Manifolds with exceptional holonomy andmirrors of their submanifolds
Kotaro Kawai
Beijing Institute of Mathematical Sciences and Applications (BIMSA)

Kotaro Kawai (BIMSA)

Exceptional holonomy, mirror submanifolds

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Holonomy group
The holonomy group Hol(g ) is one of tools to study the structure ofa Riemannian manifold (X , g ).

Theorem (Berger, 1955)
Let (X , g ) be a simply connected Riemannian manifold and it isirreducible (i.e., (X , g ) does not locally decompose into theproduct of Riemannian manifolds),
and not locally symmetric (i.e., ∇R 6= 0).
Then the holonomy group Hol(g ) is one of the following.
SO(n), U(n), SU(n), Sp(n)Sp(1), Sp(n), G2 , Spin(7).

Kotaro Kawai (BIMSA)

Exceptional holonomy, mirror submanifolds

2 / 21G2 geometry
G2 :=Aut(O)={T ∈ GL(O) | T preserves the multiplication of O}.
Identify O = R ⊕ ImO = R ⊕ R7 .
Describe the multiplication of O by φ0 ∈ Λ3 (R7 )∗ :R7 × R7 3 (u, v ) 7→ φ0 (u, v , ·)♯ ∈ R7(u ⊥ v ).

Then
G2 = {g ∈ GL(7, R) | g ∗ φ0 = φ0 } ⊂ SO(7).
def(X 7 , g ) : G2 -manifold ⇐⇒ Hol(g ) ⊂ G2 (=⇒ Ric(g ) = 0).
Hol(g ) ⊂ G2 =⇒ ∃φ ∈ Ω3 (X 7 ) s.t. ∇φ = 0.
Fixing such a φ ∈ Ω3 (X 7 ), we call (X 7 , φ, g ) a G2 -manifold.
G2 geometry is characterized by a 3-form φ.
Kotaro Kawai (BIMSA)

Exceptional holonomy, mirror submanifolds

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How to understand G2 geometry?
{G2 -manifolds}
5iu){CY 3-manifolds}{3,4-dim. manifolds}

The analogy of Calabi-Yau 3-manifolds （SU(3) ⊂ G2 ）Y 6 : a Calabi-Yau 3-mfd =⇒ S 1 × Y 6 ：a G2 -manifold
We might consider higher dimensional analogues of the theoryfor 3,4-dim. manifolds.
Flat connections on 3-mfds (=⇒ Chern-Simons theory)
ASD connections on 4-mfds (=⇒ Donaldson theory）⇝ G2 -instanton
Kotaro Kawai (BIMSA)

Exceptional holonomy, mirror submanifolds

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Calibrated geometry [Harvey-Lawson, 1982]
calibration：a closed diﬀerential form φ ∈ Ωk (X n ) on aRiemannian manifold (X n , g ) satisfying a certain condition.
calibration ⇒ calibrated submanifolds
Every compact calibrated submanifold is homologically volume
minimizing, and the volume is topological.

Hol(g ) (⊂)(X , g )calibratedsubmfds

U(n)X :Kähler
N 2k :complex
submfds2nSU(n)
X :Calabi-Yau2n

G2
X : G2 -manifold
A3 :associative
submfds
4
C :coassociative
submfds
7

Ln :special Lag.
submfds
Red objects have obstructed deformations.
Blue objects have unobstructed deformations.
Kotaro Kawai (BIMSA)

Exceptional holonomy, mirror submanifolds

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Calibrated submanifolds might be useful to understand a manifold.
Gromov-Witten invariant “counts” pseudoholomorphic curves.
⇝ Can we “count” associative submanifolds?
Casson invariant “counts” ﬂat connections.
Donaldson invariant “counts” ASD connections.

⇝ Can we “count” G2 -instantons?
Mirror symmetry for Calabi-Yau 3-manifolds⇝ Mirror symmetry for G2 -manifolds?
{G52 -manifolds}
iu){CY 3-manifolds}

Kotaro Kawai (BIMSA){3,4-dim. manifolds}

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Mirror symmetry
Strominger–Yau–Zaslow (SYZ conjecture):mirror symmetry of Calabi-Yau 3-folds would be explained interms of special Lagrangian (SL) dual T 3 -ﬁbrations (includingsingular ﬁbers).
(X 6 )∗X6fB3|f∗Smooth ﬁbers f −1 (b), (f ∗ )−1 (b) are “dual” SL T 3 .
Leung–Yau–Zaslow:
Given a SL dual T 3 -ﬁbration, SL submanifolds correspond todeformed Hermitian Yang–Mills (dHYM) connections (or LYZconnections?) via the real Fourier–Mukai transform.
Kotaro Kawai (BIMSA)

Exceptional holonomy, mirror submanifolds

7 / 21A similar argument works for G2 -manifolds.
Lee–Leung:
Given a coassociative dual T 4 -ﬁbration, (co)associative
submanifolds correspond to deformed Donaldson–Thomas(dDT) connections (or LL connections?) via the real
Fourier–Mukai transform.
calibrated submanifold
special Lagrangian(co)associative

