Discrete pluriharmonic functions as solutions of linear pluri-Lagrangian systems

Discrete pluriharmonic functions as solutions of linear pluri-Lagrangian systems

Abstract

Pluri-Lagrangian systems are variational systems with the multi-dimensional consistency property. This notion has its roots in the theory of pluriharmonic functions, in the Z-invariant models of statistical mechanics, in the theory of variational symmetries going back to Noether and in the theory of discrete integrable systems. A -dimensional pluri-Lagrangian problem can be described as follows: given a -form on an -dimensional space, , whose coefficients depend on a function of independent variables (called field), find those fields which deliver critical points to the action functionals for any -dimensional manifold in the -dimensional space. We investigate discrete 2-dimensional linear pluri-Lagrangian systems, i.e. those with quadratic Lagrangians . The action is a discrete analogue of the Dirichlet energy, and solutions are called discrete pluriharmonic functions. We classify linear pluri-Lagrangian systems with Lagrangians depending on diagonals. They are described by generalizations of the star-triangle map. Examples of more general quadratic Lagrangians are also considered.

Keywords: discrete Laplace equation, pluriharmonic function, pluri-Lagrangian system, star-triangle map, discrete complex analysis, discrete integrable systems.

1 Introduction

In the last decade, a new understanding of integrability of discrete systems as their multi-dimensional consistency has been a major breakthrough [8], [22]. This led to classification of discrete 2-dimensional integrable systems (ABS list) [1], which turned out to be rather influential. According to the concept of multi-dimensional consistency, integrable two-dimensional systems can be imposed in a consistent way on all two-dimensional sublattices of a lattice of arbitrary dimension. This means that the resulting multi-dimensional system possesses solutions whose restrictions to any two-dimensional sublattice are generic solutions of the corresponding two-dimensional system. To put this idea differently, one can impose the two-dimensional equations on any quad-surface in (i.e., a surface composed of elementary squares), and transfer solutions from one such surface to another one, if they are related by a sequence of local moves, each one involving one three-dimensional cube, like the moves shown of Fig. 1.

Figure 1: Local move of a quad-surface involving one three-dimensional cube

A further fundamental conceptual development was initiated by Lobb and Nijhoff [17] and further generalized in various directions in [18, 19, 31, 10, 12]. This develpoment deals with variational (Lagrangian) formulation of discrete multi-dimensionally consistent systems. Its main idea can be summarized as follows: solutions of integrable systems deliver critical points simultaneously for actions along all possible manifolds of the corresponding dimension in multi-time; the Lagrangian form is closed on solutions. This idea is, doubtless, rather inventive (not to say exotic) in the framework of the classical calculus of variations. However, it has significant precursors. These are:

  • Theory of pluriharmonic functions and, more generally, of pluriharmonic maps [25, 24, 13]. By definition, a pluriharmonic function of several complex variables minimizes the Dirichlet functional along any holomorphic curve in its domain . Differential equations governing pluriharmonic functions (and maps) are heavily overdetermined. Therefore it is not surprising that they belong to the theory of integrable systems.

  • Baxter’s Z-invariance of solvable models of statistical mechanics [3, 4]. This concept is based on invariance of the partition function of solvable models under elementary local transformations of the underlying planar graph. It is well known (see, e.g., [7]) that one can associate the planar graphs underling these models with quad-surfaces in . On the other hand, the classical mechanical analogue of the partition function is the action functional. This makes the relation of Z-invariance to the concept of closedness of the Lagrangian 2-form rather natural, at least at the heuristic level. Moreover, this relation has been made mathematically precise for a number of models, through the quasiclassical limit, in the work of Bazhanov, Mangazeev, and Sergeev [5, 6].

  • The classical notion of variational symmetries, going back to the seminal work of E. Noether [23], turns out to be directly related to the idea of the closedness of the Lagrangian form in the multi-time. This was further elucidated in [30].

Especially the relation with the pluriharmonic functions motivates a novel term we have introduced to describe the situation we are interested in, namely: given a -form in the -dimensional space (), depending on a function of variables, one looks for functions which deliver critical points to actions corresponding to any -dimensional manifold . We call this a pluri-Lagrangian problem and claim that integrability of variational systems should be understood as the existence of the pluri-Lagrangian structure. We envisage that this notion will play a very important role in the future development of the theory of integrable systems.

A general theory of one-dimensional pluri-Lagrangian systems has been developed in [29]. It was demonstrated that for the pluri-Lagrangian property is characteristic for commutativity of Hamiltonian flows in the continuous time case and of symplectic maps in the discrete time case. This property yields that the exterior derivative of the multi-time Lagrangian 1-form is constant, . Vanishing of this constant (i.e., closedness of the Lagrangian 1-form) was shown to be related to integrability in the following, more strict, sense:

  • in the continuous time case, is equivalent for the Hamiltonians of the commuting flows to be in involution,

  • in the discrete time case, for one-parameter families of commuting symplectic maps, is equivalent to the spectrality property [16], which says that the derivative of the Lagrangian with respect to the parameter of the family is a generating function of common integrals of motion for the whole family.

