Musings

We’ll work directly in Euclidean signature, on a 4D spin-manifold, $M$. The field content of (twisted) $N=4$ SYM is as follows

• $A$, a connection on a principal $G$-bundle, $P\to M$.
• Fermions, $\psi\in \Gamma\left(( S^+(M)\otimes V \oplus S^-(M)\otimes \overline{V})\otimes ad(P)\right)$
• Bosons, $\phi\in \Gamma\left((\wedge^2V)_\mathbb{R}\otimes ad(P)\right)$

Here $V$ is a rank-4 vector bundle associated to principal spin bundle of $M$, via some 4d representation. The “untwisted” theory corresponds to taking $V=\mathcal{O}^{\oplus 4}$, the rank-4 trivial bundle. More generally, the representations of $Spin(4)$ are real or pseudoreal, so $V\simeq \overline {V}$ and, moreover, $\wedge^2(V)$ is inevitably associated to a real representation of $Spin(4)$. $(\wedge^2V)_\mathbb{R}$ is the associated rank-6 real vector bundle.

The (would-be) supercharges, like the fermions, transform as sections of

For $V$ trivial, supersymmetry is generally broken (the above bundle has no nowhere-vanishing sections) unless $M$ is flat. The topologically-twisted versions correspond to choices of $V$, for which the bundle (1) has a trivial subbundle, and hence one or more unbroken supercharges, even on a general curved manifold $M$.

The basic observation is that, up to parity, there are three choices of $V$ with this property. They are

1. $V= S^+\oplus S^+$
2. $V= S^+\oplus S^-$
3. $V= S^+\oplus \mathcal{O}^{\oplus 2}$

The basic relations (which follow from the elementary group theory of $Spin(4)= SU(2)\times SU(2)$) are

So, for the three cases, we find

1. Fermions (supercharges) in $$(S^+\oplus S^-)\otimes V = \left(\mathcal{O}\oplus T^*_\mathbb{C} \oplus (\wedge^2 T^*_\mathbb{C})_+\right)^{\oplus 2}$$ and bosons in $$(\wedge^2V)_\mathbb{R}= \mathbb{R}^3\oplus (\wedge^2 T^*)_+$$
2. Fermions (supercharges) in $$(S^+\oplus S^-)\otimes V =\mathcal{O}\oplus \mathcal{O} \oplus T^*_\mathbb{C} \oplus T^*_\mathbb{C}\oplus \wedge^2 T^*_\mathbb{C}$$ and bosons in $$(\wedge^2V)_\mathbb{R}= \mathbb{R}^2\oplus T^*$$
3. Fermions (supercharges) in $$(S^+\oplus S^-)\otimes V =\mathcal{O}\oplus T^*_\mathbb{C} \oplus (S^+\oplus S^-)^{\oplus 2}$$ and bosons in $$(\wedge^2V)_\mathbb{R}= \mathbb{R}^2\oplus (S^+\oplus S^+)_\mathbb{R}$$

You might be forgiven if, at first, you don’t see the pattern in this list. But, if you think physically, the answer becomes clear. These topologically-twisted theories arise as the world-volume theories on (Euclidean) D3-branes wrapped on a supersymmetric 4-cycle $M\subset X$, where $X$ is a spin 10-manifold. The bosons above arise as sections of the normal bundle, $N_{M|X}$. What are the possible cases that arise? They’re all related to the manifolds of special holonomy which admit supersymmetric 4-cycles.

1. Let $X=\mathbb{R}^3\times Y$, where $Y$ is a 7-manifold, $Y=(\wedge^2T^*)_+\to M$, the bundle of self-dual 2-forms on $M$. Bryant showed that $Y$ admits a metric of $G_2$-holonomy, and, with respect to this metric, $M$ — embedded via the zero-section — is a supersymmetric cycle. The normal bundle, $N_{M|X}= R^3\oplus (\wedge^2T^*)_+$.
2. Let $X= \mathbb{R}^2\times Y$, where $Y= T^*\to M$, the total space of the cotangent bundle of $M$, which is a Calabi-Yau 4-fold, admitting a metric of $SU(4)$-holonomy. Again, $M$ — embedded via the zero-section — is a supersymmetric 4-cycle. The normal bundle is $N_{M|X}= R^2\oplus T^*$.
3. Finally, let $X= \mathbb{R}^2\times Y$, where $Y$ is the total space of $(S^+\oplus S^+)_\mathbb{R}\to M$ and admits a metric of $Spin(7)$-holonomy.

That’s the exhaustive list of manifolds of special holonomy which admit supersymmetric 4-cycles, so that’s the list of topologically-twisted version of $N=4$ SYM. The first two cases have a pair of unbroken nilpotent supercharges; the third has just a single nilpotent supercharge. Each one is interesting in its own right, but it’s case 2 which is of relevance to Geometric Langlands. I’ll explain why, on some other occasion.