Kotaro Kawai (BIMSA)“mirror”
dHYM connectiondDT connection

Exceptional holonomy, mirror submanifolds

8 / 21Deﬁnition(X 7 , φ, g ): a G2 -manifold,(L, h) → X : a smooth complex Hermitian line bundle.
A Hermitian connection ∇ of (L, h) is called a deformedDonaldson–Thomas (dDT) connection (deformed G2 -instanton) if
1 3F + F∇ ∧ ∗φ = 0,6 ∇√where F∇ = d∇ ◦ d∇ ∈ −1Ω2 is a curvature of ∇.
dDT connection can also be considered as an analogue of theG2 -instanton (DT connection): F∇ ∧ ∗φ = 0.
We expect that dDT connections will have similar properties toassociative submanﬁolds and G2 -instantons.
Examples of dDT connections:ﬂat connections (F∇ = 0)
dHYM conn. (mirror of “SL =⇒ associative”)
Kotaro Kawai (BIMSA)Exceptional holonomy, mirror submanifolds

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{dHYMO conn.}
mirrorSLsubmfds⊂⊂{dDT conn.}
iOmirror
associativesubmfds

analogueO

analogue of GW inv.?

){G2 -instantons}Ohigher dimensional
gauge theory?

Can we “count” dDT connections to deﬁne invariants?
How about the moduli (deformation) theory?
Do dDT connections behave similarly to associative submfds
and G2 -instantons?
=⇒Check if they have similar properties.
Kotaro Kawai (BIMSA)

Exceptional holonomy, mirror submanifolds

10 / 21Properties of associative submanifolds(1) The moduli space is 0-dimensional and canonically orientable if
we perturb the G2 -structure.
(2) associator equality ⇒ a G2 -structure is a calibration, and we cancharacterize associative submanifolds by the vanishing of atensor, which is useful in deformation theory.
(3) Homologically volume minimizing. The volume is topological.
(4) ciritical points of the Chern-Simons type functional.
(G2 -instantons have similar properties to the above.)

Theorem (K.-Yamamoto)
Similar properties to the above also hold for dDT connections.
(The analogy for (4) is proved by Karigiannis-Leung.)
We will state below that the “mirrors” of (2) and (3) hold true.
Kotaro Kawai (BIMSA)

Exceptional holonomy, mirror submanifolds

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“Volume” for connections(X n , g ): a compact connected oriented Riemannian manifold,(L, h) → X : a smooth complex Hermitian line bundle,
A0 = {Hermitian connections of (L, h)}.
Deﬁne the “volume functional” V : A0 → R byZV (∇) := v (∇)volg ,rX √♯v (∇) := det idTX + (− −1F∇ )s2
2F2F3= 1 + |F∇ |2 + ∇ + ∇ + · · ·2!
3!
V corresponds to the (standard) volume functional forsubmanifolds via the real FM.
V is called the Dirac-Born-Infeld (DBI) action in physics.
Kotaro Kawai (BIMSA)

Exceptional holonomy, mirror submanifolds

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Theorem (“Mirror” of associator equality, K.-Yamamoto)
Let (X 7 , φ, g ) be a G2 -manifold. For any ∇ ∈ A0 , we have1
1 + hF∇2 , ∗φi22

211+ ∗φ ∧ F∇ + F∇3 + |φ ∧ ∗(F∇ )2 |2 = v (∇)2 ,64In particular,
11 + hF∇2 , ∗φi ≤ v (∇)
2for any ∇ ∈ A0 . The equality holds if and only if ∇ is dDT.
For any dDT connection ∇, ∇ is a global minimizer of V andVR(∇) is topological.
1 + 12 hF∇2 , ∗φi volg = Vol(X ) + (−2π 2 c1 (L)2 ∪ [φ]) · [X ] .
XThis is the “mirror” of the fact that every compact associative(calibrated) submanifold is homologically volume minimizing,
and the volume is topological.
Kotaro Kawai (BIMSA)

Exceptional holonomy, mirror submanifolds

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CorollarySuppose that L is a ﬂat line bundle. Then, any dDT connection is aﬂat connection. In particular, the moduli space of dDT connections is
H 1 (X , R)/H 1 (X , Z).
Let ∇0 be a ﬂat connection and ∇ be any dDT connection. Then,sZ
Z
2 23 2FF∇∇
1 + |F∇ |2 ++volg = V (∇) = V (∇0 ) =volg ,2!
3!
XX
which implies that F∇ = 0.

Kotaro Kawai (BIMSA)

Exceptional holonomy, mirror submanifolds

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Minimal connections(X n , g ): a (not necessarily compact)
connected oriented Riemannian manifold,(L, h) → X : a smooth complex Hermitian line bundle,
A0 := {Hermitian connections of (L, h)}.
Since v (∇) ≥ 1 for ∀∇ ∈ A0 , we deﬁne the normalized volume
functional V 0 : A0 → [0, ∞] by
Z0V (∇) = (v (∇) − 1)volg .
XWe see thatV 0 (∇) = 0⇐⇒F∇ = 0.

