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September 27, 2007

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, 𝕋\mathbb{P}\mathbb{T}'. Performing a partial gauge-fixing, and integrating over the fibers of 𝕋 4\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 MM be our 4-dimensional spacetime. I’ll be deliberately hazy about its signature, but you probably want to think about the case of Euclidean 4\mathbb{R}^4. Twistor space, 𝕋=S +M \mathbb{T} = S_+\to M is the total space of the bundle of right-chiral spinors on MM. When we need to, we’ll denote the fiber coordinates by π α˙\pi_{\dot{\alpha}}. 𝕋\mathbb{P}\mathbb{T}', its projectivization, is a 1\mathbb{P}^1 bundle, 𝕋μM\mathbb{P}\mathbb{T}'\overset{\mu}{\to}M. We can compactify it to 𝕋= 3\mathbb{P}\mathbb{T}= \mathbb{P}^3 Denoting the homogeneous coordinates on 𝕋= 3\mathbb{P}\mathbb{T}= \mathbb{P}^3 by (ω α,π α˙)(λω α,λπ α˙),λ * (\omega^\alpha,\pi_{\dot{\alpha}})\simeq (\lambda \omega^\alpha,\lambda\pi_{\dot{\alpha}}),\qquad \lambda\in \mathbb{C}^* 𝕋=𝕋 1\mathbb{P}\mathbb{T}'=\mathbb{P}\mathbb{T}\setminus \mathbb{P}^1 is the open subset where π α˙0\pi_{\dot{\alpha}}\nequiv 0. Note that, in doing so, we’ve endowed 𝕋\mathbb{P}\mathbb{T}' with a complex structure. The fiber bundle structure is manifested by writing ω α=x αα˙π α˙\omega^\alpha = x^{\alpha\dot{\alpha}}\pi_{\dot{\alpha}} where x αα˙=x μσ μ αα˙x^{\alpha\dot{\alpha}}= x^\mu \sigma^{\alpha\dot{\alpha}}_\mu are the standard coordinates on MM.

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

  • AA, a (0,1) connection on PP. For present purposes, we can think of it as an (adPad P)-valued (0,1) form Ω 0,1(𝕋,adP)\Omega^{0,1}(\mathbb{P}\mathbb{T}', ad P).
  • BΩ 0,1(𝕋,adP𝒪(4))B\in \Omega^{0,1}(\mathbb{P}\mathbb{T}', ad P\otimes \mathcal{O}(-4)).

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

Our theory will have an extended gauge invariance

(1)AgAg 1+gdg 1,Bg(B+D¯β)g 1A\to g A g^{-1} + g d g^{-1},\qquad B\to g (B + \overline{D}\beta) g^{-1}

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

The action that Boels et al propose consists of two pieces

(2)S=S BF+S NLS = S_{BF} + S_{NL}

where

(3)S BF= 𝕋Ωtr(BF)S_{BF} = \int_{\mathbb{P}\mathbb{T}'} \Omega\wedge \tr ( B\wedge F)

Here ΩH 0(𝕋,Ω (3,0)(4))\Omega\in H^0(\mathbb{P}\mathbb{T}, \Omega^{(3,0)}(4)) is canonically defined (up to an overall scale). In the above coordinates, Ω =ϵ αβϵ α˙β˙π α˙dπ β˙dω αdω β =ϵ αβϵ α˙β˙π α˙dπ β˙π γ˙π δ˙dx αγ˙dx βδ˙ \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=𝕋× M𝕋S=\mathbb{P}\mathbb{T}'\times_M \mathbb{P}\mathbb{T}' be the fiber product (a 1× 1\mathbb{P}^1\times\mathbb{P}^1 bundle over MM) and ΔS\Delta\subset S be the diagonal (a divisor in SS). Let δ Δ (2)\delta^{(2)}_\Delta be the usual 2-form bump-form supported on Δ\Delta, normalized so that the integral along the fibers of S𝕋ΔS\to \mathbb{P}\mathbb{T}'\simeq \Delta is equal to 1. Let τ:S\tau: S\righttoleftarrow be the involution which exchanges the two fibers. Let ss be a holomorphic section of 𝒪(1,1)\mathcal{O}(1,1), the line bundle of bidegree (1,1), with respect to this 1× 1\mathbb{P}^1\times \mathbb{P}^1 fiber bundle, which has a simple zero on Δ\Delta. In the coordinates introduced above, s=π (1)π (2)=ϵ α˙β˙π α˙ (1)π β˙ (2) s = \langle \pi^{(1)} \pi^{(2)}\rangle = \epsilon^{\dot{\alpha}\dot{\beta}} \pi^{(1)}_{\dot{\alpha}} \pi^{(2)}_{\dot{\beta}} Similarly, let κ 12\kappa_{1 2} be a section of K 1 1/2K 2 1/2adPK_1^{1/2}\otimes K_2^{1/2}\otimes ad P with a simple pole along Δ\Delta. Here K iK_i are the vertical canonical bundles with respect to the two fibrations S𝕋S\to\mathbb{P}\mathbb{T}'. Since the fiber is a 1\mathbb{P}^1, these have a canonical square-root. κ\kappa is chosen to satisfy D¯ vertκ=δ Δ (2) \overline{D}_{\text{vert}} \kappa = \delta^{(2)}_\Delta where D¯ vert\overline{D}_{\text{vert}} is the “vertical” D¯\overline{D} operator1 on 𝕋M\mathbb{P}\mathbb{T}'\to M. When A vert=0A_{\text{vert}}=0, we can canonically identify K 1 1/2K 2 1/2𝒪(1,1)K_1^{1/2}\otimes K_2^{1/2}\simeq \mathcal{O}(-1,-1), and κ| A vert=0κ^=12πis 1\kappa|_{A_{\text{vert}}=0}\equiv\hat{\kappa} = \frac{1}{2\pi i} s^{-1}. Let κ 21=τ *(κ 12)\kappa_{2 1} = \tau^*(\kappa_{1 2}).

