Real multiplication through explicit correspondences

It is in general very subtle to integrate over the odd moduli of super Riemann surfaces in perturbative superstring computations. We study how these subtleties go away in favorable cases, including the embedding of N=0 string to N=1 string by Berkovits and Vafa, and the relation of the graviphoton amplitude and the topological string amplitude by Antoniadis, Gava, Narain and Taylor and Bershadsky, Cecotti, Ooguri and Vafa. The Poincar\'e dual of the moduli space of Riemann surfaces in the moduli space of super Riemann surfaces plays an important role.

As was shown by Harer the second homology of Mg, the moduli space of compact Riemann surfaces of genusg, is of rank 1, provided that g ≥ 3. This means that a nontrivial second de Rham cohomology class on Mg is unique up to a constant factor. But several canonical 2-forms on the moduli space have been constructed in various geometric contexts and differ from each other. In this chapter we review some of constructions in order to provide material for future research on “secondary geometry” of the moduli space Mg.

Cornalba (Ann Mat Pura Appl 149:135–151, 1987) classified the components of the singular locus of the moduli space of compact Riemann surfaces of genus $$g \ge 2$$ . Here we consider the problem of describing the intersections of these components by examining certain nerves of the cover of singular locus that the Cornalba components provide. We give a description of the 1-skeleton of such nerves which significantly extends the results of our earlier paper written together with A. Weaver where we considered a coarser cover of the singular locus. We compare the results of our earlier work with those of the present one in terms of certain natural simplicial covering maps between them.

The moduli space $$ \mathcal{M}_g $$ of compact Riemann surfaces of genus g has the structure of an orbifold and the set of singular points of such orbifold is the branch locus $$ \mathcal{B}_g $$ . In this article we present some results related with the topology of $$ \mathcal{B}_g $$ . We study the connectedness of $$ \mathcal{B}_g $$ for g ≤ 8, the existence of isolated equisymmetric strata in the branch loci and finally we stablish the connectedness of the branch locus of the moduli space of Riemann surfaces consIDered as Klein surfaces. We just sketch the proof of some of the results; complete proofs will be published elsewhere.

The moduli space $$ \mathcal{M}_g $$ of compact Riemann surfaces of genus g has the structure of an orbifold and the set of singular points of such orbifold is the branch locus $$ \mathcal{B}_g $$ . In this article we present some results related with the topology of $$ \mathcal{B}_g $$ . We study the connectedness of $$ \mathcal{B}_g $$ for g ≤ 8, the existence of isolated equisymmetric strata in the branch loci and finally we stablish the connectedness of the branch locus of the moduli space of Riemann surfaces consIDered as Klein surfaces. We just sketch the proof of some of the results; complete proofs will be published elsewhere.

International Journal of MathematicsVol. 10, No. 06, pp. 697-706 (1999) No AccessON THE TORELLI THEOREM FOR MINIMAL IMMERSIONSCLAUDIO AREZZOCLAUDIO AREZZOMathematics Institute, University of Warwick, Coventry CV4 7AL, UKThe author was supported by European Community Grant TRM N. ERBFMBICT 950079.https://doi.org/10.1142/S0129167X99000288Cited by:0 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail FiguresReferencesRelatedDetails Recommended Vol. 10, No. 06 Metrics History Received 5 August 1998 PDF download

In this paper we use deep ideas in complex geometry that proved to be very powerful in unveiling the Polyakov measure on the moduli space of Riemann surfaces and lead to obtain the partition function of perturbative string theory for 2, 3, 4 loops. Indeed a geometrical interpretation of the conformal Ward identity in two dimensional conformal field theory is proposed: the conformal anomaly is interpreted as a deformation of the complex structure of the basic Riemann surface. This point of view is in line with the modern trend of geometric quantizations that are based on deformations of classical structures. Then, we solve the conformal Ward identity by using this geometrical formalism.

The intersection numbers are defined on the moduli space of Riemann surface with s-marked points and genus g. When Riemann surface is cut and has boundary, the open intersection numbers appear. There appear open strings which touch to the boundary.

