## Abstract

We study the Desargues maps , which generate lattices whose points are collinear with all their nearest (in positive directions) neighbours. The multi-dimensional compatibility of the map is equivalent to the Desargues theorem and its higher dimensional generalizations. The nonlinear counterpart of the map is the non-commutative (in general) Hirota–Miwa system. In the commutative case of the complex field we apply the non-local -dressing method to construct Desargues maps and the corresponding solutions of the system. In particular, we identify the Fredholm determinant of the integral equation inverting the non-local -dressing problem with the *τ*-function. Finally, we establish equivalence between the Desargues maps and quadrilateral lattices provided we take into consideration also their Laplace transforms.

## 1. Introduction

Perhaps the most widely studied integrable discrete system of equations is the Hirota–Miwa system,
1.1
which is the compatibility condition of the linear equations (the adjoint of that introduced in Date *et al.* (1982))
1.2
Here and throughout the paper we use the convention that, for any function *f* defined on a multi-dimensional integer lattice , we denote by *f*_{(±i)} its shift in the *i* (positive or negative) direction of the lattice, i.e. *f*_{(±i)}(*n*_{1},…,*n*_{i},…,*n*_{N})=*f*(*n*_{1},…,*n*_{i}±1,…,*n*_{N}). Whenever it does not lead to misunderstanding, when speaking on the image *f*(*n*) of a point , we skip the argument.

In the basic case *N*=3, when the system (1.1) reduces to a single equation, it was discovered, up to a change of independent variables, by Hirota (1981), who called it the discrete analogue of the two-dimensional Toda lattice, as a culmination of his studies on the bilinear form of nonlinear integrable equations; see Zabrodin (1997) for a review of various forms of the equation and of its reductions. The general feature of Hirota’s equation was uncovered by Miwa (1982), who found a remarkable transformation which connects the equation to the Kadomtsev–Petviashvili (KP) hierarchy (Date *et al.* 1982). The Hirota–Miwa equation/system can be encountered in various branches of theoretical physics (Saito 1987; Kuniba *et al.* 1994) and mathematics (Shiota 1986; Krichever *et al.* 1998; Knutson *et al.* 2004). In the literature there are also known non-commutative versions (Nijhoff 1985; Nijhoff & Capel 1990; Nimmo 2006) of the Hirota–Miwa system.

For a while there was some activity in providing geometrical interpretation for integrable discrete systems. The idea was to transfer to a discrete level the well-known connection between geometry and integrable differential equations; see classical monographs (Darboux 1887–1896, 1910; Eisenhart 1923; Tzitzéica 1923; Bianchi 1924; Finikov 1959) written in the pre-solitonic period, and more recent works (Sym 1985; Rogers & Schief 2002; Gu *et al.* 2005). Almost immediately after the first work in this direction, which included the discrete pseudospherical surfaces (Bobenko & Pinkall 1996*a*), evolutions of discrete curves (Doliwa & Santini 1995) and discrete isothermic surfaces (Bobenko & Pinkall 1996*b*), Doliwa (1997) gave a geometric interpretation of the *N*=3 dimensional Hirota–Miwa equation in its two-dimensional Toda lattice form. The basic geometric object in Doliwa (1997) was the Laplace sequence of two-dimensional lattices made of planar quadrilaterals; see §5 for more details. Such lattices were introduced much earlier (Sauer 1937, 1970) as discrete analogues of conjugate nets on a surface.

Soon after Doliwa (1997) the multi-dimensional lattices of planar quadrilaterals, also called quadrilateral lattices for short, were considered in Doliwa & Santini (1997). In particular, it was shown there that such lattices are described by solutions of the discrete Darboux system (Bogdanov & Konopelchenko 1995). The initial boundary value problem for a multi-dimensional quadrilateral lattice is based on the following simple geometric statement (see figure 1).

*Consider points x_{0}, x_{1}, x_{2} and x_{3} in general position in , M≥3. On the plane 〈x_{0},x_{i},x_{j}〉, 1≤i<j≤3 choose a point x_{ij} not on the lines 〈x_{0},x_{i}〉, 〈x_{0},x_{j}〉 and 〈x_{i},x_{j}〉. Then there exists the unique point x_{123}, which belongs simultaneously to the three planes 〈x_{3},x_{13},x_{23}〉, 〈x_{2},x_{12},x_{23}〉 and 〈x_{1},x_{12},x_{13}〉*.

This construction scheme is multi-dimensionally compatible (Doliwa & Santini 1997) and provides an initial boundary value problem for the corresponding discrete Darboux equations in terms of *K*(*K*−1) functions of two discrete variables (also implying multi-dimensional compatibility of the system). As was shown in Doliwa *et al.* (2000), the Darboux transformations of the quadrilateral lattice (thus also the corresponding Bäcklund transformations of the discrete Darboux equations), being discrete symmetries of the quadrilateral lattice Darboux system, can be considered as a recursive augmentation of the number of independent variables. Moreover the Bianchi permutability principle of superposition of the transformations, which is often considered synonymous to integrability, is a consequence of that simple geometric scheme. Therefore, the ‘compatibility of the construction for arbitrary dimension of the lattice’ (Doliwa & Santini 1997) provides its own commuting symmetries.

