Dihedral Symmetries of Gauge Theories from Dual Calabi-Yau Threefolds

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 $X_{N,1}$.


Introduction
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. Indeed, the numerous dual approaches and formulations that are available on the string theory side provide us with a large range of tools (both computationally as well as conceptually) to explore hidden symmetries, dualities and even more sophisticated structures on the gauge theory side that would be very difficult to study otherwise. An important feature of this approach is that in many cases string theory methods give us access to non-perturbative aspects of the gauge theories and allow us to study them in an efficient manner [3][4][5]. One very rich subclass of theories which has attracted a lot of attention recently [6][7][8][9] are supersymmetric, U (M ) circular quiver gauge theories on R 5 × S 1 , which can (among other methods) be approached through F-theory compactifications on a class of toric Calabi-Yau threefolds X N,M 1 . The latter give rise to a quiver theory comprised of N nodes of type U (M ) (which we shall denote as [U (M )] N in the following). A particularity of these theories is the fact that their UV-completion in general does not only contain point-like particles, but also stringy degrees of freedom, although gravity remains decoupled. Such theories are called Little String Theories (LSTs), which have originally been introduced over a decade ago [11][12][13][14][15][16][17][18][19] and recently have received a lot of renewed interest [20, 8, 22-25, 21, 26, 27]. The fully refined, non-perturbative partition function Z N,M of this theory is captured by the (refined) topological string partition function on X N,M and can very efficiently be computed [3-6, 8, 21, 28] with the help of the (refined) topological vertex [29][30][31] (see [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 X N,M , this method provides different, but completely equivalent expansions of Z N,M , which can be interpreted as instanton expansions of different but dual gauge theories. While it is straightforward to see [4,5,8] that in this fashion the theory [U (M )] N is dual to [U (N )] M , it was argued in [25] that it is also dual to [U ( N M k )] k , where k = gcd(N, M ), thus leading to a triality of gauge theories that are engineered by X N,M .
The Calabi-Yau manifolds X N,M depend on N M + 2 independent Kähler parameters and the corresponding moduli space takes the form of a cone. The faces of the latter (which we shall call walls in the following) are (among others) comprised of singular loci where the area of one or more of the curves in the web diagram of X N,M vanish. From the perspective of the geometry of X N,M , crossing such a wall (i.e. continuing to negative area) gives rise to a new Calabi-Yau manifold, which corresponds to a different (but dual) resolution of the singularity. With the help of such flop transitions [34,35], the Kähler moduli space of X N,M can be extended to include further regions that allow the engineering of yet new gauge theories. Indeed, it was argued in [23] that the Calabi-Yau manifolds X N,M and X N ,M can be related through a series of flop transformations if N M = N M and gcd(N, M ) = gcd(N , M ). Furthermore, nontrivial checks were presented in [23] that the topological string partition functions associated with X N,M and X N ,M are the same upon taking into account the non-trivial duality map. This was shown explicitly in [21] for the cases gcd(N, M ) = 1 and a suitable basis of independent Kähler parameters was presented which is adapted to the invariance under a series of flop transformations that is instrumental in the duality X N,M ∼ X N ,M . 2 Combining this invariance of Z N,M with the triality of gauge theories proposed in [25], it was argued in [26]  . It was furthermore argued in [27] that the extended moduli space [36][37][38][39][40] of X N,M contains different decompactification regions, which engineer different five-dimensional gauge theories with various gauge structures and matter content.
While previous works have focused on interpreting different expansions of Z N,M as instanton partition functions of different gauge theories, thereby establishing a large network of dual theories, in this paper we discuss the consequences of these dualities from the perspective of symmetries of Z N,M . Focusing on the cases M = 1, rather than switching between different expansions of the partition function Z N,1 (or more concretely the free energy F N,1 ), we shall focus on one particular expansion (as a power series in a suitable basis of Kähler parameters of X N,1 ) and recast the results of [23,25,26] in the form of highly non-trivial identities among the expansion coefficients of F N,1 . From the perspective of any of the gauge theories of the type [U (M )] N , where (N , M ) are relative primes and N M = N , these correspond to generically non-perturbative symmetries that act in a highly non-trivial fashion on the spectrum of BPS states of the theory. Furthermore, since the combination of any two of these symmetries itself has to be another symmetry, they have the structure of a group G(N ) which acts naturally on the vector space spanned by the independent Kähler parameters of X N,1 .
We shall analyse G(N ) first with the help of the explicit examples N = 1, 2, 3, 4, where we can study it (or its subgroups) explicitly as a matrix group. Based on these examples, we find a pattern, which allows us to prove for generic N that G(N ) has a subgroup of the form 1) where S N ⊂ S N is a subgroup of the Weyl group of the largest simple gauge group that can be engineered from X N,1 (i.e. U (N )) and G(N ) is isomorphic to a dihedral group 3 , namely Here Dih ∞ is a finitely generated group of infinite order (while ord(Dih n ) = 2n for finite 2 ≤ n ∈ N).
In particular the group G(N ) in (1.1) combines non-trivially with other known symmetries and dualities of X N,1 : • modularity: Owing to the fact that X N,1 has the structure of a double elliptic fibration, the partition function transforms as a Jacobi form under two copies of the modular group SL(2, Z) τ and SL(2, Z) ρ . 4 Since G(N ) acts non-trivially on the modular parameters (τ, ρ) the combined symmetry group is in general larger than simply G(N ) × SL(2, Z) τ × SL(2, Z) ρ . In the simplest case N = 1, which we shall discuss in section 3, we are in fact able to analyse explicitly the resulting group and we can show that it is isomorphic to Sp(4, Z), which is the automorphism group of the genus-two curve that is the geometric mirror of the Calabi-Yau manifold X 1,1 (see [30,10]). For N > 1, the symmetry is more difficult to analyse, and we are only able to make statements about a specific region in the moduli space.
• T-duality: As mentioned above, the UV completion of the gauge theory [U (1)] N is an LST with 8 supercharges, which was called type IIb little string in [8]. The latter is T-dual to type IIa little string theory, whose low energy behaviour is described by the dual gauge theory [U (N )] 1 (see [20,41,8,22] for the discussion of T-duality of LSTs engineered from double elliptic Calabi-Yau threefolds). Denoting the partition functions of these little string theories by Z IIb and Z IIa respectively, it was proposed in [8] that the partition functions of these two little string theories are captured by Z N,1 Z IIa (τ, ρ, K) = Z N,1 (τ, ρ, K) , and Z IIb (τ, ρ, K ) = Z N,1 (ρ, τ, K ) , (1.3) where for simplicity we have only explicitly displayed the dependence on the modular parameters (τ, ρ) and only schematically indicated the dependence on the remaining Kähler parameters through K and K respectively. Furthermore, in [8] it was proposed that T-duality of the IIa and IIb LSTs simply amounts to which, from the perspective of the Calabi-Yau manifold X N,1 , corresponds to an exchange of the two elliptic curves, one in the fiber and one in the base (with a duality map relating K and K ). Since the group G(N ) in (1.1) acts non-trivially on the modular parameters (τ, ρ) (and in general mixes them in a non-trivial fashion), it extends the incarnation (1.4) of T-duality to a non-trivial group acting on the full spectrum of the LSTs. This paper is organised as follows: In section 2 we first review important aspects of the computation of the partition function Z N,1 , in particular the choice of basis of the independent Kähler parameters. Furthermore, we discuss in more detail our strategy in finding the group G(N ) in (1.1). Finally, for the sake of readability of this paper, we also give a summary of the results obtained in the subsequent sections. In sections 3 -6 we discuss in detail the examples N = 1, 2, 3, 4 respectively. For each of these cases we construct G(N ) and provide non-trivial 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. In section 7 we generalise a pattern that emerges from the previous examples and which allows us to prove (1.2) for generic N ∈ N. Finally, section 8 contains our conclusions and directions for future research. Furthermore, this paper is accompanied by two appendices, which review a particular duality transformation for the web diagrams of X N,1 and a finite presentation of the group Sp(4, Z) respectively. These technical details are relevant for the computations performed in the main body of this work.
2 Review, General Strategy and Summary of Results

