A generalized 4d Chern-Simons theory

A generalization of the 4d Chern-Simons theory action introduced by Costello and Yamazaki is presented. We apply general arguments from symplectic geometry concerning the Hamiltonian action of a symmetry group on the space of gauge connections defined on a 4d manifold and construct an action functional that is quadratic in the moment map associated to the group action. The generalization relies on the use of contact 1-forms defined on non-trivial circle bundles over Riemann surfaces and mimics closely the approach used by Beasley and Witten to reformulate conventional 3d Chern-Simons theories on Seifert manifolds. We also show that the path integral of the generalized theory associated to integrable field theories of the PCM type, takes the canonical form of a symplectic integral over a subspace of the space of gauge connections, turning it a potential candidate for using the method of non-Abelian localization. Alternatively, this new quadratic completion of the 4d Chern-Simons theory can also be deduced in an intuitive way from manipulations similar to those used in T-duality. Further details on how to recover the original 4d Chern-Simons theory data, from the point of view of the Hamiltonian formalism applied to the generalized theory, are included as well.


Introduction
A relatively new approach to integrable lattice models and 2-dimensional integrable field theories that, as a main tool, uses a gauge theory of the Chern-Simons (CS) type, has attracted a great deal of attention due to its potential in offering novel insights into the quantum integrable structure of these systems and their general properties. Such a gauge theory, known as the 4-dimensional semi-holomorphic Chern-Simons theory, or 4d CS theory for short, introduced in [1,2] and studied in detail in [3][4][5] has, in the last few years, triggered several new interesting results in the fields of integrable systems and string theory.
The 4d CS theory under consideration is defined by an action functional of the form where c is a constant 1 , M = Σ × C is a 4-dimensional manifold constructed out of the closed string world-sheet Σ and a Riemann surface C, ω is a meromorphic 1-form defined on C and CS(A) is the usual Chern-Simons 3-form for a gauge connection A on M. The theory is topological along Σ and holomorphic along C, hence the name semi-holomorphic. For an introduction to the theory and some of its properties, see the nice review [6].
What is important for the narrative of this work, is the fact that if we consider an atlas T covering the Riemann surface C, we can interpret the 4-dimensional manifold M as the total space of a trivial bundle over the base space C, with typical fiber Σ. This observation motivates us to relax such a condition in favour of a more general one, namely, that the manifold M indeed looks like the product Σ × C but only locally over any chart U ⊂ T . This causes some changes into the structure of the theory because, besides the 1-form ω defined on M, which is now seen as the pullback of a 1-form ω C defined on C, another interesting differential 1-form α (to be identified later on as a contact 1-form) defined on M can also be introduced via the pullback from C to M of a certain symplectic 2-form σ C defined on C, if the fiber bundle is taken to be inherently non-trivial. This allows to generalize the 4d CS theory action (1.1) into a new one depending on the former dynamical field A, two new non-dynamical 1-forms Ω, κ (to be specified below) constructed out of α, ω and some parameters.
The time direction in M is considered, from physical grounds, as globally defined and oriented. This means we can restrict the manifold M a bit more and consider instead a 4-dimensional space of the form M = R × M, hence the assumed non-triviality of the fiber bundle now resides entirely on the manifold M. In this work we will consider closed strings and identify the time direction with the R factor in the decomposition Σ = R × S 1 , thus the 3-dimensional manifold M now becomes the total space of a non-trivial circle bundle over the base space C. Fortunately, the structure of circle bundles of this sort was exploited heavily in the seminal paper [7], devoted to a new formulation of the conventional 3-dimensional Chern-Simons theories on Seifert manifolds which are, roughly speaking, total spaces with a circle bundle structure plus a certain technical condition over the action of an Abelian group on the total space [8]. This ultimately allowed to compute the partition function of some CS theories on these manifolds by using the method of non-Abelian localization of symplectic integrals [9]. See also [10], for the inclusion of Wilson loops within the formalism.
It is then natural to ask if the partition function of the theory defined by the action (1.1) is a candidate for using the non-Abelian localization method as well, due to the structure of the 4-dimensional manifold M = R × M that we are considering. After all, the only difference between M and M is a trivial R factor. To be more precise [7,9], if we want to show that this is the case, one first need to put the action (1.1) in the quadratic form S = ic(µ, µ) (1.2) and second, one has to show that the partition function of the theory is canonical, in the sense that it can be written as a symplectic integral of the form Here, X is a symplectic manifold with symplectic formΩ constructed out of the space A of gauge connections A defined on M. We assume that a Lie group H acts on X in a Hamiltonian fashion, with moment map µ : X → h * , where h * is the dual of the Lie algebra h of H. Also, ( * , * ) is an invariant quadratic form on h which, by the duality induced by it between h and h * , allows to define the action S in terms of µ ∈ h. The coupling constant of the theory is .
The purpose of this work is to initiate a study of the relation between the original theory (1.1), the quadratic action (1.2) and the symplectic integral (1.3). We will show below that such a relation exist but it is not a direct one and involves instead a generalization of the action functional (1.1). To do this, we follow the strategy used in [7] for computing the moment map µ in the case of the conventional 3-dimensional CS theories and make the necessary modifications in order to locate the 1-form ω and the time direction R within the general construction. As a complementary and direct approach, we shall recover the quadratic action (1.2) from a different perspective using well known field-theoretic manipulations that are usually employed in T-duality. The outcome is that the partition function of the theory takes the desired form (1.3), where X = A/S is a symplectic quotient space, is a quadratic action functional 2 generalizing (1.1) and is the path integral symplectic measure, whereΩ is a symplectic form defined on X. The Lie algebra valued function Φ and the vector field R on M will be introduced below.
In this work, we will construct (1.3) for integrable field theories of the principal chiral model (PCM) type as the main illustrative example.
The paper is organized as follows. In section (2) we construct, from general principles of symplectic geometry, an action functional that is, by definition, of the quadratic form (1.2). The construction is rather general an relies on a variant of the symplectic form originally used in [7]. In section (3), we show how this new action functional is related to the original 4d CS theory action (1.1). The quadratic action generalizing (1.1) depends on the usual 1-form ω, a contact 1-form α naturally linked to the non-triviality of the circle bundle M over C, and a pair of parameters. We also comment on a strategy devised to recover the theory (1.1), which is based on a partial gauge fixing and involves a degenerate limit in one of the parameters. In section (4), we specialize the general construction to a particular example concerning one of the simplest known Seifert manifolds M = S 3 , i.e. the 3-sphere. This case covers integrable field theories of the principal chiral model type, where the underlying circle bundle structure is that of the Hopf fibration of S 3 over C = CP 1 . We study, respectively, the Riemannian and Kähler metrics on the space of gauge connections A and on an important quotient space A, which is identified with the symplectic manifold X in the path integral expression (1.3). Then, we work out the Hamiltonian approach in order to implement the necessary partial gauge fixing and degenerate limit mentioned in section (3). After recovering the 4d CS theory for these theories, as an example, we end up by re-deriving the Lax connection for the λdeformed PCM. In section (5), we deduce the generalized 4d CS theory quadratic action (1.2) from T-duality manipulations. In section (6), we address at the formal level, the path integral measure and its relation to the symplectic formΩ defined on A, hence completing the construction of the canonical integral expression (1.3). Finally, in (7) we make some comments and provide further explanations concerning the results presented along the text.

Moment map and the quadratic action
The goal of this section is to construct an action functional S, on a 4-dimensional manifold M, that is proportional to the square of a moment map µ, associated to the Hamiltonian action of a symmetry group H on a symplectic space A, constructed out of the space A of Chern-Simons gauge connections defined on M. We follow [7] closely and proceed formally.
Warning. We are not specifying reality conditions over some quantities. As a consequence, some objects that are usually real by definition could appear as purely imaginary.
Consider a principal G-bundle P , where G is a Lie group with Lie algebra g, over a 4dimensional orientable manifold M, such that ∂M = 0. Denote by A the space of connections A on P and identify it with the vector space Ω 1 M ⊗ g of 1-forms on M taking values in the Lie algebra g. Denote by G, the group of gauge transformations acting on A. The Lie algebra of G, denoted by G Lie , consists of elements in the vector space Ω 0 M ⊗ g of Lie algebra valued functions on M.
