D ec 2 02 0 Revisiting the Multi-Monopole Point of SU ( N ) N = 2 Gauge Theory in Four Dimensions

Motivated by applications to soft supersymmetry breaking, we revisit the expansion of the Seiberg-Witten solution around the multi-monopole point on the Coulomb branch of pure SU(N) N = 2 gauge theory in four dimensions. At this point N − 1 mutually local magnetic monopoles become massless simultaneously, and in a suitable duality frame the gauge couplings logarithmically run to zero. We explicitly calculate the leading threshold corrections to this logarithmic running from the Seiberg-Witten solution by adapting a method previously introduced by D’Hoker and Phong. We compare our computation to existing results in the literature; this includes results specific to SU(2) and SU(3) gauge theories, the large-N results of Douglas and Shenker, as well as results obtained by appealing to integrable systems or topological strings. We find broad agreement, while also clarifying some lingering inconsistencies. Finally, we explicitly extend the results of Douglas and Shenker to finite N , finding exact agreement with our first calculation.

The coefficient of the logarithm is fixed by the unit charge of the massless monopole, while its branch cut ensures the correct SL(2, Z) monodromy around the monopole point. The same phenomenon occurs at the dyon point, except that the simple IR free description occurs in a different duality frame.
The monopole and dyon points of the SU(2) N = 2 theory play a crucial role in many applications of Seiberg-Witten theory. For instance, it was shown in [1] that they describe the two confining vacua of the pure SU(2) N = 1 gauge theory obtained by adding the N = 2 → N = 1 breaking superpotential d 2 θ u ∼ d 2 θ tr φ 2 via Higgsing in the IR free U(1) D gauge theory described above. In applications of N = 2 gauge theory to four-manifold topology, the monopole and dyon points give rise to the Seiberg-Witten equations [3].
In this paper we are interested in the generalization of the SU(2) monopole and dyon points to pure SU(N) N = 2 gauge theories. A systematic study of these points was initiated in [4], building on the SU(N) generalization of the Seiberg-Witten solution found in [5][6][7][8]. The Coulomb branch is now N − 1 complex dimensional and described by the gauge-invariant coordinates u n ∼ tr φ n (n = 2, . . . , N), collectively denoted by u. (As before, φ is the complex SU(N) adjoint and Lorentz scalar in the N = 2 vector multiplet.) The lowenergy effective theory at generic points is a U(1) N −1 gauge theory, and there are N − 1 dual pairs of special coordinates a k (u), a Dk (u) (k = 1, . . . , N − 1). They are the A-and B-cycle periods of a suitable meromorphic differential λ on the u-dependent Seiberg-Witten curve Σ, which now has genus N − 1. As before, the special coordinates determine the matrix τ kℓ of complexified U(1) N −1 gauge couplings via τ kℓ = ∂a Dk ∂a ℓ , and the masses of BPS states with charges (q ek , q mℓ ) ∈ Z 2(N −1) via M BPS ∼ N −1 k=1 (q ek a k + q mk a Dk ) . Changing the choice of canonical A-and B-cycles on Σ acts on the special coordinates and the matrix of couplings via an Sp(2N − 2, Z) electric-magnetic duality transformation.
As was explained in [4], the Coulomb branch of the SU(N) gauge theory has many interesting singular points, at which the Seiberg-Witten curve Σ degenerates in various ways.
The BPS dyons that become massless at such points are typically mutually non-local, i.e. they have non-vanishing Dirac pairing N −1 k=1 q ek q ′ mk − q ′ ek q mk . In particular, this means that there is no electric-magnetic duality frame in which all of them carry electric charges. Such mutually non-local massless dyons describe interacting superconformal field theories [9,10].
By contrast, the singular points that generalize the monopole and dyon points of the SU (2) theory arise when N − 1 (i.e. the maximal number of) mutually local BPS dyons simultaneously become massless [4]. This happens at precisely N distinct points on the Coulomb branch, which are related by a spontaneously broken Z 4N R-symmetry, which rotates the Coulomb branch coordinates u n by N-th roots of unity. We will collectively refer to these N points on the Coulomb branch as the multi-dyon points of the SU(N) theory. As before, it is sufficient to focus on one such point, and we choose the multi-monopole point. At the multi-monopole point the N − 1 mutually local massless dyons are electrically neutral and carry unit magnetic charge in precisely one U(1) factor of the low-energy gauge group.
As in the SU(2) theory, it is useful to pass to an S-dual magnetic description, which is  4)), as well as on the fact that the a k special coordinates at the multi-monopole point are all non-zero. 1 These were first computed in [4], a k (a Dℓ = 0) ∼ NΛ sin kπ N , (1.6) and they are indeed all non-vanishing. We will recover this result below, including a schemedependent prefactor that we omit here. 2 We now turn to applications of Seiberg-Witten theory that are sensitive to the regular terms in (1.4).

