Leech Lattice Extension of the Non-linear Schrodinger EquationTheory of Einstein Spaces

Although the nonlinear Schrodinger equation description of Einstein spaces has provided insights into how quantum mechanics might modify the classical general relativistic description of space-time, an exact quantum description of space-times with matter has remained elusive. In this note we outline how the nonlinear Schrodinger equation theory of Einstein spaces might be generalized to include matter by transplanting the theory to the 25+1 dimensional Lorentzian Leech lattice. Remarkably when a hexagonal section of the Leech lattice is set aside as the stage for the nonlinear Schrodinger equation, the discrete automorphism group of the complex Leech lattice with one complex direction fixed can be lifted to continuous Lie group symmetries. In this setting the wave function becomes an 11x11 complex matrix which represents matter degrees of freedom consisting of a 2-form abelian gauge field and vector nonabelian SU(3)x E 6 gauge fields together with their supersymmetric partners. The lagrangian field equations for this matrix wave function appear to be a promising starting point for resolving the unphysical features inherent in the general relativistic descriptions of gravitational collapse and the big bang.


Introduction
Among the known exact solutions of the Einstein's general relativity equations self-dual Einstein spaces stand out as especially interesting [1]. Our interest is these spaces was stimulated by the discovey that these solutions can also be expressed as solutions of a matrix nonlinear Schrodinger equation in 2+1 dimensions with an SU(N) Chern-Simons gauge potential [2]. This in turn led to a "superfluid" representation for the vacuum state of a space-time with no matter [3,4].
Independently of this development there are also quite general reasons [5,6] for supposing that in a quantum theory of gravity space-time can be descibed as a superfluid of "gravitons". It is encourging in this connection that, in contrast with the prediction of general relativity, there are both theoretical [7][8][9] and phenomenological [10] hints that in a quantum theory of gravity the surface of compact objects is not an event horizon, but instead represents a phase transtion in the vacuum state of space-time [11]. If it turns out that the general relativisitic description of compact objects is indeed flawed, the fault may well lie the likelihood that a consistent treatment of space-times with matter will require a quantum theory of space-time.
Unfortunately, a quantum theory of space-times with matter has remained elusive up to the present time.
In this paper we wish to draw attention to the possibility that a matrix non-linear Schrodinger equation living on a Lorentzian extension of the 24-dimensional Leech lattice may provide the needed framework for a quantum theory of gravity and elementary particles in 3+1 dimensions. Our basic idea is to replace the complex scalar wave function and gauge potential used in ref's [2,3] to encode the Kahler potential of a self-dual Einstein space by complex matrices representing discrete symmetries of the complex Leech lattice. The Leech lattice was originally constructed as a result of its connection with perfect error correcting codes [12,13]. From the point of view of constructing a quantum theory of space-time containing matter perhaps the most intriguing property of the Leech lattice is that there is a correspondence between the shapes of the minimal Leech lattice vectors (the set of such vectors centered on a point of the lattice will hereafter be referred to as the "Leech polytrope") and the types of massless fields that occur in the 10-dimensional unification of supergravity and super Yang-Mills theories [14]. As will be discussed in section 3, the degrees for freedom in our model not directly related to the metric for 3+1 dimensional space-time are related by an especially interesting discrete symmetry, M 11 , acting on the vectors of the complex Leech polytrope. The M 11 group was discovered in the 19 th century by the mathematical physicist Emil Mathieu, and was the first in a series of discoveries of sporadic finite simple groups that culminated in 1981 with Griess' construction of the Monster sporadic group. [13,15]. Perhaps the most remarkable aspect of our model for space-times with matter is that provides a link between the unification of gravity and elementary particle physics and the mathematical structure of the sporadic finite simple groups including the Moster. That this should be so is of course another illustration of Wigner's "unreasonable usefulness of mathematics in physics".
Another notable feature of our model for space-time is that the vacuum energy density will in general not be zero. Actually the numbers of bosonic and fermionic degrees of freedom for the 2-form and Yang-Mills matter fields in our theory exactly match. This means that at the level of 3+1 dimensions the contribution of zero point fluctuations of the matter fields to the ground state energy will vanish. However, the contibutions of zero point fluctuations of the bosonic and fermionic gravitational degrees of freedom to the ground state energy will not exactly cancel each other. In addition, the direct interaction of self-dual and anti-self-dual solitons in our quantum vacuum state will produce a nonzero energy density.
In Section 2 we review the gauged nonlinear Schrodinger equation theory that provides a quantum model for classical Einstein spaces. In Section 3 we exhibit the gauged nonlinear Schrodinger equation that we believe provides a framework for understanding the quantum nature of space-times wih matter.
We also exhibit a matrix generalization of the classical "Heavenly" equation for the Kahler potential for a self-dual or anti-self dual Einstein space, whose soliton-like solutions provide the building blocks for a theory of space-time with 3-form vacuum fields.

