Eigenfunctions of the Laplacian acting on degree zero bundles over special Riemann surfaces

Abstract

Abstract. We find an infinite set of eigenfunctions for the Laplacian with respect to a flat metric with conical singularities and acting on degree zero bundles over special Riemann surfaces of genus greater than one. These special surfaces correspond to Riemann period matrices satisfying a set of equations which lead to a number theoretical problem. It turns out that these surfaces precisely correspond to branched covering of the torus. This reflects in a Jacobian with a particular kind of complex multiplication.

Free compact boson on branched covering of the torus

“…Also, we should mention that in Ref. [19], there is a theorem (Theorem 3.) which states that the period matrix of the branch covering of the torus should satisfy the conditon:…”

Adinkras, Dessins, Origami, and Supersymmetry Spectral Triples

“…In the case of a branched cover p : Y → E of an elliptic curve E, it is shown in [35], [36] that it is possible to construct an infinite set of eigenvalues and eigenfunctions of the Laplacian on Y , which we recall briefly here.…”

“…We also show, by using the construction in [38] of origami curves associated to dessins d'enfant, that the Adinkra graph A of a 1D supersymmetry algebra determines uniquely an origami curve Y , in which the graph A embeds, with a choice of 2 n embeddings, where n = #E(A) is the number of edges. We then use a result of [36] showing that, for all Riemann surfaces Y that admits a branched cover p : Y → E, with E an elliptic curve (hence in particular for all origami curves), it is possible to construct a family of metrics on Y , determined by compatibility conditions on the period matrix coming from the existence of the branched cover map to E. For each metric in this family, it is shown in [36] how to obtain an infinite family of eigenvalues and eigenvectors of the corresponding Laplacian. The resulting spectrum and spectral action functional behave more like the spectral action of tori and its computation can be approached in terms of a Poisson summation formula.…”

Compactified Strings as Quantum Statistical Partition Function on the Jacobian Torus

“…This would provide a functional relation between Z stat (β) and the quantum contribution to the string partition function also in the higher genus case, generalizing the relation for g = 1. The eigenvalues λ m,n also appear in considering the Laplacian with respect to degenerate metrics [7], for which ramified covering of the torus play a crucial rôle [8]. Ramified covering of the torus correspond to a particular kind of CM (complex multiplication) satisfied by the Riemann period matrix [8].…”

“…The eigenvalues λ m,n also appear in considering the Laplacian with respect to degenerate metrics [7], for which ramified covering of the torus play a crucial rôle [8]. Ramified covering of the torus correspond to a particular kind of CM (complex multiplication) satisfied by the Riemann period matrix [8]. Remarkably, such special Riemann surfaces also appear in the null compactification of type-IIA string perturbation theory at finite temperature [9].…”