2.6 Deformed Hermitian Yang-Mills [048K]

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2.6 Deformed Hermitian Yang-Mills

Much recent research concentrates on a more nonlinear cousin of HYM, known as the deformed Hermitian Yang-Mills equation (dHYM), expertly surveyed in [18]. Although the techniques involved in dHYM are more akin to other areas of Kähler geometry, such as Kähler-Einstein metrics and the J-equation ²⁹²⁹ 29 Indeed, the recent breakthrough of Gao Chen [16] on dHYM is largely based on the methods he developed for the J-equation. The work of Collins et al [23] has strong analogy with the homogeneous complex Monge-Ampère equation important for CSCK metrics., its motivation is in part to find an improved mirror analogue of special Lagrangians, with the distant goal of constructing the Bridgeland stability on the B-side of the mirror.

The equation can be motivated differential geometrically from semiflat mirror symmetry [22]. To start, one imposes toric symmetry, so that over a local $n$ -dimensional base $D$ , the $A$ and $B$ sides are respectively the $T^{n}$ -bundles $T^{*}D/\Lambda$ and $TD/\Lambda^{\vee}$ equipped with the canonical symplectic/complex structure, where $\Lambda$ and $\Lambda^{\vee}$ are dual lattices, so that the torus fibres are naturally dual. A section $L$ of $T^{*}D/\Lambda$ over $D$ fibrewise determines a flat $U(1)$ -connection on the corresponding fibre of $TD/\Lambda$ , and the Lagrangian condition on $L$ is equivalent to these fibrewise connections fitting into a connection $A$ on a line bundle $E$ over $TD/\Lambda$ , with the integrability condition $F_{A}^{0,2}=0$ . Now a Hessian metric on $D$ induces a Kähler structure on both sides. The condition for $L$ to be a special Lagrangian of phase $\hat{\theta}$ , translates into the mirror condition on the holomorphic line bundle:

\text{Im}(e^{-i\hat{\theta}}(\omega_{X^{\vee}}+F_{A})^{n})=0.

Here the curvature $F_{A}$ of the $U(1)$ -connection is an imaginary valued $(1,1)$ -form. Up to rescaling $\omega_{X^{\vee}}$ , the global version is the dHYM equation for a Hermitian metric on a holomorphic line bundle $E$ over $X^{\vee}$ , thought of as a PDE on the potential function $\phi$ :

\text{Im}(e^{-i\hat{\theta}}(\omega_{X^{\vee}}+\sqrt{-1}\alpha_{\phi})^{n})=0,\quad[\alpha_{\phi}=\alpha+\sqrt{-1}\partial\bar{\partial}\phi]\in-c_{1}(E).

(11)

For an alternative view, we can pointwise simultaneously diagonalize $\omega_{X^{\vee}}$ and $\alpha_{\phi}$ , to extract the eigenvalues $\lambda_{i}$ ,

\omega_{X^{\vee}}=\frac{\sqrt{-1}}{2}\sum dz_{i}\wedge d\bar{z}_{i},\quad\alpha_{\phi}=\frac{\sqrt{-1}}{2}\sum\lambda_{i}dz_{i}\wedge d\bar{z}_{i}.

The dHYM equation is then

\Theta_{\omega}(\alpha)=\sum_{i}\arctan\lambda_{i}=\hat{\theta}\mod\pi\mathbb{Z}.

(12)

Thus the equation actually separates into several discrete branches depending on the choice of $\hat{\theta}$ , and changing $\hat{\theta}$ by $k\pi$ can drastically alter the behaviour of the equations. Replacing $\hat{\theta}$ by $-\hat{\theta}$ (so $E$ is replaced by its dual) does not change the problem, so without loss of generality $0\leq\hat{\theta}<\frac{n\pi}{2}$ . There are two significant limiting cases:

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When $\lambda_{i}\gg 1$ for any $i$ , so $\hat{\theta}\approx\frac{n\pi}{2}-\sum\frac{1}{\lambda_{i}}$ , the equation becomes approximately the J-equation

$\alpha_{\phi}^{n-1}\wedge\omega_{X^{\vee}}=\frac{\int\alpha^{n-1}\wedge\omega_{X^{\vee}}}{\int\alpha^{n}}\alpha_{\phi}^{n}.$

Most mathematical works on dHYM assume some lower bound such as $\hat{\theta}>\frac{(n-2)\pi}{2}$ , which may be morally interpreted as being near this large phase limit.

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When $|\lambda_{i}|\ll 1$ for any $i$ , so $\hat{\theta}\approx\sum\lambda_{i}$ , the equation is approximately the line bundle case of the HYM equation

\alpha_{\phi}\wedge\omega_{X^{\vee}}^{n-1}=\frac{\int\alpha\wedge\omega_{X^{\vee}}^{n-1}}{\int\omega_{X^{\vee}}^{n}}\omega_{X^{\vee}}^{n}.

This is also known as the large volume limit, in the sense that the Kähler metric length scale is much larger than the curvature scale of the bundle.

