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arXiv:1107.3756v2 [hepth] 20 Jul 2011
OCUPHYS 354
Baxter’s TQ equation, SU(N)/SU(2)N−3duality
and
Ωdeformed SeibergWitten prepotential
Kenji Muneyuki,a∗TaSheng Tai,a†Nobuhiro Yonezawab‡and Reiji Yoshiokab§
aInterdisciplinary Graduate School of Science and Engineering,
Kinki University, 341 Kowakae, HigashiOsaka, Osaka 5778502, Japan
bOsaka City University Advanced Mathematical Institute,
33138 Sugimoto, Sumiyoshiku, Osaka 5588585, Japan
Abstract
We study Baxter’s TQ equation under the semiclassical limit where an intriguing SU(N)/SU(2)N−3
duality is found. That is, two kinds of 4D N = 2 superconformal field theories having the above
different gauge groups are encoded simultaneously in one Baxter’s TQ equation which captures
their spectral curves.
For example, while one is SU(Nc) with Nf= 2Ncflavors the other turns out to be SU(2)Nc−3
with Nchypermultiplets (Nc> 3). In terms of Mtheory, we can interpret this as a result of
exchanging two holomorphic coordinates (u ⇔ s) of the M5brane configuration engineering the
former SU(Nc) theory. Besides, it is seen that the corresponding SeibergWitten differential
supports our proposal.
1Introduction and summary
Recently there have been new insight into the duality between integrable systems and 4D N = 2 gauge
theories. In [1, 2, 3] Nekrasov and Shatashvili (NS) have found that YangYang functions as well as Bethe
Ansatz equations of a family of integrable models are indeed encoded in a variety of Nekrasov’s partition
functions [4, 5] restricted to the twodimensional Ωbackground1. As a matter of fact, this mysterious
correspondence can further be extended to the full Ωdeformation in view of the birth of AGT conjecture
[10]. Let us briefly refine the latter point.
Recall that AGT claimed that correlators of primary states in Liouville field theory (LFT) can get re
expressed in terms of Nekrasov’s partition function ZNek of 4D N = 2 quivertype SU(2) superconformal
field theories (SCFTs). In particular, every Riemann surface Cg,n (whose doublysheeted cover is called
Gaiotto curve [11]) on which LFT dwells is responsible for one specific SCFT called Tg,n(A1) such that the
following equality
Conformal block w.r.t. Cg,n= Instanton part of ZNek
?
Tg,n(A1)
?
holds. Because of ǫ1: ǫ2= b : b−1[10] the oneparameter version of AGT conjecture directly leads to the
semiclassical LFT at b → 0. Quote the geometric Langlands correspondence [12] which associates Gaudin
integrable models on the projective line with LFT at b → 0. It is then plausible to put both insights of NS
and AGT into one unified scheme.
In this letter, we add a new element into the above 2D/4D correspondence. Starting from Baxter’s TQ
equation of XXX spinchain models we found a novel interpretation of it. That is, under the semiclassical
limit it possesses two aspects simultaneously. It describes 4D N = 2 SU(Nc) YangMills with Nf = 2Nc
∗email address: 1033310118n@kindai.ac.jp
†email address: tasheng@alice.math.kindai.ac.jp
‡email address: yonezawa@sci.osakacu.ac.jp
§email address: yoshioka@sci.osakacu.ac.jp
1See also recent [6, 7, 8, 9] which investigated XXX spinchain models along this line.
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NS5 NS5
D6
D4
D4
ξ1 ξ2 ξ3 ξ4
ℓ1ℓ2ℓ3 ℓ4
◎
◎◎
◎
◎
◎◎
◎
◎
◎◎
◎
◎
◎◎
◎
ℓ1 ξ1 ℓ2 ξ2
ℓ4 ξ4 ℓ3 ξ3
rotation
Fig. 1: Main idea: route to SU(N)/SU(2)N−3duality (ξ and ℓ indicate the location and weight of each
puncture on CP1in the last picture)
flavors T0,4(ANc−1) on the one hand and SU(2)Nc−3one with Nchypermultiplets (Nc> 3) T0,Nc(A1) on
the other hand. It is helpful to have a rough idea through Fig. 1 and 2 which also summarize Sec. 2. Then
Sec. 3 is devoted to unifying three elements: Gaudin model, LFT and matrix model as shown in Fig. 3.
