Integrable Lattice Models and Holography

We study four-dimensional Chern-Simons theory on D x \C (where D is a disk), which is understood to describe rational solutions of the Yang-Baxter equation from the work of Costello, Witten and Yamazaki. We find that the theory is dual to a boundary theory, that is a three-dimensional analogue of the two-dimensional chiral WZW model. This boundary theory gives rise to a current algebra that turns out to be an"analytically-continued"toroidal Lie algebra. In addition, we show how bulk correlation functions of two and three Wilson lines can be captured by boundary correlation functions of local operators in the three-dimensional WZW model. In particular, we reproduce the leading and subleading nontrivial contributions to the rational R-matrix purely from the boundary theory.


Introduction
A relatively novel approach to the study of integrable lattice models underlaid by the Yang-Baxter equation with spectral parameters is that of four-dimensional Chern-Simons theory, first proposed by Costello [1,2], and subsequently studied by Costello, Witten and Yamazaki [3][4][5] The theory is defined by the path integral involving the classical action where A is a complex-valued gauge field, Σ is a 2-manifold, and C is a Riemann surface endowed with a holomorphic one-form ω = ω(z)dz. It is topological along Σ (modulo a framing anomaly), but has holomorphic dependence on C, and moreover has a complex gauge group, G. As shown in [3,4], the nontrivial operators of the theory are Wilson lines, whose correlation functions realize the Yang-Baxter equation with spectral parameters, as well as the underlying Yangian algebra, quantum affine algebra, and elliptic quantum group of its rational, trigonometric and elliptic solutions, respectively. Furthermore, the boundary Yang-Baxter equation can likewise be realized by studying 4d Chern-Simons theory on an orbifold [6,7]. For C = R × S 1 , the 4d Chern-Simons action can be dimensionally reduced along S 1 to that of 3d analytically-continued Chern-Simons theory. In fact, the corresponding quantum field theories have been shown to be T-dual [8]. However, unlike ordinary 3d Chern-Simons theory, much of the work on 4d Chern-Simons relied on the path integral and Feynman diagrams alone, and no use was made of canonical quantization or holography. This was due to the infrared-free nature of 4d Chern-Simons, whereby it was straightforward to deduce a local procedure to compute the expectation values of Wilson line configurations of interest.
Nevertheless, given the importance of the 2d chiral Wess-Zumino-Witten (WZW) model dual to 3d Chern-Simons theory as a straightforward example of a holographic dual, and for describing edge modes of the nonabelian fractional quantum Hall effect, it is of interest to investigate the existence of a holographic dual of 4d Chern-Simons theory. In this work, we shall indeed derive such a dual boundary theory for 4d Chern-Simons on D × C (where D is the disk), which turns out to be a three-dimensional analogue of the 2d chiral WZW model.
We shall focus on the boundary dual of 4d Chern-Simons with C = C, which is known to give rise to rational solutions of the Yang-Baxter equation [3]. These R-matrices are intertwining operators for representations of the Yangian algebra, and thus the classical integrable lattice models of concern are equivalent to Heisenberg XXX quantum spin chains. For example, one such lattice model is the rational six-vertex model, which is equivalent to the XXX 1/2 spin chain. As we shall see, the 3d "chiral" WZW model we derive furnishes an alternative and convenient method for computing the rational R-matrices explicitly.
Given a 3d analogue of the 2d chiral WZW model, the first natural question to ask is if it admits a current algebra analogous to an affine Kac-Moody algebra. In Section 2, we shall show that this is indeed the case, i.e., the 3d "chiral" WZW model furnishes a particular limit of an "analytically-continued" toroidal Lie algebra.
One would also like to verify that the boundary theory captures the correlation functions of the bulk theory. For instance, the bulk correlator of two straight, perpendicular Wilson lines along Σ, at points z 1 and z 2 on C, and in representations R 1 and R 2 of the generators of G, realizes the R-matrix, i.e., it is computed to be to linear order in , with the full expression for the R-matrix following from general theorems [9,10]. It would be satisfying to obtain this result solely from the boundary theory, and indeed, this is what we do in Section 3 by evaluating a four-point function of local boundary operators. Furthermore, via the boundary theory, we demonstrate the topological invariance along D of the bulk correlator at order . We also explain how one may explicitly derive higher order contributions to the R-matrix, and demonstrate this by computing the 2 contribution for two crossed Wilson lines explicitly. Subsequently, we show that the boundary theory also reproduces the correlation function of a pair of parallel Wilson lines. Finally, we consider three Wilson lines, and show that their correlation function is reproduced by a six-point function in the boundary theory.

