### Twisted N=4 SYM and Special Holonomy

Luboš blogs about Witten’s ~~Loeb Lectures~~talks at Harvard on Geometric Langlands, and reminds me of one of the most beautiful (and, at least in the Mathematics community, unexploited) aspects of the subject: the connection between topological twistings of $N=4$ SYM and manifolds of special holonomy.

So I thought I’d post a little review…

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

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

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

$S^+\otimes S^+ = \mathcal{O} \oplus (\wedge^2 T^*_\mathbb{C})_+,\quad S^-\otimes S^- = \mathcal{O} \oplus (\wedge^2 T^*_\mathbb{C})_-,\quad S^+\otimes S^- = T^*_\mathbb{C}$So, for the three cases, we find

- 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 $(\wedge^2V)_\mathbb{R}= \mathbb{R}^3\oplus (\wedge^2 T^*)_+$
- 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 $(\wedge^2V)_\mathbb{R}= \mathbb{R}^2\oplus T^*$
- 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.

- 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, $N_{M|X}= R^3\oplus (\wedge^2T^*)_+$.
- 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 $N_{M|X}= R^2\oplus T^*$.
- 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.

## Re: Twisted N=4 SYM and Special Holonomy

Witten wasn’t giving the Loeb lectures (that was Hopfield). He was speaking Thursday in the joint Boston area mathematics colloquium and Friday in the Harvard math department gauge theory and topology seminar.

By the way, a character doesn’t appear correctly in this posting, both on mozilla in linux and IE on windows. Looks like its a blackboard bold C.