Novikov-Veselov Symmetries of the Two-Dimensional ( ) Sigma Model
We show that the Novikov-Veselov hierarchy provides a complete family of commuting symmetries of the two-dimensional ( ) sigma model. In the first part of the paper, we use these symmetries to prove that the Fermi spectral curve for the double-periodic sigma model is algebraic. Thus, our previous c...
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| description | We show that the Novikov-Veselov hierarchy provides a complete family of commuting symmetries of the two-dimensional ( ) sigma model. In the first part of the paper, we use these symmetries to prove that the Fermi spectral curve for the double-periodic sigma model is algebraic. Thus, our previous construction of the complexified harmonic maps in the case of irreducible Fermi curves is complete. In the second part of the paper, we generalize our construction to the case of reducible Fermi curves and show that it gives the conformal harmonic maps to even-dimensional spheres. Remarkably, the solutions are parameterized by spectral curves of turning points of the elliptic Calogero-Moser system.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 006, 37 pages
Novikov–Veselov Symmetries
of the Two-Dimensional O(N) Sigma Model
Igor KRICHEVER acd and Nikita NEKRASOV bce
a) Department of Mathematics, Columbia University, New York, USA
E-mail: krichev@math.columbia.edu
b) Simons Center for Geometry and Physics, Stony Brook University, Stony Brook NY, USA
E-mail: nnekrasov@scgp.stonybrook.edu
c) Center for Advanced Studies, Skoltech, Russia
d) Higher School of Economics, Moscow, Russia
e) Kharkevich Institute for Information Transmission Problems, Moscow, Russia
Received October 19, 2021; Published online January 24, 2022
https://doi.org/10.3842/SIGMA.2022.006
Abstract. We show that Novikov–Veselov hierarchy provides a complete family of com-
muting symmetries of two-dimensional O(N) sigma model. In the first part of the paper we
use these symmetries to prove that the Fermi spectral curve for the double-periodic sigma
model is algebraic. Thus, our previous construction of the complexified harmonic maps in
the case of irreducible Fermi curves is complete. In the second part of the paper we generalize
our construction to the case of reducible Fermi curves and show that it gives the conformal
harmonic maps to even-dimensional spheres. Remarkably, the solutions are parameterized
by spectral curves of turning points of the elliptic Calogero–Moser system.
Key words: Novikov–Veselov hierarchy; sigma model; Fermi spectral curve
2020 Mathematics Subject Classification: 14H70; 17B80; 35J10; 37K10; 37K20; 37K30;
81R12
To Leon Takhtajan on occasion
of his 70 th birthday
1 Introduction
Harmonic maps of two-dimensional Riemann surface Σ to a Riemann manifoldM are of interest
both in physics and mathematics. The harmonic maps are the critical points of the Dirichlet
functional, the sigma model action
S(X) =
∫
Σ
√
hhabgij(X)∂aX
i∂bX
j dxdy =
∫
Σ
gij(X)∂Xi∂̄Xj . (1.1)
Here the map Φ: Σ → M is represented, locally, by the pullbacks Xi(x, y) = Φ∗xi, i =
1, . . . ,dimM , of the coordinate functions (xi) on M to Σ, with x, y local real coordinates
on Σ. In (1.1) (x, y) denote the real coordinates on Σ, while ∂ = ∂z, ∂̄ = ∂z̄ denote the z and
z̄-derivatives in the complex structure determined by the conformal class of the metric hab via
habdy
adyb ∝ dzdz̄. Finally, gij(X)dXidXj is a Riemann metric on the target manifold M .
This paper is a contribution to the Special Issue on Mathematics of Integrable Systems: Classical and Quan-
tum in honor of Leon Takhtajan.
The full collection is available at https://www.emis.de/journals/SIGMA/Takhtajan.html
mailto:krichev@math.columbia.edu
mailto:nnekrasov@scgp.stonybrook.edu
https://doi.org/10.3842/SIGMA.2022.006
https://www.emis.de/journals/SIGMA/Takhtajan.html
2 I. Krichever and N. Nekrasov
Remark 1.1. AssumeM is a real slice of a complex manifoldMC whose metric gij(X) dXidXj
admits an analytic continuation to MC as a holomorphic symmetric (2, 0)-tensor field gC with
positive definite real part. Likewise, we can allow for the complex metric hC = hab dy
adyb on
the worldsheet Σ, which defines two conformal structures: one by hC(∂z, ∂z) = 0, another by
hC(∂z̄, ∂z̄) = 0. For real h these structures are complex conjugate to each other (as points
in Mg). In what follows we shall make these abstract constructions quite explicit in the case of
a two dimensional torus Σ = S1 × S1.
In case M admits a group G of isometries, extending, possibly infinitesimally, to the action
of Lie(GC) on MC preserving gC, the action (1.1) can be deformed to
ShC,AC(X) =
∫
Σ
√
hCh
ab
C (gC)ij(X)∇aX
i∇bX
j dxdy, (1.2)
where ∇aX
i = ∂aX
i+(AC)
A
a V
i
A(X) is the covariant derivative in the background gauge field AC
on Σ, a connection on a principal GC-bundle P over Σ. The action (1.2) then describes the
harmonic sections of the bundle
MC ×GC P
over Σ. We call this problem a twisted sigma model. We only consider the case of flat GC-
connections,
∂a(AC)b − ∂b(AC)a + [(AC)a, (AC)b] = 0
on a topologically trivial bundle P = Σ×GC.
Remark 1.2. This paper is a continuation of [22], where the most important case is the com-
plexified twisted one. In particular, it motivates the Fermi-curve approach we also employ
presently. Nevertheless, in this paper, mostly, we study the real untwisted case.
The zero-curvature representation of the equation of motion for the two-dimensional O(N)
sigma-model, whose target manifold is the sphere M = SN−1 with the metric induced by
a standard imbedding into RN , was discovered by Pohlmeyer in [29] where an infinite series of
conservation laws was constructed. In [32] these results were extended to a wider class of models
including the principal chiral field model – the sigma model with the target space M = G, a Lie
group with the bi-invariant metric tr
(
X−1dX
)2
.
In [16] it was shown that locally the principal chiral field equations are integrable in the
following strong sense: their solutions admit a non-linear analog of the d’Alembert representation
as a superposition of functions depending only on one variable. The problem of constructing
the global solutions, i.e., solutions defined on a compact Riemann surface is much harder.
Remark 1.3. The models studied in [16, 29, 32] were the sigma models with Minkowski source.
Accordingly, the equations of motion of these models are hyperbolic. The equations of motion
of the sigma models studied in the present paper are elliptic. The equations of motion and their
zero-curvature representation are easily transformed to the elliptic case by the simple change of
the light-cone variables ξ± = x± t to the holomorphic-antiholomorphic variables z, z̄. However,
the nature of solutions in the two cases is quite different, the compactness of the source Σ playing
a key rôle.
1.1 Worldsheet is a two-torus
Our main interest is in the double-periodic two-dimensional sigma model, i.e., we take Σ to be
a two-dimensional torus T 2 = S1×S1. Its conformal structure is parameterized by the complex
Novikov–Veselov Symmetries of the Two-Dimensional O(N) Sigma Model 3
number τ , with Im τ > 0, which usually is introduced by representing Σ ≈ C/Z⊕ τZ,
Σ = R2/Z2 = {(z, z̄) ∼ (z + ωα, z̄ + ω̄α) |α = x, y},
ωx = ω̄x = 1, ωy = τ, ω̄y = τ̄ .
(1.3)
In the complexified approach we have the complex metric
hC =
(dx+ τdy)(dx+ τ̄dy)
τ2
, (1.4)
whose conformal class is parametrized by two complex numbers τ1, τ2, with Re(τ2) > 0, or,
equivalently, by
τ = τ1 + iτ2, τ̄ = τ1 − iτ2.
We shall call the real two-torus T 2 endowed with the complex metric (1.4) with Re(τ2) > 0
a Euclidean torus T 2 in what follows.
The holomorphic vector field ∂z and the antiholomorphic vector field ∂z̄ are given by
∂z =
∂y − τ̄ ∂x
2iτ2
, ∂z̄ =
∂y − τ∂x
−2iτ2
and the (C-linear) Hodge star acts on 1-forms on Σ via
⋆ (dx+ τdy) = i(dx+ τdy), ⋆ (dx+ τ̄dy) = −i(dx+ τ̄dy),
⋆ dx = −τ1dx+ τ τ̄dy
τ2
, ⋆ dy =
dx+ τ1dy
τ2
.
In what follows we shall use the notations Tx, Ty for the translations
Tx : (x, y) 7→ (x+ 1, y), Ty : (x, y) 7→ (x, y + 1) (1.5)
of the universal cover Σ̃ = R2 of Σ = Σ̃/Z2, also known as the deck transformations.
Below, in using the derivatives along Σ, we either use specifically (1, 0) and (0, 1)-derivati-
ves ∂z and ∂z̄, or use the exterior derivative
dΣ = dx∂x + dy∂y = dz∂z + dz̄∂z̄.
The differentials dwα, dk, dΩ etc. denote the (1, 0)-differentials along the Fermi-curve.
1.2 Spectral curves from zero-curvature equations
The harmonic maps of the two-torus T 2 to S3 were constructed in [13] via the zero-curvature
representation for the principal chiral SU(2) model. The latter is the compatibility condition
for the system of two linear equations(
∂z −
U(z, z̄)
λ+ 1
)
Ψ(z, z̄, λ) = 0,
(
∂z̄ +
V (z, z̄)
λ− 1
)
Ψ(z, z̄, λ) = 0 (1.6)
with U = X−1∂zX and V = X−1∂z̄X. Since the fundamental group of a torus is abelian the
operators
Bα(λ) := (T ∗
αΨ(λ))Ψ−1(λ), α = x, y
4 I. Krichever and N. Nekrasov
of monodromy of flat connection (1.6) commute, [Bx(λ), By(λ)] = 0. Therefore, the branch
points of the two-sheeted covers (for the SU(2) case the matrices Bα are of rank 2) of the
complex λ-plane defined by the characteristic equations
Rα(µα, λ) := det (µα · I−Bα(λ)) = 0 (1.7)
coincide. The key step in [13] was the proof that the number of these branch points is finite.
The hyperelliptic curve defined by these branch points is a normalization of the analytic spectral
curves (1.7).
In general, with this construction, given the hyperelliptic curve and a point on its Jacobian, we
get a quasi-periodic harmonic map of the universal cover of Σ. Solving the periodicity constraint
is a pain. Indeed, the double-periodic maps are singled out by certain conditions on periods of
certain abelian differentials of the second kind on the hyperelliptic curve. These equations are
transcendental and so hard to control that the genus of the hyperelliptic curves in the examples
found in [13] is at most three.
For further comparison with the results of that work, we recall one more observation made
in [13]: there is only one class of harmonic maps which can not be reconstructed by the algebraic-
geometrical data, and those are the ones for which the Floquet multipliers µα are constants.
These maps are the conformal maps to a totally geodesic 2-sphere in S3 which up to isometry
of S3 is the locus{
g2 = −1 | g ∈ SU(2)
}
⊂ SU(2) ≈ S3,
i.e., the big equator two-sphere. Although for such maps the spectral curve does not exist, they
can be found explicitly [1].
Remark 1.4. Conformal harmonic maps are the maps X for which the pull-back X∗(g) of
the target space metric g is conformally equivalent to the worldsheet metric h on Σ. They
are of special interest in geometry since their images are immersed minimal surfaces. A slight
generalization of conformal harmonic maps are branched conformal harmonic maps, where X∗(g)
becomes degenerate at a finite number of points.
Remark 1.5. We thank N. Hitchin for bringing to our attention the reference [6] where non-
conformal harmonic maps T 2 −→ Sn for arbitrary n were constructed from finite-dimensional
orbits of symmetries of zero-curvature equations, while in [5], using the ideas from twistor theory
and the theory of solitons it was shown that all harmonic maps, except for one special class,
can be found in terms of some finite-dimensional orbits of commuting Hamiltonian flows. The
special class harmonic maps, the ones outside the reach of the zero-curvature representation
framework, were called pseudoholomorphic in [7], super-minimal in [2] or isotropic in [11]. We
treat these maps in the second part of the paper.
1.3 Spheres from Euclidean spaces, and complexification
Let CN be equipped with a non-degenerate quadratic form q 7→ (q,q). The group GC of linear
transformations q 7→ g · q preserving (q,q) is isomorphic to O(N,C). The hypersurface MC
defined by (q,q) = 1 has the complex metric gC induced from the complexified Euclidean metric
(dq, dq) on CN . In the basis ei, i = 1, . . . , N in CN in which (q,q) looks like
(q,q) =
N∑
i=1
(
qi
)2
the real slice M = SN−1 corresponds to qi ∈ R. There are other real slices, of de Sitter, anti-de
Sitter, or Lobachevsky geometry, where some of the qi’s are purely imaginary.
