Tao Probing the End of the World

We introduce a new type IIB 5-brane description for the E-string theory which is the world-volume theory on the M5-brane probing the end of the world M9-brane. The E-string in the new realization is depicted as spiral 5-branes web equipped with the cyclic structure which is key to uplifting to six dimensions. Utilizing the topological vertex to the 5-brane web configuration enables us to write down a combinatorial formula for the generating function of the E-string elliptic genera, namely the full partition function of topological strings on the local 1/2 K3 surface.


Introduction
Among possible interacting superconformal field theories (SCFTs) in diverse dimensions, the six-dimensional (6d) theories are less well understood. It is expected that 6d N = (1, 0) SCFTs involve tensionless strings and show very non-trivial physics. This nontrivial nature of the 6d SCFTs is deeply related to strong coupling physics of superstring theory, but its detailed feature has yet to be discovered.
The E-string theory is a typical 6d N = (1, 0) SCFT. Originally, the E-string theory has been found in the heterotic strings for small E 8 instanton [1][2][3]. There are also several other realizations of the theory. For instance, by considering M5 brane probing the end of the world M9 boundary [4,5], we can find the E-string theory on the M5 world-volume.
Recently there has been significant progress on the localization computation in 5d gauge theories. In [31,32], the localization calculation was formulated for the superconformal index of 5d N = 1 Sp(N ) gauge theories with N f flavors, and it was found that this index shows the enhancement of global symmetry SO(2N f ) × U (1) ⊂ E N f +1 as was expected from Seiberg's argument [33]. The same index was also derived from the topological string theory in [34], and the enhancement was studied in more detail in [35,36] from the perspective of topological strings on local Calabi-Yau 3-folds. In [37], it was found that Nekrasov partition functions also show the enhancement. These developments led to exact computation on the partition function and superconformal index of a 5d theory starting directly from the IR Lagrangian or from the corresponding brane web configuration/Calabi-Yau geometry. This technology is the ingredient of our approach to the E-string theory.
Thanks to the developments in localization, quantitative study on 6d N = (1, 0) SCFTs is getting more manageable. In [38,39], based on the elliptic genus computation, the E-string partition function up to four E-strings was computed. See also [40].
In this paper we propose a different description of the E-string theory based on familiar IIB 5-brane setup. In the case of 5d N = 1 gauge theories, the world-volume theory of the corresponding (p, q) 5-brane web configuration leads to a desired gauge theory [41][42][43][44].
The same web diagram specifies the toric Calabi-Yau threefold of the compactified Mtheory dual to 5-brane web, and then we calculate the 5d Nekrasov partition function [45] using the topological string method [46][47][48][49][50][51] known as the topological vertex [52][53][54][55][56] through the geometric engineering [57,58]. If these techniques are also applicable to the 6d E-string theory, they will be a very powerful method conceptually and computationally to investigate 6d dynamics. To this end, we utilize the key fact: the E-string theory of interest, which is a 6d N = (1, 0) SCFT, appears as the UV fixed point of 5d N = 1 SU (2) gauge theory with 8 flavors. Indeed, the agreement of the corresponding partition functions is checked in [39]. This means that once one finds a 5-brane web for SU (2) gauge theory with 8 flavors, one can apply all the established techniques to study the E-strings.
In this paper, we discuss such (p, q) 5-brane web description of the E-string theory. The (p, q) 5-brane web that we found is of a spiral shape with the cyclic structure associated with the spiral direction, which is the source of the hidden 6d direction accounting for the Kaluza-Klein direction. By implementing the topological vertex to this spiral web, we can write down a combinatorial expression of the full-order partition function, which is the generating function of the elliptic genera of the E-strings. This generating function is precisely the full partition function of topological strings on the local 1 2 K3 surface. Our analysis may provide a stepping stone allowing one to look for a similar description for other 6d theories as well as a new class of 5-brane web.
The paper is organized as follows. In section 2, we review IIB (p, q) 5-brane web construction for SU (2) gauge theory with eight flavors based on 7-brane monodromies, and find a spiral structure of the web diagram, which we call "Tao diagram". In sections 3 and 4, applying the topological vertex method to this Tao diagram, we compute of the partition function and compare the obtained result to the elliptic genera computed in [39]. We discuss various Tao diagrams as well as other future works in section 5. 2 Tao 5-brane web via 7-branes 2.1 7-brane background and 5d SU (2) gauge theories A large amount of 5d N = 1 gauge theories and their UV fixed point SCFTs are realized as the world-volume theories on 5-brane (p, q) webs [35,36,41,42,44,59,[62][63][64][65][66][67]. This setup is precisely the 5d version of the Hanany-Witten brane configuration. We can utilize this web realization to know the strongly-coupled UV fixed point, the BPS spectra, the Seiberg-Witten curves and so forth.
