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 dR of any irreducible complex representation of G. We show that the integer ratios {left|Gright|}^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.
Highlights
A well-known fact in finite group theory equates the sum of squares of dimensions of irreducible representations to the order of the group G
We show that the integer ratios |G|2/d2R 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)
Using some of the algebraic structures of G-TQFT2 developed in the context of open-closed topological string theory [25], we find that regarding the centre of the group algebra of G, denoted Z(C(G)) ≡ H, as a quantum mechanical Hilbert space gives a useful way to think about the one-dimensional topological quantum mechanics underlying
Summary
A well-known fact in finite group theory equates the sum of squares of dimensions of irreducible representations (irreps) to the order of the group G. We give a geometrical interpretation of these positive power sums in terms of G-TQFT2 amplitudes for entangled disconnected surfaces, where the entanglement is defined using projectors PR for the irreps living in the centre Z(C(G)) of the group algebra of G. Group representation theory of a finite group G over the complex numbers C is not a purely combinatoric subject It can involve the solution of eigenvalue equations with roots that may not be integer. For a general discussion of such problems see [3, 4], For Kronecker coefficients of symmetric groups, a lattice construction based on ribbon graphs and integer matrices arising from permutation group multiplications was recently given [10]. Described has an interpretation as a construction of the string amplitudes for a target space point from a finite set of string amplitudes for the disjoint union.
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