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Klein's Curve KLEIN'S CURVE H.W. BRADEN AND T.P. NORTHOVER Abstract. Riemann surfaces with symmetries arise in many studies of integrable sys- tems. We illustrate new techniques in investigating such surfaces by means of an example. By giving an homology basis well adapted to the symmetries of Klein's curve, presented as a plane curve, we derive a new expression for its period matrix. This is explicitly related to the hyperbolic model and results of Rauch and Lewittes. 1. Introduction Riemann surfaces with their differing analytic, algebraic, geometric and topological per- spectives have long been been objects of fascination. In recent years they have appeared in many guises in mathematical physics (for example string theory and Seiberg-Witten theory) with integrable systems being a unifying theme [BK]. In the modern approach to integrable systems a spectral curve, the Riemann surface in view, plays a fundamental role: one, for example, constructs solutions to the integrable system in terms of analytic objects (the Baker-Akhiezer functions) associated with the curve; the moduli of the curve play the role of actions for the integrable system. Although a reasonably good picture exists of the geom- etry and deformations of these systems it is far more difficult to construct actual solutions { largely due to the transcendental nature of the construction. An actual solution requires the period matrix of the curve, and there may be (as in the examples of monopoles, harmonic maps or the AdS-CFT correspondence) constraints on some of these periods. Fortunately computational aspects of Riemann surfaces are also developing. The numerical evaluation of period matrices (suitable for constructing numerical solutions) is now straightforward but analytic studies are far more difficult. When a curve has symmetries the period matrix of the curve is constrained and this may lead to simplifications, such as allowing the Baker-Akhiezer functions to be expressed in terms of lower genus expressions [BBM]. These simplifications arise (as will be described below) when an homology basis has been chosen to reflect the symmetries. For example, when studying magnetic monopoles with spatial symmetry [HMM95] such a basis enables a reduction of the associated Baker-Akhiezer functions [BE07, BE09, B]. Finding such an adapted homology basis is however both a theoretical and practical problem [G, D]. Here we shall describe a result obtained while developing new tools to investigate such adapted homology bases. The details of this software are not the focus here, though we shall point arXiv:0905.4202v2 [math.AG] 18 Oct 2010 out where we have employed this in what follows. Rather we believe the following result both illustrative and of independent interest. Theorem 1. The period matrix for Klein's curve (in the homology basis described below) takes the form 0e 1 11 1 (1.1) τ = @1 e 1A 2 1 1 e p −1+i 7 with e = 2 . To put the result in context we note that Levy [L] has assembled a very good general in- troduction to Klein's curve and various authors have studied the period matrix itself. Since 1 2 H.W. BRADEN AND T.P. NORTHOVER Riemann the construction of period matrices has been an exercise, albeit a hard one. The two ingredients, a canonical homology basis and the integration of a basis of holomorphic differentials along these, each present different difficulties. Hurwitz [H][p123], Klein and Fricke [FK][p595], and Baker [Ba] all calculated the periods of a basis of holomorphic differ- entials for a set of cycles on Klein's curve. The first three authors obtained a matrix whose imaginary part was not positive definite (and so not a period matrix) while Baker obtained a diagonal matrix (also not a period matrix). Baker explicitly noted ([Ba] footnote p267) that he was not claiming these cycles formed a canonical homology basis; in fact they are not. Rauch and Lewittes [RL], using a hyperbolic model, calculated the period matrix to be p p p 0 −1+3i 7 −1−i 7 −3+i 7 1 8 p 4p 8 p −1−i 7 1+i 7 −1−i 7 τRL = B C : @ 4 p 2 p 4 p A −3+i 7 −1−i 7 7+3i 7 8 4 8 Given the Rauch-Lewittes result it is a simple matter to establish the theorem. In terms of the symplectic matrix 0 1 1 0 1 1 0 1 − − B 0 1 1 0 1 1 C B − − C B 1 1 0 0 0 1 C AB (1.2) M = B− − C = B 0 1 0 0 0 1C CD B − C @ 0 0 0 0 0 1 A 1 0 0 1 0 0 − − we find −1 τRL = (C + Dτ)(A + Bτ) : Thus the period matrices are symplectically related and the theorem established. Of course the hard work is hidden in the construction of the symplectic transformation moving from our homology basis to that of