## December 16, 2003

### Ed’s latest

#### Posted by Arvind

Ed Witten’s latest: Is weakly coupled (N=4) Yang-Mills-theory also a perturbative string theory? He conjectures that it is, and that the string theory is a topological string theory deformed by D-instantons.

The topological B-model in question has target space ${\mathrm{CP}}^{3\mid 4}$. The correlation functions of this theory are invariant under transformations that leave the holomorphic three-form invariant. This symmetry group turns out to be PSL(4,4), the symmetry of N=4 Yang-Mills theory!

So could the action just be Yang-Mills theory? Unfortunately not. The topological model (at least the open string sector) produces a anti self-dual field A, and a self dual field G (plus some more stuff). The action turns out to be roughly $L=G\mathrm{dA}$, while the true Yang-Mills action would require an action $L=G\mathrm{dA}+ϵ{G}^{2}$ where $ϵ\sim {g}_{\mathrm{YM}}^{2}$.

EW proposes that this extra term is produced by the presence of D-instantons in the string theory (every instanton=one power of $ϵ$). As evidence, he shows that YM amplitudes with n gluons of one helicity and 2 of the opposite helicity can be reproduced in the string theory by including D-instantons (the gauge theory amplitudes are apparently well known for this case).

More generally, suppose this correspondence is true. Then there is a match between the number of D-instantons and the power of $ϵ$ in the amplitude. The terms with ${ϵ}^{r}$ can be shown to correspond to tree level processes with r gluons of negative helicity, and should be reproduced by processes with (so to speak) r-1 instantons. But an instanton number $r-1$ corresponds to a holomorphic curve with degree $d=r-1$. So this further implies that the amplitude should only be nonzero if the incoming particles all lie on this curve. (Well, not exactly; since the curve could be disconnected. But this gets into messy details.)

So we get the next conjecture: Gauge theory processes with $r$ helicity violating gluons, and at loop $l$, are nonzero only if the incoming particles lie on a curve of degree $d=r-1+l$.

This apparently works for the cases EW considers (amplitudes with 4, 5, 6 gluons). The general proof appears difficult. Even for the specific cases, many details appear to be unclear still.

I should also note that there is an important symmetry S under which the first term in the action above has S=-4, while the second has S=-8. D-instantons need to have a particular value of S in order that they contribute correctly. The required formula is presented in the paper, and a heuristic derivation is given. This is a strong indication that something is working.

On the other side, there are many issues (pointed out already in the paper). All the above only works with the open strings, and the closed string sector is completely mysterious. It is suggested that the closed strings will produce a conformal supergravity theory, and in that case the agreement between the string theory and the gauge theory is only valid at the planar level (i.e. large N). (There are issues with non-planar diagrams as well.)

Anomalies are apparently a major mystery. EW points out potential anomalies in the open string quantization (of what he calls D1-D5 and D1-D1 strings), the holomorphic anomaly of the B-model, the c-anomaly of the N-4 YM theory (how can it be coupled to conformal anything) etc. If even EW is puzzled by these anomalies, I certainly can’t say anything useful here.

Immediate developments seem clear. People will start rapidly calculating stuff in the B-model theory. Particularly to understand the coupling of closed strings in this model. There are several conjectures in section 3 of the paper which might be interesting to prove. There are also suggested extensions to other B-models.

And there is the whole issue of theories with less SUSY. These behave differently at loop level. Lots of stuff to do there potentially.

Posted at December 16, 2003 8:02 AM UTC

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Read the post Do the Twist
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Excerpt: Everyone's all a-titter about Witten's latest, which appeared today. He studies Maximal Helicity-Violating amplitudes (n-2 gluons of one chirality...
Tracked: December 16, 2003 3:04 PM

### Re: Ed’s latest

Posted by: Robert on December 16, 2003 3:43 PM | Permalink | Reply to this

### Re: Ed’s latest

I still have to read that paper. Maybe somebody can give me a head start by briefly pointing out what the topological B model is. I roughly know how to get the topological string from the superstring by doing a twist, but that’s about it, currently. :-)

Posted by: Urs Schreiber on December 18, 2003 7:10 PM | Permalink | Reply to this

### Re: Ed’s latest

Urs,

first of all you can read more than the first half of the paper without having to worry about B-model topological strings at all becaus that’s about pure and $calN=4$ gauge theory in 4d. If your question would have been “What is a twistor?” that of course would have been a different matter…

Witten himself gives a very brief explanation of the twist in the beginning of chapter 4 and there he also gives references to some of his older papers where he explains this twist in much more detail.

In very crude detail what you do is the following: You start with a theory with $calN=2$ worldsheet susy. As described on p. 50, this contains a $U\left(1\right)$ R-symmetry generated by $K$ that rotates left and right handed spinors differnetly.

In 2d, there is of course also $J$ the generator of rotations which also generates a $U\left(1\right)$. What you do now is to to mix these two symmetries and define a new Poincar'e group (and thus a new theory) that uses $J\prime =J+K/2$ to generate rotations. In this new theory the spins of the fields differ by $K/2$ from the spins in the original theory. Especiially, there is now a scalar supercharge $Q$ which has all the properties of a BRST charge. And with respect to this BRST charge the theory is topological. But all this is not really important for the paper.

An information that you might need however is that in that theory the condition for a D-brane to be supersymmetric is only to be Lagrangian and not special Lagrangian which means holomorphic is this setup. This is why considers those surfaces he calls $C$.

Posted by: Robert on December 19, 2003 8:27 AM | Permalink | Reply to this

### Re: Ed’s latest

Hi Robert,

thanks. Ok, I think that’s what I have seen before in a talk by Peter Mayr. I didn’t recall it being referred to as the “B-model”. What’s the “A-model” then? Does it have to do with “IIB or not IIB”?

Posted by: Urs Schreiber on December 19, 2003 11:41 AM | Permalink | Reply to this

### A & B

Presumably, you need to read Commun. Math. Phys. 118 (1988) 411.

Briefly, in an $N=\left(2,2\right)$ SCFT, there is both a left- and a right-moving $U\left(1\right)$ current. Making the same twist on both left and right gives you the A-model (which exists for any almost-complex manifold – i.e., for sigma models that are not even conformal). Making the opposite twist

(1)$h\to h+q/2,\overline{h}\to \overline{h}-\overline{q}/2$

on left and right gives you the B-model (which exists only for Calabi-Yau manifolds).

Posted by: Jacques Distler on December 19, 2003 3:53 PM | Permalink | Reply to this

### Re: A & B

Thanks. I’ll look at that paper as soon as I find the time.

Meanwhile, what is the physicsal interpretation of the twist? It seems to me that requiring states to be annihilated by the twisted constraint is equivalent to requiring their conformal dimension to be equal to their R-charge, or something like that.

Posted by: Urs Schreiber on December 19, 2003 6:09 PM | Permalink | Reply to this

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