We will now combine our preceding results to determine thea-periods via (2.19) and (3.2), ak=
k
X
`=1
ba`, bak=aek+a(S)k , (k= 1, . . . , N−1). (3.43) We begin by assembling the answer for bak. As explained below (3.28) and (3.42), we can use (3.23) foreak and (3.36) fora(S)k for allk= 1, . . . , N −1, as long as we set aD0 = 0 in these formulas. Substituting into (3.43) and simplifying, we find
bak=eak+a(S)k
Therefore the sum for ak in (3.43) telescopes, so that ak=−N
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Finally, we substitute δ2k=−2iNskaDk from (2.33) to obtain the final answer,
ak=−N
πsk+aDk
2πi log −iaDk 8N s3k
!
−1
! + 1
2πi
N−1
X
`=1`6=k
aD`log (ck−c`)2
(1−ck+`)2 . (3.46) 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),33 and by using trigonometric identities to simplify the argument of the second logarithm in (3.46),
ak =−2NΛ
π sk+aDk
2πi log −iaDk 16NΛs3k
!
−1
! + 1
2πi
N−1
X
`=1`6=k
aD`logsin2 (k+`)π2N
sin2 (k−`)π2N . (3.47)
Acknowledgments
The work of ED is supported in part by NSF grant PHY-19-14412. TD and EN are supported by a DOE Early Career Award under DE-SC0020421. TD is also supported by a Hellman Fellowship and the Mani L. Bhaumik Presidential Chair in Theoretical Physics at UCLA. The work of EG is in part supported by the Israel National Postdoctoral Award Program for Advancing Women in Science.
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 tk`in (1.9).
A.1 Review
First, we show that our Seiberg-Witten curve, as well as our strong-coupling scale Λ, are identical to those used in [4]. By contrast, our Seiberg-Witten differential λ differs from their differentialλe by a sign, i.e.eλ=−λ. Since ourA- andB-cycles agree with theirs (see footnotes 24–26), this means that their eak- and eaDk-periods differ from our ak- andaDk -periods by an overall sign, i.e. (eak,eaDk) = (−ak,−aDk). Note that these signs cancel inτDk`= ∂a∂aD`k =τeDk`, so that our gauge couplings agree.
To see this explicitly, let us denote the strong-coupling scale of [4] by Λ. In unitse whereΛ = 1, the Seiberg-Witten curve and differential used in [4] take the following forme (see the discussion around (2.5) in [4]),
y2=Pe(xe)2−1, Pe(xe) = 1
2xeN +O(xeN−2), λe= xde Pe
y . (A.1)
We now change variables by writing xe = 2x. Comparing with (2.1) and (2.3) we see that our conventions for the Seiberg-Witten curve match if we identify CN(x) = Pe(xe = 2x).
Substituting into (A.1) we see thateλ= 2xCN0 (x)dx/yappears to match our Seiberg-Witten
33Recall that we set Λ =12 around (2.6), and that bothakandaDk have mass-dimension one.
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differentialλin (2.4) if we set Λ = 1, so that the strong-coupling scales Λ andΛ also agree.e In fact, the two differentials differ by a sign,λe=−λ, because the authors of [4] choose the opposite branch of the square root in equation (2.12) (see their equation (2.9)), and hence the opposite sign for y. For the remainder of this appendix we work in our conventions, i.e. we use our Seiberg-Witten differential λ, and we set Λ = 12 unless otherwise indicated.
