Fiber-Base Duality and Global Symmetry Enhancement

We show that the 5D Nekrasov partition functions enjoy the enhanced global symmetry of the UV fixed point. The fiber-base duality is responsible for the global symmetry enhancement. For $SU(2)$ with $N_f\leq 7$ flavors the fiber-base symmetry together with the manifest flavor $SO(2N_f)$ symmetry generate the $E_{N_f+1}$ global symmetry, while in the higher rank case the manifest global symmetry of the two dual theories related by the fiber-base duality map generate the symmetry enhancement. The symmetry enhancement at the level of the partition function is manifest once we chose an appropriate reparametrization for the Coulomb moduli.


Introduction
Gauge theories in five dimensional (5D) spacetime are perturbatively non-renormalizable, infrared (IR) free and may have a Landau pole singularity in the ultraviolet (UV) region.
Thus, usually, they can only be thought of as the effective low energy limit of some other theory. However, in the seminal paper [1], Seiberg argued using string theory constructions that the 5D N = 1 SU(2) gauge theories coupled to N f ≤ 7 hypermultiplets 1 have UV fixed points with enhanced global symmetry E N f +1 . The presence of these non-trivial UV fixed points suggest that these gauge theories do really "exist". The IR free theory with finite coupling should be thought of as a perturbation away from the UV fixed point triggered by the relevant operator 1 g 2 F 2 µν as well as its supersymmetry partner. Seiberg's arguments were shortly thereafter extended to other 5D theories with various color groups and matter content, see [2,3] for the necessary conditions for the existence of non-trivial UV fixed points.
Until recently, there was difficulty in verifying with a field theory argument or calculation the existence of such UV fixed points with enhanced global symmetry since the corresponding UV fixed points are strongly coupled. However, the recent developments in supersymmetric localization technique open up the possibility of investigating such SCFTs directly and quantitatively. In the pioneering paper [4] the enhancement to E N f +1 symmetry was directly checked by showing that the superconformal indices of the SU(2) gauge theories have E N f +1 symmetry. In this paper, we push this even further by showing that the Nekrasov partition functions are invariant under the enhanced E N f +1 symmetry as well. 1 For N f > 8 there is a Landau pole in the Coulomb branch at a ∼ 1 g 2 cl (N f − 8) , while for N f = 8 the effective coupling is a constant and it is impossible to take the strong coupling limit.
The superconformal index is the partition function of the protected operators [5,6] of a given theory, up to a sign for the fermionic states. It counts the multiplets obeying shortening conditions, up to equivalence relations setting to zero all combinations of short multiplets that may in principle recombine into long multiplets [7]. As such it is independent of the coupling constants of the theory, is invariant under continuous deformations of the theory, S-duality [8] and can therefore be evaluated in the free field limit if a Lagrangian description is known. The superconformal index has a path integral representation [6] as the partition function of the theory on S d−1 × S 1 , twisted by various chemical potentials, and can be evaluated using localization techniques [4,[9][10][11][12].
The 5D superconformal index was computed via localization on S 4 × S 1 in [4] where it was shown that where [da] is the invariant Haar measure, PE the Plethystic exponential that gives the multi-particle index from the free single-particle index i(quiver) of a given quiver and Z 5D inst the K-theoretic instanton partition function that localizes on the north and south poles of the S 4 . The 5D theories in the IR are weakly coupled and have a Lagrangian description.
The index computed in the IR, as long as no protected states are "lost" while flowing from the UV fixed point to the IR, will reorganize itself into characters of the enhanced symmetry of the UV fixed point. Kim, Kim and Lee in [4] checked that indeed the index of SU (2) with N f ≤ 7 flavors can be expressed by characters of the groups E N f +1 . 2 Similarly, symmetry enhancement was seen in a few more cases using superconformal indices [14][15][16][17]. 3 In [18] Iqbal and Vafa pointed out that the 5D superconformal index is given by a contour integral of the square of the 5D Nekrasov partition function which can be computed using the topological string partition function (1. 2) to expect that the global symmetry structure is encoded at the spectrum of the Coulomb branch.
It is the purpose of this paper to show that this symmetry enhancement is indeed visible already in the holomorphic part of the partition function Z 5D Nek (a) and to explain that this is possible due to the the fiber-base duality [19][20][21]. Especially, for the SU (2) theories, the fiber-base duality should be interpreted as a symmetry of the theory and thus as an extension of the global symmetry, which we denote also as "fiber-base symmetry" in this paper. By combining the generators of this fiber-base symmetry with the rest of the global symmetry one obtains the full extended symmetry.
Although from a purely field theoretic point of view the fiber-base duality may seem mysterious, it can be easily understood if the gauge theory is embedded in string/Mtheory 4 : • Geometric engineering on CY 3 : exchanging the base P 1 s with the fiber P 1 s is a symmetry of the Calabi-Yau manifold [19], • (p, q) 5-brane web diagrams in type IIB: S-duality exchanges D5-branes with NS5branes [20], with the Coulomb moduli, the coupling constants and the masses being exchanged in a non-trivial manner.
The (refined) topological string partition function can be computed by using the (refined) topological vertex method [23,24]. It is read off from the toric diagram as a function of the Kähler parameters for the 2-cycles and can be interpreted as the Nekrasov partition function of the corresponding gauge theory after the appropriate identification of the Kähler parameters with the gauge theory parameters [25][26][27] has been made.
In [21,28] we studied the fiber-base duality between the low energy effective theory of the 5D N = 1 SU(N ) M −1 and the SU(M ) N −1 linear quiver gauge theories compactified on S 1 . The two different gauge theories that are related by fiber-base duality have the same toric diagram, up to a 90 degree rotation. The duality map [21] is obtained by comparing these two parametrizations for the dual theories. In addition, the same result is obtained by comparing the SW curves. For the unrefined topological string, the fiberbase duality at the level of the 5D Nekrasov partition function is immediately apparent by using the topological vertex method. For the refined case, this also holds if we assume that the partition function does not depend on the choice of the preferred direction (slicing invariance). This case was studied in a subsequent paper by Ito [29] who, compared the refined Nekrasov partition functions of the two dual theories and obtained the refined map between the gauge theory parameters. As we will explain in section 3, the refined map that he found in [29] is the same as our unrefined map [21], up to a shift of the masses by ǫ + /2.
The fiber-base duality can be further understood at the level of the integrable models that are dual to the gauge theories under consideration as a spectral duality between the two different integrable systems [30].
Furthermore, a very useful way that has been used recently to see the symmetry enhancement is through the manipulation of 7-branes [15,17,31,32], see also the works [33][34][35][36][37][38][39] as well as [40][41][42][43][44][45][46][47][48][49]. Using 7-branes, we show in section 2 that the SU (2)  An important outcome of our work is the understanding that the fiber-base duality leads to the symmetry enhancement of the 5D theory. For the simplest case of the pure SU(2) gauge theory the fiber-base duality leads to an E 1 symmetry enhancement, as was already pointed out by [20]. In section 3, we generalize for SU (2) gauge theories with N f ≤ 7 flavors. For these theories, the Weyl symmetry of SO(2N f ) is manifest in the Nekrasov partition 5 function. The duality map generates the Weyl symmetry of E N f +1 and thus the enhanced global symmetry. We will also see that the fiber-base duality plays important role to understand the enhanced global symmetry also for higher rank case 6 .
We look for their symmetry enhancement by studying the holomorphic half of the integrand of the index (1.2), the Nekrasov partition function Z 5D Nek (a). The Coulomb moduli are parameters that are going to be integrated out anyway, so we are allowed to redefine them without changing the know results and we do that in such a way so that the new Coulomb moduli are invariant under the enhanced global symmetry. Note that the number 5 Only SU(N f ) is manifest for the U (2) Nekrasov partition function. SO(2N f ) is manifest for the SP(1) = SU(2) partition function [14][15][16]50] 6 Analogous observation has been done also in [51].
of a's is the same in both dual theories. In terms of this new Coulomb moduliÃ, we expand the holomorphic piece of the integrand and we discover that the coefficients ofÃ n organize themselves in characters of the enhanced symmetry, thus the enhanced symmetry is manifest. For the SU (2) theories with N f ≤ 7 flavors we find where the coefficients 7 of the expansion are characters of the expected enhanced E N f +1 symmetry. The Nekrasov partition functions for IR gauge theories thus carry the hidden E N f +1 global symmetry at the strongly-coupled UV fixed point.

