Yang-Mills fields on CR manifolds

We review some of the most important results obtained over the years on the study of Yang-Mills fields on the four dimensional torus at the classical level.

We derive variational formulas for the total Q-prime curvature under the deformation of strictly pseudoconvex domains in a complex manifold. We also show that the total Q-prime curvature agrees with the renormalized volume of such domains with respect to the complete Einstein-K\"ahler metric. In the appendix, by Rod Gover and the first author, we study the property of CR invariant differential operator of order 2(n+3) on 2n+1 dimensional CR manifolds that appears in the secon...

In this note, we first give a criterion of pseudo-Einstein contact forms and then affirm the CR analogue of Frankel conjecture in a closed, spherical, strictly pseudoconvex CR manifold of nonnegative pseudohermitian curvature on the space of smooth representatives of the first Kohn-Rossi cohomology group. Moreover, we obtain the CR Frankel conjecture in a closed, spherical, strictly pseudoconvex CR manifold with the vanishing first Kohn-Rossi cohomology group. In particular, ...

We consider the CR Yamabe flow on a compact strictly pseudoconvex CR manifold $M$ of real dimension $2n+1$. We prove convergence of the CR Yamabe flow when $n=1$ or $M$ is spherical.

In this paper, we consider a non-degenerate CR manifold (M,H(M),J) with a given pseudo-Hermitian 1-form {\theta}, and endow the CR distribution H(M) with any Hermitian metric h instead of the Levi form L_{{\theta}}. This induces a natural Riemannian metric g_{h,{\theta}} on M compatible with the structure. The synthetic object (M,{\theta},J,h) will be called a pseudo-Hermitian manifold, which generalizes the usual notion of pseudo-Hermitian manifold (M,{\theta},J,L_{{\theta}}...

Modelled on a real hypersurface in a quaternionic manifold, we introduce a quaternionic analogue of CR structure, called quaternionic CR structure. We define the strong pseudoconvexity of this structure as well as the notion of quaternionic pseudohermitian structure. Following the construction of the Tanaka-Webster connection in complex CR geometry, we construct a canonical connection associated with a quaternionic pseudohermitian structure, when the underlying quaternionic C...

We study the minimality of an isometric immersion of a Riemannian manifold into a strictly pseudoconvex CR manifold endowed with the Webster metric hence formulate a version of the CR Yamabe problem for CR manifolds-with-boundary. This is shown to be a nonlinear subelliptic problem of variational origin.

Let $X$ be a compact connected orientable CR manifold with the action of a connected compact Lie group $G$. Under natural pseudoconvexity assumptions we show that the CR Guillemin-Strernberg map is Fredholm at the level of Sobolev spaces of CR functions. As application we study this map for holomorphic line bundles which are positive near the inverse image of $0$ by the momentum map. We also show that "quantization commutes with reduction" for Sasakian manifolds.

Inspired by the Movshev-Mason-Skinner Cauchy-Riemann (CR) ambitwistor approach, we provide a rigorous yet elementary construction of a twisted CR holomorphic Chern-Simons action on CR ambitwistor space for maximally supersymmetric Yang-Mills theory on four-dimensional Euclidean space. The key ingredient in our discussion is the homotopy algebraic perspective on perturbative quantum field theory. Using this technology, we show that both theories are semi-classically equivalent...

In this paper, we deal with a strongly pseudoconvex almost CR manifold with a CR contraction. We will prove that the stable manifold of the CR contaction is CR equivalent to the Heisenberg group model.