Pis’ma v ZhETF, vol. 117, iss. 8, pp. 632 - 633
© 2023
April 25
Gauge equivalence between 1 + 1 rational Calogero-Moser field theory
and higher rank Landau-Lifshitz equation
K.Atalikov+∗1), A.Zotov+∗×1)
+Steklov Mathematical Institute of Russian Academy of Sciences, 119991 Moscow, Russia
∗National Research Center “Kurchatov Institute”, 123182 Moscow, Russia
×National Research University Higher School of Economics, 119048 Moscow, Russia
Submitted 14 March 2023
Resubmitted 14 March 2023
Accepted 18
March 2023
DOI: 10.31857/S1234567823080128, EDN: wjlsjw
The 1 + 1 field generalization of the Calogero-Moser
transforms U-V pair for the field Calogero-Moser
model was proposed in [1, 2], see also [3]. The Hamilto-
model to the one for some Landau-Lifshitz type model:
nian is given by the following expression:
∮
ULL(z) = G(z)U2dCM(z)G-1(z) + k∂xG(z)G-1(z). (2)
H2dCM = dxH2dCM(x),
∑
For the N = 2 case explicit construction of the matrix
H2dCM(x) =
p2i (c - kqix) -
G(z) and the change of variables was derived in our pa-
i=1
per [4], and the Landau-Lifshitz model for GL2 rational
(
)2
∑
R-matrix was derived in [5]. The goal of this article is
1
−
pi (c - kqix)
-
to define the gauge transformation in glN case, describe
Nc
i=1
the corresponding Landau-Lifshitz type model and find
∑
∑
k4q2ixx
k3
qixqjxx - qjxqixx
explicit change of variables using relation (2).
-
+
-
4 (c - kqix)
2
qi - qj
Recently the 1 + 1 field generalization of the so-
i=1
i=j
called rational top model was suggested in [6]. It is given
∑
[
1
1
by Landau-Lifshitz type equation, i.e. the field variables
-
(c - kqix)2 (c - kqjx) +
2
(qi - qj)2
are arranged into N × N matrix S and(he Poisson
i=j
]
structure is linear: {Sij (x), Skl(y)} = N-1 Sil(x)δkj -
+ (c - kqix) (c - kqjx)2 - ck2 (qix - qjx)2 ,
(1)
)
Skj(x)δil δ(x - y). The construction of the Landau-
where x is the (space) field variable and k
∈ C
Lifshitz equation and its U-V pair is based on R-matrix
satisfying the associative Yang-Baxter equation [7, 8]:
is a constant parameter. The momenta pi and
coordinates qj are canonically conjugated fields:
Rℏ12Rη23 = Rη13Rℏ-η12 + Rη-ℏ23Rℏ13, Rxab = Rxab(za - zb).
Suppose rank(S) = 1, so that S2 = cS, c = tr(S). Then
{qi(x), pj(y)}
= δijδ(x - y). The model (1) is in-
tegrable in the sense that it has algebro-geometric
the Landau-Lifshitz equation reads:
solutions and equations of motion are represented in
the Zakharov-Shabat (or Lax or zero curvature) form:
∂tS = k-2c [S, ∂2xS]+2c [S, J(S)]-2k[S, E(∂xS)] , (3)
∂tU(z) - k∂xV (z) + [U(z), V (z)] = 0, where U-V pair
U2dCM(z), V2dCM(z) ∈ Mat(N, C) is a pair of matrix
(
)
2
2
valued functions of the fields pj (x), qj (x), j = 1, ..., N
where E(S) = tr2 r(0)12
S
,
S = 1N ⊗ S and J(S) =
and their derivatives. They also depend on the spectral
(
)
2
parameter z. Explicit expression for U-V pair can be
= tr2 m12(0)
S are defined through the coefficients
found in [1, 2]. It was argued in [3] that there exist
of R-matrix expansion in the classical limit Rℏ12(z) =
a gauge transformation G(z)
∈ Mat(N, C), which
= ℏ-11N ⊗ 1N + r12(z) + ℏ m12(z) + O(ℏ2) and r(0)12 is
the coefficient in the expansion r12(z) = z-1P12 + r(0)12 +
1)e-mail: kantemir.atalikov@yandex.ru; zotov@mi-ras.ru
+ O(z), where P12 is the permutation operator. Equa-
632
Письма в ЖЭТФ том 117 вып. 7 - 8
2023