Review: Partition Function and Free Energy
The web diagram for a general X N,1 is shown in Fig. 1. Each line is labelled by the area of the curve they are representing: horizontal lines are labelled by h 1,...,N , vertical lines by v 1,...,N , and diagonal lines by m 1,...,N . Not all of these areas are independent of one another, but they are subject to 2N consistency conditions (for i = 1, . . . , N ), related to the N hexagons S i of the web diagram where m i+N = m i and v i+N = v i . A general solution of these conditions is given by v i = v i+1 and m i = m i+1 for i = 1, . . . , N − 1. Another solution, which is more adapted to the computations in the remainder of this work, is provided by the blue parameters in Fig. 1, which equally represent an independent set of Kähler parameters of the Calabi-Yau manifold X N,1 . Physically, from the perspective of (one particular) gauge theory engineered by X N,1 , the parameters a 1,...,N correspond to the (affine) roots of the gauge group U (N ) (i.e. the vacuum expectation values of the vector multiplet scalars), while the parameter R is related to the coupling constant and S to the mass parameter of the matter sector. As shown in [25], however, this assignment is not unique and the Calabi-Yau manifold X N,1 in fact engineers several different gauge theories with different gauge groups 5 and possibly different matter content. In the following we will therefore not be too much concerned with the physical interpretation of the parameters ( a 1,...,N , S). Instead, we shall treat the dependence of the partition function Z N,1 ( a 1,...,N , S, R; 1,2 ) (associated with X N,1 ) on all of these parameters on equal footing. The former can be computed from the web diagram in Fig. 1 with the help of the refined topological string. Here the constants 1,2 ∈ R represent the refinement and can be thought of as a means of regularising the partition function, which would otherwise be ill defined.
An efficient method of computing Z N,1 from Fig. 1 (for arbitrary δ) was given in [21] (see also [3,5]) by computing a general building block W α 1 ,...,α N β 1 ,...,β N that depends on the Kähler parameters ( a 1,...,N , S) and is labelled by 2N integer partitions α 1,...,N and β 1,...,N , which encode how the legs of the (various) building block(s) are glued together (for details see [21]). While the formalism developed in [21] is more general and allows the computation of a much larger class of partition functions, in the present case we have with (our conventions for the normalisation of W α 1 ,...,α N β 1 ,...,β N are adapted to Fig. 1.) Here we have used the following notation: where m i=1,...,N refer to the area of the diagonal lines in Fig. 1 expressed as functions of ( α 1,...,N , S, R) with the help of the consistency conditions (2.1). Furthermore, W N ∅ is a normalisation factor (which from a physical perspective in particular encodes the perturbative contribution to the partition function) and ϑ µν is a class of theta-functions that is labelled by two integer partitions µ and ν with the further definition Finally, the arguments of the ϑ-functions can be defined as Q i,j = e −z ij andQ i,j = e −w ij where z ij and w ij are implicitly defined in Fig. 2 with respect to (part of) the web diagram (the labels on the diagonal and horizontal lines in Fig. 2 (and Fig. 1) indicate how they are glued together).
With the partition function Z N,1 , we can define the free energy as the plethystic logarithm where µ(k) is the Möbius function. We can expand the free energy in the following fashion Apart from a first order pole, F N,1 has a power series expansion in 1,2 , which allows to compute the Nekrasov-Shatashvili-limit [42,43] and the unrefined limit. For later convenience we therefore also introduce the expansion of the leading term in both parameters (which we simply denote NS) where f NS i 1 ,...,i N ,k,n ∈ Z. In [23,21,25] different duality transformations have been discussed, which involve flop transformations [34,35] of various curves of X N,1 , SL(2, Z) transformations as well as cutting and re-gluing of the web diagram. While these duality transformations were shown in [21] to leave Z N,1 (and thus also F N,1 ) invariant, they generically act in a rather nontrivial fashion on the web diagram

