Quantum Trilogy:
Discrete Toda, YSystem and Chaos
Abstract
We discuss a discretization of the quantum Toda field theory associated with a semisimple finitedimensional Lie algebra or a tamelylaced infinitedimensional KacMoody algebra , generalizing the previous construction of discrete quantum Liouville theory for the case . The model is defined on a discrete twodimensional 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 socalled Ysystem, and the theory can be welldescribed 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 .
UTF8min
Masahito Yamazaki (山崎雅人)
IPMU
Kavli IPMU (WPI), University of Tokyo, Kashiwa, Chiba 2778583, Japan
\institutionHarvard
Center for the Fundamental Laws of Nature, Harvard University, Cambridge, MA 02138, USA
Contents
1 Introduction
In celebrated papers in 1967 [1, 2], Prof. Morikazu Toda introduced the nowfamous Toda lattice. Since then, the Toda lattice became one of the most important and wellstudied 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 [3].
In this paper, we discuss a discretization of the twodimensional quantum Toda field theory associated with a symmetry algebra , which is either a finitedimensional semisimple Lie algebra or an infinitedimensional tamelylaced KacMoody algebra (the tamelylaced 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 twodimensional 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 (noncompact) quantum dilogarithm function [11], which later appeared in a number of different contexts in physics and mathematics, including quantum Teichmüller theory [12, 13, 14], complex ChernSimons theory [15, 16, 17] and finally in 3d supersymmetric theory [18], as related by the 3d/3d correspondence [19, 20] (see [21, 22] for derivation, also [23]) 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 [29]. Namely, it is a concrete dimensional lattice model saturating the conjectured bound [30] for chaos, and hence can be thought of as a higherdimensional 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 [33] 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 Ysystem (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 [39]^{1}^{1}1The latter reference also discussed the limit, to be discussed in this paper..^{2}^{2}2The 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 [14], 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 finitedimensional Lie algebra as well as an infinitedimensional tamelylaced KacMoody algebra. This will include most of the infinitedimensional affine Lie algebras. In order to simplify the presentation, for the most part we choose the symmetry algebra to be a simplylaced simple finitedimensional Lie algebra. We will comment on the nonsimplylaced case and the infinitedimensional 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
(1) 
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 twodimensional quiver by combining the Dynkin diagram for and that () for [42], see Figs. 1 and 2. This quiver is called the square product of and , and is often denoted by
(2) 
We also called this the quiver.^{3}^{3}3Such a quiver has been discussed in the context of 4d [43] and 4d [44] theories.
The definition of the square product is hopefully clear from Fig. 2, but we can give a more formal definition [42]. For this, let us first write choose a bipartite coloring of the Dynkin diagrams and ,^{4}^{4}4For 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 :
(3) 
where we defined the antisymmetric matrix (with ) by the relation
(4) 
The noncommutativity 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.^{5}^{5}5The Toda theory is known to have a symmetry (recall that is simplylaced in this subsection), as is manifest in the formula (33). This means that we should consider the socalled modular double, and should consider two copies of (3), one with and another with (see e.g. [45] for some more details). While this is important for some considerations of the Toda theory, this is not necessary for the semiclassical 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
(5) 
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 finitedimensional 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 [29].
2.1.3 Time Evolution
Let us now consider the time evolution. This is described by the following equation, which we call the quantum Ysystem (for simplylaced ):
(6) 
where () is the Cartan matrix^{6}^{6}6For 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.^{7}^{7}7Such a time evolution pattern for the classical case is called the octahedron recurrence [46], 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 simplylaced 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 nonsimplylaced, as we will comment on Sec. 2.3.
Notice that we can easily show the the causality of the system:
(8) 
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 Motivation/Derivation
2.2.1 Classical Limit and Y and TSystem
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
(9) 
where is the Cartan matrix for .
