M-theory curves from warped AdS 6 in Type IIB

: We establish a close relation between recently constructed AdS 6 solutions in Type IIB supergravity, which describe the near-horizon limit of ( p; q ) 5-brane junctions, and the curves wrapped by M5-branes in the M-theory realization of the 5-brane junctions. This provides a geometric interpretation of various objects appearing in the construction of the Type IIB solutions and a physical interpretation of the regularity conditions. Conversely, the Type IIB solutions provide explicit solutions to the equations de(cid:12)ning the M-theory curves associated with ( p; q ) 5-brane junctions.


Introduction and summary
One of the remarkable outcomes of string theory is strong evidence for the existence of interacting superconformal field theories (SCFTs) in five and six dimensions. These theories do not admit a conventional Lagrangian description, but they can be realized as lowenergy limits of string and M-theory, which allows one to study e.g. their moduli spaces and relevant deformations. In many cases, deformations can be found that do admit an effective Lagrangian description, allowing for a match to effective field theory analyses and providing further evidence for the constructions.
Five-dimensional SCFTs, which are the main concern in this work, can be realized in a variety of ways. First realizations were given in Type IIA on the worldvolume of D4-branes probing a stack of D8-branes and O8 − -planes [1][2][3]. More general classes of theories can be realized in Type IIB on the intersection point of (p, q) five-brane junctions [4][5][6], and in M-theory either on Calabi-Yau threefolds [7][8][9] or by considering the worldvolume theory of an M5-brane wrapping a holomorphic curve with one compact direction [5,[10][11][12].
The AdS/CFT correspondence provides a complementary approach, where the 5d SCFT is identified with a dual string theory on a background with an AdS 6 factor. When

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there exists a brane construction of the 5d SCFT, the dual AdS 6 solution is expected to describe the near-horizon geometry of the branes. This is the case for the gravity duals of the 5d SCFTs realized by the D4/D8 system, which have been studied extensively [2,3,[13][14][15][16][17][18]. More recently, gravity duals have also been constructed for 5d SCFTs realized by (p, q) five-brane junctions in Type IIB [19][20][21]. 1 Various aspects of the solutions and the dual SCFTs have since been studied holographically [25][26][27], and comparisons to field theory calculations supporting the proposed dualities have been presented in [28,29]. The solutions have also been extended to describe five-brane webs containing mutually local sevenbranes [30,31], and consistent truncations to 6d F (4) gauged supergravity were constructed in [32,33]. T-duals of the Type IIA solution in Type IIB [34,35] have been interpreted as Type IIB solutions with 7-branes where the brane sources are smeared [36,37].
The geometry of the Type IIB supergravity solutions of [19][20][21] is AdS 6 × S 2 warped over a Riemann surface Σ IIB , and the solutions are given in terms of a pair of locally holomorphic functions A ± on Σ IIB . For the solutions to be physically regular, Σ IIB is required to have a boundary and the functions A ± are required to satisfy certain constraints, to be reviewed below. Along the boundary of Σ IIB , the differentials ∂A ± have poles, at which the semi-infinite external five-branes of the associated 5-brane web emerge. The (p, q) charges of the emerging 5-brane are fixed by the residues of ∂A ± . The solutions are completely specified by the choice of Riemann surface Σ IIB , together with the number of poles and associated residues.
The prominent role of a Riemann surface and holomorphic functions in specifying the Type IIB supergravity solutions may seem reminiscent of the data used by Seiberg and Witten to specify 4d N = 2 theories [38,39]. Indeed, the same data can be used to specify 5d N = 1 theories engineered by (p, q) 5-brane webs in Type IIB -that is, such theories may be defined by a holomorphic curve Σ M5 , which contains one compact direction, together with a holomorphic one-form λ on that curve [5,[10][11][12]. The physical interpretation is that the 5d N = 1 theory is the worldvolume theory of an M5-brane wrapped on Σ M5 . This suggests that the Riemann surface and holomorphic data characterizing the Type IIB supergravity solutions may be related to the Riemann surface wrapped by the M5-brane in M-theory.
In this paper, we show that this expectation is indeed realized, and explicate the relationship between Σ IIB with the locally holomorphic functions A ± on the one hand, and Σ M5 with a holomorphic one-form λ on the other. More precisely, we will argue that the locally holomorphic functions A ± provide an embedding of the doubled Type IIB Riemann surfaceΣ IIB into the flat M-theory geometry, and that this embedded surface is the surface Σ M5 wrapped by the M5-brane. The Seiberg-Witten differential λ is identified with a locally holomorphic one-form A + ∂A − − A − ∂A + , which features prominently in the construction of the Type IIB solutions.
This identification between the data defining the Type IIB supergravity solutions and the data used to construct 5d SCFTs in M-theory is useful in a variety of ways. For the Type IIB solutions, it provides a geometric and physical understanding of certain aspects of the 1 Previous analyses of the BPS equations can be found in [22][23][24].