Motivation and Summary of Results
The computations described in this paper were motivated by applications of Seiberg-Witten theory that require more detailed information about the multi-monopole point than the leading logarithmic running of the couplings in (1.4) or the value of the a k -periods in (1.6).
(Two such applications are mentioned below.) Our primary interest will be the leading regular terms in (1.4), which we parametrize as follows, 3 Here t kℓ is a real, symmetric matrix that accounts for the leading threshold corrections due to massive particles that have been integrated out in the low-energy effective description on the Coulomb branch. As such we will often refer to t kℓ as the threshold matrix. Clearly the imaginary part of (1.7), which describes the matrix of gauge coupling constants at low energies, is positive definite as long as the a Dk are sufficiently close to the multi-monopole point. Note that the off-diagonal elements of the matrix t kℓ can be accessed by taking a Dk → 0 in (1.7), 1 To see this, recall from [2,4] that the monopole vev responsible for Higgsing the k-th U (1) D factor of the gauge group is set by ∂u 2 ∂a Dk at the multi-monopole point a Dk = 0. To evaluate this, it is convenient to use the renormalization group equation [11][12][13][14][15][16]. (See [17] for a simple derivation that involves promoting Λ to an N = 2 chiral background superfield.) Here F D is the dual prepotential, so that ∂F D ∂a Dk = a k . Using (1.3), we then find that It follows from (1.4) that the first term vanishes at the multi-monopole point, leaving only the term ∼ a k . 2 Rescaling Λ by a constant amounts to a change of renormalization scheme. 3 Since τ Dkℓ has non-trivial Sp(2N − 2, Z) monodromy around the multi-monopole point, we must pick a branch of the logarithm to render the matrix t kℓ in (1.7) well defined. As explained below, we will mostly work with configurations a Dk that are positive imaginary, so that −ia Dk > 0. We can then choose the principal branch of the logarithm, so that log(−ia Dk ) is real.
since the corresponding τ Dkℓ has a finite limit. 4 By contrast, the diagonal matrix elements t kk are finite threshold corrections to the divergent logarithms in τ Dkk . Thus computing them is more challenging; any such computation must regularize the logarithms by perturbing away from the multi-monopole point.
As was emphasized in [4], the structure of the threshold matrix t kℓ encodes important information about the SU(N) N = 2 gauge theory near the multi-monopole point -in particular its massive spectrum there. Upon softly breaking N = 2 → N = 1 (as reviewed below (1.4)) the threshold matrix is needed to determine the spectrum of light particles, as well as the confining string tensions. Roughly speaking, this is due to the fact that t kℓ is the matrix of gauge-kinetic terms in the low-energy U(1) N −1 D gauge theory that couples to the N − 1 massless monopole hypermultiplets at the multi-monopole point.
Our primary interest in the threshold matrix t kℓ comes from the recent observation [19] that the dynamics of non-supersymmetric adjoint QCD with gauge group G and two adjoint quarks can be analyzed by adding a certain soft supersymmetry-breaking mass term for the adjoint scalars to the pure N = 2 supersymmetric gauge theory with the same gauge group G. 5 The case G = SU(2) was analyzed in [19], where it was found that the expected confining and chiral-symmetry breaking phase of adjoint QCD emerged from the dynamics of the monopole and dyon points in the presence of the soft supersymmetry-breaking scalar mass. In upcoming work [18] we extend this to G = SU(N) for all N, where the soft supersymmetry-breaking mass deformation leads to a rich structure of phases and phase transitions that can be analyzed by focusing on the multi-dyon points. This analysis crucially depends on the detailed properties of the threshold matrix t kℓ in (1.7).
A procedure for computing t kℓ was outlined in [4], where the authors considered a particular one-parameter family a Dk (s) that approaches the multi-monopole point as s → 0.
However, this procedure was ultimately only carried out for the elements of t kℓ that dominate in the 't Hooft large-N limit of the theory emphasized in [4]. Exact results for N = 2 and N = 3 were obtained in [8]. Subsequently, the authors of [20] developed a systematic method to compute higher-order corrections to τ Dkℓ for all N, starting with the O(a D ) terms in (1.7), but they did not compute t kℓ . A formula for the off-diagonal elements of t kℓ was conjectured in [21,22], and subsequently confirmed in [23] (see also [24]), using the relationship of Seiberg-Witten theory to integrable hierarchies. More recently, the authors of [25] presented a computation of t kℓ based on (partially conjectural) topological string and matrix model 4 The physical importance of these off-diagonal terms was first stressed in [4]. They also play an important role in [18]. 5 In this embedding, the adjoint quarks are simply the two gauginos of the N = 2 gauge theory. In the supersymmetry-breaking analysis [18] we rely on the quantitative details of the threshold matrix t kℓ -not just its qualitative or large-N features. For this reason we present a detailed and direct calculation of t kℓ using standard Seiberg-Witten technology. As explained below (1.7), a full calculation of t kℓ requires regularizing the logarithmic singularities in (1.7) by perturbing away from the multi-monopole point. Here we will follow and extend the regularization method of [20], which we review in section 2. 6 Our main result (derived in section 3) is a computation of the a k periods near the multi- We find that the a k periods at the multi-monopole point are given by 8 while our result for the elements of the threshold matrix t kℓ is given by (1.10) Note that in addition to the symmetry t kℓ = t ℓk that is necessarily present (see (1.7)), the threshold matrix also satisfies This follows from the charge conjugation symmetry of the underlying SU(N) gauge theory, which is preserved at the multi-monopole point. (This will play an important role in [18].) It is tempting to speculate that the special form of t kℓ in (1.10) -a logarithm of sine functions -can be explained by appealing to the spectrum of heavy BPS states at the multi-monopole 6 See section 1.3 and appendix A for more details on the regularization method used in [4].
point, whose masses are determined by the a k ∼ sin πk N at that point (see (1.9)). 9 However, we do not know of such an explanation. 10