We remark that the discrete Darboux equations have been constructed in Bogdanov & Konopelchenko (1995) as the most general discrete system integrable by the non-local -dressing method. Moreover the differential Darboux (1910) equations give in a special limit (Bogdanov & Manakov 1988) the KP hierarchy of equations (in this context the possibility of considering an arbitrary number of independent variables is crucial). This places the quadrilateral lattice (Darboux) equations in a distinguished position among all integrable discrete systems; see also Bogdanov & Konopelchenko (1998) and Doliwa *et al.* (1999*a*,*b*) on the relation of Darboux equations, conjugate nets and quadrilateral lattices and the multi-component KP hierarchy (Date *et al.* 1983), which is often considered to be the master integrable system.

Having recognized the role of the quadrilateral lattice as the master geometric object of the integrability theory the remaining task is to study the integrable reductions of the lattice. If a geometric constraint, imposed on the initial points, propagates during the construction, the corresponding reduction of the discrete Darboux equations is called geometrically integrable. Note that such consistency of the geometric integrability scheme with the reduction in conjunction with the multi-dimensional compatibility of the quadrilateral lattice implies the multi-dimensional compatibility of the reduction. This point of view was used in Cieśliński *et al.* (1997), Doliwa (1999, 2007*a*,*b*) and Doliwa & Santini (2000) (see also the recent review by Doliwa & Santini (2006)) to select integrable reductions of the quadrilateral lattice and to find the corresponding reductions of the discrete Darboux equations.

For example, the integrability of quadrilateral lattices with elementary quadrilaterals inscribed in circles, introduced in Bobenko (1999) as a discrete analogue of orthogonal coordinate systems, was first proved in this way in Cieśliński *et al.* (1997). The integrability of the circular lattice was then confirmed by the non-local -dressing method (Doliwa *et al.* 1998), by construction of the corresponding Darboux-type transformation (Konopelchenko & Schief 1998), which satisfies (Liu & Mañas 1998; Doliwa 1999) the permutability property, by construction of such lattices using the Miwa transformation from the multi-component BKP hierarchy (Doliwa *et al.* 1999*a*), and by application of the algebro-geometric techniques (Akhmetshin *et al.* 1999). Remarkably, as described in Bazhanov *et al.* (2008), there exists a quantization procedure for circular lattices, which leads to solutions of the tetrahedron equation (the three-dimensional analogue of the Yang–Baxter equation).

In Adler *et al.* (2003) (see also Nijhoff 2002), the notion of multi-dimensional consistency as a tool to detect integrable equations has been given in the following form: a *d*-dimensional discrete equation possesses the *consistency* property, if it may be imposed in a consistent way on all *d*-dimensional sublattices of a (*d*+1)-dimensional lattice. In the terminology of Adler *et al.* (2003) the linear problem of the quadrilateral lattice is three dimensionally consistent, while the discrete Darboux system is four dimensionally consistent.1

It turns out that the geometric notion of integrability of reductions of the quadrilateral lattice very often associates them with classical theorems (Coxeter 1961) of incidence geometry. For example, integrability of the circular lattice is a consequence of the Miquel (1838) theorem. This observation makes the relationship between integrability of the discrete systems and geometry even more profound than the corresponding relation on the level of differential equations. Integrable reductions of the quadrilateral lattice come from two sources. The first are inner (i.e. invariant with respect to the full group of projective transformations of the ambient space) symmetries of the lattice. The second type of reductions arises from the postulated existence of additional structures (e.g. distinguished quadrics, hyperplanes) in the ambient space and mimics the Cayley–Klein approach to subgeometries of the projective geometry, which was the starting point of the famous Erlangen program. Such an approach to possible classification of integrable discrete systems was formulated in Doliwa (1999, 2001*a*); see also Bobenko & Suris (2007, 2009).

Apart from the geometric interpretation of the three-dimensional Hirota–Miwa equation in its two-dimensional Toda lattice form, there is known in the literature (Konopelchenko & Schief 2002) an interpretation of its Schwarzian form, the so-called Menelaus lattice. It is related to the adjoint linear problem of equation (1.2) for a map in the affine gauge, and gives the so-called discrete Schwarzian KP equation, which is related to the Hirota–Miwa equation by a non-local transformation (Konopelchenko & Schief 2002; King & Schief 2003; Schief 2003); see also §3.

An important observation (Bobenko 2009), which was one of the motivations of the present research, associates the four-dimensional consistency of the discrete Schwarzian KP equation with the Desargues configuration;2 see §2. Another fact, which was the starting point of the paper, is that there is no essential difference between the space of the algebro-geometric solutions of the Hirota–Miwa equations (; Krichever *et al.* 1998) and the quadrilateral lattice Darboux system (Akhmetshin *et al.* 1999), provided one takes their Laplace transforms (Doliwa *et al.* 2000) into consideration (Doliwa 2001*b*).

In the paper we study the maps defined by the most simple non-trivial linear condition stating that for any pair of indices *i*≠*j* the points *ϕ*, *ϕ*_{(i)} and *ϕ*_{( j)} are collinear. This is a natural geometric counterpart of the linear problem (1.2). We show in a synthetic geometry way that the multi-dimensional compatibility of the map follows from the Desargues theorem and its higher dimensional analogues.

Then, in §3 we draw algebraic consequences of the geometric definition of the Desargues maps. As the algebraic significance of the Desargues theorem suggests (Beukenhout & Cameron 1995), we consider projective spaces over division rings, which leads to the non-Abelian Hirota–Miwa system (Nimmo 2006). We also discuss various gauge-equivalent forms of the equation in the non-commutative setting.

It can be seen from both simple geometric and algebraic considerations that the Desargues maps can also be called multi-dimensional adjoint Menelaus maps (figure 2). We prefer, however, to name them in a way that reflects the projective geometric character of the lattice and captures simultaneously its integrability properties.