Introduce a pre-symplectic formΩ ∈ Ω 2 A on the space of connections A, defined bŷ where κ, Ω ∈ Ω 1 M are globally defined and everywhere non-zero 1-forms on M that satisfy, respectively, the following set of conditions: The symbolδ denotes the exterior derivative on A, d M denotes the exterior derivative on M, R ∈ X M is a vector field on M defined by the first condition (normalization condition) in (2.2) once κ is specified, i R is the interior product or contraction with R and is the Lie derivative along R. The symbol Tr, denotes an invariant quadratic form defined on the Lie algebra g. As a consequence of (2.2) and (2.3), we also have that The 2-formΩ is closed and invariant under the action of G and, in particular, under the action of the group S = U (1) × U (1) generated by the following set of independent shifts where s, s ∈ Ω 0 M ⊗ g are arbitrary and, as a consequence of this, the 2-formΩ is degenerate along elements A of the form A = sκ, A = s Ω. Thus, we take the quotient of A by the action of the group S and define the symplectic space A = A/S. Under the quotient, the pre-symplectic formΩ on A descends to a symplectic form on A, which becomes a symplectic space naturally associated to a generalization of the 4d CS theory. The action of the gauge group G on A also descends, under the quotient, to a well-defined action on A and the 2-formΩ on A is invariant under the action of G. The action of G is non-linear, contrary to the linear action of S.
The gauge group G and the group H acting on A in a Hamiltonian way are related, but are not the same. Thus, in order to identify H properly, we first consider the action of G on A and subsequently extend the algebraic structure of G Lie , until the defining condition of a Hamiltonian action on A is fulfilled. Because of we are interested in computing the square of the moment map µ associated to the action of H on A, the Lie algebra of H, denoted by h, must be endowed with a well-defined invariant inner product ( * , * ). If such an inner product exists, the action functional we are seeking is defined to be proportional to the square (µ, µ). However, before we continue, let us review some facts from symplectic geometry that are necessary for accomplishing this task.
Consider now, a Lie group H with Lie algebra h and a symplectic manifold X. The action of H preserving the symplectic formΩ on X. The action H X of H on X is said to be Hamiltonian, when there exists an algebra homomorphism from h to the algebra of functions on X under the Poisson bracket. In terms of the moment map µ : X → h * and the elements η, λ ∈ h, this statement is equivalent to the condition that µ satisfy where * , * is the dual pairing between h and h * , { * , * } is the Poisson bracket and [ * , * ] is the Lie bracket. The homomorphism from h to the algebra of functions on X is given by f = µ, η and the moment map µ, by definition, satisfy the relation where V (η) is the induced vector field on X generated by the action of η. The symbol d denotes the exterior derivative on X. The equation (2.6) also reflects, infinitesimally, the condition that the map µ commute with the action of H on X and the co-adjoint action of H on h * . The Poisson bracket of two functions on X, is given by the expression where V f 1 , V f 2 ∈ X X are the Hamiltonian vector fields associated to the functions f 1 , f 2 on X, respectively. The relation between V f and f is of the form (2.7), i.e. we have that df = −i V fΩ .
Let us return to the case at hand. Consider the action of the gauge group G on the elements of A, given by the known relation (2.9) Infinitesimally, we write g = exp η with η ∈ Ω 0 M ⊗ g. This action induces, on the space A, the following vector field The notation ( * , * , * ) with three entries will be clearer as we proceed. Now, we compute the moment map µ associated to the action of (2.10). Start by calculating the contraction on the left hand side (lhs) of the defining expression After integrating by parts, we obtain where F A = d M A + A ∧ A is the curvature of the connection A and A 0 is a constant connection with respect toδ. In what follows we take a vanishing basepoint, i.e. A 0 = 0. Using the general identity where θ ∈ Ω 1 M and s ∈ Ω 0 M ⊗ g, it is not difficult to show that (2.13) descends to a functional on A, as it is invariant under both transformations generated by the shift group S.
In order to find the Poisson algebra for two functionals of the form (2.13), we use the general result (2.8). We find that (2.15) Using the identity for arbitrary , ∈ Ω 0 M ⊗ g, we obtain where we notice the presence of a Lie algebra cocycle defined by the integral If the cohomology class of this cocycle is not zero, something we assume from now on, the action of the gauge group G on the symplectic space A is not Hamiltonian, as it violates the homomorphism condition (2.6). The cocycle (2.18) then determines a central extension 3 R of the gauge algebra G Lie which, as a vector space, is given byG Lie = G Lie ⊕ R and comes equipped with the bracket Using this result above gives where, for further reference, we have defined The first contribution in (2.25) is antisymmetric and exhibits a U (1) action generated by the vector field R on the Lie algebra elements of G Lie . Hence, the rigid action of R on M induces a natural U (1) group action on the gauge algebra G Lie , as well as on the space A, something we shall see right below. Notice that (2.26) is antisymmetric as well if d M Ω ∧ d M κ = 0 holds.
Let us understand now, from the symplectic geometry point of view, what are the implications of the action of the vector field R on the space of gauge connections A. The action of R on M, induces the following vector field on A, given by where p ∈ R. Notice that, because of κ, Ω are invariant, i.e. £ R κ = £ R Ω = 0, the U (1) action of R on A, also descends to a corresponding action on the quotient space A.
As done before with the action of the gauge group, now we compute the moment map µ associated to the action of the vector field (2.27). Compute then, the contraction on the lhs of the defining relation − i V (p,0,0)Ω =δ µ, (p, 0, 0) . and notice the following result Then, inside integrals, we can replace Using this fact above, we obtain that The latter expression is manifestly invariant under the action of S and also descends to A.
Notice that H = U (1) G and h = R ⊕G Lie because of H acts on the quotient space A in a Hamiltonian fashion and h is endowed with a well-defined invariant inner product.
A comment is in order. The condition that (2.26) vanishes is absent in [7,11], because of the cocycle used there is, roughly speaking, recovered from (2.25) by setting Ω → 1 and M → M. To show how such a condition emerges, we verify the invariance property of the inner product (2.42) under the adjoint action of h, which is given by (2.44) Then, (2.44) boils down to rd(η, λ) = pd(λ, φ) (2.45) and because of r, p, η, λ, φ are all arbitrary, we end up by enforcing that d( * , * ) = 0. We will interpret this as a condition to be imposed over the gauge parameters η ∈ Ω 0 M ⊗ g. Although an expression generalizing (2.45) will be considered below in (7.13), once we notice the existence of a second vector field R , besides R, acting on G Lie and A as well. More on this below.
Once h is equipped with an appropriate inner product, one is encouraged to use the definition (2.42) in order to dualize the moment map µ ∈ h * and by this we mean solving the following equation µ, (q, λ, b) = (µ, (q, λ, b)) , (2.46) for an element µ = (p, η, a) ∈ h on the right hand side (rhs). We quickly find that A few words concerning the latter result. Any 4-form γ ∈ Ω 4 M ⊗ g is proportional to (2.43) and can be written as for some φ ∈ Ω 0 M ⊗ g. Thus, 'dividing' by (2.43) actually means picking the term φ, so that γ This notation is unusual but it is quite useful for performing algebraic calculations.
Let us simplify the expression (2.47) further. Consider now the contracted 5-form From this follows that we can express the contraction i R A in an equivalent way Also, introduce the quantities In this way, we get a more compact expression for the moment map (2.47), i.e.
Under the shift κ A = A + κs, we have that while under Ω A = A + Ωs , we get To show this, we use the identity (2.14). Thus, the moment map µ ∈ h is, as expected, invariant under the action of the shift group S and it is a well-defined functional on the quotient space A. In particular, any object defined in terms of it is invariant too. Now we verify the equivariance property of µ under the action of h on A. The co-adjoint action of an element (q, λ, b) ∈ h on the moment map µ ∈ h * , is defined by where we have emphasized that µ depends on A. Recall that V (p, η, a) represents the action of (p, η, a) ∈ h on A. By pairing this expression against an element (q, λ, b) ∈ h and working out (2.40) to first order we find, after some algebra, that is the adjoint action of (p, η, a) on (q, λ, b), which is given by the bracket (2.38).
