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arXiv:1107.3756v2 [hep-th] 20 Jul 2011
OCU-PHYS 354
Baxter’s T-Q equation, SU(N)/SU(2)N−3duality
and
Ω-deformed Seiberg-Witten prepotential
Kenji Muneyuki,a∗Ta-Sheng Tai,a†Nobuhiro Yonezawab‡and Reiji Yoshiokab§
aInterdisciplinary Graduate School of Science and Engineering,
Kinki University, 3-4-1 Kowakae, Higashi-Osaka, Osaka 577-8502, Japan
bOsaka City University Advanced Mathematical Institute,
3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan
Abstract
We study Baxter’s T-Q 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 T-Q 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 Nchyper-multiplets (Nc> 3). In terms of M-theory, we can interpret this as a result of
exchanging two holomorphic coordinates (u ⇔ s) of the M5-brane configuration engineering the
former SU(Nc) theory. Besides, it is seen that the corresponding Seiberg-Witten 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 Yang-Yang 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 two-dimensional Ω-background1. As a matter of fact, this mysterious
correspondence can further be extended to the full Ω-deformation 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 quiver-type SU(2) superconformal
field theories (SCFTs). In particular, every Riemann surface Cg,n (whose doubly-sheeted 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 one-parameter 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 T-Q
equation of XXX spin-chain 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) Yang-Mills with Nf = 2Nc
∗e-mail address: 1033310118n@kindai.ac.jp
†e-mail address: tasheng@alice.math.kindai.ac.jp
‡e-mail address: yonezawa@sci.osaka-cu.ac.jp
§e-mail address: yoshioka@sci.osaka-cu.ac.jp
1See also recent [6, 7, 8, 9] which investigated XXX spin-chain models along this line.
1
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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 Nchyper-multiplets (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 (Seiberg-Witten differential) and shortly
discuss XYZ Gaudin models.
2XXX spin chain
Baxter’s T-Q equation [13, 14] plays an underlying role in various spin-chain 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 low-energy 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) Yang-Mills theory with Nf fundamental hyper-multiplets, its Seiberg-Witten 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 hyper-multiplets.
Meanwhile, the meromorphic SW differential λSW = udlogw provides a set of “special coordinates” through
its period integrals. The physical prepotential FSW gets extracted henceforth.
2
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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)
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