Integrable systems in 4D associated with sixfolds in
Abstract
Let be the Grassmannian of dimensional linear subspaces of an dimensional vector space . A submanifold gives rise to a differential system that governs dimensional submanifolds of whose Gaussian image is contained in . We investigate a special case of this construction where is a sixfold in . The corresponding system reduces to a pair of firstorder PDEs for 2 functions of 4 independent variables. Equations of this type arise in selfdual Ricciflat geometry. Our main result is a complete description of integrable systems . These naturally fall into two subclasses.

Systems of MongeAmpère type. The corresponding sixfolds are codimension 2 linear sections of the Plücker embedding .

General linearly degenerate systems. The corresponding sixfolds are the images of quadratic maps given by a version of the classical construction of Chasles.
We prove that integrability is equivalent to the requirement that the characteristic variety of system gives rise to a conformal structure which is selfdual on every solution. In fact, all solutions carry hyperHermitian geometry.
MSC: 37K10, 37K25, 53A30, 53A40, 53B15, 53B25, 53B50, 53Z05.
Keywords: Submanifold of the Grassmannian, Dispersionless Integrable System, Hydrodynamic Reduction, Selfdual Conformal Structure, MongeAmpère System, Dispersionless Lax Pair, Linear Degeneracy.
Department of Mathematical Physics
Faculty of Applied Mathematics
Belarussian State University
Nezavisimosti av. 4, 220030 Minsk, Belarus
Department of Mathematical Sciences
Loughborough University
Loughborough, Leicestershire LE11 3TU
United Kingdom
Department of Mathematics and Statistics
Faculty of Science and Technology
UiT the Arctic University of Norway
Tromsø 9037, Norway
emails:
[1ex] doubrov@islc.org
E.V.Ferapontov@lboro.ac.uk
boris.kruglikov@uit.no
V.Novikov@lboro.ac.uk
Contents
1 Introduction
1.1 Formulation of the problem
Let and be functions of the independent variables . In this paper we investigate integrability of firstorder systems of the form
(1) 
where and are (nonlinear) functions of the partial derivatives . The geometry behind systems (1) is as follows. Let be a 6dimensional vector space with coordinates . Solutions to system (1) correspond to dimensional submanifolds of defined as . Their 4dimensional tangent spaces, specified by the equations , are parametrised by matrices
whose entries are restricted by equations (1). Thus, equations (1) can be interpreted as the defining equations of a sixfold in the Grassmannian . Solutions to system (1) correspond to dimensional submanifolds of whose Gaussian images (tangent spaces translated to the origin) are contained in . There exist two types of integrable systems (1).
Systems of MongeAmpère type have the form
(2) 
where each equation is a constantcoefficient linear combination of the minors of . These systems were introduced in [2] in the context of ‘complete exceptionality’. Geometrically, the associated sixfolds are linear sections of the Plücker embedding . A typical example is the system
(3) 
which reduces to the first heavenly equation of Plebanski [24], , under the substitution . It governs selfdual Ricciflat 4manifolds; see Section 2.1 for further details on MongeAmpère systems.
General linearly degenerate systems correspond to sixfolds resulting as images of quadratic maps (we refer to [7] for a discussion of the concept of linear degeneracy, see also Section 1.5). As an example, let us consider the system
is a parameter. Note that this system does not belong to the MongeAmpère class (2). The elimination of leads to the secondorder equation for ,
here . Similarly, the elimination of leads to the secondorder equation for ,
Thus, one can speak of a fourdimensional Bäcklund transformation. This example can be viewed as a 4D generalisation of the Bäcklund transformation for the Veronese web equation constructed in [29]. We refer to Section 2.3 for further examples and classification results.
The main goal of this paper is to prove that systems of the above two types exhaust the list of nondegenerate integrable systems (1).
1.2 Nondegeneracy, conformal structure and selfduality
We will assume that system (1) is nondegenerate in the sense that the corresponding characteristic variety,
defines an irreducible quadric of rank 4. This is the case for all examples of physical/geometric relevance. Explicitly, the characteristic variety can be represented in the form where
The characteristic variety gives rise to the conformal structure where is the inverse matrix of ; note that nondegeneracy is equivalent to . Let denote the corresponding conformal class. Remarkably, integrability of system (1) has a natural interpretation in terms of the conformal geometry of . In 4D, the key invariant of a conformal structure is its Weyl tensor . It has selfdual and antiselfdual parts,
respectively. Here the Hodge star operator is defined as . A conformal structure is said to be selfdual if, with a proper choice of orientation, we have
(4) 