The second term in the action is

(4)S NL= Sd 4xs 4tr(B (1)κ 12B (2)κ 21)S_{NL} = \int_S d^4 x\, s^4\, \tr ( B^{(1)} \kappa_{1 2} B^{(2)} \kappa_{2 1})

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

(5)κ 12=κ^ 12+ n=3 1(n2)! S nSκ^ 13A vert (3)κ^ 34A vert (4)κ^ (n1)nA vert (n)κ^ n2 \kappa_{1 2} = \hat{\kappa}_{1 2} + \sum_{n=3}^\infty \tfrac{1}{(n-2)!} \int_{S_n\to S} \hat{\kappa}_{1 3} A^{(3)}_{\text{vert}}\hat{\kappa}_{3 4} A^{(4)}_{\text{vert}} \dots \hat{\kappa}_{(n-1) n} A^{(n)}_{\text{vert}}\hat{\kappa}_{n 2}

where the integration is over the 2nd through (n1)(n-1)st fibers of S n=𝕋× M𝕋× M× M𝕋ncopies 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 vert=0,D¯ vert B vert=0 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 MM, in the form written by Siegel and Chalmers.
  2. Instead, gauge-fixing to axial gauge η αA α=η αB α=0\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 D¯ vert=dz¯i /z¯D¯\overline{D}_{\text{vert}} = d\overline{z}\, i_{\partial/\partial\overline{z}} \overline{D}, with zz a local complex coordinate on the fiber. Since the fiber is 1-complex dimensional, D¯ vert 2=0\overline{D}_{\text{vert}}^2=0

Posted by distler at September 27, 2007 12:22 AM

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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!

Posted by: Rutger Boels on October 2, 2007 10:39 AM | Permalink | Reply to this

Re: Twistor Yang Mills

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.

I’m not sure why this should be different from obtaining Mansfield’s action.

It puzzles me that you didn’t just follow the same procedure that you followed in the Chalmers-Siegel case, to obtain a 4D action by integrating over the fibers.

The reason why this bothers me is that Ettle et al explained, in very nice fashion, the origin of certain violations of the Equivalence Theorem in the perturbation theory which follows from Mansfield’s action.

That, too, should emerge from your approach if, indeed, you are really getting the right 4D theory from twistor space.

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.

It is not, at all, a separate point. That’s exactly what I want to see emerge explicitly in your formalism. I’d like to see Ettle et al’s ϒ\Upsilon and Ξ\Xi coefficients derived explicitly from your equation (24).

Posted by: Jacques Distler on October 2, 2007 11:34 AM | Permalink | PGP Sig | Reply to this

Re: Twistor Yang Mills

Ok, I can see your worry, I think.

Obtaining the space-time action is indeed easy by performing all the fibre integrals with the help of the momentum delta-functions in this particular gauge. This gives the original CSW rules. The problem of comparing to Mansfield’s approach is more that as far as I know the vertices there have only been explicitly checked (off-shell) for 5 gluons, although nobody really doubts the general result. The ingredient which is then missing is finding the right external states. This is what worries you, right?

So let’s see: In our februari paper we were mainly concerned with deriving the MHV tree level result & assumed in effect that the ‘equivalence theorem’ held: we only evaluated eq (24) to first order to obtain the external states.

Now since that paper we’ve learned that CSW gauge on twistor space corresponds to lightcone gauge on space-time (see hep-th/0703080). With this it is easy to guess what the physical states of the gluon on space-time are in terms of twistor fields: Eq. (24) expanded further than linear order then gives you exactly the EM coefficients after doing all the fibre integrals. This has not been published anywhere in full generality yet, although I talked about it here or here for example.

Posted by: Rutger Boels on October 3, 2007 4:57 AM | Permalink | Reply to this

Re: Twistor Yang Mills

After way too many months of it slowly nearing completion, can’t resist advertising 0805.1197 [hep-th] where all possible confusion is hopefully settled.

Posted by: Rutger Boels on May 9, 2008 4:29 AM | Permalink | Reply to this

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