Canonical Homology Basis Research Articles

Free compact boson on branched covering of the torus

We have studied the partition function of a free compact boson on a n-sheeted covering of torus gluing along m branch cuts. It is interesting because when the branched cuts are chosen to be real, the partition function is related to the n-th Rényi entanglement entropy of m disjoint intervals in a finite system at finite temperature. After proposing a canonical homology basis and its dual basis of the covering surface, we find that the partition function can be written in terms of theta functions.

Computation of the topological type of a real Riemann surface

We present an algorithm for the computation of the topological type of a real compact Riemann surface associated to an algebraic curve, i.e., its genus and the properties of the set of fixed points of the anti-holomorphic involution τ \tau , namely, the number of its connected components, and whether this set divides the surface into one or two connected components. This is achieved by transforming an arbitrary canonical homology basis to a homology basis where the A \mathcal {A} -cycles are invariant under the anti-holomorphic involution τ \tau .

Duality transformations for generalized WDVV in Seiberg–Witten theory

In Seiberg–Witten theory the solutions to these equations come in certain classes according to the gauge group. We show that the duality transformations transform solutions within a class to another solution within the same class, by using a subset of the Picard–Fuchs equations on the Seiberg–Witten family of Riemann surfaces. The electric–magnetic duality transformations can be thought of as changes of a canonical homology basis on the surfaces which in our derivation is clearly responsible for the covariance of the generalized WDVV system.

Short homology bases and partitions of Riemann surfaces

The homological systole of a compact Riemann surface X is the minimal length of a simple closed non-separating goedesic curve. Since any homology basis of X must contain curves that intersect any non-separating closed curve, surfaces having small homological systoles cannot have short homology basis. It turns out that this basically the only obstruction to finding short homology basis. We show, in fact, that a compact hyperbolic genus g Riemann surface X with homological systole ε has always a canonical homology basis which consists of curves γ satisfying the length bound l(γ)⩽(g−1) 105g+4 arcsin( 4 ε )

The algorithm to calculate the period matrix of the curve $x^{m}+y^{n}=1$

We show how to take a canonical homology basis and a basis of the space of holomorphic 1-forms on the curve x m þ y n ¼ 1, and we show how to calculate its explicit period matrix.

Klein's surface of genus three and associated theta constants

This surface is famous because the conformal automorphism group Aut(R) has order 168 = 84(3 ―1), the maximum possible. In 1970, Rauch and Lewittes wrote the beautiful paper [4] in which they found the period matrix of Klein's surface.It is difficultto determine the period matrix of a non-hyperelliptic Riemann surface of genus g > 3 explicitly. Klein originally obtained his surface R in the form of the upper half-plane identified under the principal congruence subgroup of level seven, T(7), of the modular group I Rauch and Lewittes represented the surface as the unit circle uniformization of R. They found a canonical homology basis expertly and they had the matrix representations of the generators of the automorphism group Aut(R). In order to find the period matrix with respect to this canonical homology basis they used these matrix representations. The aim of our articleis to decide the proportionalities of associated theta constants of Klein's surface. In our article we represent R as a covering surface over P1. Let T be a conformal automorphim of order 7 and let be the cyclic group generated by T and R/(T} the surface obtained by identifying the equivalent points on R under . Then i?/ becomes P] conformally and so R is considered as a 7-sheet covering surface of R/(T} = P1. In the firstsection we calculate the period matrix of R by the different method from that of Rauch and Lewittes. First we choose a basis of the space

Some degenerations of a compact Riemann surface of genus 4

A number of degenerations of a compact hyperelliptic Riemann surface of genus 4 are studied, using theta function techniques. 1. Let (5, T, A) be a hyperelliptic Riemann surface of genus 4 with a canonical homology basis. Then S has a representation as a two-sheeted cover of the sphere with ten branch points. We can arrange for the surface to have branch points over 0 , 1 , «>, 1/Xj, 1/X2> * * > IA7» where its real branch points other than 0 and 1 (if any) are all greater than 1 and in ascending order (see [2]). We obtain a concrete representation of (S, T, A) which we henceforth assume is that illustrated in Figure 1.

