arXiv subjects

Rongge Xu

Publications and source records attributed to Rongge Xu.

4 recordsLinked to original sources

arXiv2026

2-Morita Theory of $E_2$-Algebras and Module Categories

Building on our previous work on 2-Morita equivalence for $E_2$-algebras using topological pictures, we develop a systematic framework for Morita equivalence of topological orders in different dimensions in terms of $n$-Morita categories $\mathrm{Mrt}_{E_n}(\mathcal{C})$. In this framework, various notions of $n$-Morita equivalence are unified as equivalences of objects in $\mathrm{Mrt}_{E_n}(\mathcal{C})$. We also compare the constructions of higher Morita categories due to Haugseng and to Gwilliam--Scheimbauer. For $n=1,2$, we prove that the functor $\mathrm{Mod}_n:\mathrm{Mrt}_{E_n}(\mathcal{C})\to \mathrm{Mrt}_{E_{n-1}}(\mathrm{LMod}^{\mathrm{rep}}(\mathcal{C}))$ is an equivalence, relating the algebraic description of higher Morita theory to its realization in terms of module categories. In this formulation, the notion of a bi-bimodule arises naturally and provides a common framework for local and confined modules. Its explicit orientation data, together with the corresponding fusion rules, clarifies the relations among defects arising from condensation in topological orders.

math-ph↗

arXiv2026

LeanCat: A Benchmark Suite for Formal Category Theory in Lean (Part I: 1-Categories)

While large language models (LLMs) have demonstrated impressive capabilities in formal theorem proving, current benchmarks fail to adequately measure library-grounded abstraction -- the ability to reason with high-level interfaces and reusable structures central to modern mathematics and software engineering. We introduce LeanCat, a challenging benchmark comprising 100 fully formalized category-theory tasks in Lean. Unlike algebra or arithmetic, category theory serves as a rigorous stress test for structural, interface-level reasoning. Our evaluation reveals a severe abstraction gap: the best state-of-the-art model solves only 12.0% of tasks at pass@4, with performance collapsing from 55.0% on Easy tasks to 0.0% on High-difficulty tasks, highlighting a failure in compositional generalization. To overcome this, we evaluate LeanBridge, a retrieval-augmented agent that employs a retrieve-generate-verify loop. LeanBridge achieves a peak success rate of 24.0% -- doubling the performance of the best static baseline. These results empirically demonstrate that iterative refinement and dynamic library retrieval are not merely optimizations but strict necessities for neuro-symbolic reasoning in abstract domains. LeanCat offers a compact, reusable testbed for tracking progress toward reliable, research-level formalization.

cs.LO↗

arXiv2025

2-Morita Equivalent Condensable Algebras and Domain Walls in 2+1D Topological Orders

We classify $E_2$ condensable algebras in a modular tensor category $\mathcal{C}$ up to 2-Morita equivalence. From a physical perspective, this is equivalent to providing a criterion for when different $E_2$ condensable algebras result in the same condensed topological phase in a 2d anyon condensation process. By considering the left and right centers of $E_1$ condensable algebras in $\mathcal{C}$, we exhaust all 2-Morita equivalent $E_2$ condensable algebras in $\mathcal{C}$ and provide a method to recover $E_1$ condensable algebras from 2-Morita equivalent $E_2$ condensable algebras. We also prove that intersecting Lagrangian algebras in $\mathcal{C} \boxtimes \overline{\mathcal{C}}$ with its left and right components generates all 2-Morita equivalent $E_2$ condensable algebras in $\mathcal{C}$. This paper establishes a complete interplay between $E_1$ condensable algebras in $\mathcal{C}$, 2-Morita equivalent $E_2$ condensable algebras in $\mathcal{C}$, and Lagrangian algebras in $\mathcal{C} \boxtimes \overline{\mathcal{C}}$. The relations between different condensable algebras can be translated into their module categories, which correspond to domain walls in topological orders. We introduce a two-step condensation process and study the fusion of domain walls. We also show that an automorphism of an $E_2$ condensable algebra may lead to a nontrivial braided autoequivalence in the condensed phase. As concrete examples, we interpret the categories of quantum doubles of finite groups. We also discuss examples beyond group symmetries. Moreover, our results can be generalized to Witt-equivalent modular tensor categories.

cond-mat.str-el↗

arXiv2024

Categorical descriptions of one-dimensional gapped phases with Abelian onsite symmetries

In this work, we analyze the macroscopic observables in the 1+1D gapped phases with Abelian onsite symmetries and show that the spacetime observables for each gapped phase form a clear structure that can be mathematically described by enriched fusion categories, which uncovers the behavior of nonlocal excitations that were blurry in traditional Landau paradigm. These categorical descriptions not only generate the known classification results for symmetry preserving and breaking phases, but also unifies lattice dualities in a broader picture. After analyzing the general lattice model together with their boundaries, we give explicit examples including nontrivial SPT phase, where nontrivial boundaries can be given directly through our classification. Using enriched categorical descriptions, the lattice dualities and their gapped phases are unified under a holographic duality between an 2d. topological order with gapped 1d boundaries and 1+1D gapped quantum liquids with a categorical symmetry, which shed light on a unified definition of all quantum phases.

cond-mat.str-el↗