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 $N\geq 4$. 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 K\"ahler 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 $X_{N,1}$.
Highlights
The engineering of supersymmetric gauge theories [1,2] in dimensions ≤ 6 through string- and M-theory constructions has been an active and fruitful field of study throughout the years
One very rich subclass of theories which has attracted a lot of attention recently [6,7,8,9] are supersymmetric, UðMÞ circular quiver gauge theories on R5 × S1, which can be approached through F-theory compactifications on a class of toric Calabi-Yau threefolds
6In the following, hEi denotes the group freely generated by the ensemble E. These matrices are symmetry transformations of the partition function ZN;1 and the free energy F N;1 in the sense of Eq (2.9), which can be checked in explicit examples
Summary
The engineering of supersymmetric gauge theories [1,2] in dimensions ≤ 6 through string- and M-theory constructions has been an active and fruitful field of study throughout the years. [32,33] for a general discussion of the topological string partition function on elliptic Calabi-Yau threefolds) Since the latter (for technical reasons) requires a choice of preferred direction in the web diagram of XN;M, this method provides different, but completely equivalent expansions of ZN;M, which can be interpreted as instanton expansions of different but dual gauge theories. III–VI we discuss in detail the examples N 1⁄4 1, 2, 3, 4, respectively For each of these cases we construct GðNÞ and provide nontrivial evidence that it is a symmetry of the F N;1 by computing the leading orders in the expansion of the former as a power series of the Kähler parameters. These technical details are relevant for the computations performed in the main body of this work
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