The integrability of conditions of selfduality by the twistor construction is due to Penrose [23], see also [10] for a direct demonstration. We will prove in Section 3 that integrability of 4D equations (1) is equivalent to the requirement that the conformal structure defined by the characteristic variety must be selfdual on every solution. Thus, solutions to integrable systems carry integrable conformal geometry. More precisely, with a suitable choice of orientation, it will be shown that the conditions of selfduality, , lead to MongeAmpère systems. Similarly, the conditions of antiselfduality, , characterise general linearly degenerate systems associated with quadratic maps . The intersection of these two classes consists of linearisable systems characterised by the conformal flatness of .
For example, the conformal structure of system (3) is given by
A direct calculation shows that is selfdual on every solution, which means that (4) holds identically modulo (3). System (3) possesses the Lax representation where are parameterdependent vector fields,
. Projecting integral surfaces of the distribution spanned by from the extended space of variables (correspondence space) to the space of independent variables one obtains a threeparameter family of totally null surfaces (surfaces) of the conformal structure . According to [23], the existence of such surfaces is necessary and sufficient for selfduality. We refer to [1, 20, 21] for a novel version of the inverse scattering transform based on commuting parameterdependent vector fields.
1.3 Dispersionless integrability in 4D
Integrability of multidimensional dispersionless PDEs can be approached based on the method of hydrodynamic reductions [17, 12, 11, 13]. In the most general setup (for definiteness, we restrict to the 4D case), it applies to quasilinear systems of the form
(5) 
where is an component column vector of the dependent variables, , and are matrices where the number of equations is allowed to exceed the number of unknowns. Note that nonlinear system (1) can be brought to quasilinear form (5) by choosing as the new dependent variables and writing out all possible consistency conditions among them, see Section 3. The method of hydrodynamic reductions consists of seeking multiphase solutions in the form
where the phases , whose number is allowed to be arbitrary, are required to satisfy a triple of consistent dimensional systems
(6) 
known as systems of hydrodynamic type. The corresponding characteristic speeds must satisfy the commutativity conditions [28],
(7) 
here . Multiphase solutions of this type originate from gas dynamics, and are known as nonlinear interactions of planar simple waves. Equations (6) are said to define an component hydrodynamic reduction of the original system (5). System (5) is said to be integrable if, for every , it possesses infinitely many component hydrodynamic reductions parametrised by arbitrary functions of one variable [13]. This requirement imposes strong constraints (integrability conditions) on the matrix elements of , see Section 3 for details.
The method of hydrodynamic reductions has been successfully applied to a whole range of systems in 3D, leading to extensive classification results. The corresponding submanifolds are generally transcendental, parametrised by generalised hypergeometric functions [22]. The results of this paper are based on a direct application of the method of hydrodynamic reductions to 4D systems of type (1). The 4D situation turns out to be far more restrictive, in particular, the integrability conditions force to be algebraic.
1.4 Equivalence group
All constructions described in the previous sections are equivariant with respect to the group acting by linear transformations on the space with coordinates . The extension of this action to is given by the formula
(8) 
where are and matrices, respectively; note that the extended action is no longer linear. Transformation law (8) suggests that the action of preserves the class of equations (1). Furthermore, transformations (8) preserve the integrability, so that can be viewed as a natural equivalence group of the problem: all our classification results will be formulated modulo this equivalence. In coordinates , the infinitesimal generators corresponding to equivalence transformations (8) are as follows:
8 translations:
19 linear generators (note the relation ):
8 projective generators:
Let us represent system (1) in evolutionary form,
(9) 
and consider the induced action of the equivalence group on the space of 1jets of functions of variables . This is a 20dimensional space with coordinates , , . One can show that the action of on has a unique Zariski open orbit (its complement consists of 1jets of degenerate systems), see Section 3.1. This property allows one to assume that all sporadic factors depending on firstorder derivatives of and that arise in the process of Gaussian elimination in the proofs of our main results in Section 3, are nonzero. This considerably simplifies the arguments by eliminating unessential branching. Furthermore, in the verification of polynomial identities involving first and secondorder partial derivatives of and one can, without any loss of generality, give the firstorder derivatives any ‘generic’ numerical values: this often renders otherwise impossible computations manageable.
