Categorical Equivalence and the Renormalization Group

Abstract

In this article we review how categorical equivalences are realized by renormalization group flow in physical realizations of stacks, derived categories, and derived schemes. We begin by reviewing the physical realization of sigma models on stacks, as (universality classes of) gauged sigma models, and look in particular at properties of sigma models on gerbes (equivalently, sigma models with restrictions on nonperturbative sectors), and decomposition, in which two-dimensional sigma models on gerbes decompose into disjoint unions of ordinary theories. We also discuss stack structures on examples of moduli spaces of SCFTs, focusing on elliptic curves, and implications of subtleties there for string dualities in other dimensions. In the second part of this article, we review the physical realization of derived categories in terms of renormalization group flow (time evolution) of combinations of D-branes, antibranes, and tachyons. In the third part of this article, we review how Landau-Ginzburg models provide a physical realization of derived schemes, and also outline an example of a derived structure on a moduli spaces of SCFTs.

arXiv comment17 pages, Contribution to Proceedings of LMS/EPSRC Durham Symposium Higher Structures in M-Theory, August 2018

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• Decomposition, Trivially-Acting Symmetries, and Topological Operators
• Orbifolds by 2-groups and decomposition
• Integrality, Duality and Finiteness in Combinatoric Topological Strings
• Decomposition and the Gross-Taylor string theory
• Decomposition in Chern-Simons theories in three dimensions
• Higher Gauge Theory
• Dilaton shifts, probability measures, and decomposition
• Decomposition squared
• Chern-Simons theory, decomposition, and the A model
• Global aspects of moduli spaces of 2d SCFTs