2d Integrable systems, 4d Chern-Simons theory and Affine Higgs bundles

In this short review we compare constructions of 2d integrable models by means of two gauge field theories. The first one is the 4d Chern-Simons (4d-CS) theory proposed by Costello and Yamazaki. The second one is the 2d generalization of the Hitchin integrable systems constructed by means the Affine Higgs bundles (AHB). We illustrate this approach by considering 1+1 field versions of elliptic integrable systems including the Calogero-Moser field theory, the Landau-Lifshitz model and the field theory generalization of the elliptic Gaudin model.


Introduction
In the nineties, we attempted to construct 2D classical integrable field theories starting with a two-dimensional WZW action [6]. The corresponding equations of motion coincide with the Zakharov-Shabat equations. These equations are the hallmark of two-dimensional integrable systems. But that approach had one essential drawback -the Lax operator did not depend on the spectral parameter. This parameter is a necessary ingredient for constructing the infinite number of commuting integrals of motion. A class of integrable theories, derived from the WZW models was considered in papers by L. Fehér et all (see the review [5]). Also, the interrelations between gauge theories and integrable systems were considered in the mid-nineties in [2,20]. Later Nekrasov and Shatashvili derived quantum integrable systems from four-dimensional gauge theories [21].
The problem with the spectral parameter was overcome in the works of Costello and Yamazaki [3] by considering the so-called four-dimensional Chern-Simons theory (4d-CS).
Here we compare 4d-CS construction with the construction of 2d integrable systems based on the Affine Higgs bundles (AHB) model proposed in [13]. The AHB model is the 2d analog of the Hitchin systems [7]. To compare the AHB theory with the 4d-CS approach we rewrite the AHB theory in the form of a special 4d CS model. It allows one to establish a correspondence between the field contents from both constructions.
The first formal difference between these two approaches is that AHB theory is free, and the nontrivial integrable models appear as a result of the symplectic reduction. The latter procedure is similar to what happens in the finite-dimensional case for the Hitchin systems. Symplectic reduction is defined by two types of constraints. The first one is given by the moment map constraints (the Gauss law analog in the YM theory). The second one is the gauge fixing conditions. After imposing these constraints we come to the symplectic phase spaces of 2d integrable systems. Using the AHB we constructed in [13] the 2d field generalization of the elliptic (spin) Calogero-Moser (CM) model. It was proved by A. Shabat (unpublished) and in [1] that this model is gauge equivalent to the Landau-Lifshitz (LL) equation [23]. The gauge transformation comes from the so-called symplectic Hecke correspondence. Another example of 2d generalization of the Hitchin systems is 2d elliptic Gaudin model. In particular, the Principal Chiral Model is reproduced in this way 1 .
Another construction similar to the AHB approach is the algebra-geometric derivation of the Zakharov-Shabat equation proposed by Krichever [11]. In particular, using the KP hierarchy he constructed the 2d version of the Calogero-Moser model. This approach can be also extended to the field version of the Ruijsenaars-Schneider models [26].
In contrast to AHB construction, the 4d-CS theory is not free. The equations of motion have the form of the moment map constraints equations, which are similar to the moment map constraints in the AHB theory. It only remains to impose some gauge fixation to come to 2d integrable systems. To compare these constructions, we rewrite the equations of motion and the moment map constraints in the AHB models in the CS form.
In the standard approach to the 2d integrable in [3,13] the 3d space has the form R × CP 1 or S 1 × CP 1 or with an elliptic curve instead of CP 1 . More generally, these 3d spaces can be replaced by an arbitrary Seifert surface [22]. The Seifert surface is a U (1) bundle over the Riemann curve Σ g of genus g. The Seifert surfaces have two topological characteristics (n, g), where n is the degree of the line bundle corresponding to the U (1) bundle. Although the moduli space of the Higgs bundles over the Seifert surfaces depends on n, the invariant Hamiltonians do not depend on it. The reason is that there exists singular gauge transformation Ξ(k) of the Lax operator L(n) such that Ξ(k) : L(n) → L(n + k).
The AHB construction allows one to define 2d analogs of the additional structures in the Hitchin systems. The first structure is the affine analog of the symplectic Hecke correspondence [1,13]. Another structure that appears in the AHB model is the affine version of the Nahm equations describing the surface defects. Both of these structures will be considered in the forthcoming publication [17].
The paper is organized as follows. In the next section we explain briefly 4d-CS construction of 2d integrable models based on the articles [3,12]. In Section 3 the AHB construction is given following notations from [13,27]. Some examples are given in Section 4. Finally, we establish the correspondence between the two construction in Section 5.