Rauch-Lewittes. In what follows we will find an homology basis that yields (1.1) directly and then relate this to Rauch and Lewittes'. Other period matrices for Klein's curve are described later in the paper including the symmetric result of [RG] obtained from a hyperbolic model. We will however begin with a plane curve model as that is how spectral curves are presented in the integrable system context. To conclude this introduction we mention the two pieces of software1 that have been written to aid calculations involving intersection numbers of paths on Riemann surfaces, extcurves and CyclePainter. Such intersection numbers arise frequently in matters relat- ing to homology and period matrices and are responsible for the matrices that appear in this paper. extcurves accepts piecewise linear descriptions of homology cycles on a surface (not passing through the branch points) and calculates their intersection number. In essence this is just a matter of keeping track of orientations and totalling the contribution from each intersection, but subtleties arise when paths don't intersect transversely. It also has periph- eral functions to deal with common applications of the intersection numbers, such as finding transformations between canonical homology bases, and to allow interaction with Maple's existing routines in algcurves. CyclePainter provides a graphical interface (resembling a basic paint program) for constructing and visualising cycles usable in extcurves, easing the burden of demanding they are piecewise linear. It provides a 2 dimensional representation of cycles, with colour corresponding to sheet information. 2. Symmetries and the period matrix Before turning to the specific setting of Klein's curve let us recall how symmetries restrict the form of the period matrix. This will outline our strategy in what follows. Let γ∗ := f g 1Located at http://gitorious.org/riemanncycles. KLEIN'S CURVE 3 a∗; b∗ be a canonical homology basis and !j be a basis of holomorphic differentials for ourf Riemanng surface . Set f g C H !j 1 Hai = A = !j τ A bi B with τ = −1 the (a-normalized) period matrix. Let σ be an automorphism of . Then σ BA C acts on H1( ; Z) and the holomorphic differentials by C ai ai AB ai ∗ k σ∗ = M := ; σ !j = !kLj ; bi bi CD bi where M Sp(2g; Z). The fundamental identity 2 I I ! = σ∗! σ∗γ γ then leads to the constraint AB (2.1) = L: CD A A B B From this equation we see that L −1 = A + Bτ; A A and consequently TrLn = Tr(A + Bτ)n: Thus we may relate the character of the automorphism σ to the trace of A + Bτ. For curves with many symmetries we may evaluate τ via character theory. Streit [St] examines period matrices and representation theory. From (2.1) we obtain the algebraic Riccati equation τBτ + τA Dτ C = 0: − − This fixed point equation may be understood as a manifestation of the Riemann bilinear relations. The strategy then is to calculate for appropriate automorphisms the matrices M and L with the aim of restricting the form of the period matrix. Such an approach goes back to Bolza [Bo] who studied the consequences of a surface admitting many automorphisms in the g = 2 setting while more recently Schiller [JS-a, JS-b] and Schindler [BS-b] have looked at the period matrices of hyperelliptic curves using this approach. When a curve has a real structure the period matrix is also constrained and Buser and Silhol [BS] construct period matrices for various real hyperelliptic curves. We also note that Bring's curve, the genus 4 curve with maximal automorphism group S5, has a period matrix essentially determined by its symmetries [RR]. 3. Klein's curve Klein's quartic curve may be expressed as the plane algebraic curve x3y + y3z + z3x = 0: It is of genus three and achieves the Hurwitz bound of 168 on the order of its (conformal) automorphism group G, which here is isomorphic to P SL(2; 7). Klein's curve also has a real structure possessing the antiholomorphic involution [x; y; z] [¯x; y;¯ z¯]. The entire symmetry group has 336 elements and is P GL(2; 7). 7! The affine projection ((x; y) [x; y; 1]) $ x3y + y3 + x = 0 has 9 branch-points in the x-plane at 0, and the points of a regular septagon centred on 0. The birational transformation 1 x3 x t 1 1 t (x; y) (t; s) = 1 + ; ; (t; s) (x; y) = − ; − ; 7! y2 − y 7! s2 s3 4 H.W. BRADEN AND T.P. NORTHOVER yields the curve s7 = t(t 1)2 − which has 3 branch points on the t plane located at 0, 1 and . In order to make the order three generator of G more manifest we perform a fractional-linear1 transformation sending these branch-points to the cube roots of unity.
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