Novikov–Veselov Symmetries of the Two-Dimensional O(N) Sigma Model 5
Turning on the twists breaks the O(N,C) symmetry down to its maximal torus. Given a flat
O(N,C)-connection AC on Σ = S1 × S1 one can find a basis ei such that the holonomies
gx = P exp
(∮
dx(AC)x
)
, gy = P exp
(∮
dy(AC)y
)
(the flatness guarantees the conjugacy classes of the path ordered exponentials do not depend
on y and x, respectively) are diagonal: for N = 2n,
gx(e2a−1 ± ie2a) = w±1
x,a(e2a−1 ± ie2a),
gy(e2a−1 ± ie2a) = w±1
y,a(e2a−1 ± ie2a), a = 1, . . . , n,
(1.8)
with some complex numbers wx,a, wy,a ∈ C×, a = 1, . . . , n, while for N = 2n + 1 (1.8) are
accompanied by additional four choices:
gxeN = ±eN , gyeN = ±eN .
Remark 1.6. Another deformation of the sigma-model breaking O(N,C) invariance is to dis-
entangle the quadratic forms (dq, dq) and (q,q). In this way on gets an ellipsoid (in modern
parlance, a squashed sphere) sigma model. Its point particle limit, i.e., the study of geodesics
on the ellipsoid, is connected [26] to the so-called winding string [22] case of the twisted sigma
model on a sphere. The connection in the general case is not known.
In the spherical O(N)-symmetric case, the embedding into the vector space allows to refor-
mulate the sigma model as a constrained linear sigma model:
S ∼
∫
Σ
(∂zq, ∂z̄q) + u((q,q)− 1), (1.9)
where u = u(x, y) is a double-periodic function on Σ, the Lagrange multiplier.
1.4 Fermi-curves
Our approach to solving the O(N) sigma model equations does not use their zero-curvature
representation. It is a further development of the scheme proposed in [18, 20] and extended
in [22]. Roughly, this approach can be described as the construction of integrable linear operators
with self-consistent potentials. The construction consists of two steps. The first step is to
parametrize a periodic linear operator by a spectral curve and a line bundle (divisor) on it.
The spectral curve parametrizes Bloch solutions of the linear equation. The second step of the
construction is the characterization of the spectral curves for which there exists a set of points
on the curve such that the corresponding Bloch solutions satisfy quadratic relation.
Unlike the zero-curvature representation, the linear equations
(−∂z∂z̄ + u)qi(z, z̄) = 0, i = 1, . . . , N, (1.10)
are the equations of motion following from (1.9) and not some auxiliary linear equation. The
potential u is the Lagrange multiplier enforcing the constraint that the vector q = (qi)Ni=1 ∈ CN
with the coordinates qi lies on the complexified unit sphere, i.e.,
(q,q) = 1. (1.11)
From the equations (1.10) and (1.11) it is easy to express u in terms of the solutions to the
linear equation:
u = −(∂zq, ∂z̄q). (1.12)
6 I. Krichever and N. Nekrasov
1.5 Novikov–Veselov hierarchy
In the next section we show that the Novikov–Veselov (NV) hierarchy is a symmetry of the O(N)
model. Recall, that NV hierarchy [30, 31] is the system of compatible linear equations
Hψ := (−∂z∂z̄ + u)ψ = 0, (1.13)
(∂tn − Ln)ψ = 0, (1.14)
with
Ln = Ln(z, z̄, ∂z) = ∂2n+1
z +
2n−1∑
i=1
wi,n(z, z̄)∂
i
z,
differential operator in the variable z with (z, z̄)-dependent coefficients. The compatibility con-
dition of the equations (1.13), (1.14) is the so-called Manakov’s triple equation
∂tnH = [Ln, H] +BnH, (1.15)
where Bn = Bn(z, z̄, ∂z) also a differential operator in the variable z with (z, z̄)-dependent
coefficients.
Remark 1.7. The complete set of times in the NV hierarchy contains twice as many variables:
(t, t̄) := (t1, t̄1, t2, t̄2, . . . ). The equation defining the evolution under the times t̄n have the
Manakov triple form with (Ln, Bn) replaced by the differential operators
L̄n = L̄n(z, z̄, ∂z̄), B̄n = B̄n(z, z̄, ∂z̄)
in the variable z̄.
To prove that the NV hierarchy is the hierarchy of symmetries of the O(N) sigma model
means to prove the constraint (1.11) remains invariant along the tn and t̄m evolution, given by
∂tnq
i = Lnq
i, ∂t̄mq
i = L̄mq
i.
As we shall see soon, this is highly nontrivial, yet purely local statement, i.e., not relying on any
assumptions on the global behaviour of solutions.
The Manakov triple-type equations assume the Lax form upon a restriction to the space of
zero modes of the operator H, i.e., onto the space of solutions of equations (1.13). Roughly
speaking, they preserve the spectral data associated with zero-eigenlevel of the operator H.
For operators H with doubly periodic potential u, the latter means that the flows defined by
the equation (1.15) preserve the Bloch–Floquet set Cu of the operator H: the set of pairs
p := (wx, wy) ⊂ C× × C× for which there exists a double Bloch solution ψp(z, z̄) = ψ(z, z̄; p) of
the equation (1.13):
T ∗
αψp = wα(p)ψp, α = x, y. (1.16)
In [19] it was shown that for a generic smooth periodic potential the locus Cu is a smooth
Riemann surface of infinite genus. Moreover, it was shown that the algebraically-integrable po-
tentials are dense in the space of all smooth periodic potentials. The latter are the potentials for
which the normalization Cu of Cu called the Fermi-curve is of finite genus. For such potentials Cu
is compactified by two smooth infinity points P±.
Another characterization of algebraically integrable potentials similar to the one that defines
the famous finite-gap potentials of the one-dimensional Schrödinger operator is as follows: they
Novikov–Veselov Symmetries of the Two-Dimensional O(N) Sigma Model 7
are such potentials, that among the flows given by the equations (1.15), only a finite number are
linearly independent.
The tn derivative of (1.10) with u given by (1.12) gives that the vector vn = Lnq satisfies
the linearized equation:
Dvn = 0,
D = ∂z∂z̄ +
(
q⊗ (∂z̄q)
t
)
∂z +
(
q⊗ (∂zq)
t
)
∂z̄ + (∂zq, ∂z̄q). (1.17)
The elliptic operator (1.17) is defined on T 2, i.e., its coefficients are double periodic. Therefore
ker(D) is finite dimensional. Hence, for any solution of the elliptic double periodic O(N) sigma
model the corresponding potential u (1.12) is algebraically integrable, i.e., the associated Fermi
curve Cu is of finite genus.
Remark 1.8. That result is an analog of the above mentioned key result in [13].
Since it was proved in [22] that a smooth irreducible Fermi curve of finite genus gives a solution
to the periodic O(N) sigma model if and only if there is a certain meromorphic function on it, it
is tempting to say that a periodic O(N) sigma model is algebraically integrable. Unfortunately,
this is only partially true. The periodicity constraints that such a curve should satisfy are given
in terms of periods of meromorphic differentials on the curve. They are transcendental and hard
to control.
In Section 5 we extend our construction to the case of reducible Fermi curves. It gives branched
conformal maps of two-torus to the real even dimensional spheres M = S2n, i.e., the periodic
solutions of the two-dimensional O(2n + 1) sigma model. It turns out that for the reducible
Fermi curve the periodicity constraints admit an explicit solution: the irreducible components
are the spectral curves of the elliptic Calogero–Moser system corresponding to its turning points.
We recall these notions in what follows.
Remark 1.9. It should be emphasized that the conformal maps to S2 are given by our con-
struction with nontrivial reducible Fermi curves. As it was mentioned above for such maps the
spectral curves arising in the framework of the zero-curvature representation are trivial. There-
fore, in general these two types of the spectral curves are different. The question: are they
interconnected in some special cases seems very interesting but at the moment is open.
2 Novikov–Veselov hierarchy
The NV hierarchy in its original form (1.15) is a system of equations on the potential u of the
Schrödinger operator H and coefficients of operators Ln. These are well-defined equations in
the sense that the number of equations is equal to the number of unknowns. Recently, their
Sato-type representation was proposed in [8]. Below, following the work of [21], we present
the hierarchy in yet another form, the most suitable for our purposes. Namely, as a system of
commuting flows on the space of special wave solutions of the Schrödinger equation.
Remark 2.1. Throughout this section z and z̄ are independent variables.
2.1 The phase space
First recall that a wave (a.k.a. formal Baker–Akhiezer) solution of the Schrödinger equation (1.13)
is a formal solution to (1.13) of the form
ψ(z, z̄; k) = ekz
(
1 +
∞∑
s=1
ξs(z, z̄)k
−s
)
. (2.1)
8 I. Krichever and N. Nekrasov
Substituting (2.1) into (1.13) gives a system of equations
∂z̄ξs+1 = Hξs ≡ −∂z∂z̄ξs + uξs, s = 0, 1, . . . , (2.2)
which recurrently determine all the ξs uniquely once the initial conditions χs(z) = ξs(z, 0) are
given.
Thus the space P of pairs (H,ψ), where H = −∂z∂z̄ + u and ψ is a formal wave solution is
identified with the space:
P ≃ {(u(z, z̄), χ1(z), χ2(z), . . . , χn(z), . . . )}, (2.3)
i.e., with the space of a function of two variables, and a sequence of functions of one variable.
For any formal series ψ of the form (2.1) define the dual formal series
ψ∗(z, z̄, k) = e−kz
(
1 +
∞∑
s=1
ξ∗s (z, z̄)k
−s
)
(2.4)
via the equations
res∞(ψ∗∂szψ)
dk
k
= −δs,0, s = 0, 1, . . . . (2.5)
Substitution of (2.1) and (2.4) into the equation (2.5) gives the equations recurrently defining ξ∗s
in terms of the coefficients of ψ:
ξ∗1 + ξ1 = 0, ξ∗2 + ξ2 + ξ1ξ
∗
1 + 2∂zξ1 = 0, . . . . (2.6)
Lemma 2.2. If ψ is a formal BA solution of the Schrödinger equation (1.13) then the dual BA
function solves the same equation
Hψ∗ = 0.
Proof. Taking the z̄ derivative of the equation (2.5) and using the equation (1.13) we get
res∞
(
(∂z̄ψ
∗)∂szψ
)dk
k
= ∂s−1
z u, s > 0. (2.7)
Then taking the z derivative of the equation (2.7) we get
res∞
(
(∂z∂z̄ψ
∗)∂szψ
)dk
k
= 0, s > 0. (2.8)
The homogeneous equations (2.8) imply that ∂z∂z̄ψ
∗ is equal to ψ∗ up to a k-independent factor,
i.e., ∂z∂z̄ψ
∗ = ũ(z, z̄)ψ∗. Comparing the leading terms and using the first equations in (2.2)
and (2.5) we get ũ = −∂z̄ξ∗1 = ∂z̄ξ1 = u. The lemma is proved. ■
Definition 2.3. A reflected formal BA solution of equation (1.13) ψσ is defined by
ψσ(z, z̄; k) := ψ(z, z̄;−k).
Lemma 2.2 can be thus formulated as follows: the map
s : ψ → (ψ∗)σ
is an involution of P, s2 = id.
Novikov–Veselov Symmetries of the Two-Dimensional O(N) Sigma Model 9
Definition 2.4. A self-dual formal BA solution of equation (1.13) is fixed point of s, i.e., a wave
solution that satisfies the constraint
ψσ(z, z̄; k) := ψ(z, z̄;−k) = ψ∗(z, z̄; k), (2.9)
equivalently
ξ∗s = (−1)sξs
for all s ≥ 1.
Our next goal is to show that the space P of pairs {(H,ψ)}, where H is a two-dimensional
Schrödinger operator and ψ its self-dual wave solution, i.e.,
P := Ps =
{
H,ψ |Hψ = 0, ψ∗ = ψσ
}
can be identified with the space
P ≃ {(u(z, z̄), χ1(z), χ3(z), . . . , χ2n−1(z), . . . )},
where χ2n+1(z) are the part of the data in (2.3) with odd indices.
Observe that the first equation in (2.6) is satisfied with ξ∗1 = −ξ1, the second equation
determines ξ2 = ξ∗2 in terms of ξ1, then the third equation is automatically satisfied by ξ∗3 = −ξ3.
It turns out that the same remains true for all odd s. Namely,
Lemma 2.5. Assume the equations
res∞
(
ψσ∂szψ
)dk
k
= −δs,0, s = 0, 1, . . . , 2n, (2.10)
are satisfied. Then
res∞
(
ψσ∂2n+1
z ψ
)dk
k
= 0 (2.11)
holds.
