Functional Equations and the Generalised Elliptic Genus

We give a new derivation and characterisation of the generalised elliptic genus of Krichever-Höhn by means of a functional equation. Introduction Functional equations provide a common thread to several investigations in mathematics and physics: our focus in this article will particularly be on the areas of topology and integrable systems where it is still unclear whether the threads before us form part of a greater fabric. In topology the German and Russian schools applied functional equations powerfully to formal group laws and genera [8, 11, 12, 13, 16, 20, 21, 22, 23]. They have arisen in the study of integrable systems in several different ways. F. Calogero, whom we honour in this volume, instigated in [18] a new use of functional equations in the study of integrable systems that is relevant here. The modern approach to integrable systems is to utilise a Lax pair. Calogero in [18], by assuming a particular ansatz for a Lax pair, reduced the consistency of the Lax pair to a functional equation and algebraic constraints. In this way he discovered the elliptic Calogero-Moser model. Similarly, by assuming an ansatz for a realisation of the generators of the Poincaré algebra, Ruijsenaars and Schneider [30] reduced the ensuing commutation relations to that of a functional equation. The Ruijsenaars-Schneider model which results from one solution to this functional equation is also integrable. For the Ruijsenaars-Schneider systems Bruschi and Calogero constructed a Lax pair, again by means of an ansatz and consequent functional equation [9, 10]. (The general solutions to the functional equations of Ruijsenaars and Schneider have now been constructed [7, 17], but it is still open whether the resulting models are completely integrable.) Later Braden and Buchstaber generalised these various Lax pair ansätze [5] and encountered a rather ubiquitous functional equation [6] that includes many functional equations arising in both cohomological computations and integrable systems. We will return to this functional equation in due course but what is of interest at this stage is that the same equations arise in both settings. This may reflect something deeper. String theory physics allows some topology changes (such as flops) [1, 34] and physical quantities such as the partition function should reflect this invariance; invariance under classical flops characterises 1 2 H. W. BRADEN † AND K. E. FELDMAN ‡ the elliptic genus [31]. The authors of [24] draw connections between the complex cobordism ring and conformal field theory. Certainly the BakerAkhiezer functions associated to the integrable systems satisfy addition formulae [14, 15] and reflect the underlying algebraic geometry [19]. The present article aims to provide a new derivation of the generalised elliptic genus of Krichever-Höhn by means of a functional equation encountered in the study of integrable systems. In the first section we will review equivariant genera of loop spaces. The following section derives the relevant functional equation which we then solve in the final section. Various remarks will be made enroute that relate this approach to existing derivations. 1. Equivariant Genera of the Loop Space Motivated by the problem of obtaining left–right asymmetric fermions in a Kaluza–Klein theory Witten in [32] suggested the study of a special twisted Dirac operator on closed spin manifolds equipped with a smooth S1-action. Witten conjectured that the character–valued index of such a twisted operator is in fact a constant and that the genus of a manifold corresponding to this Dirac operator possesses a rigidity property. To break the conjecture into simpler pieces, Landweber posed a problem on computation of a special ideal in the bordism ring of semifree S1-actions on spin manifolds. As a tool for the solution to this problem Ochanine [29] introduced an elliptic genus Q(x) = 1 2 x tanh(x/2) · ∞