3d "Chiral" WZW Model
Let us start with 4d Chern-Simons theory with complex gauge group, G, defined on Σ × C, where Σ is a disk, denoted D, and C is the complex plane, C. Its action is Here, A can be understood to be the partial connection where (r, ϕ) are polar coordinates on D and (z,z) are complex coordinates on C. Let us first vary the action to find the equations of motion. Doing so, one finds The second term of the variation is a boundary term via Stoke's theorem. In order to ensure that we have equations of motion free from boundary corrections, we shall impose the boundary condition Az = 0, whereupon the boundary term vanishes. This boundary condition is also necessary to achieve gauge invariance in the presence of boundaries. It can be shown that (2.1) is equivalent to where A has been extended to a full connection over D × C, i.e., A = A r dr + A ϕ dϕ + A z dz + Azdz. The boundary term on the RHS of (2.4) depends only on the components A ϕ , A z and Az, and vanishes using the boundary conditions Az = 0 and A z = 0. The remaining term is gauge invariant under large gauge transformations, i.e., However, we ought to restrict U such that the boundary conditions Az = A z = 0 are preserved. We shall achieve this by insisting that U tends to the identity element of G at the boundary. Now, having imposed the boundary condition Az = 0, the action (2.1) is equivalent to upon integration by parts. Varying the Lagrange multiplier field, Az, implements the constraint F rϕ = 0, which is solved by where g is a map g : D × C → G.
Changing variables from A r and A ϕ to g in the functional integral, we note that, just as in 3d Chern-Simons theory [11], no Jacobian appears when transforming the measure, i.e., 1 vol G DA r DA ϕ δ(F rϕ ) = 1 vol G Dg, (2.8) where the expression on the RHS is the relevant Haar measure, divided by the volume of the gauge group. Furthermore, substituting the solutions (2.7) into (2.6), we obtain the action (2.9) which takes the form of a three-dimensional analogue of the 2d chiral WZW model. Now, a gauge transformation (2.5) amounts to g → U g in (2.7). As a result, we may change the value of g in the interior without changing its value at the boundary, so (2.9) only depends on g at the boundary. This implies that we can divide out the volume of the gauge group to obtain the path integral Dg e iS(g) , (2.10) where g is now the map g : ∂D × C → G.