Proof. Using the identity
∂z
(
(∂izψ
σ)∂jzψ
)
=
(
(∂i+1
z ψσ)∂jzψ
)
+
(
(∂izψ
σ)∂j+1
z ψ
)
(2.12)
it is easy to show by induction in i that the equations (2.10) imply
res∞
((
∂izψ
σ
)
∂jzψ
)dk
k
= 0, 0 < i+ j ≤ 2n. (2.13)
From (2.13) with i = j = n and the identity res∞ f(k)dk = − res∞ f(−k)dk, we get
0 = ∂z
(
res∞
((
∂nz ψ
σ
)
∂nz ψ
)dk
k
)
= 2 res∞
((
∂nz ψ
σ
)
∂n+1
z ψ
)dk
k
. (2.14)
Then, by induction in i using (2.12) we get that (2.14) implies (2.11). The lemma is proved. ■
Now we are going to present the constraint (2.9) defining the self-dual wave solution in
another, equivalent, form. For that first recall, that the formal adjoint to w · ∂iz is the operator(
w · ∂iz
)∗
= (−∂z)i · w,
where w stands for the operator of multiplication by the function w(z, z̄). Below we will often
use the notion of the left action of an operator which is identical to the formal adjoint action,
i.e., the identity
(fD) := D∗f. (2.15)
10 I. Krichever and N. Nekrasov
Lemma 2.6. Let Φ denote the so-called wave operator
Φ = 1 +
∞∑
s=1
ξs(z, z̄)∂
−s
z
corresponding to the formal BA solution (2.1) of the equation (1.13). Then the constraint (2.9)
is equivalent to the operator equation
Φ∗ = ∂z · Φ−1 · ∂−1
z , (2.16)
where Φ∗ is the formal adjoint to Φ,
Φ∗ = 1 +
∞∑
s=1
(
−∂z)−s · ξs(z, z̄
)
.
Proof. By definition of Φ we have
ψ = Φekz, ψσ = Φe−kz.
Therefore, for the proof of (2.16) it is enough to show that for the function
ψ∗ =
(
e−kz∂z · Φ−1 · ∂−1
z
)
the equations (2.5) hold. The latter is an easy corollary of the identity
res∞
(
e−kzD1
)(
D2e
kz
)
dk = − res∂(D2D1) (2.17)
valid for any pseudo-differential operators D1 and D2. Indeed,
res∞
(
e−kz∂z · Φ−1 · ∂−1
z
)(
∂sz · Φekz
)dk
k
= − res∂
(
∂sz · Φ · Φ−1 · ∂−1
z
)
= − res∂ ∂
s−1
z .
The lemma is proved. ■
Corrolary 2.7. Let ψ be a self-dual formal BA function then the pseudo-differential operator
L = ∂z +
∞∑
s=1
vs(z, z̄)∂
−s
z
such that the equation
Lψ = kψ (2.18)
holds satisfy the equation
L∗ = −∂zL∂−1
z . (2.19)
To prove (2.19) note, that L = Φ · ∂z · Φ−1, then use (2.16).
Remark 2.8. The corollary 2.7 identifies the space of operators L corresponding to the self-dual
formal BA solution of the Schrödinger equation (at fixed z̄) with the phase space of the so-called
BKP hierarchy.
Novikov–Veselov Symmetries of the Two-Dimensional O(N) Sigma Model 11
2.2 The flows
Let us denote the strictly positive differential part of the pseudo-differential operator Ln by Ln
+,
i.e., if
Ln =
∞∑
i=−n
F (i)
n ∂−i
z ,
then
Ln
+ =
n∑
i=1
F (−i)
n ∂iz, Ln
− = Ln − Ln
+ = F (0)
n + F (1)
n ∂−1
z +O
(
∂−2
z
)
.
Remark 2.9. Note that this definition differs from the one used in the KP theory. There
the “+” subscript denotes the non-negative part of a pseudo-differential operator.
By definition of the residue of a pseudo-differential operator, the first leading coefficients
of Ln
− are
F (0)
n = res∂
(
Ln∂−1
z
)
, F (1)
n = res∂ Ln. (2.20)
From the equation (2.19) and the relation res∂D = −res∂D
∗ it follows that
F
(0)
2n+1 = − res∂
(
L2n+1∂−1
z
)∗
= res∂
(
∂−1
z ·
(
L2n+1
)∗)
= − res∂
(
L2n+1∂−1
z
)
= −F (0)
2n+1 = 0. (2.21)
The latter implies(
L2n+1
+
)∗
= ∂z · L2n+1
+ · ∂−1
z . (2.22)
In what follows we need the following statement (see [21, Lemma 4.3]).
Lemma 2.10. The equation
HL2n+1
+ = −F (1)
2n+1,z̄ +B2n+1H, (2.23)
where B2n+1 is a pseudo-differential operator in the variable z, holds.
Proof. Each operator D of the form D =
∑∞
i=N (a+ b∂z̄)∂
−i
z can be uniquely represented in the
form D = D1 +D2H, where D1,2 are pseudo-differential operators in the variable z. Consider
such a representation for the operator HLm = D1 +D2H. From the definition of L it follows
that HLmψ = 0. That implies D1 = 0 or the equation
HLm = D2H. (2.24)
We have the identity[
∂z̄∂z − u,Ln
+
]
= Ln
+,z,z̄ + Ln
+,z̄∂z −
[
u,Ln
+
]
+ Ln
+,z∂
−1
z · u− Ln
+,z · ∂−1
z ·H.
The first three terms are differential operators in the z variable. By definition of Ln
+ the fourth
term is also a differential operator. Therefore, the pseudo-differential operator Dn,1 in the
decomposition HLn
+ = Dn,1 +BnH is in fact a differential operator.
In the same way we get the equation
HLn
− = D̃n,1 + B̃2nH,
12 I. Krichever and N. Nekrasov
where
D̃n,1 = Ln
−,z,z̄ + Ln
−,z̄∂z −
[
u,Ln
−
]
+ Ln
−,z∂
−1
z · u. (2.25)
From the equation (2.21) it follows that the operator D̃2n+1,1 is a pseudo-differential operator
of order not greater than 0. The equation (2.24) implies HLn
+ = −HLn
− + D2H. Hence,
D2n+1,1 = −D̃2n+1,1 is a differential operator of the order 0, i.e., it is an operator of multiplication
by a function. This function is easily found from the leading coefficient of the right-hand side
of the equation (2.25). Direct computations give the equation (2.23). The lemma is proved. ■
Now we are ready to define NV hierarchy explicitly.
Theorem 2.11. The equations
∂tnu = ∂z̄F
(1)
2n+1, (2.26)
∂tnψ =
(
L2n+1
+ − k2n+1
)
ψ (2.27)
define a family of commuting flows on the space P of self-dual formal BA solutions of the
Schrödinger operators.
Proof. From (2.18) and definition of the operator L2n+1
+ it follows that(
L2n+1
+ − k2n+1
)
ψ = −L2n+1
− ψ.
The equation (2.21) implies that the operator L2n+1
− is of order at most −1. Hence, the equa-
tion (2.27) is equivalent to a well-defined system of equations on the coefficients ξs of the formal
BA function ψ.
In turn, the equation (2.23) implies that the equation (2.26) is equivalent to the operator equa-
tion (1.15) with Ln = L2n+1
+ , which is a compatibility condition for the Schrödinger equation and
the equation (2.27). Hence, the vector field defined by the right-hand sides of (2.26) and (2.27)
is tangent to the space of formal Baker–Akhiezer solutions of the Schrödinger equation. Let us
show that it is tangent to the space P of selfdual solutions, i.e., preserve constraint (2.9).
From the equation (2.27) and the same equation with k replaced by −k it follows that
∂tn
(
ψσ
(
∂szψ
))
= ψσ
(
∂sz · L2n+1
+ ψ
)
+
((
L2n+1
+ ψσ
)
∂szψ
)
. (2.28)
It is easy to show that in conjunction with the self-duality ψ∗ = ψσ the equation (2.5) implies
res∞
((
∂izψ
σ
)(
∂jzψ
)dk
k
)
= 0, i+ j > 0.
The coefficients of the operator L2n+1
+ are k-independent. Therefore, from the equations (2.28)
and (2.21) we deduce
∂tnres∞
(
ψσ
(
∂szψ
)dk
k
)
= 0, s ≥ 0,
i.e., that ∂tn is tangent to P.
The commutativity of NV flows is easy to demonstrate. Indeed, (2.27) implies[
∂tn − L2n+1
+ , ∂tm − L2m+1
+
]
ψ = 0. (2.29)
The operator in the left-hand side of the equation (2.29) is a differential operator in the variable z.
From the very form of ψ it follows that if such an operator annihilates it, then it itself must be
zero. The theorem is proved. ■
Novikov–Veselov Symmetries of the Two-Dimensional O(N) Sigma Model 13
2.3 Integrals
A direct corollary of the Schrödinger equation for a self-dual BA function is the equation
∂z̄
(
ψσψz − ψσ
zψ
)
+ ∂z
(
ψσψz̄ − ψσ
z̄ψ
)
= 0,
which is nothing but the conservation
dΣ ∗ j = 0 (2.30)
of the current
j = ψσdΣψ − ψdΣψ
σ. (2.31)
In other words the 1-form jzdz − jz̄dz̄, where
jz := ψσ∂zψ − (∂zψ
σ)ψ = 2
(
k +
∞∑
n=0
Jnk
−2n−1
)
, (2.32)
jz̄ := ψσ∂z̄ψ − (∂z̄ψ
σ)ψ = 2
∞∑
n=0
J̄nk
−2n−1 (2.33)
is closed. The current (2.31) is the Noether current corresponding to the maximal commuta-
tive subalgebra of an infinite orthogonal group, acting on the formal BA functions by linear
transformations, preserving the res∞ dk/kψσψ pairing, namely,
J(k1, k2) = ψσ(k1)dΣψ(k2)− dΣψ
σ(k1)ψ(k2) = −J(−k2,−k1)
obeys
dΣ ⋆ J(k1, k2) = 0, dΣJ(k1, k2) + res∞
dk
k
J(k1, k) ∧ J(k, k2) = 0.
By definition (2.15) of the left action of a differential operator D we have the identity
(hD)f = h(Df) +
d−i∑
i=1
∂iz
(
h
(
D(i)f
))
,
where d is the order of D and D(i) is differential operator of degree d− i whose coefficients are
explicit differential polynomials in the coefficients of the operator D. Therefore,
jz =
(
e−kz∂zΦ
−1∂−1
z
)(
∂zΦe
kz
)
+
(
e−kz∂zΦ
−1
)(
Φekz
)
= 2k + ∂zQ
(1), (2.34)
where the coefficients of the series
Q(1) =
∞∑
s=0
Q(1)
n k−2s−1
are universal polynomials in the coefficients of the self-dual BA function.
For further use, let us show that the coefficient Jn of the series (2.32) is equal to the func-
tion F
(1)
2n+1 in the right-hand side of equation (2.26). Indeed, from (2.16), (2.18), (2.17) and (2.21)
it follows that
2Jn = resk
((
ψσL2n+1
)
ψz − ψσ
z
(
L2n+1ψ
))
k−1dk
= res∂
(
Ln + ∂zLn∂−1
z
)
= 2F
(1)
2n+1. (2.35)
We conclude this section by proving that Jn are densities of the NV hierarchy integrals.
14 I. Krichever and N. Nekrasov
Lemma 2.12. The equations
∂tnJn = ∂zQn,
where Qn are explicit differential polynomials in the coefficients ξs of the self-dual BA solution
of the Schrödinger equation, hold.
Proof. The equation (2.27) and the same equation for ψσ with k replaced by −k imply
∂tnJ = ψσ
(
∂z
(
L2n+1
+ ψ
))
+
(
ψσ
(
L2n+1
+
)∗)
(∂zψ)
+
(
ψσ
(
L2n+1
+
)∗
∂z
)
ψ +
(
ψσ∂z
)(
L2n+1ψ
)
.
Therefore, from (2.35) and (2.22) it follows that ∂tnJ = ∂zQn where the coefficients of Qn
are explicit polynomials in the coefficients ξs of the self-dual BA solution ψ. The lemma is
proved. ■
3 The NV flows as symmetries of O(N) model
Our next goal is to show that the NV hierarchy defines a set of commuting symmetries of
the O(N) sigma model.
First observe that if the equations(
∂jzq, ∂
j
zq
)
= 0, 0 < j < m, (3.1)
hold then (1.10) implies ∂z̄
(
∂mz q, ∂mz q
)
= 0.
We will call the solutions of the O(N) model satisfying (3.1) wm-harmonic. Notice, that for
m > 1 they are conformal. In this way the condition of being wm-harmonic defines a filtration
on the space of conformal maps.
