# Anomalies in the self-dual Yang-Mills theory and $\mathcal{N}=2$ open-string theory

I am reading a paper, written by G. Chalmers and W. Siegel - https://arxiv.org/abs/hep-th/9606061, where they discuss the action of self-dual Yang-Mills theory, which in light-cone formalism is obtained by preserving only one of the triple vertices with the gluons of following helicity (++-), or in Lorentz invariant form the action has the following form: $$S = \int_{M} \mathcal{P} \wedge F.$$ I adopt the notation of https://arxiv.org/abs/1711.10026. There $$\mathcal{P}$$ is an anti self-dual field ($$\star \mathcal{P} = -\mathcal{P}$$), which leads to the following equation of motion for the gauge field $$A_\mu$$ : $$\star F = -F$$ Which is simply the self-duality equation for the field $$F$$. This theory has only two kind of amplitudes:

• Tree amplitudes with one field $$\mathcal{P}$$ and any number of external gluons.

• 1-loop amplitude with no field $$\mathcal{P}$$ and any number of external gluons.

The expression for the 1-loop amplitude is given by rather a simple expression: $$\mathcal{A}(1^+ \ldots n^+) = -\frac{i}{192 \pi^2}\sum \frac{[i j] \langle j k \rangle [k l] \langle l i \rangle}{\langle 1 2 \rangle \ldots \langle n 1 \rangle}$$ Which significantly resembles the famous $$\textbf{MHV}$$ amplitude.

The theory concerned is conformal on the classical level. However, at the quantum level there can be a conformal anomaly, generating the scale.

It is mentioned in the discussion section, that there is a possibility that the aforementioned one-loop contribution may by generated by a local term in the effective action by an introduction of an external field.

Do I correctly understand, that they speak about some contribution of form $$\phi T_{\mu}^{\mu}$$, where $$\phi$$ is kind of dilaton field?

The supersymmetric generalization of action of self-dual Yang-Mills action emerges as a low-energy of $$\mathcal{N}=2$$ open string theory. Then the authors say:

Explicit calculations in string theory [23], however, have indicated thevanishingof all one-loop graphs with more than three external lines in all N=2 string theories.These string results are in direct contradiction with field theory. This suggests somesubtlety was missed, possibly signalling the presence of an anomaly in the worldsheettheory describing the string.

Is there any progress in solving this issue since that time? Or any results for the anomalies in $$\mathcal{N}=2$$ string theory? I would be grateful for links or discussions in this area.

Yes, there are recent progress made in this issue. In fact, I have worked on my PhD thesis solely trying to address it. These are the two papers where we have shown that the construction of the one loop amplitudes mimick the chiral anomaly computation.

https://arxiv.org/abs/2002.11390

https://arxiv.org/abs/2110.00331

With Kirill Krasnov, I have deduced a simple formula for the series of the same helicity one-loop Yang-Mills amplitudes in terms of the Berends-Giele currents and an effective propagator.

There has been a very recent work by Ricardo Monteiro and company who showed that the anomaly results from the quantum correction of the infinite tower of symmetries in SDYM and SDGR. Here is the paper

https://arxiv.org/abs/2211.12407

Open questions remain, specifically in the context of perturbative quantum gravity. It has been shown by Bern et al., that the two loop divergence in pure gravity and the four loop divergence in $$N=4$$ supergravity may result from the anomalies in SDGR. Further, Costello showed that these anomalies can be cancelled for some very specific Lie groups by the Green Schwarz mechanism, introducing an axion field. The relevant papers are:

https://arxiv.org/abs/2111.08879 by Costello

https://arxiv.org/abs/1701.02422 by Bern and company.

I am happy to discuss all this in more details, should you be interested.