In §4 we apply the non-local -dressing method (Ablowitz *et al.* 1983; Zakharov & Manakov 1985; Konopelchenko 1993) to find large classes of solutions to the Hirota–Miwa system over the field of complex numbers. In particular we show, as one may expect from previous work (Segal & Wilson 1985; Pöppe & Sattinger 1988; Palmer 1990; Doliwa 2009), that the *τ*-function of the Hirota–Miwa system can be identified with the Fredholm determinant of the integral equation inverting the non-local -dressing problem. We also find that on the level of the non-local -dressing method the solution space of the Hirota–Miwa system is the same as in the quadrilateral lattice Darboux system (Bogdanov & Konopelchenko 1995), provided one also takes (Doliwa *et al.* 2000) the Laplace transformations of the lattice into consideration.

The three-point condition in definition of the Desargues map can be considered as a serious degeneration of the quadrilateral lattice map four-point condition. Such an approach was presented for the three-point linear problem of the Menelaus lattice in, for example, Konopelchenko & Schief (2002). In §5 we show, however, that the quadrilateral lattice theory and the Desargues lattice theory are equivalent.

After submitting the first version of the manuscript I have learnt of related recent works by Schief (2009) and Adler (2009).

## 2. Geometry of the Desargues maps

In this section, we study in detail geometric properties of the Desargues maps. After collecting basic facts on the Desargues configuration we state some genericity assumptions concerning the maps. Then we study the multi-dimensional compatibility of the Desargues maps. Here we understand this notion as a possibility of recursive augmentation of the number of independent variables preserving the geometric condition that characterizes the maps. This point of view mimics successive applications of the Darboux transformations or the recursion operator (Olver 1977). We postpone discussion of the initial boundary value problem for the Desargues maps to §5 after we show their relationship with the quadrilateral lattices.

### (a) The Desargues configuration

Among all incidence theorems in projective geometry the Desargues theorem (figure 3) plays a very distinguished role (Coxeter 1961; Beukenhout & Cameron 1995). It holds in projective spaces of dimension more than two, and is an important element in proving the possibility of introducing homogeneous coordinates taking values in a division ring; in order to introduce such coordinates on projective planes, one should add it as an axiom.

The 10 lines involved and the 10 points involved are so arranged that each of the 10 lines passes through three of the 10 points, and each of the 10 points lies on three of the 10 lines. Under the standard duality of plane projective geometry (where points correspond to lines and collinearity of points corresponds to concurrency of lines), the Desargues configuration is self-dual: axial perspectivity is translated into central perspectivity and vice versa.

At first sight it seems that the Desargues configuration has less symmetry than it really has. However, any of the 10 points may be chosen to be the centre of perspectivity, and that choice determines which six points will be vertices of triangles and which line will be the axis of perspectivity. The Desargues configuration has symmetry group *S*_{5} of order 120. It can be constructed from a five-point set, preserving the action of the symmetric group, by letting the points and lines of the Desargues configuration correspond to two- and three-element subsets of the five points, with incidence corresponding to the containment.

### Remark 2.1.

In the above interpretation of the symmetry group of the Desargues configuration, the four-element subsets give rise to five complete quadrilaterals described by the Menelaus theorem, as used in Bobenko (2009) in connection with the four-dimensional consistency of the discrete Schwarzian KP equation.

### (b) The Desargues maps

In this paper we study the following maps, the connection of which with the Desargues theorem is essential in showing their multi-dimensional compatibility.

### Definition 2.2.

By *Desargues map* we mean a map of multi-dimensional integer lattice in Desarguesian projective space of dimension M≥2, such that for any pair of indices i≠j the points *ϕ*, *ϕ*_{(i)} and *ϕ*_{( j)} are collinear.

### Remark 2.3.

Note that we consider as a directed graph.

### Remark 2.4.

The image of a Desargues map can be called a Desargues lattice. However, we would like to stress that we do not use this notion in the sense of the lattice theory as described in Birkhoff (1948) and Jonsson (1954).

Let us discuss various genericity assumptions of the map. Consider an *N*-dimensional, *N*>0, hypercube graph with a distinguished vertex labelled by ∅, its first-order neighbours labelled by {*i*}, , and other vertices labelled as follows: the fourth vertex of a quadrilateral with three other vertices *I*, , , *i*,*j*∉*I*, *i*≠*j* is .

### Definition 2.5.

A *DesarguesN-hypercube* consists of labelled vertices *ϕ*_{I} of an N-dimensional hypercube in projective space , M≥2, such that for arbitrary multi-index there exists a line *L*_{I} incident with *ϕ*_{I} and with all the points , *i*∉*I*. A Desargues *N*-hypercube is called *non-degenerate* if all its vertices are distinct. A non-degenerate Desargues *N*-hypercube is called *weakly generic* if all the lines *L*_{I} are distinct.

Given two multi-indices *I*_{1},*I*_{2}, with , the points *ϕ*_{J}, , of a weakly generic Desargues *N*-hypercube form a weakly generic Desargues (|*I*_{2}|−|*I*_{1}|)-hypercube. The space *π*_{I1,I2} spanned by the points *ϕ*_{J}, , has dimension (|*I*_{2}|−|*I*_{1}|) at most. For example, for all *i*∉*I*. We also write *π*_{I}=*π*_{∅,I}.

### Definition 2.6.

A *Desargues N-hypercube* is called generic if for all *.*

### Remark 2.7.

Note that suitable projections of generic Desargues hypercubes can produce weakly generic Desargues hypercubes.