A general theory of discrete two-dimensional pluri-Lagrangian systems has been developed in [11]. The main building blocks of the multi-time Euler-Lagrange equations for a discrete pluri-Lagrangian problem with were identified. These are the so called 3D-corner equations. The notion of consistency of the system of 3D-corner equations was discussed, and this system was analyzed for a special class of three-point 2-forms, corresponding to integrable quad-equations of the ABS list. On this way, a conceptual gap of the work by Lobb and Nijhoff was closed by showing that the corresponding 2-forms are closed not only on solutions of (non-variational) quad-equations, but also on general solutions of the corresponding multi-time Euler-Lagrange equations.

In the present paper, we study the case of quadratic Lagrangian 2-forms, leading to linear 3D-corner equations.

2 Discrete pluri-Lagrangian problem and discrete pluriharmonic functions

The following notations are used throughout the paper. The independent discrete variables on the lattice are denoted . The lattice shifts are denoted by , where is the -th coordinate vector.

  • Single superscript means that an object is associated with the edge and double superscript is used for objects associated with the plaquette .

  • Subscripts denote shifts on the lattice. For instance, the fields at the corners of the plaquette are denoted by .

In the most general form, consistent variational equations can be described as follows.

Definition 1.

(2D pluri-Lagrangian problem)

  • Let be a discrete 2-form, i.e., a real-valued function of oriented elementary squares

    of , such that . We will assume that depends on some field assigned to the points of , that is, on some . More precisely, depends on the values of at the four vertices of :

    (1)
  • To an arbitrary oriented quad-surface in , there corresponds the action functional, which assigns to , i.e., to the fields at the vertices of the surface , the number

    (2)
  • We say that the field is a critical point of , if at any interior point , we have

    (3)

    Equations (3) are called discrete Euler-Lagrange equations for the action (or just for the surface , if it is clear which 2-form we are speaking about).

  • We say that the field solves the pluri-Lagrangian problem for the Lagrangian 2-form if, for any quad-surface in , the restriction is a critical point of the corresponding action .

Definition 2.

(System of corner equations) A 3D-corner is a quad-surface consisting of three elementary squares adjacent to a vertex of valence 3. The system of corner equations for a given discrete 2-form consists of discrete Euler-Lagrange equations for all possible 3D-corners in . If the action for the surface of an oriented elementary cube of the coordinate directions (which can be identified with the discrete exterior derivative evaluated at ) is denoted by ( is the shift in the -th coordinate direction)

(4)

then the system of corner equations consists of the eight equations

(5)

for each triple . Symbolically, this can be put as , where stands for the “vertical” differential (differential with respect to the dependent field variables ).

Remark 1.

(Corner equations: maps and correspondences) Each of eight corner equations (5) relates seven fields and may be not uniquely solvable with respect to them, i.e. generically define correspondences. This leads to a substantial complication of analysis of consistency of the system discussed in the rest of this section (non-connectedness of the variety of solutions). We do not refer to these problems here and assume that the variety of solutions is connected (or one can single out a “physical” connected component). In the case of linear pluri-Lagrangian systems considered in the present paper all corner equations are linear and these problems do not appear.

As demonstrated in [11], the flower of any interior vertex of an oriented quad-surface in can be represented as a sum of (oriented) 3D-corners in . Thus, the system of corner equations encompasses all possible discrete Euler-Lagrange equations for all possible quad-surfaces. In other words, solutions of a pluri-Lagrangian problem as introduced in Definition 1 are precisely solutions of the corresponding system of corner equations.

Definition 3.

(Consistency of a pluri-Lagrangian problem) A pluri-Lagrangian problem with the 2-form is called consistent if, for any quad-surface flippable to the whole of , any generic solution Euler-Lagrange equations for can be extended to a solution of the system of corner equations on the whole .

Like in the case of quad-equations [1], consistency is actually a local issue to be addressed for one elementary cube. The system of corner equations (5) for one elementary cube is heavily overdetermined. It consists of eight equations, each one connecting seven fields out of eight. Any six fields can serve as independent data, then one can use two of the corner equations to compute the remaining two fields, and the remaining six corner equations have to be satisfied identically. This justifies the following definition.

Definition 4.

(Consistency of corner equations) System (5) is called consistent, if it has the minimal possible rank 2, i.e., if exactly two of these equations are independent.

The main feature of this definition is that the “almost closedness” of the 2-form on solutions of the system of corner equations is, so to say, built-in from the outset. One should compare the proof of the following theorem with similar proofs in [10, 29].

