The Cosmological Constant: A Categorical View Fen Zuo∗ Abstract Some theoretic results related to the cosmological constant, obtained in the 80’s and 90’s of last century, are reviewed. These results exhibit some interesting underlying pattern when viewed from a category- theoretic perspective. In doing this, we illustrate how Baez and Dolan’s Periodic Table of k-tuply monoidal n-categories can serve as a basic framework of a quantum spacetime structure. Explicitly, we show how Einstein gravity (with the cosmological constant turned on) emerges when we de-categorify certain 2-tuply monoidal 2-categories properly, as Crane proposed a long time ago. In particular, we find that some recent work on Vertex Operator Algebras and 4-manifolds fits nicely into this framework. arXiv:2007.07270v1 [hep-th] 13 Jul 2020 ∗ Email:
[email protected] 1 Contents I. Introduction 3 A. The notation 4 II. 3D 4 A. Euclidean case 4 B. Lorentzian case 7 III. 2D 8 A. Euclidean case 8 B. Lorentzian case 9 IV. 4D 10 A. Euclidean case 11 1. Barrett-Crane Model 11 2. Anyon-Condensation Picture 12 3. Gravity From Condensation? 15 4. Cosmological Constant 17 B. Lorentzian case 18 1. Identifying “Time” 18 2. Emergence of 4D Lorentz Symmetry 20 3. Cosmological Constant 20 V. Categorification 21 A. Motivation 21 B. General framework 24 1. Vertex Operator Algebras 24 2. Theory X and 4D topological invariants 24 3. VOA[M4] and 2-Tannaka-Krein duality 27 4. Gravity and Cosmological Constant 28 VI. Discussion 30 Note added: 31 2 Acknowledgments 32 References 32 I.