We study geometric consistency relations between angles of 3-dimensional (3D) circular quadrilateral lattices — lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable “ultra-local” Poisson bracket algebra defined on discrete 2D surfaces consisting of circular quadrilaterals. Quantization of this structure allowed us to obtain new solutions of the tetrahedron equation (the 3D analog of the Yang-Baxter equation) as well as reproduce all those that were previously known. These solutions generate an infinite number of non-trivial solutions of the Yang-Baxter equation and also define integrable 3D models of statistical mechanics and quantum field theory. The latter can be thought of as describing quantum fluctuations of lattice geometry.
QUANTUM GEOMETRY OF 3-DIMENSIONAL LATTICES
AND TETRAHEDRON EQUATION\FootnotePlenary talk at the XVI International Congress on Mathematical Physics, 3-8 August 2009, Prague, Czech Republic
Vladimir V. Bazhanov and Vladimir V. Mangazeev
Department of Theoretical Physics,
Research School of Physical Sciences and Engineering,
Australian National University, Canberra, ACT 0200, Australia.
Sergey M. Sergeev
Faculty of Information Sciences and Engineering,
University of Canberra, Bruce ACT 2601, Australia.
Keywords: Quantum geometry, discrete differential geometry, integrable quantum systems, Yang-Baxter equation, tetrahedron equation, quadrilateral and circular 3D lattices.
Quantum integrability is traditionally understood as a purely algebraic phenomenon. It stems from the Yang-Baxter equation [1, 2] and other algebraic structures such as the affine quantum groups [3, 4] (also called the quantized Kac-Moody algebras), the Virasoro algebra  and their representation theory. It is, therefore, quite interesting to learn that these algebraic structures also have remarkable geometric origins , which will be reviewed here.
Our approach  is based on connections between integrable three-dimensional (3D) quantum systems and integrable models of 3D discrete differential geometry. The analog of the Yang-Baxter equation for integrable quantum systems in 3D is called the tetrahedron equation. It was introduced by Zamolodchikov in [7, 8] (see also [9, 10, 11, 12, 13, 14, 15, 16] for further important results in this field). Similarly to the Yang-Baxter equation the tetrahedron equation provides local integrability conditions which are not related to the size of the lattice. Therefore the same solution of the tetrahedron equation defines different integrable models on lattices of different size, e.g., for finite periodic cubic lattices. Obviously, any such three-dimensional model can be viewed as a two-dimensional integrable model on a square lattice, where the additional third dimension is treated as an internal degree of freedom. Therefore every solution of the tetrahedron equation provides an infinite sequence of integrable 2D models differing by the size of this “hidden third dimension”. Then a natural question arises whether known 2D integrable models can be obtained in this way. Although a complete answer to this question is unknown, a few non-trivial examples of such correspondence have already been constructed. The first example  reveals the 3D structure of the generalized chiral Potts model [17, 18]. Another example  reveals 3D structure of all two-dimensional solvable models associated with finite-dimensional highest weight representations for quantized affine algebras , (where coincides with the size of the hidden dimension).
Here we unravel yet another remarkable property of the same solutions of the tetrahedron equation (in addition to the hidden 3D structure of the Yang-Baxter equation and quantum groups). We show that these solutions can be obtained from quantization of geometric integrability conditions for the 3D circular lattices — lattices whose faces are planar quadrilaterals inscribable into a circle.
The 3D circular lattices were introduced  as a discretization of orthogonal coordinate systems, originating from classical works of Lamé  and Darboux . In the continuous case such coordinate systems are described by integrable partial differential equations (they are connected with the classical soliton theory [22, 23]). Likewise, the quadrilateral and circular lattices are described by integrable difference equations. The key idea of the geometric approach [24, 19, 25, 26, 27, 28, 29, 30, 31] to integrability of discrete classical systems is to utilize various consistency conditions  arising from geometric relations between elements of the lattice. It is quite remarkable that these conditions ultimately reduce to certain incidence theorems of elementary geometry. For instance, the integrability conditions for the quadrilateral lattices merely reflect the fact of existence of the 4D Euclidean cube . In Sect.2 we present these conditions algebraically in a standard form of the functional tetrahedron equation . The latter serves as the classical analog of the quantum tetrahedron equation, discussed above, and provides a connecting link to integrable quantum systems.
In Sect. 3.1 we study
relations between edge angles on the 3D circular quadrilateral
lattices and show that these relations describe symplectic transformations of
a remarkable “ultra-local” Poisson algebra on quadrilateral
surfaces (see Eq.(21)).