Towards a Bridgeland stability condition

The existence of dHYM solutions has algebraic obstructions. Let $V$ be a $p$ -dimensional subvariety of $X^{\vee}$ , with $1\leq p<n$ . Under the large phase assumption $\hat{\theta}>\frac{(n-2)\pi}{2}$ , a pointwise consideration of eigenvalues shows [18, Prop. 3.4]

\text{Im}\big(\int_{V}e^{-\sqrt{-1}(\hat{\theta}-(n-p)\frac{\pi}{2})}(\omega_{X^{\vee}}+\sqrt{-1}\alpha_{\phi})^{n}\big)>0.

More suggestively, denote

Z_{V}(E)=-\int_{V}e^{-\sqrt{-1}\omega_{X^{\vee}}}ch(E)\in\mathbb{R}_{>0}e^{\sqrt{-1}\phi_{V}(E)},\quad\phi_{V}(E)\in(0,\pi),

(13)

so the algebraic obstruction can be rewritten as

\text{Im}(\frac{Z_{V}(E)}{Z_{X^{\vee}}(E)})>0,\quad\emph{i.e.}\quad\phi_{V}(E)>\phi_{X^{\vee}}(E)=\hat{\theta}-\frac{(n-2)\pi}{2}.

This bears some resemblance to a Bridgeland stability condition with central charge $Z_{X^{\vee}}(E)$ , even though on the technical level there are some discrepancies [18, section 3.2]. In the simplest understood examples, such as the blow-up of $\mathbb{CP}^{2}$ in a point, the obstruction criterion for dHYM seems to refine Bridgeland stability, in the sense that every dHYM stable object is Bridgeland stable, but not conversely. It is not entirely clear how to interpret this (cf. Remark 2.10).

Main achievements on dHYM

Some of the most significant results on the dHYM equation are:

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There is a well developed formal GIT picture [23, section 2][19][21], including a complexified group action on an infinite dimensional Kähler manifold, a moment map, a negatively curved symmetric space, and a Donaldson type functional which is convex along geodesics.
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The geodesic equation with prescribed boundary data have $C^{1,1}$ -solutions and viscosity solutions under a large phase assumption [18, Thm 2.7]. This leads to nontrivial algebraic obstructions [18, section 3].
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Assume large phase $\hat{\theta}>\frac{(n-2)\pi}{2}$ and the existence of a subsolution in the suitable sense, then the dHYM equation can be solved [18, Thm 5.2].
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Assume large phase $\hat{\theta}>\frac{(n-2)\pi}{2}$ , then the existence of dHYM solution can be formulated in terms of algebraic obstruction conditions [16].

Will dHYM lead to the mirror Bridgeland condition?

Despite substantial progress, many essential difficulties still need to be overcome before the dHYM equation can give rise to a Bridgeland stability condition on $D^{b}Coh(X^{\vee})$ :

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Can one relax the large phase assumption?
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Is there a generalisation of dHYM equation to higher rank vector bundles? (Currently, there is no well established PDE, let alone how to solve it.)
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Can this story be extended to complexes of vector bundles?
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How can one compare the answer with the A-side of the mirror?

Time will tell how far one can push in this program, but we would like to momentarily play the skeptic’s advocate:

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The differential geometric motivation ³⁰³⁰ 30 There is an independent physics motivation for dHYM from the Dirac-Born-Infeld action. The DBI action is however not an exact result, but depends on the assumption that certain derivative terms of the curvature can be ignored. Unlike the A-model side where the central charge $Z(L)=\int_{L}\Omega$ is believed to hold exactly, on the B-side the central charge formula $Z_{X^{\vee}}(E)=-\int_{X^{\vee}}e^{-\sqrt{-1}\omega_{X^{\vee}}}ch(E)$ is believed to be subject to worldsheet instanton corrections. of dHYM assumes semiflat ambient Kähler metrics. Such metrics only arise naturally if the manifold has toric symmetry, or as a good approximate description for degenerating Calabi-Yau metrics near the large complex structure limit/large volume limit. Near this limit, the dHYM equation may be an improvement on the HYM equation as the mirror version of special Lagrangians. But far away from such limits, the instanton corrections cannot be ignored, and indeed a general Kähler metric has no toric symmetry.
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The reliance on the large phase assumption is a possible indication of a breakdown once we move too far from a large phase limit. In the closely related special Lagrangian graph equation, relaxing the phase condition is known to result in rather severe singularities for the viscosity solutions.
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Being the section of a torus fibration is a serious assumption on the topology of a Lagrangian.
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The difficulty with higher rank vector bundles is a possible indication that $Coh(X^{\vee})$ may be not the natural abelian subcategory of $D^{b}Coh(X^{\vee})$ suited for dHYM. Furthermore, there is no a priori guarantee for arbitrary stability conditions to correspond to PDEs,³¹³¹ 31 If a Bridgeland stability condition arises from a PDE, there is no a priori guarantee that its deformations also come from PDEs. or to have any classical geometrical interpretation at all. In particular, the $B$ -side mirror to the hypothetical special Lagrangian stability condition may well be an abstract stability condition with no particular PDE interpretation.
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Due to the lack of imagination, it is hard to see how PDEs can know about the degree shifts of a complex of vector bundles, and how it can detect homotopy equivalence of complexes.

If we take this skeptic view, then we do not expect an exact comparison between the hypothetical Bridgeland stabilities on both sides of the mirror, without adding very substantial assumptions such as toric symmetry. But when HYM shares the same qualitative features as dHYM, which admit a mirror interpretation, it lends more plausibility for these features to appear also on the special Lagrangians.