Discrete Toda, Y-System and Chaos
We discuss a discretization of the quantum Toda field theory associated with a semisimple finite-dimensional Lie algebra or a tamely-laced infinite-dimensional Kac-Moody algebra , generalizing the previous construction of discrete quantum Liouville theory for the case . The model is defined on a discrete two-dimensional lattice, whose spatial direction is of length . In addition we also find a “discretized extra dimension” whose width is given by the rank of , which decompactifies in the large limit. For the case of or , we find a symmetry exchanging and under appropriate spatial boundary conditions. The dynamical time evolution rule of the model is a quantizations of the so-called Y-system, and the theory can be well-described by the quantum cluster algebra. We discuss possible implications for recent discussions of quantum chaos, and comment on the relation with the quantum higher Teichmüller theory of type .
Masahito Yamazaki (山崎雅人)
Kavli IPMU (WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan\institution
Center for the Fundamental Laws of Nature, Harvard University, Cambridge, MA 02138, USA
- 1 Introduction
- 2 Discrete Toda Theory
- 3 Quantum Chaos
- 4 Relation with Higher Teichmüller Theory
- 5 Summary and Discussion
- A Continuum Theory
- B Quantum Cluster Algebra
In celebrated papers in 1967 [1, 2], Prof. Morikazu Toda introduced the now-famous Toda lattice. Since then, the Toda lattice became one of the most important and well-studied models of integrable models. The author of the present paper himself has long been fascinated by the subject, ever since his first encountered the model in an excellent book by Toda .
In this paper, we discuss a discretization of the two-dimensional quantum Toda field theory associated with a symmetry algebra , which is either a finite-dimensional semisimple Lie algebra or an infinite-dimensional tamely-laced Kac-Moody algebra (the tamely-laced condition will be explained later in (22)). We consider the -dimensional spacetime, and discretize both the time and spatial directions. The model is thus defined on a two-dimensional lattice. When , this reduces to the discrete quantum Liouville theory, discussed in [4, 5, 6, 7, 8].
There are several motivations for the quantum discrete Toda theory.
First, such a discretization is suitable for analyzing the Toda field theory as an integrable model. While Toda (and especially Liouville) theory has been discussed extensively in the literature, most papers resort to conformal field theory (CFT) techniques. However, Toda theory is also integrable, where the integrable model techniques [9, 10] in discrete spin chains models should apply. Discrete model also serves as a UV regularization of the continuum theory.
Second, discrete Toda theory has been the birthplace of the (non-compact) quantum dilogarithm function , which later appeared in a number of different contexts in physics and mathematics, including quantum Teichmüller theory [12, 13, 14], complex Chern-Simons theory [15, 16, 17] and finally in 3d supersymmetric theory , as related by the 3d/3d correspondence [19, 20] (see [21, 22] for derivation, also ) and the gauge/YBE correspondence [24, 25, 26] (see also [27, 28]).
More recently, discrete Liouville theory has been studied in the context of quantum chaos . Namely, it is a concrete -dimensional lattice model saturating the conjectured bound  for chaos, and hence can be thought of as a higher-dimensional counterpart of the -dimensional model proposed by Sachdev, Ye, and Kitaev [31, 32]. Discrete Toda theory is a natural generalization whose semiclassical holographic dual (in the large central charge) contains particles with spin greater than , which is of interest in view of the recent constraints on such theories from causality  and quantum chaos [30, 34].
Readers should keep in mind that in the literature there has been many papers on discretizations of classical Toda equations, as early as in the seventies [35, 36]. The connection of the discrete classical Toda system with the classical Y-system (see [37, 38]) and cluster algebra, to be discussed below, is also known in the literature. Furthermore quantum aspects of discrete Liouville theory was discussed in [4, 5, 6, 7, 8], and discrete Toda theory in 111The latter reference also discussed the limit, to be discussed in this paper..222The references [40, 41] also discuss discretization of the quantum Liouville and Toda theory.
In this paper, we reformulate many of these pioneering results in the literature in modern machinery of quantum cluster algebras. Formulated this way, much of the involved computations [4, 5, 6, 7, 8, 39] are replaced by rather simple combinatorics of quivers and mutations. We discuss Toda theories associated with more general symmetry groups and with more general boundary conditions than known in the literature, discuss subtle relations with quantum higher Teichmüller theory of , and point out the connection with recent exciting developments of quantum chaos.
In the rest of this paper, in Sec. 2 we first we describe the model and comment on some of its properties. We then comment on the possible implications to quantum chaos in Sec. 3 and clarifies a relation connection with Teichmüller theory in Sec. 4, We conclude this paper with some closing remarks in Sec. 5.
2 Discrete Toda Theory
In this section we introduce and motivate the discrete Today theory (see Appendix A for continuum theory and for some notations).
As we will comment in Sec. 2.3, the formulation below of the discrete models will work both for a finite-dimensional Lie algebra as well as an infinite-dimensional tamely-laced Kac-Moody algebra. This will include most of the infinite-dimensional affine Lie algebras. In order to simplify the presentation, for the most part we choose the symmetry algebra to be a simply-laced simple finite-dimensional Lie algebra. We will comment on the non-simply-laced case and the infinite-dimensional case in Sec. 2.3.
2.1 Discrete Model
Let us now come to the definition of the statistical mechanical model. We choose to use the Hamiltonian viewpoint, first quantize the theory along a fixed time slice, and then consider its time evolution.
2.1.1 Dynamical Variables
Let us first consider the situation with a fixed time slice (say at time ). Here, we have a discrete lattice whose lattice points are labeled by a pair of integers.
First, we have an integer , labeling the “internal symmetry” of the Toda theory. Second, we have another integer for the spatial directions, which runs from to , where specifies the length of the spatial direction. This integer is taken to be infinity in the continuum limit.
In the following we sometimes combine into a single index, which we denote by , where is the index set .
For each vertex we associate a variable ; these will be the dynamical variables of the theory. One the boundary of the spatial directions, we can either consider either
(P): a periodic boundary condition, in which case is considered to be modulo : , in which case the spatial direction is a circle, or
(F): a fixed boundary condition, in which case we allow the integer to take values in and , and fix their values to be .
In each of these cases we can choose to take , in which case we have an infinite chain.
2.1.2 Commutation Relations
At the fixed time slice at time , let us next set a commutation relation between the ’s.
This is determined by the symmetry algebra , as well as another Lie algebra , which is defined to be
depending on the boundary condition. We again allow , leading to or .