Comparison with the Literature
In this subsection we compare our results (1.8), (1.9), and (1.10) to the existing literature in more detail. Along the way, we clarify some lingering inconsistencies.
Using Picard-Fuchs equations, the authors of [8] found an expansion of the dual prepotential F D (a D ) around the multi-monopole point for SU (2) and SU(3) gauge theories.
The prepotential for SU(2) is given above equation (2.11) in [8]. From it we can com- where we use Λ to denote the strong coupling scale in the conventions of [8]. Comparing the constant term a(a D = 0) in (1.13) with (1.9), we find agreement if Λ = 2Λ. By comparing (1.13) with (1.8), we then read off t 11 = log 32, in agreement with our result (1.10) for N = 2.
In the SU(3) case the prepotential F D (a D ) around the multi-monopole point is given in equations (6.13) and (6.14) of [8]. From it we can compute and an analogous formula for a 2 , which can be obtained by exchanging a D1 ↔ a D2 in (1.14). 11 9 An interpretation of t kℓ in terms of the massive BPS spectrum near the multi-monopole point must contend with the fact that this point lies on a wall of marginal stability across which the BPS spectrum jumps [1,4], while t kℓ is wall-crossing invariant. This suggests an approach along the lines of [26], where a similar puzzle was encountered and resolved. 10 As another possible hint, we record the following interesting, exact formula (inspired by [4] and appendix A) for the off-diagonal elements of t kℓ in (1.10), To show this, we write the sum over p as cos px p = − log 2 sin x 2 , valid for x ∈ R−2πZ. In turn, the latter formula follows from writing cos px in terms of exponentials and using ∞ p=1 z p p = − log(1 − z), with z = e ±ix . 11 Note that the prepotential in equations (6.13) and (6.14) of [8] is invariant under the charge-conjugation Again we use Λ to denote the strong coupling scale in the conventions of [8]. We proceed as above: by comparing the constant term a(a D = 0) in (1.14) with (1.9), we find agreement if Λ = 2 −2/3 Λ. Substituting back into (1.14) and comparing with (1.8), we can then read off t 11 = log 2 + 5 2 log 3 and t 12 = log 4, in agreement with our result (1.10) for N = 3. We now compare our results to those of [4], which apply to SU(N) gauge theories in the large-N limit. In order to keep the present discussion brief, we defer a more detailed review of [4] to appendix A, which also contains some new results (see below). As was already mentioned above, the authors of [4] considered a one-parameter scaling trajectory a Dk (s) (with real parameter s) that approaches the multi-monopole point as s → 0 (see appendix A), Substituting this into (1.7) and using our answer for the threshold matrix t kℓ in (1.10), we find that We will now compare this answer to the calculations in [4]. Although the approach outlined there in principle allows one to calculate all s-independent terms in (1.16), the authors of [4] only explicitly evaluated those terms that grow without bound in the N → ∞ limit. As reviewed in appendix A, it follows from the results of [4] that the elements of τ Dkℓ (s) that have such growing large-N contributions are 12 (1.17) Here the O(1) terms in τ Dkℓ (s) are constant as s → 0 and remain bounded at large N. This precisely agrees with (1.16) for those k, ℓ indicated in (1.17). 13 It was argued in [4] that the ∼ log N 2 threshold corrections in (1.17) are due to light particles of mass ∼ Λ N 2 , which impose a cutoff on the low-energy effective theory that vanishes in the large-N limit. In appendix A we show how to explicitly extend the computations in [4] to finite N, and we symmetry a D1 ↔ a D2 . 12 As explained in appendix A, (1.17) also applies when k = ℓ if we omit the factor (k −ℓ) 2 in the logarithm. 13 Some formulas in [4] have subsequently been extrapolated beyond the regime in (1.17), where they no longer apply. For instance, the authors of [20][21][22] appealed to [4] to argue that the diagonal elements t kk of the threshold matrix are proportional to log sin πk N , rather than our result in (1.10). Note that these two expressions do not agree in the large-N limit.
recover the full answer in (1.16).
By combining elements of [4] with insights from integrable hierarchies, the authors of [21,22] conjectured an exact (but complicated) formula for the off-diagonal elements of the threshold matrix t kℓ . 14 A simpler expression for these off-diagonal elements was subsequently obtained in [23], where they were recomputed (again within the framework of integrable hierarchies) and used to numerically verify the conjecture of [21,22] for low values of N.
The off-diagonal elements in equations (6.11) and (6.12) of [23] are easily seen to match our off-diagonal elements of τ Dkℓ in (1.7) and (1.10), as well as (1.16). The off-diagonal elements of τ Dkℓ were also examined in [24], where they were expressed in a form (see their equation (169)) that exactly agrees with our (1. 16), and shown to agree with the conjecture of [21,22].
The only complete result for the threshold matrix t kℓ (including its diagonal elements) that we are aware of was recently put forward in [25], using a dual matrix model that was motivated by appealing to conjectures in topological string theory. While the authors found agreement with [21][22][23] for the off-diagonal elements of t kℓ , they also noted disagreement for the diagonal elements t kk . We will now compare the matrix-model results of [25] to ours. Their results are expressed in terms of a matrix-model (MM) prepotential F MM D (T k ), where the T k are the dimensionless 't Hooft couplings of the matrix model, which are to be identified with the a Dk periods (see equation (4.7) of [25]). We would like to convert to a prepotential F D (a D ) from which we can compute a k = ∂F D /∂a Dk and compare to our formulas (1.8), (1.9), and (1.10). By examining the logarithmic terms, we are led to identify 15 Here Λ is a strong-coupling scale introduced for dimensional reasons, whose relation to our Λ will be fixed below. By substituting the matrix-model prepotential in equations (4.14) and (4.15) of [25] into (1.18), we find that the results of [25] imply that Comparing with (1.8) and (1.10), we see that the last term in (1.19) correctly accounts for the off-diagonal elements of our t kℓ matrix. In order to find the scheme change that relates Λ to our Λ, we compare the constant term a(a D = 0) in (1.19) with (1.9), finding agreement if Λ = 4NΛ sin π N . 16 Substituting back into (1.19), we see that the remaining terms correctly account for their counterparts in (1.8) and (1.10), including an exact match for the diagonal elements t kk of our threshold matrix.