In order to obtain the 5-brane (p, q) webs for a given theory, one often finds convenient to start from the system of the 7-branes studied in [68]. Technology for treating 7brane backgrounds was developed in [69][70][71][72][73]. The relation between 7-brane background and 5-brane configuration is as follows. Figure 1 illustrates a typical example of the correspondence between a 5-brane web system and a 7-brane configuration. The left hand side is the (p, q) 5-brane web for 5d N = 1 SU (2) pure Yang-Mills theory. There are four external legs in this configuration, whose (p, q) charges are (1, 1) and (1, −1).
We can regularize these semi-infinite external legs by terminating a (p, q) leg on a [p, q] 7-brane, and then we obtain the middle diagram in Figure 1. Without changing the world-volume theory, these 7-branes illustrated by colored circles can move freely along (p, q) line. We can therefore move them inside the 5-brane quadrilateral, and then the configuration in the right hand side of Figure 1 appears. Notice that four external legs hand side, we can move two 7-branes (blue dots) to infinity, which makes a diagram with two arms that spirally rotate infinitely many times.
are disappeared since the Hanany-Witten brane annihilation transition removes these 5-branes after the 7-branes cross the 5-brane quadrilateral. The 5-brane quadrilateral are now highly-curved because of the non-trivial metric coming from the 7-brane inside.
In summary, the (p, q) brane web for SU (2) Yang-Mills theory is related to a 5-brane loop configuration probing the following 7-brane configuration where B is [1, −1] 7-brane and C is [1, 1] 7-brane. The world-volume theory on this 5-brane loop is also SU (2) Yang-Mills theory because of the above construction. In this way, we can convert a (p, q) web into a 5-brane loop probing a 7-brane background. For 5d N = 1 SU (2) gauge theory with N f flavors, the following 7-brane background appears inside the 5-brane loopÊ Here A is a [0, 1] 7-brane. Up to 7 flavors, the 7-brane system can be recast into 5-brane web configuration by pulling out all the 7-branes to infinity by using Hanany-Witten transition [42,44,64,68]. In the language of the toric Calabi-Yau associated with the corresponding (p, q) web diagram [43], this 7-brane configuration is dual to the local P 2 blown up at N f + 1 points or local P 1 × P 1 geometry, namely the local del Pezzo surfaces  Up to the N f = 4 flavors, it is easy to see the corresponding (p, q) 5-brane web diagrams (See Appendix of [75]). It is however not so straightforward to find (p, q) 5-brane web diagrams for N f = 5, 6, 7 flavors. It was studied in [35,36,44] that the corresponding 7-brane configurations for N f = 5, 6, 7 flavors are realized in the (p, q) 5-brane web diagrams as 5d uplifts of tuned (Higgsed) T N diagrams : The 7-brane configuration for N f = 5 flavor case is exactly mapped to T 3 diagram showing E 6 symmetry; The configurations for N f = 6, 7 flavor cases are a tuned T 4 and T 6 diagram showing E 7 and E 8 symmetry, respectively.
Our focus in this paper is the 5d N = 1 SU (2) gauge theory with N f = 8 flavors and its 6d UV fixed point theory, namely the the E-string theory. Our starting point is therefore the affine 7-brane background (2.2) for N f = 8. This 7-brane configuration corresponds to 1 2 K3 surface, and M-theory or topological strings on the Calabi-Yau is dual to the E-string theory. The N f = 8 configuration is a one-point blow-up of the N f = 7 configuration. Using 7-brane monodromies one easily finds that where N is a [0, 1] 7-brane and X [p,q] is a [p, q] 7-brane. See Appendix A for more detail.
The corresponding 5-brane loop on this 7-brane background is illustrated on the left hand side of Figure 2. When we try to pull out the 7-branes, we find that the [2,1] 7-branes change their charges due to the monodromy cut created by [1, 0] 7-branes and [0, 1] 7-branes, which makes the attached 5-branes form spiral arms. After taking all 7branes to infinity, we find a spiral web diagram with multiple arms which rotate infinitely many times, as in Figure 3a. We will call such spiral diagrams "Tao diagrams" for short because its spiral structure is close to a symbolic representation of the taoism philosophy Note that Figure 3a contains multiple coincident 5-branes intersecting at one vertex.
These 5-branes actually jump over the other 5-brane at such vertex in the sense of [44].
For instance, see Figure 4. In topological vertex formalism, such jumping is realized by a degenerated Kähler parameter [36,[76][77][78][79]. We will show definite way to treat it in the next section.
Unfortunately, the above Tao web Figure 3a involves strange '(0, 2) 5-branes' which can not be any bound state. For avoiding discussing proper treatment of such 5-branes, we switch to more healthy web description in the following. In fact, a web description for a given 7-brane background is not unique because changing the ordering of 7-brane movement results in different web [64]. In the next subsection, we demonstrate in detail that there exists an another Tao web for the E-string theory, and we will use this new Tao diagram throughout this paper.

