$G_2$-manifolds

• $(X, \varphi)$

• $X$: smooth, oriented 7-manifold.

• $\varphi \in \Omega_X^3$ is nondegenerate: defines a metric, $g_\varphi$ (in a nonlinear way): $$\forall v \in T_x X : \quad (\iota_v \varphi) \wedge (\iota_v \varphi) \wedge \varphi = 6 |v|_{g_\varphi}^2 \mathrm{vol}_{g_\varphi}.$$

• Linear PDE: $\mathrm{d} \varphi = 0$.

• Nonlinear PDE: $\mathrm{d} \ast_{g_\varphi} \varphi = 0$.

• $\mathrm{Hol}^{g_\varphi} \leqslant G_2$.

• Model for $T_x X$: $\mathbb{R}^7 \simeq \mathrm{Im} (\mathbb{O})$.

The Donaldson–Segal program

• Let $\psi := \ast_\varphi \varphi$.

• $N^4 \subset X$ is a coassociative submanifold, if $\psi|_N = \mathrm{vol}_N$ ($N$ is $\psi$-calibrated).

• Donaldson–Segal: Potential invariants for $G_2$-manifolds by counting coassociatives. (Hard!)

• Conjecturally related (equivalent?) and potentially easier:
"counting" $G_2$-monopoles, $(\nabla, \Phi)$.

• Intuitively, the correspondence between coassociatives and monopoles:
$N = \Phi^{- 1} (0)$.

• In simple cases these objects are computable and agree (Oliveira JG&P '14).

• First step to prove more generally: analytic properties; e.g. asymptotics.

$G_2$-monopoles

• $\exists$ special Yang–Mills–Higgs fields, called $G_2$-monopoles: $$\ast_{g_\varphi} (F_\nabla \wedge \psi) = \nabla \Phi.$$

• $F_\nabla = F_\nabla^7 + F_\nabla^{14}$ ("vector" + "$\mathfrak{g}_2$").

• $\ast_{g_\varphi} (F_\nabla \wedge \psi) \sim F_\nabla^7$.

• intermediate energy: \begin{align} E^\psi (\nabla, \Phi) &= \int\limits_X (|F_\nabla^7|^2 + |\nabla \Phi|^2) \mathrm{vol}_{g_\varphi} \\ &= \underbrace{\int\limits_X (|\ast_{g_\varphi} (F_\nabla \wedge \psi) - \nabla \Phi|^2) \mathrm{vol}_{g_\varphi} + \mbox{boundary intergal (topological)}}_{\mbox{Bogomolny trick}} \end{align}

Asymptotically conical $G_2$-manifolds

• $(X, \varphi)$ is asymptotically conical (AC).

• AC: there is a compact manifold-with-boundary $K^7 \subset X$, such that $$(X - K, g_\varphi|_{X - K}) \sim ([R, \infty) \times \Sigma, \mathrm{d}r^2 + r^2 g_\Sigma).$$

• In particular: maximal volume growth (for a Ricci-flat manifold).

• Cheeger–Gromoll: $\Sigma$ has one component.

• $\Sigma$ is a nearly Kähler 6-manifold ($\omega \sim r^{- 2} \iota_{\partial_r} \varphi$ and $\nabla^{\mathrm{LC}} J$ is skew).

Theorem (Asymptotics of $G_2$-monopoles on AC manifolds; Fadel–N–Oliveira '20)

Let $(\nabla, \Phi)$ be a $G_2$-monopole such that
1. $E^\psi (\nabla, \Phi) < \infty$,
2. $r^2 F_\nabla^{14} \in L^\infty (X)$.

