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• W3137397313 endingPage "108515" @default.
• W3137397313 startingPage "108515" @default.
• W3137397313 abstract "We can define the “volume” V for Hermitian connections on a Hermitian complex line bundle over a Riemannian manifold X, which can be considered to be the “mirror” of the standard volume for submanifolds. This is called the Dirac-Born-Infeld (DBI) action in physics. In this paper, (1) we introduce the negative gradient flow of V, which we call the line bundle mean curvature flow. Then, we show the short-time existence and uniqueness of this flow. When X is Kähler, we relate the negative gradient of V to the angle function and deduce the mean curvature for Hermitian metrics on a holomorphic line bundle defined by Jacob and Yau. (2) We relate the functional V to a deformed Hermitian Yang–Mills (dHYM) connection, a deformed Donaldson–Thomas connection for a G2-manifold (a G2-dDT connection), a deformed Donaldson–Thomas connection for a Spin(7)-manifold (a Spin(7)-dDT connection), which are considered to be the “mirror” of special Lagrangian, (co)associative and Cayley submanifolds, respectively. When X is a compact Spin(7)-manifold, we prove the “mirror” of the Cayley equality, which implies the following. (a) Any Spin(7)-dDT connection is a global minimizer of V and its value is topological. (b) Any Spin(7)-dDT connection is flat on a flat line bundle. (c) If X is a product of S1 and a compact G2-manifold Y, any Spin(7)-dDT connection on the pullback of the Hermitian complex line bundle over Y is the pullback of a G2-dDT connection modulo closed 1-forms. We also prove analogous statements for G2-manifolds and Kähler manifolds of dimension 3 or 4." @default.
• W3137397313 created "2021-03-29" @default.
• W3137397313 creator A5083449862 @default.
• W3137397313 creator A5089479384 @default.
• W3137397313 date "2022-08-01" @default.
• W3137397313 modified "2023-10-18" @default.
• W3137397313 title "Mirror of volume functionals on manifolds with special holonomy" @default.
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• W3137397313 doi "https://doi.org/10.1016/j.aim.2022.108515" @default.
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