The equations of motion (1.10), (1.12) of the model are invariant under a conformal change of
variables z 7→ f(z), z̄ 7→ f̄(z̄). Hence if for a wm-harmonic map
(
∂mz q, ∂mz q
)
̸= 0, then without
loss of generality we may assume that the equation(
∂mz q, ∂mz q
)
= (−1)m+1 (3.2)
holds.
Theorem 3.1. Let q(z, z̄) be a complex solution of the O(N) sigma model obeying (3.1), (3.2).
Then there exists a unique up to multiplication by (z, z̄)-independent factor ρ(k), such that
ψ 7−→ ψρ(k), ρ(k) = exp
( ∞∑
s=1
ρsk
−2s+1
)
(3.3)
is a self-dual formal BA solution ψ of the Schrödinger equation with the potential u given
by (1.12) such that the constraint (1.11) is invariant under the commuting flows
∂tnq = L2n+1
+ q (3.4)
with L defined by (2.18).
Remark 3.2. Note, that the operator L defined by (2.18) is invariant under the transforma-
tion (3.3). Hence, the right-hand side is uniquely defined by q.
Novikov–Veselov Symmetries of the Two-Dimensional O(N) Sigma Model 15
Proof. Consider the moments defined by the formula
Ti(z, z̄) :=
(
q, ∂izq
)
, i = 0, 1, . . . .
The original constraint (1.11) translates to T0 = 1. By assumptions (3.1) (3.2), the next moments
equal Ti = 0, 1 < i < 2m, T2m = −1.
Taking the z̄ derivative of Ti and using the equation (1.10) we conclude that the moments
satisfy the triangular system of linear equations
∂z̄Ti = ∂z∂z̄Ti−1 − ∂z
(
q, ∂i−2
z (uq)
)
+
(
q,
[
∂i−1
z , u
]
q
)
= ∂z
(
∂z̄Ti−1 −
i−2∑
l=0
(
i− 2
l
)
Tl∂
i−l−2
z u
)
+
i−2∑
l=0
(
i− 1
l
)
Tl∂
i−1−l
z u, i > 2. (3.5)
Therefore, the moments Ti are uniquely determined by the initial conditions Ti(z, 0) for (3.5).
Now, define the Fermi-moments
f
[m]
i := res∞
(
ψσ
(
∂izψ
) dk
k2m+1
)
.
Lemma 3.3. Under the assumptions of the theorem there is a unique up to the transforma-
tion (3.3) self-dual BA solution ψ of the Schrödinger equation (1.10) such that for i > 1 the
equations
Ti = f
[m]
i (3.6)
hold.
Proof. The derivation of (3.5) uses only the fact that the corresponding quantities are given
by some bilinear form on the pair of solutions of the Schrödinger equation, i.e., the func-
tions f
[m]
i (z, z̄) defined in (3.6) for any ψ satisfy the same equations (3.5). Therefore the
equations (3.6) hold identically in (z, z̄) if they are satisfied at (z, 0). We now show by in-
duction that the latter condition allows to recover the functions χ2n−1(z) defining the self-dual
BA function.
We know from the equations (3.1) and (3.2) that the assumption holds for n = m. Now
assume the equation (3.6) holds for all i ≤ 2n − 1 for a self-dual BA function with fixed
(χ1, . . . , χ2n−3). Then
gi,j :=
(
∂jzq, ∂
i
zq
)
− res∞
dk
k2m+1
(
∂jzψ
σ
)(
∂izψ
)
= 0, i+ j ≤ 2n− 1
and
gn−i,n+i := (−1)ign,n.
From (3.5) with i = 2n (and the same equation for f
[m]
2n ) it follows that under the induction
assumption ∂z̄(g0,2n) = 0. Now we are going to show that with a proper choice of ∂zχ2n−1 the
equation (3.6) with i = 2n holds, i.e., g0,2n = 0.
Indeed, the substitution of (2.1) into the definition of f
[m]
2n gives
f
[m]
2n = 2ξ2n +R(ξ1, . . . , ξ2n−2),
16 I. Krichever and N. Nekrasov
where R is some explicit differential polynomial of its arguments. Recall, that the self-duality
of ψ gives the expression of ξ2n in terms of ξ1, . . . , ξ2n−1 of the form
0 = 2ξ2n − 2ξ1ξ2n−1 + ∂zξ2n−1 + R̃(ξ1, . . . , ξ2n−2), n > 1,
where R̃ is also a differential polynomial of its arguments.
Equation (3.6) with i = 2n restricted onto (z, 0) is equivalent to the first order differential
equation for the function χ2n−1 which defines it uniquely if the initial condition χ2n−1(0) is
given. The ambiguity in the choice of the latter corresponds to the transformation (3.3).
If equation g0,2n = 0 is satisfied then using the equation
∂zgn,n = 2gn,n+1
we get that g0,2n+1 = 0. The induction step is completed and the lemma is proved. ■
The theorem statement that the constraint (1.11) in invariant under the flows (3.4) is equiv-
alent to the equations
(
q,L2n+1
+ q
)
= 0. The latter is an easy corollary of (3.6). Indeed, since
the coefficients of L2n+1
+ are k independent equations (3.6) imply(
q,L2n+1
+ q
)
= res∞
(
ψσ,L2n+1
+ ψ
)
k−2m−1dk
= res∞
(
ψσ, ψ
)
k2n−2mdk + res∞
(
ψσ,L2n+1
− ψ
)
k−2m−1dk = 0.
For the proof of the last equation it is enough to note that the first term has no residue since
it is an odd differential. The second term is of order O
(
k−2m−2
)
and hence also has no residue.
The theorem is proved. ■
4 O(N) sigma model on a two-torus
In the previous section, the Novikov–Veselov hierarchy was defined as a system of commuting
flows on the space of self-dual solutions to the Schrödinger equations on a plane R2. Under the
assumption that the potential is periodic in one of the variables, i.e., is defined on R1 × S1, the
hierarchy can be defined as a system of flows on the space of the Schrödinger operators them-
selves. The idea of the corresponding construction goes back to [23] where the KP hierarchy
was defined as a set of commuting flows on the space of functions of two variables. The starting
point of the construction is an observation that for periodic potentials a unique formal BA func-
tion can be singled out by the condition that it is Bloch, i.e., an eigenvector of the monodromy
around S1. We refer the reader to [21, 23] where the case of two independent variables z and z̄
was considered.
In this section, we present a necessary modification of this construction to the case where the
potential u is a doubly periodic function,
u =
∑
(m,n)∈Z2
ûm,ne
2πi(mx+ny),
or, in terms of the z = x+ τy, z̄ = x+ τ̄ y coordinates:
u (z + ωα, z̄ + ω̄α) = u(z, z̄),
for α = x, y, cf. (1.3).
Novikov–Veselov Symmetries of the Two-Dimensional O(N) Sigma Model 17
4.1 The formal Bloch solutions
Lemma 4.1. Let u(z, z̄) be a double periodic function. Then there is a unique up to the trans-
formation (3.3) self-dual BA formal solution of the Schrödinger equation (1.10) of the form
ψ(z, z̄, k) = ekz+ℓ(k)z̄
(
1 +
∞∑
s=1
ζs(z, z̄)k
−s
)
(4.1)
with double periodic coefficients ζs, i.e.,
ζs
(
z + ωα, z̄ + ω̄α
)
= ζs(z, z̄), α = x, y, (4.2)
and where ℓ(k) is a formal series,
ℓ(k) =
∞∑
s=1
ℓsk
−s. (4.3)
Proof. Substitution of the equation (4.1) into the equation (1.10) gives
∂z̄ζs+1 = −ℓs+1 + uζs − ∂z∂z̄ζs −
s∑
i=1
ℓiζs−i, s = 0, 1, . . . . (4.4)
Suppose that ζj and ℓj with j ≤ s are known. The inhomogeneous first order elliptic equa-
tion (4.4) has a double periodic, i.e., satisfying (4.2), solution iff the integral of its right-hand
side vanishes, thus determining the constant ℓs+1:
ℓs+1 =
∫
Σ
dxdy
(
uζs −
s∑
i=1
ℓiζs−i
)
.
Once ℓs+1 is found, the double periodic solution ζs+1 is defined uniquely up to an additive
constant. The ambiguity in the definition of ζs+1 corresponds to the transformation ψ → ψρ̃(k),
where ρ̃(k) is a regular series in the variable k with constant coefficients.
Using the fact the both the dual ψ∗ and reflected ψσ BA functions are solutions of (1.10)
with double periodic coefficients we get the equation
ψσ = h(k)ψ∗,
for some function h(k). Under the transformation ψ 7→ ρ̃(k)ψ the identifying function h(k) gets
transformed to
h̃(k) = ρ̃(−k)−1ρ̃−1(k)h(k).
Thus, by taking ρ̃ to solve the equation ρ̃(k)ρ̃(−k) = h(k) we can transform ψ having the
form (4.1) to a self-dual BA function. By construction it is unique up to the transformation (3.3).
The lemma is proved. ■
Corrolary 4.2. The self-dual solution of the Schrödinger equation defined in Lemma 4.1 satisfies
the following monodromy properties
ψ(z + ωα, z̄ + ω̄α, k) = wα(k)ψ(z, z̄, k)
with
wx = ek+ℓ(k), wy = eτk+τ̄ ℓ(k).
18 I. Krichever and N. Nekrasov
Theorem 4.3. The equations
∂tnu = ∂z̄F
(1)
2n+1 (4.5)
with F
(1)
2n+1 = res∂ L2n+1
+ , where L is defined by the Bloch self-dual solution of (1.10), are com-
muting flows on the space of double periodic functions.
Moreover if u is given by (1.12) for some solution of O(N)-sigma model then the con-
straint (1.11) and hence the equation (1.12) are preserved by the flows
∂tnq = L2n+1
+ q.
Proof. By the Corollary 4.2 the coefficients of the operator L corresponding to the self-dual
BA solution defined in Lemma 4.1 are double periodic functions. Hence, the Bloch self-dual BA
solution are invariant under the flows (2.27), and (4.5) is just a well-defined restriction of the
flows described by Theorem 2.11.
The last statement of the theorem is a direct corollary of the following lemma.
Lemma 4.4. Let q be a solution of the O(N) sigma model on Euclidean T 2 such that the
equations (4.3) and (3.2) hold. Then there exists a unique formal series
E(k) =
∞∑
s=m
esk
−2s,
such that equations
Ti = fEi := res∞
(
ψσ
(
∂izψ
)
E(k)
dk
k
)
, (4.6)
where ψ is the self-dual BA solution defined in the Lemma 4.1, hold.
Remark 4.5. Before presenting the proof of the lemma it is instructive to compare it with the
Lemma 3.3 which was used above to single out a special self-dual solution of the Schrödinger
equation for which equations (3.6) hold. In the double periodic case that special solution is
singled out in Lemma 4.1 by the Bloch properties and in the proof of Lemma 4.4 below we will
demonstrate the validity of (4.6) for that solution. In essence we are arriving from two different
starting points at the same self-dual BA solution of the Schrödinger equation as well as the same
equations (3.6) and (4.6). In view of this, additional comments are required to clarify that, at
first glance, the latter equations have different forms.
Note that the form of the formal BA function is invariant under a change of the local coordi-
nate k 7→ k+O
(
k−1
)
. In the double periodic case the local coordinate is fixed by the periodicity
conditions (4.2) while in the Lemma 3.3 it is fixed by the condition that in this coordinate the
series E(k) equals k−2m.
Proof. Suppose that (4.6) is satisfied for 1 ≤ i ≤ 2n − 1 for the Bloch self-dual BA function
and for a series E(k) with some known (em = 1, em+1, . . . , en−1).
If equation (4.6) holds for all i ≤ 2n− 1, then
gi,j :=
(
∂jzq, ∂
i
zq
)
− res∞
((
∂jzψ
σ
)(
∂izψ
)
E(k)
dk
k
)
= 0, i+ j ≤ 2n− 1,
and
gn−i,n+i := (−1)ign,n.
Novikov–Veselov Symmetries of the Two-Dimensional O(N) Sigma Model 19
From (3.5) with i = 2n (and the same equation for fE2n) it follows that under the induction
assumption ∂z̄(g0,2n) = 0. Hence it is a constant. Therefore we can make this constant to be
equal zero, i.e., g0,2n = 0 by proper choice of the coefficient en.
If equation g0,2n = 0 is satisfied then using the equation
∂zgn,n = 2gn,n+1
we get that g0,2n+1 = 0. The induction step is completed and the lemma is proved. ■
The theorem is proved. ■
4.2 The algebraic spectral curve
Let ψ be the formal Bloch self-dual solution of the Schrödinger operator with the potential u
defined by (1.12) from a complex solution q(z, z̄) of the O(N) sigma model on Σ = T 2. Our
next goal is to show that ψ is a common eigenfunction for a ring A of commuting differential
operators in one variable, which we can take to be z, or z̄. Therefore, it is an expansion near
a marked point of the non-formal Baker–Akhiezer function on the spectral curve of A.
