent Yang-Mills vacua on S3. Since mathcal{M} vac ≅ C∞(S3, G)/G is a group manifold, the dynamics is expected to be integrable.
Highlights
Which are not unity, are not a mere redundancy of the description
We show that the classical Yang-Mills dynamics in the infrared is described by geodesic motion in the infinite-dimensional group manifold C∞(S3, G)/G of based smooth maps from S3 ⊂ dS4 into the structure group G
By exploiting the conformal invariance of Yang-Mills theory in four dimensions, we reduced Yang-Mills theory on de Sitter space dS4 in a certain adiabatic limit to a onedimensional principal chiral model with the moduli space Mvac of static gauge vacua as a target space, where in particular we identified Mvac with the infinite-dimensional Lie group GSn3 ∼= C∞(S3, G)/G
Summary
We consider the bundle of groups IntP = P ×G G (G acts on itself by internal automorphisms: h → ghg−1, h, g ∈ G) associated with P , and the bundle AdP = P ×G g of Lie algebras Both of these bundles inherit their connection A from the bundle P. Transformations (2.3) on manifolds M with boundaries ∂M are naturally split into the gauge transformations G0 = {g ∈ G | g|∂M = Id}, and physical symmetries G∂M from (2.6) The latter are sometimes called “local” or “large” gauge transformations, in this paper we shall avoid this terminology, reserving any mention of the term “gauge transformation” for the transformations in the group G0, which are identity at the boundary
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