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Integrality, duality and finiteness in combinatoric topological strings
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Journal Article
Integrality, duality and finiteness in combinatoric topological strings
2022
Overview
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.
Publisher
Springer Berlin Heidelberg,Springer Nature B.V,SpringerOpen
