Viktor Enolski

Abelian Functions for Trigonal Curves of Genus Three

We develop the theory of generalized Weierstrass sigma- and \wp-functions defined on a trigonal curve of genus three. In particular we give a list of the associated partial differential equations satisfied by the \wp-functions, a proof that the coefficients of the power series expansion of the sigma-function are polynomials of moduli parameters, and the derivation of two addition formulae.

Sigma, tau and Abelian functions of algebraic curves

We compare and contrast three different methods for the construction of the differential relations satisfied by the fundamental Abelian functions associated with an algebraic curve. We realize these Abelian functions as logarithmic derivatives of the associated sigma function. In two of the methods, the use of the tau function, expressed in terms of the sigma function, is central to the construction of differential relations between the Abelian functions.

On the Tetrahedrally Symmetric Monopole

We study SU(2) BPS monopoles with spectral curves of the form η3+χ(ζ6+bζ3−1) = 0. Previous work has established a countable family of solutions to Hitchin’s constraint that L2 was trivial on such a curve. Here we establish that the only curves of this family that yield BPS monopoles correspond to tetrahedrally symmetric monopoles. We introduce several new techniques making use of a factorization theorem of Fay and Accola for theta functions, together with properties of the Humbert variety. The geometry leads us to a formulation purely in terms of elliptic functions. A more general conjecture than needed for the stated result is given.

Dynamics on strata of trigonal Jacobians and some integrable problems of rigid body motion

We present an algebraic geometrical and analytical description of the Goryachev case of rigid body motion. It belongs to a family of systems sharing the same properties: although completely integrable, they are not algebraically integrable, their solution is not meromorphic in the complex time and involves dynamics on the strata of the Jacobian varieties of trigonal curves. Although the strata of hyperelliptic Jacobians have already appeared in the literature in the context of some dynamical systems, the Goryachev case is the first example of an integrable system whose solution involves a more general curve. Several new features (and formulae) are encountered in the solution given in terms of sigma-functions of such a curve.

On charge 3 cyclic monopoles

We determine the spectral curve of charge 3 BPS su(2) monopoles with C3 cyclic symmetry. The symmetry means that the genus 4 spectral curve covers a (Toda) spectral curve of genus 2. A well adapted homology basis is presented enabling the theta functions and monopole data of the genus 4 curve to be given in terms of genus 2 data. The Richelot correspondence, a generalization of the arithmetic mean, is used to solve for this genus 2 curve. Results of other approaches are compared.

Addition formulae over the Jacobian pre-image of hyperelliptic Wirtinger varieties

Abstract Using Frobenius-Stickelberger-type relations for hyperelliptic curves (Y. Ônishi, Proc. Edinb. Math. Soc. (2) 48 (2005), 705–742), we provide certain addition formulae for any symmetric power of such curves, which hold on the strata Wk , the pre-images in the Jacobian of the classical Wirtinger varieties. In an appendix, we give similar relations for a trigonal curve y 3 = (x – b 1)(x – b 2)(x – b 3)(x – b 4).

Periods of second kind differentials of

The problem of generalisation of classical expressions for periods of second kind elliptic integrals in terms of theta-constants to higher genera is studied. In this context special class of algebraic curves – (n, s)-curves is considered. It is shown that required representations can be obtained by comparison of equivalent expressions for projective connection by Fay-Wirtinger and Klein-Weierstrass. The case of genus two hyperelliptic curve is considered as a principle example and a number of new Thomae and Rosenhain-type formulae are obtained. We anticipate that the analysis undertaken for genus two curve can be extended to higher genera hyperelliptic curve as well to other classes of (n, s) non-hyperelliptic curves.

(n,s)

We show that the Higgs and gauge fields for a BPS monopole may be constructed directly from the spectral curve without having to solve the gauge constraint needed to obtain the Nahm data. The result is the analogue of the instanton result: given ADHM data one can reconstruct the gauge fields algebraically together with differentiation. Here, given the spectral curve, one can similarly reconstruct the Higgs and gauge fields. This answers a problem that has remained open since the discovery of monopoles.

-curves

Abstract It is known that the Jacobian of an algebraic curve which is a 2-fold covering of a hyperelliptic curve ramified at two points contains a hyperelliptic Prym variety. Its explicit algebraic description is applied to some of the integrable Henon–Heiles systems with a non-polynomial potential. Namely, we identify the generic complex invariant manifolds of the systems as a hyperelliptic Prym subvariety of the Jacobian of the spectral curve of the corresponding Lax representation. The exact discretization of the system is described as a translation on the Prym variety.