Symmetries Transformations: Strategy and Summary of Results
The transposition of G in this relation is due to the fact, that the transformation (2.8) is a passive one from the perspective of the coefficients f i 1 ,...,i N ,k,n . For given X N,1 there are in general numerous different transformations G of the type described above. Since the concatenation of two such transformations defines a new transformation, the latter form a group. In the following sections we shall determine at least a subgroup of this group for the simplest examples N = 1, 2, 3, 4, which in section 7 can be generalised to generic N ∈ N. However, before doing so and for ease of readability, we summarise our results: For generic N ∈ N, we identify a finitely generated group of symmetry transformations of the type (2.8), which can be written as (2.10) The group S N ⊂ S N is generated by simple relabellings of the web diagram of X N,1 and physically corresponds to a subgroup of the Weyl group of U (N ), which is the largest gauge group that can be engineered by X N,1 . For generic N , the group G(N ) is freely generated by two (N + 2) × (N + 2) matrices of order 2, which satisfy a specific braid relation 6 where n = 3 for N = 1, 3 and n = 2 for N = 2 but for N ≥ 4 we find n → ∞, which means that there is no braid relation in these cases. Explicitly, the generators are given by the following lower-and upper triangular matrices (2.12) 6 In the following E denotes the group freely generated by the ensemble E.
These matrices are symmetry transformations of the partition function Z N,1 and the free energy F N,1 in the sense of (2.9), which can be checked in explicit examples. In the case N = 1, combining the group G(N ) with the modular group SL(2, Z) acting on one of the modular parameters of X 1,1 generates the group Sp(4, Z). For the cases N > 1, the combination with the modular group is more difficult to analyse at a general point in the moduli space of X N,1 . However, in the region in moduli space where a 1,...,N = a in Fig. 1, this analysis is simpler to perform and we can prove that the combination of G(N ) with the modular group is a subgroup of Sp(4, Z). This is in line with the checks performed in [23] to provide evidence for the duality X

Dualities and Dih 3 Group Action
The simplest (albeit somewhat trivial) example to illustrate the idea explained in Section 2.2 is the configuration (N, M ) = (1, 1). The corresponding web diagram is shown in Fig. 4(a). Through simple SL(2, Z) transformations (as well as cutting and re-gluing) the former can also be presented (among other ways) in the form of Fig. 4(b) and Fig. 4(c).  Each diagram can be parametrised in terms of the parameters (h, v, m) or respectively ( a, S, R), ( a , S , R ) or ( a , S , R ). The latter can be expressed in terms of (h, v, m) as Inverting these relations, (h, v, m) can be expressed as linear combinations of ( a, S, R), ( a , S , R ) or ( a , S , R ) respectively These equations also furnish linear transformations between ( a, S, R), ( a , S , R ) or ( a , S , R ) The matrix G 1 is of order 3 (i.e. G 1 · G 1 · G 1 = 1 1 3×3 ) while G 2 is of order 2 (i.e. G 2 · G 2 = 1 1 3×3 ). Thus, introducing also the matrices 7 } forms a finite group, whose multiplication table is from which we can read off G(1) = {E, G 1 , G 2 , G 3 , G 4 , G 5 } ∼ = Dih 3 ∼ = S 3 . The latter can be formulated more elegantly as the free group generated by the elements furnishing the following presentation

Invariance of the Non-perturbative Free Energy
As a check of the fact that G 1,2 defined in (3.3) are indeed symmetry transformations of Z 1,1 , we can consider the coefficients in the expansion of the associated free energy F 1,1 . Indeed, for N = 1, the expansion (2.5) can be written as with Q = e − a . As explained in section 2.2, in order to be a symmetry, the coefficients f i,k,n ( 1 , 2 ) (which are functions of 1,2 with a first order pole) need to satisfy Below we tabulate examples of coefficients f i,k,n with i ≤ 8 for n = 1, i ≤ 4 for n = 2 and i ≤ 2 for n = 3 that are related by G 1,2 : Table 1 shows the relations for G 1 and Table 2 for G 2 .

Modularity and Sp(4, Z) Symmetry
The action of G(1) as presented in (3.7) combines with SL(2, Z) × SL(2, Z) into Sp(4, Z), which is (a subgroup of) the automorphism group of X 1,1 . To see this, instead of considering the action of G(1) on the vector space spanned by ( a, S, R), we consider the vector space spanned by (τ = h + v, ρ = m + v, v). Arranging the latter in the period matrix there is a natural action of Sp(4, Z), as reviewed in appendix B. The action of G 1,2 on Ω is Based on this action, we can equivalently represent the action of G(1) by G 1,2 ∈ Sp(4, Z) Table 1: where K and H are defined as in appendix B. This implies that G(1) ⊂ Sp(4, Z). Moreover, combining G(1) with the SL(2, Z) ρ symmetry 8 acting on the modular parameter 9 ρ as generates the complete action of Sp(4, Z): the generators (S ρ , T ρ ) can be expressed as S ρ = L 3 and T ρ = L 9 HL 10 H = X 2 . Furthermore, we have G 2 G 1 = L 5 KL 7 such that we can write with X 1,2,3,4,5,6 defined in (B.2). This indicates that where the last relation was shown in [47]. From (3.12), using the presentation of Sp(4, Z) given in [46], it follows that