### Definition 2.8.

A Desargues map is called (*weakly*) generic if the corresponding Desargues lattice consists of (weakly) generic Desargues N-hypercubes under identification *ϕ*_{I} with *ϕ*_{(I)} for a fixed point *ϕ* of the lattice.

Note that any weakly generic Desargues map induces a map into the Grassmann space of lines in , where *L* is the line coincident with the point *ϕ* and all the neighbouring *ϕ*_{(i)}, . Such maps are characterized by the following two properties.

— Any two neighbouring lines

*L*and*L*_{(i)}intersect.— The intersection points coincide for all 1≤

*i*≤*N*.

The maps , satisfying the first condition only, play an important role in the theory of Darboux transformations of the quadrilateral lattice (Doliwa *et al.* 2000) and are called line congruences. It is natural to also call the line congruences satisfying the second condition the Desargues congruences. Then the points of the Desargues lattice can be recovered by , .

### (c) Multi-dimensional compatibility of Desargues maps

Given point *ϕ* and its two nearest (in positive directions) neighbours *ϕ*_{(i)} and *ϕ*_{( j)}, by definition there exists a line *L* incident with the three points. Assuming the Desargues map is weakly generic, the point *ϕ*_{(ij)} can be an arbitrary point not on the line *L*. Such a choice determines the lines *L*_{(i)} and *L*_{( j)}.

#### (i) Three-dimensional compatibility and the Veblen–Young axiom

Consider a point *ϕ*_{(k)}∈*L*, *k*≠*i*,*j*. On the line *L*_{(i)} choose a point *ϕ*_{(ik)} distinct from *ϕ*_{(i)} and *ϕ*_{(ij)}, thus determining the line *L*_{(k)}. Then three-dimensional compatibility of the Desargues map, i.e. the existence of the intersection point *ϕ*_{( jk)} of lines *L*_{( j)} and *L*_{(k)}, is equivalent to the Veblen–Young axiom of the synthetic projective geometry, which in the current notation states the following (cf. figure 2):

*Given four distinct points ϕ_{( j)}, ϕ_{(k)}, ϕ_{(ij)}, ϕ_{(ik)}, if the lines L_{ϕ( j)ϕ(k)}=L and L_{ϕ(ij)ϕ(ik)}=L_{(i)} intersect, then the lines L_{ϕ( j)ϕ(ij)}=L_{( j)} and L_{ϕ(k)ϕ(ik)}=L_{(k)} intersect as well*.

There is no condition for the point *ϕ*_{(ijk)}, apart from a weak genericity assumption, which means that it should not be placed on the lines *L*, *L*_{(i)}, *L*_{( j)}, *L*_{(k)}.

#### (ii) Four-dimensional compatibility and the Desargues theorem

Add the next point *ϕ*_{(ℓ)} on the line *L*, ℓ≠*i*,*j*,*k*, and the point *ϕ*_{(iℓ)}∈*L*_{(i)}. The corresponding line *L*_{(ℓ)}, incident with *ϕ*_{(ℓ)} and *ϕ*_{(iℓ)}, intersects (by Veblen–Young) the lines *L*_{( j)}, *L*_{(k)} in the points *ϕ*_{( jℓ)} and *ϕ*_{(kℓ)}, correspondingly. The problem is to find the four points *ϕ*_{(ijk)}, *ϕ*_{(ijℓ)}, *ϕ*_{(ikℓ)} and *ϕ*_{( jkℓ)} which satisfy the Desargues map condition.

Choose a point *ϕ*_{(ijℓ)}, not on the lines *L*, *L*_{(i)}, *L*_{( j)}, *L*_{(k)}, and define therefore the lines *L*_{(ij)}, *L*_{(iℓ)} and *L*_{( jℓ)}. On the line *L*_{(iℓ)} mark a point *ϕ*_{(ikℓ)}, thus defining the lines *L*_{(ik)} and *L*_{(kℓ)}. The lines *L*_{( jℓ)} and *L*_{(kℓ)} intersect (by Veblen–Young) in the point *ϕ*_{( jkℓ)}, which gives the line *L*_{( jk)}. We have constructed two triangles in perspective from the line *L*_{(ℓ)}: the first with vertices *ϕ*_{(ij)}, *ϕ*_{(ik)}, *ϕ*_{( jk)}, and the second with vertices *ϕ*_{(ijℓ)}, *ϕ*_{(ikℓ)}, *ϕ*_{( jkℓ)}. By the Desargues theorem the three lines *L*_{(ij)}, *L*_{(ik)} and *L*_{( jk)} intersect in one point, which is by construction *ϕ*_{(ijk)} (figure 4).

#### Remark 2.9.

Note that in the generic case when the points *ϕ*, *ϕ*_{(i)}, *ϕ*_{(ij)} and *ϕ*_{(ijk)} generate the space *π*_{(ijk)} of dimension 3, then all the points whose shifts contain the index ℓ are obtained as intersections of the lines of the ‘*ijk* configuration’ with the plane generated by the points *ϕ*_{(ℓ)}, *ϕ*_{(iℓ)} and *ϕ*_{(ijℓ)}. Moreover, to keep the configuration generic we add the point *ϕ*_{(ijkℓ)} (which is not specified by the previous construction) outside the space *π*_{(ijk)}, thus generating the four-dimensional space *π*_{(ijkℓ)}.

#### (iii) The multi-dimensional compatibility for arbitrary *N*

The multi-dimensional compatibility of the Desargues map is equivalent to existence of a Desargues *N*-hypercube for arbitrary *N*, provided appropriate initial data have been prescribed. The following proposition allows us to construct generic Desargues (*N*+1)-hypercubes from generic Desargues *N*-hypercubes in spaces of the dimension large enough. It is an analogue of the well-known (mentioned in the remark above) three-dimensional proof of the Desargues theorem. By suitable projections one can therefore produce weakly generic Desargues hypercubes.