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construction that are not directly apparent in Type IIB. For example, the physical meaning of the regularity conditions is not immediately apparent in the original formulation. In the M-theory picture, on the other hand, they become the simple condition that the BPS masses associated with the punctures of Σ M5 vanish -i.e. they enforce conformality of the dual 5d theory. This gives a physical reason for the absence of Type IIB AdS 6 solutions with Σ IIB an annulus or more generally a Riemann surface with multiple boundary components or higher genus, to the extent that such solutions would allow for a similar interpretation in Mtheory. Such solutions would map to M-theory curves describing mass deformations of 5d SCFTs, and are thus not expected to have the full AdS 6 isometries. For the solutions with Σ IIB being a disc, the identification with the M-theory curve provides independent support for the identification of the solutions with the near-horizon limit of (p, q) 5-brane junctions.
For the M-theory side, the AdS 6 solutions provide explicit solutions to the polynomial equations defining the M-theory curves. We discuss this for a number of explicit classes, where the AdS 6 solutions provide simple generating functions for the polynomials defining the curves. This gives a more direct understanding of the pattern of "binomial edge coefficients," discussed in the separate context of brane tilings and their relations to dimer models in [40], and provides a simple way to compute certain multiplicities. We also discuss an interesting relation between the polynomial defining the T N theory curve and a seemingly unrelated quantity in the field of combinatorics and number theory -namely, the Wendt determinant [41,42]. We show that the polynomial defining the M-theory curve for the 5d T N theories [43], evaluated for unit arguments, coincides with the Wendt determinant. We leave further exploration to the future, where we certainly expect the connection between Type IIB solutions and M-theory curves to be mutually beneficial. For example, the M-theory perspective may help identify operators in the SCFTs dual to the Type IIB solutions [44,45]. It may also be useful for generalizing the construction of Type IIB AdS 6 solutions with 7-branes [30] to incorporate non-commuting monodromies.
The rest of this paper is organized as follows. In section 2, we review the relevant aspects of the Type IIB AdS 6 solutions as well as of the M-theory curves. In section 3, we expand upon the relation between the two pictures and formulate the concrete identification. In section 4, we verify the proposed identification for five families of supergravity solutions and M-theory curves.

Warped AdS 6 in Type IIB
The geometry of the Type IIB AdS 6 solutions constructed in [19] is a warped product

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of AdS 6 and S 2 over a Riemann surface Σ IIB . The general solution to the BPS equations is parametrized by two locally holomorphic functions A ± on Σ IIB . From these functions a locally holomorphic one-form dB on Σ IIB is defined, The SL(2, R) transformations of Type IIB supergravity are induced by a linear action of SU(1, 1) × C on the differentials (section 5.3 of [19]), with |u| 2 − |v| 2 = 1 and c ∈ C. The one-form dB is invariant under these transformations. The shifts parametrized by c leave the supergravity fields invariant, except for a gauge transformation of the two-form field. The supergravity fields are expressed in terms of A ± , B, and the composite functions [19] where w is a local coordinate on Σ. Their explicit expressions will not be needed here. Imposing global regularity conditions constrains the A ± and requires that Σ IIB have non-empty boundary. Physically regular solutions without monodromy were constructed in [20,21] for the case in which Σ IIB is a disc, or equivalently the upper half-plane. At the boundary of the Riemann surface, ∂Σ IIB , the spacetime S 2 collapses, closing off the ten-dimensional geometry smoothly. With a complex coordinate w on the upper half-plane, the A ± are given by The differentials ∂ w A ± have L ≥ 3 poles at w = r on the real line, with residues Z ± . The residues are constructed in terms of a distribution of auxiliary charges and sum to zero by construction. The locations of the poles are fixed by a set of regularity conditions These physically regular solutions admit a natural identification with (p, q) 5-brane junctions in Type IIB string theory, involving L 5-branes whose charges we denote by (p , q ) for = 1, . . . , L. At the poles r , the external (p, q) 5-branes of the associated 5-brane junction emerge, with the charges given in terms of the residues by where a D5-brane corresponds to charge (±1, 0) and an NS5-brane to (0, ±1) [28].