Dualities and Dih 2 Group Action
In this section we generalise the analysis of the previous section and, using the simplest nontrivial example (namely (N, M ) = (2, 1)), explain how the duality transformations advocated in [25,26] lead to non-trivial symmetries at the level of the set of independent Kähler parameters of X 2,1 . In the following subsection we give further evidence for this symmetry at the level of the partition function Z 2,1 . The starting point is the web diagram shown in Fig. 5 along with a parametrisation of the areas of all curves involved. The latter are not all independent of one another, but for each of the two hexagons S 1,2 , they have to satisfy the following consistency conditions: A solution for these conditions was provided in [21] in the form of the parameters ( a 1,2 , S, R) as indicated in Fig. 5 Indeed, all of the areas (h 1,2 , v 1,2 , m 1,2 ) can be expressed as a linear combination of ( a 1 , a 2 , S, R): Mirroring the diagram and performing an SL(2, Z) transformation, Fig. 5 can also be presented in the form of Fig. 6(a). After cutting the latter along the curve labelled v 1,2 and re-gluing along the curves labelled m 1,2 leads to the presentation in Fig. 6(b). The consistency conditions of this web are the same as (4.1). Furthermore, the web diagram Fig. 6(b) is of the same form as Fig. 5 and thus allows for a solution of (4.1) in terms of the parameters ( a 1 , a 2 , S , R ): Indeed, we can express the areas (h 1,2 , v 1,2 , m 1,2 ) in terms of the latter Comparing (4.3) with (4.5) gives rise to a linear relation between ( a 1 , a 2 , S, R) to ( a 1 , a 2 , S , R ): We can obtain another symmetry transformation by cutting the diagram  The latter can be parametrised by ( a 1 , a 2 , S , R ) which allows to uniquely express all areas (h 1,2 , v 1,2 , m 1,2 ) Comparing (4.8) with (4.5) gives rise to a transformation between ( a 1 , a 2 , S, R) and ( a 1 , a 2 , S , R ) Finally, cutting the diagram Fig. 7(b) along the curve labelled v 1 and re-gluing it along the line m 2 yields the diagram Fig. 8(a), which (after mirroring and performing an SL(2, Z)transformation) can also be presented in the form Fig. 8(b). This diagram is parametrised by ( a 1 , a 2 , S , R )   which provide a parametrisation of all the areas Comparing (4.11) with (4.3) provides a linear transformation between the parameters ( a 1 , a 2 , S, R) and ( a 1 , a 2 , S , R ) The matrices G 1,2,3 together with the identity matrix E = 1 1 4×4 form a discrete group of order 4, whose multiplication table is given by The latter is identical to the multiplication table of Dih 2 , i.e. the dihedral group of order 4 (which is isomorphic to the Klein four-group). We therefore have 10 (4.14) An overview over G 1,2,3 and their relation to different presentations of the web diagram Fig. 5 is given in Fig. 9 (which corresponds to the cycle graph of Dih 2 ). We remark that all other presentations of the web (including webs related by a transformation F (appendix A)) only give rise to coordinate transformations that differ from {E, which exchanges a 1 ←→ a 2 and commutes with G 1,2,3 . Since R generates the group S 2 , we can define G(2) = G(2) × S 2 as a non-trivial symmetry group of F 2,1 .
10 For further reference, we remark that G(2) can also be presented as the group freely generated by G 1,2 , i.e. Fig. 6 Fig. 7 Fig. 8 Figure 9: Presentations of web diagrams related to X 2,1 . The transformations G 1,2,3 act on the basis of independent Kähler parameters ( a 1 , a 2 , S, R). The organisation of web diagrams and transformations is reminiscent of the cycle graph of Dih 2 .

Invariance of the Non-perturbative Free Energy
It was shown in [21] that the web diagrams Fig. 5, Fig. 6 Fig. 7(b) and Fig. 8(b) give rise to the same partition function, the linear transformations G 1,2,3 in eqs. (4.6), (4.9) and (4.12) correspond to symmetries of the free energy F 2,1 ( a 1,2 , S, R; 1 , 2 ), as defined in (2.4). In this section we provide evidence for this symmetry by considering the expansion As explained in section 2.2, we have Below we tabulate coefficients f i 1 ,i 2 ,k,n with i 1 + i 2 ≤ 3 for n = 1 and i 1 + i 2 ≤ 2 for n = 2 that are related by G 1,2,3 : Table 3 shows relations for G 1 , Table 4 those for G 2 and Table 5 for G 3 .

Modularity at a Particular Point of the Moduli Space
For the case N = 1, we showed that the combination of G(1) ∼ = Dih 3 with the modular group acting as in (3.13) generates the group Sp(4, Z). The case N = 2 is more complicated. However, in the following we shall show in a particular region of the moduli space that G(2) ∼ = Dih 2 in (4.14) can be understood as a subgroup of Sp(4, Z). This region is characterised by imposing a . This region is also a fixed point of S 2 generated by R in (4.15). The remaining independent parameters can be organised in the period matrix Furthermore, the symmetry transformations G 1 in (4.6) and G 2 in (4.9) can be reduced to act on the subspace ( a, S, R) Rewriting the latter as elements of Sp(4, Z) that act like in (B.3) on the period matrix Ω in (4.18), they take the form where K, L and H are defined in appendix B. This implies that the restriction of G(2) to the particular region of the Kähler moduli space explained above is a subgroup of Sp(4, Z). However, unlike the case N = 1, we cannot conclude that the group freely generated as G

