Real Hurwitz numbers
Final Report Abstract
Real Hurwitz numbers are the real counterparts of classical Hurwitz numbers which count the number of holomorphic maps between Riemann surfaces of fixed genus and degree and with prescribed ramification behaviour. The adjective real here stands for the extra assumption that both Riemann surfaces carry anti-holomorphic involutions which are compatible with the holomorphic map. Similar to their classical counterparts, real Hurwitz numbers potentially play an important role in the still growing development connecting such enumerative theories and the related intersection-theoretic problems on moduli spaces of curves to far reaching trends in combinatorics, string theory, random matrix models, etc. The main obstacle in the study of real Hurwitz numbers is that, unlike their classical counterparts, in general these numbers depend on the position of branch points on the target surface which need to be fixed in order to correctly set up the counting problem. We encounter here a fundamental schism between real and complex enumerative problems, most prominently treated in the case of plane curves passing through points (Welschinger invariants). The main objectives of our project were as follows: 1. Prove non-triviality/lower bounds for real Hurwitz numbers, in particular, by searching for signed counts of the maps in question such that invariance under the change of branch points is restored. 2. Study the asymptotic behaviour of the lower bounds/signed counts found in the first step. 3. In analogy to classical Hurwitz numbers, study the combinatorial structure of the invariants introduced in the previous steps (piecewise polynomiality, intersections on real moduli spaces). Concerning real double Hurwitz numbers, we used the tropical approach to study so-called zig-zag covers, proved that they provide existence results and lower bounds (under some assumptions), and computed the logarithmic asymptotic growth of these bounds. The optimality of these bounds as well as their combinatorial structure are still under examination. Our investigations on simple rational functions generalize a former work, which introduces signed counts of (oriented) polynomials. Extending this approach, we introduced signed counts of simple rational functions and proved the invariance of these counts. Moreover, we proved that the generating series obtained from these counts are polynomials in terms of certain trigonometric functions and used this result to give precise criteria for vanishing/nonvanishing of the generating series and to compute the logarithmic asymptotic growth. An interesting combinatorial feature of our study is the appearance of certain generalizations of so-called alternating permutations.
Publications
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Signed counts of real simple rational functions
Boulos El Hilany, Johannes Rau
