Integrality, duality and finiteness in combinatoric topological strings

• Published in: The journal of high energy physics Vol. 2022; no. 1; pp. 1 - 59
• Main Authors: de Mello Koch, Robert, He, Yang-Hui, Kemp, Garreth, Ramgoolam, Sanjaye
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2022; Springer Nature B.V; SpringerOpen
• Subjects: Algorithms; Amplitudes; Classical and Quantum Gravitation; Eigenvalues; Elementary Particles; Fields (mathematics); Group theory; High energy physics; Holography; M(atrix) Theories; Number theory; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; String Theory; Strings; Topological Field Theories; Topological Strings; Topology; Wormholes; Topological Strings; M(atrix) Theories; Topological Field Theories
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP01(2022)071

A bstract A remarkable result at the intersection of number theory and group theory states that the order of a finite group G (denoted |G| ) is divisible by the dimension d R of any irreducible complex representation of G . We show that the integer ratios G 2 / d R 2 are combinatorially constructible using finite algorithms which take as input the amplitudes of combinatoric topological strings ( G -CTST) of finite groups based on 2D Dijkgraaf-Witten topological field theories ( G -TQFT2). The ratios are also shown to be eigenvalues of handle creation operators in G -TQFT2/ G -CTST. These strings have recently been discussed as toy models of wormholes and baby universes by Marolf and Maxfield, and Gardiner and Megas. Boundary amplitudes of the G -TQFT2/ G -CTST provide algorithms for combinatoric constructions of normalized characters. Stringy S-duality for closed G -CTST gives a dual expansion generated by disconnected entangled surfaces. There are universal relations between G -TQFT2 amplitudes due to the finiteness of the number K of conjugacy classes. These relations can be labelled by Young diagrams and are captured by null states in an inner product constructed by coupling the G -TQFT2 to a universal TQFT2 based on symmetric group algebras. We discuss the scenario of a 3D holographic dual for this coupled theory and the implications of the scenario for the factorization puzzle of 2D/3D holography raised by wormholes in 3D.