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 ($G$-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.

50 pages, 4 Figures; v2 - refs added, typos corrected

References in corpus (15)

• JT gravity as a matrix integral
• Diagonal multi-matrix correlators and BPS operators in N=4 SYM
• Branes, Anti-Branes and Brauer Algebras in Gauge-Gravity duality
• D-branes and K-theory in 2D topological field theory
• Enhanced symmetries of gauge theory and resolving the spectrum of local operators
• From Matrix Models and quantum fields to Hurwitz space and the absolute Galois group
• Integrability vs. Information Loss: A Simple Example
• Schur-Weyl duality as an instrument of Gauge-String duality
• Wormholes without averaging
• Correlators, Probabilities and Topologies in N=4 SYM
• Dijkgraaf-Witten invariants of surfaces and projective representations of groups
• The Beta Ansatz: A Tale of Two Complex Structures
• BPS states, conserved charges and centres of symmetric group algebras
• Multi-matrix models and Noncommutative Frobenius algebras obtained from symmetric groups and Brauer algebras
• Word-Induced Measures on Compact Groups

Cited by in corpus (6)

• Circuit Complexity in Topological Quantum Field Theory
• Combinatoric topological string theories and group theory algorithms
• Comments on Summing over Bordisms in TQFT
• Dilaton shifts, probability measures, and decomposition
• Row-Column duality and combinatorial topological strings
• Row and column detection complexities of character tables