The SU(N) Seiberg-Witten Solution
In this section we briefly review aspects of the Seiberg-Witten solution of the pure SU(N) gauge theory, as determined in [1,5,6]. The Seiberg-Witten curve Σ is a hyperelliptic Riemann surface of genus N −1. It can be presented in many ways that are useful for various purposes.
These presentations may differ by coordinate changes, as well as by an overall rescaling of the strong-coupling scale Λ (i.e. by a scheme change). Using this freedom, we can express the Seiberg-Witten curve in the following form, Here x, y are dimensionless complex coordinates, while C N (x) is a degree N polynomial in x whose dimensionless coefficients depend on the gauge-invariant Coulomb-branch order where the ellipsis denotes terms of order x N −3 or lower in x.
The Seiberg-Witten differential (which has mass-dimension one) is given by It is a meromorphic one-form on Σ. Once we fix a set of a canonical A-and B-cycles on Σ, 16 Note that this is an N -dependent change of scheme, though both Λ and Λ are O(1) in the large-N limit.
we can determine the special Coulomb-branch coordinates a k and a Dk by integrating λ over these cycles, Different choices of A-and B-cycles lead to special coordinates that differ by Sp(2N − 2, Z) electric-magnetic duality transformations of the low-energy U(1) N −1 gauge theory on the Coulomb branch.
Unless stated otherwise, we set the strong-coupling scale Λ (which is the only dimensionful parameter in the problem) to Λ = 1 2 , so that the Seiberg-Witten differential (2.4) simplifies to

The Multi-Monopole Point
As we reviewed in section 1.1, there are N multi-dyon points on the Coulomb branch of the SU(N) gauge theory, and we focus on the multi-monopole point. It was shown in [4] that this point occurs when C N (x) in (2.1) and (2.2) is given by a degree N Chebyshev polynomial, 17 Here and throughout the paper we use the superscript (0) to denote quantities evaluated at the multi-monopole point. The leading terms in C (0) N (x) are given by This relation can be used to analyze the branch and singular points of the Seiberg-Witten curve (2.1) at the multi-monopole point, which occur when To this end we use the following product representation for C Here, and for future use, we define the following shorthands, Note that k can be any integer, though it will typically lie in the range 1 ≤ k ≤ N − 1.
Substituting (2.10) into (2.9), we see that y 2 has N − 1 double zeroes at x = c k and two simple zeroes at x = ±1. The simple zeroes correspond to non-singular branch points of the curve, while the double zeroes indicate that the curve has N − 1 singular degeneration points reflecting the N − 1 massless monopoles, as represented in the lower panel of figure 1.
As we will review in section 2.3 below, the branch cuts of the non-singular Seiberg-Witten curve in the vicinity of the multi-monopole point can be chosen so that the singular cuts that collapse to zero length. The only remaining branch cuts of the multi-monopole curve run from +1 to +∞ and from −1 to −∞ along the real axis (see figure 1). Up to an overall choice of sign (which amounts to a choice of Riemann sheet on the Seiberg-Witten curve), this specification of the branch cuts allows us to define the square root of (2.9) as a well-defined, holomorphic function on the cut We choose the overall sign so that the following identity holds, In other words, the sign of the square root on the left-hand side varies with the sign of the The identity (2.12) extends to the entire cut x-plane, on which both sides are holomorphic functions defined by analytic continuation. Throughout the remainder of the paper we will define all square roots we encounter by ensuring compatibility with (2.12). 18 This formula can be derived by using the defining relation (2.7) for C  N (x) ′ must be at x = c k = cos kπ N , and then fixing the overall coefficient by comparing with (2.8).
, and a choice of homology basis that degenerates to the homology basis of the singular curve in the lower figure. Each branch cut (x − k , x + k ) of the regular curve degenerates to the corresponding singular point c k of the multi-monopole curve. For later use in subsection 3.2, an arbitrary point ξ ∈ (x + k , x − k−1 ) that is well separated from the endpoints of the interval has also been indicated.

The Vicinity of the Multi-Monopole Point
In order to explore the neighborhood of the multi-monopole point, we deform (2.7) by adding to the Chebyshev polynomial C (2.13) The N − 1 complex coefficients of P N −2 (x) describe the N − 1 Coulomb branch directions along which we can approach the multi-monopole point by taking these coefficients to be sufficiently small (we will make this precise below). It is convenient to trade these N − 1 coefficients for the values P k of P N −2 (x) at N − 1 distinct points, which we take to be x = c k , Conversely, we can express P N −2 (x) in terms of the constants P k using the Lagrange interpolation formula, which we can in turn write in terms of the Chebyshev polynomial C The addition of P N −2 (x) in (2.13) deforms the zeroes of the curve y 2 = (C N (x)) 2 − 1.
Recall from the discussion below (2.11) that the singular curve y 2 = C If all P k are non-zero, every one of these zeroes is simple and corresponds to a branch point of the (everywhere non-singular) Seiberg-Witten curve. We choose the branch cuts to run from +∞ to x 0 , from x + k to x − k , and from x N to −∞ in the complex x-plane, as shown in figure 1.
If we scale towards the multi-monopole point by taking all P k → 0, then all δ's in (2.16) vanish, i.e. the simple zeroes at x 0 , x N approach +1, −1 respectively, while the branch cuts 20 To see this, we approximate the curve as y 2 ≃ C    N (x) and its first two derivatives at these zeroes, which can be done using the defining relation (2.7).
connecting the simple zeroes x + k and x − k collapse to singular double zeroes at c k . The length of these cuts tracks the vanishing monopole masses as we approach the multi-monopole point (see section 2.5 below). Below, we will always choose the P k to be non-vanishing, but sufficiently small to ensure that the cuts from x + k to x − k (whose length is 2|δ k |) are much shorter than their distance to the nearest branch point.
We will evaluate the special Coulomb-branch coordinates a k and a Dk as a function of the P k , to leading order in small P k , by explicitly integrating the Seiberg-Witten differential λ in (2.6) over suitable A-and B-cycles (specified below) as in (2.5). Since λ is a holomorphic one-form, the periods a k and a Dk are locally holomorphic functions of the P k . (Globally they may be branched and can undergo monodromy.) We can therefore simplify our computations by taking the P k = (−1) k |P k | to be small real numbers of alternating sign, so that the δ k in (2.16) are small, real, and positive, i.e. δ k > 0. Using (2.15) we can further check that these sign choices imply δ 0 , δ N < 0. In summary, This leads to the simplified cut complex x-plane depicted in the upper panel of figure 1, since all branch cuts now run along the real axis.
Before we can compute the a k and a Dk periods we must choose a set of canonical A-and B-cycles. Since we would like to associate the a Dk with the light monopoles, we choose the cycles B k (k = 1, . . . , N − 1) to encircle the short branch cuts connecting x ± k once, in the counterclockwise direction (see figure 1). 21 Note that these cycles do not cross any branch cuts, so that the a D -periods a Dk = 1 2πi B k λ in (2.5) can be evaluated on a single sheet. This computation was carried out in [20] and will be reviewed in section 2.5.
In order to define a suitable basis of A-cycles A k (k = 1, . . . , N − 1) conjugate to the B k defined above, we first define a simpler basis A k of one-cycles that encircle the first N −1 pairs of branch points in a counterclockwise direction, i.e. A 1 encircles The way in which the A k cycles traverse the first and second sheets, as well as their intersections with the B k cycles, are shown in figure 1. Explicitly, A 1 intersects B 1 in a negative sense, while A k (k = 2, . . . , N − 1) intersects B k−1 in a positive sense and B k in a negative sense.
Thus the A k cycles are not themselves conjugate to the B k cycles. However they can be used 21 This matches the conventions of the B k cycles in [20], which agree with the α k cycles in [4]. 22 These A k agree with the γ k cycles in [4].
to construct a basis of conjugate A k cycles as follows, This cycle intersects B k in a negative sense, without intersecting any of the other B-cycles. 23 We can therefore compute the a-periodes a k = 1 2πi A k λ in (2.5) as follows, The computation of the a k , and hence the a k , will be described in section 3.