M5-branes on holomorphic curves
Consider a (p, q) 5-brane web in Type IIB in the (x 5 , x 6 ) plane. All 5-branes extend in the field theory directions x 0 , . . . , x 4 . Compactifying x 4 on a circle with radius R 4 and T-dualizing leads to Type IIA compactified on the T-dual circle with radiusR 4 = α /R 4 and g IIA = √ α g IIB /R 4 . This is equivalent to M-theory compactified on a torus with coordinates (x 4 , x 10 ) and R 10 = √ α g IIA = g IIBR4 . Decompactified Type IIB corresponds to the limit of vanishing volume,R 4 R 10 → 0, with fixed R 10 /R 4 .
In M-theory, the 5-brane web corresponds to a single M5-brane wrapping x 0 , . . . , x 3 and a complex curve Σ M5 ⊂ M 4 , where M 4 = R 2 × T 2 is the space spanned by (x 5 , x 6 , x 4 , x 10 ). Using complex coordinates s, t, defined by the curve is an algebraic variety defined by The polynomial P (s, t) can be constructed in an algorithmic way from the brane web, as will be reviewed shortly, and Σ M5 is directly related to the Seiberg-Witten curve of the 4d theory obtained by compactifying x 4 [39]. Supersymmetry requires Σ M5 to be a calibrated submanifold. The calibration is given by 10) and the primitive yields the Seiberg-Witten differential, e.g.
The Type IIB SL(2, Z) duality is realized in M-theory as the SL(2, Z) acting on the (x 4 , x 10 ) torus via The Seiberg-Witten differential in (2.11) is invariant under these SL(2, Z) transformations.

M-theory curves and grid diagrams
The polynomial P (s, t) defining Σ M5 is obtained from the grid diagram associated with a given 5-brane web [5]. The grid diagram is constructed by placing one vertex in each face of the web and connecting vertices in adjacent faces by a line that crosses the intermediate 5-brane perpendicularly. This gives a convex polygon ∆(P ) ⊂ Z 2 . 2 One may read off the polynomial P (s, t) from ∆(P ) as follows: for each point in ∆(P ) with coordinates (α i , β i ) ∈ Z 2 , one adds a monomial s α i t β i with an arbitrary coefficient, resulting in Explicit examples will be shown in section 4. 2 The grid diagram is also referred to as the Newton polygon.

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Now consider one of the asymptotic 5-branes with charges (p, q), in all-ingoing convention. Supersymmetry requires the slope of this brane in the (x 5 , x 6 )-plane to be This is the condition that there be zero force at the vertices of the web. In M-theory, holomorphicity demands that this constraint be completed to an analogous constraint on s and t. The imaginary part of the holomorphic constraint is Interpreting τ as the modular parameter of the M-theory torus, this fixes the M5-brane to be oriented along the (p, q) cycle of T 2 . Without loss of generality, we set the asymptotic value of the axio-dilaton scalar to τ ∞ = i. 3 The embedding of the (p, q) 5-brane into the (x 5 , x 6 )-plane is then given by where the intercept, m, corresponds to a mass parameter and T s is the string tension [5].
The projection of the M5-brane curve onto the (x 5 , x 6 )-plane should approach this embedding asymptotically. In the s, t coordinates, (2.16) becomes exp(m/R 4 T s )|s| −q |t| p = 1, while the asymptotic region corresponds to −px 5 , −qx 6 → ∞, or |s| −p , |t| −q → ∞. In summary, the M-theory curve should behave as with |A| = exp(m/R 4 T s ). Requiring that P (s, t) = 0 exhibits this behavior puts constraints on the coefficients c i . For a group of N external 5-branes with charges (p, q), the constraint is In the conformal limit, these 5-branes are coincident, and the M5-brane curve is expected to approach this stack of coincident branes. The boundary condition then becomes where α is a phase, i.e. |α| = 1, which encodes the asymptotic behavior of the M-theory curve in the (x 4 , x 10 ) directions.