E-string theory via Tao (p, q) web
As we have observed in the previous subsection, N f = 8 theory leads to spiral 5-brane configuration and opens up new dimension associated with the cyclic nature. Such Tao web would give intuitive and effective description of the 6d E-string theory. In this subsection, we give more simple and useful Tao diagram which describes the E-string theory. The starting point is the affine 7-brane background again We will consider 5-brane loop configuration on this configuration to construct a dual description of the local 1 2 K3 surface as was done in the cases of the local del Pezzo surface [35,36,44,64,68]. The resulting Tao web is therefore generalized toric description of the local 1 2 K3 surface. As is explained in Appendix A, reordering these 7-branes leads to an another expression of theÊ 9 configurationÊ 9 = ANC ANC ANC ANC. to move the 7-branes away from the middle 5-brane loop, however a 7-brane soon hits a branch-cut coming from an another 7-brane. The shape of the resulting web configuration therefore depends on the ordering of 7-brane motion. Let us consider the 7-brane motion illustrated in Figure.6. Continuing to move 7-branes with such an ordering, we finally find the spirally-growing 5-brane web Figure.7. This is precisely Tao brane web which describes SU (2) gauge theory with N f = 8 flavors, that is the E-string theory. This Tao web Figure 7 has six external legs making spiral cycles, and each leg has period 6 of its joints. In addition, there are six straightly out-going legs consist of bundles of 5-branes of the same (p, q) type. A bundle increases the number of the constituent 5-branes by one each time the bundle crosses a spiral leg. This structure comes from the Hanany-Witten effect that occurs when one moves all remnants of 7-branes to infinity after creating six spiral legs in Figure 6. This local structure is a degenerated version of the T N →∞ 5-brane web. From this point of view, we can construct the Tao web by gluing six T ∞ sub-webs. Treatment of such a degenerated toric diagram was studied in [36,[76][77][78][79]. We will explain it in the next section.

What will happen when N f ≥ 9?
It is claimed in [33]  A way to see the N f = 8 case is critical is that if one successively applies 7-brane monodromy to move a [p, q] 7-brane through other 7-branes, then it comes back to the original configuration after one rotation which is a necessary condition for spiral shape of 5-brane web. For instance, 7-brane configuration corresponding to the middle diagram in Figure 2 is given by and two [2, 1] 7-branes (green dots in the diagram) are rotating clockwise. This means that these two 7-branes undergo the following monodromies due to the branch cuts from the remaining 7-branes hence the charge of the rotating 7-branes changes as it passes through each cut. For one rotation, the monodromy matrix that two [2, 1] 7-branes go through is Hence, the charge of two [2, 1] 7-branes remain the same after one rotation.
One can apply the same logic to N f = 9 flavors. A configuration for N f = 9 that one can find is one which adds one more flavor [1, 0] 7-branes to Figure 2 is If we take the [3, 1] 7-brane and let it go through clockwise all the branch cut then it is easy to see that the charge changes due to monodromy can never be same as any of 7-branes above. In a similar way, the charge of [3, 1] 7-brane becomes [10,3] after one rotation is given by Note that the direction that the resultant charge points is inward rather than outward.
The more it goes around, the more it points inwardly. The 5-brane attached to this inwardly rotating 7-brane eventually cross into the 5-brane loop in the middle, and thus its shrinks to the origin rather than going away from the origin. In other words the configuration with N f = 9 flavors never makes a proper 5-brane web, it all collapses. In a similar fashion, one finds collapsing of brane configuration for higher flavors N f > 9.
This is consistent with the known result and it is a geometric account why N f = 8 configuration is critical.