Let q(z, z̄, t), t = (t1, t2, . . . ) be a an orbit of q(z, z̄, 0) under the flows of the NV hierarchy.
As we have shown above, the constraint (1.11), and thus the equation (1.12) are invariant under
the flows. Taking the tn derivative of (1.10) and using the equation
∂tnu = −
(
∂z(L2n+1
+ q), ∂z̄q
)
−
(
∂zq, ∂z̄
(
L2n+1
+ q
))
(4.7)
we get the equation (1.17) with Ln = L2n+1
+ . The space of solutions to an elliptic equation on Σ
is finite dimensional. Hence for all but a finite number of integers n there are constants cn,m
such that for the operator
L̃n := L2n+1
+ +
n−1∑
i=0
cn,mL2m+1
+ , (4.8)
the equation
L̃nq = 0 (4.9)
holds. We call the finite set I of indices n for which there are no such constants the gap set.
Lemma 4.6. The operators L̃n defined above commute with each other[
L̃n, L̃m
]
= 0, n,m /∈ I.
Proof. Consider the linear combination of the vector fields ∂tm
∂t̃n = ∂tn +
n−1∑
m=0
cn,m∂tm
with cn,m as in equation (4.8). From the equations (4.7), (4.9) it follows that ∂t̃nu = 0.
Then (2.26) implies ∂z̄F̃n = 0, where
F̃n := F
(1)
2n+1 +
n−1∑
m=0
cn,mF
(1)
2m+1
20 I. Krichever and N. Nekrasov
with F
(1)
2n+1 given by the equation (2.20). Therefore F̃n is a constant. From the equation (2.34)
it follows that F̃n is z-derivative of some double periodic function. The latter implies F̃n = 0.
Then from equation (2.23) it follows that
H(Lnψ) = 0. (4.10)
The coefficients of the operator Ln are double periodic functions. Therefore, from (4.10) it
follows that k−2n+1Lnψ is a formal Bloch solution of the Schrödinger equation. From (2.22)
it follows that it is a self-dual Bloch solution. Then the uniqueness up to multiplication by
a constant series of the latter implies the equation
L̃nψ = an(k)ψ, an(k) = k2n+1 +
∞∑
s=−n
an,sk
−2s−1, (4.11)
where an,s ∈ C are some constants.
The equation (4.11) implies that
[
L̃n, L̃m
]
ψ = 0. From the form of ψ it easily follows that ψ
is not in a kernel of any nonzero ordinary differential operator in the variable z. The lemma is
proved. ■
Recall the fundamental fact of the theory of commuting linear ordinary differential operators
[3, 4, 14, 15, 25]:
Lemma 4.7 (Burchnall–Chaundy). Let Ln and Lm be commuting ordinary linear differential
operators of orders n and m, respectively. Then there exists a polynomial R in two variables
such that the equation
R(Ln, Lm) = 0 (4.12)
holds.
Recall that the affine curve defined by (4.12) is compactified by one smooth point P+. The
corresponding algebraic curve Γ is called the spectral curve. The maximal commutative ringAz of
ordinary differential operators in the z-variable containing the operators Ln and Lm is isomorphic
to the ring A(Γ, P+) of meromorphic functions on Γ having the only pole at P+.
As shown in [14, 15] for a generic pair of commuting operators of co-prime orders the spectral
curve is smooth and the common eigenfunction ψ of the commuting operators is the Baker–
Akhiezer function:
10. As a function of p ∈ Γ it is meromorphic on Γ \ P+ with z-independent divisor D of poles
of degree equal to the genus of Γ.
20. In the neighborhood of P+ it has the form (2.1).
A priori, there are two spectral curves in the problem under consideration: the one, which
we have just discussed, and the other, with a smooth marked point P−, corresponding to the
commuting differential operators in the variable z̄. In fact, these spectral curves coincide. Indeed,
the operators L̄n satisfy the equations
[
L̄n, H
]
= B̄nH where B̄n is a differential operator in the
variable z̄. Then from the uniqueness of the Bloch BA solution up to a transformation of the
form (3.3) the equation
L̄nψ = ān(k)ψ, ān(k) =
∞∑
s=0
ān,sk
−2s−1 (4.13)
follows. The series ān(k) in (4.13) is an expansion in the neighborhood of P+ of the function
ān ∈ A(Γ, P−) having pole of order n at the second marked point P−.
Novikov–Veselov Symmetries of the Two-Dimensional O(N) Sigma Model 21
Definition 4.8. A Schrödinger operator is called algebraic-geometric (finite-gap) if it is sta-
tionary for all but a finite number of the NV hierarchy flows.
Theorem 4.9. The spectral curve Γ corresponding to an algebraically integrable operator H is
a curve with involution σ fixing the smooth marked points P±. In the generic case when Γ is
smooth: the points P± are the only fixed points of σ; the common eigenfunction ψ(z, z̄; p), p ∈ Γ,
of the operators Ln, L̄m satisfying the equation Hψ = 0 is the two-point BA function on an
algebraic spectral curve Γ, i.e.,
10. In the neighborhoods of P± it has an essential singularity of the form
ψ+(z, z̄; p) = ek+z
( ∞∑
n=0
ξ+n (z, z̄)k
−n
+
)
, p→ P+,
ψ−(z, z̄; p) = ek−z̄
( ∞∑
n=0
ξ−n (z, z̄)k
−n
−
)
, p→ P−
(4.14)
with
ξ±0 = 1.
20. Outside marked points P± it is meromorphic with (z, z̄)-independent divisor of poles D
satisfying the constraint
D +Dσ = K + P+ + P−, (4.15)
where K is the canonical class, i.e., the equivalence class of the zero divisor of a holomor-
phic differential on Γ.
The involution σ of Γ is just an avatar of the self-duality of the formal Bloch solution of the
Schrödinger equation or equivalently of the equation (2.22). The statement that in a generic
case Γ is smooth and P± are the only fixed points of σ is corollary of the NV reconstruction
of the algebraically integrable potentials [30, 31] from a smooth algebraic curve with involution
having two fixed point and a divisor D satisfying (4.15).
The proof of the constraint (4.15) follows the line of arguments in the proof of Lemma 2.3
in [18] (see more details in [22]). The Γ-derivative dψ is a solution of the same Schrödinger
equation (1.10). Define the new current
β = ψσdΣ(dψ)−
(
dΣψ
σ
)
dψ,
a 2-form on Σ× Γ, with one leg along Σ, one leg along Γ. From (1.10) we derive
dΣ ⋆ β = 0. (4.16)
Since ψ is a Bloch function, its Γ-differential has the following monodromy properties (here
q ∈ Γ is a point on the Fermi-curve, not to be confused with the vector q ∈ CN ):
dψ(z + ωα, z̄ + ω̄α, q) = wα(q)(dψ(z, z̄, q) + dpα(q)ψ(z, z̄, q)),
where α = x, y,
dpα := d logwα,
or, in terms of the currents β and j:
T ∗
α(⋆β)− ⋆β = dpα ∧ ⋆j, (4.17)
22 I. Krichever and N. Nekrasov
where Tα, α = x, y, are the deck transformations (1.5). Now, integrate (4.16) over a fundamental
domain in Σ̃ for Z2, use Stokes theorem to write it as an integral over the boundary, and
use (4.17), to write
dpy
∮
A
⋆j − dpx
∮
B
⋆j = 0. (4.18)
In components (4.18) reads
dpy⟨jz − jz̄⟩x = dpy⟨τjz − τ̄ jz̄⟩y,
or, equivalently
(τ1⟨jx⟩x − ⟨jy⟩x)dpy = (τ τ̄⟨jx⟩y − τ1⟨jy⟩y)dpx, (4.19)
where we denote by ⟨. . . ⟩α the averages
⟨O⟩x(y) =
∫ 1
0
dxO(x, y), ⟨O⟩y(x) =
∫ 1
0
dyO(x, y).
A priori, in the equation (4.19) the averages are taken at the specific values of y and x, respec-
tively, corresponding to the boundaries of the chosen fundamental region. However, the current
conservation (2.30) implies the contour integral
∮
γ ⋆j does not change under the homotopy of γ.
Thus, the specific linear combinations of ⟨jβ⟩α with α, β = x, y we see in (4.19) are independent
of x, y.
The equation (4.19) implies that the zeroes of the meromorphic function ⟨τ τ̄ jx − τ1jy⟩y are
the zeros of the differential dpy, while the zeros of the meromorphic function ⟨τ1jx − jy⟩x are
the zeros of the differential dpx. Hence, the differential
dΩ :=
−2iτ2dpx
⟨τ1jx − jy⟩x
=
−2iτ2dpy
⟨τ τ̄ jx − τ1jy⟩y
is holomorphic on Γ away of the marked points P±, where it has simple poles with residues ∓1.
Its zeros are the poles of the functions ψ and ψσ. That completes the proof of (4.15).
Remark 4.10. We call the divisors D satisfying (4.15) admissible. They are parameterized by
points of the Prym P(Γ) variety which is the subvariety of the Jacobian J(Γ) that is odd with
respect to involution of J(Γ) induced by the involution σ of the curve.
Remark 4.11. For singular spectral curves the differential dΩ is a meromorphic section of the
dualizing sheaf with simple poles at P±, i.e.,
dΩ ∈ H0(Γ, ωΓ + P+ + P−). (4.20)
In more practical terms (4.20) means that the preimage of dΩ on the normalization of a singular
curve may have simple poles at the preimages of nodes with opposite residues.
4.3 The periodicity constraint
The reconstruction of an algebraically integrable potential u(z, z̄) from a smooth algebraic
curve Γ with involution having two fixed points P± and an admissible divisor D gives in gen-
eral u that is a quasi-periodic function of its arguments. The curves corresponding to double
periodic functions are singled out as follows.
It follows from the non-degeneracy of the imaginary part of the matrix of b-periods of normal-
ized holomorphic differentials that for a curve Γ with a fixed first jet
[
k−1
±
]
1
of local coordinates
Novikov–Veselov Symmetries of the Two-Dimensional O(N) Sigma Model 23
at the two marked points P± there exist unique meromorphic differentials dpα, with α = x, y
having a pole of order 2 at P+ of the form
dpx = dk+
(
1 +O
(
k−2
+
))
, dpy = τdk+
(
1 +O
(
k−2
+
))
,
and a pole of order 2 at P− of the form
dpx = dk−
(
1 +O
(
k−2
−
))
, dpy = τ̄dk−
(
1 +O
(
k−2
−
))
,
holomorphic away from P±, and such that their periods are imaginary, i.e., Re
(∮
cdpα
)
= 0 for
any 1-cycle c on Γ. The locus of spectral curves corresponding to the double periodic potentials
is singled out by the constraint that the periods are integer multiple of 2πi,
nα,c :=
1
2πi
∮
c
dpα ∈ Z, ∀c ∈ H1(Γ,Z). (4.21)
Indeed, if equations (4.21) hold then the functions wα(p) =
∫ p
dpα are single valued on Γ. Then
the uniqueness of the Baker–Akhiezer function implies
ψ(z + ωα, z̄ + ω̄α; pt) = wα(p)ψ(z, z̄; p), p ∈ Γ,
since both sides of the equation have the same analytical properties on Γ.
Choosing a basis of cycles on Γ one can extend it to a basis of cycle on the curves in a neigh-
borhood of G. Then for fixed set of nα,c equations (4.21) identify the spectral curves of periodic
potentials with the level set of 4g real analytic functions on the moduli space of curves with
fixed first jets of local coordinates near two marked points.
A smooth genus g algebraic curve with involution having 2n + 2 fixed points is uniquely
defined by a factor-curve Γ0 = Γ/σ and a choice of 2n+ 2 points of it. Hence the space of such
curves is of dimension 3g0+2n− 1, where g0 is the genus of Γ0. The Riemann–Hurwitz formula
implies g = 2g0 + n. If the first jets of local coordinates near two fixed points of involution are
odd, i.e., σ∗dpα = −dpα. Hence, their periods over cycles that are even with respect σ vanishes.
The latter implies that the locus of the algebraic data
{
Γ, σ, P±,
[
k−1
±
]
1
}
, with σ having 2(n+1)
fixed points, corresponding to periodic potentials is of dimension g0 + 1.
