Yang–Mills theory for bundle gerbes

Abstract

Given a bundle gerbe with connection on an oriented Riemannian manifold of dimension at least equal to 3, we formulate and study the associated Yang-Mills equations. When the Riemannian manifold is compact and oriented, we prove the existence of instanton solutions to the equations and also determine the moduli space of instantons, thus giving a complete analysis in this case. We also discuss duality in this context.2000 Mathematics Subject Classification. 70S15, 81T13. Key words and phrases. bundle gerbe, abelian Yang-Mills theory.

Higher abelian gauge theory associated to gerbes on noncommutative deformed M5-branes and S-duality

“…Then, as in [24], the total moduli space M = B∈B M B is diffeomorphic to a torus T b 2 (M ) -bundle over the affine space B.…”

“…Then, with this assumption and in our case of an even dimension these take the form (see e.g. [32]) 24) which is part of the expansion of the heat kernel, namely the sections u n p ∈ C ∞ (M 6 θ , End(Λ n M 6 θ )), n = 0, 1, · · · appear in the coefficients of the short time asymptotic expansion of the heat kernel of the Laplacians on p-forms These coefficients can be calculated in terms of the various curvatures of the manifold, which we assume extend to deformed case (M 6 θ , g). These can be found explicitly in [19].…”

“…with the parameter Θ set to zero. This then reduces to the Yang-Mills gerbe theory described by [24]. Furthermore, we will first work at the level of differential forms, where the cohomology class of the gerbe is trivial.…”

“…Let G be an abelian gerbe on The moduli space of the solutions to the equation of motion of the system can be described explicitly in the classical case [24]. Our current discussion is a straightforward extension to the deformed case.…”

“…[5,6,28] for extensive description of gerbes). On the other hand, an action principle for gerbes was studied in [24] S(B) = where B is the B-field of the gerbe whose 3-curvature is H B = dB with Dixmier-Douady class [H B ] ∈ H 3 (M; Z). This theory only has a Z 2 -symmetry given by B → −B.…”