7-branes and enhanced symmetry
Type IIB 7-branes are magnetic sources of the IIB dilaton-axion and can carry (p, q) charges. The usual D7-brane carries an (1, 0) charge while a (p, q) 7-brane is obtained by acting on a D7 brane with an SL(2, Z) transformations. In this paper X (p,q) will denote a (p, q) 7-brane, the (p, q) charge of which is defined up to an overall sign ambiguity X (−p,−q) ≡ X (p,q) . The symplectic inner product between two (p, q) charges is defined as Since a 7-brane is charged under the IIB dilation and axion, a 7-brane X (p,q) induces a branch cut in the transversal 2-plane. So a 7-brane changes its charge if it crosses a branch cut coming from another 7-brane. Let us consider adjoining two 7-branes X z 1 X z 2 . In our convention the branch cuts go downward. Under their reordering, one of these 7-branes will pass a branch cut and then change its charge as These 7-branes are key to understanding the enhancement of the global symmetry of 5D SUSY theories. Let us consider the gauge symmetry of the world-volume theory on coinciding 7-branes. Not all 7-brane configurations are collapsible and the possible gauge symmetries on 7-branes are already classified [38,39]. Since a 7-brane is a source of the dilaton-axion, this scalar experiences a monodromy transformation when moving around a 7-brane. A 7-brane X (p,q) develops the following monodromy 3) The possible 7-brane configurations are classified by using this monodromy. The necessary condition [38] that the set of 7-branes X (p 1 ,q 1 ) · · · X (pn,qn) can be collapsed to a point 8 is On these coinciding 7-branes, a mysterious enhanced gauge symmetry appears. Up to an overall SL(2, Z) transformation and reordering, all the collapsible 7-brane configurations and the corresponding gauge symmetry are where the world-volume of an H 0 7-brane configuration carries no symmetry, H 1 carries SU(2) and H 2 carries SU(3). Probing these 7-brane gauge symmetries, we can explain the enhancement of the flavor symmetry in 5d supersymmetric theories.
In the following, we demonstrate this method for various examples.
The SU(2) gauge theory with 4 flavors given by the 5-brane configuration of part (a) of figure 2 is a more interesting case. We can recast this web into the 5-brane loop on the 7-brane backgroundÊ 5 = X 4 (1,0) X (1,−1) X (1,1) X (1,−1) X (1,1) , but these 7-branes are not collapsible coincidentally. We can find a D 5 configuration in this background [31] where ≃ implies equality of the corresponding products of K's, up to a conjugation by K (1,0) . The UV fixed point is therefore described by figure 2(c), and the enhanced symmetry in this case is D 5 ∼ = SO(10). The SU (2) theories with other N f flavor can be also discussed analogously [31] and their global turns out to be E N f +1 as expected.

SU(3) N f = 6 SQCD
As in the case of SU(2) theories, the 7-brane technique works well for the SU(3) N f = 6 SQCD depicted in figure 3. This web can be modified by using two types of 7-branes without changing the world-volume 5d gauge theory and, by moving them inside of the 5-brane loop, we obtain the 7-brane configuration illustrated in the middle line of figure 4.

SU(N ) gauge theory
Since a (p, q) 7-brane can pass though (p, q) 5-branes without Hanany-Witten effect, we can recast this configuration into the right side of figure 6. This system has four stacks Figure 6: The web diagram for SU(N ) M−1 linear quiver (Left). By moving the (1, 0) and the (0, 1) 7-branes inside the 5-brane multi-loop, we obtain the right hand side. The shape of 5-brane multi-loop is actually warped nontrivially because of the background metric coming from 7-branes. of collapsible 7-branes, which are two X N (1,0) and two X M +1 (0,1) . The non-Abelian part 9 of the enhanced symmetry for N ≥ 3 case is therefore SU(N ) 2 × SU(M ) 2 . This symmetry is naturally consistent with the fiber-base duality.
The case of SU(2) quivers is a little exceptional. The configuration for N = 2 is one of M 5-brane rooms standing side by side, so that the (0, 1) 7-branes can be moved to the same 5-brane room without experiencing the Hanany-Witten effect. This means that two X M (0,1) 7-brane stacks are now recombined in a single stack as X 2M (0, 1) , and that therefore the enhanced symmetry for the SU(2) quiver is SU(2) 2 × SU(2M ). This symmetry, of course, coincides with that of the SQCD with gauge group SU(M ) and N f = 2M .

