These last two days, two papers appeared on related subjects. Broadly speaking, they deal with $\mathcal{N}=(2,2)$ theories in two dimensions. Another common feature is that Seiberg is one of the authors in both articles:

  • The long flow to freedom, Ofer Aharony, Shlomo S. Razamat, Nathan Seiberg, Brian Willett. arXiv:1611.02763.
  • Shortening Anomalies in Supersymmetric Theories, Jaume Gomis, Zohar Komargodski, Hirosi Ooguri, Nathan Seiberg, Yifan Wang. arXiv:1611.03101.

Today I will talk briefly about the second of these articles, emphasizing what I consider to be a very neat explanation of a principle that might seem counter-intuitive, if not paradoxal.

The paper deals with (super)conformal theories. Such theories have a conformal manifold, parametrized by exactly marginal couplings. For $\mathcal{N}=(2,2)$ theories, this manifold factorizes locally as
\mathcal{M} = \mathcal{M}_c \times \mathcal{M}_{tc}
where the two factors are Kähler manifolds parametrized by the coupling constants of the exactly marginal operators constructed from the chiral and twisted chiral rings. But this seems to contradict the old fact that the conformal manifold of $\mathcal{N}=(4,4)$ SCFTs is locally of the form $$\frac{O(4,n)}{O(4)\times O(n)} , $$ which does not factorize and is not even Kähler! This seems contradictory, because of course $\mathcal{N}=(4,4)$ SCFTs can be seen as particular $\mathcal{N}=(2,2)$ SCFTs.

This situation is a very nice example where more symmetry invalidates generic results. The memo would be:

A more symmetric situation is not merely a particular case of a less symmetric situation.

Here $R$-symmetry is responsible for this unusual behavior: in a generic $\mathcal{N}=(2,2)$ SCFT the $R$-symmetry is $U(1)$, and in more symmetric models it is enhanced to a non-abelian group. It means that more symmetry adds new fields (here, the currents associated to the enhanced symmetry), and these fields can spoil properties of the less symmetric and more generic theories, like the factorization above. Finally, we obtain the strange-sounding, but now understandable, statement:

The factorization (\ref{fac}) breaks down only if the supersymmetry is larger than $\mathcal{N}=(2,2)$.

Another example of this principle can be found in four-dimensional supergravity. The target space of $\mathcal{N}=1$ supergravity is Kähler, but for $\mathcal{N}=2$ it is quaternionic and not Kähler. There is no contradiction because the latter theory has additional multiplets that are not present in a general $\mathcal{N}=1$ theory.

One of the main ideas of the paper is that the usual Seiberg trick of promoting coupling constants in the Lagrangian to background superfields might not work with a bigger symmetry because in that case one has to be careful about anomalies.