Dualities and Dih 3 Group Action
Following the previous example of X (δ=0) 2,1 , we can also analyse X  The starting point is the web diagram shown in Fig. 10, which includes labels for the areas of all curves. The consistency conditions associated with the three hexagons S (0) 1,2,3 take the form A solution of these conditions is provided by the parameters ( a such that the areas (h 1,2,3 , v 1,2,3 , m 1,2,3 ) can be expressed as the linear combinations The web diagram of X (δ=0) 3,1 allows various other presentations: mirroring the diagram and performing an SL(2, Z) transformation, the web can be drawn in the form of Fig. 11(a). Furthermore, cutting the diagram along the lines labelled v 1,2,3 and re-gluing them along the lines labelled m 1,2,3 one obtains Fig. 11(b). The latter is again a web diagram with δ = 0, which can thus be parametrised by ( a 1,2,3 , S (1) , R (1) ), as indicated in Fig. 11 Table 4: Action of G 2 : the indices are related by (i 1 , i 2 , k , n ) T = G 2 · (i 1 , i 2 , k, n) T .   such that the areas can be expressed in the following manner Moreover, as explained in section 2.2, comparing (5.5) with (5.3) gives rise to a symmetry of the partition function as a linear transformation relating ( a In order to obtain another symmetry generator we first perform a transformation F as explained in appendix A. The corresponding geometry is of the type X and a parametrisation of the various curves through an independent set of Kähler parameters is shown in Fig. 12.  Figure 12: Web diagram after a transformation F of Fig. 10. The blue parameters are the same as defined in eq. (5.2).
The duality map of F is explicitly given by As was shown in [21] for generic X (δ) N,1 , the independent parameters ( a (0) 1,2,3 , S (0) , R (0) ) are invariants of F in the sense that the parameters appearing in Fig. 12 are the same as the ones defined in (5.2). 12 While the transformation F itself therefore does not generate a new non-trivial symmetry transformation, one can consider different presentations of Fig. 12. Indeed, mirroring the latter and performing an SL(2, Z) transformation, on obtains Fig. 13(a). Cutting the latter along the lines labelled −h 1,2,3 and re-gluing them along the lines labelled 12 The only δ-dependence (and thus dependence on F) appears in the coefficient of S (0) in the defining equation of R (0) (see the generic parametrisation of X  m 1,2,3 yields the presentation Fig. 13(b). The set of independent parameters ( a 1 + a 2 + a (2) . (5.9) Comparing (5.12) with (5.3) gives rise to a symmetry of the partition function as a linear transformation relating ( a 1,2,3 , S (2) , R (2) ) to ( a (5.10) One can find another symmetry transformation by cutting the diagram Fig. 12 along the line labelled −h 1 and re-gluing it along the line labelled v 3 . After mirroring the diagram, it can also be presented in the form of Fig. 14, which corresponds to a web diagram of the form X (δ=1) 3,1 . The latter can thus be parametrised by ( a (3) 1,2,3 , S (3) , R (3) ), as shown in Fig. 8: Indeed, the areas (h 1,2,3 , v 1,2,3 , m 1,2,3 ) can be expressed in terms of ( a Since the partition functions computed from Fig. 14 and Fig. 10 are the same [21], comparing eq. (5.12) to eq. (5.3) gives rise to a linear transformation that is a symmetry of Z 3,1 . Figure 14: Presentation of the web diagram obtained by cutting Fig. 12 along the line −h 1 and gluing along the line v 3 .

Explicitly, one finds
where the 5 × 5 matrix G 3 is given by (5.14) From Fig. 12 one can extract yet another symmetry generator. Indeed, cutting the diagram along the curves v 1,2,3 and re-gluing it along the lines m 1,2,3 one obtains Fig. 15(a). Cutting furthermore along the line labelled −h 1 and re-gluing along the line m 3 one obtains Fig. 15(b) after performing an SL(2, Z) transformation.  An independent set of parameters is given by which allows to express (h 1,2,3 , v 1,2,3 , m 1,2,3 ) in the following fashion Comparing eq. (5.16) to eq. (5.3) gives rise to the following linear transformation The matrix G 4 is of order 3, which means that G 5 = G 4 · G 4 is a new symmetry element. It can also be associated to a particular presentation of the web diagram of X 3,1 . To see this, we first perform a transformation F on the web diagram in Fig. 12 to obtain Fig. 16. Figure 16: Web diagram after a transformation F of Fig. 12. The blue parameters are the same as defined in eq. (5.2).
Since F leaves the partition function invariant, the parameters ( a 1,2,3 , S (0) , R (0) ) are the same as introduced in eq. (5.2). Furthermore, we have introduced the areas where we have used the definitions (5.7). Next, we cut the diagram Fig. 16 along the lines labelled v 1,2,3 and re-glue it along the lines labelled m 1,2,3 to obtain Fig. 17(a). Cutting the diagram again along the line −v 3 , it can also be presented in the form of Fig. 17(b), which is a diagram with shift δ = 0. It can be parametrised by ( a

Comparing (5.19) to (5.3) indeed gives rise to the following symmetry transformation
Other presentations of the web diagram of X 3,1 do not give rise to other symmetries than G 1,2,3,4,5 , apart from a permutation of the parameters a 1,2,3 . These latter symmetries form the group S 3 , which, from the point of view of the gauge theory engineered by X 3,1 , corresponds to the Weyl group of the gauge group U (3). Factoring out this S 3 , the 5 × 5 identity matrix  E = 1 1 5×5 and the linear transformations G 1,2,3,4,5 form a finite group of order 6, which commute with S 3 and whose multiplication table is given by This table is the same as the one of the dihedral group Dih 3 , such that we have  R (4) − 2S (4) web diagram in Fig. 15( Fig. 17 Fig. 11 (2) web diagram in Fig. 13 Figure 18: Presentations of web diagrams related to X 3,1 . The transformations G 1,2,3,4,5 act on the basis of independent Kähler parameters ( a that is freely generated by G 2 and G 3

Invariance of the Non-perturbative Free Energy
As in the previous example, following the result of [21], the linear transformations G 1,2,3,4,5 in eqs. (5.6), (5.10), (5.14), (5.17) and (5.20) correspond to symmetries of the free energy F 3,1 ( a 1,2,3 , S, R; 1 , 2 ), as defined in (2.4). In this section we provide evidence for this symmetry, however, for simplicity we limit ourselves to checking the leading limit in 1,2 of the free energy.
To this end, we introduce the following expansion where f NS i 1 ,i 2 ,i 3 ,k,n ∈ Z and Q i = e − a i (for i = 1, 2, 3), Q S = e −S and Q R = e −R . As explained in section 2.2, the fact that the (shifted) web diagrams in Fig. 9 all give rise to the same partition functions implies In Tables 6, 7 and 8 we tabulate coefficients f NS i 1 ,i 2 ,i 3 ,k,n with i 1 + i 2 + i 3 ≤ 7 for n = 1 and n = 2 that are related by G 1,2,3,4,5 . 13

Modularity at a Particular Point of the Moduli Space
Similarly to the case N = 2 above, we can analyse how the group G(3) is related to Sp(4, Z) at the particular region in the moduli space, which is characterised by a Using the parametrisation (5.23) of G(3), it is sufficient to analyse the relation of the generators G 2 and G 3 to Sp(4, Z). The restriction of these generators to the subspace ( a, S, R) can be    Table 7: Action of G 3 : the indices are related by (i 1 , i 2 , i 3 , k , n ) T = G T 3 · (i 1 , i 2 , i 3 , k, n) T .