Current Algebra via Canonical Quantization
We are now interested in computing a Poisson bracket involving the expression J ϕ = − 1 π ∂ ϕ gg −1 , which we shall eventually use to obtain a quantum mechanical commutation relation in the form of a current algebra. In what follows, we shall takez to be the (complex) time direction. Now, given an arbitrary action that is first order in the time derivative with dynamical variables φ i , i.e., where ω ij = ∂ ∂φ i A j − ∂ ∂φ j A i is the symplectic structure on the classical phase space. The Poisson bracket of any two functions X and Y on the phase space is then defined by where ω jk ω kl = δ j l [12]. For the 3d "chiral" WZW model, the variation (2.11) implies that its phase space symplectic structure is given by ω = 1 g ⊗ (−1) π ∂ ∂ϕ ⊗ 1 z , where 1 g acts on the Lie algebra index, (−1) π ∂ ∂ϕ acts on the ϕ coordinate, and 1 z acts on the z coordinate. Its inverse is therefore Let us now compute the Poisson bracket of X = TrA ∂g ∂ϕ g −1 (ϕ, z) and Y = TrB ∂g ∂ϕ ′ g −1 (ϕ ′ , z ′ ), where A and B are arbitrary generators of the group G. In the notation of (2.15), this can be done by evaluating δXδY = ∂X ∂φ i ∂Y ∂φ j δφ i δφ j , and subsequently replacing δφ i δφ j by ω ij . Proceeding in this manner, we find (2.19) To obtain the Poisson bracket, we ought to replace (g −1 δg(ϕ, z)) a (g −1 δg(ϕ ′ , z ′ )) b (where a and b are Lie algebra indices) by (2.21) In the quantum theory, this Poisson bracket corresponds to the canonical commutation relation This is in fact equivalent to the current algebra To express this algebra in a more familiar form, we expand the currents in terms of their Fourier modes along S 1 , and utilize the orthogonality of these modes, which leads us to (2.25) This algebra has the form of a Kac-Moody algebra with generators having holomorphic dependence on the Riemann surface, C. Note that there is no quantization condition on here, unlike the current algebra derived from the boundary theory of ordinary 3d Chern-Simons theory.
To further understand this algebra, we shall write z = ǫt + iθ, compactify the θ direction to be valued in [0, 2π], and subsequently take ǫ → 0. Upon doing so, we may perform another expansion in Fourier modes, i.e., Then, employing the orthogonality of these modes, the resulting algebra is This is known as a two-toroidal Lie algebra (or a centrally-extended double loop algebra), which, in particular, arises as the current algebra of the four-dimensional WZW model [13]. Hence, the algebra (2.25) that we obtained can be understood to be an 'analytical continuation' of the two-toroidal Lie algebra (2.27), with one of the two 'loop directions' decompactified. This is not surprising, considering the fact that 4d Chern-Simons theory for C = R × S 1 can be understood to be 3d Chern-Simons theory for the loop group, but with the 'loop direction' complexified [14].

Wilson Lines and Boundary Local Operators
We shall now describe Wilson lines in 4d Chern-Simons theory in terms of local operators of the boundary theory. This is possible due to the flatness condition that restricts the gauge field components on D to be pure gauge configurations, as shown in (2.7). To see this, note that a Wilson line along a curve C, starting at t i and ending at t f , and in a representation R, satisfies If t f and t i are points on ∂D = S 1 , this implies that a bulk Wilson line operator can be completely described in terms of a pair of local boundary operators. In fact, such boundaryanchored Wilson lines are automatically gauge invariant, since all gauge transformations are trivial at the boundary. This suggests that correlation functions of Wilson line operators in 4d Chern-Simons theory can be captured by correlation functions of local operators in the 3d "chiral" WZW model. This includes correlators of crossed Wilson lines which compute the R-matrices of integrable lattice models. We shall now attempt to verify this.
In the bulk computation by Costello, Witten, and Yamazaki [3], the order contribution to the R-matrix was found by performing perturbation theory around the trivial field configuration, A = 0, and computing free-field propagators between Wilson lines. In the same vein, we shall consider perturbation theory around the field configuration g = 1, and we shall use a free-field propagator to compute the relevant correlation function involving operators appearing on the right hand side of (3.2), i.e., The relevant operators are depicted in Figure 1. Now, we may expand the field, g, as g = e φaT a = 1 + φ a T a + . . .
The free part of the 3d WZW action is then where we have performed integration by parts after expanding the field g.
Next, from the free action, one may construct the generating functional where x = (ϕ, z,z), y = (ϕ ′ , z ′ ,z ′ ), and ∆ ab is the propagator which obeys and is given explicitly by ∆ ab (x) = δ ab 1 2πi Moreover, the propagator satisfies ∆ ab (x − y) = ∆ ba (y − x). The two point function can be found from (3.5) to be Before we proceed, we first note that the propagator (3.7) has no dependence on thē z coordinate. This can be understood from the fact that there is a gauge redundancy in the 3d "chiral" WZW model, namely the invariance of the parametrizations (2.7) under transformations generated by Ω in (2.14). This redundancy can be fixed by setting B(z,z) = 1 in (2.13). As a result, the operator g has noz-dependence, and therefore correlation functions involving it should not havez-dependence either.
Another pertinent point to note is that the propagator is a multi-valued function, as it includes the expression on the RHS of(3.8) as a factor. Hence, to obtain a single-valued propagator, we ought to define it with a branch cut. We shall pick the branch cut to be from r = 0 to (r = R, ϕ = π), where R is the radius of D. This effectively restricts ϕ in (3.8) to take values in (−π, π). In this manner, we obtain a well-defined, single-valued propagator. Now, to compute the RHS of (3.3), we shall expand each operator to linear order in φ (3.11) We then only keep terms of quadratic or lower order in the fields, while taking selfcontractions (i.e., correlators of operators with the same value of z) to be zero via regularization. Also note that 1-point functions can be shown to be zero using (3.5). Hence, we find (3.12) Here, we have used the fact that ∆ π 2 = 1 2π and likewise ∆ − π 2 = − 1 2 . We have thus obtained, from our 3d "chiral" WZW model, the exact order correlation function for a pair of perpendicular Wilson lines that Costello, Witten and Yamazaki [3] computed via the bulk 4d Chern-Simons path integral.

