We study pseudo Yang-Mills fields on a compact strictly pseudoconvex CR
manifold.Comment: 52 page

Barletta E.

Beem J. K.

Dragomir S.

Elisabetta Barletta

Hajime Urakawa

Jerison D.

Kobayashi S.

Lee J. M.

O’Neill B.

Sorin Dragomir

Tanaka N.

Urakawa H.

Webster S. M.

English

arXiv

Crossref

Yang–Mills fields on CR manifolds

We study pseudo Yang–Mills fields on a compact strictly pseudoconvex CR manifold $M$, i.e., the critical points of the functional $\mathcal{P Y M}(D)= \frac{1}{2} \int_M ||\pi_H R^D ||^2 \theta \wedge (d \theta )^n$ where $D$ is a connection in a Hermitian CR holomorphic vector bundle $(E,h) \to M$. Let $\Omega = \{ \varphi < 0\} \subset \mathbb{C}^n$ be a smoothly bounded strictly pseuodoconvex domain and $g$ the Bergman metric on $\Omega$.  We show that boundary values $D_b$ of Yang-Mills fields $D$ on $(\Omega , g)$ are pseudo Yang-Mills fields on $\partial \Omega$,  provided that $i_T R^{D_b}=0$ and $i_N R^D = 0$ on $H(\partial \Omega)$.
If $S^1 \to C(M) \to M$ is the canonical circle bundle and $\pi^* D$ is a Yang-Mills field with respect to the Fefferman metric $F_\theta$ of $(M, \theta)$, then $D$ is a pseudo Yang-Mills field on $M$. The Yang-Mills equations $\delta^{\pi^* D} R^{\pi^* D} = 0$  project on the Euler-Lagrange equations $\delta_b^D R^D = 0$ of the variational principle $ \delta \mathcal{P Y M} (D) = 0$, provided that $i_T R^D =0$. 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 $\{d^2 \mathcal{P Y M} (D^t)/ d t^2\}_{t=0} = \int_M \langle S_b^D (\varphi) , \varphi \rangle \theta \wedge (d \theta )^n$, $D^t = D + A^t$ [provided that $D$ is a pseudo Yang-Mills field and $\varphi \equiv \{ d A^t/dt\}_{t=0} \in \mathrm{Ker} (\delta^D)$], and show that  $S_b^D (\varphi) \equiv \Delta_b^D \varphi + \mathcal{R}_b^D (\varphi )$ $\varphi \in \Omega^{0,1} [\mathrm{Ad}(E)]$, is a subelliptic operator

BARLETTA, Elisabetta

DRAGOMIR, Sorin

H. URAKAWA

Archivio della Ricerca - Università della Basilicata

Yang-Mills fields on CR manifolds

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.240.3742

Yang-Mills fields on CR manifolds

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Archivio della Ricerca - Università della Basilicata