Finally, in Sec. 4 we complete our proposal by examining λSW (SeibergWitten differential) and shortly
discuss XYZ Gaudin models.
2XXX spin chain
Baxter’s TQ equation [13, 14] plays an underlying role in various spinchain models. It emerges within the
context of quantum inverse scattering method (QISM) or algebraic Bethe Ansatz. On the other hand, it
has long been known that the lowenergy Coulomb sectors of N = 2 gauge theories are intimately related to
a variety of integrable systems [15, 16, 17, 18, 19]. Here, by integrable model (or solvable model) we mean
that there exists some spectral curve which gives enough integrals of motion (or conserved charges). In the
case of N = 2 SU(Nc) YangMills theory with Nf fundamental hypermultiplets, its SeibergWitten curve
[20, 21] is identified with the spectral curve of an inhomogeneous periodic Heisenberg XXX spin chain on
Ncsites:
w +QNf(u)
w
= PNc(u). (2.1)
Here, two polynomials PNcand QNfencode respectively parameters of N = 2 vector and hypermultiplets.
Meanwhile, the meromorphic SW differential λSW = udlogw provides a set of “special coordinates” through
its period integrals. The physical prepotential FSW gets extracted henceforth.
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2.1 Baxter’s TQ equation and u ⇔ s symmetry
Indeed, (2.1) arises from
det?w − T(u)?= 0
detT(u) = QNf(u) =
→
Nc
?
w2− trT(u)w + detT(u) = 0,T(u) : monodromy matrix,
i=1
(u − m−
i)(u − m+
i).
trT(u) = t(u) = PNc(u), transfer matrix, encodes the quantum vev of the adjoint scalar field Φ. In fact,
(2.1) belongs to the conformal case where Nf= 2Ncbare flavor masses are indicated by m±
TQ equation:
i. Quote Baxter’s
t(u)Q(u) = △+(u)Q(u − 2η) + △−(u)Q(u + 2η). (2.2)
• η is Plancklike and ultimately gets identified with one of two Ωbackground parameters ǫ1in Sec. 4.
• (2.2) boils down to (2.1) as η → 0 with m±
i= (ξi± ℓi). For ξ and ℓ, see Sec. 2.2.
Quite curiously, however when we look into its λSW this time another theory emerges. In other words,
by viewing the Nf= 2Ncbrane configuration as the pure N = 2 YangMills one perturbed by NfD6branes
which introduce flavors, “another theory” means its π/2rotated version. Pictorially, it looks like a doubly
sheeted cover of CP1with Ncstrands attached. In particular, ℓ (ξ) becomes the weight (location) assigned
to each strand. Because this viewpoint is closely related to N = 2 Gaiotto curves, a series of quivertype
SU(2) SCFTs T0,n(A1) discovered by Gaiotto [11] is hence made contact with.
Baxter’s TQ eq
Bethe Ansatz eqO(η)
Fuchs eq
O(η)
Two aspects of
Baxter’s TQ eq
Lamé eq
Gaudin spectral curve
Fig. 2: Mathematical description of Fig. 1
One may try to figure out this roughly through Mtheory. In Mtheory [22] exchanging two holomorphic
coordinates u = x4+ ix5and s = x6+ ix10(w = exp(−s/R), R: radius of Mcircle x10) is pictorially done
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0, 1, 2, 3
◦
◦
◦
u

◦

6


◦
7, 8, 9
◦


D6
NS5
D4
Table 1: Type IIA D6NS5D4 brane configuration
by a π/2rotation (u ⇔ s) of the corresponding Type IIA NS5D4 brane system2(see Fig. 1). In terms of
λSW that this symmetry says merely −uds ≈ sdu.
2.2More detail
Let us refine the above argument below. Consider a quantum spinchain built over H = ⊗N
at each site labeled by n we assign an irreducible (ℓn+ 1)dimensional space Vnw.r.t. Lie algebra sl2(ℓn:
highest weight). Within the context of QISM, monodromy and transfer matrices are defined respectively by
n=1Vn. Namely,
T(u) =
?
AN(u)
CN(u)
BN(u)
DN(u)
?
= LN(u − ξN)···L1(u − ξ1),(2.3)
?t(u) = AN(u) + DN(u).
Notice that the nth Lax operator Lnacts on Vn. By inhomogeneous we mean that the spectral parameter
u has been shifted by ξ. Conventionally,?t(u) or its eigenvalue t(u) is called generating function because a
arises just from the celebrated YangBaxter equation.
