Dihedral Symmetries of Gauge Theories from Dual Calabi-Yau Threefolds

Abstract

Recent studies (arXiv:1610.07916, arXiv:1711.07921, arXiv:1807.00186) of six-dimensional supersymmetric gauge theories that are engineered by a class of toric Calabi-Yau threefolds $X_{N,M}$, have uncovered a vast web of dualities. In this paper we analyse consequences of these dualities from the perspective of the partition functions ${\mathcal{Z}}_{N,M}$ (or the free energy ${\mathcal{F}}_{N,M}$) of these theories. Focusing on the case $M=1$, we find that the latter is invariant under the group $\mathbb{G}(N)\times S_N$: here $S_N$ corresponds to the Weyl group of the largest gauge group that can be engineered from $X_{N,1}$ and $\mathbb{G}(N)$ is a dihedral group, which acts in an intrinsically non-perturbative fashion and which is of infinite order for $N\ge 4$. We give an explicit representation of $\mathbb{G}(N)$ as a matrix group that is freely generated by two elements which act naturally on a specific basis of the Kähler moduli space of $X_{N,1}$. While we show the invariance of ${\mathcal{Z}}_{N,1}$ under $\mathbb{G}(N)\times S_N$ in full generality, we provide explicit checks by series expansions of ${\mathcal{F}}_{N,1}$ for a large number of examples. We also comment on the relation of $\mathbb{G}(N)$ to the modular group that arises due to the geometry of $X_{N,1}$ as a double elliptic fibration, as well as T-duality of Little String Theories that are constructed from $X_{N,1}$.

arXiv comment58 pages, v2: minor corrections

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• Revisiting the classifications of 6d SCFTs and LSTs
• From Little String Free Energies Towards Modular Graph Functions
• M5 branes and Theta Functions
• Diagrammatic Expansion of Non-Perturbative Little String Free Energies
• Symmetric Orbifold Theories from Little String Residues
• Non-perturbative Symmetries of Little Strings and Affine Quiver Algebras