Fakultät für Mathematik und Physik - Universität Hannover

Department of Mathematics and Physics
Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover/Germany

Graduiertenkolleg 1463 "Analysis, Geometrie und Stringtheorie"
Research Training Group 1463 "Analysis, Geometry and String Theory"

Winter Semester 2009/2010

Short course 

Herr Prof. Dr. Samuel Grushevsky, Stony Brook University

"String scattering amplitudes and modular forms"
18.01.2010  16:15-17:15 Uhr   Herrmann-Windel-Saal, Gebäude 1105, Welfengarten 1a

The (super)string measure is one of the central quantities involved in the formulation of string theory. Various expressions for the bosonic string measure were proposed in 1980s. The question of finding the superstring measure is much harder than for the bosonic measure, as the supermoduli are present in the theory. D'Hoker and Phong started the modern program of computing the superstring measure from factorization constraints (restrictions to lower genera). In this talk we discuss the recently proposed ansatze for the superstring measure, using their ideas, and a detailed study of the moduli spaces of Riemann surfaces, and appropriate modular forms. Knowledge of string theory or of modular forms will not be required to understand this talk.

"The physics of the problem of computing the (super)string measure"
19.01.2010  10:15-11:15 Uhr   Herrmann-Windel-Saal, Gebäude 1105, Welfengarten 1a

We will discuss the physical formulation of the problem of computing the string measure. We will explain the Belavin-Knizhnik results for the bosonic string and for the superstring, explain that the measure is a volume form on an appropriate moduli space, and will discuss the physically expected properties it is expected to satisfy. We will then survey the 1980s work on the bosonic string measure.

"Moduli of curves and abelian varieties, and modular forms"
20.01.2010  12:00-13:30 Uhr   Herrmann-Windel-Saal, Gebäude 1105, Welfengarten 1a

We will introduce the moduli space of abelian varieties and develop the theory of Siegel modular forms. We will define theta functions and use them to consider projective embeddings of the moduli, and will survey what is known about the rings of modular forms (i.e. the geometry of these projective embeddings), and the mathematical open questions in this study.

"An ansatz for the superstring measure"
21.01.2010  16:30-17:30 Uhr  
Herrmann-Windel-Saal, Gebäude 1105, Welfengarten 1a

We will use the language of modular forms and theta constants to propose an explicit ansatz for the superstring scattering measure in genus up to 4, and a conjectural ansatz - for higher genera. We will show that this ansatz satisfies the factorization constraints, as expected by D'Hoker and Phong, and formulate mathematically the further physical properties that one would expect this ansatz to satisfy.

"Further results and questions on superstring scattering amplitudes"
22.01.2010  10:00-11:00 Uhr  
Hauptgebäude 1101, Welfengarten 1

We will study further physical properties of the proposed ansatz for the superstring measure. We will show that the ansatz gives a vanishing cosmological constant, and a vanishing two-point function for genus up to 3. We will then discuss the situation in genus 5, obtaining an explicit ansatz, with a modification needed to ensure the vanishing of the cosmological constant - which turns out to be related to the classical Schottky problem in mathematics. We then survey further open questions and approaches to superstring scattering amplitudes.