As far as the inhomogeneous periodic XXX spin chain is concerned, its TQ equation reads
series of conserved charges can be extracted from its coefficients owing to [?t(u),?t(v)] = 0. The commutativity
t(u)Q(u) = △+(u)Q(u − 2η) + △−(u)Q(u + 2η),
K
?
Q(u) =
k=1
(u − µk),
△±=
N
?
n=1
(u − ξn±?ℓnη),(2.4)
where each Bethe root µksatisfies a set of Bethe Ansatz equations (η?ℓ = ℓ):
△+(µk)
△−(µk)=
n=1
N
?
(µk− ξn+?ℓnη)
(µk− ξn−?ℓnη)
=
K
?
l(?=k)
µk− µl+ 2η
µk− µl− 2η.
(2.5)
Here, the η dependence enables us to carry out a semiclassical limit later on. Through
t(u)
?△+△−
=Q(u − 2η)
Q(u)
?
△+
△−
+Q(u + 2η)
Q(u)
?
△−
△+
(2.6)
and
w ≡
?
△+
△−(1 − 2ηQ′
Q)(2.7)
as η → 0 and ℓ stays fixed we arrive at
t(u)
?△+(u)△−(u)
= w +1
w
(2.8)
2In [23] this symmetry has been notified in the context of Todachain models because two kinds of Lax matrices exist there.
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which is nothing but (2.1). From now on we call λSW of (2.8) “ηdeformed” SW differential λη
as in [24, 25] because
?Ψ′
SW≡ udlogw
λη
SW≃ 2ηud
Ψ
?
+ O(η2), Ψ =
1
Q(u)
?
n
(u − ξn)?ℓn/2. (2.9)
It is this form that makes contact with the π/2rotated counterpart. As η → 0 the classical version of (2.5)
reads
N
?
n=1
?ℓn
2(µk− ξn)=
K
?
l(?=k)
1
(µk− µl).(2.10)
Two comments on (2.8) follow.
• In Mtheory D6branes correspond to singular loci of xy = △+(u)△−(u). This simply means that one
incorporates flavors via replacing a flat R4on (u,s) by a resolved A2Nc−1type singularity.
• Without the perturbation by flavors one sees that?λη
any treelevel potential which may bring N = 2 pure YangMills to N = 1 descendants.
Surely, this intuition is noteworthy because (2.10) manifests itself as the saddlepoint condition within
the context of matrix models. To pursue this interpretation, one should regard µ’s as diagonal elements of
a Hermitian matrix M (of size K × K). Besides, the treelevel potential there now obeys
SWreduces to a logarithm of the usual Vandermonde
measure evaluated over Bethe roots. This sounds like the standard DijkgraafVafa story [26, 27, 28] without
W′(x) =
N
?
n=1
ℓn
(x − ξn).
In other words, we are equivalently dealing with “N = 2” Pennertype matrix models which have been
heavily investigated recently in connection with AGT conjecture due to Dijkgraaf and Vafa [29]. In what
follows, our goal is to show that λη
SWdoes reproduce the ǫ1deformed SW prepotential w.r.t. T0,N(A1). This
is accomplished by means of the chart drawn in Fig. 3. We thus refer to this phenomenon as the advertised
SU(N)/SU(2)N−3(N > 3) duality.
3 XXX Gaudin model
Now we turn our attention to another wellstudied integrable model: XXX Gaudin model. Eventually, it
turns out that λη
SWnaturally emerges as the holomorphic oneform (2.9) of Gaudin’s spectral curve which
captures Gaiotto’s curve of T0,N(A1) (last picture of Fig. 1).
The essential difference between Heisenberg and Gaudin models amounts to the definition of their gen
erating functions. Below we explain following Fig. 3 two important aspects of Gaudin’s spectral curve.
3.1RHS of Fig. 3
Expanding around small η, we yield
Ln(u) = 1 + 2ηLn+ O(η2),
T(u) = 1 + 2ηT + η2T(2)+ O(η3),
t(u) = 1 + η2trT(2)+ O(η3),
τ(u) ≡1
(3.1)
(3.2)
(3.3)
2trT2,
T =
?
n
Ln=
?
A(u)
C(u)
B(u)
−A(u)
?