Physical parameters of Tao web
In this section, we go back to the Tao diagram, which corresponds to N f = 8. It is discussed in [43] that (p, q) 5-brane web can be reinterpreted as toric diagram. There are various supports [36,37,65] that this claim works also for "toric-like diagram" 1 introduced in [44], which contains 5-branes jumping over other 5-branes. Therefore, we expect that it is also the case for our Tao diagram even if it extends to infinity. In other word, we expect that our Tao diagram gives toric-like description for local 1 2 K3. Assuming this, we compute the E-string partition function using the topological vertex in the next section.
As a preparation, we need to study the relation among the Kähler parameters of the corresponding geometry as well as their relation with the gauge theory parameters. 1 It was denoted as "dot diagram" in their paper.
For each segment E ∈ {edges} in the web diagram, we associate the length parameter t E and exponentiated one Q E = exp(−t E ). In the language of toric-like geometry, this parameter is the Kähler parameter of the two-cycle corresponding to the segment. Unlike usual web diagram, there are infinitely many segments and their Kähler parameters in our diagram. They are however highly constrained since the six radial legs trigger the turns of the spirals. We can solve such constraint equations and finally find only ten free parameters.
We introduce a notation for organizing infinitely many Kähler parameters. In the Tao web there are six spiral external legs, and we label them by integers = 1, 2, · · · , 6 in counterclockwise order. Infinite number of straight line segments compose a spiral leg. To describe this spiral curve, we labels Kähler parameters Q i( ) with i being the turn number of the -th arm. It is convenient to introduce the "distance" ∆ between adjacent arms where = 1, 2, · · · , 6, and i = 1, 2, · · · . We define so that (3.1) is satisfied also for = 0 and/or = 6. This parametrization is depicted in Figure 8. By iteration, one finds that any Kähler parameter can be expressed as one also finds that the -th arms are aligned parallel every six turns We thus have twelve parameters: ∆ and Q 1( ) . On the other hand, as there are eight flavors and one gauge coupling, one only needs nine physical parameters. Hence, the twelve parameters are not all independent, but are subject to the following three constraints: (iii) constraint on the origin of the Coulomb branch The last constraint requires some explanation. If we perform local deformation to make the vanishing Coulomb moduli (See the right hand side of Figure 8), then the origin of the Coulomb branch is supposed to be the starting point of the Kähler parameters δ and δ. By taking horizontal projection associated with δ, one finds that Likewise, one obtainsδ From the vertical projection, it is easy to see that It follows from (3.9),(3.10), and (3.11) that one finds (3.8). Therefore, the parameters ∆ and Q 1( ) , constrained by (i), (ii) and (iii), can be a set of building blocks describing all the Kähler parameters for the Tao spiral.
Since only ten parameters are independent in the Tao web, all the Kähler parameters can be written in terms of Q mf , Q F and Q B in Figure 8.
, the instanton factor q and the Coulomb moduli A = e −Ra introduced through the relation The first and the second relations of (3.12) are straightforward to understand if we carefully follow the sequence of Hanany-Witten transition from Figure 2 to Figure 6 and study which parameter in one diagram corresponds to which parameter in the other diagram. We will see that Q mf corresponds to the distance between one of the color D5 branes and one of the flavor D5-branes while Q F corresponds to the distance between the two color D5-branes in Figure 2. We need further explanation for the third relation. It is natural that the horizontal distance Q B is proportional to the instanton factor q since the distance between two NS5-branes corresponds to the gauge coupling constant. On top of that, the pre-factor in front of the right hand side of Q B appears as follows: When we compute the topological string partition function, the factor in the form (1 − q n Q) typically appears, where Q is a certain product of the Kähler parameters. When we rewrite this factor in the form of sinh, we obtain the prefactor √ q n Q. In order for the topological string amplitude to agree with the Nekrasov partition function, we need to absorb the collection of such factor into Q B and regard it as the instanton factor q.
For 0 ≤ N f ≤ 7, it is explicitly checked that such factor 2 is given by (the inverse of) We assumed that it is also the case for N f = 8. From Figure 8, one finds that ∆ are expressed in terms of Q mf , Q F , and Q B , This leads to the period (3.4) to be the instanton factor squared In a similar fashion, all Q 1( ) are expressed in terms of y i , A, and q, and so do all Kähler parameters. The result for the other parameters are summarized in Appendix C. Since it is known that the instanton factor of the 5d Sp(1) gauge theory with N f = 8 flavor is identified as the modulus of the compactified torus of E-string, we find that the period of our Tao diagram is also given by this modulus. This is consistent with the intuition that the spiral structure will corresponds to the KK mode of the E-string and thus, the cyclic structure of the spiral in our Tao diagram is a key to the uplift to 6d. In the next section, we compute the partition function and give a support for this claim quantitatively.
Here, we briefly comment the subtle difference between the parameters used in Nekrasov partition function for the 5d Sp(1) gauge theory with 8 flavors and those used in E-string partition function, which is clarified in [39]. Our parameters y f , q and A are those for the 5d Nekrasov partition function. If we would like to obtain the E-string partition function in anÊ 8 manifest form, we need to use other parameters y 8 and A defined as while the other parameters are identical. Since our formula of partition function will be given in a closed form and in a way that does not depend on the detail of the parametrization of the Kähler parameters, it should be possible to interpret either as the 5d Nekrasov 2 For instance, see [59] for N f = 4 flavors. partition function or as the E-string partition function depending on which parametrization we use. However, when we compare our formula with the know result [32,39], we use the parameters for the 5d Nekrasov partition function.