M-theory curves from Type IIB AdS 6
In this section we discuss the connection between AdS 6 solutions in Type IIB and the holomorphic curves wrapped by M5-branes in M-theory. Our main result is a relation between the Riemann surface Σ IIB appearing in the supergravity solution and the M-theory curve Σ M5 . Detailed evidence for the proposed relation will be presented in section 4.

A ± and algebraic equations
Before discussing the identification in detail, we rewrite the locally holomorphic functions A ± in (2.5) in a more suggestive way. Using the relation between residues and 5-brane charges (2.7), as well as the conjugation relations spelled out below (2.5), we have where the combinations s and t are defined as and we have introduced a constant a defined by A 0 + ≡ 3 4 α a. With these definitions, the locally holomorphic one-form dB defined in (2.2) takes the form while κ 2 and G of (2.4) are given by The first claim, which we will verify for a number of explicit examples in section 4, is that the Riemann surface Σ IIB with the locally holomorphic functions A ± provides a solution to equation (2.9) defining the associated M-theory curve, via the identification An immediate consequence of this identification is that the holomorphic one-form dB in (3.3) is directly related to the Seiberg-Witten differential λ in (2.11), via

Global structure
We have claimed that the functions A ± on the Riemann surface Σ IIB provide a solution to the equation defining the M-theory curve, Σ M5 . We now address this identification at the global level. The relation (3.1) with (3.6) and (2.8) in fact suggests a more direct identification of A ± with the coordinates in M-theory as follows,

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That is, the functions A ± provide an embedding of Σ IIB into the four-dimensional space M 4 = R 2 × T 2 spanned by the M-theory coordinates (x 5 , x 6 , x 4 , x 10 ). An apparent challenge to a direct identification of Σ IIB and Σ M5 is the fact that, being a disc or the upper half-plane, Σ IIB has a boundary, while Σ M5 does not. We note that with Θ the Heaviside function. Consequently, for integer charges p k , q k , Thus, the segments of the boundary of Σ IIB in between poles are mapped to curves in planes of constant x 4 and x 10 . The embedding of Σ IIB into M 4 is illustrated in figures 2 and 5 for the T 1 and + 1,1 solutions, respectively. A natural interpretation for the boundary in Σ IIB can be obtained as follows. We recall that the regularity conditions in Type IIB supergravity have two branches of solutions (section 5.4 of [19]), These two branches are mapped into one another by complex conjugation. The regular solutions discussed above with Σ IIB being the upper half-plane realize the branch R + . For any such regular solution in the upper half-plane, the extension of the A ± into the lower half-plane provides an equivalent regular solution, realizing the second branch of regularity conditions R − . The two solutions are separated at the boundary of Σ IIB , where κ 2 = G = 0.
Since the 10d spacetime in Type IIB is closed off smoothly at ∂Σ IIB by the collapsing S 2 , the solutions in the upper and lower half-planes are two realizations of equivalent Type IIB solutions. But for the identification of Σ IIB with the M-theory curve, it is natural to consider the full, doubled, Riemann surfaceΣ IIB . 4 The precise relation we propose is then (3.12) That is, the embedding of the doubled Type IIB Riemann surfaceΣ IIB into the fourdimensional part of the M-theory geometry, with the embedding functions given by A ± via (3.8), is the M-theory curve Σ M5 . The doubled Type IIB Riemann surfaceΣ IIB is a closed surface with punctures at the poles r . Suppose we encircle one of the poles r . Then ln(w − r ) → ln(w − r ) + 2πi, and consequently (3.13)

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With the identifications (2.7) and (3.8), this means that x 10 → x 10 + 2πR 10 p . (3.14) This is indeed the desired behavior: the (p, q) 5-brane charges become the winding numbers of the M5-brane, with the winding on the M-theory circle x 10 encoding the D5 charge and the winding on x 4 encoding the NS5 charge. This furthermore implies that the curve defined by the embedding (3.8) is smooth across the boundary of Σ IIB , despite the fact that the A ± are not single-valued in the doubled Riemann surfaceΣ IIB (noting that the differentials ∂ w A ± are single-valued onΣ IIB ). That is, since mapping from the upper half-plane Σ IIB into the lower half-plane ofΣ IIB induces the following map on the M-theory curve, w →w : x 10 → −x 10 mod 2πR 10 . (3.16) Then due to (3.10), the boundary of Σ IIB is mapped to fixed points of this action on the torus.

