# Physics behind the Kobayashi-Hitchin correspondence

Let $$X$$ be a $$d$$-dimensional Kähler manifold with Kähler metric $$\omega$$. Now consider the following setups:

1. Suppose $$E \rightarrow X$$ is hermitian vector bundle with hermitian connection $$A$$. In this context is interesting to study the hermitian Yang-Mills equations $$F^{(2,0)}=0,$$ $$F^{(1,1)} \wedge\omega^{d-1}=r\omega^{d},$$ over $$X$$. The interest comes from the fact that they can be shown to be the relevant instantons for the Donaldson-Thomas theory. Also see What, to a physicist, are instantons and the Donaldson invariants? to read about the importance of Donaldson theory (and its analogues) in physics.

2. Now suppose $$E \rightarrow X$$ is equipped with the structure of a holomorphic vector bundle.It was indicated in this answer how to beautifully interpret the physics of a vector bundle stability condition as sort of a "BPS condition". In a nutshell, the idea is to interpret the slope $$\mu(E):=\frac{deg(E)}{rank(E)}$$ as a "charge density" $$\rho(E)=\frac{Q(E)}{M(E)}$$; see 'Bridgeland stability' for details. Now, if we identify $$\mu(E) = \rho(E)$$ and by definition BPS states maximize the radio $$\rho(E)=\frac{Q(E)}{M(E)}$$, then stability condition for $$E$$ follows immediately, namely that any sub-object $$E^{\prime} < E$$ obeys $$\mu(E^{\prime}) < \mu(E)$$; therefore there is no decay channel such as $$E \rightarrow E^{\prime} +$$ something else.

The famous Kobayashi–Hitchin correspondence relates the two setups from above over arbitrary compact complex manifolds as follows:

Theorem: A holomorphic vector bundle $$E \rightarrow X$$ over a compact complex manifold with metric 2-form $$\omega$$ admits a Hermite–Einstein metric if and only if it is slope polystable (a direct sum of stable bundles of the same slope).

Question: Is there any way to understand what this theorem says in physical terms?

Concretely, what intrigues me is how the analogy with supersymmetry works so beautifully in the second case, but there is no obvious (at least to me) analogies with supersymmetry in the first one. In other words, all I want is an intuition on why the Donaldson-Thomas instantons are automatically 'BPS'.