Critical dimension of ${\cal N}=2$ strings

Question

In "A tour through ${\cal N}=2$ strings" by Neil Marcus (https://arxiv.org/abs/hep-th/9211059) the following problem - among others - is noted:

The critical dimension of the ${\cal N}=2$ string is 4, as it can be seen from the central charge $$c=-26+2 \cdot 11-2=-6$$ or from the partition function $$Z_{string}=\frac1{4\pi}\int_{\mathcal{M}} \frac{d^2\tau}{\tau_2^{D/2}} $$ which is modular invariant in $D=4$ only.

On the other hand, the corresponding field theory calculation yields $$Z=\frac12 \frac1{(4\pi)^{D/2}}\int_0^\infty \frac{ds}{s^{1+D/2}}e^{-sm^2}$$ which suggests that the critical dimension is $D=2$.

This review is now almost 30 years old, so, has this been resolved by now? It appears to me that the interest in ${\cal N}=2$ strings has pretty much died. If true, why?

Answer 1

It think there are no rational arguments about why people stopped to be interested in $\mathcal{N}=2$ strings. They are arguably as important as the $\mathcal{N}=1$ ones.

However, I will enumerate some possible "sociological" reasons of why this unfortunate lack of interest.

1. The precise relation with $\mathcal{N}=1$ superstrings is not clear. Despite of the fact that N=1 strings can be proved to be twisted versions of the N=2 case (see The N=1 superstring as a topological field theory and On the uniqueness of string theory); it's not clear at all what is the physical significance of the N=2 vacua. The reason for this is that N=2 strings induce self-dual gravity as its target space gravity and this seems to be unrelated to our observations about gravity. This is connected with the fact that N=2 strings can only consistently propagate on target spacetimes with (++--) or (++++) signatures, and of course, it is difficult to make physical sense of this.
2. N=2 strings have been of enormous importance for the topological string development but topological string theory is not as "popular" as the physical superstring. N=2 strings were important for the discovery of S-dualties among topological A and B models and many interesting developments in integrability and unification in the context of non-critical string theories (see N=2 strings and the twistorial Calabi-Yau and S-duality and topoligical strings). The problem is that not much people is aware of the significance of topological strings in the context of physical superstrings and again, its difficult to decide whether N=2 strings are on the same footing that the N=1 strings or not.

Despite of the aforementioned reasons, there are also important and widely overlooked hints and programs that indicate that N=2 and N=4 superstrings may be of fundamental importance to understand string theory as a whole.

1. I highly recommend the wonderful blog post Worldsheets and spacetimes: kinship and cross-pollination for some speculations on the importance of N=2 strings to understand string theory in its full generality.
2. A tour through N=2 strings
3. New Principles for String/Membrane Unification