Non-perpendicular Wilson Lines
We can generalize the calculation above to the case of non-perpendicular Wilson lines. As an example, we shall start with two perpendicular Wilson lines, and rotate the vertical Wilson line clockwise by an angle, δ, as shown in Figure 2. The four-point function we should compute is (3.14) Expanding each operator to linear order in φ as in (3.11), we find Now, note that (3.8) can be rewritten as This implies that sin(k π 2 )cos(kδ) + cos(k π 2 )sin(kδ) k , (3.17) and We find that the sums over k above can be separated into two types of sums, each having the form of a Fourier series, namely, the Fourier series for a square wave, ∞ k=1 sin( kπ 2 )cos(kx) k = π 4 sign(cos(x)), (3.19) and the Fourier series for a sawtooth wave, for x ∈ R. However, single-valuedness of the propagators involved in the computation (3.15) requires that − π 2 < δ < π 2 , implying From here we find that ∆ π 2 +δ = 1 2 for − π 2 < δ < π 2 . As a result, (3.15) is in fact independent of the angle δ, and agrees precisely with the result we found for perpendicular Wilson lines. Hence, we once again find agreement with the results of Costello, Witten and Yamazaki [3].

Arbitrarily Crossed Wilson Lines
We can generalize the preceding calculations further to more general configurations of crossed Wilson lines, for which we expect to obtain the same result as (3.12) due to the topological invariance of 4d Chern-Simons along D. For instance, we can consider both Wilson lines rotated from perpendicularity, as shown in Figure 3. The corresponding four-point function is unaffected by the additional rotation, i.e., we find (where − π 2 < −α + δ < π 2 to ensure single-valued propagators) with the use of the identity (3.23). A different generalization is that of perpendicular Wilson lines crossing at a point that is not the origin, r = 0, as shown in Figure 4. The four-point function in this case is also independent of the angles β and ρ shown in the figure, i.e., we have (where − π 2 < β + ρ < π 2 and − π 2 < β − ρ < π 2 to ensure single-valuedness of propagators) using the identity (3.23). Note that the allowed ranges of β + ρ and β − ρ mean that the result is valid only when the Wilson lines are crossed.
Let us now study the most general case. We shall show that the four-point function corresponding to any arbitrary configuration of crossed Wilson lines has the same expression. Such a configuration, as depicted in Figure 5, is determined by four angles, namely α, β, γ, and ρ. The four-point function is then (3.26) Here, to ensure single-valuedness of propagators, we require − 3π However, to ensure that we are considering only crossed Wilson lines, we require the stronger conditions − π 2 < β−ρ−α+δ < π 2 , − π 2 < −β+ρ−α+δ < π 2 , − π 2 < β+ρ−α+δ < π 2 and − π 2 < −β − ρ − α + δ < π 2 . These conditions in turn allow us to use (3.23), whereby we find that (3.26) is We have thus shown that topological invariance along Σ of the bulk correlation function of two crossed Wilson lines is reflected in the dual four-point function of the boundary 3d "chiral" WZW model, at least up to order .

