Claude Itzykson

Conformal invariance and applications to statistical mechanics

This volume contains Introductory Notes and major reprints on conformal field theory and its applications to 2-dimensional statistical mechanics of critical phenomena. The subject relates to many different areas in contemporary physics and mathematics, including string theory, integrable systems, representations of infinite Lie algebras and automorphic functions.

Quantum intersection rings

Within the broadly defined subject of topological field theory E. Witten suggested in [1] to study generalized “intersection numbers” on a compactified moduli space \({\bar M_{g,n}}\) of Riemann surfaces. These are computed by integrating pullbacks of appropriate forms on a target Kahler manifold obtained through holomorphic maps of marked surfaces. The corresponding axioms, discussed by R. Dijkgraaf and E. and H. Verlinde [2], were investigated by M. Kontsevich and Y. Manin [3] and lead for a subclass of targets to surprising results on the enumeration of rational curves. Our purpose here is to study a few illustrative examples and to check some of them using the most primitive tools of geometry.

COVARIANT DIFFERENTIAL EQUATIONS AND SINGULAR VECTORS IN VIRASORO REPRESENTATIONS

Abstract We give explicit expressions for the singular vectors in highest weight representations of the Virasoro algebra using a precise definition of fusion.

LAUGHLIN'S WAVE FUNCTIONS, COULOMB GASES AND EXPANSIONS OF THE DISCRIMINANT

In the context of the fractional quantum Hall effect, we investigate Laughlins ansatz for the ground state wave function at fractional filling of the lowest Landau level. Interpreting its normalization in terms of a one-component plasma, we find the effect of an additional quadrupolar field on the free energy, and derive estimates for the thermo dynamically equivalent spherical plasma. In the second part of the paper, we present various methods for expanding the wave function in terms of Slater determinants, and obtain sum rules for the coefficients. We also address the apparently simpler question of counting the number of such Slater states using the theory of integral polytopes.

The Branching Rules of Conformal Embeddings

After recalling the main properties of a conformal embedding of Lie algebrasg⊃p, which is defined by the equality of the Sugawara central charges on both sides, we launch a systematic study of their branching rules. The bulk of the paper is devoted to the proof of a general formula in the casesu(mn)1⊃su(m)n⊕su(n)m. At the end we give some applications to the construction of modular invariant partition functions.

Polynomial averages in the Kontsevich model

We obtain in closed form averages of polynomials, taken over hermitian matrices with the Gaussian measure involved in the Kontsevich integral, and prove a conjecture of Witten enabling one to express analogous averages with the full (cubic potential) measure, as derivatives of the partition function with respect to traces of inverse odd powers of the external argument. The proofs are based on elementary algebraic identities involving a new set of invariant polynomials of the linear group, closely related to the general Schur functions.

Fuchsian triangle groups and Grothendieck dessins. Variations on a theme of Belyi

According to a theorem of Belyi, a smooth projective algebraic curve is defined over a number field if and only if there exists a non-constant element of its function field ramified only over 0, 1 and ∞. The existence of such a Belyi function is equivalent to that of a representation of the curve as a possibly compactified quotient space of the Poincaré upper half plane by a subgroup of finite index in a Fuchsian triangle group. On the other hand, Fuchsian triangle groups arise in many contexts, such as in the theory of hypergeometric functions and certain triangular billiard problems, which would appear at first sight to have no relation to the Galois problems that motivated the above discovery of Belyi. In this note we review several results related to Belyis theorem and we develop certain aspects giving examples. For preliminary accounts, see the preprint [Wo1], the conference proceedings article [BauItz] and the “Comptes Rendus” note [CoWo2].

Singular vectors of the Virasoro algebra

Abstract We present an explicit construction of the singular (or null) vectors in highest weight Verma modules.

Comments on the links between SU(3) modular invariants, simple factors in the Jacobian of Fermat curves, and rational triangular billiards

We examine the proposal made recently that the su(3) modular invariant partition functions could be related to the geometry of the complex Fermat curves. Although a number of coincidences and similarities emerge between them and certain algebraic curves related to triangular billiards, their meaning remains obscure. In an attempt to go beyond the su(3) case, we show that any rational conformal field theory determines canonically a Riemann surface. Comment: 56 pages, 4 eps figures, LaTeX, uses epsf

Lattice Models With a Solvable Symmetry Group

Abstract A class of lattice models with a global symmetry characterized by a solvable group are shown to be equivalent to ones having an Abelian symmetry, to which the Kramers-Wannier dual transformation can be applied. Examples are afforded by permutation groups S 3 and S 4 .