Now that we have a pair of Lie algebras and , we can define a two-dimensional quiver by combining the Dynkin diagram for and that () for , see Figs. 1 and 2. This quiver is called the square product of and , and is often denoted by
The definition of the square product is hopefully clear from Fig. 2, but we can give a more formal definition . For this, let us first write choose a bipartite coloring of the Dynkin diagrams and ,444For periodic boundary condition, we assume to be even, so that is bipartite. and we orient their edges so that all the arrows are starting from (ending at) white (black) vertices. Then we first form the tensor product of and , whose the vertices are the pair of the vertices () for (), and the number of arrows from to is
zero if and
equals the number of arrows from to if
equals the number of arrows from to if
Then the square product is obtained by reversing all arrows in the full subquivers of the form and where is a sink of and is a source of .
Since a vertex of the Dynkin diagram of (of ) are labeled by an index (), a vertex of is labeled by , the set labeling the ’s.
We can now state the commutation relation amongst ’s at time :
where we defined the anti-symmetric matrix (with ) by the relation
The non-commutativity parameter in (3) is the quantum parameter, namely plays the role of the Planck constant. This parameter is to be identified with the same parameter appearing in Sec. A for the continuum theory. Let us here remark that the semiclassical limit in the continuum theory is , namely , which is also the semiclassical limit of the discrete model.555The Toda theory is known to have a symmetry (recall that is simply-laced in this subsection), as is manifest in the formula (33). This means that we should consider the so-called modular double, and should consider two copies of (3), one with and another with (see e.g.  for some more details). While this is important for some considerations of the Toda theory, this is not necessary for the semi-classical consideration of the theory , including the application to quantum chaos discussed in Sec. 3.
Note that in logarithmic variables (), the commutation relation (3) can be written as
After some linear change of basis this reduces to the canonical commutation relations, which we can easily quantize following the standard canonical quantization procedure.
Note that the commutation relation (3) allows for a finite-dimensional cyclic representation when is a root of unity; we then have a quantum mechanical model where everything is regularized to be finite. This can be still consistent with the semiclassical limit when we take with a large integer .
2.1.3 Time Evolution
Let us now consider the time evolution. This is described by the following equation, which we call the quantum Y-system (for simply-laced ):
where () is the Cartan matrix666For example, for we have (7) for (for ) and we have defined .
This equation means that is determined by a finite number of -variables at times and . As shown in Fig. 3, for the case of the time evolution is determined by the “octahedron rule”, namely the unknown dynamical variable at a vertex of the cube is determined once we know those variables at all other vertices of the cube.777Such a time evolution pattern for the classical case is called the octahedron recurrence , see also literature on discrete integrable systems, such as [47, 48]. This is also reminiscent of tensor networks.
Note that the fixed boundary condition (F) stipulates that the terms involving and do not appear on the right hand side of (6).
For the simply-laced case discussed here, the Dynkin diagram is bipartite and the dynamics of the ’s for white vertices (“white dynamics”) and those for black vertices (“black dynamics”) decouple. For this reason we can choose to keep only one of them, and in fact in the formulation of the discrete Liouville theory in [4, 5, 6, 7, 8] only one such copy is retained. However, such a decoupling of black and white dynamics is no longer present when is non-simply-laced, as we will comment on Sec. 2.3.
Notice that we can easily show the the causality of the system:
where the distance is defined as the number of edges in the shortest path (graph geodesic) connecting two vertices and . Note that in this definition we disregard the orientation of the graph, and use only the resulting unoriented graph.
The minimal set of ’s needed for the future/past time evolution is shown in Fig. 4. We can think of this as the discretization of the Cauchy surface.
2.2.1 Classical Limit and Y- and T-System
In order to motivate the time evolution rule (6), let us consider the classical limit . Then the variables mutually commute, which therefore we denote by . The time evolution rule reduces to
where is the Cartan matrix for .
This set of equations is known as the (classical) Y-system of type .888For fixed boundary condition (), this is also known the Y-system of type at level . See  for survey on Y- and T-system. It is worth mentioning that Y-system also appeared in [51, 52]. 999In the literature on Y- and T-systems it is most common to denote the Y-system (T-system) in terms of capitalized letters (). In this paper we use an uncapitalized letters (), to better match with the cluster algebra notation. For the case of the fixed boundary condition, we set at the endpoints, just as in the quantum case.
The Y-system can be derived from a discretization of the Toda equation of the Hirota-type , known as the T-system. This reads
Let us first rescale the arguments by a factor of , e.g. where and are kept finite while is send to be zero in the continuum limit. Expanding the resulting equation with respect to quadratic order in , we obtain
where we defined the light-cone coordinates by . This is the Hirota bilinear form of the two-dimensional Toda equation.
There is one subtlety in the T-system, which is that it allows for some “gauge ambiguity”. Let us for example consider the case , for which case the T-system reads
This has an ambiguity of the form
While it is possible to first quantize and then divide by this gauge ambiguity, it is often more economical to first mod out by this gauge transformation and then quantize. In this case, we should consider the gauge-invariant combination, which leads to
or more generally for simply-laced
We can verify that this satisfies the classical Y-system, and thus coincides with the classical limit of the quantum variables .
2.2.2 Quantization à la Cluster Algebra
Having explained the rule in the limit, let us now come to the quantum case. Basically, what should be done is to promote the classical variables into non-commutative variables obeying (3). In particular, when we replace the classical -system (10) by its quantized version, we need to include appropriate power of (which we do not see in the classical limit) in such a way that the resulting expression is consistent with the commutation relation (3).
For some simple cases we can play around with the expressions, and after some trial and errors we can arrive at the expression (6). However, we can also appeal to the more general mathematical theory of the so-called quantum cluster algebras, which is a quantization of the classical cluster algebras .101010Our notation mostly follows those of [54, 45, 55]. This makes it possible to borrow some machineries developed for theory there. In this paper we do not provide a detailed explanation of the quantum cluster algebra, and interested readers are referred to App. B and e.g. to [56, 45].
In the cluster algebra we have two defining ingredients.
First, we have a quiver , which determines the algebra at a fixed time slice (this corresponds to Sec. 2.1.1 and Sec. 2.1.2. For our case, the quiver is the quiver (2), already shown in Fig. 2. We associate dynamical variable , the so-called quantum -variable111111The -variable corresponds to the Y-system, whereas T-system corresponds to the so-called cluster -variables. [57, 58], to each vertex of the quiver .121212If is root of unity (as discussed above to make the Hilbert space finite), then the quantum -variable reduces to the cyclic cluster variable, as discussed in .
