### Twistor Yang Mills

In a previous post, I promised I would say something about Boels et al. Aside from generally being busy with other matters, I’ve been rather confused about their paper.

The idea that they want to sell is that there’s a certain (nonlocal) 6-dimensional field theory, living on projective twistor space, $\mathbb{P}\mathbb{T}'$. Performing a partial gauge-fixing, and integrating over the fibers of $\mathbb{P}\mathbb{T}'\to \mathbb{R}^4$, we obtain either conventional 4D Yang Mills, or Mansfield’s Lagrangian, depending on which gauge choice we make.

This sounds very plausible; it’s the details that I’m hazy about.

Let’s recall a few things about twistors. Let $M$ be our 4-dimensional spacetime. I’ll be deliberately hazy about its signature, but you probably want to think about the case of Euclidean $\mathbb{R}^4$. Twistor space, $\mathbb{T} = S_+\to M$ is the total space of the bundle of right-chiral spinors on $M$. When we need to, we’ll denote the fiber coordinates by $\pi_{\dot{\alpha}}$. $\mathbb{P}\mathbb{T}'$, its projectivization, is a $\mathbb{P}^1$ bundle, $\mathbb{P}\mathbb{T}'\overset{\mu}{\to}M$. We can compactify it to $\mathbb{P}\mathbb{T}= \mathbb{P}^3$ Denoting the homogeneous coordinates on $\mathbb{P}\mathbb{T}= \mathbb{P}^3$ by $(\omega^\alpha,\pi_{\dot{\alpha}})\simeq (\lambda \omega^\alpha,\lambda\pi_{\dot{\alpha}}),\qquad \lambda\in \mathbb{C}^*$ $\mathbb{P}\mathbb{T}'=\mathbb{P}\mathbb{T}\setminus \mathbb{P}^1$ is the open subset where $\pi_{\dot{\alpha}}\nequiv 0$. Note that, in doing so, we’ve endowed $\mathbb{P}\mathbb{T}'$ with a complex structure. The fiber bundle structure is manifested by writing $\omega^\alpha = x^{\alpha\dot{\alpha}}\pi_{\dot{\alpha}}$ where $x^{\alpha\dot{\alpha}}= x^\mu \sigma^{\alpha\dot{\alpha}}_\mu$ are the standard coordinates on $M$.

Let $P=\mu^* P'$ be a $G$-principal bundle on $\mathbb{P}\mathbb{T}'$, which is pulled back from a $G$-principal bundle on $M$. The fields of our theory are

- $A$, a (0,1) connection on $P$. For present purposes, we can think of it as an ($ad P$)-valued (0,1) form $\Omega^{0,1}(\mathbb{P}\mathbb{T}', ad P)$.
- $B\in \Omega^{0,1}(\mathbb{P}\mathbb{T}', ad P\otimes \mathcal{O}(-4))$.

Note that $\overline{D}=\overline{\partial} +A$ is typically *not* closed: $\overline{D}^2 = F\in \Omega^{0,2}(\mathbb{P}\mathbb{T}', ad P)$.

Our theory will have an extended gauge invariance

where $\beta\in \Gamma(\ad P \otimes\mathcal{O}(-4))$.

The action that Boels et al propose consists of two pieces

where

Here $\Omega\in H^0(\mathbb{P}\mathbb{T}, \Omega^{(3,0)}(4))$ is canonically defined (up to an overall scale). In the above coordinates, $\begin{aligned} \Omega &= \epsilon_{\alpha\beta}\epsilon^{\dot{\alpha}\dot{\beta}} \pi_{\dot{\alpha}}d\pi_{\dot{\beta}} d\omega^\alpha d\omega^\beta\\ &= \epsilon_{\alpha\beta}\epsilon^{\dot{\alpha}\dot{\beta}} \pi_{\dot{\alpha}}d\pi_{\dot{\beta}} \pi_{\dot{\gamma}}\pi_{\dot{\delta}} d x^{\alpha\dot{\gamma}} d x^{\beta\dot{\delta}} \end{aligned}$

