arXiv:1107.3756v2 [hep-th] 20 Jul 2011
Baxter’s T-Q equation, SU(N)/SU(2)N−3duality
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
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
. 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 ) on which LFT dwells is responsible for one specific SCFT called Tg,n(A1) such that the
Conformal block w.r.t. Cg,n= Instanton part of ZNek
holds. Because of ǫ1: ǫ2= b : b−1 the one-parameter version of AGT conjecture directly leads to the
semiclassical LFT at b → 0. Quote the geometric Langlands correspondence  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: email@example.com
†e-mail address: firstname.lastname@example.org
‡e-mail address: email@example.com
§e-mail address: firstname.lastname@example.org
1See also recent [6, 7, 8, 9] which investigated XXX spin-chain models along this line.
ξ1 ξ2 ξ3 ξ4
ℓ1 ξ1 ℓ2 ξ2
ℓ4 ξ4 ℓ3 ξ3
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
= 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.
2.1 Baxter’s T-Q equation and u ⇔ s symmetry
Indeed, (2.1) arises from
det?w − T(u)?= 0
detT(u) = QNf(u) =
w2− trT(u)w + detT(u) = 0,T(u) : monodromy matrix,
(u − m−
i)(u − m+
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±
i. Quote Baxter’s
t(u)Q(u) = △+(u)Q(u − 2η) + △−(u)Q(u + 2η). (2.2)
• η is Planck-like and ultimately gets identified with one of two Ω-background 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 Yang-Mills one perturbed by NfD6-branes
which introduce flavors, “another theory” means its π/2-rotated 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 quiver-type
SU(2) SCFTs T0,n(A1) discovered by Gaiotto  is hence made contact with.
Baxter’s TQ eq
Bethe Ansatz eqO(η)
Two aspects of
Baxter’s TQ eq
Gaudin spectral curve
Fig. 2: Mathematical description of Fig. 1
One may try to figure out this roughly through M-theory. In M-theory  exchanging two holomorphic
coordinates u = x4+ ix5and s = x6+ ix10(w = exp(−s/R), R: radius of M-circle x10) is pictorially done
0, 1, 2, 3
7, 8, 9
Table 1: Type IIA D6-NS5-D4 brane configuration
by a π/2-rotation (u ⇔ s) of the corresponding Type IIA NS5-D4 brane system2(see Fig. 1). In terms of
λSW that this symmetry says merely −uds ≈ sdu.
Let us refine the above argument below. Consider a quantum spin-chain 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
= LN(u − ξN)···L1(u − ξ1),(2.3)
?t(u) = AN(u) + DN(u).
Notice that the n-th 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 Yang-Baxter equation.
As far as the inhomogeneous periodic XXX spin chain is concerned, its T-Q 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),
(u − µk),
(u − ξn±?ℓnη),(2.4)
where each Bethe root µksatisfies a set of Bethe Ansatz equations (η?ℓ = ℓ):
µk− µl+ 2η
µk− µl− 2η.
Here, the η dependence enables us to carry out a semiclassical limit later on. Through
=Q(u − 2η)
+Q(u + 2η)
△−(1 − 2ηQ′
as η → 0 and ℓ stays fixed we arrive at
= w +1
2In  this symmetry has been notified in the context of Toda-chain models because two kinds of Lax matrices exist there.
which is nothing but (2.1). From now on we call λSW of (2.8) “η-deformed” SW differential λη
as in [24, 25] because
+ O(η2), Ψ =
(u − ξn)?ℓn/2. (2.9)
It is this form that makes contact with the π/2-rotated counterpart. As η → 0 the classical version of (2.5)
Two comments on (2.8) follow.
• In M-theory D6-branes 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−1-type singularity.
• Without the perturbation by flavors one sees that?λη
any tree-level potential which may bring N = 2 pure Yang-Mills to N = 1 descendants.
Surely, this intuition is noteworthy because (2.10) manifests itself as the saddle-point 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 tree-level potential there now obeys
SWreduces to a logarithm of the usual Vandermonde
measure evaluated over Bethe roots. This sounds like the standard Dijkgraaf-Vafa story [26, 27, 28] without
(x − ξn).
In other words, we are equivalently dealing with “N = 2” Penner-type matrix models which have been
heavily investigated recently in connection with AGT conjecture due to Dijkgraaf and Vafa . In what
follows, our goal is to show that λη
SWdoes reproduce the ǫ1-deformed 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 well-studied integrable model: XXX Gaudin model. Eventually, it
turns out that λη
SWnaturally emerges as the holomorphic one-form (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),