E-string partition function via topological vertex
In the previous section, we found new web description of the 6d E-string theory. In this section, we compute the E-string partition function by applying the topological vertex computation to our web. We will see that our partition function precisely agrees with partition function computed by completely different method [32,39]. For simplicity, we concentrate on the self-dual Ω-background ε 1 = −ε 2 in this paper.

Combinatorial expression of E-string partition function
In this section, we compute the topological string partition function of the Tao web Figure 7. This Tao web contains a number of 5-branes jumping another 5-branes as we commented in the previous section. In the left hand side of Figure 9, such a configuration is illustrated. This jumping is realized by degenerating the corresponding Kähler parameters as the middle diagram in the topological vertex formalism, and these jumping 5-branes are decoupled from the nontrivial trivalent vertex as the right hand of Figure   9. We therefore need to take only this nontrivial vertex into account in the topological vertex computation.
In the topological vertex computation of Nekrasov partition functions, we first decompose a toric web into basic building blocks [48][49][50]. Following this idea, we consider the basic spiral block Figure 10 of the Tao diagram. On the two internal edges located at the starting point of the spiral, two generic Young diagrams R 1 and R t 2 are assigned. This is because we need to glue six such building blocks for reconstructing the original Tao diagram in the topological vertex formalism. This gluing procedure is performed by summation over all such Young diagrams associated with the legs we want to glue together. All the other external legs without label are associated with the empty Young diagram ∅. The partition function for this sub-diagram is then given by the topological vertex as In the original Tao diagram, the exponentiated Kähler parameters Q i are stronglycorrelated with each other because of the 7-brane construction of the web, but we temporarily assign generic values to all these Kähler parameters just for simplicity. By substituting the definition of the topological vertex function C R 1 R 2 R 3 (q), we can write down this sub-diagram explicitly in terms of Shur and skew-Shur functions. The result Notice that the topological vertex function has the cyclic symmetry The generating function of the elliptic genera of the E-strings, which is the topological string partition function for the local 1 2 K3, therefore takes the following combinatorial form where R 7 := R 1 and Q 0( ) s are (4.5) The Kähler parameters I for the six sides of the central hexagon in Figure 7 are This is a simple and closed expression for the generating function of the E-string elliptic genera. By expanding it in terms of the Coulomb branch parameter A, we can obtain the n E-string elliptic genus as the coefficient of A n .
Here, we comment on the discrete symmetry that our partition function enjoys. It is straightforward to see that the expression (4.3) is invariant under the following transfor- where Q i(7) = Q i(1) and I 7 = I 1 . This transformation is generated by the "π/3 rotation" of the Tao diagram in Figure 8, which transforms one of the arms to the next one.

