The DT-Instanton Equation on Almost Hermitian 6-Manifolds

Abstract

This article investigates a set of partial differential equations, the DT-instanton equations, whose solutions can be regarded as a generalization of the notion of Hermitian–Yang–Mills connections. These equations owe their name to the hope that they may be useful in extending the DT-invariant to the case of symplectic 6-manifolds. In this article, we give the first examples of nonabelian and irreducible DT-instantons on non-Kähler manifolds. These are constructed for all homogeneous almost Hermitian structures on the manifold of full flags in $\mathbb {C}^3$. Together with the existence result we derive a very explicit classification of homogeneous DT-instantons for such structures. Using this classification we are able to observe phenomena where, by varying the underlying almost Hermitian structure, an irreducible DT-instanton becomes reducible and then disappears. This is a non-Kähler analogue of passing a stability wall, which in string theory can be interpreted as supersymmetry breaking by internal gauge fields as in Anderson et al. (J High Energy Phys 09:026, 2009).

Appendix A. The topology of the bundles \(P_{\beta }\)

Recall that the bundles $P_{\beta }$ are constructed via $P_{\beta }= \mathrm{SU}(3) \times _{(T^2, \lambda _{\beta })} \mathrm{SO}(3)$, where

$$\begin{aligned} \lambda _{\beta } = \mathrm {diag}( e^{\frac{i}{2}\beta } , e^{-\frac{i}{2}\beta } ) \in \mathrm{SU}(2)/\mathbb {Z}_2 . \end{aligned}$$

Let $V_{\beta }=P_{\beta } \times _{SO(3)} \mathbb {R}^3$ be the vector bundle associated with respect to the standard representation SO(3)-representation, and consider the $\mathrm{U}(2)$ bundle

$$\begin{aligned} E_{\beta }=\mathrm{SU}(3) \times _{(T^2 , \tilde{\lambda }_{\beta } )} \mathbb {C}^2 , \end{aligned}$$

where

$$\begin{aligned} \tilde{\lambda }_{\beta } = \mathrm {diag}( e^{i\beta } , 0 ) \in \mathrm{U}(2), \end{aligned}$$

and the ismorphism. This has the property that the $\mathrm{U}(2)$-adjoint bundle of $E_{\beta }$ splits as $\mathfrak {u}_{E_{\beta }} \cong \underline{\mathbb {R}} \oplus V_{\beta }$ and

$$\begin{aligned} w_2(V_{\beta })= c_1(E_{\beta }) \mod 2, \ \ \ p_1(V_{\beta }) = c_1(E_{\beta })^2 - 4 c_2(E_{\beta }) . \end{aligned}$$

We shall now compute the Chern classes of the bundles $E_{\beta }$ using Chern-Weyl theory. For this we must equip $E_{\beta }$ with a connection which we choose to be the standard invariant connection given by

$$\begin{aligned} A_{\beta }= \beta \otimes \mathrm {diag}(i,0). \end{aligned}$$

This has curvature $F_{\beta } = d\beta \otimes \mathrm {diag}(i,0)$ and so

$$\begin{aligned} c_1(E_{\beta })=-\frac{1}{2\pi }[d \beta ] , \ \ c_2(E_{\beta })= \frac{1}{4 \pi ^2} [d \beta ] \cup [d \beta ]. \end{aligned}$$

Furthermore, a computation using the Maurer–Cartan equations shows that

$$\begin{aligned} d\beta _1^2 + d\beta _2^2 + d \beta _1 \wedge d\beta _2 = d \mathop {\mathrm {Im}}((\eta _1+i\theta _1) \wedge (\eta _1+i\theta _1) \wedge (\eta _1+i\theta _1)), \end{aligned}$$

and so in $H^4(\mathbb {F}_2, \mathbb {Z})$ we have

$$\begin{aligned}{}[d\beta _1] \cup [d \beta _2 ]= - [d\beta _1] \cup [d \beta _1 ] - [d\beta _2] \cup [d \beta _2 ]. \end{aligned}$$

So, writing $\beta = k \beta _1 + l \beta _2$ we compute that

$$\begin{aligned} w_2(V_{\beta })=-\frac{1}{2\pi }\left( k[d \beta _1] + l [d \beta _2] \right) \mod 2 , \end{aligned}$$

while

$$\begin{aligned} p_1(V_{\beta })= & {} \frac{1}{4 \pi ^2} [d \beta ] \cup [d \beta ] \\= & {} \frac{1}{4 \pi ^2} \left( k^2 [d \beta _1] \cup [d \beta _1 ] + 2kl [d \beta _1] \cup [d \beta _2] + l^2 [d \beta _2] \cup [d \beta _2] \right) \\= & {} \frac{1}{4 \pi ^2} \left( k(k-2l) [d \beta _1] \cup [d \beta _1 ] + l(l-2k) [d \beta _2] \cup [d \beta _2] \right) . \end{aligned}$$

In particular, when $\beta$ is one of the roots $r_1$, $r_2$, $r_3$ we respectively obtain

$$\begin{aligned}&w_1(P_{r_1}) = -\frac{1}{2\pi } [d \beta _1] \mod 2 , \ \ \ p_1(V_{r_1})= -\frac{3}{4 \pi ^2} [d \beta _1] \cup [d \beta _1 ] , \\&w_1(P_{r_2}) = \frac{1}{2\pi }[d \beta _2] \mod 2 , \ \ \ p_1(V_{r_2})= -\frac{3}{4 \pi ^2} [d \beta _2] \cup [d \beta _2] , \\&w_1(P_{r_3}) = -\frac{1}{2\pi }\left( [d \beta _1] - [d \beta _2] \right) \mod 2 , \\&p_1(V_{r_3})= \frac{1}{4 \pi ^2} \left( 3 [d \beta _1] \cup [d \beta _1 ] + 3 [d \beta _2] \cup [d \beta _2] \right) , \end{aligned}$$

so these three bundles are all topologically different.