Deligne Categories in Lattice Models and Quantum Field Theory, or Making Sense of $O(N)$ Symmetry with Non-integer $N$

Abstract

When studying quantum field theories and lattice models, it is often useful to analytically continue the number of field or spin components from an integer to a real number. In spite of this, the precise meaning of such analytic continuations has never been fully clarified, and in particular the symmetry of these theories is obscure. We clarify these issues using Deligne categories and their associated Brauer algebras, and show that these provide logically satisfactory answers to these questions. Simple objects of the Deligne category generalize the notion of an irreducible representations, avoiding the need for such mathematically nonsensical notions as vector spaces of non-integer dimension. We develop a systematic theory of categorical symmetries, applying it in both perturbative and non-perturbative contexts. A partial list of our results is: categorical symmetries are preserved under RG flows; continuous categorical symmetries come equipped with conserved currents; CFTs with categorical symmetries are necessarily non-unitary.

59 pages, many figures; v3 typos fixed, version submitted to journal; v4 note 21 added about a terminological clash

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• Gentle introduction to rigorous Renormalization Group: a worked fermionic example
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• Superconformal Boundaries in $4-\epsilon$ dimensions
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• The F-theorem in the melonic limit
• Navigating through the O(N) archipelago
• Homoclinic RG flows, or when relevant operators become irrelevant
• The search for the universality class of metric quantum gravity
• Multiple Mass Hierarchies from Complex Fixed Point Collisions
• The tricritical Ising CFT and conformal bootstrap
• Chaotic RG Flow in Tensor Models
• Coupled minimal models revisited II: Constraints from permutation symmetry
• On Exotic Consistent Anomalies in (1+1)$d$: A Ghost Story
• Factorized lightcone expansion of conformal blocks
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• Disturbing news about the $d=2+\epsilon$ expansion
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• Emergent Supersymmetry at Large $N$
• On currents in the $O(n)$ loop model
• Bifundamental Multiscalar Fixed Points in $d=3-\epsilon$
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