Fiber-base symmetry and symmetry enhancement for SU(2) theory
The goal of this section is to demonstrate that for the SU(2) gauge theories with N f < 8, the fiber-base duality can be combined with the manifest SO(2N f ) flavor symmetry to generate the full enhanced E N f +1 symmetry seen in the superconformal index. Furthermore, we show that the enhanced symmetry is present in the partition functions as well by expanding them in a power series of the appropriately defined invariant Coulomb modulusÃ.

Pure SU(2)
We first demonstrate our idea with the simplest example of the pure SU(2) super Yang-Mills theory. The pure SU(2) theory has enhanced E 1 = SU(2) global symmetry at the UV fixed point [1,4]. The relevant deformation of the UV fixed point that will drive 10 it 9 We discuss in section 4 that the actual global symmetry is SU(N ) 2 × SU(M ) 2 × U(1) if we also include the Abelian part. 10 At the UV fixed point [1] we have 1 g 2 = 0 . To trigger an RG flow we need to add a coupling constant deformation (the supersymmetrization of 1 g 2 F 2 µν ). To achieve that we need to begin with the UV fixed to the IR respects the Weyl symmetry of E 1 = SU(2) and thus it is natural to believe that the holomorphic part of the partition function will enjoy the enhanced symmetry. In what follows we will try to understand this from the view point of the fiber-base duality.
The pure SU(2) theory is realized using the local P 1 ×P 1 geometry that is the canonical line bundle over P 1 × P 1 . The toric diagram of the geometry is depicted in figure 7 and can be seen as a P 1 fibered over the base P 1 . We denote the Kähler parameter for the base P 1 as Q B and the Kähler parameter for the fiber P 1 as Q F . At this level it is clear that we can exchange the role of base and the fiber without changing the geometry. We will have the same Calabi-Yau manifold with the Kähler parameters exchanged This is known as fiber-base duality and will refer to the relation (3.1) and it's generalizations as the "duality map". Roughly speaking, the fiber-base duality is simply understood as just rotating the toric diagram by 90 degrees. As discussed in [22], the toric diagram can be reinterpreted as a (p, q) 5-brane web. In this language, the fiber-base duality is translated as the S-duality which exchanges the D5-branes and the NS5-branes [20].
The "duality map" can be checked/derived quantitatively by using the Nekrasov partition function which can be derived using the topological A model on the local toric Calabi-Yau as the topological string partition function. The unrefined topological vertex formalism computes the 5D Nekrasov partition function with self-dual Ω-deformation parameters ǫ 1 = −ǫ 2 = . The vertex function in the unrefined topological vertex formalism has cyclic invariance C µνλ = C νλµ = C λµν . Using the cyclic invariance together with the duality map (3.1) is enough to show that fiber-base duality is a symmetry of the unrefined point theory and gauge the Cartan part of the global symmetry and then give a vacuum expectation to the scalar of the new vector multiplet. This is also some times referred to the mass deformation of the non-Lagrangian theory. See section 5 in [52]. partition function 11 : The unrefined vertex C µνλ ≡ C µνλ (q) is a function of the topological string coupling At the level of the refined topological vertex formalism, the fiber-base duality is less trivial/obvious. The 5D Nekrasov partition function with generic Ω-deformation is given by the refined topological vertex formalism. The refined topological vertex function (A.7) is a function of two Ω-deformation variables C µνλ ≡ C µνλ (q, t) = C µνλ (t, q) and is not cyclic invariant as one of its legs is special. The direction of these special legs has to be parallel with each other and is called the preferred direction. In the refined case, the partition function reads Nevertheless, it is conjectured 12 that the topological amplitude does not depend on the choice of the preferred direction. Combining this conjecture together with the duality map (3.1) one can show that fiber-base duality is a symmetry of the refined partition function [29].
Comparing the topological string amplitude on P 1 × P 1 with the Nekrasov partition function for the pure SU(2) theory we obtain the relation between the Kähler parameters of the CY on one hand and the Coulomb moduli parameter a and the gauge coupling constant q = e 2πiτ on the other: where β is the circumference of the compactified S 1 . The duality map (3.1) can then be rewritten in the language of gauge theory parameters as In what follows we interpret the first equation in (3.5) is invariant under the duality map (3.1). SinceÃ is given by the exponential of the new Coulomb moduli parameter a new := a − log(q) 4β that is defined via a shift, we also refer to it as "the shifted Coulomb moduli parameter".
In summary, we learn that in order to see the global symmetry enhancement we should parametrize the Kähler parameters in terms of the shifted Coulomb moduli parameter as We introduce the new variable u 2 = q so as to simplify our formulas.
The topological string amplitude (3.3) can be taken from [15], rewritten using the function of (A.13) and the parametrization of (3.7) where we remind that u 2 = q. The functionsZ and S µλ are defined in equation (A.1) in the appendix A. Inserting (3.7) and expanding Z N f =0 in the modulusÃ leads to the expression 13 We thus discover that it organizes in terms of characters of the enhanced E 1 global symmetry. 13 In what follows characters are labeled by their dimension.
To obtain the expansion (3.9) we have assumed that the coefficient ofÃ 2n is completely determined by k instanton contributions with k ≤ n in (3.8). We can check this assumption experimentally by expanding to a few more orders but we also justify it in the appendix B.
Let us finish this subsection by making some remarks on the superconformal index.
The index is defined as the contour integral over the Coulomb modulus [18] where ǫ > 0 is small enough so that the contour integral only picks up the residue at zero, we have set q = xy, t = y x and M (x, y) is the refined McMahon function 14 . The "complex conjugation" acts by inverting 15Ã , u, x and y. One notes that the expansion of Z N f =0 in powers ofÃ does not commute with the expansion in powers of x due to the presence in the sum over partitions µ i of (3.8) of terms like 1 −Ã ux 2 −1 . Thus, we cannot extract the directly by plugging the expansion (3.9) into (3.10). In the above, all characters of E 1 are functions of the variable u. In order to get the result (3.11), we need to expand Z N f =0 in a power series in x first.

