In this section we change gears in order to discuss flat Higgs bundles in the context of a larger theory: Kapustin-Witten systems on a four-manifold.
We show that flat Higgs bundles describe an invariant subspace of a natural map on the moduli space of solutions of the Kapustin-Witten equations. More gen- erally, we argue that the moduli space of a family of extended Bogomolny theories parametrized by a parameter qP S1 can be given the structure of a S1-bundle over the moduli space of solutions to the Acharya-Pantev-Wijnholt system.
Let M “ QˆR be an oriented Riemannian four-manifold, A a connection
on a G-bundle on E Ñ M, F its curvature, and Φ P Ω1pAdpEqq. Consider the
Kapustin-Witten equations [KW06]:
pF ´Φ^Φ`qDAΦq` “0
pF´Φ^Φ´q´1D
AΦq´ “0
DA‹M Φ“0 (7.3.1)
where for a two-form α, α˘ denotes its selfdual and anti-selfdual components, ‹
M
is the Hodge star operator on M and q PR.
We refer to 7.3.1 as the KWq system and q is called a twisting parameter. In [GW11], Gaiotto and Witten study the dimensional reduction of the KW1 system
down to two dimensions, and show the moduli space of solutions provides knot invariants. In section 7 below I will sketch an approach to relate the discussion on this section to [GW11] and also the work of Aganagic and Vafa [AV01].
The KWq system is a deformation of the Hermitian-Yang-Mills equations. We proved in section 4 that the APW system is a dimensional reduction of HYM. Hence APW is also a dimensional reduction of KWq. The precise way in which this happens is given by the following lemma:
Lemma 7.3.1. Let pQ : M Ñ Q be the projection to Q. The, for q ‰ ˘i, the
pullback of connections by pQ defines an injective mapι :MAP WpQqãÑMKWqpMq
Proof. Start from 7.3.1 and perform dimensional reduction, assuming moreover that
A “π˚A and Φ “π˚φ (i.e., A
0 “φ0 “0). Let’s see what happens with the first
equation: let G “F´Φ^Φ`qDAΦ. Then the equation is:
G“ ´ ‹M G (7.3.2)
But G has no dt components, and ‹MG has only components with a dt, so they must both vanish. Thus we get:
F ´Φ^Φ`qDAΦ“0 (7.3.3)
An analogous argument holds for the other two equations. We get:
F ´φ^φ`qDAφ “0
F ´φ^φ´q´1DAφ“0
DA‹φ“0 (7.3.5)
Ifq‰ ˘i, the first and second equations combined imply thatDAφ “0. There- fore we obtain the flat Higgs bundles equations.
So far this is hardly interesting, as it is well-known that most of the solutions to KWq are flat connections. The interesting idea comes when we compare the dimensionally reduced KWq for different values of q. For q “ 1, we get equations 10.35 in [KW06]: F ´ rφ, φs “ ‹ ˆ DAφ0´ rA0, φs ˙ DAφ“ ‹ ˆ DAA0` rφ0, φs ˙ DA‹φ“ ‹rA0, φ0s (7.3.6)
We now perform the dimensional reduction of the KW0 system. The equations
are:
pF ´Φ^Φq` “0
DA‹Φ“0 (7.3.7)
Lett be a coordinate on R,tx1, x2, x3ucoordinates on Qand write A“A0dt`
Aidxi, Φ“φ0dt`φidxi.6 We write A“Aidxi and φ “φidxi and think of these as a connection and a Higgs field on Q, respectively. Let F be the curvature of A.
We assume the matrices do not depend ont. The first equation in 7.3.7 becomes a set of three equations:
Fij ´ rφi, φjs “ p´1qk
ˆ
DkA0` rφ0, φks
˙
piăj, i‰k ‰jq (7.3.8)
These can be rewritten using the Hodge star ‹on Q:
F ´ rφ, φs “ ‹
ˆ
DAA0` rφ0, φs ˙
(7.3.9)
(Notice that A0 and φ0 are just matrix-valued functions on Q, soDAA0 and rφ0, φs
are matrix-valued one-forms on Q, and are mapped to matrix-valued two-forms by
‹).
Next, the second equation in 7.3.7 becomes:
DAiφj “ rA0, φks ´DAkφ0 piăj, i‰k ‰jq (7.3.10)
which can be put together as:
Finally, the last equation in 7.3.7 is equivalent to:
rA0, φ0sdtdx1dx2dx3` p´1qiDAiφi “0 (7.3.12)
which can be rewritten as:
DA‹φ“ ‹rA0, φ0s (7.3.13)
Putting all together we get the following set of equations:
F ´ rφ, φs “ ‹ ˆ DAA0` rφ0, φs ˙ DAφ“ ‹ ˆ ´DAφ0` rA0, φs ˙ DA‹φ“ ‹rA0, φ0s (7.3.14)
for a G-connection A on E ÑQ with curvature F, an element φ P Ω1pQ, AdpEqq
and two fields A0, φ0 P Ω0pQ, AdpEqq. Notice we used the fact that‹2 “1.
When φ0 “A0 “0 both sets of equations 7.3.6 and 7.3.14 reduce to:
F ´φ^φ“0
DAφ “0
DA‹φ“0 (7.3.15)
Let KW1 denote the space of solutions to 7.3.6 and KW0 the space of solutions to 7.3.14. There is an “anti-involution”: KW0 ÑKW1 pA0, φ0q Ñ pφ10,´A 1 0q (7.3.16)
that matches the APW subspaces of KW1 and KW0. These subspaces parametrize
the same objects, so we do not distinguish them.
In fact, one can perform the dimensional reduction for general parameter q, and from the general formula one can see that q defines a S1-action on KW :“ \qPS1KWq leaving APW invariant. More precisely, if one considers the family over AP W whose fiber overpA, φ, hqispq, A0pqq, φ0pqqq, thenS1 acts by rotation on the
fibers.
The relationship betweenKW and AP W seems more surprising when one con- siders the analogy with instantons on four-manifolds. If 7.3.1 is analogous to the anti-selfduality equations onM, then 7.3.14 correspond to the Bogomolny equations on Q and 7.3.15 are analogous to the rather boring solutions given by flat connec- tions on Q. Of course, in the classical situation one works with a real gauge group, while in the present case APW systems are actually equivalent to flat complex connections. Hence the analogy is stronger than it seems at first.
such as a natural connection?
2. In [PW11] Pantev and Wijnholt introduced a Morse-Novikov complex count- ing solutions to 7.3.15. In [Wit12], Witten argues that counting certain solu- tions to 7.3.1 should give rise to Khovanov homology. Can the Morse-Novikov homology be realized as a subcomplex, and if so, what kind of knot invariants does it give rise to?
In general, we expect that APW is some sort of “invariant subspace” for a flow of time-independent solutions KWq ÑKWr. In other words, if (2) holds, the Morse-Novikov complex should be an invariant subspace for the differentials computing Khovanov homology.