4d Chern-Simons model and integrable systems
Let us describe the field content of 4d Chern-Simons model. Consider a Riemann curve C and the space time M = R 2 × C with the local coordinates (x, t), (z.z) 2 . On R 2 ∼ C introduce the complex coordinates w = x + t,w = x − t. Let G be a complex simple Lie group. Consider a principal G bundle P over M and equip it with the connections Let ω be a one form on Σ (ω = ϕ(z)dz). It is a section of the canonical class K C on C. The four-dimensional CS action is defined as where CS(A) is the standard CS action and A is the defined above connection (2.1).
Beyond the points where the form ω vanishes the equations of motion corresponding to (2.2) take the form: These equations are invariant under the gauge transformations Let f be the gauge transformation fixing the gauge as A f z = A 0 z . We identify A f w = L(w,w, z) with the Lax operator, and A fw = M (w,w, z) with the evolution operator M . Then the first equation in (2.3) turns into the Zakharov-Shabat type equation for some 2d integrable system: In the most part of the paper [3] it is assumed that there is a gauge choice or, put it differently, that the moduli space of holomorphic bundles over C is empty. It is indeed true if C is a rational curve, but almost never true in the general case. For example, if C is an elliptic curve this is possible for the topologically non-trivial bundles. If it is the case, then the equations 2 and 3 from (2.3) mean that A w and Aw are holomorphic on C and in this way they are constants. Therefore, we are left with the Zakharov-Shabat equation, where the operators L and M are independent on the spectral parameter z.
In order to come to meaningful cases with L and M depending on the spectral parameter one should consider higher genus curves. One more possibility is to consider additional degrees of freedom by introducing surface defects in the 4d-CS model. The surface defects come from the poles and zeros of the meromorphic 1-form ω in (2.2). The zeros of ω mean that the Lax operator has poles at this points and the corresponding coefficients (residues) define additional degrees of freedom in the theory. These defects are called the disorder defects.
The poles of ω lead to restrictions of the gauge fields at these poles and also add degrees of freedom. These defects are called the order defects. Below we consider these defects in terms of AHB theory in greater detail.

Three-dimensional space
Consider a principal U (1)-bundle W over Riemann curve Σ: The total space of the bundle is called the Seifert surface. Let (z,z, θ) be local coordinates on W and Ω (m,n,k) (W ) the space of corresponding (m, n, k)-forms. Redefine the one forms as Here n is the degree of the U 1 -bundle andμ(z,z) ∈ Ω (0,−1,1) is the Beltrami differential. Consider Ω (1,0) (Σ)-form dz on Σ and let π * (dz) ∈ Ω (1,0,0) (W ). Define two vector fields on W , which annihilate the form π * dz: The first field ∂ θ acts along the S 1 fibers and thereby annihilates the form π * dz. For the second field ∂μ z this condition means that Consider a line bundle L over Σ g , which is a complexification of the U (1)-bundle. Let D z ⊂ Σ g be a small disc with the center z = 0 and D ′ z ⊂ D z The degree n of the bundle is defined by a holomorphic non-vanishing transition function f (z) on D z \D ′ z . The degree can be changed by the multiplication f (z) → f (z)w(z) in the following way.
This procedure is called the modification of the U (1)-bundle.
If the bundle W is trivial then one can take n = 0.
In the examples below we assume n = 0.
Let G be a complex Lie group and P is a principle G-bundle over W . We define preliminary, the affine Higgs bundle (AHB) over W as a pair of connections (3.7) The first component ∂μ z +Az defines the complex structure on the sections of P in (z, θ) direction. The precise definition of the AHB is given below (3.17). The second component is the Higgs connection. It is an affine analogue of the Higgs field introduced by Hitchin [8].