The scaling trajectory of [4] is given by Pe(s)(xe) = esPe(0)(e−s/Nxe), where Pe(0)(xe) = cos(Narccosex2). (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 CN(s)(x) =esPe(0)e−Ns2xe=esCN(0)e−Nsx . (A.2) Here CN(0)(x) = cos(Narccosx) is the Chebyshev polynomial in (2.7) that describes the singular curve at the multi-monopole point. Expanding (A.2) to first order in sand com-paring with (2.13), we find that the degree-(N−2) polynomial describing the approach to the multi-monopole point ass→0 is given by
PN−2(x) =s
CN(0)(x)− x
NCN(0)(x)0+O(s2) . (A.3) Note that the leadingO(xN) term cancels out, so thatPN−2(x) does indeed have degreeN− 2. From this we computePk=PN−2(ck) = (−1)ks. Substituting into (2.31), we find that
aDk = isk
N s+O(s2) = 2iΛsk
N s+O(s2) . (A.4)
Here we have restored Λ, which was previously set to Λ = 12. This establishes the for-mula (1.14) quoted in the introduction.34
The authors of [4] compute the magnetic gauge coupling matrix τDk`(s) along their scaling trajectory. They show that this matrix is exactly diagonalizable in a basis of sine functions, so that35
τDk`(s) = 2 N
N−1
X
p=1
τD(p, s)spksp` (A.5) Here the eigenvalues τD(p, s) are given in (5.12) of [4] (up to the overall factor ofi, which is missing there),
τD(p, s) = i 2 sin2Nπp
F(p, s)
G(p, s), (p= 1, . . . , N −1), (A.6) where the functions F(p, s) and G(p, s) are defined via the following integrals in (5.9) and (5.10) of [4],
F(p, s) = 1 π
Z b
−bdθcos (1−Np)θ
√
e−2s−sin2θ, b= arcsine−s, G(p, s) = 1
π Z a
−adθ cos (1− Np)θ
√cos2θ−e−2s, a= arccose−s .
(A.7)
34Note that setting Λ = 1 in (A.4) should reduce to minus equation (5.4) in [4]. It does so up to an overall factor of (−2) that is missing in [4].
35Here we invert equation (5.11) in [4] usingPN−1
p=1 spksp`= N2δk`.
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In [4] these integrals were only evaluated for small sand small Np, F(p, s) = 1 whereO(1) refers to the expansion in smallNp. This agrees with (5.14) in [4] once the answer there is consistently expanded in small Np (and again including a missing factor ofi).
To compute τDk`(s), we substitute (A.9) back into (A.5) (and use footnote 35),36 τDk`(s) =− i 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= Np and convert the sum over p to an integral overρ,
(1) If either Nk or N` vanish asN → ∞then the corresponding sine functions in the inte-grand of (A.11) vanish at the lower limit of the integral and can be Taylor expanded there. This cancels the ρ1 pole and renders the integral finite in the large-N limit. A reliable computation of this finite contribution requires knowledge of theO(1) terms in (A.11).
(2) If bothkand`areO(N) then both sine functions in (A.11) approach non-zero O(1) constants at ρ = N1. The integral is therefore dominated by the ρ1 divergence there, which can be reliably computed without knowing the O(1) terms in (A.11),
2i
t cost is the cosine integral function, which is bounded away from x = 0, but diverges as Ci(x) = logx+O(1) when x → 0. Here we have performed
36Note that some of the formulas below, e.g. (A.10) or (A.12), are similar to (1.11) in footnote 11.
However, the latter formula for the off-diagonal elements oftk`is exact, while the former equations are only large-N approximations.
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the computation for k > `; the answer for k < ` can be inferred by symmetry, and when k=` the first cosine integral function in (A.12) is replaced by log N1.
Sincekand`are bothO(N) (see above), the second cosine integral function in (A.12) isO(1) in the large-N limit. The only way the first cosine integral function can avoid a similar fate is if k−`N vanishes at large N, so that Ciπ(k−`)N = logk−`N +O(1).
Substituting back into (A.10) we thus find that τDk`(s) =− i
2πδk`logs+ i
2πlog N2
(k−`)2 +O(1) +O(s) . (A.13) 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−`N → 0 as N → ∞. (As explained above, a special case is k=`, where we retain the logN2 but omit the factor (k−`)2 in the denominator of the logarithm.) If insteadk−`=O(N), then this logarithm becomes part of theO(1) terms, which were not computed in [4].37