The case of SU(2) with N f ≤ 4 fundamental flavors
We now proceed to the cases with fundamental flavors. As we will see, our idea will generalize straightforwardly, up to two additional features/points that will play an important role.
The first point is concerned with the mass parametrization in the refined case. As was already discussed in [53], starting with the N = 2 * Nekrasov partition function, one has to shift the mass parameter for the adjoint hypermultiplet as Only then does one get the correct Nekrasov partition function for N = 4 SYM (which is 1) by sending to zero the mass deformation m new = 0. This shift is convenient also for the fundamental hypermultiplets due to the following reason. The explicit duality map for the theory with matter in the refined case is discussed in [29], where it is pointed out that the dependence on the Omega deformation parameter (in the combination of ǫ + := ǫ 1 + ǫ 2 or q/t) appears when we parametrize the Kähler parameters in terms of gauge theory parameters. However, such dependence disappears when we shift the masses as in (3.12) and the parametrization (the duality map) becomes exactly the same as in the unrefined case [21]. Moreover, this shift of the mass parameters (3.12) is motivated from the Weyl symmetry that acts as which is more natural than the transformation to the original mass parameter m old → ǫ + − m old . Therefore, the new shifted mass parameters should be used and the duality map for the refined case is exactly the same as in the unrefined case.
The second point is about flopping. The procedure of flopping involves sending the Kähler parameter Q of one of the branes of the web diagram to Q −1 while changing the geometry of the web diagram and the Kähler parameters Q i of all the branes adjacent to the one being flopped are sent to Q i Q, see figure 8. In our previous article [15], we use the (3.14) As we show in appendix A, under flopping 16 the new function behaves as  (2) with N f = 1 Figure 9: The left hand side is the original SU(2) one flavor case, while for the right diagram we have flopped Q 1 .

SU
In figure 9 we see on the left hand side the original flavor SU(2) dual toric diagram while the right one is the flopped one. The expected global symmetry is E 2 = SU(2)×U(1), whose Weyl transformation is given by which are related to the instanton factor q and the fundamental mass m 1 as (4.10) in [4] The fiber-base duality map is, as before, given by the exchange 17 which can be read off from either the original or the flopped diagram, see figure 9. The parametrization can also be read off from the diagram and is given by where we used (3.17). The duality map (3.18) leads to the transformation rules Again, we can identify the first two as the Weyl transformation of the enhanced symmetry (3.16). From the transformations (3.20), we identify the invariant Coulomb modulus as where c is an arbitrary constant. For reasons that we shall explain in subsection 3.4, here we choose it to be c = − 1 /7, indicating thatÃ = q 2 7 e −βa in which case the Kähler parameters are rewritten as The frame in which the flavor symmetry is apparent corresponds to the right diagram of figure 9 and the corresponding topological string partition function can be computed in two ways. The first is to directly apply the refined topological string formalism on the right diagram of figure 9. The second way is to take the result of [15], equation (4.27), that was computed for the the left diagram of figure 9, rewrite the results using the functions S instead of R and then use the flopping rule (3.15), which in this case implies . The final result obtained using either method reads Using the expression (3.23), setting the correct Coulomb modulus (3.22) and expanding in a power series inÃ, we obtain the expression (3.24) In the above the dependence of the fugacity u 1 is contained in the SU(2) characters χ SU (2) dim . Thus, in this frame the enhanced global symmetry E 2 ∼ = SU(2) × U(1) is apparent.
The enhanced symmetry in the case of two flavors is E 3 = SU(2) × SU (3), whose Weyl group is S 2 × S 3 with transformations that act on the fugacities u and y j by either u → u −1 or by permuting the y j . We define these fugacities to be the following functions where q is the instanton factor and m i are the masses of the fundamental flavors 18 . By construction, the fugacities obey the constraint y 1 y 2 y 3 = 1. The parametrization of the Figure 10: The left hand side is the SU(2) with two flavor, while the right hand side is the flopped version. We choose to put the two exterior branes diagonally opposite to each other because that way there is no reason to remove contributions coming from parallel exterior branes.
moduli can be taken from figure 10 and reads The fiber-base duality map again acts by exchanging Q F and Q B while leaving Q 1 and Q 2 invariant. This translates to the following map In order to find the invariant Coulomb modulus, we consider the Weyl transformations of the flavor symmetry SO(4) = SU(2) × SU(2), given by m 1 ↔ m 2 or m 1 ↔ −m 2 . By (3.25), they are equivalent to and together with the first three in (3.27) generate the full E 3 Weyl symmetry. We thus define the Coulomb modulusÃ which is invariant under the fiber-base duality and the flavor Weyl symmetry and is thus also invariant under the complete set of E 3 Weyl transformations. We can then express the Kähler parameters as In figure 10 we see on the left hand side the original two flavor SU(2) dual toric diagram and on the right hand side the flopped version. For the flopped diagram, we obtain using the parametrization (3.30) where we remind (3.25) that q = y 3 2 1 is the instanton factor. Since there are no exterior parallel branes, there is no full spin content to remove. Expanding (3.31) in the invariant Coulomb modulus, we get (3.32) In the above, the characters are again labeled by its dimension. Note that the invariance under the Weyl transformations (3.28) of the flavor symmetry SO(4) = SU(2) × SU (2) is directly visible in the Nekrasov partition function (3.31). Checking the invariance under the exchange u ↔ u −1 requires using (A.5), relabeling the partitions and using the fact that the topological amplitude is invariant under t ↔ q.
In the case of three hypermultiplets, the enhanced symmetry is E 4 = SU (5), whose Weyl transformations are given by permutations y i ↔ y j of the fugacities y i subject to i y i = 1.
They are parametrized by where as before q is the instanton factor and m i are the masses of the fundamental flavors.
These fugacities were chosen in such a way that the obvious flavor symmetry exchanging the masses only affects the y i for i = 2, . . . , 5. The Weyl transformation involving y 1 will, as we shall see in the following, arise from the fiber-base duality. The parametrization of the moduli is read off from figure 11 and can be expressed using the fugacities as The fiber-base duality map now acts by exchanging not just Q B and Q F but also Q 2 and Q 3 , which translates to that involve y 1 . One easily see that the invariant Coulomb modulus should be defined as leading to the following parametrization of the moduli Taking the flopped three flavor diagram, computing the topological string partition function and dividing out [15,16,50] the decoupled non-full spin content M( Using (3.38), we obtain the following expansion of the topological string partition function where the characters are labeled by their dimensions.