Deﬁnition
Critical points of V 0 (or V ) are called minimal connections.
Every dDT conn is a global minimizer of V =⇒ minimal.
Kotaro Kawai (BIMSA)

Exceptional holonomy, mirror submanifolds

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Proposition∇ ∈ A0 is minimal ⇐⇒ δ∇ F∇ = 0 =⇒ (dδ∇ + δ∇ d) F∇ = 0.
|{z}=:∆∇δ∇ : Ωk → Ωk−1 : ∇-dependent diﬀerential operator like thecodiﬀerential d ∗ .
This is a similar characterization to Yang–Mills connections.
To “count” dDT connections, we need to consider thecompactiﬁcation of the moduli space.
To do this, we should know how bubbles occur. For Yang–Millsconnections, it is known from
(1) Price’s monotonicity formula,(2) ε-regularity theorem of Uhlenbeck-Nakajima.

Can we show these analogies for minimal connections?
=⇒ (1) is (probably) OK.
Kotaro Kawai (BIMSA)

Exceptional holonomy, mirror submanifolds

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Theorem (K., Monotonicity formula)(X n , g ): an oriented Riemannian manifold, withdim X = n = 2m + 1 and Ric(g ) ≥ 0. Fix p ∈ X .
(L, h) → X : a smooth complex Hermitian line bundle.
Then ∃a = a(n, p, g ) ≥ 0, 0 < ∃rp′ < injg (p), ∃ a function
Θ : [0, ∞) → R s.t. for any minimal connection ∇(0, rp′ ] → R,e aρ
ρ 7→ρ

2Z
(v (∇) − 1)volg + 2aΘ(ρ)
Bρ (p)

is non-decreasing.

Kotaro Kawai (BIMSA)

Exceptional holonomy, mirror submanifolds

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(Outline of the proof)• We ﬁrst show the “integration by parts formula” for min. conn ∇.
ZZ(∆∇ f1 ) · f2 · v (∇)volg =f1 · (∆∇ f2 ) · v (∇)volg ,XX

where f1 , f2 ∈ Ω0 , one of which is compactly supported.
• Set f1 = 1, f2 = “cut oﬀ function” and compute ∆∇ f2 .
• After some calculations, we see that the monotonically is obtained
if the following is satisﬁed:(1) 0 < ∃rp′ < injg (p), ∀τ ∈ [0, rp′ ],ZZZ∂nnvolg ,ωn τ ≥volg ≥ τvolg .
∂τ Bτ (p)Bτ (p)Bτ (p)(2) (trG∇−1 − 1)v (∇) − n + 1 ≥ 0.
(1) is satisﬁed if Ric(g ) ≥ 0 (relative volume comparison theorem).
(2) is an algebraic condition. It is satisﬁed if dim X = n = 2m + 1.
Kotaro Kawai (BIMSA)

Exceptional holonomy, mirror submanifolds

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Corollary
Let (L, h) −→ R2m+1 be a (necessarily trivial) smooth complex
Hermitian line bundle over (R2m+1 , g0 ), where g0 is the standard ﬂatmetric.
If ∇ is minimal with V 0 (∇) < ∞, then ∇ is ﬂat. (i.e. F∇ = 0.)(proof) We can take a = 0 and rp′ = ∞ for (R2m+1 , g0 ).
If F∇ 6= 0, ∃p ∈ R2m+1 , ∃R0 > 0 s.t.
Z
1(v (∇) − 1)volg0 > 0.
R0 BR0 (p)By monotonicity formula, for ∀R ≥ R0 ,ZZ11(v (∇)−1)volg → 0 (R → ∞),0<(v (∇)−1)volg ≤R0 BR0 (p)R BR (p)which is a contradiction.
Kotaro Kawai (BIMSA)

Exceptional holonomy, mirror submanifolds

19 / 21Future work
Roughly, we could show
2e aρρκZ(v (∇) − 1)volgBρ (p)

is non-decreasing for κ = 1.
I am not sure κ = 1 the best for the monotonicity. That is, wemight be able to prove the monotonicity for κ > 1.
In fact, for dDT connections on a G2 -manifold, (recall thatG2 -manifolds are 7-dim.) we can take κ = 13/7 > 1.
For Yang–Mills connections, there is an analogous monotonicity
formula. In that case, κ is taken to be “scaling invariant” (in acertain sense). There are no such a property for our case.

Can we show “ε-regularity theorem” to study “blowup set”?
Can we construct nontrivial examples of minimal/dDTconnections?
Kotaro Kawai (BIMSA)

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I would like to thank the organizers and everyone
involved for holding such a great conference.

Thank you so much!

Kotaro Kawai (BIMSA)

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