Type IIB regularity conditions
The asymptotic behavior of the M5-brane curve is constrained by the conditions (2.17).
We will now discuss how this behavior is realized by the identification (3.6), and obtain a geometric perspective on the Type IIB regularity conditions (2.6). Consider the limit in which With the explicit expressions in (3.2), we find that in this limit corresponding to the asymptotic region where 5-branes with charges (p k , q k ) are, as expected. Furthermore, in this limit the explicit expressions in (3.2) give which is finite, as required by (2.17). As seen from (2.17), the mass parameter associated with the external 5-branes is given by (3.20)

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Using the identification of the residues with the 5-brane charges (2.7), as well as the definition of the constant a below (3.2), the Type IIB regularity conditions in (2.6) are precisely the statement that m 2 k = 0 for all k. The Type IIB regularity conditions are therefore interpreted from the M-theory perspective as the requirement that the 5-branes within each group of like-charged external 5-branes are coincident, with the associated mass parameter vanishing, such that the field theory is conformal. 5 The identification of dB with the Seiberg-Witten differential allows for an additional physical interpretation of the regularity conditions (2.6) from the 4d perspective. Of the L conditions in (2.6) only L − 1 are independent, due to the fact that the Z ± sum to zero by construction, implementing charge conservation at the 5-brane junction. These conditions may be formulated more concisely in the upper half-plane as where C k denotes a curve connecting two points on the boundary ∂Σ IIB to either side of the pole r k . In this formulation, charge conservation amounts to the fact that the sum of the cycles C k is contractible.
In the doubled surfaceΣ IIB , the addition of the complex conjugate on the left hand side in (3.21) can be implemented by closing the contour C k in the lower half-plane, such that the pole is encircled completely. Denoting byĈ k a closed contour around the pole p k inΣ IIB , the regularity conditions become With the identification ofΣ IIB as the Seiberg-Witten curve of the 5d theory compactified on x 4 , and of dB as the Seiberg-Witten differential via (3.7), the regularity conditions (3.22) again become the statement that the BPS masses associated with the punctures vanish.

Σ IIB of general topology
The identification ofΣ IIB with the M-theory curve Σ M5 gives an interesting perspective on potential AdS 6 solutions in Type IIB where Σ IIB is a Riemann surface with multiple boundary components or higher genus. From the Type IIB perspective, it is not a priori clear whether such solutions should exist. The construction used in [21] of imposing the global regularity conditions on the general local solution to the BPS equations and reducing them to a finite number of constraints in principle works for Riemann surfaces of arbitrary topology. This was spelled out explicitly in section 6 of [21]. But solutions to these constraints were only found for the upper half-plane. For the annulus, an explicit search was conducted, but no solutions were found. 5 A different way to realize a conformal theory can be to send certain combinations of the m k to infinity.
In the 5-brane web this corresponds to moving some of the 5-branes off to infinity, leaving behind a 5-brane junction with a different set of external branes, and an associated supergravity solution with a different set of poles, residues and regularity conditions. From the perspective of the associated M-theory curve, assuming that the identification ofΣ IIB with Σ M5 extends to Σ IIB of more general topology, Σ IIB with multiple boundaries or higher genus would correspond to M-theory curves Σ M5 of higher genus. Such curves are associated to 5-brane webs with open faces, i.e. mass deformations. These webs describe renormalization group flows, as opposed to renormalization group fixed points, and are therefore not expected to have an AdS 6 dual. This gives a physical interpretation for the absence of annulus solutions in Type IIB, and suggests more generally the absence of AdS 6 solutions for Riemann surfaces with multiple boundary components or higher genus.

Case studies
In this section, we verify the relation between the Type IIB AdS 6 solutions and M-theory curves discussed in section 3 for a number of explicit examples.