(3.4)
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Bethe Ansatz equation ↔
Lamé equation ↔
Gaudin spectral curve
Freefield conformal block
Pennertype matrix model
Gaudin Hamiltonian
Classical Liouville/Goper
Langlands duality
Ωdeformed
prepotential
Fig. 3: Flow chart of Sec. 3
where
A(u) =
N
?
n=1
Jz
n
u − ξn,B(u) =
N
?
n=1
J−
n
u − ξn,C(u) =
N
?
n=1
J+
n
u − ξn. (3.5)
Instead of trT(2)(trT = 0) the generating function one would like to adopt is (s =?ℓ/2 = ℓ/2η)
τ(u) =
(u − ξn)2
N
?
n=1
?η2sn(sn+ 1)
+
cn
u − ξn
?
,cn=
N
?
i?=n
2?Jn·?Ji
ξn− ξi,
?Jn·?Jn= η2sn(sn+ 1). (3.6)
Notice that?J = (Jz,J±) are generators of sl2Lie algebra while cn’s are called Gaudin Hamiltonians which
commute with one another as a result of the classical YangBaxter equation. In fact, the Nsite Gaudin
spectral curve is expressed by Σ :x2= τ ⊂ T∗C, a doublysheeted cover of C = CP1\{ξ1,··· ,ξN}.
According to the geometric Langlands correspondence3, cn’s give exactly accessory parameters of a Goper:
D = −∂2
z+
N
?
n=1
δn
(z − ξn)2+
N
?
n=1
? cn
z − ξn,δ = s(s + 1),c = η2? c
defined over C = CP1\{ξ1,··· ,ξN}. The nonsingular behavior of D is ensured by imposing
N
?
n=1
? cn= 0,
N
?
n=1
(ξn? cn+ δn) = 0,
N
?
n=1
(ξ2
n? cn+ 2ξnδn) = 0.
Certainly, one soon recalls that this τ(u) is nothing but the holomorphic LFT stresstensor as the central
3See [30, 31] for more detail.
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charge 1 + 6Q2goes to infinity (or b → 0). Namely,
1
2∂2
zϕcl−1
4(∂zϕcl)2=
N
?
n=1
δn
(z − ξn)2+
N
?
n=1
? cn
z − ξn.
Here, ϕclwhich satisfies Liouville’s equation stands for the unique saddlepoint w.r.t. Stot[ϕ] being specified
below. Meanwhile, Polyakov conjectured that ? cngets obtained by computing LFT correlation functions of
n
precisely,
primary fields ZL= ?
?
Vαn? subject to b → 0 with Vα= exp(2αφ) (∆α= α(Q − α), Q = b + b−1). More
? cn= −∂Stot[ϕcl]
∂ξn
,
? αn= bαn= sn+ 1
where on a large disk Γ
Stot=
?
Γ
d2z
?1
4π∂zφ2+ µe2bφ?
+ boundary terms,
Stot[φ] =1
b2Stot[ϕ].
3.2LHS of Fig. 3
As shown in [32], τ(u) has another form in terms of a(u), eigenvalue of A(u):
τ(u) = a2− ηa′− 2η
?
k
a(u) − a(µk)
u − µk
,a(u) =
N
?
n=1
ηsn
(u − ξn)
(3.7)
with {µk} being Bethe roots. This expression is extremely illuminating in connection with Pennertype
matrix models. Borrowing Q(u) from (2.4) and defining
ℜ(u) ≡ Q(u)exp
?
−1
η
?u
a(y)dy
?
=
?
k
(u − µk)
?
n
(u − ξn)−sn,(3.8)
we can verify that there holds
ηx′+ x2= τ,x(u) = ηℜ′(u)
ℜ(u)= −a +
?
k
η
u − µk.(3.9)
This is the socalled Lam´ e equation in disguise. Equivalently, ℜ(u) satisfies a Fuchstype equation with N
regular singularities on the projective line, i.e.
Imposing the semiclassical limit η → 0 with ηsn= O(η0) resembles that in LFT when ? α = bα is kept
consequently arrive at Gaudin’s spectral curve
?η2∂2
u− τ(u)?ℜ(u) = 0 (see Fig. 2).
fixed during b → 0. Note that ǫ1 = η ≈ b has been claimed by AGT4. By omitting the term ηx′, we
x2= τ.(3.10)
Owing to (3.9), it is tempting to introduce φKS, KodairaSpencer field of ZMshown in (3.13), i.e.
2x ≡ ∂φKS= −W′+ 2η tr?