Crossed Wilson Lines at Order 2
Having found the expected O( ) contribution to the four-point function of arbitrarily inserted crossed Wilson lines in (3.26), we may improve on this and compute higher order contributions by using (free-field) n-point functions defined from the generating functional (3.5), for even n. We shall demonstrate this explicitly for the O( 2 ) contribution to the correlation function of perpendicular Wilson lines.
Firstly, from the generating functional (3.5), the (free-field) four-point function can be found. Expanding the operators in the RHS of (3.3) to quadratic order in φ as we then find via (3.28) that, to order 2 , (3.3) is 30) where we have used the notation ∆ ab ϕ = ∆ ϕ δ ab for brevity. In a similar manner, one can compute contributions to the R-matrix of order 3 and above. Note that these contributions are not expected to remain invariant under moves of the local boundary operators that correspond to rotations and translations of the bulk Wilson lines, due to the framing anomaly that arises in the bulk theory at order 2 for non-perpendicular Wilson lines [3]. This framing anomaly ought to be computable in our boundary WZW model as well, by taking into account its interaction terms when computing correlation functions of local operators.
A slightly more involved calculation shows that the result of (3.30) holds, modulo the framing anomaly, for arbitrarily inserted Wilson lines (as depicted in Figure 5), assuming the same constraints on the angles given below (3.26).

Parallel Wilson Lines
The OPEs of parallel Wilson lines in 4d Chern-Simons theory do not have the same singular behaviour as correlation functions of crossed Wilson lines. In this subsection, we shall consider such correlation functions of parallel Wilson lines and show how they are captured by correlation functions of boundary operators. We shall focus on the free-field limit, at order , and retrieve the expected behaviour in this regime.
Using (3.2), the correlation function of the operators we are interested in (depicted in Figure 6) is  Expanding each operator to linear order in φ and keeping only terms of quadratic or lower order in the fields (as in the discussion below (3.11)), we have which is non-singular for z 1 = z 2 at order , as expected. Here, we have used the previously derived fact that ∆ π 2 = 1 2 , as well as ∆ 3π 2 = 1 2 , which follows from (3.17), (3.19) and (3.20), and (3.33) Also, note that we must define the propagator here with a different branch cut from that of previous sections, namely, from r = 0 to (r = R, ϕ = 0). This effectively restricts ϕ in (3.8) to take values in (0, 2π).

Three Wilson Lines
We next consider correlation functions of three Wilson lines, all crossing each other but otherwise inserted arbitrarily (c.f. Figure 7), which corresponds to the following boundary correlator: 34) where the dependence on C has been suppressed for brevity. Expanding each operator in (3.34) to linear order in φ we find where various constraints on the angles are necessary for single-valuedness of propagators and to ensure that the Wilson lines are all crossed. Once again, there is agreement with the bulk 4d Chern-Simons computation. We may further compute the correlation function (3.34) to order 2 by expanding the operators in (3.34) to quadratic order in φ. Doing so, we find the following expression (modulo the framing anomaly): (3.36) This result agrees with the bulk 4d Chern-Simons computation. To see this, let us consider the configurations in Figure 8. From the bulk theory, we know that the equivalence of these two configurations gives rise to the Yang-Baxter equation where (3.38) We shall now make use of the identity which follows since the classical r-matrix, r ij = T a R i ⊗T R j a z i −z j (i, j = 1, 2, 3, where j > i), obeys the classical Yang-Baxter equation. Using (3.39), the last three terms at order 2 of (3.36) can be shown to be 40) whereby (3.36) agrees with the LHS of (3.37). Alternatively, we can use (3.39) such that the last three terms at order 2 of (3.36) are 41) whereby (3.36) agrees with the RHS of (3.37). Thus, the boundary six-point function (3.36) is in agreement with the bulk correlation function of three Wilson lines up to order 2 , modulo the framing anomaly. We expect that this will hold at higher orders of as well.