Theorem 1.

For any triple of the coordinate directions , the action over an elementary cube of these coordinate directions is constant on solutions of the system of corner equations (5):

Proof.

On the connected six-dimensional manifold of solutions, the gradient of considered as a function of eight variables, vanishes by virtue of (5). ∎

Definition 5.

(Discrete pluriharmonic functions)

If all are quadratic forms of their arguments, then the action functional is called the Dirichlet energy corresponding to the quad-surface . We call a solution of the pluri-Lagrangian problem in this case a discrete pluriharmonic function.

We are aware that our Definition 5 is not an immediate discretization of the classical notion of pluriharmonic functions on , and might be therefore misleading. However, we believe that these two notions are close in spirit and we hope that the further developments will establish a closer relation between the classical and the discrete pluriharmonicity, including approximation theorems for harmonic functions through discrete harmonic functions, cf. [14].

In the context of Definition 4, functional independence is replaced by linear independence, and the rank is understood in the sense of linear algebra.

Theorem 2.

The 2-form is closed on pluriharmonic functions. Let be a discrete pluriharmonic function, and let , be two quad-surfaces with the same boundary, then the Dirichlet energies of harmonic functions and coincide.

Proof.

The constants from Theorem 1 can be evaluated on the trivial pluriharmonic function , which gives . ∎

3 Example: discrete complex analysis

We recall general scheme of discrete complex analysis following [20, 21] (see also [7, 14, 15, 28]). Let be a quad-graph. Consider a generic face with the vertices ordered cyclically counterclockwise, and let , be the diagonals , , respectively. Assume that the weights are defined on diagonals of all quads, such that

Definition 6.

(Discrete holomorphic and discrete harmonic functions)

  • A function on the vertices of a quad-graph is called discrete holomorphic if on every face of it satisfies

    (6)
  • A function is called discrete harmonic if it is a real part of a discrete holomorphic function, .

One may think about planar quads with the vertex coordinates in the complex plane and the weights defined as .

Theorem 3.

A function is discrete harmonic if and only if it satisfies with the following discrete Laplace operator

(7)

or, equivalently, if it is critical for the Dirichlet energy

(8)

(where ).

Note that

where , which shows that the Dirichlet energy is well defined.

Lemma 4.

Let be a discrete harmonic function on a simply connected quad-graph . Then there exists a function such that is a a discrete holomorphic function. Such a function is unique up to an additive imaginary bi-constant (function taking one constant value on all black vertices and another constant value on all white vertices of ), is harmonic with the same weights and is called conjugate to .

Proof.

The conjugate function on a quad is defined by the formulas

(9)

Closedness of the so defined 1-form for is equivalent to harmonicity of the function (see (7)). In turn, Cauchy–Riemann equation (6) for is equivalent to (9). ∎

Recall (see, e.g., [7]) that the quad-graph can be interpreted as a quad-surface in . We are interested in extending all objects from to . Suppose that the complex weights are assigned to the diagonals of elementary squares of , so that the weights are assigned to the diagonals .

Definition 7.
  • A function is called discrete holomorphic if it satisfies equation

    (10)

    on every elementary square of .

  • A function is called discrete pluriharmonic if its satisfies the discrete Laplace equation , i.e., if it is critical for the Dirichlet energy

    (11)

    on every quad-surface in .

Like in Lemma 4, for a discrete pluriharmonic function , there exists a conjugate pluriharmonic function such that is a discrete holomorphic function on . It is defined by the formulas analogous to (9):

(12)

Existence of discrete pluriharmonic functions is a condition on the weights . It can be locally reformulated as a condition on a 3D cube.

Theorem 5.

Complex weights correspond to a consistent discrete Laplace operator if and only if they satisfy the star-triangle equation

(13)
Proof.

This follows from the consistency of discrete Cauchy-Riemann (called also discrete Moutard) equations (see [9]). ∎

One can see that if projections of a quad-surface on all two-dimensional coordinate planes are injective then this quad-surface can be chosen as a support of initial data for the map (13). In this case the weights on a quad-surface can be chosen arbitrarily and extended to the whole of via the map (13). Thus, any discrete Laplace operator on such a quad-surface is integrable in the sense that it can be extended to a consistent Laplace operator on the whole lattice.

4 Classification of discrete pluriharmonic functions with Lagrangians depending on diagonals

4.1 From discrete pluriharmonic functions to vector Moutard equations

Consider Lagrangians

Let be a pluriharmonic function (i.e., let it satisfy all eight corner equations for each cube).

Lemma 6.

Equations

are consistent (by virtue of the corner equations for ) and define the function , called conjugate pluriharmonic function, up to a bi-constant (constant on black and on white points separately).

Proof.

