[Paper Review] Twisted Gauge Fields

Jordan François|arXiv (Cornell University)|Jul 19, 2019

Black Holes and Theoretical Physics35 references4 citations

TL;DR

This paper introduces a generalized gauge theory framework based on twisted associated bundles constructed via group action cocycles rather than standard group representations. It defines twisted connections and covariant derivatives, generalizing Yang-Mills and Cartan geometries, and shows that conformal tractors, twistors, and anomalies in quantum field theory arise naturally within this formalism, revealing a deeper geometric structure underlying known physical theories.

ABSTRACT

We propose a generalisation of the notion of associated bundles to a principal bundle constructed via group action cocycles rather than via mere representations of the structure group. We devise a notion of connection generalising Ehresmann connection on principal bundles, giving rise to the appropriate covariant derivative on sections of these twisted associated bundles (and on twisted tensorial forms). We study the action of the group of vertical automorphisms on the objects introduced (active gauge transformations). We also provide the gluing properties of the local representatives (passive gauge transformations). The latter are generalised gauge fields: They satisfy the gauge principle of physics, but are of a different geometric nature than standard Yang-Mills fields. We also examine the conditions under which this new geometry coexists and mixes with the standard one. We show that (standard) conformal tractors and Penrose's twistors can be seen as simple instances of this general picture. We also indicate that the twisted geometry arises naturally in the definition and study of anomalies in quantum gauge field theory.

Motivation & Objective

• To generalize the notion of associated bundles beyond standard group representations by using group action cocycles.
• To define a new class of connections—twisted connections—that generalize Ehresmann connections and induce covariant derivatives on twisted bundles.
• To unify the geometric description of known structures like conformal tractors and Penrose twistors within a single framework.
• To explore the interplay between twisted geometry and standard gauge theory, particularly in mixed settings involving both standard and twisted fields.
• To establish the relevance of this formalism for anomalies in quantum gauge field theory and potential extensions to supergeometry and quantization.

Proposed method

• Constructs twisted associated vector bundles using a principal bundle P with structure group H and a cocycle C: H × H → G, leading to a G-bundle Q = P ×C(H) G.
• Defines a twisted connection on P as a distribution of horizontal subspaces satisfying cocycle-equivariance, generalizing the Ehresmann connection.
• Derives the covariant derivative on sections of twisted bundles and computes its curvature using the twisted connection form.
• Introduces active gauge transformations as actions of vertical automorphisms on twisted bundles, and passive transformations via local gluing rules.
• Develops a mixed geometry combining standard and twisted bundles, with mixed connections and curvature, allowing coexistence of both structures.
• Applies the formalism to known geometric objects: conformal tractors and local twistors are shown to be instances of twisted gauge fields.

Experimental results

Research questions

• RQ1 How can the standard construction of associated bundles be generalized beyond group representations to include group action cocycles?
• RQ2 What is the geometric structure of connections and covariant derivatives in this twisted setting, and how do they generalize Yang-Mills and Cartan connections?
• RQ3 In what way do conformal tractors and Penrose twistors emerge as special cases of this twisted gauge theory?
• RQ4 How can twisted and standard gauge fields coexist and interact in a unified geometric framework?
• RQ5 What is the role of this twisted geometry in the context of anomalies in quantum field theory, particularly in relation to Wess-Zumino terms?

Key findings

• Conformal tractors and Penrose twistors are shown to be natural instances of twisted gauge fields, providing a unified geometric interpretation.
• The standard associated bundle Q = P ×H G is isomorphic to the twisted bundle P ×C(H) V, but the twisted construction reveals a more fundamental geometric role for cocycles.
• Twisted connections induce covariant derivatives on twisted tensorial forms, generalizing the minimal coupling of matter fields to gauge interactions.
• Mixed gauge theories—combining standard and twisted fields—can be consistently formulated, with mixed connections and curvature defined via compatible structures.
• The framework naturally accommodates anomalies in QFT, particularly Wess-Zumino terms, which are shown to be sections of twisted line bundles requiring twisted connections for proper covariant differentiation.
• The BRST formalism can be extended to twisted gauge theories, suggesting that path integral and canonical quantization methods remain applicable, though new cohomological structures may emerge.