Year: 2006
Type: article
Abstract: We study pseudo Yang–Mills fields on a compact strictly pseudoconvex CR manifold M, i.e., the critical points of the functional PYM(D)=12∫M∥πHRD∥2θ∧(dθ)n, where D is a connection in a Hermitian CR holomorphic vector bundle (E,h)→M. Let Ω={φ<0}⊂Cn be a smoothly bounded strictly pseuodoconvex domain and g the Bergman metric on Ω. We show that boundary values Db of Yang–Mills fields D on (Ω,g) are pseudo Yang–Mills fields on ∂Ω, provided that iTRDb=0 and iNRD=0 on H(∂Ω). If S1→C(M)→πM is the canonical circle bundle and π*D is a Yang–Mills field with respect to the Fefferman metric Fθ of (M,θ) then D is a pseudo Yang–Mills field on M. The Yang–Mills equations δπ*DRπ*D=0 project on the Euler–Lagrange equations δbDRD=0 of the variational principle δPYM(D)=0, provided that iTRD=0. When M has vanishing pseudohermitian Ricci curvature the pullback π*D of the (CR invariant) Tanaka connection D of (E,h) is a Yang–Mills field on C(M). We derive the second variation formula {d2PYM(Dt)∕dt2}t=0=∫M⟨SbD(φ),φ⟩θ∧(dθ)n, Dt=D+At [provided that D is a pseudo Yang–Mills field and φ≡{dAt∕dt}t=0∊Ker(δD)], and show that SbD(φ)≡ΔbDφ+RbD(φ), φ∊Ω0,1[Ad(E)], is a subelliptic operator.
Source: Journal of Mathematical Physics
DOI: 10.1063/1.2222082
Authors: Elisabetta Barletta, Sorin Dragomir, Hajime Urakawa
Institutions: University of Basilicata, Tohoku University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1965131359