For completeness, it is important to study how µ behaves under the action ofG ⊂ H, i.e. under finite gauge transformations. After some algebra, we obtain cf. [11] µ ( g A) , (q, λ, b) = Ad * (0,g,a) µ(A), (q, λ, b) = µ(A), Ad (0,g,a) (q, λ, b) (2.61) where I = g −1 d M g. To first order in η with g = e η , we have in agreement with (2.58) for p = 0. Furthermore, by setting ν = (q , λ , b ) ≡ Ad (0,g,a) (q, λ, b) we may ask if the norm of ν = (q, λ, b) is preserved under the Adjoint action ofG. From the inner product (2.42), we find 65) where we have used £ R I = g −1 d M £ R gg −1 g and integrated by parts with ∂M = 0. Now, using the contracted 5-form where we have introduce a 3-form χ(g) ∈ Ω 3 M , defined by Then, we obtain the equivalent form The second term on the rhs right above is related to the 'obstruction' (2.26), while the third term resembles the behavior of a Chern-Simons theory under the action of finite gauge transformations. It is a modified Wess-Zumino (WZ) term. This will be verified later on when we consider the quadratic expression (µ, µ) for the dualized moment map µ. To first order in η the inner product is then invariant if (2.26) vanishes.
After this digression on symmetry properties, we now proceed to simplify the quantity a defined in (2.54). Let us write (2.72) Now, consider the contracted 5-form (2.74) Now, inserting both results above into their respective positions, gives (2.75) We can simplify this expression even further by considering the contracted 5-form Using this result and the definitions introduced above, allows to write being an element Ω 3 M , is the well known Chern-Simons 3-form. Armed with these results, now we are able to compute the square of the moment map (2.47) with respect to the inner product (2.42). As an element of h, the moment map is of the form µ = (p, η, a), thus At this point, we see that the 1-form Ω corresponds to a generalization of the 'twist' form ω entering the definition of the conventional 4d CS theory (1.1). The action (2.81) is also invariant under the following changes (rescalings) The S-invariant action functional on the quotient space A, is then defined by (2.83) The action (2.83) can be interpreted as a quadratic completion of the 4d CS theory defined in terms of the Lagrangian density L ∼ ω ∧ CS(A). To find its functional variation, we use the following results , where we have used (2.14).
To simplify this expression further, consider the contracted 5-form From this result, it follows that Then, the general variation takes the form and from this we get the equations of motion (eom) of the theory, which are given by To understand better the rôle played by the object Φ introduced before in (2.52), we rewrite such a definition in the equivalent form This means that Φ is actually determined by wedging the eom of the theory against the 1-form κ. Later on, we shall see how (2.90) raises as a genuine eom from a well-defined variational problem applied to an extended action functional S(A, Φ).
Now, let us we analyze the behavior of the quadratic expression (2.80) under the action of the gauge group G. Using (2.9), we get the transformation properties of the CS 3-form under the action of finite gauge transformations, namely, Using the contracted 5-form with B(J) given by the second relation of (2.52) with J in the place of A.
Putting all together gives, cf. (2.69) The last contribution on the rhs right above is again related to the obstruction (2.26), while the second term on the rhs is a generalization of the usual WZ shift χ(g) common to CS theories. Thus, for a path integral formulation of the theory it is reasonable to consider that further analytic constraints over the gauge group elements g must, in principle, be imposed in order to obtain an honest gauge theory. In this paper we will adopt this point of view.
The eom (2.89) also reflects a lack of gauge covariance. If we demand that the eom (2.89) are preserved under gauge transformations, i.e. if then an equation for g emerge. Indeed, using and (2.89), we find that (2.96) holds, provided g ∈ G satisfy the following equation This equation can be solved ∀A, if we choose g to be such that B(J) = 0. Notice that (2.26) can be written alternatively as Then, if B(j) = 0 for j = I = J = 0, we have that (2.99) vanishes as well. In this case, (2.97) reduces to g Φ = g −1 Φg (2.100) and the field Φ then transforms in the adjoint. Below, we will see how (2.26) extends the usual 'boundary' condition imposed over the gauge parameters η in conventional 4d CS theories, that they must vanish at the set of poles of the twist 1-form ω.
We now proceed to verify if the quadratic action (2.83) is a real number. To show it, we invoke the same reality conditions used in [12], when applied to the conventional 4d CS theories. In the present case, these conditions impose suitable properties to be obeyed by the 1-forms Ω, κ and the gauge connection A. Notice that all manipulations done above have been at the formal level and, as a consequence of this, it should not come as a surprise to find that some objects are purely imaginary instead of real. This is because we have not specified any reality conditions on the Lie algebra g, the gauge connection A, the central extensions, the manifold M and so on.
Let us first declare that the Lie algebra g is actually a complex Lie algebra. Let τ : g → g be an anti-linear involutive automorphism. It provides g with an action of the cyclic group Z 2 . Its fixed point subset is a real Lie subalgebra g R of g, regarded itself as a real Lie algebra. The anti-linear involution τ is compatible with the bilinear form on g, in the sense that for any a, b in the Lie algebra g or by extension in G Lie . Denote by x a set of local coordinates on the manifold M and endow it with a complex structure. The complex conjugation x → x defines an involution ν : M → M, which also provides M with a Z 2 action. We then require γ ∈ Ω p M and ρ ∈ Ω p M ⊗ g to be equivariant under this action of Z 2 , i.e.
For instance, γ represents any of the differential forms Ω, κ and their exterior derivatives and ρ represents the gauge connection A the Lie algebra valued functions Φ and the contraction i R A. The rest of the proof goes exactly like in [12] (see §2.5) after noticing the fact that ν has the effect of conjugating the complex structure on M and thus also of reversing its orientation. Let us perform a sample computation to see how this works. Consider the first term in (2.81) and conjugate it, then (2.103) After repeating for each contribution in (2.81) (or in the form (3.23) below), we end up with As a consequence, the action (2.83) obeys S = S and S ∈ R. The gauge transformations are also required to preserve the conditions (2.102) for ρ = A, which is equivalent to restricting the gauge elements to the ones satisfying the condition τ (g) = ν * g. In the latter expression, τ also denotes the lift of the antilinear automorphism τ acting on G Lie to the group G. With this, the last contribution to (2.95) also flips sign. After repeating the same procedure for the pre-symplectic form (2.1), we conclude thatΩ = iΩ is real.
For future reference, now we introduce a second vector field R ∈ X M besides R, defined by the following set of conditions As a consequence of (2.105) and (2.106), we also have that The obstruction (2.26) can be written in terms of the vector field R as well. In fact, by using the contracted 5-form A possible solution to the condition B(d M g) = 0, may be taken to be of the form 5 Not to be confused with the γ used above.

Relation to the 4d Chern-Simons theory
We now solve explicitly the conditions (2.2) and (2.3) imposed over κ and Ω in a simple way.
To do so, we take the 4-dimensional manifold M to be of the form M = R × M, where the R factor is identified with the time direction and where M is taken to be the total space of a non-trivial circle bundle of degree n = 0 over a Riemann surface C. The S 1 fiber is seen as a closed string and locally, the product Σ = R × S 1 having the topology of a cylinder, is identified with the closed string world-sheet. Thus, we have that M is defined by This is a natural generalization of the space M = Σ × C originally used to define the 4d CS theories, in which M = S 1 × C is a trivial S 1 fiber bundle over the base manifold C. In the present case, the non-trivial bundle structure in (3.1) provides extra room for introducing geometric structures that can be exploited in order to generalize the 4d CS theories. In particular, the circle bundle (3.1) admits the existence of invariant contact structures defined on the total space M. We follow [7] closely.