1
u − M
?. (3.11)
Then (3.10) becomes precisely the spectral curve of ZM.
4Bear in mind η = ?/b such that when b → 0 η is kept fixed and small.
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Because of (3.11) using
?
∂φKSdu = −
?
λη
SW
(3.12)
we can obtain the treelevel free energy F0of ZM:
ZM=
?
DMexp
?
−1
ηW(M)
?
,
W′=
N
?
n=1
ℓn
(u − ξn).(3.13)
Of course, the saddlepoint of ZMis dictated by (2.10). We want to show next in Sec. 4 that F0thus yielded
does characterize the SCFT T0,N(A1).
Remark that ∂φKSdu = −λη
Mtheoretical u ⇔ s symmetry (x,u) ∈ C×C∗here is no longer (s,u) ∈ C2said before. We introduce v = xu
such that xdu here and the former udlogw look more symmetrical. Besides, the proposed SU(N)/SU(2)N−3
duality does not rigorously mean just a π/2rotation as noted in Fig. 1 which naively leads instead to an
SU(N)/SU(2)N−1duality5.
Rewriting ZMin terms of a multiintegral over diagonal elements of M is another crucial step:
?
i<j
i,n
SWup to a total derivative term. This fact though reflects the aforementioned
ZM⇒
dz1···
?
dzK
?
(zi− zj)2?
(zi− ξn)−?ℓn?
n<m
(ξn− ξm)?ℓn?ℓm/2.
One is able to refer to this expression as the freefield realization of an Npoint conformal block of primary
states in (possibly Wickrotated) LFT under b → 0. Then we have (ℓn= η?ℓn, η = ?b−1):
lim
b→0log
?
Vℓ1/2(ξ1)···VℓN/2(ξN)
?
free= η2?F,
?F = logZM,ZM= exp(η−2F0+ ···).
Notice that the (Wickrotated) free propagator?φ(z1)φ(z2)?
operators
at small η. Next, we will identify them with the Ωdeformed SW prepotential of T0,N(A1) such that our
proposal is completed.
free= log(z1− z2)1/2and K heavy screening
?dz exp−2b−1φ(z) were utilized6. Obviously, the treelevel free energy F0 gets equal to η2?F
4Application and discussion
By examining a concrete example, we want to present that F0and λη
ǫ1deformed SW prepotential and differential of T0,N(A1), respectively.
Let us focus only on N = 4 and quote the known τ(u) from [33] with δ = ? α(1 − ? α):
u2+
SW= −∂φKSdu are indeed the very
1
η2τ(u) =δ1
δ2
(u − q)2+
δ3
(1 − u)2+δ1+ δ2+ δ3− δ4
u(1 − u)
+
q(1 − q)? c(q)
u(u − q)(1 − u)
(4.1)
where q represents the crossratio of M¨ obiustransformed (0,1,q,∞) on CP1. Using Polyakov’s conjecture
as well as the residue of τ around u = 1 we have (v = xu)
?
?2?Fδ,δn(q) = (δ − δ1− δ2)logq +(δ + δ1− δ2)(δ + δ3− δ4)
6One should be aware of the validity of the freefield approximation for LFT correlators. It is valid only when the “momentum
conservation” bM = Q −?αn (for generic b) is respected.
q? c(q) = −η−2
ux2du = −1
2η−2
?
vλη
SW= −q∂
∂q
?Fδ,δn(q),(4.2)
2δ
x + O(q2).
5We thank Yuji Tachikawa for his comment on this point.
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Certainly, without quoting Polyakov’s conjecture the equality c = −∂qF0 arises directly once the stress
tensor nature of the spectral curve (∂φKS)2= 4τ in Hermitian matrix models is taken into account. Here,
?F is named classical conformal block in the pioneering paper of Zamolodchikov and Zamolodchikov [33].
?trΦ2?ǫ1= 2? q∂? qW.
Another ingredient we need is the deformed version of Matone’s relation [34, 35, 36] proposed in [37, 24, 25]:
(4.3)
For T0,4(A1), (4.2) and (4.3) together manifest λη
1
b2?Fδ,δn(q) =
SWas a deformed SW differential if there holds
?
m : bare flavor mass
1
ǫ1ǫ2W(ǫ1) ≡ lim
ǫ2→0logZNek
a,m,ǫ1,ǫ2, ? q = exp(2πiτUV)
?