SU(2) with N f = 4
In the case of four flavors, the enhanced symmetry is E 5 = SO(10), whose Weyl transformations are y i ↔ y j and y i ↔ y j −1 . The five independent fugacities y i are given by The parametrization of the moduli is given by as illustrated in figure 12. The fiber-base duality map for our parametrization is which is translated into These together with the first two in (3.44) generates the E 5 Weyl symmetry. If we definẽ which is invariant under the E 4 Weyl transformation, we find the following parametrization In the four flavor case, as in the previous cases, we go to the frame that allows for an expansion in positive powers of the Coulomb modulus. This frame is depicted in figure 12 and its partition function is given, after dividing out the decoupled non-full spin content Setting the parameters (3.46) in (3.47), then expanding Z N f =4 inÃ leads to where E 5 = SO(10) and the characters are normalized such that the character of the fundamental representation is χ E 5 10 = 5 i=1 (y i + y −1 i ).

The case of SU(2) with N f ≥ 5 fundamental flavors
Starting at five flavor, we have the problem that we cannot choose a frame that would simultaneously allow for an expansion in only positive powers of the invariant Coulomb modulus and that would also be treatable using the standard refined topological formalism.
The issue being that in the frame in which only positive powers of the invariant Coulomb modulus appear, we cannot choose a preferred direction, see figure 13 for an illustration in the N f = 5 case. In principle, one could use the new vertex [58] and we leave that for future research.
In [16,59], it is checked that the topological string partition function for the E N f +1 theory, whose corresponding toric diagram is given in [60] in the language of (p, q) 5brane web, agree with the Nekrasov partition function for SP (1) theory with N f flavor. To compute the topological string amplitude, we need the new topological string vertex introduced by Iqbal and Kozçaz [58], depicted here by the red circles.
Taking this result into account, we will now use the results of [13], equation (3.38ff), to show the symmetry enhancement of the Nekrasov partition function for N f = 5, 6 and 7.
Furthermore, for each case, we will show by using an appropriate (p, q) web diagram that the fiber-base duality is part of the enhanced symmetry.
In order to compute the instanton contribution, we first need to include the contribution from the singlet field (the SP(1) antisymmetric tensor) to compute the ADHM quantum mechanics and then, divide by the extra factor which is discussed in section 3.4 in their paper to obtain the instanton Nekrasov partition function. 19 As for the perturbative part, we can easily read off from the computation of the superconformal index. The perturbative contribution from the vector multiplet to the superconformal index is given in [4,13] as where we have definedã = e −βa . The numerator of the second factor corresponds to the constant map, given by the MacMahon function, which we have omitted in this paper.
The denominator of the second factor cancels with the Haar measure of the Coulomb moduli integral. The last factor is the unimportant diverging constant, which we discard by hand. Then, the perturbative contribution to the corresponding topological string 19 The computation in this subsection is based on the explicit result of the instanton computation, which Chiun Hwuang, one of the author of [13], gave us by Mathematica file. We appreciate his kindness. partition function is "chiral half" of the first part, which we give Here we took the positive power ofã so that it agrees with the topological string result which we have used in the previous subsection. 20 Also for the hypermultiplet contribution, we choose the chiral half of the perturbative part of the superconformal index in such a way that the only the positive power ofã appears.
Definingm i = e −βm i , we can give the perturbative part of the partition function for which is consistent with what we have used in the previous subsection.

SU(2) with N f = 5
For the case of SU(2) with five flavors, we start with the left side of figure 14 and by the Hanany-Witten effect obtain the right hand side which is the T 3 web diagram, after we set the 7-brane to infinity. We use the left hand side of figure 14 to read off the parametrization and the right hand side to derive the action of the fiber-base duality. We introduce the fugacities y i = e −βm i , y j = e βm j , y 6 = q −2 .  The fiber-base duality map is easily read off from figure 14 and leaves Q 1 invariant while transforming the rest as combined with the exchange y 3 ↔ y 5 and y 2 ↔ y 4 . We would like to see this fiber-base duality as a Weyl reflection of E 6 . For this purpose, let us number the simple roots of E 6 as in figure 15 and, using a system of weights h i subject to (h i , h j ) = δ ij , given by One easily sees that the scalar products of the above expressions are (e i , e i ) = 2 for all i and We can now turn our attention to the partition function. The instanton contribution Multiplying the above with the perturbative part (3.51) and using the invariant Coulomb modulus (3.57), we arrive at 21 The precise character identities needed to derive (3.59) from the instanton and perturbative parts are contained in appendix C.

SU(2) with N f = 6
For the case of SU(2) with six flavors, we start with the left side of figure 16 and by Hanany-Witten effect obtain the right hand side which can also be understood by taking 21 We use the LieART [61] convention of what constitutes V and what is V . the T 4 web diagram and Higgsing one side. We use the left hand side of figure 16 to read off the parametrization and the right hand side to derive the action of the fiber-base duality.
The duality map is easily taken from the right hand side of figure 16 and reads together with the exchange y 2 ↔ y 5 and y 3 ↔ y 4 . Let us now see how to interpret this transformation as arising from a Weyl reflection of E 7 . To construct the root system of E 7 , we start with seven basic weights h i obeying the scalar products (h i , h j ) = δ ij and then build the simple roots of the Lie algebra E 7 , numbered as shown on the Dynkin diagram of figure 15, as The instanton contribution for N f = 6 is given by multiplying with the perturbative part, we find The final case to consider is the one of SU(2) with seven flavors. The web diagram is shown in figure 17, with the right hand side showing the configuration for which the fiber-base symmetry is apparent. 22 As in the previous subsections, we introduce the fugacities so that the Kähler parameter are given by From the right hand side of figure 17, we see that the fiber-base duality map acts on the Kähler parameters as  figure 15 and the roots themselves are given as We now direct our attention to the Nekrasov partition function. The instanton con- Combining the above with the perturbative part (3.51) leads to thus explicitly showing the E 8 invariance of the partition function.