Rewriting the Seiberg-Witten Differential
The expansion of the a-and a D -periods around the multi-monopole point is substantially complicated by the fact that the point around which we are expanding is singular. Following [20], this problem can be alleviated by a judicious rewriting of the Seiberg-Witten differential, which involves stripping off a locally exact one-form. To this end, we first introduce a family C We can then decompose the Seiberg-Witten differential λ in (2.6) as follows, Here λ is a locally defined meromorphic one-form given by the parametric integral

22)
23 Our A k cycles agree with the β k cycles of [4].
while S is a locally defined scalar function, Neither λ nor S are globally well defined on the Seiberg-Witten curve. (In particular, λ is not a valid Seiberg-Witten differential.) The reason is that both λ and S involve functions whose branch points do not coincide with the zeroes (2.16) of the Seiberg-Witten curve. We must therefore carefully define the branch cuts of the functions appearing in (2.22) and (2.23), which we will do below.
Let us outline the derivation of the decomposition (2.21). It is straightforward to verify the identity Integrating from µ = 0 to µ = 1 and recalling the definitions of λ, λ in (2.6), (2.22) gives Note that the second term on the left-hand side is (minus) the Seiberg-Witten differential of the multi-monopole curve.
We pause here to discuss the branch cuts of the functions appearing in (2.24) and (2.25).
The zeroes of the function C with the δ's given in (2.16). We therefore choose the branch cuts of C We now continue to simplify (2.25), starting with the Seiberg-Witten differential of the multi-monopole curve on the left-hand side. Using (2.12), we obtain Note that our choice of branch cuts in (2.12) implies that the branch cuts of (2.27) run from ±1 to ±∞, with no branch cut between −1 and 1. Finally, we can carry out the definite µ integral in (2.25) by changing variables to µ = C (0) . Most of our calculations below will stay away from these cuts. An exception occurs in section 3.5.

Expanding the a D -Periods Around the Multi-Monopole Point
We now review the computation of the periods a Dk = 1 2πi B k λ in (2.5) to leading order in small P N −2 (x), as described in [20], where the calculation was also carried out to higher orders. Along the B k -cycles the one-form λ and the scalar function S in (2.22) (2.29) We now use (2.12) to simplify the square root in the denominator, N (x) ′ has simple zeroes at x = c ℓ (ℓ = 1, . . . , N − 1) (see e.g. (2.10)), and only the zero at x = c k is encircled by the cycle B k , we can use (2.30) to evaluate (2.29) by residues, 24 As expected, a Dk vanishes as P k → 0. Note that the alternating sign choices for P k in (2.17) translate into the statement that all a Dk ∈ iR + .
For future reference, we substitute (2.31) into (2.15) (and use footnote 24) to express the polynomial P N −2 (x) in terms of the a Dk , We can similarly express δ k in (2.16) directly in terms of a Dk , The reason is that A k cycles necessarily cross branch cuts as they traverse the two sheets of the Seiberg-Witten curve (see the upper panel of figure 1.) Consequently, they cannot be evaluated using residues. A related complication is that the decomposition λ = λ + dS introduced in (2.21) is more delicate, because λ and S are not single valued along the A k cycles. In particular, the differential dS (though locally exact) contributes to the integral. 24 As in footnote 20, we evaluate C We begin by converting the period integral over A k into an ordinary real integral connecting neighboring branch points of the Seiberg-Witten curve. Taking into account the counterclockwise orientation of the A k cycles (see figure 1), which leads to a minus sign, and the fact that both the cycles and the Seiberg-Witten differential λ are odd under the hyper-elliptic involution (x, y) → (x, −y), which leads to a factor of 2, we can write The locations of the branch points x 0 , x ± k are given by (2.16) (see also the upper panel of figure 1). In the remainder of the paper, we explain how to evaluate the definite integrals in (3.1) to leading order in small P N −2 .
Despite the aforementioned subtleties, it is useful to decompose λ = λ + dS as in (2.21), with λ and S given by (2.22) and (2.23). This leads to a corresponding decomposition of a k , Here a k is the contribution obtained by replacing λ in (3.1) with λ defined in (2.22), Analogously, the contribution a (S) k in (3.2), which is due to the exact differential dS, reduces to a set of boundary contributions from the limits of the definite integrals in (3.1),