,(4.4)
a : UV vev of Φ,
under q = ? q and ǫ1= η. In fact, the first equality has already been verified in [38].
SW differential of T0,N(A1). This remarkable property arises schematically from Fig. 1.
To conclude, we have checked that λη
SWfrom Baxter’s TQ equation (2.8) turns out to be the ǫ1deformed
Other Gaudin models
There are still two other Gaudin models, say, hyperbolic and elliptic ones. Let us briefly discuss the elliptic
type because it sheds light on N = 2∗T1,1(A1). Now Bethe roots satisfy the following classical Bethe Ansatz
equation:
N
?
n=1
snθ′
θ11(µk− ξn)
11(µk− ξn)
= −πiν +
?
l(?=k)
θ′
θ11(µk− µl),
11(µk− µl)
ν ∈ integer.(4.5)
Regarding it as a saddlepoint condition, we are led to the spectral curve analogous to (3.10)
x2=
?N
n=1
?
N
?
snθ′
θ11(u − ξn)
11(u − ξn)
−
K
?
k=1
θ′
θ11(u − µk)
N
?
11(u − µk)
?2
=
n=1
℘(u − ξn)η2sn(sn+ 1) +
n=1
Hnζ(u − ξn) + H0
where
Hn=
N
?
N
?
i?=n
3
?
3
?
a=1
wa(ξn− ξi)Ja
nJa
i,
H0=
n=1a=1
?
−℘
?ω5−a
?
2
?
Ja
nJa
n
+
i?=n
wa(ξi− ξn)
?
ζ
?
ξn− ξi+ω5−a
2
?
− ζ
?ω5−a
2
??
Ja
nJa
i
?
.
(4.6)
Notice that ℘(u) and ζ(u) respectively denote Weierstrass ℘ and ζfunction. Periods of ℘(u) are (see
Appendix A for wa)
ω1= ω4= 1,ω2= τ,ω3= τ + 1.
(4.7)
Hn’s (?Hn= 0) are known as elliptic Gaudin Hamiltonians [39, 40]. Since all these are elliptic counterparts
here plays the role of λη
of those in the rational XXX model, according to the logic of Fig. 3 it will be interesting to see that xdu
SWin the ǫ1deformed N = 2∗SW theory when n = 1.
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Acknowledgments
TST thanks Kazuhiro Sakai and Hirotaka Irie for encouragement and helpful comments. RY is supported
in part by GrantinAid for Scientific Research No. 23540316 from Japan Ministry of Education. NY and
RY are also supported in part by JSPS Bilateral Joint Projects (JSPSRFBR collaboration).
ADefinition of wn
In Appendix A, wnthat appears in (4.6) is defined according to [39, 40]. We choose periods of ℘(u) as in
(4.7). Weierstrass σfunction is defined by
σ(u) = σ(u;ω1,ω2)
= u
?
n,m∈Z
(n,m)?=(0,0)
?
1 −
u
nω1+ mω2
?
exp
?
u
nω1+ mω2
+1
2
?
u
nω1+ mω2
?2?
.
(A.1)
Note that σ(u) satisfies
ζ(u) =σ′(u)
σ(u),℘(u) = −ζ′(u).(A.2)
We introduce constants eaand ηarelated to ωa/2 as
ea= ℘(ωa/2),ηa= ζ(ωa/2),ςa= σ(ωa/2),(a = 1,2,3).
(A.3)
Using ηaand ςa, we define σ00, σ10and σ01as
σ00(u) =exp[−(η1+ η2)u]
ς3
σ
?
u +ω3
2
?
,
σ10(u) =exp(−η1u)
ς1
σ
?
?
u +ω1
2
?
?
,
σ01(u) =exp(−η2u)
ς2
σu +ω2
2
.
(A.4)
We define Jacobi ϑfunction as
ϑ00(u) = ϑ(u;τ) = ϑ(u)
?
ϑ01(u) = ϑ
=
∞
n=−∞
?
exp?πin2τ + 2πinu?,
u +1
2
?1
?1
?
,
ϑ10(u) = exp
4πiτ + πiu
4πiτ + πi(u +1
?
ϑ
?
u +1
?
2τ
?
?
u +1
,
ϑ11(u) = exp
2)
ϑ
2+12τ
?
,
(A.5)
from which Weierstrass σfunctions are listed as follows:
ω1exp
?η1
ω1u2
? ϑ11
?u
11(0)
ω1
?