1.5 Linearly degenerate systems
The definition of linear degeneracy is inductive: a multidimensional system is said to be linearly degenerate (completely exceptional [2]) if such are all its traveling wave reductions to two dimensions. Thus, it is sufficient to define this concept in the 2D case,
Setting and differentiating by one can rewrite this system in twocomponent quasilinear form,
or, in matrix notation,
Recall that the matrix is said to be linearly degenerate if its eigenvalues (assumed real and distinct) are constant in the direction of the corresponding eigenvectors. Explicitly, , no summation, where denotes Lie derivative in the direction of the eigenvector , and . For quasilinear systems, the property of linear degeneracy is known to be related to the impossibility of breakdown of smooth initial data [26]. In terms of the original functions and , the conditions of linear degeneracy reduce to a pair of secondorder differential constraints [7],
Requiring that all traveling wave reductions of a multidimensional system to 2D are linearly degenerate in the above sense, we obtain differential characterisation of linear degeneracy:
Proposition 1 [7]. System (9) is linearly degenerate if and only if the functions and satisfy the relations
(10) 
where Sym denotes complete symmetrisation over . Note that conditions (10) are invariant under the equivalence group .
The key observation is that secondorder overdetermined system (10) is not in involution: its differential prolongation results in the two branches characterised by additional secondorder differential constraints. The first branch leads to MongeAmpère systems (10 additional secondorder constraints). The second branch corresponds to general linearly degenerate systems (4 additional secondorder constraints), see Section 3.2 for the details of this analysis.
1.6 Summary of the main results
Our results imply that several seemingly different approaches to integrability described above lead to one and the same class of systems (1).
Theorem 1
Under the nondegeneracy assumption, the following conditions are equivalent:
(a) System (1) is integrable by the method of hydrodynamic reductions.
(b) Conformal structure defined by the characteristic variety of system (1) is selfdual on every solution.
(c) System (1) is linearly degenerate.
(d) The associated sixfold is either a codimension two linear section of the Plücker embedding , or the image of a quadratic map .
Theorem 1 and the results of [3] imply that any integrable system (1) possesses a Lax representation in parameterdependent commuting vector fields. Integral surfaces of these vector fields give rise to surfaces of the conformal structure .
Examples of integrable systems (1) are discussed in Section 2. The proof of Theorem 1 is given in Section 3. All calculations are based on computer algebra systems Mathematica and Maple (these only utilise symbolic polynomial algebra over , so the results are rigorous). The programmes are available from the arXiv supplement to this paper.
2 Examples and classification results
In this section we discuss examples of 4D systems which, as will be demonstrated in Section 3, exhaust the list of all integrable systems of type (1).
2.1 MongeAmpère systems
Systems of MongeAmpère type correspond to sixfolds that can be obtained as codimension two linear sections of the Plücker embedding of the Grassmannian. Recall that is an 8dimensional algebraic variety of degree 14 embedded into . All 2component systems of MongeAmpère type are integrable. They were classified in our recent paper [8].
Proposition 2 [8]. In four dimensions, any nondegenerate system of MongeAmpère type is equivalent to one of the following normal forms:
All these systems can be reduced to various heavenlytype equations. Introducing the potential such that one obtains the linear ultrahyperbolic equation , the second heavenly equation [24], the first heavenly equation [24], and the Husain equation [18], respectively. All of them originate from selfdual Ricciflat geometry. Their integrability by the method of hydrodynamic reductions was established in [12, 13].
Representing system (1) in evolutionary form (9) one obtains a differential characterisation of the MongeAmpère property.
Proposition 3 [8]. The necessary and sufficient conditions for system (9) to be of MongeAmpère type are equivalent to the following secondorder relations for and ,
(11) 
where . Equations for can be obtained by the simultaneous substitution and (30 secondorder relations altogether).
Table 1 below contains the (Lie algebra) structure of the stabilisers of MongeAmpère systems under the action of the equivalence group (note that different cases are distinguished by the dimensions of the stabilisers).
Table 1: types of isotropy algebras of MongeAmpère systems in 4D
System of equations  dim()  Levi decomposition of the algebra 

