Musings

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” $\overline{D}$ operator¹ 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

1. 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.
2. 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 …