By using the explicit parametrization of the Kähler moduli parameters summarized in
Appendix C, the transformation in (4.7) is rewritten in terms of the mass parameters y i and the instanton factor q as combined with a certain SO(16) Weyl transformation 3 acting on y i (i = 1, · · · 8.) Or, if we use the parameters for the E-string (3.15), it is rewritten as This transformation can be interpreted as a part of expected E 9 symmetry. This is analogous to the "fiber-base duality map" studied in [37,59].

Comparing with elliptic genus approach
Now that we have the all-order generating function of the elliptic genera (4.3) that is the topological string partition function for the local 1 2 K3 in the self-dual Ω-background, we can derive various results. One of surprising applications is reproducing the known partial results on E-strings from our partition function. The expansion of the generating function in A Z E-string T ao (y, q, A) = Z extra (y, q) M (q) 1 + ∞ n=1 A n Z n-string (y, q) , (4.10) should lead to the known elliptic genera Z n-string (y i , q; q) of the E-string theory [12,39].
In this expression we introduce the extra factor Z extra as the A-independent coefficient which comes from the additional contribution that is not contained in the E-string theory.
The E-string partition function is therefore defined by removing the extra contribution from E-string Tao partition function (4.10). The normalized partition function Z E-string corresponds to the E-string theory Z E-string T ao (y, q, A) = Z extra (y, q) Z E-string (y, q, A). The E-string partition function can be expressed as the plethystic exponential form Likewise, the extra factor 4 can also be expressed as In what follows, we compute the E-string Tao partition function as the instanton expansion. At each order of the expansion we determine the extra factor and the E-string partition function (4.14)

Perturbative and instanton parts
Now we expand the partition function in terms of the instanton factor and compare to the 5d Nekrasov partition function or the E-string partition function studied in [32,39].
Although our Tao diagram spirally extends to infinity and thus (4.3) includes infinitely many Young diagram sums, it turns out that we can truncate the arm at finite place if 4 The extra factor is an analogue of what is called the contribution coming from effective classes in [60], the missing state/sector in [61], non full spin content in [36], U(1) factor in [35], and stringy contributions in [32].
we compute only the finite order of instanton expansion.

Perturbative
First, we start from computing the perturbative part. In the topological vertex computation of 5d Nekrasov partition functions, the perturbative contribution to a partition function is given by the sub-diagrams that correspond to turning off the instanton factor q [48][49][50]. Indeed, we see that by taking into account the parametrization in Appendix C, setting q = 0 makes most Young diagram summations in (4.3) become trivial (only empty Young diagram contribute). Figure 11 is two sub-pieces, which survives after setting q = 0.
In practice, it is more convenient to use the technique developed in [80] rather than directly using (4.3). In this computation only the Young diagram sums for the horizontal lines corresponding to D5-branes remain. We summed over these Young diagrams up to total ten boxes for each of the two sub pieces, which corresponds to the expansion of the partition function in terms of y 3 and y 7 up to degree ten.
The topological vertex method then leads to the following expression of the perturbative partition function Z pert+extra = PE q (1 − q) 2 1 − (y 1 + y 3 + y 4 + y 5 + y 7 + y 8 )A −1 −(y 1 + y 2 + y 2 −1 + y 3 + y 4 + y 5 + y 6 + y 6 −1 + y 7 + y 8 )A + 2A 2 +E 0 (y) + O(y 3 11 , y 7 11 ) (4.15) Taking into account that y 3 and y 7 to the power more than one do not appear up to 10, we expect that higher order correction O(y 3 11 , y 7 11 ) may also vanish. Here, E 0 (y) is the factor which does not depend on A and is given by E 0 (y) = q (1 − q) 2 y 1 y 2 −1 + y 1 y 3 + y 2 y 3 + y 5 y 6 −1 + y 5 y 7 + y 6 y 7 . We should regard such factor as the extra factor and should be removed by hand. If we suitably use the analytic continuation with a certain regularization:  It should be emphasized that when we define the instanton contribution, we should divide the E-string Tao partition function by the perturbative part (4.15) that analytic continuation is not performed (4.20)

One-instanton
When we compute the one instanton contribution, the Kähler parameters including high power of instanton factor q should be all truncated. When we consider the sub-diagram which includes instanton factor with power up to 1, we obtain the subdiagram Figure 12.