The Integrals a k≥2
We start by evaluating a k≥2 in (3.3), to leading order in small P N −2 . For convenience, we recall some formulas from (2. 16) and (2.20), Although the quantity P N −2 in which we would like to expand appears in the numerator of the integrand, it is not legal to set it to zero in the limits of the integral and under the square root in the denominator. To see this, and to get some intuition for how to proceed, we study the singularities of the integral (3.5) in more detail.
At the endpoint µ = 1 of the µ-integral, one simple zero of the polynomial (3.7) collides with the endpoint at x + k of the x-integral, while another simple zero of (3.7) collides with the other endpoint at x − k−1 . Although the resulting square root singularities are integrable, they modify the expansion of the integral in the small perturbation P N −2 .
In order to treat these two singularities, it is convenient to temporarily separate them by introducing a midpoint ξ ∈ (x + k , x − k−1 ) (which is arbitrary but chosen to be well separated from either endpoint, as shown in the upper panel of figure 1) and splitting the x-integral in (3.5) as follows, In the first integral (over x ∈ [x + k , ξ]) only the singularity at x + k = c k + δ k is relevant, so that we can set all other δ's in (3.7) to zero, (3.9) Here we have used (2.10) to obtain the second line. Analogously, only the singularity (3.8), which can therefore be evaluated by approximating (3.10) We must now take the square roots of (3.9) and (3.10), whose sign is fixed by comparing with (2.12). Since x − c k > 0 and x − c k−1 < 0 have opposite signs on the interval [x + k , x − k−1 ], we obtain the following two approximations,

(3.11)
We can use the approximations on the first and second line to simplify the first and second integrals in (3.8), respectively. Substituting the representation (2.32) for P N −2 (x) into these integrals, we find that the polynomial C N (x) ′ cancels, so that .

(3.12)
In order to further simplify this integral, we collect terms I(a Dk ) whose numerators are proportional to a Dk , terms J(a D,k−1 ) whose numerators are proportional to a D,k−1 , and a remainder R(a D,ℓ =k,k−1 ), 25 a k≥2 ≡ I(a Dk ) + J(a D,k−1 ) + R(a D,ℓ =k,k−1 ) .

(3.13)
We now proceed to define, simplify, and evaluate the integrals I(a Dk ), J(a D,k−1 ), and R(a D,ℓ =k,k−1 ): (1.) The integral I(a Dk ) in (3.13) contains all terms in (3.12) whose numerator is proportional to a Dk , . (3.14) Since the numerator of the second integral has a simple zero at x = c k−1 , and we are only working to leading order in small a Dℓ , it is permissible to take δ k−1 → 0 in the upper limit of this integral, while approximating its integrand as follows, To leading order in small a Dℓ , the two integrals in (3.14) thus combine into a single integral, which no longer depends on the auxiliary midpoint ξ, .

(3.16)
As explained in appendix B.2, this integral can be evaluated explicitly, 26 with the following result, (3.13) consists of all terms in (3.12) whose numerator is proportional to a D,k−1 , .

(3.18)
In exact analogy with the discussion around (3.15), we take δ k → 0 in the first integral and rewrite its integrand so that it can be combined with the second integral. In total, .

(3.19)
By comparing with (3.16), we see that the integrals J(a D,k−1 ) and I(a D,k ) are related by a suitable redefinition of parameters. This redefinition is explained in appendix B.3, where we show that The remainder R(a D,ℓ =k,k−1 ) in (3.13) consists of all terms in (3.12) whose numerators do not contain a Dk or a D,k−1 . Making approximations analogous to those we applied to I(a Dk ) and J(a D,k−1 ) above, we can express It is straightforward to evaluate this integral using substitution (see appendix B.1), Finally, we substitute (3.17), (3.20), (3.22) into (3.13) to obtain a formula for a k≥2 , (3.23)

The Integral a 1
We now evaluate a 1 in (3.3), to leading order in small P N −2 . Recall from (2.16), (2.17) that and from (3.7) that C Following the same logic as for the a k≥2 in section 3.2 above, we can now approximate the square root in the denominator using the first line of (3.11) for k = 1 and all x ∈ [x + 1 , 1]. As before, we then substitute (2.32) for P N −2 (x) to obtain the following simplification of the integral (3.24), .

(3.26)
We can set δ 1 → 0 in the second term, so that Comparing with (3.16), we see that the first integral in (3.27) is exactly I(a D1 ), while comparing with (3.21) shows that the second integral in (3.27) is R(a D,ℓ =1 ), i.e. both can be obtained by setting k = 1 in (3.16) and (3.21). Evaluating these integrals by setting k = 1 in (3.17) and (3.22), we find that Note that this coincides with (3.23) evaluated at k = 1, as long as we declare that a D0 = 0.

