Self-duality and self-similarity of little string orbifolds

We study a class of $\mathcal{N}=(1,0)$ little string theories obtained from orbifolds of M-brane configurations. These are realized in two different ways that are dual to each other: either as $M$ parallel M5-branes probing a transverse $A_{N-1}$ singularity or $N$ M5-branes probing an $A_{M-1}$ singularity. These backgrounds can further be dualized into toric, noncompact Calabi-Yau threefolds $X_{N,M}$ which have double elliptic fibrations and thus give a natural geometric description of T-duality of the little string theories. The little string partition functions are captured by the topological string partition function of $X_{N,M}$. We analyze in detail the free energies ${\mathrm{\Sigma }}_{N,M}$ associated with the latter in a special region in the Kähler moduli space of $X_{N,M}$ and discover a remarkable property: in the Nekrasov-Shatashvili limit, ${\mathrm{\Sigma }}_{N,M}$ is identical to $NM$ times ${\mathrm{\Sigma }}_{1,1}$. This entails that the Bogomol’nyi-Prasad-Sommerfield (BPS) degeneracies for any $(N,M)$ can uniquely be reconstructed from the $(N,M)=(1,1)$ configuration, a property we refer to as self-similarity. Moreover, as ${\mathrm{\Sigma }}_{1,1}$ is known to display a number of recursive structures, BPS degeneracies of little string configurations for arbitrary $(N,M)$ as well acquire additional symmetries. These symmetries suggest that in this special region the two little string theories described above are self-dual under T-duality.

Figure 1

M5 branes separated along $x^6$.

Figure 2

The M-brane configuration for orbifolded little string theories.

Figure 3

The $(N,M)$-brane web diagram. It is also the toric diagram of the Calabi-Yau threefold $X_{N,M}$. The length of the blue lines represents the mass deformation $m$, which is considered to be the same throughout the whole diagram.

Figure 4

Various dual descriptions of the configuration $(N,M)=(1,1)$.