Quantization of this structure allows one
to obtain all currently known solutions
of the tetrahedron equation. They are presented
in Sect. 4, namely
including their interaction-round-a-cube and vertex forms. Additional details on the corresponding solvable 3D models, in particular, on their quasi-classical limit and connections with geometry, can be found in .
2 Discrete differential geometry: “Existence as integrability”
In this section we consider classical discrete integrable systems associated with the quadrilateral lattices. There are several ways to extract algebraic integrable systems from the geometry of these lattices. One approach, developed in [35, 36, 25, 37, 27], leads to discrete analogs of the Kadomtsev-Petviashvili integrable hierarchy. Here we present a different approach exploiting the angle geometry of the 3D quadrilateral lattices.
2.1 Quadrilateral lattices
Consider three-dimensional lattices, obtained by embeddings of the integer cubic lattice into the -dimensional Euclidean space , with . Let , denote coordinates of the lattice vertices, labeled by the 3-dimensional integer vector , where and . Further, for any given lattice vertex , the symbols , , etc., will denote neighboring lattice vertices. The lattice is called quadrilateral if all its faces are planar quadrilaterals. The existence of these lattices is based on the following elementary geometry fact (see Fig. 1) ,
Consider four points in general position in , . On each of the three planes , choose an extra point not lying on the lines , and . Then there exist a unique point which simultaneously belongs to the three planes , and .
The six planes, referred to above, obviously lie in the same 3D subspace of the target space. They define a hexahedron with quadrilateral faces, shown in Fig. 1. It has the topology of the cube, so we will call it “cube”, for brevity. Let us study elementary geometry relations among the
angles of this cube. Denote the angles between the edges as in Fig. 2. Altogether we have angles, connected by six linear relations
which can be immediately solved for all “’s”. This leaves 18 angles, but only nine of them are independent. Indeed, a mutual arrangement (up to an overall rotation) of unit normal vectors to six planes in the 3D-space is determined by nine angles only. Once this arrangement is fixed all other angles can be calculated. Thus the nine independent angles of the three “front” faces of the cube, shown in Fig.2a, completely determine the angles on the three “back” faces, shown in Fig.2b, and vice versa. So the geometry of our cube provides an invertible map for three triples of independent variables
Suppose now that all angles are known. To completely define the cube one also needs to specify lengths of its three edges. All the remaining edges can be then determined from simple linear relations. Indeed, the four sides of every quadrilateral are constrained by two relations, which can be conveniently presented in the matrix form
Assume that the lengths , , on one side of the two pictures in Fig.4 are given. Let us find the other three lengths , on their opposite side, by iterating the relation (3). Obviously, this can be done in two different ways: either using the front three faces, or the back ones — the results must be the same. This is exactly where the geometry gets into play. The results must be consistent due to the very existence of the cube in Fig. 1 as a geometric body.
However, they will be consistent only if all geometric relations between the two sets of angles in the front and back faces of the cube are taken into account. To write these relations in a convenient form we need to introduce additional notations. Note, that Fig.3 shows two thin lines, labeled by the symbols “” and “”. Each line crosses a pairs of opposite edges, which we call “corresponding” (in the sense that they correspond to the same thin line). Eq.(3) relates the lengths of two adjacent edges with the corresponding lengths on the opposite side of the quadrilateral.
Consider now Fig.4a which contains three directed thin lines
connecting corresponding edges of the three quadrilateral faces.
By the analogy with
the 2D Yang-Baxter equation, where similar arrangements
occur, we call them “rapidity”
where are defined in (3) and their dependence on the angles is implicitly understood. It follows that
the lengths are defined as in Fig.4, and the superscript “” denotes the matrix transposition. Performing similar calculations for the back faces in Fig.4b and equating the resulting three by three matrices, one obtains
This matrix relation contains exactly nine scalar equations where the LHS only depends on the front angles (7), while the RHS only depends on the back angles (9). Solving these equations one can obtain explicit form of the map (2). The resulting expressions are rather complicated and not particularly useful. However the mere fact that the map (2) satisfy a very special Eq.(8) is extremely important. Indeed, rewrite this equation as
where is an operator acting as the substitution (2) for any function of the angles,
where both sides are compositions of the maps (2), involving six different sets of angles. Algebraically, this equation arises as an associativity condition for the cubic algebra (10). To discuss its geometric meaning we need to introduce discrete evolution systems associated with the map (2).
