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.
), and by Leung–Yau–Zaslow[2] using mirror symmetry from the corresponding equations of motion for D-branes in the A-model of string theory.
In this section we present the dHYM equation as explained in the mathematical literature by Collins-Xie-Yau.
[3] The deformed Hermitian–Yang–Mills equation is a fully non-linear partial differential equation for a Hermitian metric on a line bundle over a compact Kähler manifold, or more generally for a real
The dHYM equation may also be written in local coordinates.
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.
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.
, then there exists a solution of the dHYM equation in the class
, the blow-up of complex projective space, Jacob-Sheu show that
, 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.
[8] 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
In general it is conjectured that the existence of solutions to the dHYM equation for a class
[5][6] 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.
[9] 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
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
[10] 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
This Chern connection arises from a choice of Hermitian metric
(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
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).
[12] 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
, which in string theory corresponds to a spacetime with no background higher electromagnetic field.
, and is derived from the corresponding equations of motion for A-branes through mirror symmetry.
[1][2] Mathematically the A-model describes D-branes as elements of the Fukaya category of
equipped with a flat unitary line bundle over them, and the equations of motion for these A-branes is understood.
