Self-dual solutions to pseudo Yang-Mills equations. (English) Zbl 1320.32043
Summary: We study pseudo Yang-Mills fields on a compact \(5\)-dimensional strictly pseudoconvex CR manifold \(M\), i.e., critical points to the functional
\[
\mathcal{Y M}_b(D) = \frac{1}{2} \int_M \| \Pi_H R^D \|^2 \theta \wedge(d \theta)^2
\]
on the space \(\mathcal{C}(E, h)\) of all connections \(D\) on a Hermitian vector bundle \((E, h)\) over \(M\), such that \(D h = 0\). If
\[ \mathcal{A} = \big\{D \in \mathcal{C}(E, h) : \xi \rfloor R^D = 0, G_\theta^\ast(\operatorname{Tr}(R^D), d \theta) = 0 \big\} \]
and \(D \in \mathcal{A}\) is an absolute minimum to \(\mathcal{Y M}_b : \mathcal{A} \to \mathbb{R}\), then (i) \(\Delta_b \operatorname{Tr}(R^D) = 0\) and (ii) \(D\) is self-dual or anti-self-dual according to the sign of
\[ c_2(\theta, D) = \int_M \theta \wedge \Big\{\mathbf{P}_2(D) - \frac{m - 1}{2 m} \mathbf{P}_1(D) \wedge \mathbf{P}_1(D) \Big\} \] [where \(\mathbf{P}_k(D)\) is the \(k\)-th Chern form of \((E, D)\)] and provided \(c_2(\theta, D)\) is constant on \(\mathcal{A}\).
on the space \(\mathcal{C}(E, h)\) of all connections \(D\) on a Hermitian vector bundle \((E, h)\) over \(M\), such that \(D h = 0\). If
\[ \mathcal{A} = \big\{D \in \mathcal{C}(E, h) : \xi \rfloor R^D = 0, G_\theta^\ast(\operatorname{Tr}(R^D), d \theta) = 0 \big\} \]
and \(D \in \mathcal{A}\) is an absolute minimum to \(\mathcal{Y M}_b : \mathcal{A} \to \mathbb{R}\), then (i) \(\Delta_b \operatorname{Tr}(R^D) = 0\) and (ii) \(D\) is self-dual or anti-self-dual according to the sign of
\[ c_2(\theta, D) = \int_M \theta \wedge \Big\{\mathbf{P}_2(D) - \frac{m - 1}{2 m} \mathbf{P}_1(D) \wedge \mathbf{P}_1(D) \Big\} \] [where \(\mathbf{P}_k(D)\) is the \(k\)-th Chern form of \((E, D)\)] and provided \(c_2(\theta, D)\) is constant on \(\mathcal{A}\).
MSC:
| 32V20 | Analysis on CR manifolds |
| 53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |
| 58E30 | Variational principles in infinite-dimensional spaces |
References:
| [1] | Barletta, E.; Dragomir, S.; Urakawa, H., Yang-Mills fields on CR manifolds, J. Math. Phys., 47, 8 (2006), 083504. pp. 41 · Zbl 1112.53014 |
| [3] | Dragomir, S.; Ichiyama, T.; Urakawa, H., Yang-Mills theory and conjugate connections, Differential Geom. Appl., 18, 229-238 (2003) · Zbl 1040.53033 |
| [4] | Dragomir, S.; Tomassini, G., (Differential Geometry and Analysis on CR Manifolds. Differential Geometry and Analysis on CR Manifolds, Progress in Mathematics, vol. 246 (2006), Birkhäuser: Birkhäuser Boston, Basel, Berlin) · Zbl 1099.32008 |
| [5] | Dragomir, S.; Urakawa, H., On the inhomogeneous Yang-Mills equation \(d_D^\ast R^D = f\), Interdiscip. Inform. Sci., 6, 1, 41-52 (2000) · Zbl 0959.58020 |
| [9] | Gharib, G. M., Canonical reduction of the self-dual Yang-Mills equations to inhomogeneous nonlinear Schrödinger equation and exact solutions, Appl. Math. Sci., 2, 49, 2431-2442 (2008) · Zbl 1187.58034 |
| [10] | Graham, R. C., On Sparling’s characterization of Fefferman metrics, Amer. J. Math., 109, 853-874 (1987) · Zbl 0663.53050 |
| [11] | Contact metric 5 manifolds, CR twistor spaces, and integrability, J. Math. Phys., 44, 1, 366-368 (2003), Erratum |
| [12] | Itoh, M., Sasakian manifolds, Hodge decomposition and Milnor algebras, Kyushu J. Math., 58, 121-140 (2004) · Zbl 1076.53056 |
| [13] | Jost, J., Riemannian Geometry and Geometric Analysis (2002), Springer: Springer Berlin, Heidelberg, New York · Zbl 1034.53001 |
| [14] | Kostant, B., (Quantization and Unitary Representations. Quantization and Unitary Representations, Lecture Notes in Math., vol. 170 (1970), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York), 87-208 · Zbl 0223.53028 |
| [15] | LeBrun, C., Twistor CR manifolds and three-dimensional conformal geometry, Trans. Amer. Math. Soc., 284, 2, 601-616 (1984) · Zbl 0513.53006 |
| [16] | Lee, J. M., Pseudo-Einstein structures on CR manifolds, Amer. J. Math., 110, 157-178 (1988) · Zbl 0638.32019 |
| [17] | Lee, J. M.; Melrose, R., Boundary behaviour of the complex Monge-Ampère equation, Acta Math., 148, 159-192 (1982) · Zbl 0496.35042 |
| [19] | Ohkubo, T., CR Weyl geometry and connections of the canonical bundle, Kodai Math. J., 29, 113-143 (2006), CR Weyl geometry and connections of the canonical bundle, Preprint · Zbl 1104.53042 |
| [20] | Ohkubo, T.; Sakamoto, K., CR Einstein-Weyl structures, Tsukuba J. Math., 29, 2, 309-361 (2005) · Zbl 1104.53043 |
| [21] | Tanaka, N., A Differential Geometric Study on Strongly Pseudo-Convex Manifolds (1975), Kinokuniya Book Store Co., Ltd.: Kinokuniya Book Store Co., Ltd. Kyoto · Zbl 0331.53025 |
| [22] | Vaisman, I., New examples of twisted cohomologies, Boll. Unione Mat. Ital., 7-B, 7, 355-368 (1993) · Zbl 0781.53029 |
| [23] | Webster, S. M., Pseudohermitian structures on a real hypersurface, J. Differential Geom., 13, 25-41 (1978) · Zbl 0379.53016 |
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