#### Proposition 2.10.

*Given a generic Desargues N-hypercube in , where N<M, on the N lines L_{∅},L_{{1}},L_{{1,2}},L_{{1,2,…,N−1}}, choose N points ϕ_{{N+1}},ϕ_{{1,N+1}},ϕ_{{1,2,N+1}},…,ϕ_{{1,2,…,N−1,N+1}} in generic position, correspondingly, in such a way that the N−1 dimensional subspace U of*
i

*s not incident with any vertex of the hypercube. Then the unique intersection points of the hyperplane with the lines of the*.

*N*-hypercube, and the points of the initial hypercube supplemented by a point*ϕ*_{{1,2,…,N,N+1}}∉*π*_{{1,2,…,N}}give a generic Desargues (*N*+1)-hypercube

#### Proof.

By the assumption of the proposition the lines *L*_{I}, , are not contained in the hyperplane *U*, thus all the points are well defined. Having then all the vertices of the (*N*+1)-hypercube we will check that it satisfies the desired properties.

Given multi-index there are two possibilities.

—

*N*+1∉*I*. When |*I*|=*N*then*I*={1,2,…,*N*} and define*L*_{I}as the unique line incident with*ϕ*_{{1,2,…,N}}and*ϕ*_{{1,2,…,N,N+1}}. If |*I*|<*N*then take as the line*L*_{I}the line of the*N*-hypercube, and by construction.—

*N*+1∈*I*. There exists*i*∈{1,2,…,*N*},*i*∉*I*. Set*L*_{I}as the unique line incident with*ϕ*_{I}and . To conclude the proof of the Desargues property we will show that*L*_{I}is independent of a particular choice of such an index*i*. When |*I*|=*N*then there is nothing to prove because there is only one index*i*∉*I*. If 1≤|*I*|<*N*set*J*=*I*∖{*N*+1}, there exists*j*∈{1,2,…,*N*},*i*≠*j*, and*j*∉*J*. Consider the plane , which contains the three lines*L*_{J}, and and is not contained in*U*. Then , which also shows that , thus the index*j*can also be used to define*L*_{I}.

Finally, to prove genericity of the Desargues (*N*+1)-hypercube, note that for all the intersection is the point *ϕ*_{I2} only. This implies that , and . ▪

### (d) The adjoint Desargues maps

To obtain the analogous geometric meaning of the adjoint of the linear problem of the Hirota–Miwa system, define the adjoint Desargues maps (or multi-dimensional Menelaus maps, if one restricts to affine part of ) as maps such that for any pair of indices *i*≠*j* the points *ϕ**_{(i)}, *ϕ**_{( j)} and *ϕ**_{(ij)} are collinear. This leads to the following definition of an adjoint Desargues *N*-hypercube.

### Definition 2.11.

An *adjoint DesarguesN-hypercube* consists of labelled vertices of an N-dimensional hypercube in projective space , *M*≥2, such that for an arbitrary multi-index , |I|>1, and for any pair of distinct indices i,j∈I the vertex *ϕ**_{I} is incident with a line passing through *ϕ**_{I∖{i}} and *ϕ**_{I∖{j}}.

One can note that given the Desargues map , its superposition *ϕ*°ı with the arrows inversion map , ı(*n*)=−*n*, is an adjoint Desargues map (and vice versa). Similarly, any Desargues *N*-hypercube gives rise to the adjoint Desargues *N*-hypercube under identification . The geometric theory of the adjoint Desargues map follows from that identification.

## 3. Desargues maps and various non-commutative discrete KP equations

In this section we study the algebraic consequences of the geometric definition of the Desargues map . Since to prove its multi-dimensional compatibility we use only the Desargues theorem then the natural coordinates of the projective space are elements of a division ring . This leads to the corresponding non-commutative nonlinear equations which we formulate first in the arbitrary gauge, i.e. keeping the freedom in rescaling the homogeneous coordinates by a non-zero factor. Two basic specifications of the gauge are discussed in §3*b*.

### (a) The linear problem for the Desargues maps and its compatibility conditions

In the homogeneous coordinates (we consider right vector spaces) the map can be described in terms of the linear system 3.1 where are certain non-vanishing functions.

### Proposition 3.1.

*The compatibility of the linear system (3.1 ) is equivalent to equations*
3.2*and*
3.3
*where the indices i,j,k are distinct*.

### Proof.

From the linear problem (3.1), for the pair (*i*,*k*) find *ϕ*_{(k)} in terms of ** ϕ** and

*ϕ*_{(i)}. Similarly, find

*ϕ*_{(k)}from the equation for the pair (

*j*,

*k*). Comparing the resulting relation between

**and**

*ϕ*

*ϕ*_{(i)}and

*ϕ*_{( j)}with the linear problem (3.1) for the pair (

*i*,

*j*) gives, after some elementary algebra, the first equation.

The compatibility of the linear problem (3.1) shifted in *k* direction with two other similar equations involving three distinct indices *i*,*j*,*k* gives rise to a linear relation between ** ϕ**,

*ϕ*_{(k)}and

*ϕ*_{(ij)}. Their linear independence implies the vanishing of the corresponding coefficients 3.4 3.5 3.6

Equations (3.4) and (3.6) directly lead equation (3.3).