Second, we specify a sequence of operations called “mutations” (see (34) in Appendix), where each mutation is labeled by a vertex of the quiver and is denoted by . Then a sequence of mutations is labeled by a sequence of vertices .
Such a sequence of mutations corresponds to a non-trivial time evolution (this is a counterpart of Sec. 2.1.3). It turns out that one step of the time evolutions of the discrete Toda theory (from time to ) corresponds to mutations at all the vertices of the quiver, colored either black or white in Fig. 2.131313Since none of the white vertices are connected with each other, the ordering of such mutations does not matter. Namely, if we define
then the time evolution from time to corresponds to mutations or , depending on whether is even or odd. It turns out that the order of products in (16) do not matter, since consecutive mutations at non-adjacent vertices is known to commute with each other and since the quiver is bipartite.
There is yet another advantage of the cluster-algebraic reformulation. Namely, we can write a time-evolution operator
so that in the Schrödinger picture the state evolves by :
This is because such an operator for each mutation , satisfying
has already been constructed explicitly in the literature [58, 56, 54, 45], see (36). More concretely, such an operator can be written in terms of the quantum dilogarithm function (whose argument contains an operator ), a linear operator mixing among ’s, and a permutation operator. We can then define our time-evolution operator to be one of the following, depending whether is even or odd:
In this sense, the time evolution of the discrete Toda theory has already been solved.
For example, suppose that the spatial direction is periodic. We can choose an initial state and a final state , and compute the transition amplitude as an integral expressions [54, 45]. We then have the initial and final states at the past and future boundary circles of the annulus. By conformal transformation this can be mapped into a vacuum correlation function on a sphere. Such a transition amplitude (and its trace) is known as the cluster partition function [54, 45, 55] (see also ).141414The discussion in these references are limited to the case where is simply-laced.
If the final state coincides with the time evolution of initial state (namely if ), then the overlap amplitude is known as the survival amplitude (or Loschmidt echo).
For a typical thermal system, a small disturbance of the initial state by a local operator is expected to thermalize quickly and to be washed away, in time scale of the dissipation time, of order the inverse temperature.
The situation in our model seems to be very different, for the case of the fixed boundary condition (F). It turns out that the time evolution is periodic, with period of order (which can be approximately regarded as inverse temperature of the system):
where is the dual Coxeter number of . This is known as the periodicity of the Y-system, as conjectured in [60, 61, 62] and proven later in [63, 64, 42] (including the non-simply-laced cases in [65, 66]), see  for more references.151515Typically, periodicity of the Y-system is stated for the classical Y-system, however periodicity of the classical Y-system is actually equivalent with that for the quantum Y-system, as proven more generally in . 161616 One consequence (21) is that product of the time-evolution operator from to is trivial. This gives rise to the quantum dilogarithm identity (see  for more details).
While a quantum mechanical system with a discrete spectrum in general is known to show a quantum recurrence phenomenon , this case is very special since the recurrence time here grows linearly in the degrees of freedom, not in double exponentially.171717Another difference from the general case is that the state here come back to exactly the same state, whereas in general cases the state comes back only infinitely close to the original state. We might interpret this short-time revival as a signature of integrability of the model. We will comment more on this in Sec. 3. Note that the period goes to infinity in the long-strip limit . A similar periodicity is not known for the case of the periodic boundary condition (P).
2.3 More General
Let us now comment on the case where is a more general algebra. This includes three different generalizations;
A non-simply-laced finite-dimensional simple Lie algebra
A Lie superalgebra
An infinite-dimensional tamely-laced Kac-Moody algebra
The basic story stays the same in all of these cases. First, classical T- and Y-systems for these cases are already known in the literature , which we can regard as the discretization of the the Toda equation. Then these equations can be reformulated in the language of classical cluster algebras, and by following the quantization procedure of quantum cluster algebra we obtain the discretization of the quantum Toda theory, as we wanted.
For the case of an infinite-dimensional Kac-Moody algebra, we need to impose a technical condition that the Kac-Moody algebra is tamely-laced [68, 69]. Recall that a Kac-Moody algebra is defined from a generalized Cartan matrix . This Kac-Moody algebra is called tamely-laced  if is symmetrizable and satisfies
where is a diagonal matrix symmetrizing . This includes most of the affine Lie algebras, except for and .
The quiver for the with a fixed boundary condition () is shown in Fig. 5. This quiver is not bipartite, and the vertices are labeled by , with and , with . Such a difference arises since () is a long (short) root.
The time evolution (quantum Y-system) is given by the quantization of the classical Y-system known in the literature :
Note that for -variable corresponding to the short root (), the time-evolution step is half that for the long root (). However, it is also the case that the range of is doubled, and hence the spatial spacing is also reduced by half. This means that that we still have the causality (8) with the same velocity of light, as long as we modify the definition of distance accordingly; such a definition of the distance is natural when we draw the quiver on a two-dimensional plane, as down in Fig. 5.
2.4 Exchange Symmetry
An unexpected feature of our construction is that the quiver in Fig. 2 has an obvious symmetry exchanging and . This symmetry is also observed in our time evolution rules.
Since is taken to be either or , this symmetry is present if
Fixed boundary condition (F) and
Periodic boundary condition (P) and
In either case, the symmetry exchanges and , and the corresponding indices and :181818T-system is mapped as .
This symmetry is known as the level-rank duality of the Y- (and T-) system, as noted in the context of the RSOS model [70, 71].191919Such a symmetry has been discussed in a rather different context in [43, 72, 44].
While this symmetry is trivial from the standpoint of the Y-system, this exchange symmetry (24) is rather surprising from the standpoint of the discrete Toda theory, since the role of the symmetries and are completely different– is the size of the spatial direction, while is the rank of the symmetry algebra of the theory, and a-priori there is nothing to indicate the symmetry between the two. It is also interesting that the massive deformation of a CFT, replacing by , is translated into the change of the boundary condition, from fixed to periodic.
Notice that for a general choice of (e.g. the non-simply-laced ) the choice now breaks the exchange symmetry mentioned above. This might motivate us to consider a more general theory, where specifies the type of the Toda lattice and the spatial direction. For exceptional the length of the Dynkin diagram is bounded by above, and hence we have trouble taking the continuum limit. Moreover, if we wish to obtain a periodic spatial directions then , which has a circular affine Dynkin diagram, is the only possible option, at least when is either a finite or affine Lie algebra.
3 Quantum Chaos
3.1 Bounds on Quantum Chaos
As stated in introduction, discrete Liouville theory was recently proposed to be maximal chaotic .
