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In [[mathematics]] and [[theoretical physics]], and especially [[gauge theory]], the '''deformed Hermitian–Yang–Mills (dHYM) equation''' is a [[differential equation]] describing the [[equations of motion]] for a [[D-brane]] in the [[Topological_string_theory#B-model|B-model]] (commonly called a '''B-brane''') of [[string theory]]. The equation was derived by Mariño-Minasian-[[Greg_Moore_(physicist)|Moore]]-[[Andrew_Strominger|Strominger]]<ref>Marino, M., Minasian, R., Moore, G. and Strominger, A., 2000. Nonlinear instantons from supersymmetric p-branes. Journal of High Energy Physics, 2000(01), p.005.</ref> in the case of [[Abelian]] gauge group (the [[unitary group]] <math>\operatorname{U}(1)</math>), and by Leung-[[Shing-Tung_Yau|Yau]]-[[Eric_Zaslow|Zaslow]]<ref>Leung, N.C., Yau, S.T. and Zaslow, E., 2000. From special lagrangian to hermitian-Yang-Mills via Fourier-Mukai transform. arXiv preprint math/0005118.</ref> using [[mirror symmetry]] from the corresponding equations of motion for D-branes in the [[Topological_string_theory#A-model|A-model]] of string theory.
== Definition ==
The deformed Hermitian–Yang–Mills equation is a fully non-linear partial differental equation for a [[Hermitian metric]] on a [[line bundle]] over a [[Compact_space|compact]] [[Kähler manifold]], or more generally for a real [[Complex_differential_form|<math>(1,1)</math>-form]]. Namely, suppose <math>(X,\omega)</math> is a Kähler manifold and <math>[\alpha] \in H^{1,1}(X,\mathbb{R})</math> is a class. The case of a line bundle consists of setting <math>[\alpha]=c_1(L)</math> where <math>c_1(L)</math> is the first [[Chern class]] of a [[holomorphic line bundle]] <math>L\to X</math>. Suppose that <math>\dim X = n</math> and consider the topological constant
:<math>\hat z([\omega], [\alpha]) = \int_X (\omega + i \alpha)^n.</math>
Notice that <math>\hat z</math> depends only on the class of <math>\omega</math> and <math>\alpha</math>. Suppose that <math>\hat z\ne 0</math>. Then this is a complex number
:<math>\hat z([\omega], [\alpha]) = r e^{i \theta}</math>
for some real <math>r>0</math> and angle <math>\theta\in [0,2\pi)</math> which is uniquely determined.
Fix a smooth representative [[differential form]] <math>\alpha</math> in the class <math>[\alpha]</math>. For a smooth function <math>\phi: X \to \mathbb{R}</math> write <math>\alpha_{\phi} = \alpha + i \partial \bar \partial \phi</math>, and notice that <math>[\alpha_{\phi}] = [\alpha]</math>. The '''deformed Hermitian–Yang–Mills equation''' for <math>(X,\omega)</math> with respect to <math>[\alpha]</math> is
:<math>\begin{cases}\operatorname{Im}(e^{-i\theta} (\omega + i \alpha_{\phi})^n) = 0\\
\operatorname{Re}(e^{-i\theta} (\omega + i \alpha_{\phi})^n) > 0.\end{cases}</math>
==References==
[[Category:Geometry]]
[[Category:String theory]]
[[Category:Differential geometry]]
[[Category:Partial differential equations]]
== Definition ==
The deformed Hermitian–Yang–Mills equation is a fully non-linear partial differental equation for a [[Hermitian metric]] on a [[line bundle]] over a [[Compact_space|compact]] [[Kähler manifold]], or more generally for a real [[Complex_differential_form|<math>(1,1)</math>-form]]. Namely, suppose <math>(X,\omega)</math> is a Kähler manifold and <math>[\alpha] \in H^{1,1}(X,\mathbb{R})</math> is a class. The case of a line bundle consists of setting <math>[\alpha]=c_1(L)</math> where <math>c_1(L)</math> is the first [[Chern class]] of a [[holomorphic line bundle]] <math>L\to X</math>. Suppose that <math>\dim X = n</math> and consider the topological constant
:<math>\hat z([\omega], [\alpha]) = \int_X (\omega + i \alpha)^n.</math>
Notice that <math>\hat z</math> depends only on the class of <math>\omega</math> and <math>\alpha</math>. Suppose that <math>\hat z\ne 0</math>. Then this is a complex number
:<math>\hat z([\omega], [\alpha]) = r e^{i \theta}</math>
for some real <math>r>0</math> and angle <math>\theta\in [0,2\pi)</math> which is uniquely determined.
Fix a smooth representative [[differential form]] <math>\alpha</math> in the class <math>[\alpha]</math>. For a smooth function <math>\phi: X \to \mathbb{R}</math> write <math>\alpha_{\phi} = \alpha + i \partial \bar \partial \phi</math>, and notice that <math>[\alpha_{\phi}] = [\alpha]</math>. The '''deformed Hermitian–Yang–Mills equation''' for <math>(X,\omega)</math> with respect to <math>[\alpha]</math> is
:<math>\begin{cases}\operatorname{Im}(e^{-i\theta} (\omega + i \alpha_{\phi})^n) = 0\\
\operatorname{Re}(e^{-i\theta} (\omega + i \alpha_{\phi})^n) > 0.\end{cases}</math>
==References==
[[Category:Geometry]]
[[Category:String theory]]
[[Category:Differential geometry]]
[[Category:Partial differential equations]]
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