We now analyze the supersymmetry condition for a configuration of n D6-branes wrapping Q. If we work with any Calabi-Yau, the moduli space of such configura- tions receive corrections from holomorphic disks bounded by the branes. However,
if we assume X is near the large volume limit, such corrections are small and we can replace X by the symplectic linearization T˚Q. We will argue then that the
brane configuration is described by a triplepE, Ac, hqconsisting of a rankncomplex vector bundle, a “stable” flat SLpn,Cq-connectionAconQ, and a harmonic metric h on E. The equation determining h is a moment map condition that selects a prefered gauge orbit of Ac under complexified gauge transformations. In the next section we will give an interpretation of this data by using Corlette’s theorem to introduce a moduli space of “flat Higgs bundles” on Q.
Remark 4.2.1. It is known [Wit96] that a configuration of n D6-branes on a fixed special Lagrangian Q is described by a SUpnq-connection on Q. In our setup, the
extra complexified directions parametrize K¨ahler deformations that keepQ special Lagrangian.
The unbroken supersymmetry (BPS) condition for IIA string theory on T˚Q
with n D6-branes is given by theHermitian-Yang-Mills equations:
$ ’ ’ & ’ ’ % F2,0 “0 ΛF “0 (4.2.1)
HereF is the curvature of aSUpnq-connectionAon a holomorphic vector bundle
E over T˚Q endowed with a hermitian metric, and Λ is the Lefschetz operator of
contraction by the K¨ahler form. Note that becauseAis hermitian, the first equation implies F0,2 “F2,0 “0.
given by a flat connection on a complex line bundle over Q. This is a dimensional reduction of a Up1q Hermitian-Yang-Mills connection on a line bundle over T˚Q.
Thus, to find the condition on Q, we need to compute the dimensional reduction of equations 4.2.1 down to Q.4
Since we are working on T˚Q, choose local coordinates x
j on Q and yj on the fiber,j “1,2,3. The complex structureJ adapted to the semi-flat metric is chosen such that zj “xj `iyj are holomorphic coordinates (i.e., Q is totally real). Now, the hermitian condition allows us to write A1,0 “A0,1, and it follows that:
A1j,0 “Aj `iAj`3
A0j,1 “Aj ´iAj`3 (4.2.2)
Let us write A1,0
“ ř3j“1Ajdxj `iθjdyj. Assume that A and θ do not depend on the fiber directions yj. It is clear that:
A:“ 3 ÿ
j“1
Ajdxj (4.2.3)
becomes a well-definedsupnq-valued one-form onQ. Moreover, becauseQis special Lagrangian, NQ{T˚Q –T Qso the a priori vertical supnq-valued one-form
4Another way to see this is by noting that the HYM equation describes a “spacetime filling
brane” onT˚Q, and the wrapping condition is obtained by performing three T-dualities along the
3 ÿ
j“1
θjdyj (4.2.4)
can be be dualized (via the metric) to a supnq-valued one-form θ. In this semi-flat setup, the dualization map is the action of the complex structure Jkjpdyjq “ ´dxk.
Letd“ B ` B. By assumption, BA“ Bθ“0. We have:
F2,0
“ BA1,0`A1,0^A1,0
“ BpA`iθq ` pA`iθq ^ pA`iθq (4.2.5)
and
F1,1 “ BA0,1`A1,0^A0,1 (4.2.6)
The first equation becomes:
FA“θ^θ
DAθ “0 (4.2.7)
and the second equation is:
DA‹θ“0 (4.2.8)
where the “bundle Hodge star” on Ω1pAdpE|
Qqqis a combination of the Hodge star on the base Q and the Hodge star induced by the hermitian metric on E. One
can also define the “adjoint” of DA by D:A :“ ‹DA‹ and write equation 4.2.8 as D:Aθ “0.
We will refer to equations 4.2.7 and 4.2.8 as the Acharya-Pantev-Wijnholt sys- tem, or APW for short. The reason we choose this name is that, as far as the author is aware, the recognition that these equations describe the supersymmetric gauge theory associated to a system of D6-branes wrapping a three-cycle appears
first in work of Acharya [Ach98], and the Higgs bundle/spectral cover interpreta- tion, which we will discuss next, first appeared in the work of Pantev and Wijnholt [PW11]. Recently, the system has been studied more carefully in [BCHSN18] and [BCHLTZ18]. We note, however, that these equations have appeared long before (in a different context) in the Mathematics literature in the works of Donaldson [Don87] and Corlette [Cor88]. In fact, theorem 5.1.8 below establishes that solutions to these equations are essentially described by the well-known Donaldson-Corlette theorem 5.1.7.