Recall that in classical systems the chaos refers to the sensitivity of the system to the initial condition; small perturbations of the initial conditions grows exponentially in time. In quantum systems it is more subtle to define chaos. The recent proposal is to use the out-of-time correlator of the form , where and are generic operators of the theory,202020The operators and need to be smeared in the time direction, to avoid singularities. In practice this is naturally incorporated in the -prescription for the analytic continuation from Euclidean to Lorentzian signature . and the subindex refers to the evaluation at a thermal state with temperature . The quantum chaos is then characterized by the exponential growth of this correlator, as a function of time :
Here is some time scale after which the exponential growth begins. Also, the exponent is called the Lyapunov exponent; quantum chaos is characterized by , and the larger the value of the more chaotic the system is.
In classical system the value of the Lyapunov exponent can be arbitrary large. However, in quantum systems there exists a conjectured bound 212121This assumes some hierarchy between dissipation time and scrambling time.
where is the inverse temperature and we have set .
For us, the interesting fact is that the quantum discrete Liouville theory saturates this bound, so that we have .
3.2 Quantum Chaos in Discrete Toda Theory
Since our model is a natural generalization of the discrete Liouville theory, it is natural to ask if our model adds anything to these discussions. In this section we take to be a simple finite-dimensional Lie algebra, in particular , hence the Toda theory in the continuum is conformal.
The most direct method to tackle this problem is to evaluate the out-of-time correlator explicitly in our model; we can try to take and to be for example and . Here the inverse temperature can be identified with the length of the spatial direction. Such a computation seems to be involved, and has not been done, even in the simplest case of .
Instead let us here appeal to the fact that our model reduces to the Toda field theory in the continuum. Since we expect the Lyapunov exponent to be a characterization of the effective theory and to be UV-insensitive, we expect that the exponent for the discrete and continuum theories coincide.
The continuum theory, i.e. the Toda theory, is a two-dimensional CFT with symmetry, and by taking the central charge to be large we expect that gravity is semiclassical in the holographic dual (recall that the Newton constant in the bulk is inversely proportional to the central charge).222222For existence of semiclassical holographic dual, we also need to take into account the sparseness of the spectrum for the semiclassical holographic dual [74, 75]. Such a bulk theory is known to be the Chern-Simons theory, which contains particles with spin greater than .
One might therefore conclude that for this result violates the bound (26), and hence the quantum discrete Toda theory is inconsistent, at least in the continuum limit. However this is not correct—the derivation of (27) assumes the dominance of the vacuum identity block, which does not hold in the quantum Toda theory. That quantum Toda theory is consistent is far from trivial, since its bulk dual, namely Chern-Simons theory, contains a finite number of higher spin particles, which in general is known to violate causality (after suitable coupling to matters) .
It is therefore an important problem to compute the value of the Lyapunov exponent for quantum Toda theory, both for discrete and continuum cases. Note that one existent argument for the value of the Lyapunov exponent for the case [29, section 2] relies on some results in Liouville theory [76, 77, 78], whose Toda () counterpart seems to be unknown in the literature.
We conjecture that the Lyapunov exponent is positive () and hence is chaotic, for all values of .
For better understanding of our theory, one possibility is to consider the large limit, so that we have at we have an infinite number of higher spin particles in the bulk, where the apparent discrepancy between (26) and (27) is sharpest. The “extra dimension” has width , hence decompactifies in the large limit. This is a version of the dimensional oxidation, where the two-dimensional lattice is turned into a three-dimensional lattice.232323This depends on the order of the two limits; the large limit and the continuum limit. Most naively, we should first take a continuum limit, and then take the large limit. However, that will give a diverging contribution to the Lyapunov exponent from the vacuum block (see (27)), signaling the need for resummation. This might motivate taking the large limit first. This is somewhat reminiscent of the situation in , where we first need to resum the global block into the Virasoro conformal block before taking the Regge limit. Notice that thanks to the exchange symmetry between the spatial length and rank , the decompactified dimension is on equal footing with the spatial direction.
Such a large limit should be compared with the case of the two-dimensional CFT with the symmetry, which is dual to the Vasiliev theory [79, 80] in theory. This theory has a vanishing Lyapunov exponent (), and hence is not chaotic . What happens there, at least schematically, is that we have the re-sum the infinite series representing the infinite spins, and the result has a effective spin not larger than , making the theory consistent. One possible reason behind such a miraculous resummation is the integrability of the CFT. In this respect one should keep in mind that Liouville/Toda theory is also integrable. However, integrability in itself does not necessarily guarantee that the system is non-chaotic —many of the integrable charges are non-local, and the time evolution could happen in the basis where the charges are not conserved. One could also turn on a small non-integrable deformation of the system, to ensure that the system is chaotic .
4 Relation with Higher Teichmüller Theory
In this section, let us discuss the relation of our discrete model with the higher Teichmüller theory. The higher Teichmüller theory in question will be defined on an annulus, and we choose the periodic boundary condition for the discrete Toda theory242424Essentially the same argument can be repeated for the higher Teichmüller theory on a strip and the discrete Toda theory with fixed spatial boundary condition. However, one subtlety in this case is that we need to modify the definition of the Dehn twist on the boundary of the strip. depending on the fixed or periodic boundary condition.
The reference  pointed out the equivalence between discrete Liouville theory and the Teichmüller theory on annulus. It was also pointed out that the evolution operator (recall (17)) in the former coincides with the geometrical Dehn-twist operator of the latter.
The argument of  required some complicated computations involving quantum dilogarithm functions. However, from a modern perspective there is no need to go through such complicated computations, to establish the equivalence mentioned above—this equivalence follows from the simple observation that (a) the discrete Liouville theory with periodic spatial boundary condition and (b) Teichmüller theory on annulus, are both described by the same quantum cluster algebra datum, namely by the same quiver and the mutation sequence.
To explain the cluster algebra structure for the quantum Liouville theory on annulus, let us first consider a triangulation of the annulus, as in Fig. 6.252525In Teichmüller theory, we are supposed to consider an ideal triangulation, namely a triangulation where all the vertices are located on the punctures (or marked points on the boundary) of the surface. This means we actually have an annulus with marked points, with each for the upper and lower circular boundaries of the annulus.
Given a triangulation of the annulus, we can change the triangulation, by applying the operation of Fig. 7 (call a flip) to one of the squares. By repeating this flip as in Fig. 8, and then changing the relative positions of the two boundaries of the annulus, we can realize the so-called Dehn twist, as applied to the triangulation (Fig. 8).