Closedness of the so defined 1-form for is equivalent to the corner equations for the function . ∎

The above equations can be put as

or, in the matrix form,

(14)

(vector Moutard equation), with the matrix coefficients

(15)

whose entries are related to the coefficients of the Lagrangians via

(16)

We say that vector Moutard equation (14) is consistent if arbitrary initial data and () can be extended to the fields at all vertices of a 3D cube so that (14) is fulfilled on all six faces. Valid initial data for a consistent vector Moutard equation consist of 8 numbers, e.g.,

  • the values and all for .

These data can be put into the one-to-one correspondence with the following initial data for a pair of conjugate pluriharmonic functions within one elementary cube:

  • values of at 6 vertices, which define by corner equations the values of at the remaining two vertices, and the values of at two vertices (one white and one black).

Thus, the task of the classification of consistent pluri-Lagrangian systems with quadratic Lagrangians depending on diagonals turns out to be equivalent to the task of the description of consistent systems of vector Moutard equations.

4.2 Vector Moutard equations and non-commutative star-triangle relation

The solution of the latter task is described in the following theorem.

Theorem 7.

Matrices of the form (15) assigned to plaquettes in serve as coefficients of a consistent system of vector Moutard equations if and only if their entries satisfy

(17)

for some fixed triple , where , and their evolution is expressed (in the generic case, when ) through a solution of coupled star-triangle relations

(18)

via the following relations:

(19)

and

(20)

where , are the two roots of the quadratic equation .

We will prove this theorem by considering one 3D cube. It will be convenient to simplify the notations: we will denote the matrix coefficients assigned to the faces of the cube by , , where, for instance,

and other ones obtained by cyclic shifts of indices (the entries will carry the same indices).

Proposition 8.

Vector Moutard equation is consistent if and only if the matrix coefficients satisfy

(21)

and then the matrices are given by non-commutative star-triangle relations

(22)

where is any cyclic permutation of . Note that the cyclic permutation of indices in (21) leads to an equivalent equation.

Proof.

Moutard equations on the three faces adjacent to the vertex 123 yield three different linear expressions for in terms of the initial data. Identifying the corresponding coefficients, we obtain:

(23)

The first equations in each of these three lines allow us to express, say, and linearly through in two different ways, e.g.,

These two expressions agree if condition (21) is satisfied. Now the second equations in each of the three lines (4.2) yield linear equations for each of , which result in (22). ∎

For matrices of the form , as in (15), we will give a convenient resolution of the constraint (21), which will allow us to give a complete solution of the system consisting of (21), (22) on .

Lemma 9.

Matrix equation (21) is equivalent to

(24)

i.e., there exist such that

(25)
Proof.

We have:

This has to be equal to the same matrix with . A direct inspection shows that the off-diagonal terms give no conditions, while the diagonal ones give only one condition:

which is nothing but (24). ∎

Proof of Theorem 7. Formula (17) describes propagation of condition (25) to the lattice . To show that this condition propagates under flipping the 3D cubes, consider an elementary cube with (25) satisfied on three faces adjacent to one vertex. We have to show that for the opposite three faces there follows

(26)

From

we derive:

We further compute:

(we added and subtracted , , and ). Analogously, .

To parametrize matrix formulas (22) by a system of coupled scalar star-triangle relations, we need a more precise computation. We start with

(27)

where and . Upon condition (25), we find:

(28)

and, similarly,

(29)

Now the latter formula can be put as

and upon using the expressions from (27), we put this as

(30)

By transforming the right-hand side of the latter equation, one systematically eliminates in favor of and , according to (25). In particular, one easily finds that . Thus, we find:

In the last transformation, we have taken into account that and . Plugging the latter result along with (28) into (30), we finally arrive at

which demonstrates (one of) equations (18). ∎

Remark. The oscillating sign in (37) can be explained in the following way. The matrices in (14) can be interpreted as mapping from the diagonal to the diagonal of an elementary square . A more geometric way would be to consider matrices mapping black diagonals (connecting points with even) to white diagonals (connecting points with odd). These are matrices

For these matrices, constraint (25) holds on all faces of , without changing the sign of .

Example 1. , , so that , and . Since , it is more convenient to replace definitions (19), (20) by the “non-oscillating” ones, namely,

Thus, we can set

where the complex-valued coefficients evolve according to the star-triangle equation:

(31)

Example 2. , , so that , and . Like in the previous example, we replace definitions (19), (20) by the “non-oscillating” ones, namely,

Thus, we can set

where the coefficients , evolve according to two (uncoupled) star-triangle equations:

(32)

Example 3. , , thus . The parametrization of Theorem 7, taken literally, does not work, but the matrix star-triangle relations (22) for triangular matrices can be easily solved. Indeed, from (27) with we find directly:

(33)