Period: 2021-2023.

Financed by: ANCS-UEFISCDI contract 117/2021

Budget: 1198032 lei

Grant director: Sergiu Moroianu

Team members:

Rezumat.

Spatiul de moduli de suprafete Riemann este intim legat de geometria hiperbolica. Metrica Riemann pe acest spatiu este definita de produsul scalar L^2 fata de unica metrica hiperbolica dintr-o clasa conforma. Mai mult, forma Kähler pe partea regulata a spatiului de moduli admite diverse potentiale Kähler pe acoperirea universala in sens orbifold, adica pe spatiul Teichmüller. Unul dintre aceste potentiale este determinantul Laplacianului fata de metrica hiperbolica. Un altul este volumul renormalizat al unei metrici hiperbolice pe o 3-varietate de volum infinit care corespunde unui cobordism hiperbolic in dimensiune 2+1. Vom extinde aceste obiecte - si rezultate - la frontiera spatiului Teichmüller. Vom investiga determinantul operatorului Dirac pe suprafete hiperbolice, limita sa adiabatica (colapsarea unei familii de geodezice simple disjuncte) si sa construim in acest fel un potential Kähler care se extinde continuu la unele din componentele de bord, adica la spatiilor Teichmüller ale suprafetelor punctate. Este cunoscut ca determinantul Laplacianului devine singular la toate fetele de bord ale spatiului Teichmüller space corespunzatoare limitei adiabatice. Anticipam ca formula de urma Selberg va juca un rol in analiza noastra, dar intentionam de asemenea sa folosim algebre de operatori pseudodiferentiali adaptate acestei limite geometrice, pentru a controla nucleul Schwartz al puterilor complexe ale operatorului Dirac.

Abstract.

The moduli space of Riemann surfaces is intimately related to hyperbolic geometry. The Riemann metric on this space is defined by the scalar product L^2 with respect to the only hyperbolic metric in a conformal class. Moreover, the Kähler form on the regular part of the moduli space admits various Kähler potentials on the universal covering in the orbifold sense, i.e. on the Teichmüller space. One of these potentials is the determinant of the Laplacian against the hyperbolic metric. Another is the renormalized volume of a hyperbolic metric on a 3-manifold of infinite volume that corresponds to a hyperbolic cobordism in dimension 2+1. We will extend these objects - and results - to the frontier of Teichmüller space. We will investigate the determinant of the Dirac operator on hyperbolic surfaces, its adiabatic limit (the collapse of a family of simple disjoint geodesics) and in this way construct a Kähler potential that extends continuously to some of the board components, i.e. to the Teichmüller spaces of the dotted surfaces. It is known that the determinant of the Laplacian becomes singular at all edges of the Teichmüller space corresponding to the adiabatic limit. We anticipate that the Selberg trace formula will play a role in our analysis, but we also intend to use algebras of pseudo-differential operators adapted to this geometric limit, to control the Schwartz kernel of complex powers of the Dirac operator.

Our results improve the understanding of the moduli spaces of Riemann surfaces of finite volume. Each compact Riemann surface, together with a topological choice of a spin structure, determines an entire function Z(s), called the Selberg zeta function, displaying a symmetry around $s=1/2$ like the famous Riemann zeta function. The value at s=1/2 is of particular interest since it is related to the "determinant" of the Dirac operator. By one of the results of our project, the family of Selberg zeta functions, so in particular the central value, have a limit after renormalization by a constant factor when the Riemann surface "degenerates'' along a simple closed curve.

At the limit, we obtain a noncompact hyperbolic Riemann surface with two cusps. We proved that this noncompact surfaces also admits a Selberg zeta function, which equals the renormalizd limit of the zeta functions of the degenerating surfaces.

Updated: November 23, 2023