2.2 Discrete evolution systems: “Existence as integrability”
Consider a sub-lattice of the 3D quadrilateral lattice, which only includes points with . The boundary of this sub-lattice is a 2D discrete surface formed by quadrilaterals with the vertices having at least one of their integer coordinates equal to zero and the other two non-negative. Assume that all quadrilateral angles on this surface are known, and consider them as initial data. Then repeatedly applying the map (2) one can calculate angles on all faces of the sub-lattice , defined above (one has to start from the corner ). The process can be visualized as an evolution of the initial data surface where every transformation (2) corresponds to a “flip” between the front and back faces (Fig. 2) of some cube adjacent to the surface. This makes the surface looking as a 3D “staircase” (or a pile of cubes) in the intersection corner of the three coordinate planes, see Fig. 5 showing two stages of this process.
Note, that the corresponding evolution equations can be written in a covariant form for an arbitrary lattice cube (see Eq.(23) below for an example). It is also useful to have in mind that the above evolution can be defined purely geometrically as a ruler-and-compass type construction. Indeed the construction of the point in Fig. 1 from the points (and that is what is necessary for flipping a cube) only requires a 2D-ruler which allows to draw planes through any three non-collinear points in the Euclidean space.
Similar evolution systems can be defined for other
quadrilateral lattices instead
of the 3D cubic lattice considered above.
Since the evolution is local (only one cube is flipped at a
time) one could consider finite lattices as well. For example, consider
six adjacent quadrilateral faces covering the front surface of the
The functional tetrahedron equation (12) states that the results will be the same. Thereby it gives an algebraic proof for the equivalence of two “ruler-and-compass” type constructions of the back surface of the dodecahedron in Fig. 6. Can we also prove this equivalence geometrically? Although from the first sight this does not look trivial, it could be easily done from the point of view of the 4D geometry. The required statement follows just from the fact of existence of the quadrilateral lattice with the topology of the 4D cube . The latter is defined by eight intersecting 3-planes in a general position in the 4-space. The two rhombic dodecahedra shown in Fig. 6 are obtained by a dissection of the 3-surface of this 4-cube, along its 2-faces, so these dodecahedra must have exactly the same quadrilateral 2-surface. Thus the functional tetrahedron equation (12), which plays the role of integrability condition for the discrete evolution system associated with the map (2), simply follows from the mere fact of existence of the 4-cube, which is the simplest 4D quadrilateral lattice. For a further discussion of a relationship between the geometric consistency and integrability see .
3 Quantization of the 3D circular lattices
3.1 Poisson structure of circular lattices
The 3D circular lattice [19, 28, 29] is a special 3D quadrilateral lattice where all faces are circular quadrilaterals (i.e., quadrilaterals which can be inscribed into a circle). The existence of these lattices is established by the following beautiful geometry theorem due to Miquel  (see Fig. 7)
Miquel theorem. Consider four points in general position in , . On each of the three circles , choose an additional new point . Then there exist a unique point which simultaneously belongs to the three circles , and .
It is easy to see that the above six circles lie on the same sphere. It follows then that every elementary “cube” on a circular lattice (whose vertices are at the circle intersection points) is inscribable into a sphere, see Fig. 7. The general formulae of the previous subsection can be readily specialized for the circular lattices. A circular quadrilateral has only two independent angles. In the notation of Fig. 3 one has
Due to the Miquel theorem we can simply impose these restrictions on all faces of the lattice without running to any contradictions. The two by two matrix in (3) takes the form
where we have introduced new variables
instead of the two angles . Note that the new variables are constrained by the relation
Conversely, one has
Let the variables , , , correspond to the front and back faces of the cube. The map (2) then read explicitly
At this point we note that exactly the same map together with the corresponding equations (8) and (12) were previously obtained in . Moreover, it was discovered that this map is a canonical transformation preserving the Poisson algebra
where . Note that variables on different quadrilaterals are in involution. The same Poisson algebra in terms of angle variables reads
This “ultra-local” symplectic structure trivially extends to
any circular quad-surface of initial data, discussed above.