Then
1. $|\Phi| \rightarrow m > 0$, (the mass of the monopole)
2. $|\nabla \Phi| = O (r^{- 6})$, (sharp)
3. $|[F_\nabla, \Phi]|$ decays exponentially,
4. $(\nabla, \Phi)|_{\Sigma_R} \rightarrow (\nabla^\infty, \Phi^\infty)$, (in some gauge)
5. $\nabla^\infty \Phi^\infty = 0$,
6. $\nabla^\infty$ is pseudo-Hermitian–Yang–Mills: $$\Lambda F_{\nabla^\infty} = 0, \quad \& \quad F_\nabla^{0,2} = 0.$$
Corollary: Finite mass, monopole class, energy expressions, decay for (co)kernel elements of the linearization, and Fredholm theory.

Ideas of the proof

• Finite mass: $\Delta |\Phi|^2 = - 2 |\nabla \Phi|^2 \ \oplus$ Taubes.

• Decay: $\epsilon$-regularity and Moser iteration: if $x \in X$ is "far", that is $|x| = \mathrm{dist} (x, x_0) \gg 1$, then $$|\nabla \Phi (x)| \leqslant \frac{C}{|x|^{7/2}} \int\limits_{B_{|x|/2} (x)} |\nabla \Phi|^2 \mathrm{vol}_{g_\varphi}.$$

• Key "trick": bound $\int_X r^{2 \alpha} |\nabla \Phi|^2 \mathrm{vol}_{g_\varphi}$, using a (sharp) Hardy-type inequality and Agmon's trick.
(Stolen from Mark Stern, who in turn learned it from a paper of Agmon.)

Agmon's trick

• We prove the following: $$\int\limits_X f^2 \mathrm{vol}_{g_\varphi} \leqslant \frac{4}{25} \int\limits_X |\mathrm{d} (r f)|^2 \mathrm{vol}_{g_\varphi} + \mbox{small stuff}, \: \& \: \int\limits_X |\nabla (\chi \Psi)|^2 \mathrm{vol}_{g_\varphi} = \int\limits_X |\mathrm{d} \chi|^2 |\Psi|^2 \mathrm{vol}_{g_\varphi} + \int\limits_X \chi^2 \langle \Psi | \nabla^* \nabla \Psi \rangle \mathrm{vol}_{g_\varphi}.$$

• ...and use them with $f \approx r^\alpha |\nabla \Phi|$, $\chi = r^{\alpha + 1}$, and $\Psi = \nabla \Phi$.

• This gives (after a little more magic): $\int\limits_X r^{2 \alpha} |\nabla \Phi|^2 \mathrm{vol}_{g_\varphi} \leqslant \frac{C}{5 - 2 \alpha} \| \nabla \Phi \|_{L^2}^2$.

• Combining with Moser iteration yields: $|\nabla \Phi| = O (r^{- 6 + \epsilon})$.

• Then $\epsilon = 0$ can be achieved via an "infinite maximum principle" argument.

• The rest is standard...

Current work

• Theorem, Fadel–Oliveira: (Informal version) As $m \rightarrow \infty$, the curvature concentrates around a coassociative ("monopole bubbling") and an associative ("instanton bubbling").

• Bubbling data: Coassociative, $N^4 \subset X$ $\oplus$ Fueter section from $N$ into the BPS moduli $\oplus$ singular Dirac monopole around $N$.

• Indices equal: "Callias = gluing data" $$\underbrace{\tfrac{1}{6} \alpha^3 \left[ \Sigma \right] - \tfrac{1}{24} \left( \alpha \cup p_1 \left( \Sigma \right) \right) \left[ \Sigma \right]}_{\mbox{AS on the boundary}} + \underbrace{\mathrm{def}_{\mathrm{weight}}}_{\mbox{zero by the main theorem}} = \underbrace{\left( b_0 \left( N \right) + b_2^+ \left( N \right) - b_1 \left( N \right) \right)}_{\mbox{Fueter}} - \underbrace{1}_{\mbox{gauging Dirac}}.$$

• Current focus: glue-in.

Future wish list

• Generalization to "ALX" $G_2$-manifolds, especially ALC ones (lots of examples).

• "Finiteness"/"Compactness".

• Enumerative invariants.

• Doing the same for Calabi–Yau-monopoles and special Lagrangians.