Effective coupling
In this section we show that the fiber-base invariant Coulomb moduli parameterÃ that we introduced in the previous sections should be understood as the effective coupling constant of the theory.
The 5D theories that we have been studying are non-renormalizable and should be viewed as field theories with a cutoff. Even if in the classical theory the cubic term in the prepotential vanishes, it can be generated at one loop [42]. Since the coefficient of the cubic term is a finite quantity, it is independent of the cutoff. Following Seiberg [1], the effective coupling constant for the 5D SU (2) theory with N f flavor is given by 23 where φ is the vacuum expectation value of the real scalar in the 5D vector multiplet. In our article, we consider the parameter region where |Ã| ≪ 1, and the situation φ ≫ m i . In this case, the effective coupling reduces to In order to identify this effective coupling constant as the "invariant Coulomb moduli parameter", we divide by the factor 4 − N f /2 and obtaiñ Since φ is the real part of a, this implies that the invariant Coulomb moduli parameter should be defined asÃ where q is the instanton factor. The invariant Coulomb moduli parameters introduced in the previous subsections, see (3.6), (3.21), (3.29), (3.37), (3.45), (3.57), (3.65) and (3.73) all agree with the general solution (3.79).

On the perturbative part of the partition functions
In this subsection we focus on a subtle difference between perturbative part in the Nekrasov partition function and the perturbative part of the topological string partition function.
So far, we have identified the topological string partition function divided by the contribution from the non-full spin content, which we denote the normalized topological string partition function, as the full Nekrasov partition function. Indeed, it is known that the instanton part of the Nekrasov partition function is perfectly reproduced from the normalized 23 This is equation (3.5) in [1] but with different conventions.
topological string partition function [15,16]. However, there is a subtle difference between the perturbative part in the Nekrasov partition function and the perturbative part of the topological string partition function. The tree level contribution is not included in the topological string partition function and moreover the 1-loop piece is also slightly different.
In the following, we clarify these difference at the level of the effective coupling constant and check that this difference is also invariant under the enhanced global symmetry.
Since we have already checked that the normalized topological string partition function is invariant under the enhanced global symmetry, we discuss that the full Nekrasov partition function is also invariant.
For simplicity, we will illustrate our point only for the pure SU(2) case partially following discussion in [62]. The extension to the cases with matter is straightforward. Following [18], we have found in (3.8) that the perturbative contribution computed from the topological string partition function is given by where we used (A.1). We can compute its contribution to the prepotential by taking the logarithm and it is where we have used the standard convention (A.6). From this F top pert that we obtain from topological string we go ahead and compute the 4D 1-loop effective coupling corrected by the contribution from the Kaluza-Klein tower. We find where we used that Q F = e −2βa . As mentioned above, the tree level contribution is not included. Moreover, there is discrepancy, already discussed in [62,63], between (3.82) and the 1-loop result computed from the field theory side [64]. In order to reproduce the correct result known from perturbation theory, including the tree level part, we need to add to the calculation (3.82) from topological string τ missing (a) = −2 ln(Q F Q B ) = 8βa − 2 ln q. Finally, we check that the perturbative contribution (3.84), and not just (3.82) from the topological string, reproduces the expected 4D/5D limit. When we take decompactifying 5D limit, we should substitute the 5D coupling constant in the instanton factor as and then take the limit β → ∞. Assuming Re(a) > 0, we get which is exactly the correct answer for the effective coupling constant for the flat 5D SYM theory that appears already in [1]. On the other hand, when we take 4D limit, the instanton factor should be identified as q = (βΛ) 4 . which is the correct 4D effective coupling.

Fiber-base duality for higher rank gauge theory
In this section, we consider the case with the higher rank gauge group: the SU(N ) M −1 linear quiver theory with N + N fundamental flavor, which include SU(N ) with 2N fundamental flavor as a special case. Unlike SU(2) case, the theory is not self dual under the fiber-base duality for generic N and M . In this case, the fiber-base duality map is not part of the enhanced symmetry any more. However, we will see that the fiber-base duality still plays an important role to understand the global symmetry enhancement. Especially, by using the fiber-base duality map, we derive the invariant Coulomb moduli parameter, which is invariant under the Weyl symmetry of the enhanced global symmetry.
The relation between the fiber-base duality and the global symmetry enhancement for slightly different theory has been studied in [51] in the context of superconformal index.

Notation and the duality map
We begin by introducing our parametrization of the web diagram of the SU(N ) M −1 linear quiver with N f = N + N . Here and in the following we follow [21] as closely as possible and work with exponentiated distances. We begin with the exponentiated positionsã Bα can be written as follows The moduli Q (i) and Q (i) ′ are related via the constraint and the gauge couplings are given by the geometric average 3) The number of independent quantities is easy to count: we have (M + 1)N positionsã (i) α subject to one condition setting the center of mass, as well as M − 1 couplings q (i) , leaving us with exactly (M + 1)(N + 1) − 3 independent parameters.
In the topological string computations it is convenient to use the parametersã Bα . However, in order to relate the theory to its dual, it is convenient to introduce new variables: the "traceless" positionŝ for α = 1, . . . , N .
In (3.42) of [21], we derived the following map 24 between the parameters of the original 24 We remark that due to a difference in the convention in which distances are measured between the current article and [21], we need to apply the transformationã The label d at the left hand side of the equation indicates that they are the variables of the dual theory.

Invariant Coulomb moduli: a special case
We now want to derive the set of parameters which make the enhanced symmetry apparent.
However, as we discussed in section 2, the expected global symmetry in the case M = 2, N ≥ 3 is given by SU(2N ) × SU(2) × SU(2). We will see that the duality map helps us to understand this global symmetry enhancement.
In the following, we use the duality map (4.8) with M = 2 is substituted. First, using (4.10), we find that that the first line of (4.8) for α = 0 and N can be written as Thus, from the view point of the dual theory, the U(1) N × U(1) N −1 symmetry is enhanced to SU(2N ).
Combining the result above, from the point of view of either theory, we find that the global symmetry is enhanced to SU(2N ) × SU(2) × SU(2) as expected. It would be convenient to useM α , Q (1) and Q (1)′ as the corresponding fugacities.
Finally, we derive the invariant Coulomb moduli parameter using the maps corresponding to the remaining moduli parameters. From the the first line of (4.8) it is straightforward to derive the following set of equations: Since SU(2N ) part is identified as part of the manifest global symmetry of the original SU(N ) theory, it is natural to assume that the Coulomb moduli parameterâ Thus, we can identify them as the invariant Coulomb moduli parameters.
In summary, our conclusion is that we should define the invariant Coulomb moduli parameters for the SU(N ) gauge theory with N f = 2N as (4.14) where α = 2, · · · , N − 2.