Dualities and Dih ∞ Group Action
Continuing the previous examples, we next consider X (δ=0) 4,1 , whose web diagram is shown in Fig. 19. While the method we employ to study it is the same as in the previous cases, we shall encounter a novel twist. The consistency conditions stemming from the web diagram are  Figure 19: Web diagram of X 4,1 . An independent set of Kähler parameters is shown in blue.
while a solution is provided by the parameters ( a (0) 1,2,3,4 , S (0) , R (0) ) a (0) The dihedral groups found in the previous examples were generated by two transformations. The latter can in fact be obtained in a simple fashion by considering two diagrams that are obtained from Fig. 19 through a rearrangement and a flop transformation respectively:

2) transformation F:
Another symmetry transformation can be obtained after performing a transformation F on Fig. 19, as shown in Fig. 21. 3 Figure 21: Web diagram after a transformation F of Fig. 19. The blue parameters are the same as defined in eq. (6.2).
Here we have introduced the variables The parameters ( a (0) 1,2,3,4 , S (0) , R (0) ), shown in blue in Fig. 21, are the same as those appearing in Fig. 21, such that the flop transformation alone does not lead to a nontrivial symmetry transformation. However, starting from the web diagram Fig. 21, we can present it in the form of Fig. 22. The parametrisation in terms of the variables ( a  in Fig. 22(b) can be related to ( a (0) 1,2,3,4 , S (0) , R (0) ) in Fig. 19 through the transformation where The matrix G 2 has det G 2 = 1 but does not have finite order. 14 This implies that the matrices 14 Indeed, by complete induction one can show that which only resembles the identity matrix for n = 0.  Figure 23: Web diagram after two transformations F of Fig. 19. The blue parameters are the same as defined in eq. (6.2).
We have seen in the previous section that the symmetry transformation G 2 is of infinite order, which is markedly different than what we have seen in the previous examples. While we will present explicit checks that G 2 is indeed a symmetry of the free energy in the next subsection, we first want to provide an intuitive explanation of what makes the case (N, 1) = (4, 1) different than all preceding ones. Indeed, we will provide some indication that the extended moduli space of X 4,1 contains many more regions that are represented by (a priori) very different looking web diagrams. While this will not prove that G 2 is of infinite order (as we have already done in the previous section by purely algebraic means), it will indicate the novel aspect of X 4,1 (in comparison to the previous examples).
Returning to Fig. 22(b), the latter is a web diagram of the form X (δ=2) 4,1 . Another way of obtaining such a diagram is to perform two transformations of the form F on Fig. 19, as is shown in Fig. 23, with the new parameters as well as Notice that even upon imposing the consistency conditions (6.1), the parametrisation of the web diagram Fig. 23 is different than the one of the web diagram Fig. 22(b). 15 Thus, there is a duality transformation that transforms the web X (2) 4,1 −→ X (2) 4,1 , however, with a non-trivial duality map D acting on the areas of all curves involved. The duality D can be repeatedly applied to X (2) 4,1 in Fig. 22(b), thus producing an infinite number of diagrams of the type X (2) 4,1 , each one with an a priori different parametrisation of individual curves.
Moreover, since the blue parameters ( a 1,2,3,4 , S (0) , R (0) ) in Fig. 23 are the same as in Fig. 19, the duality map D from the perspective of the independent Kähler parameters precisely corresponds to the symmetry transformation G 2 . Therefore, the transition from Fig. 23 to Fig. 22(b) gives (a new) geometric representation of G 2 at the level of web diagrams, which readily allows to also compute arbitrary powers of G 2 .
Finally, notice that the above discussion does not generalise to the cases N = 2, 3 (but can be extended to N > 4). Indeed, web diagrams with shifts δ ≥ 2 for N = 2, 3 can readily be related (possibly through simple cutting and re-gluing operations) to web diagrams with δ ∈ {0, 1}, which only gave rise to symmetry transformations of finite order. 16 In other words, in the cases N = 2, 3, the equivalents of the diagrams Fig. 22 and Fig. 23 are of the type δ ≤ 1, which we have seen to provide only transformations of finite order.

Invariance of the Non-perturbative Free Energy
As non-trivial check for the fact that G 1 and G 2 are indeed symmetries of Z 4,1 , we consider the non-perturbative free energy associated with the latter. For simplicity, we restrict ourselves to the leading term in 1,2 . To this end, we define lim 1,2 →0 1 2 F 4,1 ( a 1,2,3,4 , S, R; 1 , 2 ) = ∞ n,ia=0 k∈Z where f NS i 1 ,i 2 ,i 3 ,i 4 ,k,n ∈ Z and Q i = e − a i (for i = 1, 2, 3, 4), Q S = e −S and Q R = e −R . In the same manner as explained in section 2.2, the symmetry transformations G 1 and G 2 act in the following manner on the coefficients f NS We can explicitly check the relations (6.9) by computing the relevant expansions of the free energies. However, since the matrix G 1 in (6.3) contains very large numbers, the relations are easier to check for the matrices G 1 · G 2 and G 2 with (6.10) In Table 9 and Table 10 we tabulate examples of coefficients f NS i 1 ,i 2 ,i 3 ,i 4 ,k,n with i 1 +i 2 +i 3 +i 4 ≤ 6 for n = 1 and n = 2 that are related by G 1 · G 2 and G 2 respectively.  Table 9: Action of G 1 · G 2 : (i 1 , i 2 , i 3 , i 4 , k , n ) T = (G 1 · G 2 ) T · (i 1 , i 2 , i 3 , i 4 , k, n) T .  Table 10: Action of G 2 : (i 1 , i 2 , i 3 , i 4 , k , n ) T = G T 2 · (i 1 , i 2 , i 3 , i 4 , k, n) T .