Affine holomorphic bundles
The affine Higgs bundles are the cotangent bundles to the affine holomorphic bundles, which we are going to define.
In the previous subsection we introduced the connection acting on the sections Γ(P) (3.7): Consider, in addition, a line bundle L over Σ with the connection (∂z +kz) ⊗ dz. The antiholomorphic connection on P ⊕ L is the pair of operators ∇Ā ,μ,k = DĀ ,μ (∂z +k(z,z)) ⊗ dz .
It can be considered as a map of the spectral curve Σ to the loop group The structure group of the bundle P ⊕ L (the gauge group) is defined by replacing L(G) with its central and co-central extensions (A.7): More precisely, Consider its infinitesimal action on ∇Ā ,μ,k . As a vector space the Lie algebra Lie(Ĝ G ) has three components: Their action on ∇Ā takes the form: (3.12) The moduli of holomorphic structure on P(M ) ⊕ L is the quotient space where we fix the gauge asĀ →Ā f =L, i.e. (3.14) One can fix the action of the abelian subgroups {exp(ε 3 )}, {exp(ε 3 (z,z)∂)} onμ andk (3.12) in a similar way. We preserve the notations for the gauge transformed variablesμ andk.

Affine Higgs bundles
Introduce the Higgs field Φ(z,z, θ). Let K be a canonical class of Σ. Then the Higgs field is The affine Higgs bundle is the pair The connection form A θ in (3.7) is related to the Higgs field Φ as The fields of the Higgs bundles have the following dimensions: Table 1: The cotangent bundle structure of the AHB comes from the pairing (A. 10) Define the symplectic form Ω on H af f (G) The form is invariant under the action of the gauge groupĜ (3.10). Along with (3.12), the corresponding Hamiltonian vector fields are as follows: The action ofĜ is generated by the moment maps m j : More explicitly, The quotient of C af f under the action of the gauge groupǦ (3.10) is the moduli space of the affine Higgs bundles: We can first fix the gauge and then solve the moment map equations. In this respect M af f (G) is defined as the set of solutions of equations

Parabolic structures. The order defects.
To introduce the parabolic structure we attach the coadjoint orbits a ) of the loop group L(G) (A.14) to the marked points z a ∈ Σ, a = 1, . . . , n. It means that we add the order defects in the theory. The disorder defects correspond to the reducing the gauge groupǦ (3.10) to the subgroupǦ(× a F l a ) ⊂Ǧ, which preserves the affine flags F l a at the marked points. It was proved in [14] that these construction are equivalent. Here we follow the order defects description.
The affine parabolic Higgs bundle has the following field contents: The coadjoint orbits (A.14) are equipped with the Kirillov-Kostant symplectic form (A.15). Thereby, the symplectic form on the reduced parabolic Higgs bundle H af f par (G) is equal to where Ω is the form (3.19) and ω a are the Kirillov-Kostant forms (A. 15). Due to the presence of new terms in the form, the moment map constraints (3.22) are upgraded as It means that ν is not a constant in (3.27) but a meromorphic (1, 0)-form on Σ with the first order poles at z = z a : In other words, ν = const implies that we deal with orbits without central extension only, i.e.
Since n a=1 c (0) a = 0, in the case of a single marked point (likewise it happens for the Landau-Lifshitz equation) the orbit has the form (3.30) and ν = ν 0 is a constant.
Next, we pass to the symplectic quotient (the moduli space). Let us fix a gauge as in (3.14) and The moment map constraint equation (3.27) with m 1 = 0 is modified as Solutions of this equation along with (3.28) define the moduli space of the affine parabolic bundles as the symplectic quotient space It is a phase space of 2d integrable systems. The symplectic form (3.26) on M af f, par (G) turns into (see (3.26))