Consider the following solutions κ, Ω ∈ Ω 1 M to the conditions (2.2) and (2.3), given by We have chosen α τ = 0 ∈ R and ζ > 0 ∈ R to be non-zero constants, α ∈ Ω 1 M to be a contact 6 1-form on the 3-dimensional manifold M and ω = π * ω C ∈ Ω 1 M , with ω C ∈ Ω 1,0 C to be a (1, 0)-form on the Riemann surface C. The 'twist' 1-form ω C is a meromorphic differential with a set of zeroes and poles on C denoted, respectively, by z and p. Notice that, in order for the solutions (3.2) to exists, the manifold M must admit a contact structure, hence (M, α) is required to be a contact manifold 7 . Furthermore, over (3.2) we have imposed the conditions ∂ τ α = 0 and ∂ τ ω = 0. Concerning the rescalings (2.82), we notice that κ → tκ, imply and from this follows that Ω is invariant, as initially assumed in (2.82). Split now the exterior derivative where d M denotes the exterior differential on M. 6 Any compact, orientable 3-manifold possesses a contact structure [13]. Recall that M = R × M is assumed to be orientable and such that ∂M = 0. 7 For a comprehensive description of contact manifolds, the reader is referred to the references [14][15][16] The total space M admits a free U (1) action arising from the rotations of the fibers S 1 and also admits a natural contact structure, which is invariant under the action of U (1). Such a contact 1-form α is introduced via the Boothby-Wang construction [17], see also [15,16]. This means that α is defined by the relation and where n > 0 (after a suitable choice of orientation) is the degree of the bundle M. The geometric meaning of the relation (3.6) is that α defines a U (1)-connection on M regarded now as the total space of a U (1) principal bundle over C, that is induced by the symplectic form σ C and has a non-trivial curvature given by d M α. The contact 1-form α constructed in this way is U (1) invariant, globally defined on M and satisfies the contact condition 8 that the top-form α ∧ d M α ∈ Ω 3 M is nowhere vanishing all over M. The action of U (1) along the fibers is generated by the Reeb vector field R ∈ X M . It is a non-vanishing vector field, globally defined over M and canonically associated to α by the normalization condition α(R) = i R α = 1. The 1-form α is invariant because, as a connection 1-form on M, it separates any tangent space T p M over p ∈ M into vertical and horizontal spaces. R is vertical and d M α is a horizontal 2-form, thus £ R α = 0. Furthermore, the integral of α over any fiber S 1 is normalized to one, i.e.
Hence, after a fiber integration, we get that is never zero because, by assumption, the bundle M is non-trivial.
Returning to the solutions (3.2), for the vector field R ∈ X M , we have taken All conditions written in (2.2) and (2.3) are satisfied after noticing that ω is a horizontal 1-form, i.e. it is obtained by pulling ω C back from C to M, with vanishing dτ component. Actually, both conditions are solved in general by first considering for α τ , f τ ∈ Ω 0 M arbitrary non-zero functions. The normalization condition i R κ = 1, requires that f τ = 1/α τ and i R (d M κ) = 0 is equivalent to ∂ τ α − dα τ = 0. The condition i R Ω = 0 can be solved by taking such that i R ω = 0 and with Ω τ ∈ Ω 0 M being an arbitrary non-zero function. Finally, the invariance condition £ R Ω = 0 imply i R (d M Ω) = 0, which is equivalent to Taking α and ω as defined above and restricting Ω τ to be a constant, requires α τ to be a constant too. Thus, in (3.3) we have chosen a simple non-trivial solution. The explicit form of the vector field R ∈ X M introduced above in the last section, will be given in due course. Now, we consider a key result concerning the 4-form (2.43) introduced above. Notice that because of ω C ∧ σ C ∈ Ω 2,1 C is a (2, 1)-form on C, hence it vanishes by dimensionality reasons. Then, the result (3.14) imply that the 4-dimensional top-form (2.43) takes the more explicit form This expression is globally defined on M and nowhere vanishing. It is basically proportional to the volume form dVol M of the manifold M. If we integrate (3.15) over M, we get where we have integrated the τ direction over a finite interval of size ∆τ . At this point, it is interesting to compare (3.15) with the top-form obtained from the symplectisation [16] R × M, σ = d M e ζτ /2 α of the contact manifold (M, α), which is given by The factor e ζτ can be absorbed into σ but this spoils the closedness of the symplectic form σ.
In what follows, we denote d M = d in order to avoid clutter.
To see how the action (2.83) generalize the conventional 4d Chern-Simons theories, we proceed by interpreting ζ as a deformation parameter. Exhibiting ζ explicitly in the expres- we find that Right above, we get and Now, inserting these results into the action (2.81), we obtain the following expansion where (3.23) The first term in the rhs of the first line above matches perfectly with the 4d Chern-Simons theory (1.1). Thus, if we are interested in recovering the 4d CS theories it is desirable to find a way to do so. Fortunately, inspired by (3.22) we can implement the following two-step strategy: • Step I, we gauge fix the κ-shift symmetry by imposing the gauge fixing condition Φ ≈ 0.
This step simplifies drastically the expressions (3.23) and we end up with an expansion (3.22) involving only the powers ζ 0 = 1 and ζ 1 = ζ in the deformation parameter ζ.
Because of κ Φ = Φ + s, this gauge fixing condition is accessible. Yet, we still need to verify if it is a good gauge fixing condition. This requires running the Dirac algorithm. The partially gauged fixed theory is still invariant under the Ω-shifts. • Step II, we take ζ → 0 at the end. This is a degenerate limit rendering several expressions ill-defined. For instance, from (3.18) we realize that in this limit, the 2-form Ω ∧ κ vanishes at the set of zeroes z of the twist 1-form ω and the term d M Ω ∧ κ localizes at the set of poles p of ω. In the first case, the pre-symplectic form (2.1) vanishes at the set z and from (3.15), we have that the first contribution to the inner product (2.42) is absent. Thus, ζ can be interpreted also as a regularizing parameter.
Notice that these steps make no sense if performed in reverse order. However, by assuming everything is fine we get, after setting Φ = ζ = 0, the partially gauge fixed action When ζ → 0, Ω = ω and the Ω-shift symmetry is reduced to ω A = A + sω. There is a new 'boundary' term in (3.24) that is not present in the original theory 9 . However, by demanding that this contribution vanish, we can fix part of the analytic structure of the connection A that later on will define the Lax connection L of an integrable field theory associated to the 4d CS theory. We will work out this explicitly in the next section. Then, if the field A satisfies the condition

Principal Chiral Model type theories
In this section, we specialize the construction introduced above to a particular situation and consider an example corresponding to the description of integrable field theories of the Principal Chiral Model (PCM) type, i.e. we take M = S 3 , C = S 2 . We first gather some basic results concerning the Hopf fibration of S 3 . Then, following [7] we show that the induced metric on the space A is Kähler with respect to the symplectic formΩ and a complex structure J to be defined below. We also perform the Hamiltonian analysis in this case, where we implement a partial gauge fixing for the action of the shift group S by means of the condition Φ ≈ 0 and subsequently, take the limit ζ → 0, where we recover the known 4d CS theory action. Finally, as an example of a solution to (3.25), we re-derive the Lax connection for the lambda deformed PCM.

A contact form and the Hopf fibration
This case corresponds to the Riemann surface C = CP 1 , which is the spectral space associated to integrable field theories of the principal chiral model type [5]. Thus, we have a S 1 bundle over S 2 and is the Hopf fibration. The degree of this bundle is n = 1 [18]. Furthermore, it is known that S 3 is one of the simplest Seifert manifolds [8].
Now we gather some basic facts concerning the Hopf fibration of S 3 and then move to the explicit construction of the contact 1-form α and its associated Reeb vector field R. Here, for sake of completeness, we try to be as self-contained as possible.
Consider S 3 as the unit sphere in C 2 with coordinates (z 0 , z 1 ) and CP 1 as the quotient space of S 3 under the equivalence relation (z 0 , z 1 ) ∼ λ (z 0 , z 1 ), for any λ ∈ S 1 . Define the projection π : S 3 → CP 1 by the natural map where [z 0 , z 1 ] are the homogeneous coordinates of CP 1 . Let be the coordinate charts of CP 1 . From (4.3), we have that the local coordinates on U 0 and U 1 are, respectively, given by w = z 1 /z 0 and z = z 0 /z 1 , so that w = 1/z .
