title: Quantum field theoretic and combinatorial perspectives on intersection numbers on the moduli space of complex curves
title: Explicit calculations in the Quartic Kontsevich Model &amp; logarithmic concavity
titleType: Subtitle
Creators: name: Jost, Finn Bjarne
Publisher: name: University of Muenster, Institutional Repository MIAMI
Year: 2025
Subjects: subject: 516 Geometry
subject: 530 Physics
subject: 511 General principles of mathematics

description: In dieser Thesis werden Invarianten des Modulraum komplexer Kurven, welche intersection numbers genannt werden, mit dem quartischen Kontsevich-Modell sowie dem LSZ-Modell in Verbindung gesetzt. Darüber hinaus wird logarithmische Konkavität in dem Modulraum komplexer Kurven mithilfe der kombinatorischen Ehrhart Theorie identifiziert. Im Rahmen des universellen Prinzips der Topologischen Rekursion werden explizite Ausdrücke für Korrelationsfunktionen niedrigen topologischen Typs gefunden, welche deren Abhängigkeit zu den Schnittzahlen beschreiben. Mithilfe der Ehrhart Theorie aus der Kombinatorik, welche die Zählung von Gitterpunkten in Polytopen beschreibt, werden in einem eingeschränkten Rahmen werden natürliche Fragen bezüglich der zu Schnittzahlen korrespondieren den Polytopklassen angegangen. Dabei wird nicht nur eine Interpretation der charakterisierenden Daten durch so genannte konsekutiv-geordnete Partitionsfolgen, sondern auch Muster wie logarithmische Konkavität entdeckt.
description: In this thesis, invariants of the moduli space of complex curves, known as intersection numbers, are related to the quartic Kontsevich model and the LSZ model. Moreover, logarithmic concavity is identified on the moduli space of complex curves using combinatorial Ehrhart theory. Within the universal framework of topological recursion, explicit expressions for correlation functions of low topological types are derived, describing their dependence on intersection numbers. Using Ehrhart theory, which deals with counting lattice points in polytopes, natural questions regarding the polytope classes corresponding to intersection numbers are addressed within a restricted framework. This approach not only provides an interpretation of the characteristic data through so-called order-consecutive partition sequences but also reveals novel features such as logarithmic concavity.
description: Introduction ..... iii
1 Field Theories On Non-commutative Space ..... 1
1.1 Axiomatic approach to QFTs ..... 2
1.2 Non-commutative setting ..... 3
1.3 Solution strategy of Quartic Kontsevich Model ..... 9
1.4 LSZ model ..... 16
2 The Moduli Space Of Curves and Topological Recursion ..... 17
2.1 Formal introduction ..... 17
2.1.1 Cohomology on the moduli space of curves ..... 24
2.1.2 Virasoro constraints ..... 28
2.2 Topological recursion ..... 30
2.2.1 Definition of topological recursion ..... 30
2.2.2 Topological recursion and intersection numbers ..... 35
2.2.3 Application to the LSZ model ..... 40
2.2.4 Blobbed topological recursion ..... 43
2.2.5 Blobbed topological recursion and intersection numbers ..... 49
2.2.6 Application to Quartic Kontsevich Model ..... 67
3 Logarithmic Concavity ..... 73
3.1 Short exposition of Ehrhart theory ..... 74
3.1.1 Relation of f∗- and h∗-vector ..... 77
3.1.2 Ehrhart theory background ..... 78
3.2 Ehrhart theory and intersection numbers ..... 81
3.3 Combinatorial definitions ..... 84
3.4 Logarithmic concavity ..... 87
3.4.1 Generating function for OCPS ..... 91
3.4.2 Proof of logarithmic concavity of OCPS ..... 94
3.4.3 Abstract theory of Pólya frequency sequences ..... 95
3.5 Proof of Theorems 3.2.4 and 3.2.5 ..... 97
3.5.1 f∗-expansion ..... 98
3.5.2 h∗ expansion ..... 99
Conclusions ..... 101
Scientific Appendices
A Complementary Definitions ..... 105
A.1 Hermitian and self-adjoint operators ..... 105
A.2 Singular homology and cohomology ..... 106
B Calculations and Data ..... 113
B.1 Expansion data in topological recursion ..... 113
B.1.1 Closed expression for z´,p,l in terms of x´,n ..... 113
B.1.2 Closed expression for y´,n and x´,n in terms of t´,k and z´,k,l ..... 114
B.1.3 First t´,k and ˆt´,k in terms of x´,n and y´,k ..... 115
B.2 Data supporting conjecture about logarithmic concavity of all normalized
Ehrhart polynomials associated to intersection numbers ..... 116
B.2.1 Numerical methods ..... 119
Formal Appendices
C Curriculum Vitae ..... 123
D Illustration Note ..... 125
List of Figures .....127
Bibliography ..... 129