The Boundary Terms a (S) k≥2
We begin by evaluating the boundary contributions a The function S(x) was defined in (2.23), which we repeat here, . The location of the branch points x ± k is given by (2.16) (see also (2.17)), Since the x ± k are zeros of C (1) N (x ± k ) = ±1. We can also use (2.7) to show that C (0) N (c k ) = (−1) k . Since these expressions must agree as P N −2 → 0, we obtain the following exact statement, Substituting into (3.30), we find that where the function f k (x) is defined as follows, Here we must use (2.12) to fix the branch of the square root. It is now straightforward to expand this function around x = c k , 27 which in turn gives Substituting into (3.29) and using δ 2 k = − 2i N s k a Dk (see (2.33)), we obtain The last contribution we will need is a In (3.35) we have already evaluated We must now calculate S(x 0 ), where x 0 = 1 − δ 0 , with δ 0 = 1 (2.16) and (2.17)). Since it follows from (2.33) that P N −2 (x) -and hence δ 0 -is linear in the a Dk , we are free to drop terms beyond first order in δ 0 . In fact, we will show that S(x 0 ) vanishes to this order, To see this we must -for the first and only time in this paper -explicitly contend with the branch cuts of S(x). As explained below (2.28), these branch cuts lie entirely inside the intervals (1, +∞), (x − k , x + k ) and (−∞, −1), 28 but so does the point x 0 = 1 − δ 0 > 1. If 27 We use (2.7) to compute derivatives of C (0) In (3.35) we evaluated S(x) at the branch points x = x ± k , which lie at the boundary of these intervals. Hence S(x) is single valued there.
we naively proceed as in section 3.4 above and attempt to evaluate S(x 0 ) by expanding the function S(x) in (3.30) around x = 1, we find 29 (3.40) As we are evaluating S(x) on one of its branch cuts, it is not surprising that we encounter sign ambiguities. Since δ 0 < 0, we choose the first square root in (3.40) to be positive, √ −2δ 0 > 0.
As we will see, the relative sign between the two square roots in (3.40) is then fixed so that the second square root √ 2δ 0 = i √ −2δ 0 exactly cancels the first one, leading to (3.39).
Importantly, this cancellation must occur on physical grounds: the contributions to the aperiods computed in (3.23) and (3.28) already saturate the required monodromies around the multi-monopole point (see (1.8)). Therefore all other contributions must be analytic in the a Dk , which would not be the case if the square roots in (3.40) did not cancel. To see how this cancellation comes about explicitly, we must reexamine the origins of the two square roots in turn.
The first square root term −N √ −2δ 0 < 0 in (3.40) comes from expanding the logarithmic term in (3.30). As explained in section 2.4, this term arises from the integral in (2.28).
It follows that the square root that appears in the denominator of that integral must be positive. To see this explicitly, we examine the first integral (2.17)), it follows that the square root in the denominator of the integral must be positive. Equivalently, we can analyze the second form of the integral x C (1) N (x 0 ) = 1 and C (3.41) 29 We use (3.32) for k = 0 to compute C N (x 0 ) = C N (x) ′ is positive over the integration region (which can be shown using footnote 29), we conclude that the sign of (3.41) is set by the sign of the square root in the denominator of the integrand on the right-hand side. However, this sign is not independent. Rather, it must coincide with the signs of the square roots that appear in the integrands of the two integrals in (2.28). (This ultimately follows from the identity (2.24), from which all results in section 2.4 follow.) As we explained above, the latter signs must be positive to render the first term in (3.40) negative. We thus conclude that the square root in the integrand of (3.41), and hence the whole integral, is in fact positive. This completes the proof that the (3.40) is positive and cancels the negative first square-root term, which leads to (3.39).

Final Result for the a k
We will now combine our preceding results to determine the a-periods via (2.19) and (3.2), We begin by assembling the answer for a k . As explained below (3.28) and (3.42), we can use (3.23) for a k and (3.36) for a (S) k for all k = 1, . . . , N − 1, as long as we set a D0 = 0 in these formulas. Substituting into (3.43) and simplifying, we find (3.44) Therefore the sum for a k in (3.43) telescopes, so that Finally, we substitute δ 2 k = − 2i N s k a Dk from (2.33) to obtain the final answer, In order to make contact with the formulas in the introduction, we restore the strong coupling scale Λ by suitably inserting 1 = 2Λ into (3.46), 30 and by using trigonometric identities to simplify the argument of the second logarithm in (3.46), .

A. Comparison with Douglas and Shenker
In this appendix we review some of the results obtained in [4] in our conventions. We then extend these results to obtain an alternative derivation of the threshold matrix t kℓ in (1.10).

A.1. Review
First, we show that our Seiberg-Witten curve and differential, as well as our strongcoupling scale Λ, are identical to those used in [4]. Let us denote the strong-coupling scale 30 Recall that we set Λ = 1 2 around (2.6), and that both a k and a Dk have mass-dimension one.
of [4] by Λ. In units where Λ = 1, the Seiberg-Witten curve and differential used in [4] take the following form (see the discussion around (2.5) in [4]), if we set Λ = 1. Therefore the strong-coupling scales Λ and Λ also agree. For the remainder of this appendix we set Λ = 1 2 unless otherwise indicated, as we have done throughout. The scaling trajectory of [4] is given by (See the discussion below equation (5.1) in [4].) Comparing with (2.7) and the discussion above, we see that the scaling trajectory in our conventions is given by Here C Note that the leading O(x N ) term cancels out, so that P N −2 (x) does indeed have degree N −2.
From this we compute P k = P N −2 (c k ) = (−1) k s. Substituting into (2.31), we find that Here we have restored Λ, which was previously set to Λ = 1 2 . This establishes the formula (1.15) quoted in the introduction. 31 The authors of [4] compute the magnetic gauge coupling matrix τ Dkℓ (s) along their scaling trajectory. They show that this matrix is exactly diagonalizable in a basis of sine functions, so that 32 Here the eigenvalues τ D (p, s) are given in (5.12) of [4] (up to the overall factor of i, which is missing there), where the functions F (p, s) and G(p, s) are defined via the following integrals in (5.9) and (5.10) of [4], , a = arccos e −s . (A.7) In [4] these integrals were only evaluated for small s and small p N , Substituting into (A.6) gives where O(1) refers to the expansion in small p N . This agrees with (5.14) in [4] once the answer there is consistently expanded in small p N (and again including a missing factor of i). To compute τ Dkℓ (s), we substitute (A.9) back into (A.5) (and use footnote 32), 33 The leading logarithm exactly agrees with the one in (5.16) of [4]. We must now analyze the subleading terms in (A.10), which approach a finite constant as s → 0. Following [4] we show that the sum over p can be reliably evaluated in the large-N limit. To this end, we let ρ = p N 32 Here we invert equation (5.11) in [4] using N −1 p=1 s pk s pℓ = N 2 δ kℓ . 33 Note that some of the formulas below, e.g. (A.10) or (A.12), are similar to (1.12) in footnote 10. However, the latter formula for the off-diagonal elements of t kℓ is exact, while the former equations are only large-N approximations. and convert the sum over p to an integral over ρ, We distinguish two cases: 1.) If either k N or ℓ N vanish as N → ∞ then the corresponding sine functions in the integrand of (A.11) vanish at the lower limit of the integral and can be Taylor (A.12) Here Ci(x) = Here the O(1) terms are s-independent and finite in the large-N limit. The second logarithm in (A.13) is only reliable if k−ℓ N → 0 as N → ∞. (As explained above, a special case is k = ℓ, where we retain the log N 2 but omit the factor (k − ℓ) 2 in the denominator of the logarithm.) If instead k − ℓ = O(N), then this logarithm becomes part of the O(1) terms, which were not computed in [4]. 34