ϑ′
= σ(u),exp
?η1
ω1u2
? ϑab
?u
ab(0)
ω1
?
ϑ′
= σab(u)(ab = 0),
(A.6)
10
Page 11
where
ϑ′
ab(t) =
d
dtϑab(t).
(A.7)
From these, wa(u) is defined as
w1(u) =
cn(u√e1− e3;
sn(u√e1− e3;
dn(u√e1− e3;
sn(u√e1− e3;
?
e2−e3
e1−e3)
?
?
e2−e3
e1−e3)
=σ10(u)
σ(u)
=ϑ′
11(0)
ϑ10(0)
ϑ10(u)
ϑ11(u),
w2(u) =
e2−e3
e1−e3)
?
?
e2−e3
e1−e3)
=σ00(u)
σ(u)
=ϑ′
11(0)
ϑ00(0)
ϑ00(u)
ϑ11(u),
w3(u) =
1
sn(u√e1− e3;
e2−e3
e1−e3)
=σ01(u)
σ(u)
=ϑ′
11(0)
ϑ01(0)
ϑ01(u)
ϑ11(u).
(A.8)
References
[1] N. A. Nekrasov and S. L. Shatashvili, “Quantum integrability and supersymmetric vacua,” Prog. Theor.
Phys. Suppl. 177 (2009) 105119. [arXiv:0901.4748[hepth]].
[2] N. A. Nekrasov and S. L. Shatashvili, “Quantization of Integrable Systems and Four Dimensional Gauge
Theories,” [arXiv:0908.4052[hepth]].
[3] N. Nekrasov, A. Rosly and S. Shatashvili, “Darboux coordinates, YangYang functional, and gauge
theory,” [arXiv:1103.3919 [hepth]].
[4] N. A. Nekrasov, “SeibergWitten Prepotential from Instanton Counting,h Adv. Theor. Math. Phys. 7
(2004) 831864. [hepth/0206161].
[5] N. Nekrasov and A. Okounkov, “SeibergWitten Theory and Random Partitions,h [hepth/0306238].
[6] Y. Zenkevich, “Nekrasov prepotential with fundamental matter from the quantum spin chain,”
[arXiv:1103.4843 [mathph]].
[7] N. Dorey, S. Lee and T. J. Hollowood, “Quantization of Integrable Systems and a 2d/4d Duality,”
[arXiv:1103.5726 [hepth]].
[8] H. Y. Chen, N. Dorey, T. J. Hollowood and S. Lee, arXiv:1104.3021 [hepth].
[9] D. Gaiotto and E. Witten, “Knot Invariants from FourDimensional Gauge Theory,” [arXiv:1106.4789
[hepth]].
[10] L. F. Alday, D. Gaiotto and Y. Tachikawa, “Liouville Correlation Functions from Fourdimensional
Gauge Theories,” Lett. Math. Phys. 91 (2010) 167197. [arXiv:0906.3219 [hepth]].
[11] D. Gaiotto, “N=2 dualities,” [arXiv:0904.2715[hepth]].
[12] B. Feigin, E. Frenkel and N. Reshetikhin, “Gaudin model, Bethe ansatz and critical level,” Comm.
Math. Phys. 166 (1994) 27. [hepth/9402022].
[13] R. J. Baxter, “Partition function of the eightvertex lattice model,” Ann. Phys. 70 (1972) 193228.
[14] R. J. Baxter, “Eight vertex model in lattice statistics and onedimensional anisotropic Heisenberg chain,”
Ann. Phys. 76 (1973) 124; 2547; 4871.
[15] A. Gorsky, I. Krichever, A. Marshakov, A. Mironov and A. Morozov, “Integrability and SeibergWitten
Exact Solution,” Phys. Lett. B355 (1995) 466474. [hepth/9505035].
[16] P. C. Argyres, M. R. Plesser and A. D. Shapere, “The Coulomb phase of N=2 supersymmetric QCD,”
Phys. Rev. Lett. 75 (1995) 16991702. [hepth/9505100].
[17] R. Donagi and E. Witten, “Supersymmetric YangMills Systems And Integrable Systems,” Nucl. Phys.
B460 (1996) 299334. [hepth/9510101].
11
Page 12
[18] A. Gorsky, A. Marshakov, A. Mironov and A. Morozov, “N=2 Supersymmetric QCD and Integrable
Spin Chains: Rational Case Nf< 2Nc,” Phys. Lett. B380 (1996) 7580. [hepth/9603140].