Non-linear Schrodinger equation for Einstein spaces
The coherent state wave function for a 2-dimensional quantum fluid of anyons interacting via Chern-Simons gauge potentials satisfies the non-linear Schrodinger equation [2][3][4]: where σ H is the "Hall conductivity". Neglecting spatial variations in the electric field, the usual Guass' law will be replaced by the Chern-Simons equation where B is the strength of an effective magnetic field whose direction is perpendicular to the layer, ρ is the charge per unit area, and € 1/κ is an inverse length with σ H = κ. It is interesting that these equations also appear in a tight binding model for the motion of charged particles on a hexagonal lattice [16]. It was shown some time ago [17] that the time independent version of Eq. (1) in conjunction with Eq's (1-2) can be solved analytically if one assumes that € g = ±e 2  /mcκ .
where the wave function Φ and potentials The in-plane electric field E α will also be a diagonal matrix satisfying the Hall effect equation:: where Time independent analytic solutions to eq. (5-7) can be found for any value of N if Eq.(4) is satisfied. These analytic solutions satisfy the 2-dimensional self-duality condition D α Φ = ±iε αβ D β Φ and represent zero energy ground states for a stack of N planes (or N hexagonal lattices).
In the limit N → ∞ the analytic solutions of Eq. 5 take a particularly simple form such that the effective magnetic field seen by the jth soliton has the simple form: where R jk  Actually the effective action (10) for chirons suggests a connection with the Kosterlitz-Thouless condensation of vortex and anti-vortex pairs in the 2-dimensional XY model [18]. It is an elementary identity that the right hand side of (9) can be rewritten in the form which is similar in form to a configuration of 2-dimensional XY vortices. In the XY model the phase variations in a 2-dimensional condensate can be described by a partition function of the form where Θ is a periodic coordinate whose period is 2π and K is a constant. It can be shown that a discrete version of this theory interpolates between the low and high temperature phases of the XY model. Indeed evaluating the exponential in (13)  specifically applies to space-times with no matter, we will argue in the fillowing that a very similar construction may be used to construct space-times containing certain kinds of vacuum fields.

Extension to include matter
In this section we will outline how the "superfluid" description of Einstein spaces introduced in Section 2 might be extended to space-times with matter consisting of certain kinds of massless elementary particles or vacuum fields. Since the solutions of the 2-dimensional nonlinear Schrodinger equation that were used to construct self-dual Einstein spaces were either holomorphic or anti-holomorphic functions, we will assume that it is actually the complex Leech lattice, which is a 12-dimensional lattice whose coordinates are Eisenstein integers, rather than the 24dimensional Leech lattice with real coordinates that is the natural setting for our theory.
The coordinates x i where i = 1,2 refer to the 2-dimensions of the distinguished hexagonal section.
The presence of the gauge potential indicates that our set up is covariant with respect to continuous gauge transformations.
Although at this stage it appears that it might be possible to find a generalization of Einstein spaces for any choice of a continuous gauge symmetry, in the following we will focus on the continuous symmetries that arise from the action of M 11 on the vectors of the complex Leech polytrope. As in the SU(N) case we will assume that the matrix gauge potential is not independent of the wave function, but satisfies the nonabelian Chern-Simons equations: where α,β,γ =0,1,2 and the gauge fields are 11x11 matrices. We will also assume that the wave function satisfies self-duality condition D α Ψ= ±iε αβ D β Ψ. The fact that the self-duality and where the index a runs over the adjoint representations of SO (11) or E 6 as well as SU (3) where K ab is the Cartan matrix for an SO (11) or SU(3) x E 6 Lie algebra associated with the orbits of M 11 . This matrix equation generalizes the classical "Heavenly" equation satisfied by the Kahler potential for a self-dual Einstein space [1]. Just as holomorphic and anti-holomorphic solutions of the classical "Heavenly" equation were used in [3,4] to construct quantum models for Einstein spaces, we also expect that holomorphic and anti-holomorphic solutions of the matrix Eq. 20 can be used to construct quantum models for space-times with 3-form or SU (3) x E 6 vacuum fields. How this works in detail for space-times of interest, e.g. gravitational collapse, will be considered in future papers.
We close with a few comments about the general character It follows from Eqs. (17) that the 3-form and Yang-Mills vacuum fields will be strongly by the dynamics of space-time, as well as visa versa. Because our self-duality constraint can only be satisfied for some particular configuration of vacuum 3-form and Yang-Mills field strengths, it follows that the vacuum energy will not be zero for general configurations of the vacuum fields.
The author and Nick Manton pointed out some time ago [21] that a "geometric" Higgs potential could appear in the 3- In summary we believe that the lagrangian field equations (15)(16)(17)(18) constitue a first step towards filling the void that arises from the fact that the way matter is treated in classical general relativity doesn't take into account possible changes in the structure of the space-time vacuum. We believe that it is this defect which prevents classical general relativity from providing a physical description of either gravitational collapse or the origin of the big bang.
As a final note we would like to mention that it was Richard Slansky [22] who originally pointed out that in many ways E 6 is the nicest Yang-Mills gauge symmetry for a grand unified theory of elementary particles.