Using equation (3.3) we can replace equations (3.4) and (3.5) by 3.7and 3.8

We will show that equation (3.7) follows from condition (3.2). Indeed, starting from the identity and using equation (3.2), we get equivalent to equation (3.7). Also equation (3.8) is a direct consequence of condition (3.2). ▪

### Corollary 3.2.

*For any three distinct indices i,j,k we can write down three distinct equations of the form (3.2 ). However, any two of them imply the third one*.

### Proof.

Equation (3.2), where the indices *i* and *k* enter symmetrically, is equivalent to
3.9
Adding equation (3.9) to the similar one with the indices *k* and *j* exchanged one obtains equation (3.2) with the indices *i* and *j* exchanged
▪

### Corollary 3.3.

*Equation (3.3) implies the existence of the potentials ρ*_{i} : ,*unique up to functions of single variables n_{i}, such that*
3.10

### (b) Gauges

We are still left with the possibility of applying the gauge transformation
3.11
where is an arbitrary non-vanishing function. Then satisfies the linear problem (3.1) with the coefficients
3.12
By fixing properties of *G* one can arrive at the relationship between the coefficients of the linear problem. We will discuss two gauges. The first gauge, which because of the geometric interpretation can be called the affine gauge, gives in the commutative case the discrete modified KP system. The second gauge in the commutative case leads to the Hirota–Miwa system.

#### (i) The modified discrete KP gauge

#### Proposition 3.4.

*When the gauge function is a non-vanishing solution of the linear problem (3.1 ) then the coefficients* *are constrained by the relation*
3.13

#### Remark 3.5.

When the solution of the linear problem is taken as the last coordinate *ϕ*^{M+1} of the homogeneous representation of the map, then we obtain the standard transition to the non-homogeneous coordinates.

#### Remark 3.6.

In the affine gauge the algebraic compatibility system (3.2) consists, for any triple of distinct indices *i*,*j*,*k*, of one independent equation.

It is convenient (we follow the reasoning presented in Schief (2003) in the commutative case) to rewrite the linear problem (3.1) subject to condition (3.13) as 3.14 where 3.15

Then the algebraic compatibility takes the form 3.16 which allows for introduction of a potential such that 3.17

The second part of the compatibility condition then takes the form of the non-commutative discrete mKP system (Nijhoff & Capel 1990) 3.18

Finally, note that owing to the compatibility of the system
3.19
each coordinate of ** ϕ** satisfies the generalized lattice spin system (Nijhoff & Capel 1990)
3.20
also called the non-commutative Schwarzian discrete KP system (Bogdanov & Konopelchenko 1998; Konopelchenko & Schief 2005).

#### (ii) The Hirota–Miwa system gauge

In order to introduce the second gauge we need the following result.

#### Lemma 3.7.

*There exists non-vanishing function G defined as a solution of the system*
3.21

#### Proof.

The algebraic compatibility of equation (3.21) for three pairs of indices *i*,*j*,*k* has the form
3.22

It can be proved by application of the algebraic compatibility condition (3.2) starting from the identity ▪

#### Proposition 3.8.

*The linear system (3.1) is gauge equivalent to the discrete linear problem of the non-Abelian Hirota–Miwa system* (Date *et al.* 1982; Nimmo 2006 )
3.23

#### Proof.

Take the gauge function *G* as in lemma 3.7 above, which gives (we skip tildes)
3.24
and set . ▪

In this gauge the compatibility conditions (3.2–3.3) reduce to the following systems for distinct triples *i*,*j*,*k*
3.25
and
3.26

This allows us to introduce potentials such that 3.27 see Nimmo (2006) for further properties of the system.

## 4. Application of the non-local -dressing method

In this section the division ring is replaced by the field of complex numbers. By application of the non-local -dressing method (Ablowitz *et al.* 1983; Zakharov & Manakov 1985; Konopelchenko 1993) we construct solutions of the Hirota–Miwa system and the corresponding solutions of the linear problem.

Consider the following integro-differential equation in the complex plane
4.1
where *R*(*λ*,*λ*′) is a given datum, which decreases quickly enough at in *λ* and , and the function *η*(*λ*), the normalization of the unknown *χ*(*λ*), is a given rational function, which describes the polar behaviour of *χ*(*λ*) in and its behaviour at

We remark that the dependence of *χ*(*λ*) and *R*(*λ*,*λ*′) on and will be systematically omitted, for notational convenience.

Owing to the generalized Cauchy formula, the non-local problem (4.1) is equivalent to the following Fredholm integral equation of the second kind
4.2
with the kernel
4.3
Recall (e.g. Smithies 1965) that the Fredholm determinant *D* is defined by the series
4.4
where

For a non-vanishing Fredholm determinant the solution of equation (4.2) can be written in the form 4.5 where the Fredholm minor is defined by the series 4.6

Let , *i*=1,…,*N* be distinct points of the complex plane. Consider the following dependence of the kernel *R* on the variables
4.7
or equivalently
4.8
where is independent of *n*. We assume that *R*_{0} decreases at *λ*_{i} and in poles of the normalization function *η* fast enough such that *χ*−*η* is regular in these points (Bogdanov & Manakov 1988; Bogdanov & Konopelchenko 1995).

## Remark 4.1.

In this paper we always assume that the kernel *R* in the non-local problem is such that the Fredholm equation (4.2) is uniquely solvable. Then, by the Fredholm alternative, the homogeneous equation with *η*=0 has only the trivial solution.