A.2. Some New Results
We now explain how to extend the results of [4] reviewed above to exactly compute the constant terms in τ Dkℓ (s) at small s. To this end, we must expand the function F (p, s) in (A.7) at small s, but work exactly in p. To this end we expand To get our bearings, we begin by substituting this into (A.7) and naively expanding both the limits of the integral and the integrand, Note that all cosines in the denominators of the integrand diverge at the upper endpoint of the integral when s → 0. Let us estimate this divergence by considering where we have changed variables to χ = π 2 − θ. Since the divergence arises from the vanishing sine in the denominator as s → 0, we can extract the leading divergence by Taylor expanding 34 Note that (A.13) should agree with equation (5.16) in [4] as long as k = αN + k and ℓ = αN + ℓ with α = O(1) and k N , ℓ N → 0 as N → ∞. Expanding (5.16) in [4] in this regime yields the integrand around χ = 0, so that (A. 16 where the ellipsis denotes all subleading terms of the form s n (cos θ) −k with k < 2n + 1.
Substituting back into (A.15), we can carry out the χ integral over all n ≥ 1 terms in (A. 18) using (A.17), The sum over n can be performed using Mathematica and evaluates to log 2. 35 The remaining integral in (A.19) must be expanded up to and including O(1) for small s.
(Note that evaluating its leading divergence using (A.17) only captures the logarithmically divergent piece of the integral.) This can also be done using Mathematica, (A.20) 35 To see this analytically, we can again use (B.12) to express ∞ n=1 Γ(n + 1 2 ) Γ( 1 2 )n!2n To show that the integral indeed evaluates to log 2, we replace its lower limit by ε > 0 and take ε → 0 at the end. Using (B.2) and (B.7), we evaluate Combining the two integrals and taking ε → 0 we obtain log 2.
Here γ is Euler's constant and ψ(x) is the digamma function. Using Gauss' digamma theorem (see for instance equation (29) on page 19 of [27]), we can evaluate where c pq = cos πpq N , following the notation of (2.11). Substituting back into (A.20), we find that (A.19) simplifies to We now substitute (A.22) into (A.6) to obtain Finally we are in a position to substitute this into (A.5) and compute τ Dkℓ (s). To this end we need the sum in footnote 32, as well as the following more complicated sum, This leads to The remaining sum over q evaluates to We begin by evaluating the integral (3.21), which we repeat here, We need the following basic integral, Let us define the following sign factor, By comparing with the integral (B.2), we see that σ = sign(I). We proceed to evaluate this integral using several substitutions: • Substituting u = 1 x−c , we find that • Changing variables to w = s 2 u − c, we find that • Note that the sign of the integration variable w in (B.5) is given by sign(w) = σ. We can thus change variables one more time, to w = σ cosh η with η > 0, and evaluate We can further simplify (B.6) by using the fact that cosh −1 v = log(v + v 2 − 1), as long as v ≥ 1. Since this is indeed the case for the arguments of the cosh −1 functions in (B.6), we can finally express the integral in the following form, We can now apply this to evaluate R(a D,ℓ =k,k−1 ) = − 1 πi ℓ =k,k−1 s ℓ a Dℓ I(c k , c k−1 ; c ℓ ) in (B.1), for which we need where the integral I(a Dk ) that we must evaluate is given by .

(B.11)
Note that this integral is manifestly positive. We will not retain terms in I(a Dk ) that vanish faster than a Dk . For this reason, we can drop terms in I(a Dk ) that vanish when a Dk → 0, or equivalently when δ k → 0 (see (2.33)).
We will directly evaluate the integral (B.11) by expanding both inverse square roots in absolutely convergent power series and integrating term by term. 36 To this end, we expand the first inverse square root via 1 1 − x 2 = ∞ n=0 Γ(n + 1 2 ) Γ( 1 2 )n!
x 2n , (B. 12) and similarly for the second inverse square root. After substituting into (B.11), we can carry out the µ integral. We then simplify the x integral by shifting x → x + c k and expanding the numerator using the binomial formula, so that I(a Dk ) = The remaining x-integral is trivial, (B.14) Substituting back into (B.13), we now drop all terms that vanish as δ k → 0. The only remaining terms are the n = 0 logarithmic terms and the n = 0 polynomial terms from the upper limit of the x-integral (B.14), as well as the ℓ = 2m polynomial terms from the lower limit of the same integral. Paying attention to the restrictions on summation indices that result from (B.14), we can now express (B.13) as a sum of three terms, where f 1,2,3 are given by the following series expressions, Γ(m + 1 2 )Γ(n + 1 2 ) Γ( 1 2 ) 2 m!n! c 2m k 2n(n + 1) .

(B.16)
The sums over m in f 1 and f 3 can be evaluated using (B.12), while the remaining sum in f 3 can be performed using Mathematica. This gives To evaluate f 2 in (B.16), we define the function f 2 (x) via Differentiating (B.18) term by term and summing the resulting series using (B.12), we find . (B.20) We now integrate this equation from a to x, where c k < a, x < c k−1 . The first term on the right-hand side leads to an integral of the form (B.2), while the second term integrates to a logarithm, (B.28) Comparing this with (B.13), we see that we can compute J(a D,k−1 ) from I(a Dk ) by substituting c k → −c k−1 , c k−1 → −c k , and δ k → δ k−1 . 38 Substituting these replacements into (B.24), 37 A naive continuation that gives wrong answers is J(a D,k−1 ) = − I(a Dk ) c k ↔c k−1 ,δ k →−δ k−1 . (One way to see that this cannot be correct is that the two sides have opposite signs.) This continuation fails because flipping the sign of δ extends the x-integral past a branch point of the square root in the denominator. 38 Note that these continuations do not run afoul of the same problems as the ones in footnote 37.