Categorical Equivalence and the Renormalization Group

TL;DR

This paper explores how categorical equivalences are realized through renormalization group flow in various physical models, linking advanced mathematical structures like stacks and derived categories to physical theories such as sigma models, D-branes, and Landau-Ginzburg models.

Contribution

It provides a comprehensive review of the physical realization of categorical structures via renormalization group flow in string theory and related models, highlighting new insights into stacks, derived categories, and derived schemes.

Findings

Sigma models on gerbes decompose into disjoint unions of theories

Derived categories are realized through RG flow of D-branes and tachyons

Landau-Ginzburg models realize derived schemes and moduli space structures

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.

Tables

	$(\text{principal }G-\text{bundle }E\to Y,$				(1)
			$G-\text{equivariant map }\mathrm{Tot}\left(E\right)\to X).$

	${\int }_{\partial \Sigma }dx[\overline{\eta }d\eta +i{\psi }^i\left({\partial }_{i}P\right)\eta +i{\psi }^{\overline{ı}}\left({\partial }_{\overline{ı}}\overline{P}\right)\overline{\eta }$
			$+i{\psi }^i\left({\partial }_{i}Q\right)\overline{\eta }+i{\psi }^{\overline{ı}}\left({\partial }_{\overline{ı}}\overline{Q}\right)\eta$
			$-i|P{|}^2-i|Q{|}^2],$

Equations

$$X=\frac{{\text{mathbbm}C}^2\times {\text{mathbbm}C}^{\times }}{{\text{mathbbm}Z}_2},$$

$$(x,y,t)\in {\text{mathbbm}C}^2\times {\text{mathbbm}C}^{\times }\mapsto (-x,-y,-t),$$

$$1⟶K⟶G⟶H⟶1,$$

$$\mathrm{QFT}\left([X/G]\right)=\mathrm{QFT}\left(\coprod _{\hat{G}}(Y,B)\right),$$

$$H^2(Y,Z(G))\stackrel{Z(G)\to U(1)}{⟶}{H}^2(Y,U(1)).$$

$$\int [D\varphi ]\exp (-S)\sum _B\exp \left(\int {\varphi }^{\ast }B\right)$$

$$1⟶{\text{mathbbm}Z}_2⟶D_4⟶{\text{mathbbm}Z}_2\times {\text{mathbbm}Z}_2⟶1,$$

$$\mathrm{CFT}\left([X/D_4]\right)=\mathrm{CFT}\left([X/{\text{mathbbm}Z}_2\times {\text{mathbbm}Z}_2]\coprod [X/{\text{mathbbm}Z}_2\times {\text{mathbbm}Z}_2]\right),$$

$$D_4$$

$${\text{mathbbm}Z}_2\times {\text{mathbbm}Z}_2$$

$$Z\left([X/D_4]\right)=\frac{1}{|D_4|}\sum _{gh=hg}{Z}_{g,h},$$

$$Z\left([X/D_4]\right)$$

$$=$$

$$Z\left([X/D_4]\right)=Z\left([X/{\text{mathbbm}Z}_2\times {\text{mathbbm}Z}_2]\coprod [X/{\text{mathbbm}Z}_2\times {\text{mathbbm}Z}_2]\right)$$

$$\tau \mapsto \frac{a\tau +b}{c\tau +d},$$

$$\left[\begin{array}[]{cc}a&b\\ c&d\end{array}\right]\>\in\>SL(2,{\mathbbm{Z}}).$$

$$(\tau ,z)\mapsto \left(\frac{a\tau +b}{c\tau +d},\frac{z}{c\tau +d}\right),$$

$$\mathrm{d}z\mapsto \frac{\mathrm{d}z}{c\tau +d}.$$

$${\mathcal{M}}_{1,1}=[\mathfrak{h}/SL(2,\text{mathbbm}Z)].$$

$$(\tau ,z,|0⟩)\mapsto \left(\frac{a\tau +b}{c\tau +d},\frac{z}{c\tau +d},\pm \frac{|0⟩}{\sqrt{c\tau +d}}\right).$$

$${\wedge }^{\bullet }TX\otimes \sqrt{K_X},$$

$$\psi |+⟩=0,\psi |-⟩=|+⟩,\overline{\psi }|-⟩=0.$$

$$|\pm ⟩\mapsto \pm \exp \left(i\pi /2\right)|\pm ⟩.$$

$$SO(2,2;\text{mathbbm}Z)$$

$$\delta \eta$$

$$\delta \overline{\eta }$$

$$\int_{\partial\Sigma}\Big{[}-\alpha\psi^{\overline{\imath}}\partial_{\overline{\imath}}\left(\overline{P}\overline{Q}\right)-\tilde{\alpha}\psi^{i}\partial_{i}\left(PQ\right)\Big{]}.$$

$${\mathcal{E}}_0={\oplus }_i{\mathcal{A}}_i,{\mathcal{E}}_1={\oplus }_j{\mathcal{B}}_j.$$

$$P_i:{\mathcal{A}}_i⟶B_{i+1},$$

$$Q_i:{\mathcal{B}}_i⟶{\mathcal{A}}_{i+1}.$$

$$\cdots ⟶{\mathcal{A}}_i\stackrel{P_i}{⟶}{\mathcal{B}}_{i+1}\stackrel{Q_{i+1}}{⟶}{\mathcal{A}}_{i+2}\stackrel{P_{i+2}}{⟶}{\mathcal{B}}_{i+3}\stackrel{Q_{i+3}}{⟶}\cdots .$$

$$0⟶\mathcal{O}(-D)⟶\mathcal{O}⟶0$$

Full text

Categorical Equivalence and the Renormalization Group

LMS/EPSRC Durham Symposium on Higher Structures in M-Theory

Eric Sharpe111Corresponding author e-mail: ersharpe@vt.edu aa Department of Physics, MC 0435, 850 West Campus Drive, Virginia Tech, Blacksburg, VA 24061, USA

\shortabstract

1 Introduction

Over the last twenty years, we have gained a much better appreciation of how many abstract mathematical concepts play a role in various aspects of modern physics. In particular, there seems to be a general story that in physical realizations of categorical structures, notion of homotopy are often realized by the renormalization group. We illustrate the relationship in Table 1.