The geometry of $G_2$-monopoles and the Donaldson–Segal program

Ákos Nagy

University of California, Santa Barbara

Joint with Saman Habibi Esfahani (MSRI/Duke), Daniel Fadel (Unicamp), Lorenzo Foscolo (UCL), & Gonçalo Oliveira (IST Austria)

Parts of this talk is based on 2009.06788; to appear in the Memoirs of the AMS, joint with Fadel & Oliveira.

New four-dimensional gauge theories, Mathematical Sciences Research Institute

Thursday, October 27, 2022.

this presentation can be viewed at akosnagy.com/talks/G2-mono_MSRI/G2-monopoles.html

$G_2$-manifolds

$(X, \varphi)$

$X$: smooth, oriented 7-manifold.

Model for $\left( T_x X, \varphi_x \right)$: $\: \mathrm{Im} \left( \mathbb{O} \right) \cong \mathbb{R}^7$.

$\varphi \in \Omega_X^3$ is nondegenerate: defines a metric, $g_\varphi$ (in a nonlinear way): \begin{equation} \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}. \end{equation}

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

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

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

The Donaldson–Segal program

Let $\psi := \ast_\varphi \varphi \in \Omega_X^4$, which is a calibration on $X$.

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

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 is similar to Taubes' "GR = SW" Theorem:

$N \approx \Phi^{- 1} (0)$.

In simple cases these objects can be computed and they agree (Oliveira JG&P '14).

First step to prove more generally: analytic properties; e.g. asymptotics and Fredholm theory.

$G_2$-monopoles

$\exists$ special Yang–Mills–Higgs fields, called $G_2$-monopoles: \begin{equation} \ast (F_\nabla \wedge \psi) = \nabla \Phi. \end{equation}

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

$\ast (F_\nabla \wedge \psi) \sim F_\nabla^7$.

intermediate energy: \begin{align} E^\psi (\nabla, \Phi) &= \int\limits_X (|F_\nabla^7|^2 + |\nabla \Phi|^2) \mathrm{vol} \\ &= \underbrace{\int\limits_X (|\ast (F_\nabla \wedge \psi) - \nabla \Phi|^2) \mathrm{vol} + \mbox{boundary integral (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 \begin{equation} (X - K, g_\varphi|_{X - K}) \sim ([R, \infty) \times \Sigma, \mathrm{d}r^2 + r^2 g_\Sigma). \end{equation}

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).

Main Theorem (Fadel–N–Oliveira '20)

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

$|\Phi| \rightarrow m > 0$, (the mass of the monopole)

$\underbrace{|\nabla \Phi| = O (r^{- 6})}_{\mbox{sharp}}$ and $|[F_\nabla, \Phi]|$ decays exponentially,

In some gauge: $(\nabla, \Phi)|_{\Sigma_R} \rightarrow (\nabla^\infty, \Phi^\infty)$.

$\underbrace{\Lambda F_{\nabla^\infty} = 0, F_\nabla^{0,2} = 0}_{\mbox{pseudo-Hermitian–Yang–Mills}}$ and $\underbrace{\nabla^\infty \Phi^\infty = 0}_{\mbox{reduction of the bundle to $L \oplus L^*$}}$.

$E^\psi (\nabla, \Phi) = 4 \pi m \left\langle \: \alpha \cup \psi_\infty \: \middle| \: \left[ \Sigma \right] \: \right\rangle$, where $\alpha = c_1 (L)$ is the monopole class.

Corollaries: Finite mass, monopole class, 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 \begin{equation} |\nabla \Phi (x)| \leqslant \frac{C}{|x|^{7/2}} \int\limits_{B_{|x|/2} (x)} |\nabla \Phi|^2 \mathrm{vol}_{g_\varphi}. \end{equation}

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

Recall: $\int\limits_X |\nabla (\chi \Psi)|^2 \mathrm{vol}_{g_\varphi} = \int\limits_X |\mathrm{d} \chi|^2 |\Psi|^2 \mathrm{vol}_{g_\varphi} + \int\limits_X \chi^2 \langle \Psi | \nabla^* \nabla \Psi \rangle \mathrm{vol}_{g_\varphi}$
We prove a version of Hardy's inequality: $\int\limits_X f^2 \mathrm{vol}_{g_\varphi} \leqslant \mathbf{\frac{4}{25}} \int\limits_X |\mathrm{d} (r f)|^2 \mathrm{vol}_{g_\varphi} + \mbox{small stuff}$.

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

After a little more magic, this yields: $\int\limits_X r^{2 \alpha} |\nabla \Phi|^2 \mathrm{vol}_{g_\varphi} \leqslant \frac{C}{5 - 2 \alpha} \| \nabla \Phi \|_{L^2}^2$.

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, $N \subset X$,
- a Fueter section, $\Psi$, from $N$ into the BPS moduli,
- and a singular Dirac monopole, $\left( D, \phi \right)$, around $N$.

Indices equal: "Callias = gluing data"
Theorem: \begin{equation} \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 via 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}}. \end{equation}

Current goal: glue-in.

Glue-in (joint with Esfahani, Fadel, Foscolo, and Oliveira)

Fix:
- Coassociative, $N \subset X$,
- a Fueter section, $\Psi$, from $N$ into the BPS moduli (with mass $m \gg 1$),
- and a singular Dirac monopole, $\left( D, \phi \right)$, around $N$.

Approximate solution: $\left( \nabla^0, \Phi^0 \right)$ close to $N$ is $\Psi$ and close to infinity is $\begin{pmatrix} \left( D, i m + \phi \right) & 0 \\ 0 & \left( D^*, - i m - \phi \right) \end{pmatrix}$.

Look for solutions in the form of $\left( \nabla, \Phi \right) = \left( \nabla^0, \Phi^0 \right) + \mathcal{L}_{\left( \nabla^0, \Phi^0 \right)} \left( \dot{A}, \dot{\Phi} \right)$.

Problem: $\mathcal{L} \mathcal{L}^* = \nabla^* \nabla + \mathrm{ad}_\Phi^* \mathrm{ad}_\Phi + \mathcal{R}_\nabla$, where $\mathcal{R}_\nabla$ is indefinite.

Heuristically, $\mathcal{L}$ is surjective when $b_1 (N) = 0$.

Future wish list

Generalization to ALC $G_2$-manifolds (lots of examples by Foscolo).

Generalization to other "ALX" $G_2$-manifolds; most of our work should generalize to nonparabolic manifolds.

Calabi–Yau-monopoles and special Lagrangians.

"Finiteness"/"Compactness"/Enumerative invariants.