Period Relations of Schottky Type on Riemann Surfaces

It was recognized in Riemann's work more than one hundred years ago and proved recently by Rauch, cf. [R2], that g(g + 1)/2 unnormalized periods of normal differentials of first kind on a compact Riemann surface S of genus g > 2 with respect to a canonical homology basis are holomorphic functions of 3g 3 complex variables, the moduli, which parametrize space of Riemann surfaces near S and, hence, that there are (g 2)(g 3)/2 holomorphic relations among those periods. Eighty years ago, Schottky [S1] exhibited relation for g = 4 as vanishing of an explicit homogeneous polynomial in Riemann theta constants. Sixty years ago, Schottky and Jung [SJ] conjectured a result which implies Schottky's earlier one and some generalizations for higher genera. Here, we formulate Schottky and Jung's conjecture precisely and, on basis of a recent result of Farkas [F3], [F4], prove it. We then derive Schottky's result (we believe for first time correctly) and exhibit a typical relation of this kind for g = 5 (we show how to do this for any genus). We do not prove that our relations imply all relations, but there are some indications that they do, indications to be dealt with in subsequent publications. The present paper is a consolidation and expansion of notes [RF1] and [FR]. Our main result is formulated in ? 3 as Theorem 1 which asserts proportionality of squares of one set of theta constants, Schottky constants, to certain two-term products of another set, Riemann theta constants, both sets attached to S with a definite canonical homology basis and both defined in ? 2. Section 2 also contains other essential preliminary definitions and lemmata. In ? 5 we attain principal object of whole investigation by showing how, for g ? 4, substitution of proportionalities of Theorem 1 into suitable identities for general (g 1)-theta constants, in particular, for Schottky theta constants leads immediately to relations among Riemann

On the Schottky Relation and Its Generalization to Arbitrary Genus

If S is a compact Riemann surface of genus g>2 together with a canonical homology basis (F, A) r = at, *.., y, A = al .**, 3g and if de1, **,9 denotes the basis of abelian differentials of first kind on S dual to (F, A); i.e., 4 there must be these relations. Another way of looking at the problem is the following: The totality of g x g matrices which are symmetric and have positive definite imaginary part is called the Siegel upper half plane of degree g, denoted by ?Rg. Not all elements of g9 are Riemann matrices for some (S, r, A). As a matter of fact for g > 4 the elements of 9g which are Riemannmat rices for (S, F, A) form a set of positive codimension. The problem we are considering is to determine this set. The first person to make a break-through in this direction was F. Schottky [12]. In the case g = 4, Schottky showed that for the set in question in e, the associated even theta constants satisfy a special relation of the form 1/ r, ?+ Vr2 + V r3 = 0 where ri is a product of 8 theta constants. Rationalizing this expression we have an explicit homogeneous polynomial in the Riemann theta constants which of course gives the one relation for g = 4 among the 10 periods. Subsequently, Schottky and Jung in a joint note [13] indicated a way of re-deriving the genus 4 result and generalizing it to arbitrary g. Their idea was to establish certain relations between the Riemann theta constants and what was referred to in [10] as the Schottky theta constants; however, to our knowledge they never establish these relations. These relations were established for the case g = 2 by Rauch and the author in [10, 11] and the relations for g = 2 were seen to be a consequence of the vanishing

TWO KINDS OF THETA CONSTANTS AND PERIOD RELATIONS ON A RIEMANN SURFACE

It was recognized in Riemann's work more than one hundred years ago and proved recently by Rauch (cf. Bull. Am. Math. Soc., 71, 1-39 (1965) that the g(g + 1)/2 unnormalized periods of the normal differentials of first kind on a compact Riemann surface S of genus g >/= 2 with respect to a canonical homology basis are holomorphic functions of 3g - 3 complex variables, "the" moduli, which parametrize the space of Riemann surfaces near S and, hence, that there are (g - 2)(g - 3)/2 holomorphic relations among those periods. Eighty years ago, Schottky exhibited the one relation for g = 4 as the vanishing of an explicit homogeneous polynomial in the Riemann theta constants. Sixty years ago, Schottky and Jung conjectured a result which implies Schottky's earlier one and some generalizations for higher genera.Here, we formulate Schottky and Jung's conjecture precisely and, on the basis of a recent result of Farkas (these PROCEEDINGS, 62, 320 (1969)), prove it. We then derive Schottky's result (we believe for the first time correctly) and exhibit a typical relation of this kind for g = 5 (we can do this for any genus). We do not prove that our relations imply all relations, but there are some indications that they do.