Equations of motion
The gauge invariant integrals are generated by the traces of the monodromies of the Higgs field A θ . We take the Hamiltonian in the form: Consider equations of motion on the "upstairs" space H af f (G) (3.25). They are derived by means of the symplectic form (3.26) and the Hamiltonians (3.36). In this way we obtain the following free system: Recall that after the symplectic reduction we come to the fieldsL (3.14) and L (3.31). For simplicity, we keep the same notation for the coadjoint orbits variables S α , so they are transformed as in (A.17). This yields H(L, ν) = Σ ω(z,z) tr exp 1 ν(z,z) S 1 dθL(z,z, θ) . Its solution L has the same form as for n = 0, but the angle parameter θ is replaced withθ. The corresponding monodromy matrix is conjugated to the original monodromy matrix where the gauge transformation assumes the form In this way, as we claimed in the Introduction, the invariants of the monodromy matrix and, in particular, the Hamiltonian are independent of n.
It follows from the moment map equation (3.33) that for the parabolic bundles the Lax operator L has the first order poles at the marked points z a . Let w a = z − z a . The generating function of the Hamiltonians (3.41) has the expansion: Consider the set of times T a,j = {t a,j } corresponding to the Hamiltonians H a j . The onedimensional spaces T a,j are isomorphic to R. Let ∂ a,j = {H a j , } be the Poisson vector field on the moduli space M af f, par (G) (3.34). Assume that the gauge transformation f comes from the gauge fixation (3.14). Define the connection form M a,j = ∂ a,j f f −1 . From (3.31) we have Φ = −ν∂f f −1 + f Lf −1 . Plugging it into (3.37) we come to the Zakharov-Shabat equation

Conservation laws
The matrix equation (1.3.47) allows one to write down the conservation laws. The eigenvalues of the monodromy matrix of solutions Ψ are the gauge invariant. Represent solutions of (1.3.47) as the P-exponent where R is periodic in θ. The monodromy of Ψ(θ, z) is Consider the monodromy in a neighborhood of a pole z a ∈ Σ of L/ν with a local coordinate w a = z − z a . If

The Hamiltonains
H a j ∼ tr V exp ı There is a recurrence procedure to define the matrices S a j . Details can be found in [13,19].

The action
Consider the 4d action on the space corresponding to the Hamiltonian system defined above: 3 Here H a j are the Hamiltonians (3.36) and S W ZW is the Wess-Zumino-Witten action To come to the action on the moduli space of the affine Higgs bundles H af f, par (G) (3.25) we need to impose the moment map constraints (3.27) and fix the gauge. To do it one should introduce in the action the terms containing the ghost and the anti-ghost fields. Instead, we first fix the gauge and rewrite the action in terms of the fields L andL. The action takes the form

Examples
In all examples we consider the trivial S 1 bundles and putμ = 0.
Let us perform the gauge transformation with f defined as follows: Then the Lax matrix L is transformed into The linear problem where ψ is the Bloch wave function ψ = exp{−i χ}, leads to the Riccati equation: The decomposition of χ(z) provides densities of the conservation laws (see [4]): H k ∼ dθχ k−1 .
Then the Lax operator of the LL equation is defined as It satisfies the moment map equation and has the quasi-periodicities as the Higgs field Φ in Table 2.
To write it down we use the Kronecker elliptic function related to the curve Σ τ : where ϑ(z) is the theta-function ϑ(z|τ ) = q The Kronecker function has the following quasi-periodicities: and has the first order pole at z = 0 It is related to the Weierstrass function ℘ as follows: Let The Lax operator assumes the form The symplectic form Ω (3.26) is reduced ro the symplectic form on the orbit O(p (0) , 0) (A.15): The Hamiltonian H LL 2 (4.8) assumes the form where ℘ α are the values of the Weierstrass functions at the half-periods. It is the Hamiltonian of the Euler-Arnold top on the group L(G) defined by the inverse inertia tensor The corresponding equations of motion (see (A.16)) are the Landau-Lifshitz equations: (4.18)
The Hamiltonian of the elliptic Calogero-Moser (ECM) field theory is the integrable 2d continuation of the standard two-particle ECM Hamiltonian (a motion of particle in the Lame potential) where ℘(2u) is the Weierstrass function. In the field case we have the canonical Poisson bracket {v(θ), u(θ ′ )} = δ(θ − θ ′ ). From (4.8) and (4.9) one finds where h = u 2 θ + l 2 . For v and u it is the Hamiltonian (4.22). The equations of motion produced by H CM 0 are of the form: ) .

(4.24)
There exists a transformation Ξ of the Lax operators : such that solutions of (4.24) become solutions of the LL equation. (u, v) → (S α , α = 1, 2, 3) [1]. It was called the symplectic Hecke correspondence for integrable systems [13] and can be described in terms of solutions of the extended Bogomolny equation [10,15]. In the 2d case one should define the affine version of the extended Bogomolny equation. We will come to this point in a separate publication.