[19] I. M. Krichever and D. H. Phong, J. Diff. Geom. 45 (1997) 349389. [hepth/9604199].
[20] N. Seiberg and E. Witten, “ElectricMagnetic Duality, Monopole Condensation, And Confinement In N
= 2 Supersymmetric YangMills Theory,” Nucl. Phys. B 426 (1994) 1952, Erratumibid. B 430 (1994)
485486. [hepth/9407087].
[21] N. Seiberg and E. Witten, “Monopoles, Duality and Chiral Symmetry Breaking in N=2 Supersymmetric
QCD,” Nucl. Phys. B431 (1994) 484. [arXiv:hepth/9408099].
[22] E. Witten, Nucl. Phys. B 500 (1997) 342. [hepth/9703166].
[23] A. Gorsky, S. Gukov and A. Mironov, “Multiscale N=2 SUSY field theories, integrable systems and
their stringy/brane origin I,” Nucl. Phys. B 517 (1998) 409461. [hepth/9707120].
[24] R. Poghossian, “Deforming SW curve,” JHEP 1104 (2011) 033. [arXiv:1006.4822 [hepth]].
[25] F. Fucito, J. F. Morales, D. R. Pacifici and R. Poghossian, “Gauge theories on Ωbackgrounds from non
commutative SeibergWitten curves,” JHEP 1105 (2011) 098. [arXiv:1103.4495 [hepth]].
[26] R. Dijkgraaf and C. Vafa, “Matrix Models, Topological Strings, and Supersymmetric Gauge Theories,h
Nucl. Phys. B 644 (2002) 3. [hepth/0206255].
[27] R. Dijkgraaf and C. Vafa, “On Geometry and Matrix Models,h Nucl. Phys. B 644 (2002).
[arXiv:hepth/0207106].
[28] R.Dijkgraafand C.Vafa,“APerturbativeWindowintoNonPerturbativePhysics,h
[arXiv:[hepth/0208048]].
[29] R. Dijkgraaf and C. Vafa, “Toda Theories, Matrix Models, Topological Strings, and N=2 Gauge Sys
tems,” [arXiv:0909.2453 [hepth]]. [30]
[30] J. Teschner, “Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands
correspondence I,” [arXiv:1005.2846 [hepth]].
[31] E. Frenkel, “Lectures on the Langlands program and conformal field theory,” [hepth/0512172]
[32] O. Babelon and D. Talalaev, “On the Bethe Ansatz for the JaynesCummingsGaudin model,” J. Stat.
Mech. 0706 (2007) P06013. [hepth/0703124].
[33] A. B. Zamolodchikov and A. B. Zamolodchikov, “Structure constants and conformal bootstrap in Li
ouville field theory,” Nucl. Phys. B 477 577605 (1996). [hepth/9506136].
[34] M. Matone, “Instantons and recursion relations in N=2 Susy gauge theory,” Phys. Lett. B 357 (1995)
342348. [hepth/9506102].
[35] J. Sonnenschein, S. Theisen and S. Yankielowicz, “On the Relation Between the Holomorphic Pre
potential and the Quantum Moduli in SUSY Gauge Theories,” Phys.Lett. B 367 145150 (1996).
[hepth/9510129].
[36] T. Eguchi and SK. Yang, “Prepotentials of N=2 Supersymmetric Gauge Theories and Soliton Equa
tions,” Mod.Phys.Lett. A 11 131138 (1996). [hepth/9510183].
[37] R. Flume, F. Fucito, J. F. Morales and R. Poghossian, “Matone’s Relation in the Presence of Gravita
tional Couplings,” JHEP 0404 008 (2004). [hepth/0403057].
[38] T. S. Tai, “Uniformization, CalogeroMoser/Heun duality and Sutherland/bubbling pants,” JHEP 1010
107 (2010). [arXiv:1008.4332 [hepth]].
[39] E. K. Sklyanin, T. Takebe, “Algebraic Bethe Ansatz for XYZ Gaudin model,” Phys. Lett.A 219,
217225 (1996). [arXiv:qalg/9601028].
[40] E. K. Sklyanin, T. Takebe, “Separation of Variables in the Elliptic Gaudin Model,” Comm. Math. Phys.
204 (1999) 1738. [arXiv:solvint/9807008].
12