The bundle structure is introduced via the local trivializations with inverses given by For j = i, one has that so the transition map is given by Let us write some expressions in a more explicit way. Define S 2 and S 3 by the elements (u 1 , u 2 , u 3 ) ∈ R 3 and (x 1 , x 2 , x 3 , x 4 ) ∈ R 4 obeying u 2 1 + u 2 2 + u 2 3 = 1 and x 2 1 + x 2 2 + x 2 3 + x 2 4 = 1, respectively. In terms of the complex coordinates of C 2 introduced above, we set z 0 = x 1 +ix 2 and z 1 = x 3 + ix 4 . The latter defining the complex structures to be considered here.
Let (x, y) be the stereographic projection coordinates of a point in the southern hemisphere U 1 of S 2 from the north pole. We have that and In a similar way, the stereographic coordinates (u, v) of the northern hemisphere U 0 projected from the south pole are (4.10) Then, On the equator of S 2 , u 3 = 0, |z 0 | = |z 1 | = 1/ √ 2 and the transition function t 01 become From (4.4) and (4.5), we write for i = 1, the bundle coordinate relations Now we use the local trivialization coordinates (4.12) over the chart U 1 ⊂ CP 1 in order to compute the contact form α and mainly to run the Hamiltonian analysis of the theory below 10 .
Notice that π : (z 0 , z 1 ) → z = z 0 /z 1 . Consider the symplectic form on CP 1 , given by the Kähler form (4.13) To find the pull-back π * σ C , simply take z = z 0 /z 1 in order to obtain a local expression on M, which is given by where we have used |z 0 | 2 + |z 1 | 2 = 1 and d |z 0 | 2 + |z 1 | 2 = 0 to reach the final form. The contact 1-form is then given by Now, the Reeb vector field satisfying the condition α(R) = 1, is and its integral curves, given by with t ∈ R are, not surprisingly, the S 1 fibers of the Hopf bundle. Then, In real coordinates, we have and from this follows that where is the volume form of R 4 and i is the inclusion map S 3 → R 4 . The 2π 2 is the 3-dimensional surface volume of a 3-sphere of unit radius. The integral of α ∧ dα over M is one, cf. (3.9).
In the local bundle coordinates (z, σ), which are the ones that we will use later on to run the Dirac algorithm, we have (4.22) The contraction (2.51), is equivalent to where we have used the decomposition Then, in these coordinates the Reeb vector field takes the simple form and from (3.10), we get that Alternatively, the expression (4.26) can be obtained from (4.16) and (4.12).
In what follows we will set α τ = α σ . Locally, the product Σ = R × S 1 is identified with the closed string world-sheet cylinder and the integral curves of the vector field R are spirals drawn on Σ. After introducing light-cone coordinates via the definitions σ ± = τ ± σ, ∂ ± = 1 2 (∂ τ ± ∂ σ ), we have that R = 4π∂ + . Then, (3.10) can be understood as a global definition of the light-cone vector ∂ + ∈ T Σ. The global counterpart of ∂ − , denoted by R and announced before, will be introduced later on when needed. Now, we verify explicitly the conditions (2.102) for κ, Ω with x = (τ , σ, z) and x = (τ , σ, z). The reality conditions for κ are trivially satisfied, while for Ω they imply that ϕ(z) = ϕ(z). The latter condition is equivalent to the statement that the zeroes and the poles in the sets z, p, are either real of coming in complex conjugate pairs [6,12]. In what follows, we will assume that this is the case.
Let us consider the expression (2.26) in more detail. After writing we get that Then, the condition for the inner product (2.42) to be invariant under the adjoint action of h, requires restricting the elements η ∈ Ω 0 M ⊗g of the gauge algebra to depend on the world-sheet coordinates τ , σ only through the light-cone coordinate σ + . In general they may be chosen to satisfy the global condition £ R η = 0, see (2.109). In the degenerate limit ζ → 0, it suffices instead to impose the condition η| p = 0, that the gauge parameters vanish at the set of poles p of the twist 1-form ω. This is already a well known fact in the literature. We also verify explicitly that d M Ω ∧ d M κ = 0 in both situations, so the expression (2.26) is anti-symmetric.

Riemannian and Kähler metrics on A and A
We work now in the Hopf coordinates, 0 ≤ η ≤ π/2, 0 ≤ ξ 1 , ξ 2 ≤ 2π, which are defined by z 0 = e iξ 1 sin η, z 1 = e iξ 2 cos η. In these coordinates, the round metric of M = S 3 takes the form We want to relate g M to the pull-back of the Kähler metric g C on C associated to the Kähler form (4.13) given by and to the contact form α. In order to compute π * g C , we set z = z 0 /z 1 as before and obtain π (π * g C ) = dη ⊗ dη + sin 2 η cos 2 η (dξ 1 − dξ 2 ) ⊗ (dξ 1 − dξ 2 ) . (4.34) The metric g M , then takes the compact form In this guise, we see that the action of the vector field R is an isometry of the metric g M and this follows from as can be seen from the expression for dα written in (4.31).

Now, we introduce a Riemannian metric g M on the 4-dimensional manifold
where ρ is a real positive constant to be fixed below.
In matrix form, we have that In order to proceed, we need an explicit expression for the second vector field R ∈ X M , satisfying the conditions (2.105) and (2.106). We quickly find that Thus i R Ω = 1, because of i R ω = 0. In the local bundle coordinates (z, σ), the vector field (4.38) becomes R = (1/Ω τ )∂ − and we see that (4.38) corresponds, up to a constant, to a global definition of the world-sheet light-cone tangent vector ∂ − ∈ T Σ. Its integral curves are also spirals drawn on Σ. Thus, the vectors ∂ ± span T Σ locally, as expected.
Consider now the contracted 5-form for any γ ∈ Ω 1 M . This result allows to write the contraction with R as Using a similar result for i R γ, see (2.51), we get the basic contractions (4.42) From these results, we find that Thus, R and R are light-like Killing vectors. They are not orthogonal under g M and the explicit value for g M (R, R ) will not be important in what follows.
Once g M have been defined, we consider now a Riemannian inner product on the space A of gauge connections defined by the expression for A, A ∈ A and use it to implement an orthogonal decomposition of A in terms of the quotient space A = A/S and space S of Abelian shifts. The Hodge duality operator * to be used in what follows is defined in terms of the metric introduced in (4.37), the coordinates x µ = (τ , η, ξ 1 , ξ 2 ), with µ = 1, 2, 3, 4 and the orientation dτ We also have the useful expressions where we have used * κ = √ ρ with ω = m dη +i sin η cos η (dξ 1 − dξ 2 ) , * ω = m √ ρ sin η cos ηdτ ∧dξ 1 ∧dξ 2 −2πidτ ∧dη ∧α We find that As a consequence, the norm of any element A can be decomposed in the form Now, we proceed to relate the first term on the rhs in (4.55) to the pre-symplectic form (2.1) in order to define a Kähler metric structure on A. Using and the contracted 5-form we have that In a similar way, we use and the contracted 5-form to find that where we have introduced two 'effective' Hodge duality operators defined by The expression (4.63) can be simplified a bit more if now we introduce the quantity Thus, after using the fact that * 2 κ = * 3 κ = 0, we get , (4.65) Explicitly, we find that where we have taken and used (4.31) with α = α 1 dξ 1 + α 2 dξ 2 . Notice that the 1-forms dη and (dξ 1 − dξ 2 ) span the space ker i R = Ω 1 M,Hor of horizontal 1-forms defined by the vector field R, cf. (4.36) above. On the one hand, we obtain that showing that the second contribution on the rhs in (4.63) is absent. On the other hand, from the explicit expressions * 2 (dτ ) = 0, we find that the 'effective' 2-dimensional Hodge duality operator * 2 actually defines a complex structure on the space Ω 1 M,Hor , i.e. it obeys * 2 2 = −1, provided we fix the constant ρ to be It satisfies the basic relation where we have used the defining relation (3.6) with n = 1.
As a consequence of these results, we get the desired relation withΩ being the pre-symplectic form defined in (2.1). Recall that nowΩ| A is symplectic.