In order to make contact with this geometrical picture with the more algebraic setup in cluster algebras, let us first associate a quiver of Fig. 9 to each triangle of the triangulation. Here the quiver has three vertices, each of which is shown as a square box. This means that the corresponding vertices (and the quantum cluster -variables associated with them) are non-dynamical (“frozen” in the terminology of cluster algebras).
Now in the triangulation of an annulus the triangles are glued together along their edges. Whenever two triangles are glued together, we first concatenate the quivers by identifying the quiver vertices associated with the glued edge, and then promote that vertex (and the associated quantum cluster -variables ) to be dynamical. We denote such a dynamical vertex by a circle in Fig. 10. By repeating this procedure you obtain the quiver for the annulus, as shown in the bottom of Fig. 10.262626Such a construction can be sort of as an open analog of the Gaiotto’s construction for gauging , as emphasized in 
Now the first non-trivial observation is that the resulting quiver for the annulus, once we disregard the non-dynamical (squared) vertices and the edges beginning/ending on them, coincides with the quiver introduced in (2).
We can moreover match the time evolutions, namely the mutation sequence of the quiver. In the gluing rule of Fig. 10, the flip of the triangulation (Fig. 6) is turned into a change of the quiver as shown in Fig. 11. This is nothing but a mutation of the quiver (see Appendix), for a vertex on the flipped edge (as represented as a crossed vertex in the figure). Once we establish this, we can translate the flips for a realization of the Dehn twist (Fig. 8) into a sequence of mutations, as in Fig. 12. The result is to mutate all the even (or odd) vertices. This coincides with our previous discussion (see explanation around (16)). This establishes what we wanted to show. Notice that for this purpose graphical/combinatorial manipulations are enough, and no complicated computations are necessary.
That there exists a correspondence between the quantum Liouville theory and quantum Teichmüller theory is in itself not surprising. This is because classical Liouville theory originated in the study of the uniformization of a Riemann surface, which deals with the Teichmüller space. Such an equivalence was conjectured to persist at the quantum level , which equivalence was later proven in [84, 85].
However, what is shown here is more dramatic, namely we have a direct relation between quantum discrete Liouville theory and quantum Teichmüller theory in the continuum. This is a rare example where “a discretization of a theory reproduces the original theory”. As we will see next, it turns out that this is a special feature of the case, and does not really hold for the case.
Let us now come to the case of . The generalization of the quantum Teichmüller theory for this is the higher Teichmüller theory of .
The part about the triangulation of the annulus, as shown in Figs 6, 7 and 8, stay the same. The difference comes for the rule for the quiver (Fig. 9), which for the case is given in Fig. 13. This quiver has frozen vertices on the boundary edge, which are regarded as non-dynamical. Note the quiver also has dynamical vertices in the interior of the triangle.
The gluing rule, previously shown in Fig. 10, stay essentially the same. The only difference is that whenever we glue an edge of the two triangles vertices are turned dynamical for general .
Let us first glue two triangles, to obtain a square. In this case, we obtain a quiver in Fig. 14, which looks very different from the quiver in Fig. 2. One might therefore conclude that the relation between the two theories is lost completely.
It turns out, however, that there is a sequence of mutations relating the two (see Fig. 14), and hence the two quantizations are simply related by some unitary transformation :
where () denotes the variables of the discrete Toda (higher Teichmüller) theory.272727One should not that such a unitary transformation is not local on the lattice, since mutations mixes variables on neighboring vertices of the quiver. This means that the causality (8), which holds for , does not hold for , for the case with .
While this is encouraging, such a nice story ceases to exist once we begin to glue two squares (hence four triangles). Indeed, the resulting quiver as required by the quantum Teichmüller theory is shown in Fig.15, and even after mutations still is different from the quiver (2) of Fig. 2. The basic reason for this is that when we glue two squares we turn the non-dynamical vertices into dynamical vertices, and the structures of the non-dynamical vertices are different between the two theories, already for a square (two triangles). This means that the direct relation between discrete Toda theory and continuum higher Teichmüller theory does not hold, for .
This is not a surprising statement, since as we discussed before there is no a-priori reasoning to guarantee an equivalence between the two. Nevertheless it would be interesting to explore further if there is anything we can extract by the similarities of the two subjects, even in the case of . For example, the discrepancy between the two comes from gluing edges, which are locate on one-dimensional edges and hence would be suppressed compared with those degrees of freedom on the interior of the triangles, in the large limit. This could be another indication that large limit has some special properties.
5 Summary and Discussion
In this paper, we formulated the discrete Toda theory from the quantum Y-system associated with the quiver . Here is a symmetry algebra of the theory, which can be a finite-dimensional semisimple Lie algebra or an infinite-dimensional tamely-laced Kac-Moody algebra. Another algebra is either or , depending on the spatial boundary conditions. Our formulation naturally generalizes the quantum Liouville theory in the literature, however as we discussed in Sec. 4 the direction relation with the higher Teichmüller theory on annulus seems to be lost for . We also commented on possible implications to chaos.
Let us here comment on some more open questions which are not touched in the main text.
First, it would be interesting to see if the discretized model in this paper helps to solve the Toda field theory in the continuum limit, for example to compute the three-point structure constants and the four-point conformal blocks. While the details of the computation might be involved, our discretized/regularized model is ‘solved’ already, and might give rise to a systematic method to solve the continuum Toda theory.
As commented in the main text, our model corresponds to particular examples of the quantum cluster algebra, and the discussion naturally generalizes to more general choices of quivers and mutation sequences (cf. [86, 87]). It would be interesting to identity (if any) the two-dimensional CFT in the continuum limit, and compute their Lyapunov exponents. The hope is that this generalization provides a rich landscape of discrete quantum mechanical systems to explore quantum chaos.
In Sec. 4 we worked on the relation between discrete Toda theory and the higher Teichmüller theory, for the case. While the conclusion was negative overall, we also obtained a positive result, namely that some structures of the higher Teichmüller theory (namely the dynamical part of the quiver for a square) can be extracted from the corresponding Y-system. It would be interesting to verify this for more general case . Note that the general version of the higher Teichmüller theory is being developed only recently [88, 89].
In our model, we discussed two types of spatial boundary conditions, fixed and periodic. This is motivated partly by simplicity, and partly by those often used in the literature of classical Y-system. However, we have not tried to find exhaustive list of boundary conditions consistent with integrability of the model. Note that the classical boundary conditions preserving integrability are highly constrained in the continuum in the continuum affine Toda theories [90, 91]. In this respect, taking the general algebra in the -quiver (say for or ) could realize some interesting integrable boundary conditions.
