Integrality, duality and finiteness in combinatoric topological strings
Abstract
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 (GCTST) of finite groups based on 2D DijkgraafWitten topological field theories (GTQFT2). The ratios are also shown to be eigenvalues of handle creation operators in GTQFT2/GCTST. 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 GTQFT2/GCTST provide algorithms for combinatoric constructions of normalized characters. Stringy Sduality for closed GCTST gives a dual expansion generated by disconnected entangled surfaces. There are universal relations between GTQFT2 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 GTQFT2 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.
 Publication:

Journal of High Energy Physics
 Pub Date:
 January 2022
 DOI:
 10.1007/JHEP01(2022)071
 arXiv:
 arXiv:2106.05598
 Bibcode:
 2022JHEP...01..071D
 Keywords:

 M(atrix) Theories;
 Topological Field Theories;
 Topological Strings;
 High Energy Physics  Theory;
 Mathematics  Combinatorics;
 Mathematics  Group Theory
 EPrint:
 50 pages, 4 Figures