Yang-Mills fields on CR manifolds

Journal Article · Tue Aug 15 04:00:00 EDT 2006 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.2222082· OSTI ID:20860767

Barletta, Elisabetta; Dragomir, Sorin; Urakawa, Hajime ^[1]

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1. Dipartimento di Matematica, Universita degli Studi della Basilicata, Campus Macchia Romana, 85100 Potenza (Italy)

We study pseudo Yang-Mills fields on a compact strictly pseudoconvex CR manifold M, i.e., the critical points of the functional PYM(D)=(1/2)M parallel {pi}{sub H}R{sup D} parallel {sup 2}{theta} and (d{theta}){sup n}, where D is a connection in a Hermitian CR holomorphic vector bundle (E,h){yields}M. Let {omega}={l_brace}{phi}<0{r_brace} subset of C{sup n} be a smoothly bounded strictly pseuodoconvex domain and g the Bergman metric on {omega}. We show that boundary values D{sub b} of Yang-Mills fields D on ({omega},g) are pseudo Yang-Mills fields on {partial_derivative}{omega}, provided that i{sub T}R{sup D{sub b}}=0 and i{sub N}R{sup D}=0 on H({partial_derivative}{omega}). If S{sup 1}{yields}C(M){yields}{sup {pi}}M is the canonical circle bundle and {pi}*D is a Yang-Mills field with respect to the Fefferman metric F{sub {theta}} of (M,{theta}) then D is a pseudo Yang-Mills field on M. The Yang-Mills equations {delta}{sup {pi}}{sup *D}R{sup {pi}}{sup *D}=0 project on the Euler-Lagrange equations {delta}{sub b}{sup D}R{sup D}=0 of the variational principle {delta}PYM(D)=0, provided that i{sub T}R{sup D}=0. When M has vanishing pseudohermitian Ricci curvature the pullback {pi}*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 {l_brace}d{sup 2}PYM(D{sup t})/dt{sup 2}{r_brace}{sub t=0}=M<S{sub b}{sup D}({phi}),{phi}>{theta} and (d{theta}){sup n}, D{sup t}=D+A{sup t} [provided that D is a pseudo Yang-Mills field and {phi}{identical_to}{l_brace}dA{sup t}/dt{r_brace}{sub t=0}(set-membership sign)Ker({delta}{sup D})], and show that S{sub b}{sup D}({phi}){identical_to}{delta}{sub b}{sup D}{phi}+R{sub b}{sup D}({phi}), {phi}(set-membership sign){omega}{sup 0,1}[Ad(E)], is a subelliptic operator.

• OSTI ID: 20860767
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 8 Vol. 47; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
CONVEX MANIFOLDS
LAGRANGE EQUATIONS
MATHEMATICAL OPERATORS
METRICS
RICCI TENSOR
VARIATIONAL METHODS
VECTORS
YANG-MILLS THEORY