## Remark 4.2.

The structure of the function *E*(*λ*;*n*) mimics the analytic structure of the Baker–Akhezer wave function used in Krichever *et al.* (1998) to solve the Hirota–Miwa system by the algebro-geometric techniques, where the role of the Fredholm alternative is played by the Riemann–Roch theorem.

## Lemma 4.3.

*The evolution (4.7 ) of the kernel R implies the following evolution of the determinants in the series defining the Fredholm determinant D*

## Proof.

The evolution (4.7) of the kernel *R* implies that the kernel *K* of the integral equation (4.2) is subject to the equation
4.9
and the conclusion is reached by basic linear algebra. ▪

## Proposition 4.4.

*Let χ(λ;n) be a solution of the*

*problem (4.1 ) with the canonical normalization*4.10

*η*=1 then the function*ψ*(*λ*;*n*)=*χ*(*λ*;*n*)*E*(*λ*;*n*) satisfies the linear system (3.23 ) with the potentials

## Proof.

The combination (*λ*−*λ*_{i})*χ*_{(i)}(*λ*;*n*)−(*λ*−*λ*_{j})*χ*_{( j)}(*λ*;*n*) satisfies the Fredholm equation with constant (in *λ*) normalization thus must be proportional to *χ*(*λ*;*n*). By evaluating both sides in *λ*_{i} we find the coefficient of proportionality. Multiplication of both sides by *E*(*λ*;*n*) gives the statement. ▪

## Corollary 4.5.

*The form of U_{ij} given above implies that the potentials r_{i}, defined by equation (3.27 ), read*
4.11

## Theorem 4.6.

*Within the considered class of solutions of the Hirota–Miwa system the τ-function is given by*
4.12

## Proof.

Evaluation of equation (4.5) at *λ*_{i} for *χ*(*λ*;*n*) as in proposition 4.4 gives
4.13
From lemma 4.3 we obtain that
4.14
Comparison of both equations shows that
4.15
which owing to equations (3.28) and (4.11) gives the statement. ▪

The above result provides the ‘determinant interpretation’ of the *τ*-function within the class of solutions which can be obtained by application of the non-local -dressing method.

Recently, within the same approach the *τ*-function of the quadrilateral lattices has been studied (Doliwa 2009). As can be deduced from Bogdanov & Konopelchenko (1995) and Doliwa *et al.* (2000), the structure of the datum in the non-local -dressing method which leads to the quadrilateral lattices and all the lattices generated by their Laplace transforms is as follows. Let , be pairs of distinct points of the complex plane, let be points of the integer lattice and let , , be a point of the *A*_{K−1} root lattice. The function *E*(*λ*;(*m*,ℓ)) which should replace the function *E*(*λ*;*n*) in equation (4.8) reads
4.16

The variable *m* is the quadrilateral lattice discrete parameter, while the Laplace transformation is given by ℓ_{i}↦ℓ_{i}+1, ℓ_{j}↦ℓ_{j}−1. After the proper identification of 2*K* points , , with the points , we obtain the change of variables discussed in §5.

## 5. Desargues maps and quadrilateral lattices

This section is devoted to the study of the relationship between Desargues maps and quadrilateral lattices. We will show that the theory of quadrilateral lattices can be embedded in the theory of the Desargues maps, and for odd *N*=2*K*−1 this embedding is one-to-one (the case of even *N*=2*K* can be treated as a dimensional reduction of 2*K*+1). The relation described below generalizes the relation, known on the *τ*-function level, between the Hirota–Miwa equation and its version in the discrete two-dimensional Toda lattice form (Zabrodin 1997). The relation between the discrete two-dimensional Toda lattice and the two-dimensional quadrilateral lattice was the subject of Doliwa (1997, 2000).

Recall that the condition of planarity of elementary quadrilaterals of written in the non-homogeneous coordinates gives the following linear problem: 5.1 where are certain functions which should satisfy the corresponding compatibility condition (a version of the discrete Darboux system).

The Laplace transformation of *ψ* is constructed (Doliwa 1997; Doliwa *et al.* 2000) via intersection of the tangent lines 〈*ψ*,*ψ*_{(i)}〉 with their *j*th negative neighbours 〈*ψ*_{(−j)},*ψ*_{(i,−j)}〉 (figure 5). In the non-homogeneous coordinates we have
5.2

The Laplace transforms of quadrilateral lattices are quadrilateral lattices again, and the following relations hold (Doliwa *et al.* 2000):

They allow us to parametrize the quadrilateral lattices generated from one quadrilateral lattice via the Laplace transformations by points of the root lattice of the type *A*_{K−1} (see also the discussion in Doliwa *et al.* (1999*b*)). This suggests considering the Laplace transformation directions as new variables. In order to place all variables on an equal footing we change the variables as suggested in §4.

Consider the following change of variables between the integer lattice and , where is the *A*_{K−1} root lattice
here, for convenience, we have also defined *n*_{2K}=−*n*_{1}−*n*_{2}−⋯−*n*_{2K−1}.

For fixed ℓ∈*Q*(*A*_{K−1}) define the map given by *ψ*^{ℓ}(*m*)=*ϕ*(*n*), where the relationship between *n* and *m* and ℓ is as given above. Then we have
5.3
and for *i*≠*K*
5.4
where *e*_{i} is the element of the canonical basis of having 1 as the *i*th component and 0’s elsewhere.

## Proposition 5.1.

*The maps* *are quadrilateral lattice maps. Moreover ψ^{ℓ+ei−ej} is the Laplace transform*

*of ψ*

^{ℓ}.