Let Sg0,n of curves Γ be the locus of algebraic geometrical data satisfying periodicity con-
straints for some fixed set n = {nα,c}. If we choose a basis of a and b cycles on G0 with the
canonical matrix of intersection, then the (local) coordinates on Sg0,n can be defined similarly
to those for the families of the Seiberg–Witten curves, namely
A0 = resP+ pxdpy, Ai =
∮
π∗(ai)
pxdpy, i = 1, . . . , g0, (4.22)
where π : Γ → Γ0 is the projection. Although the Abelian integral px of dpx is multi-valued
the expressions (4.22) are well-defined. Indeed, a shift of px by a constant does not change A0
since dpy has no residue. It does not change Ai either, since dpy is odd with respect to the
involution σ while the cycle π∗(ai) is even.
Note that in [22] it was shown that
A0 =
∫
Σ
u(z, z̄)dz ∧ dz̄.
We conclude this section by identifying algebraic spectral curve with the Fermi curve. More
precisely,
Theorem 4.12 ([19, Theorem 3.1]). Let Γ be a smooth spectral curve corresponding to a double
periodic potential u then it is a normalization of the Bloch–Floquet locus Cu defined in (1.16),
i.e., there exits a map Γ → Cu which is one-to-one outside the preimage of the set of singular
points of Cu.
24 I. Krichever and N. Nekrasov
5 w∞-harmonic maps to spheres
In this section we generalize the construction of Novikov and Veselov to the case of reducible
spectral curves, and show that it provides solutions for the O(2n + 1)-model. Moreover the
periodicity constraint for the potential is effectively solved in terms of the spectral curves of the
elliptic Calogero–Moser system.
Remark 5.1. We present here only the basic case when each of the irreducible components Γ±
is smooth. The generalization of the construction to the case of singular curves follows the
standard line of arguments and will be presented elsewhere.
Let Γ± be a smooth genus g± algebraic curve with the holomorphic involution
σ : Γ± 7−→ Γ±,
with 2(n+ 1) fixed points
σ(P±) = P±, σ
(
pi±
)
= pi±.
Let us fix the σ-odd local parameters k−1
± in the neighborhoods of the marked points P±,
k±(σ(p)) = −k±(p).
The projection
π : Γ± 7−→ Γ0
± = Γ±/σ
represents Γ± as a two-sheet covering of the quotient-curve Γ0
± with 2(n+1) branch points P±, p
i
±,
the involution σ permuting the sheets. From the Riemann–Hurwitz formula it follows that the
genus of Γ± equals
g± = 2g0± + n,
where g0± is the genus of Γ0
±.
Let dΩ±(p) be a third kind meromorphic differential on Γ0
± with the divisor of poles at the
branching locus Γσ
± with residues ∓1 at the marked points P±. The differential dΩ± has
#Γσ
± + 2g0± − 2 = g± + n
zeros that we denote by γ0,±s , s = 1, . . . , g± + n,
dΩ±
(
γ0,±s
)
= 0. (5.1)
For each zero γ0,±s we choose one of its preimages on Γ±, i.e., a point γ±s on Γ± such that
π
(
γ±s
)
= γ0,±s , s = 1, . . . , g± + n
(there are 2g±+n such choices). Below D± = γ±1 + · · · + γ±g±+n will be called the admissible
divisor.
Lemma 5.2. The generic data consisting of a matrix G ∈ O(2n+ 1,C),
GtG = 1
and a pair
(
Γ±, σ, P±, k±, p
i
±, γ
±
s
)
, defines the unique pair of functions ψ±(z, z̄; p) on Γ± with
the following analytic properties in the variables p ∈ Γ±, respectively:
Novikov–Veselov Symmetries of the Two-Dimensional O(N) Sigma Model 25
10. Outside P± the only singularities of the function ψ± are the poles at γ±s , s = 1, . . . , g±+n.
The poles are simple if all of the γ±s ’s are distinct.
20. In the neighborhoods of P± the function ψ± has an essential singularity of the form (4.14)
with
ξ±0 = 1.
30. The gluing equations
y+(z, z̄) = Gy−(z, z̄), (5.2)
where y± =
(
yi±
)2n+1
i=1
∈ C2n+1 are the vectors with the coordinates
yi± = ri±ψ±
(
z, z̄; pi±
)
, i = 1, . . . , 2n+ 1, (5.3)
with (
ri±
)2
= ∓ resp±i dΩ± (5.4)
hold.
Proof. The first step of the proof is a simple counting of parameters and equations. According
to [14] the space of functions ψ± satisfying (4.14) is n+ 1-dimensional. Therefore, the (2n+ 1)
gluing equations (5.2) cut out a one-dimensional space of pairs ψ = (ψ+, ψ−). The condition
ξ+0 = 1 further normalizes ψ. It remains to show that the second normalization ξ−0 = 1 is
satisfied automatically.
Consider the differentials dΩ̃± = ψσ
±ψ±dΩ± on Γ±, respectively. By definition of the admis-
sible divisor dΩ̃± is a meromorphic differential on Γ± with only the simple poles at P±, p
i
±.
Then by the residue theorem we have(
ξ+0
)2
=
2n+1∑
i=1
respi+
dΩ̃+ = −yt
+y+ = −yt
−y− = resP−dΩ̃− =
(
ξ−0
)2
, (5.5)
where in the middle we used the equations (5.3), (5.4) and the assumption G ∈ O(2n + 1,C).
Since at z = 0, z̄ = 0 we have ξ±0 (0, 0) = 1 the equation (5.5) implies the equality ξ+0 = ξ−0 . The
lemma is proved. ■
In what follows we call the pair of functions ψ := (ψ+, ψ−) the BA function on Γ := Γ+
⊔
Γ−.
Given ψ and a matrix H ∈ O(2n+ 1,C) introduce the vector q(z, z̄) ∈ C2n+1:
q = Hy+. (5.6)
Theorem 5.3. The BA function ψ(z, z̄; p) on Γ satisfies the equation
(∂z∂z̄ − u(z, z̄))ψ(z, z̄; p) = 0,
with the potential
u(z, z̄) = ∂z̄ξ
+
1 = ∂zξ
−
1 .
Moreover, the 2n+ 1-dimensional vector q(z, z̄) defined in (5.6) satisfies the equations:
(q,q) = 1, (5.7)(
∂izq,q
)
= 0, i > 0. (5.8)
Proof. The proof of the first statement of the theorem is standard and based on the uniqueness
of the BA function. Equation (5.7) is just the first equation in (5.5).
The differential
(
∂izψ−
)
ψσ
−dΩ− has no residue at P− for i > 0. Hence the sum of its residues
at the points pi−, i = 1, . . . , 2n+ 1 is zero. The latter is equivalent to the equation (5.8) due to
the gluing conditions (5.2) and orthogonality of H. ■
26 I. Krichever and N. Nekrasov
5.1 Real and regular solutions
In general the potential u(z, z̄), and the functions qi(z, z̄), given by the construction above,
may have poles. The following theorem describes sufficient conditions for qi(z, z̄) to be real and
regular.
Theorem 5.4. Suppose the data in Lemma 5.2 satisfies the following conditions:
(a) there is an anti-holomorphic bijection τ : Γ+ −→ Γ− commuting with σ, such that
τ(P+) = P−, τ
(
pi+
)
= pi−,
(b) the matrix G, which is be definition orthogonal, is Hermitian, i.e.,
Ḡ = G−1 = Gt, (5.9)
(c) the admissible divisors are τ -invariant, e.g., τ(D+) = D−.
Then the potential defined by the corresponding BA function is real, u = ū.
Moreover, if
G = H−1H̄ (5.10)
for some H ∈ O(2n+ 1,C), then the vector q(z, z̄) given by (5.6) is real and therefore regular.
Remark 5.5. In this case the curve Γ− can be seen as Γ+ with the opposite complex structure,
so that Γ = Γ ∪ Γ̄ has a real structure.
Proof. For a generic effective divisor D± of degree g± + n equation (5.1) uniquely defines the
corresponding meromorphic differential dΩ±. Therefore, under the assumption (c) the equation
dΩ+(p) = −dΩ−(τ(p))
holds. The latter implies the equation
(
ri±
)2
=
(
ri∓
)2
for
(
ri±
)2
defined in (5.4). It is assumed
that the square roots of the both sides of the equation are chosen consistently, i.e., that the
equations
ri± = r̄i∓ (5.11)
hold.
The equations (5.9), (5.11) imply that the gluing equations (5.2) hold for the pair(
ψ̄−(z, z̄; τ(p)), ψ̄+(z, z̄; τ(p))
)
of BA functions. The uniqueness of BA function then implies
ψ±(z, z̄; p) = ψ̄∓(z, z̄; τ(p)). (5.12)
The reality of u is a direct corollary of (5.12). The reality of q is an easy corollary of (5.12)
and (5.10). The theorem is proved. ■
Novikov–Veselov Symmetries of the Two-Dimensional O(N) Sigma Model 27
5.2 Periodicity constraint
The potential u(z, z̄) defined by data in Lemma 5.2 is periodic if and only if there are func-
tions wα,± on Γ± such that equations
ψ±(z + ωα, z̄ + ω̄α; p) = w±
α (p)ψ±(z, z̄; p) (5.13)
hold. The first part of the constraints that single out the corresponding algebraic-geometrical
data are similar to that in Section 4.3.
Namely, denote by dp+α the unique differentials on Γ+ with a single pole at P+ of the form
dp+x = dk+
(
1 +O
(
k−2
+
))
, dp+y = τdk+
(
1 +O
(
k−2
+
))
and denote by dp−α the unique differentials on Γ− with a single pole at P− of the form
dp−x = dk−
(
1 +O
(
k−2
−
))
, dp−y = τ̄dk−
(
1 +O
(
k−2
−
))
,
such that all their periods are imaginary.
The functions
w±
α (p) = exp
(∫ p
dp±α
)
(5.14)
are single-valued on Γ± iff the periods of dp±α are integral multiples of 2πi
1
2πi
∮
c±
dp±α = n±
α,c± ∈ Z ∀c± ∈ H1(Γ±,Z). (5.15)
At first glance the equations (5.15) are identical to the equations (4.21).
5.2.1 Enters elliptic Calogero–Moser system
It turns out that in the case of reducible Fermi curves the constraints (5.15) are solved by the
spectral curves of elliptic Calogero–Moser (eCM) system!
Theorem 5.6 ([12]). The equations (5.15) are satisfied iff Γ± is the normalization of the spectral
curve CN± of N±-particle eCM system.
Recall that the N -particle eCM is the Hamiltonian system on
XN = T ∗(EN\diag
){
(ρi, zi)
N
i=1 | ρi ∈ C, zi ∈ E, zi ̸= zj , i ̸= j
}
,
E = C/Zωx ⊕ Zωy, which is governed by the Hamiltonian
H2 =
1
2
N∑
i=1
ρ2i + ν2
∑
i<j
℘(zi − zj).
This is an algebraic integrable system with the Lax operator [17]
L(α) =
∥∥∥∥ρiδij + ν
σ(α+ zi − zj)
σ(zi − zj)σ(α)
(1− δij)
∥∥∥∥N
i,j=1
. (5.16)
The flows of eCM linearize on the Jacobian of the spectral curve
CN ⊂M = C× (E\{α = 0}),
CN : Det(k − L(α)) = kN +
N∑
j=1
kN−jcj(α) = 0. (5.17)
28 I. Krichever and N. Nekrasov
The coefficients cj(α), j = 1, . . . , N of the characteristic polynomial (5.17) are meromorphic
functions on E with poles at α = 0 of order j, so that c1(α) = H1, c2(α) = H2− ν2N(N−1)
2 ℘(α),
etc.
In what follows the value of ν is immaterial, so we set it to be equal to ν = 1. As shown
in [17], near α = 0 the polynomial R(k, α) admits a factorization of the form
R(k, α) =
N∏
i=1
(
k + aiα
−1 + hi +O(α)
)
,
with a1 = 1 − N and ai = 1 for i > 1, for some hi ∈ C. This implies that the closure of the
affine curve defined by (5.17) is obtained by adding one point (∞, 0), at which N − 1 branches
are tangent to each other (corresponding to a2 = · · · = aN = 1), and one branch is transverse
to them. Thus if we blow up the point (∞, 0) ∈ P1 × E, we would get a smooth point P
corresponding to the first branch, and a point p′ contained in the remaining N − 1 branches.
Thus, generically, after the second blow up we get a compact curve whose projection onto the
elliptic curve has N preimages. We call the marked point the preimage of P on the second
blowup.
Due to the degenerate nature of the residue of L(α) at α = 0 there are only N independent
linear parameters in the coefficient functions. An explicit form of these parameters proposed
in [9] is based on an observation that a polynomial R(k, α) has a unique representation of the
form
R(k, α) := f(k + ζ(α), α), (5.18)
where
f(p, α) =
1
σ(α)
σ
(
α+
∂
∂p
)
H(p) =
1
σ(α)
N∑
n=0
1
n!