Modularity at a Particular Point of the Moduli Space
Similarly to the cases N = 2, 3, we can analyse how the group G(4) is related to Sp(4, Z) at the particular region in the moduli space, which is characterised by a . We can introduce the period matrix Using the parametrisation (6.7) of G(4), it is sufficient to analyse the relation of the generators G 1 and G 2 = G 2 · G 1 to Sp(4, Z). The restriction of these generators to the subspace ( a, S, R) can be written in the form Rewriting them furthermore to act as elements of Sp(4, Z) in the form of (B.3) on the period matrix Ω in (6.11), they take the form , S ρ , T ρ , S τ , T τ is isomorphic to Sp(4, Z).

Symmetry Transformations of Generic Webs
We can summarise all previous examples by introducing the following matrices as well as The matrices G 2 (N ) and G ∞ (N ) for the examples previously studied are given explicitly as eq. (4.6) and eq. (4.12) 3 G 3 G 3 · G 2 eq. (5.10) and eq. (5.14) 4 G 1 · G 2 G 2 eq. (6.3) and eq. (6.5) where the equation numbers refer to the definitions of the matrices in the individual cases. The matrices G 2 and G ∞ (N ) furnish two symmetry relations for a web diagram of the type (N, 1). To see this, in the following we shall check explicitly the combinations of G ∞ (N ) · G 2 (N ) and G ∞ (N ), which at the level of the web diagrams are generated by the same transformations we already discussed in the example of (N, 1) = (4, 1) and which can be generalised for generic N :

1) rearrangement:
We first verify that G ∞ (N ) · G 2 (N ) is a symmetry. To this end, we start from the configuration shown in Fig. 1 for δ = 0, which (after mirroring and performing an SL(2, Z) transformation) can be presented as in Fig. 24(a). The latter in turn can alternatively be presented in the form. Fig. 24(b). The matrix G ∞ (N ) · G 2 (N ) (defined in (7.1) and (7.2) respectively) relates the parameters in the web diagram Fig. 1 to those in Fig. 24(b) in the following way which proves (7.3).

2) transformation F:
In a similar fashion we can show that G ∞ (N ) is a symmetry transformation. To this end, we first consider a transformation of the type F acting on the web diagram Fig. 1 for δ = 0 which results in the web digram shown in Fig. 25, representing X are the same as in Fig. 1, while we also have introduced Cutting the diagram Fig. 25 along the lines v 1,...,N −1 and re-gluing it along the lines m 1,...,N we obtain the web diagram shown in Fig. 26(a). Cutting the latter diagram furthermore along the line −h N it can also be represented in the form Fig. 26(b), which corresponds to a staircase diagram with shift δ = N − 2. The set of independent Kähler parameters ( a 1,...,N , S , R ) can be related to ( a 1,...,N , S, R) in the following manner ( a 1 , . . . , a N , S , R) T = G ∞ (N ) · ( a 1 , . . . , a N , S , R ) T , (7.7) To show this, we use (7.2) and (7.5) along with which matches (7.6) and therefore shows that G ∞ (N ) is a symmetry transformation.

Modularity at a Particular Point of the Moduli Space
Using the general parametrisation of the group G(N ) in (7.14), we once again ask the question how the latter is related to Sp(4, Z) at the particular region in the moduli space, which is characterised by a Using the parametrisation (7.14) of G(4), it is sufficient to analyse the relation of the generators G 2 (N ) and G 2 (N ) to Sp(4, Z). The restriction of these generators to the subspace ( a, S, R) can be written in the form or on the space (τ, ρ, v) 19 S N is a subgroup of S N and for N ≥ 3 is isomorphic to Dih N . For N = 2 we have S 2 ∼ = S 2 . (N ), S ρ , T ρ , S τ , T τ is isomorphic to Sp(4, Z).