Let us now consider the gauge group G and use (2.110) to write (4.74) Using g A = g −1 Ag + I and (4.74) in (4.64), gives Then Π ( g A) ⊂ ker i R ∩ ker i R and the quotient space A is preserved by the action of a 'restricted' gauge group defined by the elements g satisfying (2.110).
Notice that d M Ω ∼ dη ∧ (dξ 1 − dξ 2 ). In this case, in order to get d M Ω ∧ γ = 0, we propose a linear combination of the form This choice in (4.75) provide the usual gauge field transformations with g obeying £ R g = 0, namely Furthermore, the WZ-type term (2.67) becomes The expression (4.74) exhibits an orthogonal decomposition, under the inner product (4.44), of the current I ∈ Ω 1 M ⊗ g along im i R and ker i R ⊂ ker i R while keeping I ∈ ker i R . This follows from the fact that im ⊥ i R = ker i R , because of Alternatively, in the local bundle coordinates (z, σ) over U 1 , we have that d M Ω ∼ dz ∧ dz and this time we propose instead Using this in gives for the gauge field transformations. The condition £ R g = 0, now takes the form This is precisely the condition ∂ − g = 0 we found before. The WZ-type term takes the same form as in (4.80), but with the volume form dτ ∧ dσ ∧ dz ∧ dz.
Now, we consider the WZ-type term. Using (4.74) in (4.80), allows to write 11 with I = g −1 dg being the contribution along M in the decomposition I = I τ dτ + I. In principle, the WZ-type term contribution can be set to zero if further analytic restrictions are imposed over the elements g ∈ G. For example, by enforcing the holomorphicity condition ∂ z g = 0 when using the local bundle coordinates (z, σ). Also notice the possibility of taking the limit ζ → 0 as well. It is important to emphasize that at this level of analysis it is not clear if (4.87) could, alternatively, be related to some quantization condition as occurs in conventional CS theories.
Finally, the result (4.73) shows that the induced metric by the Riemannian metric (4.44) on the quotient space A is Kähler, with respect to the symplectic formΩ| A and complex structure This is an important result and we will come back to it later on when considering the symplectic measure of the generalized 4d CS theory path integral.

Hamiltonian analysis and 4d Chern-Simons theory
In order to perform the Hamiltonian analysis, we first need to isolate the time differential dτ from all expressions. The exterior derivative d M was already introduced in (3.5) and now we write it in terms of the local bundle coordinates (z, σ). Thus, Some quantities of interest to be used below are: i) The curvature of the connection A under the decomposition (4.24), namely, ii) The Lie algebra value field Φ defined previously in (2.52), which now takes the form and iii) The Chern-Simons 3-form Also, introduce the following variables They are invariant under κ-shifts and will be useful for writing results coming from the Hamiltonian analysis in a more compact way.
From the expression (2.81), we quickly obtain the Lagrangian of the theory (4.95) Because of the Lagrangian is quadratic in Φ, we see from (4.91), that the theory is not only linear in the velocities ∂ τ A but also quadratic on them.
To find the canonical momenta, we compute the variation of the Lagrangian (4.95) with respect to all field components in δ(∂ τ A). Using the following result where the 2-form P ∈ Ω 2 M ⊗ g is given by The components in the expansion are, actually, the usual canonical momenta defined by where we have used the local decompositions Also, notice that From the absence of the term ∂ τ A τ in the Lagrangian and the very form of P , we easily detect the presence of three primary constraints given by P τ ≈ 0,κ ∧ P + icΩ ∧κ ∧ A ≈ 0,Ω ∧ P ≈ 0. (4.103) In components, the last two constraints can be written in the form

105) and
φΩ = Ω z P z + Ω z P z + Ω σ P σ ≈ 0. (4.106) Then, the three primary constraints are Let us now introduce the compact notation The canonical Poisson bracket of the theory is defined by where i, j = τ , σ, z, z. Then, the phase space coordinates satisfy the usual relations where C 12 = η AB T A ⊗ T B is the tensor Casimir of the Lie algebra g, δ σσ = δ(σ − σ ) and δ zz = δ(z − z ) are Dirac delta distributions. The latter can be written locally in the form Define now the quantities where s, s ∈ Ω 0 M ⊗ g are arbitrary and φ κ = κ τ P τ + φκ, φ Ω = Ω τ P τ + φΩ.
under the bracket (4.110) and Poisson commute between.
The canonical Hamiltonian is given by the Legendre transformation and the total Hamiltonian is given by where u τ , u, u ∈ Ω 0 M ⊗ g are arbitrary Lagrange multipliers. The variation δH must depend [19] only on the variations (δA, δP ) and (δA τ , δP τ ). In order to find it, we first compute the variation of the P -dependent contribution to H, then use (4.98), (4.102) and finally subtract δL. We find that (4.117) To compute δL correctly, we calculate the variation of L without performing any integration by parts involving the derivative ∂ τ along the τ direction. Thus, by using the identity (4.118) with X = Φ being an element of Ω 0 M ⊗ g, we find from (4.95), that Then, only depends on the phase space coordinates variations, as required.
Before we continue, as a consistence test, let us find the Lagrangian eom of the theory. After integrating by parts the term ∂ τ (δA) and simplifying, we obtain The eom extracted from (4.121) also derive from (2.89) and both are perfectly equivalent. Now, we return to the expression (4.120) and find the functional derivatives of H with respect to the phase space coordinate fields. We get that where we have used (4.124) Armed with these expressions, now we are able to verify the time preservation of the primary constraints. Starting with P τ , we find that and from this we obtain a secondary constraint given by Now, we consider S κ (s) and obtain (4.127) By using (4.92), (3.2), we get that and from (4.23), (4.25), we have For S Ω (s ), we follow a similar calculation to obtain Not surprisingly, the canonical Hamiltonian of the theory is invariant under the two independent shifts in S.
There is a single secondary constraint γ(η). From the expression (4.126), it is not difficult to verify that it is invariant under both shift transformations as well, hence It remains to verify if its time preservation introduce a tertiary constraint or if instead it determines some of the Lagrange multipliers. From the expression we realize that this is actually a condition over the Lagrange multiplier u τ , provided that δγ(η) δAτ = 0 and it is not difficult to check that this is indeed the case. The explicit form for u τ is not required in what follows, hence we do not need to perform the calculation explicitly.
It is clear from (4.126) that the computation of the Poisson algebra for two γ s is rather tedious. Fortunately, in the present work, we are only interested in implementing the step one mentioned above, see (3), in the Hamiltonian formulation. Thus, we gauge fix the first class constraint that generates the κ-shift symmetry by choosing the following gauge fixing condition Φ ≈ 0. (4.135) Now, the pair φ κ ≈ 0 , Φ ≈ 0 of constraints become a second class set and we impose them strongly by means of a Dirac bracket. Notice that Then, (4.135) is also a good gauge fixing condition.
The Dirac bracket is given by  and equals the canonical Poisson bracket if we restrict to phase space functionals that are κ-shift invariant.
Step one is accomplished.
The partially gauge fixed theory is no longer quadratic in the velocity ∂ τ A but linear and this restructures completely the whole set of Hamiltonian constraints allowing to 'restart' the Dirac algorithm again. To see this, consider a theory with a Lagrangian that can be decomposed in the form L = L 0 + L 1 + L 2 , where L 0 , L 1 and L 2 are the terms independent, linear and quadratic in the velocities. The Hamiltonian energy function is H = L 2 − L 0 , i.e. the quadratic term is preserved, the linear term is absent and the velocity-independent term flips sign. As the gauge fixed theory is now linear in the velocities, the Hamiltonian energy function and the canonical Hamiltonian coincide when restricted to the constraint surface defined by (4.98), after taking Φ = 0. Thus, H = −L 0 and (4.98) leads now to three new 'primary' constraints.
Explicitly, the gauge fixed action is invariant under the remaining Ω-shifts, it is given by and has the following Lagrangian (4.140) To verify the invariance of (4.139) under Ω-shifts it is useful to notice that We restart the Dirac procedure by taking (4.139) as the new action functional. For the canonical momentum, we have now P τ = 0, P = icΩ ∧ A, (4.142) or in components, leading to the existence of four primary constraints, given by The total Hamiltonian is now where We expect the first class constraint associated to the Ω-shift symmetry to re-emerge as a particular linear combination of the four primary constraints (4.144).