T N solutions
As a first example we discuss the 5d T N theories [43]. These are realized by triple junctions of N D5, N NS5, and N (1, 1) 5-branes (figure 1a). The polynomial P (s, t), obtained from the grid diagram (figure 1b), is given by (4.1) The boundary conditions, in the conformal limit, are

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with |α i | = 1. This fixes the coefficients c k,N −k , c k,0 and c 0,k for k = 0, . . . , N to be binomial. The remaining coefficients encode Coulomb branch parameters. Without loss of generality, we fix c 0,0 = 1. Then for N = 1, one finds Consistency of the boundary conditions requires α 1 = α 2 α 3 . The remaining freedom in α 2 , α 3 corresponds to translations in the compact directions. The Type IIB supergravity solutions corresponding to triple junctions of D5, NS5, and (1, 1) 5-branes were discussed in detail in [28,29], including comparisons of holographic results to field theory computations. The functions A ± are given by (section 4.3 of [28]) This realizes the T N charges in all-ingoing convention. Via (3.1) this yields For N = 1, these solve (4.3) with α 2 = −α 3 = 1 via s = s and t = t. More generally, s and t satisfy Solving this equation for either s in terms of t or t in terms of s yields N branches of solutions. These are realized in (4.5) by the fact that solving for w in terms of s or t yields N branches of solutions. Evaluating the expression for the remaining one of s or t for these w gives N branches for s in terms of t and t in terms of s. Eq. (4.6) can be converted to a polynomial equationP T N (s, t) = 0 with the same roots. The result is This is indeed a polynomial in s and t for each N , where each term has combined degree at most N , as in (4.1); all fractional powers of s and t drop out. This shows that the subspace in M 4 defined by (3.8) is indeed an algebraic variety. That the polynomial satisfies the boundary conditions spelled out in (4.2) for general N can be verified directly by inspecting P T N in (4.6). It also follows from the general discussion in section 3.3, which showed that s and t extracted from regular supergravity solutions automatically realize the appropriate asymptotic behavior. Some explicit forms of the coefficientsc ij ofP T N (s, t) = ijc ij s i t j for small N arec The coefficients which are not fixed by the boundary conditions (4.2) are tuned to specific values, corresponding to the origin of the Coulomb branch. This is the expected result for the curve extracted from a Type IIB supergravity solution with an AdS 6 factor, describing the conformally invariant vacuum state. We now discuss the mapping of the Type IIB Riemann surface Σ IIB to the M-theory curve. With the identification of s, t given in (4.5) with s, t and their relation (2.8) to the M-theory coordinates (x 5 , x 6 , x 4 , x 10 ) on M 4 = R 2 × T 2 , we obtain the embedding ofΣ IIB into M 4 as The poles at r 1 , r 2 , r 3 correspond to the NS5, D5, and (1, 1) 5-branes, respectively. The geometry of the curve for N = 1 is illustrated in figure 2. The curve for generic N is obtained by a simple rescaling. We note that eq. (4.7) is precisely the formula quoted in (3.13) of [40], which made use of earlier results in [46]. The context of that result was a proposed correspondence between brane tilings and dimer models. Though we have not been considering brane tilings in the current work, the curves wrapped by the NS5-branes in the brane tiling construction are of the same form as the curves being wrapped by the M5-brane here. In the current context, the formula of [40] appears more naturally in the form (4.1), coming directly from the warped AdS 6 solutions. The pattern of binomial coefficients on the edges (cf. (4.8)), which JHEP11(2018)175 was traced back in [40] to the expression (4.7), implements the boundary conditions on the curve as discussed in section 3.3.
We also note an interesting relation between the polynomial defining the T N theory curve and a seemingly unrelated quantity in the field of combinatorics and number theory. Namely, this is the Wendt determinant [41,42], given by where ζ m is a primitive m-th root of unity. To make the relation to the polynomialP T N (s, t) transparent, we note the alternative expressioñ This expression shows that the Wendt determinant W n is obtained by evaluating the polynomial for s = t = 1, The first terms in the sequence are given by The relation of the Wendt determinant to circulant matrices with all binomial coefficients may provide an interesting perspective on the conformal invariance of the curve. We leave further investigation of this relation to the future. For each theory obtained by wrapping an M5-brane on a holomorphic curve, there is an alternative interpretation as M-theory on a (singular) Calabi-Yau threefold. In the particular case of rank 1 SCFTs with toric realizations (i.e. theories with grid diagrams with a single internal dot), this threefold is a complex cone over F 0 or a del Pezzo surface dP n , n ≤ 3 [7][8][9]. This may be seen by interpreting the brane web as the toric skeleton defining the geometry [47]. In the case of the T 1 theory, the corresponding Calabi-Yau threefold is simply C 3 . The higher rank T N theories correspond to orbifolds of C 3 , i.e. C 3 /(Z N × Z N ) with the orbifold action given by [40]   the large-N limits of the two theories. The curves are likewise closely related, as we will discuss now. We start with the supergravity picture in this case, and compare to the construction of the curve via the grid diagram associated with the brane web at the end. The functions A ± are given by from which we extract, via (3.1), (4.16) They satisfy This can be understood from the result for the T N solution as follows. We first note that s, t for the Y N solution are related to s, t for the T N solution by This may be interpreted as the Y N solution being obtained from the T N solution by an SL(2, R) transformation with a = 0, c = −1/b = −1/d = √ 2, acting as in (2.12), combined with a charge rescaling N → √ 2N . As a consequence of (4.18), we have We now compare to the polynomial equation obtained from the grid diagram of the Y N junctions. A sample grid diagram is shown in figure 3b, and the resulting polynomial takes the form (4.20)

