Geometry of quadrilateral nets: second Hamiltonian form

The circular net [1] -a special type of three-dimensional quadrilateral net [2] -is an example of geometrically integrable (see [3] and references therein) system endowed by a discrete space-time Hamiltonian structure [4] what brings together geometrically integrable and completely integrable Hamiltonian systems. A class of analytical equations describing the three-dimensional quadrilateral nets is usually refereed to as discrete Darboux-Manakov-Zakharov systems [5][6][7]. In this paper we discuss another special type of quadrilateral net whose geometry is described by the second Hamiltonian form of DMZ systems [8].

Following [2], the 3D quadrilateral net is a Z 3 lattice imbedded into a multidimensional linear space,

such that each quadrilateral, e.g.

(

is the planar one. A local cell (hexahedron) of quadrilateral net is shown in Fig. 1. Geometric integrability is based on the axiomatic statement [2]: given the points x 1 , x 2 , x 3 , x 12 , x 13 , x 23 of the hexahedron, its corners x and x 123 can be obtained uniquely by a two-dimensional ruler (ruler which draws a plane via three non-collinear points). The circular net is the quadrilateral net such that each its hexahedron can be inscribed into a sphere. In this paper, instead of the circular condition, suppose firstly that the target space is four-dimensional Euclidean space,

Q. net:

Each hexahedron is an element of a three-dimensional hyperplane. In the framework of discrete-differential geometry [3], the quadrilateral net can be viewed as a planar mesh of three-dimensional manifold embedded into four-dimensional space. The hyperplanes are more general objects then quadrilaterals since such net is not necessarily quadrilateral.

In some sense this condition is analogues to extra condition for the circular net.

It is convenient to label the “octants” on the left of Fig. 2

Numerical coefficients u 1 , w 1 , κ 1 in ( 9) are associated with the edge e 1 which is orthogonal to all n c , n e , n h and n d . Analogous relations for edges e 2 and e 3 are respectively ( 10)

and such equations for outgoing edges e ′ i are ( 11)

All numerical coefficients u # i , w # i and κ # i can be expressed in terms of angular data as follows. Let θ ce be an angle between n c and n e , (12) (n c , n e ) = cos θ ce .

Let further ϕ 1,e be a dihedral angle of hyperplane n e for the edge e 1 . In terms of unit vectors of Fig. 2, ϕ 1,e is the dihedral angle between planes (e 1 , e ′ 2 ) and (e 1 , e ′ 3 ). We extend straightforwardly these self-explanatory notations to whole dual graph of the junction, Fig. 2. Then the coefficients in relation ( 9) are given by ( 13)

and similarly for all other relations and their coefficients. The geometry of junction without condition ( 7) provides ( 14)

Since there are at most four linearly independent vectors among eight n a , …, n h , the consistency of equations (9-11) relates the fields u ′ i , w ′ i on outgoing edges and fields u i , w i on tensor cube of proper representations of local Weyl algebras such that

For instance, the modular representation [11] of the local Weyl algebra is given by ( 23)

where x, p is the self-conjugated Heisenberg pair

and “physical” regime for b is

Modular partner to ( 23) is

Form of the map (22) for ũi , wi , κi coincides with that for u i , w , κ i ; in the strong coupling regime 0 < η < 1 partner equations are Hermitian conjugated. Kernel of the intertwiner (22) in the coordinate representation of Heisenberg pairs ( 24) is (27)

, where function ϕ is the non-compact quantum dilogarithm [11] (28) ϕ(z)

Positive “geometric” signs can be obtained by the analytical continuation λ i → λ i + iη and non-unitary gauge transformation w → e -2πηx we 2πηx = -q -1 w. In that case the kernel of R-matrix (27) has the semi-classical (b → 0 and e 2πbx → u) asymptotic ( 29)

where the generating function is given by (21). It worth mentioning the cyclic representations of Weyl algebra with q 2N = 1. The cyclic representation is a Z N fiber over the base of centers ũ = u N , w = w N [12]. Equations of motion for C-valued centers follow from quantum map (15), they just coincide with classical equations of motion. It is natural then to identify the evoluting centers ũi , wi directly with the geometric data (13) and pose quantum problems in Hilbert space (30) H = Z ⊗(size of net’s section) N in the presence of external classical geometry. The structure of Z ⊗3 N intertwiners and modified tetrahedron equations are discussed in more details in e.g. [13,14]. Homogeneous point ũ′ i = ũi , w′ i = wi of the Zamolodchikov-Bazhanov-Baxter model [15] is complex one, it is not a geometrical regime.

-λ 1 )(x 1 -x ′ 1 )+(λ 3 -iη)(x 2 -x ′ 1 )}

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