∂nασ(α)
∂nH
∂pn
, (5.19)
and H is the monic degree N polynomial
H(p) = pN +
N∑
l=1
Ilp
N−l, (5.20)
whose coefficients I = (I1, . . . , IN ) ∈ B = CN are the integrals of motion of the N -particle eCM
system.
For generic I ∈ B the corresponding curve CN is smooth of genus N . For I belonging to
certain loci in B the curves CN degenerate.
Definition 5.7. We call a smooth genus g algebraic curve an N -particle eCM curve if it is
a normalization of a curve defined by the equation (5.17).
By this definition we always have g ≤ N . As shown in [12], for fixed g, the N -particle eCM
curves become dense in the moduli space of all smooth genus g algebraic curves, as N → ∞.
The number N of eCM particles can be expressed through the differentials dpx, dpy on
a smooth genus g algebraic curve whose periods satisfy the periodicity constraints (with gen-
eral ν) (5.15):
N =
∣∣∣∣ 1
2πiν
resP (pxdpy)
∣∣∣∣ .
For the eCM curves the multipliers (5.14) are quite explicit: for p = (k, α), with ζ = σ′/σ
wx(p) = ek−ζ(α)+2ζ( 1
2
)α, wy(p) = eτ(k−ζ(α))+2ζ( τ
2
)α.
Novikov–Veselov Symmetries of the Two-Dimensional O(N) Sigma Model 29
5.2.2 Spectral curves of the turning points of the eCM system
Our next goal is to single out the eCM curves with involution. Notice, that since the Weierstrass
σ-function is odd, the eCM curves defined by the equations (5.18)–(5.20) with Iodd = 0 are
invariant under the involution
σ̃ : (k, α) 7→ (−k,−α), (5.21)
which leaves the marked point P fixed. The involution σ̃ descends to the involution of E fixing
α = 0 and the half-periods α = ωj , j = 1, 2, 3, where
ω1 =
1
2
, ω2 =
τ
2
, ω3 =
τ + 1
2
.
Hence σ̃ induces an involution of the fibers on the eCM curve over the half-periods of E.
Therefore, for even N there is at least one additional fixed point of σ̃ among the preimages
of α = 0, besides P . The same parity argument shows that for odd N there are at least four
fixed points of σ̃ - the marked point and one over each of the non-zero half-periods.
The dimension of the space Btp ⊂ B of the eCM curves invariant under the symmetry (5.21)
is [N/2]. For generic values of I2i, i = 1, . . . , [N/2], the corresponding curves are smooth of
genus N on which the fixed points of σ̃ described above are the only fixed points, i.e., generically,
σ̃ has 2 and 4 fixed points for N even and odd, respectively.
Now we are going to show that:
Theorem 5.8. A spectral curve CN of the eCM system admits a holomorphic involution σ̃
under which the marked point P is fixed, i.e., σ̃(P ) = P , iff it corresponds to a turning point
(0, zi)
N
i=1 ∈ XN .
The if part of the statement of the theorem is obvious. Indeed, if ρi = 0, for all i = 1, . . . , N,
then the Lax operator (5.16) obeys (cf. [28, equations (7.53) and Section 7.5]):
L(−α) = −Lt(α)
implying that the corresponding spectral curve is invariant under the symmetry (5.21). The
only if part of the theorem is a corollary of the following new identity for the Riemann theta-
function corresponding to a smooth genus g algebraic curve with at least one point fixed, which
is an algebraic-geometric incarnation of the characterization of the CKP tau function in terms
of turning points of the KP hierarchy (see [24, Theorem 2.2]).
Theorem 5.9. Let Γ be a smooth genus g algebraic curve with involution σ̃ under which the
marked point P is fixed, P = σ̃(P ). Then for any point Z0 of the Jacobian such that
Z0 + σ̃(Z0) = K + 2A(P ) ∈ Jac(Γ) (5.22)
(K is the canonical class) the identity
∂t∂z ln θ(Uz + V t+ Z0|B)|t=0 = 0 (5.23)
holds. Here θ(Z|B) is the Riemann theta-function defined by the matrix B of b-periods of a basis
of normalized holomorphic differentials on Γ; the vectors U and V have the coordinates
U j =
1
2πi
∮
bj
dΩ2, V j =
1
2πi
∮
bj
dΩ3,
where dΩ2 and dΩ3 are the normalized meromorphic differentials on Γ with the only pole at P
of the second and the third order, respectively.
30 I. Krichever and N. Nekrasov
Proof. Recall that a smooth genus g algebraic curve Γ with fixed local coordinate in the
neighborhood of a point P ∈ Γ, k−1(P ) = 0 and a generic effective degree g divisor D =
γ1 + · · · + γg defines a unique function ϕ(z, t; p) with the following analytic properties with
respect to p ∈ Γ (with z, t considered as fixed parameters):
10. Outside P the singularities of ϕ are poles at the divisor D.
20. In the neighborhood of P the function ϕ has the form
ϕ(z, t; p) = ekz+k2t
(
1 +
∞∑
s=1
ξs(z, t)k
−s
)
, k = k(p).
The explicit formula for ϕ in theta functions is [14]:
ϕ(z, t; p) =
θ(A(p) + zU + tV + Z|B)θ(Z|B)
θ(zU + tV + Z|B)θ(A(p) + Z|B)
ezΩ2(p)+tΩ3(p). (5.24)
Remark 5.10. In [14] the basic BA function ϕ(t; p) was defined as the function of an infinite
set of variables t = (t1, t2, . . . ). Here we only use its dependence on the first two times z = t1,
t = t2, while setting to zero the rest: ti = 0, i > 2. The subspace of Z for which (5.22) holds
is one of the connected components of the odd part of the Jacobian. It is invariant under the
flows corresponding to the odd times t2j+1 of the KP hierarchy.
The function ϕ satisfies the linear equation [14]:(
∂t − ∂2z + u(z, t)
)
ϕ(z, t; p) = 0 (5.25)
with
u = 2∂2z ln θ(Uz + V t+ Z|B) + 2c,
where c is a constant equal to the first coefficient of the Laurent expansion dΩ2 = dk
(
1+ ck−2+
· · ·
)
.
The substitution of the equation (5.24) into the equation (5.25) gives the sequence of equations
∂tξs − 2∂zξs+1 − ∂2zξs + uξs = 0, s = 0, 1, . . . (5.26)
of which the first, s = 0, gives an expression of u in terms of ξ1: u = 2∂zξ1.
Suppose now that the curve Γ admits an involution σ̃ under which P is fixed and k is odd,
k(p) = −k(σ̃(p)). Suppose also that the divisor D + σ̃(D) is the zero-divisor of a meromorphic
differential dΩ∗ with the only pole at P . The latter is equivalent to (5.22). Then the differential
dΩ̃∗ := (∂zψ(z, 0; p))ψ(z, 0; σ̃(p))dΩ∗
is a meromorphic differential on Γ with the only pole at P . Hence, it has no residue at P .
Computing this residue in terms of the coefficients of the expansion (5.24) we get
2ξ2(z, 0)− ξ21(z, 0) + ∂zξ1(z, 0) + c1 = 0, (5.27)
where c1 is a constant defined by the Laurent expansion of dΩ∗ at P .
Taking the z-derivative of (5.27) and using (5.26) with s = 1 we get the equation
0 = ∂z
(
2ξ2(z, 0)− ξ21(z, 0) + ∂zξ1(z, 0)
)
= ∂tξ1|t=0,
which is equivalent to the equation (5.23). The theorem is proved. ■
Novikov–Veselov Symmetries of the Two-Dimensional O(N) Sigma Model 31
We are now ready to present the full set of constraints on the geometric data above which
produce the solutions of the double-periodic O(2n+ 1)-model.
Theorem 5.11. The potential u(z, z̄) defined by the data in Lemma 5.2 is double-periodic if
and only if:
(i) Γ± is a normalization of the spectral curve of a turning point of the eCM system;
(ii) the matrix G satisfies the equation
G =W+
α GW
−
α , (5.28)
where W±
α are two diagonal matrices
W±
α = diag
(
w±
α
(
pi±
))
. (5.29)
The part (i) of the theorem statement was proved above. The part (ii) easily follow from the
compatibility of the monodromy equations (5.13) and the gluing conditions (5.2).
Remark 5.12. Note, that under the involution σ̃ the functions w±
α transform to
(
w±
α
)−1
.
Therefore, at the branch points w±
α equal 1 or −1. Thus, if the condition (i) is satisfied then
the corresponding potential u(z, z̄) is double periodic with periods 2 and 2τ .
5.3 Example: O(3) sigma model
Let Γ be the spectral curve corresponding to a turning point of N -particle eCM system. As
we mentioned above, for N = 2ℓ and generic values of the Ieven (and Iodd = 0) the curve Γ is
smooth of genus g = N and has only two fixed points of the involution (5.21). Our construction
of solutions for the O(3) sigma model requires the curves with involution having at least 4 fixed
points.
Our next goal is to identify the singular 2ℓ-particle eCM curves whose normalization has
four fixed points of the involution (5.21). Consider the function F (p, α) := σ(z)f(p, α) with f
defined by the equation (5.19). For even N it is odd with respect to the involution (5.21). The
equations
∂αF (0, 0) = 0, ∂pF (0, 0) = 0 (5.30)
cut out an (ℓ− 2)-dimensional linear subspace in the space of parameters Ieven. If the equa-
tions (5.30) hold, then the expansion of F near the point (p, α) = (0, 0) has the form:
F = b1p
3 + b2p
2α+ b3pα
2 + b4α
3 + · · · ,
where bi = bi(I) are some linear functions of the parameters Ieven. For generic values of these
parameters the equation
b1g
3 + b2g
2 + b3g + b4 = 0
has three distinct roots gi, i = 1, 2, 3. That implies that the equation F (p, α) = 0 has three
distinct solutions of the form
p = giα+
∞∑
s=1
gi,sα
(2s+1), i = 1, 2, 3.
In other words: under the constraints (5.30) the equation R(k, α) has three smooth branches
given by
k = −ζ(α) + p(α),
32 I. Krichever and N. Nekrasov
which represent the expansion of k near three points on the normalization of the singular spectral
curve. Since gi are distinct the branches are invariant under the involution. The normalization of
the singular spectral curve has genus (2ℓ−3). The corresponding quotient-curve is of genus (ℓ−2).
By Theorems 5.3 and 5.11 a pair of curves with involution having four fixed points on each
of them, a pair of admissible divisors and an orthogonal 3× 3 matrix G gives a solution to O(3)
sigma-model: the equations (5.28) are trivially satisfied, since the matricesW±
α are unit matrices
as all the branch points are among the preimages of one point α = 0. Our next goal is to present
this solution to O(3) model explicitly in theta functions.
On Γ the corresponding BA function has g+1 poles at the points γ1, . . . , γg+1 of the admissible
divisor D, i.e., the divisor such that D and σ(D) are preimages of the zeros of a differential dΩ on
the factor-curve Γ/σ with simple poles at the marked point P and three other points p1, p2, p3.
The space of the BA function, i.e., the functions having exponential singularity at P with
g+1 poles is two-dimensional. Hence on Γ the Baker–Akhiezer function in Lemma 5.2 is a linear
combination
ψ(z, z̄; p) = ϕ1(z, p) + c(z, z̄)ϕ2(z, p) (5.31)
of two basic functions. The first one equals
ϕ1 = ezΩ2(p) θ(A(p)−A(p3)− Z)θ(A(γ1) + Z)θ(A(γ2) + Z)
θ(A(p3) + Z)θ(A(p)−A(γ1)− Z)θ(A(p)−A(γ2)− Z)
× θ(A(p) +A(p3)−A(γ1)−A(γ2) + zU − Z)
θ(A(p3)−A(γ1)−A(γ2) + zU − Z)
, (5.32)
with
Z =
g+1∑
s=3
A(γs)−K.
ϕ1 is the unique function with exponential singularity at the marked point P with poles at D
and zero at p3, normalized so that its regular factor at P equals 1.
The function ϕ2 is defined by the similar conditions: the poles are at D, its regular factor
at P equals 0 and ϕ2(z, p3) = 1:
ϕ2 = ez(Ω2(p)−Ω2(p3)) θ(A(p)− Z)θ(A(p3)−A(γ1)− Z)θ(A(p3)−A(γ2)− Z)
θ(A(p3)− Z)θ(A(p)−A(γ1)− Z)θ(A(p)−A(γ2)− Z)
× θ(A(p)−A(γ1)−A(γ2) + zU − Z)
θ(A(p3)−A(γ1)−A(γ2) + zU − Z)
. (5.33)
Denote by
f1(z) = ϕ1(z, p1), f2(z) = ϕ1(z, p2),
g1(z) = ϕ2(z, p1), g2(z) = ϕ2(z, p2) (5.34)
the values of the basic functions at the points p1 and p2 (their values at p3 are 0 and 1, respec-
tively). The vanishing of the sum of the residues of matrix ∥ϕaϕσ̃b dΩ∥, for a, b = 1, 2 implies
r21f
2
1 + r22f
2
2 = 1, (5.35)
r21g
2
1 + r22g
2
2 = −r23, (5.36)
r21f1g1 + r22f2g2 = 0, (5.37)
Novikov–Veselov Symmetries of the Two-Dimensional O(N) Sigma Model 33
where
r2i = − respi dΩ, i = 1, 2, 3.