Gaudin field theory and principal chiral model
The Gaudin models in classical mechanics are described by the Higgs fields (i.e. the Lax matrices) with a set of simple poles at punctures on a base curve with local coordinate z. For elliptic models the latter is the elliptic curve Σ τ with punctures z a . Then the Lax matrix is fixed by a chose of coadjoint orbits S a = Res z=za L(z) attached to punctures together with some boundary conditions (or the quasi-periodic behaviour). See [24] for a review of models related to SL-bundles and [18] for a generic complex Lie group G. Similarly, in the 1+1 field case the Gaudin type models are generalizations of the previously given examples to a multi-pole Higgs field.
Principal chiral model. The rational 2d field Gaudin model corresponding to Riemann sphere with two punctures is the widely-known principal chiral model. Indeed, consider the Zakharov-Shabat equation 4 The we have equations of motion 27) which are generated by the Poisson brackets and the Hamiltonian Here S 1 dθ P a is the shift operator in the loop algebraŝl(N, C): The substitution S 1 = 1 2 (l 0 + l 1 ) and S 2 = 1 2 (l 0 − l 1 ) transforms (4.27) into equation of the principal chiral model: Also, by changing the coordinates (θ, t) to the "light-cone" coordinates ξ = t+θ 2 , η = t−θ 2 , one gets (4.32) Elliptic 1+1 Gaudin model: first flows. Let us proceed to the elliptic case. The multipole extensions of the (spin) Calogero-Moser field theory were studied in [13]. Here we briefly review results of [27] on the multi-pole generalization ofŝl(2, C)-valued Lax matrix (4.16) with the quasi-periodic properties (4.10): Using (4.6)-(4.8) one gets the following "first flow" Hamiltonians: Here and below we use the following notations for the linear operators: The Hamiltonians (4.35) generate dynamics described by the following equations: (4.37) These equations are equivalent to Zakharov-Shabat equation (4.25) with L(z) (4.33) and . (4.39) or by analogy with (4.32): where λ a are the eigenvalues of S a (i.e. spectrum of S a is diag(λ a , −λ a )), and it is assumed that ∂ θ λ a = 0. Equation of motion take the form In the case of a single marked point (n = 1) we get the Landau-Lifshitz equation in the form: Similarly, one can write down in the trigonometric and rational cases. For example, in the straightforward rational limit (related to XXX 6-vertex R-matrix) the above equations provide the model of coupled Heisenberg magnets. The rational 11-vertex deformation was described in [16]. Trigonometric 6-vertex and 7-vertex models are described in the same way.

Correspondence between 4d-CS and AHB
Let us pass to the following new field: It follows from (3.32) that the system (3.46) assumes the form: (5.5) The delta-functions in the r.h.s of (3.5.5) mean that the connection form (i.e. L) has the first order poles. Equations (5.5) are the equations of motion for the 4d-CS action on the 4d spaces M a,j (3.52) where A a,j = (D M a,j , DL′ a,j , D A θ ) and CS(A a,j ) := tr A a,j ∧ dA a,j + 2 3 A a,j ∧ A a,j ∧ A a,j . Thereby, we rewrite the equations (3.46) of the AHB theory in the Chern-Simons form (2.2).
Comparing the system (5.5) with the system (2.3) in 4d-CS theory we come to the following relations between the fields in these two constructions: Thus, we established the equivalence of two constructions at the classical level in the case when the surface defects correspond to the first order poles and the W bundles (3.1) are trivial.
We assume that p (0) is a semi-simple element in the Cartan subalgebra h C ⊂ g. Its centralizer is the Cartan subgroup H C . The invariants defining the orbit O(p (0) , c (0) ) are the conjugacy classes of the monodromy operator corresponding to the connection c (0) ∂ + S along a contour in C * . In fact, there is a one-to-one correspondence between the set of L(G)-orbits and the set of conjugacy classes in the group G. The orbit is the coset space O(p (0) , c (0) ) ∼ L(G)/H C for c (0) = 0, and O(p (0) , 0) ∼ L(G)/L(H C ), where H C is the Cartan subgroup of G.