The general case
Since (4.15) The remaining U(1) fugacity is given by M k=1m We can explicitly check that the following set of Coulomb moduli parameters are invariant under the enhanced global symmetrỹ where as in (4.14) the index α runs over 2, . . . , N − 2.

Partition functions for higher gauge groups
The purpose of this section is to construct the partition functions for the general linear quiver theories SU(N ) M −1 . Furthermore, we will demonstrate for the case of SU(3) with six flavors that the expansion in the invariant Coulomb moduli makes the enhanced symmetry SU(6) × SU(2) 2 , whose presence we derived in section 2 using D7 branes, manifest.

The strip decomposition for linear quivers
In [15], we computed the topological string amplitude of the T N junction by cutting the web diagram in N strips. We can perform a similar procedure for the quiver theories, but the strips are different from the ones in [15] due to the fact that the number of flavor branes on the left and on the right is the same. The geometry is depicted in figure 19. Setting λ 0 = λ L = ∅, we can resum the expression for the topological string partition function as By using (5.1), we can write the partition function of the SU(N ) M −1 linear quiver as with theã (r) α read off from figure 18. To obtain the proper partition function, we then need to divide by the decoupled non-full spin content.

Flopping for SU(3)
In order to be able to see the symmetry on the level of the partition function, we need to  diagram is given by a sum over the product of two strips (5.1) and equals 25 The flopping is illustrated in figure 20. We can use (dropping phases) the flopping equation (A.14) to flop the parts (5.5) 25 Compared to section 4, here we use the notation Q1 ≡ Q (1) and Q2 ≡ Q (1) ′ .
We thus get into the frame, depicted in figure 21, that tell us that the full flavor symmetry should contain SU(6) × SU(2) × SU (2). After removing the non full-spin content the partition function becomes We now want to make the flavor symmetry in the partition function apparent. First, using translation invariance, we get the parametersã i to obey the conditionã 1ã2ã3 = 1.
Furthermore, we see that We introduce new mass parameters subject to the relationM i with 6 i=1M i = 1 via the equationM They should make the SU(6) part of the flavor symmetry manifest. The SU(2) × SU (2) part is contained in the Q i parameters. Furthermore, as we shall see in (4.14), we need to define the invariant Coulomb moduli asÃ 1 = Q 1ã 1 a 2 andÃ 2 = Q 2ã 2 a 3 . We can also write the inverse relatioñ Expanding to first order inÃ i and second order in t and q, we get (6)  (Q 2 ) + · · · + · · · , (5.10) where the characters of SU(6) are labeled by their dimensions. While we did not succeed in obtaining the exact t, q dependence of the coefficients in theÃ i expansion, (5.10) still demonstrates the appearance of the enhanced symmetry SU(6) × SU(2) 2 at the level of the partition function.

Conclusions
In this paper, we have discussed how the fiber-base duality of some Calabi Yau geometries affects the global symmetry enhancement of the corresponding 5D N = 1 gauge theory.
We have clarified how the masses, the gauge coupling, and the Coulomb moduli parameters are mapped to each other by the fiber-base duality by using the idea discussed in [28]. In addition, there is a multitude of other possible directions for future works. One of them is the study of the relation between our E N f +1 manifest expansion of the Nekrasov partition function and the elliptic genus of the E-string or Nekrasov-type partition function that was recently studied for instance in [65][66][67][68][69][70]. Taking into account that the elliptic genus of the E-string is also written in an E 8 manifest way, it is natural to expect that proper dimensional reduction and/or mass decoupling limit will directly reproduce our E N f +1 results.
Since we have shown that the fiber-base duality, which translates to S-duality in the brane setup picture, is part of the enhanced E N f +1 symmetry, it would be also interesting to consider whether there is any relation between this observation and [71] in which the duality between del-Pezzo surfaces and U-dualities was discussed. Our result may also be related to the theta function arising in the BPS spectrum of M5-branes wrapped on del-Pezzo surfaces [72].
Recently, an interesting observation was made 26 by [73] that the fiber-base duality can be essentially seen as the 5D uplift of the 4D "N = 2 dualities" of [75]. However, there are several subtleties that should be studied further. In [73], using the superconformal index the authors showed that SU(3) with 6 flavors and with Chern-Simons level 1 is dual to SU(2) × SU(2) with 3 flavors coupled to the first color group and 1 flavor to the second, denoted as 3 + SU(2) × SU(2) + 1. In [28] and in this paper, we showed that the fiber-base dual of SU(3) with 6 flavors and with Chern-Simons level 0 is 2 + SU(2) × SU(2) + 2. Ideas used in this paper will definitely be useful to further investigations in this direction.
The fiber-base duality implies, as we already suggested in [28], that gauge theories with different gauge groups are equivalent, meaning that the correlation functions of different

A.1 Definitions and basics
For the reader's convenience, we collect here the definitions of the special functions used in the main text. First we mention those that are given by finite products In the above, the notation (i, j) ∈ µ means that i runs over the rows of µ, i.e. i = 1, . . . , ℓ(µ), while j runs over the columns, i.e. j = 1, . . . , µ i . Then we also need M that is given by an infinite product or by a Plethystic exponential as , where the product converges for all Q if |t| < 1 and |q| < 1 and the Plethystic exponential converges for all t and all q provided that |Q| < q −1+θ(|q|−1) t θ(|t|−1) with θ(x) = 1 if x > 0 and zero otherwise. The function M can be defined for all Q, t and q if we require that .
We shall often use the combined function The following exchange properties can be useful ; t, q), R λµ (Q; q, t) = R µλ (Q; t, q). (A.5)