From (4.146), we get the functional derivatives where this time we have defined which is nothing but (4.126) with Φ = 0. We also find that are actually conditions over the Lagrange multipliers u σ , u z , u z . Concerning the time preservation of the secondary constraint (4.150), a similar argument leading to (4.134) holds, hence no new constraints are produced.
By introducing the constraint 2-form we recover the Ω-shift symmetry generator S Ω (s) = Tr(s, Ω τ P τ +Ω ∧ φ) (z,σ) , At this stage it is necessary to separate the set of constraints found so far as first or second class constraints. However, we will not continue along this line but rather implement the step two mentioned above, see (3), which corresponds to taking the degenerate limit ζ → 0. Recall that here we are interested in recovering the conventional 4d CS theories from the Hamiltonian theory point of view and not in pursuing a thorough Hamiltonian analysis of the generalized 4d CS theory.
In the ζ → 0 limit, we have that where the component Ω z = ϕ is identified with the twist function of the underlying integrable field theory. We also supplement the limit with the condition (3.25). The action functional (4.139), the Lagrangian (4.140) and the canonical Hamiltonian (4.146) are those of the 4d CS theory and are given, respectively, by (3.26) and (4.156) Step two is accomplished.
Concerning the Hamiltonian constraints, we have from (4.144) and (4.150), that and The constraint φ z ≈ 0, is actually an identity reflecting the fact that the field component A z completely decouples from the theory. The rest of the Hamiltonian analysis follows exactly the lines considered in [21,22], to which the reader is referred for further details. Thus, we will not repeat their results here, but instead gather some relevant facts to be used later.
The constraint P τ ≈ 0 is first class and can be gauged fixed by choosing a gauge fixing condition of the form i.e. the component A τ is chosen to be a function of the gauge field component A σ evaluated at the set of poles p of the twist 1-form ω. The constraint φ z together with the component A z , can be ignored. The constraints φ z and φ σ form a second class pair and (4.158) is a first class constraint, provided we restrict the gauge parameters to satisfy the condition η| p = 0, which is a possible way for canceling the obstruction (4.29). The latter constraint, i.e. γ(η) ≈ 0, then reduces to F A ≈ 0 and as a gauge fixing condition, we can choose A z ≈ 0, (4.160) together with the restriction ∂ z g = 0 over the gauge parameters. The 4d CS theory is completely recovered but apparently it is not gauge invariant because of the WZ-type term is now proportional to ω ∧ χ(g), as is well known in the literature.

Recovering the lambda-PCM Lax connection
Here, we quickly explore the implications of the condition (3.25) in determining the analytic structure of the lambda-PCM Lax connection.
In the gauge Φ = 0, the equations of motion (2.89) reduce to and (2.90) becomes a trivial identity. The gauge fixed action functional is (4.139) and in the ζ → 0 limit, it becomes (3.24), i.e.
The eom (4.161) reduce to In this case, the action and the eom are given, respectively, by Notice that, despite of the fact that we are imposing the condition (4.164), both expressions in (4.165) are invariant under the residual ω-shift symmetry given by ω A = A + sω. This follows from the fact that dω ∧ ω = π * (d C ω C ∧ ω C ) = 0, where d C is the exterior derivative on the base manifold C. Locally, we have ω = ϕ(z)dz, thus the gauge field component A z dz decouples from the theory, a prominent characteristic of the action (1.1).
In components, the eom in (4.165) are equivalent to the set of equations ϕF zµ = ω zz A µ , ϕF τ σ = 0, (4.166) where µ = τ , σ and dω = ω zz dz ∧ dz. The lambda deformed PCM is specified by the twist function [23] ϕ where a, b ∈ R. The zeroes z and the poles p of ω on the chart U 1 are located at z = ±1 and z = z ± = ±a, respectively 12 and all are real numbers. In local coordinates around each pole in the chart U 1 , we have that where we have used the expression (4.111). Then, The first equation in (4.166), in the gauge A z = 0 (cf. (4.160)), imply Thus, we have with H µ (z) holomorphic. Consistency, i.e. A µ (z)| z=z ± = A µ (z ± ), fixes H µ (z) and we end up with the result . (4.172) This connection interpolates between the set of poles p.
Now we solve the condition (4.164), which is equivalent to This equation can be solve by taking the following linear combination (cf. (4.159)) Then, (4.174) imply that The solutions are where s = ±1 and s = ±1 are sign functions. Thus, (4.172) for µ = τ , becomes In the light-cone coordinates the Lax connection L ± (z), now identified with A ± (z), has the following analytic structure for some world-sheet currents I ± , i.e. the Lax connection has poles at the zeroes of the twist 1-form ω. Using the solutions (4.172) with µ = σ, (4.178) and comparing with (4.180), we find that s = s = −1 and identify The latter are nothing but the lambda deformed PCM currents [24,31]. which are the lambda deformed PCM eom. They are also the conventional PCM eom. It is interesting to notice the instrumental rôle played by the condition (4.164) in deriving the Lax connection of the associated integrable field theory.

Quadratic action from duality
In this section, by using 'T-duality', we construct from zero a generalization of the 4d Chern-Simons theory. The argument follows the same logic used in [7] but adapted now to the present case, where we have two shift transformations in S. The original 4d CS theory need to be modified in an specific way in order for 'T-duality' to be consistently defined. We also show how the quadratic action (2.83) emerges naturally as a dual model.
We start with the 4d CS theory on M = Σ × C, with action (1.1) The difficulty in dualizing (5.1) comes from the structure of the 'twist' form ω and the 4d manifold M, which turns the action ill-defined for implementing 'T-duality' via gauging. In what follows, we shall break the strategy for constructing the dual model in the correct way into two simple steps.
The first step consist in regularizing the theory. Instead of (5.1) we consider its Ωshift invariant extension (4.139), which is defined now on the manifold M = R × M, with a non-trivial circle bundle space M and with action functional defined by where we have used (2.91), (2.92) and defined The last contribution on the rhs of (5.5) is related to the obstruction (2.26), while the second term is the WZ-type term found before.
The second step consist in introducing the κ-shift symmetry. To do so, we consider a Lie algebra valued field Φ ∈ Ω 0 M ⊗ g, which transforms like for an arbitrary s ∈ Ω 0 M ⊗ g under κ-shifts and like Φ → t −1 Φ under the arbitrary rescalings (2.82). We also demand that it is invariant under Ω-shifts, i.e. Ω Φ = Φ. The key idea now [7] is to notice that the combination A − κΦ is invariant under κ-shifts and rescalings. Thus, a double-shift invariant action functional is obtained from (5.2) after making the substitution A → A − κΦ. We find that 13 We have not specified the behavior of Φ under gauge transformations. However, in order to be consistent with the transformation of Φ, as defined in (2.52), under gauge transformations (2.97) we demand that g Φ = g −1 (Φ + B(J)) g. (5.9) Then, we get for some W ∈ Ω 0 M ⊗ g. Again, there is a term related to (2.26) and a WZ-type contribution. Even if we set g Φ = g −1 Φg, we obtain the same type of expression with a change W → W for another W ∈ Ω 0 M ⊗ g, whose explicit form is not relevant in what follows. Now we consider the 'T-duality' manipulations. Using the κ-shift symmetry we can reach a gauge where Φ = 0, in this case we recover the regularized theory (5.2). Alternatively, if we calculate the Φ field eom and put them back into the action (5.8), we find a dual action that is classically equivalent to (2.83). Indeed, the Φ eom is nothing but the first expression defined in (2.52), i.e.
and fulfills all the required transformation properties, see for instance (2.55) and (2.56). At this point we can see an advantage of the regularized theory, as the denominator in the expression right above never vanish, making the variational problem for the quadratic field Φ well-defined over the manifold M. Otherwise, we would have a contribution of the form ω ∧ dα = 0, see (3.14), in the second term on the rhs of (5.8) turning the theory linear in the field Φ. The effective action obtained after replacing (5.11) in (5.8) gives the dual action S dual = ic(µ, µ), (5.12) which is precisely the S-invariant quadratic action constructed in (2.83). Once in the form (5.12), we can recover (5.2) if we gauge fix the κ-shift symmetry with the gauge fixing condition Φ = 0. The original theory (5.1) is finally recovered by taking the degenerate limit ζ → 0 and by imposing the boundary conditions (4.164) on the gauge connection A. This further clarifies the why of the two step strategy introduced before in section (3). From (5.11), we find that Φ(A = sΩ) = 0, Φ(A = sκ) = s (5. 13) and from this result we obtain S(A = sΩ) dual = 0 and S(A = sκ) dual = 0. This confirms that (5.12) descends to the quotient A, where a second component of A manifestly decouples from the theory. The number of components of A, as a Lie algebra valued 1-form is two, see (4.66) and (4.84). Recall that we still have the action of the gauge group G on A.