Two-instanton
The two-instanton contribution including the extra factor is where the extra factor E 2 is given by + y 1 y 3 + y 2 y 3 + y 5 y 6 + y 6 y 5 + 1 y 5 y 7 + 1 y 6 y 7 + y 5 y 7 + y 6 y 7 . (4.26) The two-instanton contribution takes the following form The coefficient functions c k are functions of q and A such that they are expressed in terms of the SU (2) characters of q, for instance, χ 2 (q) = q −1 + q, χ 3 (q) = q −2 + 1 + q 2 and χ 4 (q) = q −3 + q −1 + q + q 3 . It is worthy noting that the coefficient functions are so that where a i (q) is the coefficient of the i-th power in A which is expressed in terms of linear combinations of χ dim (q). The coefficient functions c k are given as follows: 32) (4.33) (4.34) (4.36)

Three-instanton
The three-instanton contribution including the extra factor is where the extra factor E 3 is given by y 1 y 4 y 5 y 6 y 2 y 3 y 7 y 8 + y 1 y 2 y 5 y 8 y 3 y 4 y 6 y 7 . (4.39) We do not have clear understanding why E 3 is very similar to E 1 but slightly different.

Four-instanton
The four-instanton contribution including the extra factor is where the extra factor E 4 turned out to be identical to E 2 : The main part F 4 is hard to write down explicitly. Here, we write the expansion of F 4 to order A 2 , + (5χ 4 (q) + 6χ 3 (q) + 11χ 2 (q) + 8)χ (2) + (4χ 3 (q) + 4χ 2 (q))χ (4) + (3χ 2 (q) − 2)χ (6) To compare to the partition function computed in [32,39], we expand our result in terms of the Coulomb modulus A (which corresponds to w in [39]) and obtain wherẽ Heref 1 andf 2 are precisely the ones that are used in [39]. We further checked fiveinstanton contribution by inputing y i with some specific values and found that F 5 is also consistent 6 . Hence, the result obtained from topological vertex amplitude based on the Tao diagram agrees with the result obtained from the instanton calculus as well as elliptic genus of E-strings in [32,39] 7 .

Discussion
In this article, we computed the partition function for 5d SU (2) gauge theory with eight flavors based on a Tao diagram which is a (p, q) 5-brane web that has spirally rotating arms with the period accounting for the instanton configuration of the theory.
As shown in [39], the Nekrasov function coincides with the elliptic genera of the E-string theory. Applying the topological vertex to the Tao web diagram, we reproduced the same partition function as [39] up to four instantons. Thus this gives a new way of studying 6d (1, 0) theories via Tao diagrams with IIB perspectives. We note that there are extra 6 As an example, we substituted y 1 = 1, y 2 = 2, y 3 = 1, y 4 = 3, y 5 = 1, y 6 = 2, y 7 = 1, y 8 = 3, q = 5.
We did not consider simpler massless case because zero appears in the numerator at the middle of the computation if we substitute y 1 = y 2 = · · · = y 8 = 1 from the beginning. 7 Taking into account our setup 1 + 2 = 0 and the notation convention introduced earlier, one easily finds that t and u in [39] correspond to 1 and q, respectively, in our convention. The higher dimensional irreducible representations used in [32,39] can be expressed in terms of the fundamental weights. For  factors in our computation which do not depend on the Coulomb modulus which we mode out by hand to reproduce the correct partition function.
Although we have considered a simple Tao diagram like Figure 7 in the main text, various Tao-type diagrams are possible which are of spiral webs with cyclic structure.
We propose to call a collection of Tao diagrams 'class T (ao)'.
A typical example of theories of class T would be the higher rank E-string theory. An easy way to find such a web would be exploring consistent black-white grid (or toric-like) diagrams with spiral belts in the triangulated diagram. For instance, see Figure 13 which may describe 5d Sp(N ) gauge theory with massless antisymmetric hypermultiplet. The diagram in Figure 13 has N normalizable deformations, namely N -dimensional Coulomb branch, as there are N internal points in the toric-like diagram. Moreover, the web is a collection of coinciding N E-string Tao webs that are essentially decoupled 8 to each other in this web realization. This E-string like diagram with multidimensional Coulomb branch like Figure 13 is a candidate for 5-brane web description of the rank-N E-string theory.
We can also consider another higher-rank generalization of our E-string Tao web.
Based on the fact that the rank-1 E-string theory is the UV fixed point of 5d SU (2) With these junctions, it describes 5d SU (N ) gauge theory with 2N + 4 flavors and makes a spiral web given on the right hand side of Figure 14 as 7-branes (with arrows) across the branch cuts created by other 7-branes. The way it makes spiral shape is the same as in Figure 6. It was classified that SU (N ) gauge theory exists up to 2N flavors, and the UV fixed point disappears if the number of hypermultiplets exceeds the limit N f ≤ 2N [74].
Our Tao-like generalization yielding to 2N + 4 flavors, on one hand, seems to be out of the classification. On the other hand, from the existence of consistent web diagram with spiral-direction, one may expect to find a 6d fixed point in UV 9 . Finding F-theoretic or another realization of this 6d SCFT is an interesting direction to pursue further.
The idea to find new Tao-diagram is also applicable to linear quiver and 5d T N theories [44,83]. Figure 15  is therefore the linear quiver theory SU (N ) M with additional two pairs of fundamental hypermultiplets of SU (N ) at each end of the quiver. Moving 7-branes outside, we find a consistent web diagram of class T , and thus there might be a 6d fixed point theory associated with this brane configuration. The diagram (b) is the T N geometry, which is C 3 /Z N × Z N , blown up at three points. This is an example of class T created from 5d T N theory.
One can create infinitely many webs of class T from conventional 5-brane webs and identify the global symmetry of such webs by using the collapsing 7-branes as was studied in [37,68]. Determining the global symmetry may lead to a clue for identifying the corresponding 6d theory associated with a given web. It would be interesting to see whether our construction is brane-web counterpart of the F-theoretic classification of 6d SCFTs studied recently in [23].
There are many other future directions. For instance, refinement 1 + 2 = 0 of our Tao partition function would be an important direction. In this paper, we present a generic 5-brane configuration but the explicit computation is based on the unrefined case. Generalization to 1 + 2 = 0 may involve further complication. Deriving the Seiberg-Witten curve of the E-string theory by using Tao web is also a fruitful direction.
In [35,75], a systematic way of computing 5d Seiberg-Witten curve starting from the toric-like diagram was developed. This method might be applicable to Tao web, leading to the Seiberg-Witten curve obtained in [14,16].