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The boundary conditions in the conformal limit demand that the coefficients on the edges be binomial. More precisely, the requirements are Consistency of the boundary conditions requires α 1 α 2 = α 2 3 . Eq. (4.17), which is satisfied by s and t obtained from the supergravity solution, may again be converted to a polynomial equation, 0 =P Y N (s, t), as follows. Eq. (4.17) for N = 1 is equivalent to This is again a polynomial in s and t, and takes precisely the form in (4.20). Moreover, the edge coefficients are binomial, reflecting the fact that the curve obtained from the supergravity solution automatically satisfies the correct boundary conditions. Some explicit forms for small N arẽ As before, the coefficients corresponding to Coulomb branch deformations are tuned to particular values for the conformally invariant vacuum state. The supergravity solution again provides an explicit solution to the equation defining the M-theory curve, with A ± providing the embedding as discussed in section 3. The Y 1 theory may also be obtained by considering M-theory on C × C 2 /Z 2 . The Y N theories are obtained via orbifolds thereof.

+ N,M solutions
The next example is a quartic junction of N D5-branes and M NS5-branes, as shown in figure 4a. This configuration has been discussed already in [5]. An example for the associated grid diagram is shown in figure 4b. The polynomial P (s, t) defining the Mtheory curve is given by  The boundary conditions in the conformal limit are, with |α i | = 1. Consistency of the boundary conditions requires α 1 α 3 = α 2 α 4 .
We again show that the functions A ± of the corresponding supergravity solution provide an explicit parametrization of the curve. They are given by (section 4.2 of [28])  (4.31) The binomial form of the edge coefficients again implies that the correct boundary conditions are satisfied. The curve obtained from the supergravity solution is shown in figure 5. The + 1,1 theory may also be obtained by considering M-theory on the conifold C. The + N,M theories are obtained by considering M-theory on C/(Z N × Z M ), with the orbifold action given in (4.14), but with ν M = 1.

X N,M solutions
The X N,M theories are defined by quartic junctions of N (1, −1) 5-branes and M (1,1) 5-branes, as in figure 6a. They are closely related to the + N,M theories, in a very similar way to how the Y N theories are related to the T N theories.
The quantities s and t extracted from the supergravity solution (as discussed in section 4.2.2 of [28]) via (3.1) are These are related to the complex coordinates of the + N,M theory by In Type IIB, the two configurations are related by an SL(2, R) rotation, together with a rescaling of charges. However, the two configurations are not related by SL(2, Z) in the full string theory description, as can be seen by comparing (4.33) to (2.12). Using These are generally polynomials of precisely the form implied by the grid diagram (figure 6b), with binomial edge coefficients implementing the boundary conditions. The X 1,1 theory may be described as M-theory on the cone over F 0 = P 1 × P 1 .

+ N solutions
As a final example we consider the + N theories, which are realized by sextic junctions of NS5, D5, and (1,1) 5-branes as shown in figure 7a. The polynomial P (s, t) obtained from the grid diagram takes the form P (s, t) = 0≤i , j≤2N N ≤i+j≤3N c i,j s i t j . with |α i | = 1. For consistency, we require that α 1 α 2 α 3 = α 4 α 5 α 6 . The supergravity solution has been discussed in section 4.5 of [28]. Via (3.1), s and t are found to be s = 1 where r 1 = −r 2 = −2 + √ 3 , r 4 = −r 5 = 2 + √ 3 , r 3 = −r 6 = 1 . The + 1 theory may be obtained from M-theory on the cone over dP 3 . The + N theory is obtained by a Z N × Z N orbifold of this geometry.