Deformed Hermitian Yang–Mills equation
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 B-model (commonly called a B-brane) of string theory. The equation was derived by Mariño-Minasian-Moore-Strominger in the case of Abelian gauge group (the unitary group formula_1), and by Leung–Yau–Zaslow using mirror symmetry from the corresponding equations of motion for D-branes in the A-model of string theory. Definition. In this section we present the dHYM equation as  explained in the mathematical literature by Collins-Xie-Yau. 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 Kähler manifold, or more generally for a real formula_2-form. Namely, suppose formula_3 is a Kähler manifold and formula_4 is a class. The case of a line bundle consists of setting formula_5 where formula_6 is the first Chern class of a holomorphic line bundle formula_7. Suppose that formula_8 and consider the topological constant Notice that formula_10 depends only on the class of formula_11 and formula_12. Suppose  that formula_13. Then this is a complex number for some real formula_15 and angle formula_16 which is uniquely determined. Fix a smooth representative differential form formula_12 in the class formula_18. For a smooth function formula_19 write formula_20, and notice that formula_21. The deformed Hermitian Yang–Mills equation for formula_3 with respect to formula_18 is The second condition should be seen as a positivity condition on solutions to the first equation. That is, one looks for solutions to the equation formula_25 such that formula_26. This is in analogy to the related problem of finding Kähler-Einstein metrics by looking for metrics  formula_27 solving the Einstein equation, subject to the condition that formula_28 is a Kähler potential (which is a positivity condition on the form formula_27). Discussion. Relation to Hermitian Yang–Mills equation. The dHYM equations can be transformed in several ways to illuminate several key properties of the equations. First, simple algebraic manipulation shows that the dHYM equation may be equivalently written In this form, it is possible to see the relation between the dHYM equation and the regular Hermitian Yang–Mills equation. In particular, the dHYM equation should look like the regular HYM equation in the so-called large volume  limit. Precisely, one replaces the Kähler form formula_11 by formula_32 for a positive integer formula_33, and allows formula_34. Notice that the phase formula_35 for formula_36 depends on formula_33. In fact, formula_38, and we can expand Here we see that and we see the dHYM equation for formula_32 takes the form for some topological constant formula_43 determined by formula_44. Thus we see the leading order term in the dHYM equation is which is just the HYM equation (replacing formula_12 by formula_47 if necessary). Local form. The dHYM equation may also be written in local coordinates. Fix formula_48 and holomorphic coordinates formula_49  such that at the point formula_50, we have Here formula_52 for all formula_53 as we assumed formula_12 was a real form. Define the Lagrangian phase operator to be Then simple computation shows that the dHYM equation in these local coordinates takes the form where formula_57. In this form one sees that the dHYM equation is fully non-linear and elliptic. Solutions. It is possible to use algebraic geometry to study the existence of solutions to the dHYM equation, as demonstrated by the work of Collins–Jacob–Yau and Collins–Yau. Suppose that formula_58 is any analytic subvariety of dimension formula_50 . Define the central charge formula_60 by When the dimension of formula_62 is 2, Collins–Jacob–Yau show that if formula_63, then there exists a solution of the dHYM equation in the class formula_64 if and only if for every curve formula_65 we have In the specific example where formula_67, the blow-up of complex projective space, Jacob-Sheu show that formula_18 admits a solution to the dHYM equation if and only if formula_69 and for any formula_58, we similarly have It has been shown by Gao Chen that in the so-called supercritical phase, where formula_72, algebraic conditions analogous to  those above imply the existence of a solution to the dHYM equation. This is achieved through comparisons between the dHYM and the so-called J-equation in Kähler geometry. The J-equation appears as the *small volume limit* of the dHYM equation, where formula_11 is replaced by formula_74 for a small real number formula_75 and one allows formula_76. In general it is conjectured that the existence of solutions to the dHYM equation for a class formula_77 should be equivalent to the Bridgeland stability of the line bundle formula_78. This is motivated both from comparisons with similar theorems in the non -deformed case, such as the famous Kobayashi–Hitchin correspondence which asserts that solutions exist to the HYM equations if and only if the underlying bundle is slope stable. It is also motivated by physical reasoning coming from string theory, which predicts that physically realistic B-branes (those admitting solutions to the dHYM equation for example) should correspond to Π-stability. Relation to string theory. Superstring theory predicts that spacetime is 10-dimensional, consisting of a Lorentzian manifold of dimension 4 (usually assumed to be Minkowski space or De sitter or anti-De Sitter space) along with a Calabi–Yau manifold  formula_62 of dimension 6 (which therefore has complex dimension 3). In this string theory open strings must satisfy Dirichlet boundary conditions on their endpoints. These conditions require that the end points of the string lie on so-called D-branes (D for Dirichlet), and there is much mathematical interest in describing these branes. In the B-model of topological string theory, homological mirror symmetry suggests D-branes should be viewed as elements of the derived category of coherent sheaves on the Calabi–Yau 3-fold formula_62. This characterisation is abstract, and the case of primary importance, at least for the  purpose of phrasing the dHYM equation, is when a B-brane consists of a holomorphic submanifold formula_81 and a holomorphic vector bundle formula_82 over it (here formula_83 would be viewed as the support of the coherent sheaf formula_84 over formula_62), possibly with a compatible Chern connection on the bundle. This Chern connection arises from a choice of Hermitian metric formula_86 on formula_84, with corresponding connection formula_88 and curvature form formula_47. Ambient on the spacetime there is also a B-field or Kalb–Ramond field formula_90 (not to be confused with the B in B-model), which is the string theoretic  equivalent of the classical background electromagnetic field (hence the use of formula_90, which commonly denotes the magnetic field strength). Mathematically the B-field is a gerbe or bundle gerbe over spacetime, which means formula_90 consists of a collection of two-forms formula_93 for an open cover formula_94 of spacetime, but these forms may not agree on overlaps, where they must satisfy cocycle conditions in analogy with the transition functions of line bundles (0-gerbes). This B-field has the property that when pulled back along the inclusion map formula_95 the gerbe is trivial, which means the B-field may be  identified with a globally defined two-form on formula_83, written formula_97. The differential form formula_12 discussed above in this context is given by formula_99, and studying the dHYM equations in the special case where formula_100 or equivalently formula_77 should be seen as "turning the B-field off" or setting formula_102, which in string theory corresponds to a spacetime with no background higher electromagnetic field. The dHYM equation describes the equations of motion for this D-brane formula_103 in spacetime equipped with a B-field formula_90, and is derived from the corresponding equations of motion for A-branes through mirror symmetry . Mathematically the A-model describes D-branes as elements of the Fukaya category of formula_62, special Lagrangian submanifolds of formula_62 equipped with a flat unitary line bundle over them, and the equations of motion for these A-branes is understood. In the above section the dHYM equation has been phrased for the D6-brane formula_107.