In this paper we study the moduli space of the tropicalizations of Riemann surfaces. We first tropicalize a smooth pointed Riemann surface by its pair of pants decomposition. Then we can construct the moduli space of tropicalizations based on a fixed regular tropicalization, and compactify it by adding strata parametrizing weighted contractions. This, in the covering level, is analogous to adding the frontier set, subordinate to a pants decomposition, to the Teichmüller space. We show that this compact moduli space is also Hausdorff. In the end, we compare this moduli space with the moduli space of Riemann surfaces, establishing a partial order-preserving correspondence between the stratifications of these two moduli spaces.

In the moduli space {\mathcal M}_{g} of Riemann surfaces of genus g \geq 2 there is important, so-called, real locus {\mathcal R}_{g} , consisting of points representing Riemann surfaces having symmetries, by which we understand antiholomorphic involutions. {\mathcal R}_g {itself} is covered by the strata {\mathcal R}_g^{k} , each being formed by the points representing surfaces having a symmetry of given topological type k . These strata are known to be real analytic varieties of dimension 3(g-1) . Also, their topological structure is pretty well known. Natanzon–Seppälä have realized that they are connected orbifolds homeomorphic to the factors of \mathbb{R}^{3g-3} with respect to actions of some discrete groups and Goulden–Jackson–Harer and Harer–Zagier have found their Euler characteristics, expressing them through the Riemann zeta function. However, topological properties of the whole real locus {\mathcal R}_{g} were less studied. The most known fact is its connectivity, proved independently by Buser–Seppälä–Silhol, Frediani and Costa–Izquierdo. This paper can be seen as a further contribution to the study of topology of {\mathcal R}_g , {which was possible through the notion of the nerve \mathcal{N}_g , associated to {\mathcal R}_{g} and called the real nerve . We find upper bounds for its geometrical and homological dimensions and we show their sharpness for infinitely many values of {odd g . Precise values of these dimensions for even g have been found by the authors in an earlier paper.

We will present an algebraic surface in the moduli space of triply punctured tori with the remarkable property that it is totally geodesic for Teichmuller metric. Our example reveals a surprising connection between Teichmuller theory and the geometry of cubic curves and is the first of its kind not arising from a covering construction. This is joint work with C. McMullen and A. Wright.

This paper gives a construction, using heat kernels, of differential forms on the moduli space of metrised ribbon graphs, or equivalently on the moduli space of Riemann surfaces with boundary. The construction depends on a manifold with a bundle of Frobenius algebras, satisfying various conditions. These forms satisfy gluing conditions which mean they form an open topological conformal field theory, i.e. a kind of open string theory. If the integral of these forms converged, it would yield the purely quantum part of the partition function of a Chern-Simons type gauge theory. Yang-Mills theory on a four manifold arises as one of these Chern-Simons type gauge theories.

We derive an analog of Mirzakhani’s recursion relation for hyperbolic string vertices and investigate its implications for closed string field theory. Central to our construction are systolic volumes: the Weil-Petersson volumes of regions in moduli spaces of Riemann surfaces whose elements have systoles L ≥ 0. These volumes can be shown to satisfy a recursion relation through a modification of Mirzakhani’s recursion as long as L ≤ 2 sinh−1 1. Applying the pants decomposition of Riemann surfaces to off-shell string amplitudes, we promote this recursion to hyperbolic string field theory and demonstrate the higher order vertices are determined by the cubic vertex iteratively for any background. Such structure implies the solutions of closed string field theory obey a quadratic integral equation. We illustrate the utility of our approach in an example of a stubbed scalar theory.

We investigate functorial properties of two hermitian line bundles over the moduli space of flat SU(n)-connections on a closed oriented surface; that is, of the Chern–Simons line bundle and the determinant line bundle. We investigate actions of cyclic subgroups of the mapping class group on them. As a consequence, we show that if we modify the determinant line bundle by the Hodge bundle over the moduli space of Riemann surfaces, then these line bundles are functorially isomorphic. This implies two quantum Hilbert spaces defined by the Chern–Simons line bundle and the modified determinant line bundle are functorially isomorphic.

The study of the mapping class group Mod( S ) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. A Primer on Mapping Class Groups begins by explaining the main group-theoretical properties of Mod( S ), from finite generation by Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod( S ) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.