This set of equations is known as the (classical) Ysystem of type [49].^{8}^{8}8For fixed boundary condition (), this is also known the Ysystem of type at level . See [50] for survey on Y and Tsystem. It is worth mentioning that Ysystem also appeared in [51, 52]. ^{9}^{9}9In the literature on Y and Tsystems it is most common to denote the Ysystem (Tsystem) 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 Ysystem can be derived from a discretization of the Toda equation of the Hirotatype [36], known as the Tsystem. This reads
(10) 
where .
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
(11) 
where we defined the lightcone coordinates by . This is the Hirota bilinear form of the twodimensional Toda equation.
There is one subtlety in the Tsystem, which is that it allows for some “gauge ambiguity”. Let us for example consider the case , for which case the Tsystem reads
(12) 
This has an ambiguity of the form
(13) 
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 gaugeinvariant combination, which leads to
(14) 
or more generally for simplylaced
(15) 
We can verify that this satisfies the classical Ysystem, 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 noncommutative 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 socalled quantum cluster algebras, which is a quantization of the classical cluster algebras [53].^{10}^{10}10Our 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 socalled quantum variable^{11}^{11}11The variable corresponds to the Ysystem, whereas Tsystem corresponds to the socalled cluster variables. [57, 58], to each vertex of the quiver .^{12}^{12}12If 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 [59].
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 nontrivial 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.^{13}^{13}13Since none of the white vertices are connected with each other, the ordering of such mutations does not matter. Namely, if we define
(16) 
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 nonadjacent vertices is known to commute with each other and since the quiver is bipartite.
The time evolution rule (6) in the quantum theory then follows from the transformation rules (35) of quantum variables.
There is yet another advantage of the clusteralgebraic reformulation. Namely, we can write a timeevolution operator
(17) 
so that in the Schrödinger picture the state evolves by :
(18) 
This is because such an operator for each mutation , satisfying
(19) 
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 timeevolution operator to be one of the following, depending whether is even or odd:
(20) 
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 [56]).^{14}^{14}14The discussion in these references are limited to the case where is simplylaced.
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):
(21) 
where is the dual Coxeter number of . This is known as the periodicity of the Ysystem, as conjectured in [60, 61, 62] and proven later in [63, 64, 42] (including the nonsimplylaced cases in [65, 66]), see [50] for more references.^{15}^{15}15Typically, periodicity of the Ysystem is stated for the classical Ysystem, however periodicity of the classical Ysystem is actually equivalent with that for the quantum Ysystem, as proven more generally in [58]. ^{16}^{16}16 One consequence (21) is that product of the timeevolution operator from to is trivial. This gives rise to the quantum dilogarithm identity (see [56] for more details).
While a quantum mechanical system with a discrete spectrum in general is known to show a quantum recurrence phenomenon [67], this case is very special since the recurrence time here grows linearly in the degrees of freedom, not in double exponentially.^{17}^{17}17Another 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 shorttime 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 longstrip 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 nonsimplylaced finitedimensional simple Lie algebra

A Lie superalgebra

An infinitedimensional tamelylaced KacMoody algebra
The basic story stays the same in all of these cases. First, classical T and Ysystems for these cases are already known in the literature [50], 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 infinitedimensional KacMoody algebra, we need to impose a technical condition that the KacMoody algebra is tamelylaced [68, 69]. Recall that a KacMoody algebra is defined from a generalized Cartan matrix . This KacMoody algebra is called tamelylaced [68] if is symmetrizable and satisfies
(22) 
where is a diagonal matrix symmetrizing . This includes most of the affine Lie algebras, except for and .
In order to highlight some subtleties in these generalizations, let us here take the simplest nonsimplylaced example, namely (see [65, 66] for more details on the nonsimplylaced cases).
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 Ysystem) is given by the quantization of the classical Ysystem known in the literature [66]:
(23) 
Note that for variable corresponding to the short root (), the timeevolution 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 twodimensional 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 :^{18}^{18}18Tsystem is mapped as .