Acknowledgement
We would like to thank Giulio Bonelli,

A Conventions
A.1 7-branes Given a 7-brane labeled by a pair of two coprime integers [P, Q], which is the magnetic source of dilaton-axion scalar τ = χ+ie −φ , τ undergoes a monodromy around the 7-brane, which is SL(2, Z) invariant, and the monodromy matrix K [P,Q] is given by In our convention, [1, 0] 7-brane is the familiar D7-brane. When a (p, q) 5-or [p, q] 7brane is counterclockwise crossing a branch cut of a [P, Q] 7-brane, then 5-or 7-brane experiences the K [P,Q] monodromy due to the [P, Q] 7-brane and the charge (p, q) is altered to be another (p , q ) given by When clockwise crossing, the brane experiences the monodromy K −1 [P,Q] . For example, consider a 7-brane associated with the flavor branes, which is of [1,0] or [−1, 0] branch cut, then the monodromy matrix is given by When a [2,1]  For convenience, let us use a short hand notation for frequently appearing 7-branes: A.2 7-brane rearrangement In this paper we make full use of a special ordering of the 7-branes (2.5) associated with the N f = 8 theory. Let us derive this configuration. Consider affine 7-brane background E 9 associated with SU (2) N f = 8 theory, namely the local 1 2 K3 surfacê This leads to a (p, q) 5-brane web with N f = 8: E 9 configuration given in Figure 5. In addition, it follows from (A.9) and (A.6) that

B Topological vertex
The topological vertex formalism is a powerful method to compute the all-genus topological string partition functions for the toric Calabi-Yau threefolds. The basic building block is the vertex function C R 1 R 2 R 3 (q) = q κ R 1 2 S R 3 (q ρ ) P S R t 1 /P (q R 3 +ρ )S R 2 /P (q R t 3 +ρ ), (B.1) where S R and S R/Q are the Schur and skew Schur functions. Using the vertex function, a topological string partition function can be calculated as a Feynman-diagram-like quantity associated with the toric web diagram of interest [52,80].
The convention related to Young diagram used in the topological vertex formalism is as follows: |Y | := i Y i is the norms of Y , and we introduce S Y (q R+ρ ) represent the Shur function S Y (x 1 , x 2 , x 3 , · · · ) for the special arguments See [100] for more on the Schur functions.