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We presents a simple model in which the formation of ocular dominance columns in striate cortex emerges spontaneously. The model generalizes Kohonen's original self-organizing model while incorporating a continuous variable corresponding to ocularity. We demonstrates in this model that the column formation is a sort of phase transition due to (local) spontaneous breaking of left-right ocular symmetry in the distribution of nerve cells in the cortex observed at early stage. In particular, he derives an exactly (locally) periodic columnar structure with the hyper column width given by ∼4√3σ where σ is the lateral correlation length of the model. This result does not depend on the details of the generalization of Kohonen's model. Generalization to formations of another columnar structure is straightforward.

A four-dimensional analogue of residual η- and ϵ-symmetries in the light-cone gauge formulation of 2D gravity a la Knizhnik- Polyakov-Zamolodchikov is considered. We argue that residual symmetries in the metric for self-dual gravity by Plebanski et al. precisely correspond to this. We also show that the η-symmetry is nothing but a Kahler gauge symmetry in the gauge used by Alekseev and Shatashvili for the derivation of the induced 2D gravity action.

We show that in the BRST analysis of the Ŵ ∞ algebra the BRST charge is nilpotent at the central charge c =−2.

It is well known that the second hamiltonian structures for the (N−1)th generalized KdV hierarchy (of An type) are isomorphic to the classical WN algebra for finite N(⩾2). Here we construct an analogue of the first structures for the KP hierarchy in its pseudo-differential operator formalism L = D+D −1 ν 1 +D −2 ν 2 +D −3 ν 3 +… , and show that the corresponding Poisson brackets are given by the W1+∞ algebra with no central charge. Its implication to the quantization of 4D self-dual gravity is briefly discussed.

We report on a study of the expectation values of Wilson loops in D=3 Chern-Simons theory. The general skein relations (of higher orders) are derived for these expectation values. We show that the skein relations for the Wilson loops carrying the fundamental representations of the simple Lie algebras SO(n) and Sp(n) are sufficient to determine invariants for all knots and links and that the resulting link invariants agree with Kauffman polynomials. The polynomial for an unknotted circle is identified to the formal characters of the fundamental representations of these Lie algebras.

In this Letter, we report on a study of the expectation values of Wilson loops in D=3 topological Chern-Simons theory associated with the fundamental representation of the simple Lie algebras SO(n) and Sp(n). The skein relations satisfied by these expectation values are derived by conformal field-theory techniques. New hierarchies of invariant polynomials for knots in S 3 can be derived from these relations (at least) up to ten crossings. The N=3 Akutsu-Wadati polynomials are a special case with G=SO(3). The expectation value of the Wilson loops for a couple of simple unknotted circles is identified to the Weyl character.

A three-dimensional generalization of the wave function of Girvin {ital et} {ital al}. is constructed by taking the {ital N}{r arrow}{infinity} limit of a solution of the two-dimensional massless Dirac equation with an SU({ital N}) gauge field. The resulting wave function is closely related to instanton solutions of the self-dual Einstein quations, and can be used to construct a multiparticle wave function with remarkable holonomy properties and a ground-state wave function with off-diagonal long-range order.

Because of its relevance to lower-dimensional conformal geometry, known as a generalized Weierstrass inducing, the modified Veselov-Novikov (mVN) hierarchy attracts renewed interest recently. It has been shown explicitly in the literature that an extrinsic string action \`a la Polyakov (Willmore functional) is invariant under deformations associated to the first member of the mVN hierarchy. In this note we go one step further and show the explicit invariance of the functional under deformations associated to all higher members of the hierarchy., Comment: 12 pp LaTeX2e. To be published in Physics Letters B

A homogeneous-space construction of superconformal symmetry is proposed for the case when three species of Majorana-Weyl fermions exist, each of which transforms according to the adjoint representation of a semisimple Lie group G. The energy-momentum tensor is given by the difference of two current-current products which correspond to a group Γ and the group G with Γ satisfying G ⊂ Γ ⊆ SO(3 dim G), and some appropriate conditions. The symmetry group in this case is given by the centralizer of G in the homogeneous space Γ/G, and is equal to (SO(3))4 for a purely fermionic D = 4 string with Γ = SO(3 dim G).