The author would like to thank Richard Eager, Simeon Hellerman, Ivan Ip, Atsuo Kuniba and Herman Verlinde for illuminating discussion and correspondence. The author benefited from his presentation on discrete Liouville theory in May 2016 at IAS, and he would like to thank the audience for feedback. This research is supported by WPI program (MEXT, Japan), by JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers, by JSPS KAKENHI Grant No. 15K17634, and by JSPS-NRF Joint Research Project.
Appendix A Continuum Theory
In this appendix we review the two-dimensional Toda field theory in the continuum limit, and set up some notations.
The conformal Toda field theory, or Toda field theory for short, is a two-dimensional CFT associated with a simple Lie algebra .282828In Lie algebra notation this should be denoted by . We here follow the literature of the Y-system and use the symbol for algebra.. When we denote the rank of by , the theory has scalar fields parametrizing the Cartan subalgebra of , and the Lagrangian is given by
In this expression, is the canonical pairing (Killing form) in , with which we identify the elements of and its dual, and denotes a positive simple root.
In order for this theory to be conformal, the parameter (background charge) should be related to another parameter as
where is the Weyl vector:
with the set of positive roots. This model has a central charge
where is the (dual) Coxeter number of . For the case , we have and this reduces to
We can also choose to be an infinite-dimensional (untwisted or twisted) affine Lie algebra. In this case, we have an extra affine simple root, , and the summation over in (29) should now include . Physics in this case is very different, since the theory is a massive perturbation of a CFT and is non-conformal.
Appendix B Quantum Cluster Algebra
For the convenience of the reader we here include minimal summary of quantum cluster algebras. The contents of this section is a simplified version of the appendices B and C in .292929There is one difference in notation: here is in . We have chosen this convention to remove the square roots from the time evolution rules (6).
Let us begin with a quiver , i.e., a finite oriented graph. We denote its vertices by . Let us define an anti-symmetric matrix as in (4). The quivers discussed in this paper has no loops and oriented -cycles, and hence the quiver can be identified with the anti-symmetric matrix .
Given a vertex , we define a new quiver (mutation of at vertex ) by
Given a quiver , we associate quantum -variable for each vertex , and we impose the commutation relation (3). The non-commutativity parameter is , with “Planck constant” .
Mutation acts on these quantum -variables as
This can be represented as an operator
Here is a quantum dilogarithm function satisfying the difference equation
and the hermitian operator give a transformation properties of (the logarithm of) the so-called tropical version of -variables:
-  M. Toda, “Vibration of a Chain with Nonlinear Interaction”, J. Phys. Soc. Jpn. 22, 431 (1967).
-  M. Toda, “Wave Propagation in Anharmonic Lattices”, J. Phys. Soc. Jpn. 23, 501 (1967).
-  M. Toda, “Nonlinear waves and solitons”, Kluwer Academic Publishers Group, Dordrecht; SCIPRESS, Tokyo (1989), Translated from the Japanese.
-  L. D. Faddeev and L. A. Takhtajan, “Liouville model on the lattice”, Lect. Notes Phys. 246, 166 (1986), in: “De Vega, H.J. and Sanchez, N. (ed.): Field Theory, Quantum Gravity and Strings”, pp. 166-179.
-  L. D. Faddeev and A. Yu. Volkov, “Abelian current algebra and the Virasoro algebra on the lattice”, Phys. Lett. B315, 311 (1993), hep-th/9307048.
-  L. D. Faddeev and A. Yu. Volkov, “Algebraic quantization of integrable models in discrete space-time”, hep-th/9710039, in: “Discrete integrable geometry and physics (Vienna, 1996)”, pp. 301–319, Oxford Univ. Press, New York (1999).
-  L. D. Faddeev, R. M. Kashaev and A. Yu. Volkov, “Strongly coupled quantum discrete Liouville theory. 1. Algebraic approach and duality”, Commun. Math. Phys. 219, 199 (2001), hep-th/0006156.
-  L. D. Faddeev and R. M. Kashaev, “Strongly coupled quantum discrete Liouville theory. 2. Geometric interpretation of the evolution operator”, J. Phys. A35, 4043 (2002), hep-th/0201049.
-  L. D. Faddeev and L. A. Takhtajan, “Hamiltonian Methods in the Theory of Solitons”, Springer, Berlin (1987).
-  L. D. Faddeev, “How algebraic Bethe ansatz works for integrable model”, hep-th/9605187, in: “Relativistic gravitation and gravitational radiation. Proceedings, School of Physics, Les Houches, France, September 26-October 6, 1995”, pp. pp. 149-219.
-  L. D. Faddeev and R. M. Kashaev, “Quantum Dilogarithm”, Mod. Phys. Lett. A9, 427 (1994), hep-th/9310070.
-  L. Chekhov and V. Fock, “Quantum Teichmuller space”, Theor. Math. Phys. 120, 1245 (1999), math/9908165.
-  R. M. Kashaev, “Quantization of Teichmüller spaces and the quantum dilogarithm”, Lett. Math. Phys. 43, 105 (1998).
-  V. Fock and A. Goncharov, “Moduli spaces of local systems and higher Teichmüller theory”, Publ. Math. Inst. Hautes Études Sci. 103, 1 (2006).
-  K. Hikami, “Generalized Volume Conjecture and the A-Polynomials: The Neumann-Zagier Potential Function as a Classical Limit of Quantum Invariant”, J. Geom. Phys. 57, 1895 (2007), math/0604094.
-  T. Dimofte, S. Gukov, J. Lenells and D. Zagier, “Exact Results for Perturbative Chern-Simons Theory with Complex Gauge Group”, Commun. Num. Theor. Phys. 3, 363 (2009), arxiv:0903.2472.
-  J. E. Andersen and R. Kashaev, “A TQFT from Quantum Teichmüller Theory”, Commun. Math. Phys. 330, 887 (2014), arxiv:1109.6295.
-  N. Hama, K. Hosomichi and S. Lee, “SUSY Gauge Theories on Squashed Three-Spheres”, JHEP 1105, 014 (2011), arxiv:1102.4716.
-  Y. Terashima and M. Yamazaki, “SL(2,R) Chern-Simons, Liouville, and Gauge Theory on Duality Walls”, JHEP 1108, 135 (2011), arxiv:1103.5748.
