Gauge invariant composite fields out of connections, with examples

Abstract

In this paper we put forward a systematic and unifying approach to construct gauge invariant composite fields out of connections. It relies on the existence in the theory of a group valued field with a prescribed gauge transformation. As an illustration, we detail some examples. Two of them are based on known results: the first one provides a reinterpretation of the symmetry breaking mechanism of the electroweak part of the Standard Model of particle physics; the second one is an application to Einstein's theory of gravity described as a gauge theory in terms of Cartan connections. The last example depicts a new situation: starting with a gauge field theory on Atiyah Lie algebroids, the gauge invariant composite fields describe massive vector fields. Some mathematical and physical discussions illustrate and highlight the relevance and the generality of this approach.Comment: 22 pages, revised version to appear in Int. J. of Geometric Methods in Modern Physic

Artificial versus Substantial Gauge Symmetries: A Criterion and an Application to the Electroweak Model

“…According to (C2), the Lorentz gauge symmetry SO is then substantial. 12 10 It happens that the abelian U(1) Higgs model can be treated via dressing along the same line, as is shown explicitly in [34]. This is directly relevant to our discussion of the electroweak model in the next section.…”

“…Locally, i.e so long as one works on U ⊂ M seen both as a coordinate patch and a trivializing open set for the Lorentz bundle, one can decompose the tetrad as e = ut, where u = u a b ∈ S O(1, 3) and t = t b µ has the same d.o.f as the metric field and is such that g = t T ηt. This decomposition relies on the Schweinler-Wigner orthogonalization procedure, see [34] section 4.3 for details en references. Therefore, one has on the one hand u γ = γ −1 u, so that u is a (minimal) local SOdressing field.…”

A note on Weyl invariance in gravity and the Wess–Zumino functional

“…It is like a gauge fixing on the conformal component of the metric field. Rather, it might be viewed as a change of variables within the field space which g and τ belong to [23]. Hence, the field τ carries 1 as Weyl conformal weight and will play the rôle of dilaton as we shall see.…”

“…In fact, a u may be interpreted as a change of variable in the field space of the (a, u)'s; see discussion in [23,29, see in particular section 2 in both of those references] on what we called the dressing field method, a construction which goes back to Dirac [30] and which, in turn, enters in the construction of the Wess-Zumino functional.…”

Weyl gravity and Cartan geometry

“…If one doesn't want to be restricted to a torsion free geometry, and nevertheless wants to restore full gauge-invariance, then the so-called dressing field method is the way forward. See [26][27][28] for details, and the following for a brief recap. where vi is the anticommuting ghost field associated with infinitesimal conformal boosts.…”