The second term is a bit more mysterious. Let $S=\mathbb{P}\mathbb{T}'\times_M \mathbb{P}\mathbb{T}'$ be the fiber product (a $\mathbb{P}^1\times\mathbb{P}^1$ bundle over $M$) and $\Delta\subset S$ be the diagonal (a divisor in $S$). Let $\delta^{(2)}_\Delta$ be the usual 2-form bump-form supported on $\Delta$, normalized so that the integral along the fibers of $S\to \mathbb{P}\mathbb{T}'\simeq \Delta$ is equal to 1. Let $\tau: S\righttoleftarrow$ be the involution which exchanges the two fibers. Let $s$ be a holomorphic section of $\mathcal{O}(1,1)$, the line bundle of bidegree (1,1), with respect to this $\mathbb{P}^1\times \mathbb{P}^1$ fiber bundle, which has a simple zero on $\Delta$. In the coordinates introduced above,
$s = \langle \pi^{(1)} \pi^{(2)}\rangle = \epsilon^{\dot{\alpha}\dot{\beta}} \pi^{(1)}_{\dot{\alpha}} \pi^{(2)}_{\dot{\beta}}$
Similarly, let $\kappa_{1 2}$ be a section of $K_1^{1/2}\otimes K_2^{1/2}\otimes ad P$ with a simple pole along $\Delta$. Here $K_i$ are the vertical canonical bundles with respect to the two fibrations $S\to\mathbb{P}\mathbb{T}'$. Since the fiber is a $\mathbb{P}^1$, these have a canonical square-root. $\kappa$ is chosen to satisfy
$\overline{D}_{\text{vert}} \kappa = \delta^{(2)}_\Delta$
where $\overline{D}_{\text{vert}}$ is the “vertical” $\overline{D}$ operator^{1} on $\mathbb{P}\mathbb{T}'\to M$. When $A_{\text{vert}}=0$, we can canonically identify $K_1^{1/2}\otimes K_2^{1/2}\simeq \mathcal{O}(-1,-1)$, and $\kappa|_{A_{\text{vert}}=0}\equiv\hat{\kappa} = \frac{1}{2\pi i} s^{-1}$. Let $\kappa_{2 1} = \tau^*(\kappa_{1 2})$.

The second term in the action is

This is invariant under the extended gauge symmetry (1), but nonlocal in the fiber directions. It also depends nonpolynomially on $A_{\text{vert}}$, the “vertical” component of the connection

where the integration is over the 2^{nd} through $(n-1)$^{st} fibers of
$S_n = \overset{n\, \text{copies}}{\overbrace{\mathbb{P}\mathbb{T}'\times_M \mathbb{P}\mathbb{T}'\times_M\dots \times_M \mathbb{P}\mathbb{T}'}}$

Now, the claim is two-fold

- Doing a partial gauge-fixing of (1) $A_{\text{vert}}=0,\quad \overline{D}^\dagger_{\text{vert}}B_{\text{vert}}=0$ and integrating over the fibers, one obtains the ordinary Yang-Mills action on $M$, in the form written by Siegel and Chalmers.
- Instead, gauge-fixing to axial gauge $\eta^\alpha A_\alpha =\eta^\alpha B_\alpha=0$, integrating out the vertical components of the fields, and then integrating along the fibers, one obtains Mansfield’s Lagrangian.

Unfortunately, I can’t follow this latter computation at all. What I’ve tried to do here is clean up the presentation of the Lagrangian (3),(4) to the point where I hoped I might see my way through to the result.

Alas …

^{1} $\overline{D}_{\text{vert}} = d\overline{z}\, i_{\partial/\partial\overline{z}} \overline{D}$, with $z$ a local complex coordinate on the fiber. Since the fiber is 1-complex dimensional, $\overline{D}_{\text{vert}}^2=0$

## Re: Twistor Yang Mills

Dear Jacques,

I fear that we/I must have generated some confusion, so let’s see: In our published work we did not make contact with Mansfield’s approach (see below though). Instead, we did the following

1. As you say, we can reduce to the space-time action in Chalmers and Siegel form by doing a partial gauge-fixing of the twistor action with the requirement that the (0,1)-forms to be harmonic up the fibres and integrating out the fibre directions etc..

2. Instead, we can obtain the MHV diagram rules of Cachazo, Svrcek and Witten directly as the Feynman rules of this action by gauge-fixing to an axial gauge which is inaccessible from space-time.

This axial gauge is obtained by choosing an anti-holomorphic direction on twistor space that is horizontal over space-time and corresponds to the choice of a constant spinor in the MHV diagram formalism. In this gauge, momentum eigenstates are particularly simple being delta functions up the fibre, allowing us to do all the fibre integrals which then yields the MHV diagram rules.

As a separate point, it can also be shown that the ‘Ettle-Morris’ coefficients of Mansfield’s field transformation are exactly equivalent to the space-time projection (a la eq. 24 in our last paper) of the twistor fields in this particular gauge. This shows that their non-linear canonical transformation on space-time is our linear gauge transformation on twistor space.

Hope to have cleared some of the confusion!