-  T. Dimofte, D. Gaiotto and S. Gukov, “Gauge Theories Labelled by Three-Manifolds”, Commun. Math. Phys. 325, 367 (2014), arxiv:1108.4389.
-  S. Lee and M. Yamazaki, “3d Chern-Simons Theory from M5-branes”, JHEP 1312, 035 (2013), arxiv:1305.2429.
-  C. Cordova and D. L. Jafferis, “Complex Chern-Simons from M5-branes on the Squashed Three-Sphere”, JHEP 1711, 119 (2017), arxiv:1305.2891.
-  J. Yagi, “3d TQFT from 6d SCFT”, JHEP 1308, 017 (2013), arxiv:1305.0291.
-  Y. Terashima and M. Yamazaki, “Emergent 3-manifolds from 4d Superconformal Indices”, Phys. Rev. Lett. 109, 091602 (2012), arxiv:1203.5792.
-  M. Yamazaki, “Quivers, YBE and 3-manifolds”, JHEP 1205, 147 (2012), arxiv:1203.5784.
-  M. Yamazaki, “New Integrable Models from the Gauge/YBE Correspondence”, J. Statist. Phys. 154, 895 (2014), arxiv:1307.1128.
-  V. V. Bazhanov, V. V. Mangazeev and S. M. Sergeev, “Faddeev-Volkov solution of the Yang-Baxter equation and discrete conformal symmetry”, Nucl. Phys. B784, 234 (2007), hep-th/0703041.
-  V. P. Spiridonov, “Elliptic beta integrals and solvable models of statistical mechanics”, Contemp. Math. 563, 181 (2012), arxiv:1011.3798.
-  G. Turiaci and H. Verlinde, “On CFT and Quantum Chaos”, JHEP 1612, 110 (2016), arxiv:1603.03020.
-  J. Maldacena, S. H. Shenker and D. Stanford, “A bound on chaos”, JHEP 1608, 106 (2016), arxiv:1503.01409.
-  S. Sachdev and J. Ye, “Gapless spin fluid ground state in a random, quantum Heisenberg magnet”, Phys. Rev. Lett. 70, 3339 (1993), cond-mat/9212030.
-  A. Kitaev, “A simple model of quantum holography”, Talks at KITP, April 7, 2015 and May 27, 2015.
-  X. O. Camanho, J. D. Edelstein, J. Maldacena and A. Zhiboedov, “Causality Constraints on Corrections to the Graviton Three-Point Coupling”, JHEP 1602, 020 (2016), arxiv:1407.5597.
-  E. Perlmutter, “Bounding the Space of Holographic CFTs with Chaos”, JHEP 1610, 069 (2016), arxiv:1602.08272.
-  M. J. Ablowitz and J. F. Ladik, “Nonlinear differential-difference equations and Fourier analysis”, J. Mathematical Phys. 17, 1011 (1976).
-  R. Hirota, “Nonlinear partial difference equations. I. A difference analogue of the Korteweg-de Vries equation”, J. Phys. Soc. Japan 43, 1424 (1977).
-  I. Krichever, O. Lipan, P. Wiegmann and A. Zabrodin, “Quantum integrable systems and elliptic solutions of classical discrete nonlinear equations”, Commun. Math. Phys. 188, 267 (1997), hep-th/9604080.
-  A. Kuniba, Y. Ohta and J. Suzuki, “Quantum Jacobi-Trudi and Giambelli formulae for from analytic Bethe ansatz”, J. Phys. A28, 6211 (1995), hep-th/9506167.
-  R. M. Kashaev and N. Reshetikhin, “Affine Toda field theory as a three-dimensional integrable system”, Commun. Math. Phys. 188, 251 (1997), hep-th/9507065.
-  A. Bytsko and J. Teschner, “The Integrable structure of nonrational conformal field theory”, Adv. Theor. Math. Phys. 17, 701 (2013), arxiv:0902.4825.
-  C. Meneghelli and J. Teschner, “Integrable light-cone lattice discretizations from the universal R-matrix”, arxiv:1504.04572.
-  B. Keller, “The periodicity conjecture for pairs of Dynkin diagrams”, Ann. of Math. (2) 177, 111 (2013).
-  S. Cecotti, A. Neitzke and C. Vafa, “R-Twisting and 4d/2d Correspondences”, arxiv:1006.3435.
-  J. J. Heckman, C. Vafa, D. Xie and M. Yamazaki, “String Theory Origin of Bipartite SCFTs”, JHEP 1305, 148 (2013), arxiv:1211.4587.
-  D. Gang, N. Kim, M. Romo and M. Yamazaki, “Aspects of Defects in 3d-3d Correspondence”, arxiv:1510.05011.
-  D. P. Robbins and H. Rumsey, Jr., “Determinants and alternating sign matrices”, Adv. in Math. 62, 169 (1986).
-  A. Doliwa and P. M. Santini, “Multidimensional quadrilateral lattices are integrable”, Phys. Lett. A 233, 365 (1997).
-  V. E. Adler, A. I. Bobenko and Y. B. Suris, “Classification of integrable equations on quad-graphs. The consistency approach”, Comm. Math. Phys. 233, 513 (2003).
-  F. Ravanini, R. Tateo and A. Valleriani, “Dynkin TBAs”, Int. J. Mod. Phys. A8, 1707 (1993), hep-th/9207040.
-  A. Kuniba, T. Nakanishi and J. Suzuki, “T-systems and Y-systems in integrable systems”, J. Phys. A44, 103001 (2011), arxiv:1010.1344.
-  N. Gromov, V. Kazakov and P. Vieira, “Exact Spectrum of Anomalous Dimensions of Planar N=4 Supersymmetric Yang-Mills Theory”, Phys. Rev. Lett. 103, 131601 (2009), arxiv:0901.3753.
-  L. F. Alday, J. Maldacena, A. Sever and P. Vieira, “Y-system for Scattering Amplitudes”, J. Phys. A43, 485401 (2010), arxiv:1002.2459.
-  S. Fomin and A. Zelevinsky, “Cluster algebras. I. Foundations”, J. Amer. Math. Soc. 15, 497 (2002).
-  Y. Terashima and M. Yamazaki, “3d N=2 Theories from Cluster Algebras”, PTEP 2014, 023B01 (2014), arxiv:1301.5902.
-  D. Gang, N. Kim, M. Romo and M. Yamazaki, “Taming supersymmetric defects in 3d-3d correspondence”, J. Phys. A49, 30LT02 (2016), arxiv:1510.03884.