Conclusions
In this paper, we studied the consequences of the web of dualities among certain supersymmetric quiver gauge theories on R 5 × S 1 which are engineered by a class of toric Calabi-Yau threefolds X N,M . These dualities have been established in [23,21,25,26], here, however, rather than focusing on the different physical theories, we have analysed their consequences from the perspective of the partition function Z N,M . For the sake of simplicity, our analysis has been limited to the case M = 1. We found that the partition function Z N,1 associated to the geometries X N,1 is invariant under the group G(N ) ∼ = G(N ) × S N which acts on the vector space spanned by a maximal set of independent Kähler parameters. Here S N ⊂ S N has an intuitive interpretation as a subgroup of the largest gauge group that can be engineered by the given geometry, which is U (N ) in this case. The group G(N ), was shown to depend on N as derived in (7.14) and was found by exploiting the fact that X N,1 can be related to various other geometries (that are part of the same extended Kähler moduli space) trough flop-and symmetry transformations. These geometries are characterised by giving rise to the same topological string partition function (i.e. the same Z N,1 ), but they are described by web diagrams which have their Kähler parameters related trough a non-trivial duality map to the ones of the initial geometry. By studying a collection of these 'self-duality' maps we showed that they form the group G(N ).
A notable feature is the appearance of the infinite dihedral group for N ≥ 4. By using the matrix representations of the generating elements, we have explicitly shown in section 7 that for the cases N ≥ 4, the group G(N ) is generated by two matrices of order 2, which have no nontrivial braid relations (implying the existence of a group element of infinite order). An intuitive understanding of the appearance of the infinite order generator can be gained by looking at the behaviour under the series of flop transformations F, reviewed in appendix A. They can be used to relate web diagrams that look identical but have a non-trivial mapping between their Kähler parameters. By iterating this procedure, it is thus possible to generate an infinite series of inequivalent web diagrams, thus giving an intuitive argument for the appearance of an infinite order group. For the cases with N ≤ 3 there is no such iterative procedure for producing non-trivially related geometries, due to the simpler nature of the diagram. Furthermore, we showed that G(N ) combines non-trivially with other known symmetry groups of the partition function. For the case N = 1, we showed explicitly that G(1) ∼ = Dih 3 together with the modular group SL(2, Z) freely generate Sp(4, Z), which is known to be the automorphism group associated to the mirror curve of X 1,1 [30,10]. For N > 1, we showed that in a particular region of the Kähler moduli space, G(N ) corresponds to a subgroup of Sp(4, Z). Similarly, the group G(N ) mixes non-trivially with the T-duality (as specifically proposed in [8]) that relates the IIa and IIb Little String Theories that are engineered by X N,1 . In both cases, it would be interesting to extend this analysis and to characterise the full (non-perturbative) U-duality group of the LSTs. We leave this point for future work.
From the perspective of the various gauge theories engineered by X N,1 , the symmetry group free energy F N,1 in the NS-limit is fully captured by F 1,1 .
• In [24] it was argued that in the NS limit a particular part of Z N,M (called the reduced partition function) can be be written as the partition function of a symmetric orbifold CFT, giving rise to numerous Hecke like relations between various terms in the corresponding free energies.
• In [9] it was demonstrated at a large number of examples that (in the unrefined limit) for a particular choice of some of the Kähler parameters, the partition function Z N,M can be written as the sum over the weights of a single integrable representation of the affine Lie algebra a N −1 associated with the gauge group U (N ).
It is important that in all these cases, it was necessary to choose particular values for (some of) the Kähler moduli and/or the regularisation parameters 1,2 , in one way or another. The elements of the group G(N ) we found in the current work, are more general in the sense that they are symmetries of Z N,1 (or the corresponding free energy F N,1 ) at a generic point in the Kähler moduli space of X N,1 and for generic values of 1,2 . 20 In the future, it will be interesting to analyse, how G(N ) combines with the additional symmetries mentioned above in the respective regions of the moduli space. At a generic point in the moduli space, it would be interesting to analsye how G(N ) combines with other symmetries of the partition function (such as the modular groups SL(2, Z) τ and SL(2, Z) ρ ) to form an even larger symmetry group. As the symmetries discussed in this work impose severe constraints on the structure of Z N,1 , it would be interesting to investigate how much perturbative information (from the perspective of either one of the gauge theories engineered by X N,1 ) on the spectrum is required to recover the whole non-perturbative partition function. Questions of this type have recently been considered, e.g. in [44], where the authors showed that the partition function can be reconstructed by using information from the 2d world-sheet theories of the little string in combination with T-duality.
Another interesting implication of the symmetries discussed in this work concerns the consequences at the level of the gauge theories themselves. For example, in [45], the authors used the well known fiber-base duality of (a limit of) X N,1 in order to argue for an enhancement of the global symmetry group of a certain class of five-dimensional theories at their superconformal fixed point. They showed explicitly the appearance of characters of the enhanced global symmetry group when expanding the Nekrasov partition function in a specific set of Coulomb branch parameters that are invariant under fiber-base duality. While the theories we analysed here are six-dimensional and also do not have a superconformal fixed point (rather their UV completions are LSTs), one might hope to gain information about some enhanced global symmetry. We leave some of these points for future work.

A Duality Transformation F
Since it is frequently used in the main body of this paper, in this appendix we review a particular duality transformation (called F) that was first proposed in [23] (see also [21]) and which acts on a shifted web diagram as shown in Fig. 1 by changing δ → δ + 1. We specifically recall the duality map.
Starting from the web diagram in Fig. 1 with shift δ ∈ {0, . . . , N − 1}, the duality trans-· · · · · · a a 1 2 δ + 1 formation F is comprised of flop transformations on the curves with areas {h 1 , . . . , h N }, along with SL(2, Z) transformations and cutting and re-gluing of the web diagram. The resulting web diagram can again be presented in the form of a shifted 'staircase' diagram with shift δ + 1, as shown in Fig. 27.
It is important to notice that the independent Kähler parameters ( a 1,...,N , S, R) (shown in blue in Fig. 27) are in fact the same parameters as in Fig. 1, which in [21] were indeed shown to be invariant under the duality transformation. Similarly, these parameters are a solution of the consistency condi-tions imposed by the hexagons S 1,...,N , the latter being equivalent to the conditions (2.1) stemming from the hexagons S 1,...,N in the web diagram in Fig. 1. While the basis of the Kähler parameters ( a 1,...,N , S, R) is invariant under F, the individual curves (h 1,...,N , v 1,...,N , m 1,...,N ) are not invariant under the transformation F. Indeed, with respect to Fig. 27 we have the following duality map where h i+N = h i for i = 1, . . . , N .

B Presentation of Sp(4, Z) and Modularity
In [46] a presentation of Sp(4, Z) in terms of 2 generators (satisfying 8 defining relations) has been given. The latter are of order 2 and 12 respectively where H = KL 5 KL 7 K. We also mention that another presentation [47] (in terms of 6 generators and 18 defining relations) is given by X 1,2,3,4,5,6 , which can be expressed in terms of L and K as follows X 1 = L 5 KL , X 2 = L 9 HL 10 H , X 3 = L 8 KL 10 , X 4 = HL 9 HL 10 , X 5 = HL 6 , X 6 = L 9 HL 6 H . For convenience, we provide the action of some of the generators on the period matrix Ω