(24) 
This symmetry is known as the levelrank duality of the Y (and T) system, as noted in the context of the RSOS model [70, 71].^{19}^{19}19Such a symmetry has been discussed in a rather different context in [43, 72, 44].
While this symmetry is trivial from the standpoint of the Ysystem, 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 apriori 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 nonsimplylaced ) 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 [29].
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 outoftime correlator of the form , where and are generic operators of the theory,^{20}^{20}20The 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 [73]. 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 :
(25) 
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 [30]^{21}^{21}21This assumes some hierarchy between dissipation time and scrambling time.
(26) 
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 [29].
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 finitedimensional 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 outoftime 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 UVinsensitive, we expect that the exponent for the discrete and continuum theories coincide.
The continuum theory, i.e. the Toda theory, is a twodimensional 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).^{22}^{22}22For 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 ChernSimons theory, which contains particles with spin greater than .
In this case, we can argue that the conformal block for the identity operator contributes to the Lyapunov exponent as (see [34], which builds on the discussion for the case [73])
(27) 
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 ChernSimons theory, contains a finite number of higher spin particles, which in general is known to violate causality (after suitable coupling to matters) [33].
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 twodimensional lattice is turned into a threedimensional lattice.^{23}^{23}23This 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 [73], 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 twodimensional 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 [34]. What happens there, at least schematically, is that we have the resum 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 nonchaotic —many of the integrable charges are nonlocal, and the time evolution could happen in the basis where the charges are not conserved. One could also turn on a small nonintegrable deformation of the system, to ensure that the system is chaotic [29].
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 theory^{24}^{24}24Essentially 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.
4.1
The reference [8] 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 Dehntwist operator of the latter.
The argument of [8] 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.^{25}^{25}25In 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 socalled 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 nondynamical (“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.^{26}^{26}26Such a construction can be sort of as an open analog of the Gaiotto’s construction for gauging [81], as emphasized in [82]
Now the first nontrivial observation is that the resulting quiver for the annulus, once we disregard the nondynamical (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 [83], 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.
4.2
Let us now come to the case of . The generalization of the quantum Teichmüller theory for this is the higher Teichmüller theory of [14].
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 nondynamical. 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 :
(28) 
where () denotes the variables of the discrete Toda (higher Teichmüller) theory.^{27}^{27}27One 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 nondynamical vertices into dynamical vertices, and the structures of the nondynamical 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 apriori 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 onedimensional 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 Ysystem associated with the quiver . Here is a symmetry algebra of the theory, which can be a finitedimensional semisimple Lie algebra or an infinitedimensional tamelylaced KacMoody 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 threepoint structure constants and the fourpoint 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 twodimensional 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 Ysystem. 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 Ysystem. 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.
Acknowledgments
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 JSPSNRF Joint Research Project.
Appendix A Continuum Theory
In this appendix we review the twodimensional 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 twodimensional CFT associated with a simple Lie algebra .^{28}^{28}28In Lie algebra notation this should be denoted by . We here follow the literature of the Ysystem 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
(29) 
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
(30) 
where is the Weyl vector:
(31) 
with the set of positive roots. This model has a central charge
(32) 
where is the (dual) Coxeter number of . For the case , we have and this reduces to
(33) 
We can also choose to be an infinitedimensional (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 nonconformal.
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 [45].^{29}^{29}29There is one difference in notation: here is in [45]. 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 antisymmetric 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 antisymmetric matrix .
Given a vertex , we define a new quiver (mutation of at vertex ) by
(34) 
with .
Given a quiver , we associate quantum variable for each vertex , and we impose the commutation relation (3). The noncommutativity parameter is , with “Planck constant” .
Mutation acts on these quantum variables as
(35) 
This can be represented as an operator
(36) 
Here is a quantum dilogarithm function satisfying the difference equation
(37) 
and the hermitian operator give a transformation properties of (the logarithm of) the socalled tropical version of variables:
(38) 
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