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 Z N,M (or the free energy F N,M ) of these theories. Focusing on the case M = 1, we find that the latter is invariant under the group G(N )×S N : here S N corresponds to the Weyl group of the largest gauge group that can be engineered from X N,1 and G(N ) is a dihedral group, which acts in an intrinsically non-perturbative fashion and which is of infinite order for N ≥ 4. We give an explicit representation of 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 Z N,1 under G(N ) × S N in full generality, we provide explicit checks by series expansions of F N,1 for a large number of examples. We also comment on the relation of 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 .

Symmetries in A-type little string theories. Part I. Reduced free energy and paramodular groups

“…In this paper we have revisited the symmetries of a particular subsector of the BPS counting function of a system of parallel M5-branes with M2-branes stretched between them that engineers a little string theory with N = 2 supersymmetry. While this sector has been studied earlier in the literature, notably in [18], here we have taken a slightly different approach, making use of a recent better understanding [16] of symmetries that act directly on the Fourier coefficients of the free energy. The latter can be understood as a consequence of a web of dual low energy theories engineered from the same M5-M2-brane setup.…”

“…where T is an (N +2)×(N +2) matrix, which can be decomposed as T = G ∞ (N ) · S. Here G ∞ (N ) and S are two elements in the nonperturbative symmetry group G(N ) ∼ = G(N ) × Dih N found in [16], which we shall discuss in detail in Subsection 2.3. Explicitly, we have for S ∈ Dih N…”

“…and we refer the reader to [12] for a precise definition of the class of theta-functions ϑ α i ,α j and to [16] for a definition of z ij and w ij . 3 After the definition of the partition function, we can also introduce the free energy as the plethystic logarithm of Z N,1 (see [20]).…”

“…The web of dual (quiver) gauge theories is very astonishing from a purely field theoretic point of view, since the duality maps are generically non-perturbative in nature, exchanging coupling constants, gauge-and mass parameters in a rather non-trivial fashion. In [16] the question was raised, whether this web also has sizeable implications for an individual gauge theory, such as additional symmetries. Focusing on the case gcd(N, M ) = 1 (and thus working in the case M = 1) and guided once more by the structure of the extended moduli space of X N,1 , we argued for the existence of a symmetry group of the form G(N ) ∼ = G(N ) × S N , where S N ⊂ S N is (a subgroup of) the Weyl group of the U (N ) gauge group (i.e.…”

Symmetries in A-type little string theories. Part II. Eisenstein series and generating functions of multiple divisor sums

“…This was shown explicitly for M = 1 in [11] and for generic (N, M ) (but in the limit of vanishing 1,2 ) in [18], using a more geometric approach. Focusing on M = 1, it was furthermore argued in [7] that this duality also leads to additional symmetries for any single such theory. More concretely, it was demonstrated (and verified explicitly in a number of examples) that Z N,1 (ω, 1,2 ) is invariant under the symmetry group G(N ) ∼ = G(N ) × S N where S N ∼ = Dih N ⊂ S N acts on the gauge parameters, while…”