Integrality, Duality and Finiteness in Combinatoric Topological Strings

TL;DR

This paper explores the mathematical structure of combinatoric topological strings related to finite groups, revealing new algorithms and dualities, and discusses implications for holography and wormhole models in theoretical physics.

Contribution

It introduces finite algorithms for constructing ratios of group representation dimensions from topological string amplitudes and uncovers dualities and universal relations in these models.

Findings

Ratios ${|G|^2 / d_R^2}$ are combinatorially constructible.

Eigenvalues of handle creation operators relate to these ratios.

Universal relations between amplitudes are captured by Young diagrams.

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 ($G$-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.