Deformed Hermitian Yang–Mills equation


In mathematics and theoretical physics, and especially gauge theory, the deformed Hermitian Yang–Mills equation is a differential equation describing the equations of motion for a D-brane in the B-model of string theory. The equation was derived by Mariño-Minasian-Moore-Strominger in the case of Abelian gauge group, 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 -form. Namely, suppose is a Kähler manifold and is a class. The case of a line bundle consists of setting where is the first Chern class of a holomorphic line bundle. Suppose that and consider the topological constant
Notice that depends only on the class of and. Suppose that. Then this is a complex number
for some real and angle which is uniquely determined.
Fix a smooth representative differential form in the class. For a smooth function write, and notice that. The deformed Hermitian Yang–Mills equation for with respect to 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 such that. This is in analogy to the related problem of finding Kähler-Einstein metrics by looking for metrics solving the Einstein equation, subject to the condition that is a Kähler potential.

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 by for a positive integer, and allows. Notice that the phase for depends on. In fact,, and we can expand
Here we see that
and we see the dHYM equation for takes the form
for some topological constant determined by. Thus we see the leading order term in the dHYM equation is
which is just the HYM equation.

Local form

The dHYM equation may also be written in local coordinates. Fix and holomorphic coordinates such that at the point, we have
Here for all as we assumed 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. 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 is any analytic subvariety of dimension. Define the central charge by
When the dimension of is 2, Collins-Jacob-Yau show that if, then there exists a solution of the dHYM equation in the class if and only if for every curve we have
In the specific example where, the blow-up of complex projective space, Jacob-Sheu show that admits a solution to the dHYM equation if and only if and for any, we similarly have
It has been shown by Gao Chen that in the so-called supercritical phase, where, 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 is replaced by for a small real number and one allows.
In general it is conjectured that the existence of solutions to the dHYM equation for a class should be equivalent to the Bridgeland stability of the line bundle. 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 should correspond to Π-stability.

Relation to string theory

predicts that spacetime is 10-dimensional, consisting of a Lorentzian manifold of dimension 4 along with a Calabi-Yau manifold of dimension 6. 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, 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. 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 and a holomorphic vector bundle over it, possibly with a compatible Chern connection on the bundle.
This Chern connection arises from a choice of Hermitian metric on, with corresponding connection and curvature form. Ambient on the spacetime there is also a B-field or Kalb–Ramond field , which is the string theoretic equivalent of the classical background electromagnetic field. Mathematically the B-field is a gerbe or bundle gerbe over spacetime, which means consists of a collection of two-forms for an open cover 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. This B-field has the property that when pulled back along the inclusion map the gerbe is trivial, which means the B-field may be identified with a globally defined two-form on, written. The differential form discussed above in this context is given by, and studying the dHYM equations in the special case where or equivalently should be seen as turning the B-field off or setting, 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 in spacetime equipped with a B-field, 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, special Lagrangian submanifolds of 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.