To resolve an apparent ambiguity in naming of the angles,
this surface must be equipped with oriented rapidity lines, similar to
those in Fig. 4
Thus, the evolution defined by the map (18) is a symplectic transformation. The corresponding equations of motion for the whole lattice (the analog of the Hamilton-Jacobi equations) can be written in a “covariant” form. For every cube define
where stands for , where is such that coincides with the coordinates of the top front corner of the cube (vertex in Fig.1). Let be the shift operator . Then
where is an arbitrary permutation of and
Note that Eq.(23) also imply
Remarks. The equations (23) have been previously obtained in , see Eq.(7.20) therein. The quantities in (23) should be identified with the rotation coefficients denoted as in . The same equations (23) are discussed in §3.1 of , where one can also find a detailed bibliography on the circular lattices (we are indebted to A.I.Bobenko for these important remarks).
3.2 Quantization and the tetrahedron equation
In the next section we consider different quantizations of the map (18) and obtain several solutions of the full quantum tetrahedron equation (see Eq.(33) below). In all cases we start with the canonical quantization of the Poisson algebra (21),
where is the quantum parameter (the Planck constant) and is a numerical coefficient, introduced for a further convenience. The indices label the faces of the “surface of initial data” discussed above. Since the commutation relations (26) are ultra-local (in the sense that the angle variables on different faces commute with each other), let us concentrate on the local Heisenberg algebra,
for a single lattice face (remind that the angles shown in Fig. 3 are related by (13)). The map (18) contains the quantities , defined in (15), which now become operators. For definiteness, assume that the non-commuting factors in (15) are ordered exactly as written. Then the definitions (15) give
where the elements and generate the Weyl algebra,
The operators (28) obey the commutation relations of the -oscillator algebra,
In the same paper  it was also shown that
there exists a quantum version of the map (18), which acts as an automorphism of the tensor cube of the -oscillator algebra (30). The formulae (18) for the quantum map stay exactly the same, but the relation (16) should be replaced by either of the two relations on the second line of (30), for instance, . In particular, (19) should be replaced with
It follows then from (12) that the linear operator satisfies the quantum tetrahedron equation
where each of the operators , , and act as (32) in the three factors (indicated by the subscripts) of a tensor product of six -oscillator algebras and act as the unit operator in the remaining three factors.
4 Solutions of the tetrahedron equation
4.1 Fock representation solution
is the -deformed Gauss hypergeometric series. This 3D -matrix satisfies the constant tetrahedron equation (33). In matrix form this equation reads
where the sum is taken over six indices and
Note that Eq.(38) does not contain any spectral parameters. Originally the -matrix (35) was obtained in  in terms of a solution of some recurrence relation, which was subsequently reduced to the -hypergemetric function in .
4.2 Modular double solution
In this subsection we set in (26)
where is a free parameter, . Here it will
convenient to work with a slightly modified version
Consider a non-compact representation  of the -oscillator algebra (30) in the space of functions on the real line admitting an analytical continuation into an appropriate horizontal strip, containing the real axis in the complex -plane (see  for further details). Such representation essentially reduces to that of the Weyl algebra
realized as multiplication and shift operators
The generators in (30) are expressed as
As explained in  the representation (42) is not, in general, irreducible. Therefore, the relation (32) alone does not unambiguously define the linear operator in this case. Following the idea of  consider the modular dual of the algebra (41),
acting in the same representation space
We found that if the relation (32) is complemented by its modular dual
The dual version of the map is defined by the same formulae (18), where quantities , are replaced by their “tilded” counterparts . The value of does not, actually, enter the map (18), but needs to be taken into account in the relations between the generators of the -oscillator algebra. Thus, the linear operator in this case simultaneously provides the two maps and (given by (18) with ). The explicit form of this operator is given below.
and define a special function
where is the non-compact quantum dilogarithm 
The values of are assumed to be such that poles of numerator in the integrand of (49) lie above the real axis, while the zeroes of the denominator lie below the real axis. For other values of the integral (49) is defined by an analytic continuation. Note that for the integral can be evaluated by closing the integration contour in the upper half plane (see eq.(78) in ), which is very convenient for numerical calculations.
This -matrix satisfies the constant tetrahedron equation (33). Its matrix form is given by (38) where is substituted by , , and the sums are replaced by the integrals over along the real lines . One can verify that these integrals converge. The solution (51) was obtained in .
The “interaction-round-a-cube” formulation of the modular double solution
Note that due to the presence of two delta-functions in (51) the edge spins are constrained by two relations and at each vertex of the lattice. Here we re-formulate this solution in terms of unconstrained corner spins, which also take continuous values on the real line. Figure 8 shows an elementary cube of the lattice with the corner spins “” arranged in the same way as in . The corresponding Boltzmann weight reads