## Proof.

Assume that *i*<*j*<*K*. The point and the points *ϕ*=*ψ*^{ℓ} and belong to the line containing (positive) neighbours of *ϕ*_{(−2i)}. Similarly, the same point and the points and belong to the line containing (positive) neighbours of *ϕ*_{(−2i,2j−1,−2j)}. This shows that the lines and intersect in . Therefore the four points *ψ*^{ℓ}, , and are coplanar, and .

For *j*<*i*<*K* the reasoning is similar. The details of the case when one of the indices *i* or *j* is equal to *K* is left to the reader. ▪

Let us illustrate the above reasoning (still *i*<*j*<*K*) in making a simple calculation in the affine gauge (3.13). Collinearity of *ψ*^{ℓ}, and gives
5.5

Similarly, collinearity of , and gives in the affine gauge 5.6

Elimination of from the above equations implies that *ψ*^{ℓ} satisfies equation (5.1) with the coefficients
5.7and
5.8

Equation (5.5) gives 5.9 which because of the identification (5.8) agrees with equation (5.2).

## Remark 5.2.

The reverse identification from *K*-dimensional quadrilateral lattice *ψ* and all quadrilateral lattices generated via the Laplace transformations to the corresponding 2*K*−1 Desargues lattice is based on the observation (Doliwa *et al.* 2000) that for the fixed direction *i* of the quadrilateral lattice the 2*K* points are collinear. The corresponding lines (in the present notation they are denoted *L*_{(−2i)}) form the *i*th tangent congruence of the lattice *ψ*^{ℓ}.

## Remark 5.3.

It is known (Doliwa & Santini 1997) that the *K*-dimensional quadrilateral lattice is uniquely determined from a system of *K*(*K*−1)/2 quadrilateral surfaces intersecting along the *K* initial discrete curves which have one point in common. The successive application of the Laplace transformations then generates a (2*K*−1)-dimensional Desargues lattice. Because a quadrilateral surface is uniquely determined from two initial curves by two functions of two discrete variables, a solution of a (2*K*−1)-dimensional Hirota–Miwa equation is determined given *K*(*K*−1) functions of two (appropriate) variables.

## Remark 5.4.

The Desargues lattices of even *N*=2*K* dimension can be obtained as a dimensional reduction of 2*K*+1 Desargues lattices (set *n*_{2K+1}=0). Equivalently, it is generated by the Laplace transformations from a *K*-dimensional quadrilateral lattice and focal lattices of a congruence conjugate to the lattice (see Doliwa *et al.* (2000) for explanation of the terms used).

## 6. Conclusion and final remarks

In this paper we studied an elementary geometric meaning of the celebrated Hirota–Miwa system. The multi-dimensional compatibility of the corresponding map relies on the Desargues theorem and its higher dimensional generalizations. Since the Desargues theorem is valid in projective spaces over division rings, we are automatically led to the non-commutative Hirota-Miwa system of equations. Note that the division ring context of the Hirota–Miwa equation should not be considered just as a curiosity. It is known (Panov 1994; Cliff 1995; Jordan 1995) that the standard quantum algebras (Jimbo 1985; Drinfeld 1987; Woronowicz 1987; Reshetikin *et al.* 1990) admit division rings of quotients. In view of recent developments on quantization of the discrete Darboux equations (Bazhanov & Sergeev 2006; Bazhanov *et al.* 2008) this aspect of integrable discrete geometry deserves deeper studies.

Although the linear problem for the Desargues maps seems to be a strong degeneration of the linear problem for the quadrilateral lattice map, surprisingly both theories are equivalent, as suggested by their equivalence on the level of the algebro-geometric solutions, and those obtained by the non-local -dressing method. We also found the meaning of the *τ*-function of the Hirota–Miwa equation for that class of solution as a Fredholm determinant.

We would like to stress that the above-mentioned equivalence becomes elementary and visible on the level of discrete systems. On the level of differential equations the situation is much more subtle. It is however known (Konopelchenko & Martínez Alonso 1994) that one component of KP hierarchy can been reformulated, after the transition to the so-called Miwa coordinates, as a system of infinite number of (partial differential) Darboux equations.

The theory of discrete integrable systems is richer (e.g. Suris 2003; Grammaticos *et al.* 2004) but also, in a sense, simpler than the corresponding theory of integrable partial differential equations. In the course of a limiting procedure, which gives differential systems from the discrete ones, various symmetries and relations between different discrete systems are lost or hidden. The present paper gives new examples supporting this claim, and shows once again the superior role of the (non-Abelian) Hirota–Miwa equation in the integrable systems theory.

## Acknowledgements

I acknowledge discussions with Jarosław Kosiorek and Andrzej Matraś on the role of the Desargues configuration in foundations of geometry, and an important warning by Mark Pankov on a terminological confusion with the lattice theory. I would also like to thank a referee for his comments on the manuscript which helped me to improve presentation. It is my pleasure to thank the Isaac Newton Institute for Mathematical Sciences for hospitality during the programme *Discrete Integrable Systems*.

## Footnotes

↵1 Note a terminological confusion: following Doliwa & Santini (1997), we speak about ‘compatibility of the construction for arbitrary dimension of the lattice’, while the word ‘consistency’ is used in the context of integrable reductions of the quadrilateral lattice (e.g. Doliwa

*et al.*(2000), where we say that the ‘geometric integrability scheme is consistent with the circular reduction’).↵2 In Bobenko (2009) we find reference of this fact in relation to the work of King & Schief (2003), where, however, this result is not mentioned. I have learned that this important observation is due to W. K. Schief (2009, personal communication).

- Received June 4, 2009.
- Accepted November 11, 2009.

- © 2009 The Royal Society