Pointed Drinfeld center functor

TL;DR

The paper develops a unified, higher-categorical framework for centers of algebras in fusion categories by introducing the pointed Drinfeld center functor, which combines the Drinfeld center and the full center into a single, symmetric monoidal equivalence. It proves that this pointed center yields a 3-equivalence between appropriate higher-categorical structures, and uses it to formulate a precise boundary-bulk relation for 1+1D rational conformal field theories, including a concrete fusion formula for fusing walls along a nontrivial 1+1D bulk. The results generalize prior work on centers, Morita invariants, and domain walls, and provide a robust mathematical language for understanding boundary conditions and defects in RCFTs and 2+1D topological orders. The work also interprets these constructions physically via 3-functors and a topological Wick rotation, clarifying when spatial fusion anomalies vanish and how different chiral symmetries couple across boundaries.

Abstract

In this work, using the functoriality of Drinfeld center of fusion categories, we generalize an earlier result on the functoriality of full center of simple separable algebras in a fixed fusion category to all fusion categories. This generalization produces a new center functor, which involves both Drinfeld center and full center and will be called the pointed Drinfeld center functor. We prove that this pointed Drinfeld center functor is a symmetric monoidal equivalence. It turns out that this functor provides a precise and rather complete mathematical formulation of the boundary-bulk relation of 1+1D rational conformal field theories (RCFT). In this process, we solve an old problem of computing the fusion of two 0D (or 1D) wall CFT's along a non-trivial 1+1D bulk RCFT.

Figures (5)

• Figure 1: This figure illustrates the physical meaning of the image of the 2-truncated fully faithful functor $\widehat{\mathfrak{Z}}$, where $F\in \mathop{\mathrm{Fun}}\nolimits_{\EuScript{L}|\EuScript{M}}(\EuScript{X},\EuScript{X}')$.
• Figure 2: The picture depicts the boundary-bulk relation of 2+1D topological orders. The arrows indicate the orientation of the boundaries or walls and the order of tensor product of topological excitations on the boundaries or walls.
• Figure 3: This figure depicts the 1+1D world sheet of three 1+1D bulk CFT's $A_\mathrm{bulk}^{(i)}$ separated by two 1D domain walls, each of which consists of three wall CFT's $[x,x],[y,y],[z,z]$ (resp. $[p,p],[q,q],[r,r]$) separated by two 0D domain walls $[x,y],[y,z]$ (resp. $[p,q],[q,r]$) for $x,y,z\in\mathop{\mathrm{Fun}}\nolimits_{\mathop{\mathrm{Mod}}\nolimits_V}(\EuScript{M}_1,\EuScript{M}_2), p,q,r\in \mathop{\mathrm{Fun}}\nolimits_{\mathop{\mathrm{Mod}}\nolimits_V}(\EuScript{M}_2,\EuScript{M}_3)$, where $\EuScript{M}_i$ is the category of boundary conditions canonically associated to $A_\mathrm{bulk}^{(i)}$ for $i=1,2,3$. All internal homs live in $\mathfrak{Z}(\mathop{\mathrm{Mod}}\nolimits_V)$.
• Figure 4: This figure depicts the 1+1D world sheet of three 1+1D bulk CFT's separated by two 1D domain walls (depicted as two vertical lines).
• Figure 5: This figure illustrate the physical meaning of the image of 2-truncation of $\widehat{\mathfrak{Z}}$, where $\EuScript{W}:= \mathop{\mathrm{Fun}}\nolimits_{\EuScript{L}|\EuScript{M}}(\EuScript{X},\EuScript{X}')$ and $F\in\EuScript{W}$, and $f_r$ and $f_l$ are defined in (\ref{['eq:f-r']}) and (\ref{['eq:f-l']}), respectively.

Theorems & Definitions (90)

• Remark 2.1
• Proposition 2.2: kz1
• Theorem 2.3: dsskz1
• Definition 2.4: kz1
• Remark 2.5
• Theorem 2.7: kz1
• Remark 2.8
• Theorem 2.9: kz1
• Definition 2.10
• Corollary 2.11