A.2 Topological vertex
We use the Ω deformation parameters as where ǫ ± = ǫ 1 ± ǫ 2 . The refined topological vertex is given by We remind that for a partition ν, the vector t −ρ q −ν is given by The framing factors are defined as In our previous article [15], we use the functions defined in (A.1) to write the topological string partition functions. In order to make the invariance under flopping Q → Q −1 as nice as possible, we will now introduce a new one.
First, let us consider the function N λµ . Using the identities (i,j)∈λ i = 1 2 ||λ t || 2 + |λ| and (i,j)∈λ µ i = min{ℓ(λ),ℓ(µ)} i=1 λ i µ i we obtain after a straightforward computation the Up to the zeta function regularization, we have Setting t = q in (A.11), we reproduce formula (42) in [54] for the unrefined case. Combining both (A.10) and (A.11), we arrive at the following compact expression for the functions R λµ of (A.1): We see that if we define a new function as

B. On the Coulomb moduli expansion
In order to show the symmetry enhancement, we expanded in section 3 the Nekrasov partition function in terms of the invariant Coulomb moduli parameterÃ. For simplicity, we first assumed that the coefficients ofÃ m are completely determined by the k instanton and k ≤ 4m for N f = 7. We checked this assumption experimentally by expanding to a few more orders with Mathematica. In this appendix, we give an analytical justification of this assumption, which is valid at least up to N f ≤ 4.

B.1 Pure SYM
We warm up with the pure SYM case, for which it is straightforward to prove our assumption. We want to study theÃ expansion of the topological string amplitude (3.2) that is given by an expression of the form G(q, t; µ 1 , µ 2 , ν 1 , ν 2 ) . (B.1) Figure 22: Subamplitude Using the parametrization (3.7) of Q B and Q F in terms ofÃ 2 , we see that theÃ 2n coefficient is obtained from the terms with Performing the summation over the Young diagram ν 1 and ν 2 in (B.1), we reproduce the standard form of the Nekrasov partition function, which is an expansion in terms of the instanton factor. Since the instanton factor is included only in Q B , the k-instanton contribution to the Nekrasov partition function comes from the terms in (B.1) satisfying The terms satisfying (B.2) include only the terms satisfying (B.3) with k ≤ n. Therefore, in order to obtainÃ 2n terms from the Nekrasov partition function, it is enough to consider the k-instanton contribution with k ≤ n.

B.2 The four flavor case
The generalization of the statement above for the cases with up to two flavors is straightforward. The situation is slightly different for N f ≥ 3. Here we will discuss the case with N f = 4 as an example. The main difference stems from the fact that the toric diagram has external lines that are parallel.
(B.5) and the refined topological vertex is given in (A.7). These functions correspond to the amplitudes of the two toric diagrams depicted in figure 22.
Such topological amplitudes, but without the framing factors f η (q, t) andf η (q, t) (defined in (A.9)), were computed in [56] in the context of the refined Hopf link and its relation to the S-matrix of the refined Chern-Simons theory [89]. Using a slightly modified version of the computation in [56], we can show that (B.4) are also degree |Y 1 | + |Y 2 | polynomials of Q even though the summation for η in (B.5) is taken over all possible partitions. Higher order terms cancel against each other and the Taylor expansion terminates at a finite order.
Knowing this property, we see that we need only the terms in the sums of (B.5) with in order to compute (B.4) explicitly.
It is straightforward to see graphically that the toric diagram for SU(2) with four flavor, depicted in the left of figure 12, can be constructed by combining the sub-diagrams in figure 22, and that it reads We use the "normalized" amplitude (B.4) rather than (B.5) itself because, as we have discussed in [28], we should remove the decoupled degrees of freedom, referred to as the "non-full spin content", coming from the parallel external legs.
Since theÃ dependence of the parameters are Q −1 i ∝Ã, and Q F , Q B ∝Ã 2 as given in (3.46), theÃ m contribution in (B.7) comes from the terms with

B.3 Computation of the normalized partition function Z =
In the previous subsection B.2, we used the property that the sub-amplitude Z = Y 1 ,Y 2 (Q; q, t) defined in (B.4) is a finite polynomial in Q of order |Y 1 | + |Y 2 |. In this subsection, we show this property following the discussion in [90]. Writing down Z ′= Y 1 ,Y 2 (Q; q, t) explicitly, by using the definition of the refined topological vertex, we obtain As in [90], we moreover define the open string partition functioñ where we divided by the prefactor q which implies that the normalized version obeys Z = (Q; t, q; x, y) ≡Z ′= (Q; t, q; x, y) We can immediately expand this product in powers of x i and y j and find that their coefficients are polynomials of Q of degree less or equal to the sum of the power of x i and y j .
This means that when we expandZ = in terms of Schur functions exactly as in (B.10), the coefficient Z = Y 1 ,Y 2 (Q; q, t) that multiplies s Y 1 (x)s Y 2 (y) has to be a polynomial in Q of degree |Y 1 | + |Y 2 |.

B.4 Computation of the normalized partition function Z ||
Using the definition of the refined topological vertex (A.7), the amplitude defined in (B.5) is explicitly given by Factoring out the overall factor, we concentrate on where we absorbed the framing factor f η (t, q) −1 in the Macdonald polynomial P η (t −ρ ; q, t).

(B.15)
Here, x(p) indicates that the Macdonald function, which is symmetric in the variable x, should be rewritten in terms of the power sum function p n (x) = i x n i . The derivative operator p * is defined as p * n = 1 − q n 1 − t n n ∂ ∂p n . (B.16) The inner product * , * t is given as while the other inner product * , * ′′ r,t is given in (2.5) of [91] as 27 See also [90].
The Schur function is expanded in terms of the elementary symmetric function e µ as s λ (x) = µ≤λ V λ,µ e µ T (x) (B.20) where e λ ≡ e λ 1 e λ 2 · · · . Then it is straightforward to show that where we defined H λ = H λ 1 H λ 2 · · · and p n (t −ρ ) = t n 2 /(1 − t n ). Taking into account that H λ is a linear combination of derivatives ∂ ∂pn , we find that the normalized amplitude Z || Y 1 ,Y 2 (Q; q, t) is a degree |Y 1 | + |Y 2 | polynomial in Q.

C. On character formulas
In this appendix, for the convenience of the reader we summarize the formulas for the characters as well as our conventions that we have used in section 3.3.
We begin with the N f = 5 case. We define the SO(10) characters as