Also from the residue theorem we have 1 = r21 + r22 + r23.
The equations (5.35)–(5.37) imply
g1 =
h
r21
f2, g2 = − h
r22
f1, h =
√
−1r1r2r3. (5.38)
From (5.38) it follows that the values ψj = ψ(z, z̄; pj), for j = 1, 2, 3 are equal to
ψ1 = f1 + c
h
r21
f2, ψ2 = f2 − c
h
r22
f1, ψ3 = c. (5.39)
It is easy to verify, using (5.39), (5.42), that the vector y = (yj = rjψj)
3
j=1, obeys, first
y21 + y22 + y23 = 1,
and second
v :=
y1 + iy2
1 + y3
= r1f1(z) + ir2f2(z). (5.40)
The functions f1(z) and f2(z) are meromorphic functions of z, so is v = v(z).
Remark 5.13. It is easy to verify that the ansatz
q1 + iq2 =
2v
1 + vv̄
, q1 − iq2 =
2v̄
1 + vv̄
, q3 =
1− vv̄
1 + vv̄
with ∂z̄v = ∂z v̄ = 0, obeys (5.8), i.e., provides a w∞-harmonic map.
The function c(z, z̄) in the equation (5.31) is found from the equation (5.2). Explicitly, this
is the condition that the values ψj = ψ(z, z̄;ωj), for j = 1, 2, 3 are equal to the corresponding
values of the BA function on the curve Γ̄. In the real case the conditions have the form
y = Gȳ
with matrix G such that
G−1 = Gt = Ḡ = 1. (5.41)
If G = 1, then the gluing condition for ψ3 is just the equation c = c̄, while the reality equations
for y1 and y2 give
c = −r
2
1
h
f1 − f̄1
f2 + f̄2
=
r22
h
f2 − f̄2
f1 + f̄1
. (5.42)
Remark 5.14. We stress that the formulae above hold for any solutions of the O(3)-model
given by the Theorem 5.3. In general they are quasi-periodic.
If the periodicity constraints are satisfied, then the functions ψj are periodic or anti-periodic.
For the curves constructed at the beginning of this section, i.e., the curves which are the normal-
izations of the singular spectral curves defined by the equation (5.30) the functions ψj are elliptic
functions. Hence the function v(z) is an elliptic function. In other words we have produced an
34 I. Krichever and N. Nekrasov
instanton family of the solutions of the O(3) model. Let us compute its instanton charge – the
degree of the corresponding map Σ → S2.
From the equations (5.32), (5.33) we see that the poles of the functions f1(z), f2(z) are the
zeros of the function
θ(zU + Z0) = 0, Z0 = A(p3)−A(γ1)−A(γ2)− Z.
These are exactly the coordinates zi of the turning point. Hence, f1 and f2 have 2ℓ common
poles.
Notice that the equation (5.35) implies
v−1(z) = r1f1(z)− ir2f2(z). (5.43)
Hence, the function v(z) has poles at exactly half of poles of the functions fi. The other half
are zeros of v(z). Hence we get that the instanton number of solutions defined by spectral curve
corresponding to the turning points of 2ℓ-particle eCM curves satisfying the constraints (5.30)
is equal to ℓ and v(z) has the form
v(z) = A
ℓ∏
i=1
σ(z − ai)
σ(z − bi)
, (5.44)
where the parameters ai, bi ∈ E obey
ℓ∑
i=1
(ai − bi) = 0.
These parameters are determined by the curve Γ and the vector Z in (5.32), (5.33).
Remark 5.15. The NV flows change the parameters A, ai, bi in a complicated manner. In
terms of Γ, Z these flows preserve Γ and change Z linearly.
Let us count the number of parameters in our construction. First, we have ℓ − 2 complex
parameters for the curves satisfying (5.30). The space of differentials dΩ on a factor curve with
the simple poles at the branch points defining the admissible divisors is of dimension ℓ. Note,
that the corresponding potential u(z, z̄) depends only on the equivalence class of the admissible
divisor. Indeed, if D̃ is equivalent to D, then the relation between the BA functions is just
ψ̃(z, z̄; p) = χ(p)ψ(z, z̄; p), where χ(p) is the unique function with poles at D̃, zeros at D and
normalized by the equation χ(P ) = 1. Notice that the corresponding vectors ỹ(z, z̄) and y(z, z̄)
defined in (5.3) are equal, thanks to the equation
dΩ̃ = ff σ̃dΩ.
Thus, overall we have 2ℓ− 3 complex parameters for the curve and the equivalence class of the
admissible divisor. In addition, we have 3 real parameters for the matrix G satisfying (5.41),
i.e., the space of constructed potentials is of real dimension 4ℓ− 3.
Recall the well-known formula
u(z, z̄) = ∂z∂z̄ ln(1 + vv̄)
for the ℓ-instanton potentials, where v(z) is of the form (5.44). Although v(z) depends on 2ℓ
complex parameters, the space of the corresponding potentials u(z, z̄) is of real dimension 4ℓ−3,
since u is invariant under the SU(2)/{±1} ≈ SO(3) fractional linear transformations
ṽ(z) =
cv(z) + d
−d̄v(z) + c̄
, |c|2 + |d|2 = 1. (5.45)
Hence, we reproduce all instanton potentials.
Novikov–Veselov Symmetries of the Two-Dimensional O(N) Sigma Model 35
For G fixed, the solution of the O(3)-model is defined by a matrix H obeying (5.10). The
ambiguity in the choice of H is the transformation, equivalent to (5.45):
H ′ = hH, h ∈ O(3,R).
5.4 Other components
Consider now the (2ℓ−1)-dimensional locus of turning points for which the spectral curve passes
through the point k = 0, α = ωi for some i = 1, 2, 3. The space of the corresponding spectral
curves is of dimension ℓ− 1 and in terms of the parameters Ieven is defined by the equation
f(ζ(ωi), ωi) = 0, (5.46)
where the function f is defined by the equation (5.19).
Note that by the parity arguments the number of branches of a spectral curve passing through
the point (k = 0, α = ωj) is even, i.e for generic values of Ieven satisfying (5.46) the corresponding
curve is singular and has one node. Let Γ be a normalization of that curve. It is a smooth curve
of genus 2ℓ− 1 with involution σ having four fixed points. Two, including the marked point P
over α = 0 and two over α = ωj that are preimages of the node. In that case the monodromy
matrices W±
α in the equation (5.29) are not-trivial, cf. (5.47). This implies that the matrix G
of the Lemma 5.2 is diagonal.
The corresponding solutions of O(3)-model are given by the same formulae as above: yj =
rjψj with ψj defined in the equations (5.39) and (5.31)–(5.33). The parameters of the solutions
are: ℓ−1 complex parameters for the spectral curve, ℓ−1 complex parameters for the equivalence
class of an admissible divisor. Overall we have 4ℓ− 4 real parameters.
The twisting parameters of the solutions depend on the choice of ωj . For example, for j = 1
the functions f1(z), f2(z) defined in the equation (5.34) have the following monodromy properties
f1(z + ω1) = f1(z), f1(z + ω2) = −f1(z),
f2(z + ω1) = f2(z), f2(z + ω2) = −f2(z).
(5.47)
Hence, the function v(z) defined in (5.40) has the following monodromy properties:
v(z + ω1) = v(z), v(z + ω2) = −v(z).
The functions f1(z), f2(z) has N = 2ℓ poles in the fundamental domain with periods ω1, ω2.
From the equation (5.43) it follows that w(z) has poles only at half of them. Therefore, it is of
the form
v(z) = C
ℓ∏
i=1
σ(z − ci)
σ(z − di)
for some ci, di satisfying the equation
2
ℓ∑
i=1
(ci − di) = ω2.
5.5 O(2n + 1) sigma model
We conclude the section by describing the curves giving the solutions to O(2n + 1)-model for
an arbitrary n. Recall that the spectral 2ℓ-particle eCM curve with involution is defined by the
equation, cf. (5.19):
f(k + ζ(α), α) = 0,
36 I. Krichever and N. Nekrasov
with Iodd = 0. For generic values of the parameters Ieven it is a smooth curve of genus 2ℓ
with two fixed points of involution over α = 0. Suppose now that these parameters satisfy the
equations
∂iα∂
j
p(σ(α)f(p, α)) = 0, i+ j ≤ 2n. (5.48)
Of course, in order for (5.48) to have solutions we should take ℓ sufficiently large.
Then the point (k = 0, α = 0) is a singular point of the spectral curve through which pass
2n + 1 branches. For generic Ieven the system (5.48) of (n + 1)(2n + 1) linear equations this is
the only singular point of the spectral curve, thus the genus of the normalization of the spectral
curve equals 2ℓ− 2n2 − n− 1.
Remark 5.16. As in the case of the quantum mechanical models [27] whose classical limit is an
algebraic integrable system, our solutions have a mysterious connection to N = 2 supersymmet-
ric gauge theories in four dimensions. The spectral curves are the Seiberg–Witten curves, the
differential pxdpy is essentially the Seiberg–Witten differential. In particular, the periodic solu-
tions of the O(N) model corresponding to reducible Fermi curves are connected to the N = 2∗
theory [10]. The origin and the implications of this connection remain to be seen.
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1 Introduction
1.1 Worldsheet is a two-torus
1.2 Spectral curves from zero-curvature equations
1.3 Spheres from Euclidean spaces, and complexification
1.4 Fermi-curves
1.5 Novikov–Veselov hierarchy
2 Novikov–Veselov hierarchy
2.1 The phase space
2.2 The flows
2.3 Integrals
3 The NV flows as symmetries of O(N) model
4 O(N) sigma model on a two-torus
4.1 The formal Bloch solutions
4.2 The algebraic spectral curve
4.3 The periodicity constraint
5 w_infty-harmonic maps to spheres
5.1 Real and regular solutions
5.2 Periodicity constraint
5.2.1 Enters elliptic Calogero–Moser system
5.2.2 Spectral curves of the turning points of the eCM system
5.3 Example: O(3) sigma model
5.4 Other components
5.5 O(2n+1) sigma model
References
|
| id | nasplib_isofts_kiev_ua-123456789-211539 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T12:06:12Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Krichever, Igor Nekrasov, Nikita 2026-01-05T12:30:14Z 2022 Novikov-Veselov Symmetries of the Two-Dimensional ( ) Sigma Model. Igor Krichever and Nikita Nekrasov. SIGMA 18 (2022), 006, 37 pages 1815-0659 2020 Mathematics Subject Classification: 14H70; 17B80; 35J10; 37K10; 37K20; 37K30; 81R12 arXiv:2106.14201 https://nasplib.isofts.kiev.ua/handle/123456789/211539 https://doi.org/10.3842/SIGMA.2022.006 We show that the Novikov-Veselov hierarchy provides a complete family of commuting symmetries of the two-dimensional ( ) sigma model. In the first part of the paper, we use these symmetries to prove that the Fermi spectral curve for the double-periodic sigma model is algebraic. Thus, our previous construction of the complexified harmonic maps in the case of irreducible Fermi curves is complete. In the second part of the paper, we generalize our construction to the case of reducible Fermi curves and show that it gives the conformal harmonic maps to even-dimensional spheres. Remarkably, the solutions are parameterized by spectral curves of turning points of the elliptic Calogero-Moser system. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Novikov-Veselov Symmetries of the Two-Dimensional ( ) Sigma Model Article published earlier |
| spellingShingle | Novikov-Veselov Symmetries of the Two-Dimensional ( ) Sigma Model Krichever, Igor Nekrasov, Nikita |
| title | Novikov-Veselov Symmetries of the Two-Dimensional ( ) Sigma Model |
| title_full | Novikov-Veselov Symmetries of the Two-Dimensional ( ) Sigma Model |
| title_fullStr | Novikov-Veselov Symmetries of the Two-Dimensional ( ) Sigma Model |
| title_full_unstemmed | Novikov-Veselov Symmetries of the Two-Dimensional ( ) Sigma Model |
| title_short | Novikov-Veselov Symmetries of the Two-Dimensional ( ) Sigma Model |
| title_sort | novikov-veselov symmetries of the two-dimensional ( ) sigma model |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211539 |
| work_keys_str_mv | AT kricheverigor novikovveselovsymmetriesofthetwodimensionalsigmamodel AT nekrasovnikita novikovveselovsymmetriesofthetwodimensionalsigmamodel |