We construct the nonlocal (parafermionic) SU(3) current algebra with Z N -symmetry in two dimensions, following the Z-algebra a la Lepowsky and Wilson. By computing the characters of the representation, we show that the simplest N =2 algebra (central charge of the conformal sector c = 6 5 ) is equivalent to a combined system of the tri-critical ( c = 7 10 ) and the critical ( c = 1 2 ) Z 2 -Ising models.

Quantum corrections to the vacuum energy are evaluated at one-loop order in eleven-dimensional supergravity on the round seven-sphere S7 and are shown to vanish. The cancellation is also shownfor all ultraviolet poles at z = 11 2 , 10 2 , … , corresponding to divergences of eleventh and lower powers of momentum cut-off Λ

Based on an ordinary formulation of BRST the BRST charge concerning the affine Kac-Moody algebra g with the central extension is investigated. The nilpotency is satisfied at the value of central charge = -(quadratic Casimir of Ad ( G ), which shows the theory to fall into the unphysical lowest weight representation sector without lower bound in energy. No improvement is made by a semi-direct coupling to the Virasoro algebra, in which the central charge of the Virasoro algebra gets a modification: C − 26 − 2 dim G .

A detailed calculation of the spectrum of the lowest energy, E0, is presented for vector, axial vector, tensor, and scalar fluctuation modes (still incomplete) in eleven-dimensional supergravity compactified on the squashed 7-sphere. The general correspondence is also given between the generalised Laplacian and the quadratic G-Casimir invariant in an arbitrary homogeneous space G/H.

We evaluate all spectra of the vector harmonics on a squashed seven-sphere in the manifestly SO(5) × SU L (2) covariant formalism.

We quantize the nonlinear σ-model with a Wess-Zumino term in 1 + 1 dimensions by requiring both conformal and chiral symmetries, following the method of Faddeev and Shatashvili. On this basis we consider a string model moving on a group manifold; the critical dimension is found to be 26.

Degenerate representations are theoretically constructed with the Neveu-Virasoro algebra. The differential relation is derived from the correlation functions in these representations.

The extended Virasoro algebra with spin-3 current is derived in the framework of the KP hierarchy. A speculative form of a further extension of the Virasoro algebra with spin-4 current is given.

We discuss a spontaneous compactification scheme of eleven-dimensional supergravity on the exotic spheres Σ7 and on the ordinary S7 with the Hopf-fibering. These spheres are the nontrivial fibration of S4 × SO(4)S3 with the SO(4)-instantons. In particular two classical solutions are explicitly obtained for the Hopf-fibering of S7 in the eleven-dimensional supergravity. The ratios of the curvatures in SAdS4 × S4′ × S3 are (−1 : 1 : 1 4 ) and ( −9 5 : 1 : 5 4 ), and the corresponding super-symmetries are N = 8 and N = 1, respectively. They are gauge equivalent to the round and the squashed S7.

A consistency problem for the Kaluza-Klein ansatz is shown to arise when the kinetic term of extra gauge fields depends on the internal coordinates. Consequences on cosmological solutions are discussed. A position-dependent kinetic term implies either a very special condition to be satisfied by it, or the time-dependent size of internal space and the maximally symmetric universe in the vacuum.

The 1 + 1 dimensional analogue of the quantizaton procedure of the gravitational constant by Witten and Bagger in a generalized non-linear σ-model is studied. By making use of the tensor calculus in 1 + 1 dimensions, the model is constructed and found to be manifestly independent of the Kahler gauge transformation. Therefore, the topological properties are not produced in the 1 + 1 dimensional models.

It is shown that the magnetic g factor is unity for any massive spinning particle in the Kaluza-Klein theory of five dimensional space-time.

Properties of Killing spinors on spheres and hyperbolic manifolds are investigated with an emphasis on the relations to Killing vectors, conformal Killing vectors, and solutions of Maxwell’s equations.