-  R. M. Kashaev and T. Nakanishi, “Classical and Quantum Dilogarithm Identities”, SIGMA 7, 102 (2011), arxiv:1104.4630.
-  V. V. Fock and A. B. Goncharov, “Cluster ensembles, quantization and the dilogarithm”, Ann. Sci. Éc. Norm. Supér. (4) 42, 865 (2009).
-  V. V. Fock and A. B. Goncharov, “The quantum dilogarithm and representations of quantum cluster varieties”, Invent. Math. 175, 223 (2009).
-  I. C.-H. Ip and M. Yamazaki, “Quantum Dilogarithm Identities at Root of Unity”, Int. Math. Res. Not. 2016, 669 (2016), arxiv:1412.5777.
-  A. B. Zamolodchikov, “On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories”, Phys. Lett. B253, 391 (1991).
-  A. Kuniba and T. Nakanishi, “Spectra in conformal field theories from the Rogers dilogarithm”, Mod. Phys. Lett. A7, 3487 (1992), hep-th/9206034.
-  F. Gliozzi and R. Tateo, “ADE functional dilogarithm identities and integrable models”, Phys. Lett. B348, 84 (1995), hep-th/9411203.
-  F. Gliozzi and R. Tateo, “Thermodynamic Bethe ansatz and threefold triangulations”, Int. J. Mod. Phys. A11, 4051 (1996), hep-th/9505102.
-  E. Frenkel and A. Szenes, “Thermodynamic Bethe ansatz and dilogarithm identities. I”, Math. Res. Lett. 2, 677 (1995), hep-th/9506215.
-  R. Inoue, O. Iyama, B. Keller, A. Kuniba and T. Nakanishi, “Periodicities of T-systems and Y-systems, dilogarithm identities, and cluster algebras II: types , , and ”, Publ. Res. Inst. Math. Sci. 49, 43 (2013).
-  R. Inoue, O. Iyama, B. Keller, A. Kuniba and T. Nakanishi, “Periodicities of T-systems and Y-systems, dilogarithm identities, and cluster algebras I: type ”, Publ. Res. Inst. Math. Sci. 49, 1 (2013).
-  P. Bocchieri and A. Loinger, “Quantum Recurrence Theorem”, Phys. Rev. 107, 337 (1957).
-  D. Hernandez, “Drinfeld coproduct, quantum fusion tensor category and applications”, Proc. Lond. Math. Soc. (3) 95, 567 (2007).
-  A. Kuniba, T. Nakanishi and J. Suzuki, “T-systems and Y-systems for quantum affinizations of quantum Kac-Moody algebras”, SIGMA 5, 108 (2009), arxiv:0909.4618.
-  V. Bazhanov and N. Reshetikhin, “Restricted Solid on Solid Models Connected With Simply Based Algebras and Conformal Field Theory”, J. Phys. A23, 1477 (1990).
-  A. Kuniba, T. Nakanishi and J. Suzuki, “Ferromagnetizations and antiferromagnetizations in RSOS models”, Nucl. Phys. B356, 750 (1991).
-  S. Cecotti and M. Del Zotto, “ systems, systems, and 4D supersymmetric QFT”, J. Phys. A47, 474001 (2014), arxiv:1403.7613.
-  D. A. Roberts and D. Stanford, “Two-dimensional conformal field theory and the butterfly effect”, Phys. Rev. Lett. 115, 131603 (2015), arxiv:1412.5123.
-  I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, “Holography from Conformal Field Theory”, JHEP 0910, 079 (2009), arxiv:0907.0151.
-  T. Hartman, C. A. Keller and B. Stoica, “Universal Spectrum of 2d Conformal Field Theory in the Large c Limit”, JHEP 1409, 118 (2014), arxiv:1405.5137.
-  J. A. Teschner, “On quantization of Liouville theory and related conformal field theories”, http://lss.fnal.gov/archive/other1/desy-95-118.pdf.
-  J. Balog, L. Feher and L. Palla, “Coadjoint orbits of the Virasoro algebra and the global Liouville equation”, Int. J. Mod. Phys. A13, 315 (1998), hep-th/9703045.
-  H. Dorn and G. Jorjadze, “Boundary Liouville theory: Hamiltonian description and quantization”, SIGMA 3, 012 (2007), hep-th/0610197, in: “Non-perturbative and symmetry methods in field theory: Proceedings, O’Raifeartaigh Symposium, LOR 2006, Budapest, Hungary, June 22-24, 2006”, pp. 012.
-  M. A. Vasiliev, “Consistent equation for interacting gauge fields of all spins in (3+1)-dimensions”, Phys. Lett. B243, 378 (1990).
-  M. A. Vasiliev, “Higher spin gauge theories: Star product and AdS space”, hep-th/9910096.
-  D. Gaiotto, “N=2 dualities”, JHEP 1208, 034 (2012), arxiv:0904.2715.
-  D. Xie and M. Yamazaki, “Network and Seiberg Duality”, JHEP 1209, 036 (2012), arxiv:1207.0811.
-  H. L. Verlinde, “Conformal Field Theory, 2- Quantum Gravity and Quantization of Teichmuller Space”, Nucl. Phys. B337, 652 (1990).
-  J. Teschner, “An analog of a modular functor from quantized Teichmüller theory”, math/0510174, in: “Handbook of Teichmüller theory. Vol. I”, pp. 685–760, Eur. Math. Soc., Zürich (2007).
-  J. Teschner and G. S. Vartanov, “Supersymmetric gauge theories, quantization of , and conformal field theory”, Adv. Theor. Math. Phys. 19, 1 (2015), arxiv:1302.3778.
-  R. Inoue and T. Nakanishi, “Difference equations and cluster algebras I: Poisson bracket for integrable difference equations”, RIMS Kokyuroku Bessatsu B28, 63 (2011), arxiv:1012.5574.
-  T. Nakanishi, “Periodicities in cluster algebras and dilogarithm identities”, arxiv:1006.0632.
-  I. Le, “Cluster Structures on Higher Teichmuller Spaces for Classical Groups”, arxiv:1603.03523.
-  C. K. Zickert, “Fock-Goncharov coordinates for rank two Lie groups”, arxiv:1605.08297.
-  P. Bowcock, E. Corrigan, P. E. Dorey and R. H. Rietdijk, “Classically integrable boundary conditions for affine Toda field theories”, Nucl. Phys. B445, 469 (1995), hep-th/9501098.
-  G. W. Delius, “Soliton preserving boundary condition in affine Toda field theories”, Phys. Lett. B444, 217 (1998), hep-th/9809140.