1: linear ultrahyperbolic 
graded by  
13  
is selfnormalizing  
2: 2nd heavenly 
graded by  
11  
is selfnormalizing  
3: 1st heavenly 
graded by  
10  
is not selfnormalizing  
4: Husain system 
semisimple  
9  
is not selfnormalizing 
Notes:
(1) The factors are irreducible representations of the corresponding
(same for the factor in ) in cases 13.
(2) Lie algebra structure of the nilradical of in case 2:
,
(equivariance fixes the brackets uniquely).
(3) We indicate real forms of the equations in the lefthand side. Since the classification is over ,
the corresponding complex forms should be taken, e.g. in case 4.
(4) Normalizers of in cases 3, 4 both have dimensions 11
(extension of the factor to in case 3 and
of to the tracefree part of in case 4).
2.2 Linearisable systems
In this section we characterise systems (1) which can be linearised by a transformation from the equivalence group . Note that linearisable systems are necessarily of MongeAmpère type.
Theorem 4. Under the nondegeneracy assumption, the following conditions are equivalent:
(a) System (1) is linearisable by a transformation from the equivalence group .
(b) System (1) is invariant under a dimensional subgroup of .
(c) The characteristic variety of system (1) defines a conformal structure which is flat on every solution: .
Proof. Equivalence : Consider a nondegenerate linear system, say (note that all nondegenerate linear systems of type (1) are equivalent). This system is invariant under a 13dimensional subgroup of with the following infinitesimal generators (we use the notations of Section 1.4):
(12) 
This Lie algebra is isomorphic to the semidirect product , where is the tensor product of the standard representation of , and the representation of . Here (resp. ) acts on the first (resp. second) factor of .
To establish the converse, let be the symmetry group of system (1). We can always assume that the point , specified by , belongs to the sixfold corresponding to our system. Let be the stabiliser of this point in . Note that , as takes to itself. The stabiliser of the point is spanned by infinitesimal generators . Since the system is nondegenerate, we can bring it to a canonical form
(13) 
This form (together with the point ) is stabilised by 7 elements of listed in the last two lines of (12). Thus, so that . The equality holds only if . However, the generator acts by nontrivial rescalings on terms of order 2 and higher in (13). Hence, for , all higherorder terms must vanish identically, leading to a linear system.
Equivalence : Let us represent system (1) in evolutionary form (9) and take the corresponding conformal structure . Conformal flatness is equivalent to the vanishing of the Weyl tensor
(14) 
where is the curvature tensor, is the Schouten tensor, is the Ricci tensor, and is the scalar curvature. Calculating (14) and using equations (9) along with their differential consequences to eliminate all higherorder partial derivatives of and containing differentiation by , we obtain expressions that have to vanish identically in the remaining higherorder derivatives (no more than thirdorder derivatives are involved in this calculation). In particular, equating to zero coefficients at the remaining thirdorder derivatives of and we obtain 34 secondorder relations for and that contain 30 relations (11) governing MongeAmpère systems, plus 4 extra (more complicated) relations. The easiest way to finish the proof is to note that according to Proposition 2 of Section 2.1, any 4D system of MongeAmpère type is equivalent to one of the four normal forms, and direct verification shows that conformal structures defined by characteristic varieties of the last three (nonlinearisable) normal forms are not flat on generic solutions. Thus, the above 34 secondorder relations are nothing but the linearisability conditions. This finishes the proof of Proposition 4.
2.3 Systems associated with quadratic maps
In this section we classify integrable systems (1) which correspond to sixfolds resulting as images of quadratic maps . These maps come from the following geometric construction.
Consider two vector spaces and . Let and be two linear maps. The collection of 2planes , , defines a subvariety of , the image of a quadratic map . In the particular case this construction goes back to Chasles [4] who considered the locus of lines spanned by an argument and the value of a projective transformation; see also [5], p. 556. Quadratic maps result from the above construction when . This gives a map , leading by duality to a quadratic map .
In coordinates, this reads as follows. Consider projective space with homogeneous coordinates . Let and be two matrices representing the corresponding linear maps. Introduce the matrix of linear forms on ,
where and . The Plücker coordinates define a quadratic map . By duality, this gives a sixfold , and the corresponding system (1). Explicit parametric formulae can be obtained from the factorised representation,
which gives , . Eliminating ’s, we obtain two relations among , which constitute the required system .
Tables 2–6 below comprise a complete list of resulting systems (1) labelled by JordanKronecker normal forms [16] of the matrix pencil (see the end of this section for an illustrative calculation leading to the first case of Table 2). Note that and are defined up to transformations , where the matrix is responsible for a change of basis in and the matrix corresponds to the action of the equivalence group . Modulo these transformations, and must have exactly one Kronecker block of the size , for (the cases of a single Kronecker block, as well as of more than one Kronecker blocks, lead to either degenerate or linear systems). We group systems according to the size of the Kronecker block. Within each table, systems are labelled by Serge types of the remaining Jordan block. In all cases (with the exception of the most generic system from Table 6) we have chosen canonical forms which, via elimination of , imply secondorder equations for . We also present the associated dispersionless Lax pairs in the form of two commuting dependent vector fields, .
Table 2: canonical forms with one Kronecker block
Segre type  Canonical form  Equation for  Lax pair 

[1111] 

[211] 

[22] 

[31] 

[4] 


Table 3: canonical forms with one Kronecker block
Segre type  Canonical form  Equation for  Lax pair 

[111] 

[21] 