The following diagram roughly summarizes our findings: Steps I and II supplemented with a solution to the condition (3.25).

Path integral and non-Abelian localization
In this section we comment on the second ingredient involved in the formulas of non-Abelian localization, i.e. the path integral symplectic measure. The first one, discussed extensively above, being the quadratic form of the action functional. See [7], for the original 3-dimensional CS theory formulation.
Consider the theory defined formally by the path integral where the action in the exponential is given by (5.8), N is defined by N = N × 1 Vol(S κ ) × Vol(S Ω ) × Vol(G) (6.2) and N is a normalization constant. As we have shown in (2.55), (2.56), (5.13), the field Φ belongs to the orbit generated by the action of the κ-shift group S κ , while it is a fixed point under the action of the Ω-shift group S Ω . Then, the integral associated to the measure DΦ is over S κ , while the integral associated to DA is, as usual, over the whole space A of gauge connections.
The translation-invariant measure DΦ is defined independently of any metric on M by the invariant, quadratic form This quadratic form is, up to scale, used to formally define the volume of S κ . Similar expressions are used to define S Ω and G, as anticipated before in (2.42) and (4.50). This is to be complemented with the formal definition of the translation-invariant measure DA induced by the norm (4.72) and its orthogonal decomposition between the spaces A and S.
On the one hand, using the κ-shift symmetry, we can fix Φ = 0 trivially with unit Jacobian, and the resulting integral over S κ produces a formal factor of Vol(S κ ). Hence, the theory is equivalent to (5.2), i.e. to the Ω-shift invariant extension of the 4d CS theory (1.1). The resulting action functional is valued in the quotient A/S Ω and DA integrates over A/S Ω × S Ω , where the integral over S Ω produces a formal factor or Vol(S Ω ). Then, in principle, we get On the other hand, because of the field Φ appears only quadratically in the action (5.8), we can perform the path integral over Φ directly. Integrating out Φ, produces a contribution The resulting action functional is given by the dual quadratic expression (5.12) and is valued in A. Thus DA integrates over A × S. Here, we make use of the results (4.72), (4.73) related to the fact that the quotient space A is symplectic and equipped with a Kähler metric in order to write the measure along A in the form (see (1.5)) DA| A = expΩ. (6.6) As we showed above, this measure is to be taken over elements Π(A) ∈ A of the form (4.64). An important consequence of the fact that the metric on A is Kähler is that the Riemannian measure DA on A is actually the same as the symplectic measure defined byΩ. Indeed, if X is a symplectic manifold of dimension 2n with symplectic formΩ, then the symplectic measure on X is given by the top-formΩ n /n!. This measure can be represented by the expression expΩ, where we implicitly pick out from the series expansion of the exponential the term which is of top degree on X. Consequently, because of the Riemannian and the symplectic measures on A agree, we can formally replace DA over A in the path integral by the expression (6.6) above and write instead where we have used Vol(S) = Vol(S κ ) × Vol(S Ω ) and defined = /2c. This integral takes the canonical form (1.3) with X = A, as required by the non-Abelian localization method. The normalization constant N can be adjusted to an specific valued if needed.
A consequence of an expression like (6.7) is that it suggests an interesting relationship between the quantum integrable structure of the 4d CS theory and the geometry of the symplectic quotient space A. In principle, the 4d CS theories could be explored via localization.

Concluding remarks
Clearly, the pre-symplectic form (2.1) plays a crucial rôle in the construction of the path integral (1.3), as it specifies the moment map µ used to define the quadratic action S = ic(µ, µ), as well as the path integral symplectic measure e Ω over the quotient space A. Thus, a comment on what inspired its definition is in order.
Consider the original pre-symplectic form defined in [7], which in the present notation takes the formΩ = − 1 2 M α ∧ Tr δ A ∧δA . (7.1) The circle fibers, i.e. the 'strings', correspond to the integral curves of the Reeb vector field R satisfying the normalization condition α(R) = 1. After introducing the time direction, we extend M to M = R × M. Locally, the manifold M looks like M = Σ × C and the light-cone tangent vectors to the string world-sheet Σ, are given by ∂ ± ∼ ∂ τ ± ∂ σ . It is then natural to extend them, respectively, to their global counterparts (3.10), (4.38) and to introduce two 1-forms κ and Ω such that κ(R) = 1 and Ω (R ) = 1, see (3.2), (4.39). Because of R defines vertical and horizontal directions in the tangent space of the total space M, we can add any horizontal 1-form ω with no dτ term to κ or to Ω , without spoiling the normalization conditions and this is because i R ω = i R ω = 0. We choose to add it to Ω and define ω as the pull-back, by the projection map π, of the twist 1-form ω C on C that specifies the associated integrable field theory. There is some room to introduce an arbitrary parameter, which we call ζ. Then, a natural generalization of (7.1) to four dimensions, in which we include the time direction, is given bŷ Now we show how the 1-forms Ω, κ of (3.2) and used in (7.3) are constructed. Let us start with the interpolating expressions and where s ∈ (0, 1) and ζ ∈ R. They satisfy the normalization conditions i R κ = i R Ω = 1. By demanding that Ω ∧ κ ∧ d M κ = ζdτ ∧ α ∧ dα, we find The final step is to verify if the inner product just defined is invariant, which is equivalent to having, cf. (2.44), [(p, η, a) , (q, λ, b)] , (r, φ, c) = (p, η, a) , [(q, λ, b) , (r, φ, c)] , (7.12) wherer = r + r . This conditions boils down tõ (7.13) We solve this by taking d( * , * ) = 0 and p = q = r = 0. Thus, the vector field (7.7), the bracket (7.10) and the inner product (7.11) reduce to the ones considered before. As a consequence, the quadratic action remains unaltered. The vector fields R and R have different uses in the formulation of the generalized 4d CS theory, at least as implied by the solutions to (7.13) chosen in the present paper. It would be interesting to consider other possible solutions and their implications.
For integrable field theories on (semi)-symmetric spaces, the formulation presented here requires to consider non-trivial S 1 bundles over the base space C = CP 1 /Z 4 . Coset spaces of the form CP 1 /Z T were first considered in [26] on an approach devised to reformulate Z Tgraded coset σ-model as dihedral affine Gaudin models. The particular case T = 4, was also studied in [27], where the symmetric-space λ-model exchange algebra was recovered from the point of view of the conventional 4d CS theory. It is then desirable to study the generalized 4d CS theory on non-trivial circle bundles over spaces of this type, due to their relation to important non-ultralocal integrable field theories like the σ-models on (semi)-symmetric spaces and their integrable deformations [28][29][30][31][32][33][34] too. We expect to consider this in the near future.
The generalized theory presents a behavior, under the action of finite gauge transformations, that is similar to the conventional 4d CS theory. Thus, the last couple of terms in (2.95) must be properly handled first in order for the expression (6.7) to make perfect sense at the quantum level. In this work we have adopted the strategy of imposing restrictions over the gauge elements g ∈ G in order to cancel both contributions, turning the generalize theory gauge invariant. We do not know if this approach is the only way to do it or if there is some gauge group structure that can be exploited instead. We expect to consider this subtle issue in a more systematic way elsewhere.