Welcome to my homepage
About Me š¼
I am now an assistant professor at Tsinghua University, Yau Mathematical Sciences Center.
My name in Chinese: å½ę/ęøę
Email:
binguimath(at)gmail(dot)com
bingui(at)tsinghua(dot)edu(dot)cn
Research Interests š§
I am a mathematical analyst working on vertex operator algebras (VOAs), a mathematical model of 2d conformal field theory. This means that I am interested in analysis problems in VOAs and their representation categories, and that I like to solve VOA problems using analytic methods. These methods can be divided into two parts:
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Real analysis in unitary VOAs: Using functional analysis and especially von Neumann algebras, I study the unitarity of VOAs, the unitarity of VOA modules and their fusion products, the relation between VOAs and von Neumann algebras (more precisely: conformal nets).
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Complex analysis in (possibly non-unitary) VOAs: Using complex analysis, I study conformal blocks and their complex-analytic properties, factorization of conformal blocks, C_2 cofinite VOAs and their representations.
Courses šŖ
I have written LaTeX lecture notes for the following courses. See this page for the courses I taught using my old lecture notes.
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(2022 Spring) Vertex Operator Algebras, Conformal Blocks, and Tensor Categories
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(2023 Fall & 2024 Spring) Analysis I & II for Qiuzhen college
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(2024 Fall) Topics in Operator Algebras: Algebraic Conformal Field Theory
Publications and Preprints šµāš«
The number preceding each title indicates the year the paper was submitted to arXiv. Note that the preprints on this website might be more updated than the arXiv versions.
See this page for the full publications.
Recent papers
2023 (Joint with Hao Zhang) Analytic Conformal Blocks of C2-cofinite Vertex Operator Algebras I: Propagation and Dual Fusion Products, submitted, arXiv:2305.10180 Preprint Slides
2023 Geometric Positivity of the Fusion Products of Unitary Vertex Operator Algebra Modules, Comm. Math. Phys., Vol. 405 (2024). arXiv:2306.11856, Preprint Slides
2024 (Joint with Hao Zhang) Analytic Conformal Blocks of C2-cofinite Vertex Operator Algebras II: Convergence of Sewing and Higher Genus Pseudo-q-traces, submitted, arXiv:2411.07707 Preprint. (A talk with slides)
2025 (Joint with Hao Zhang) Analytic Conformal Blocks of C2-cofinite Vertex Operator Algebras III: The Sewing-Factorization Theorems, submitted, arXiv:2503.23995 Preprint.
We completely solve the problem of formulating and proving a factorization theorem for finite logarithmic chiral CFTs. In particular, our result naturally connects with the topological modular functors. Prior to our work, factorization had been established mainly for rational chiral CFTs, while for finite logarithmic chiral CFTs, no existing work had successfully proven a factorization result that aligns with the perspectives from TQFT and tensor categories, even in low-genus cases such as genus one.
Earlier representative works
2017 Unitarity of The Modular Tensor Categories Associated to Unitary Vertex Operator Algebras, II, Comm. Math. Phys., (2019) 372: 893-950. arXiv:1712.04931 Preprint
This is the second (and final) paper in a series where I initiated the systematic study of unitarity in the tensor categories of VOA representations.
2018 Categorical Extensions of Conformal Nets, Comm. Math. Phys., 383, 763-839 (2021). arXiv:1812.04470 Preprint Errata (A talk with slides)
This is the first paper studying the equivalence of the representation tensor categories associated to unitary completely rational VOAs and their conformal nets.
2020 Unbounded Field Operators in Categorical Extensions of Conformal Nets. First submitted in Oct. 2020, and later to another journal in Feb. 2025. arXiv:2001.03095 Preprint
This is the second paper studying the equivalence of representation tensor categories. I solved the conjecture that every unitary affine VOA (i.e. simply-connected WZW model) and its associated loop group conformal net have unitarily equivalent representation tensor categories.
Advice for young mathematicians: Donāt hesitate to submit your favorite work to a top journalāif you donāt mind the risk of losing your job because the paper gets rejected after more than four years of review.
2021 Genus-zero Permutation-twisted Conformal Blocks for Tensor Product Vertex Operator Algebras: The Tensor-factorizable Case, To appear in Commun. Contemp. Math., arXiv:2111.04662 Preprint Slides
Although this paper focuses on VOAs, its motivation originates from conformal nets and operator algebras. One of its main goals is to address the question: Why are conformal nets, which are defined and studied in the genus-0 setting, capable of describing higher-genus chiral CFTs? The answer lies in what I refer to as the āpermutation-twisted/untwisted correspondenceā: genus-0 permutation-twisted chiral CFTs correspond to higher-genus untwisted chiral CFTs via branched coverings. This observation represents, in my view, a significant contribution of operator algebras to the understanding of VOAs. Unfortunately, the importance of this correspondence has yet to be widely recognized within the VOA community.
Notes š„±
These are the lecture notes for the course Analysis I & II offered to undergraduates at Qiuzhen College (ę±ē书é¢) of Tsinghua university in the fall of 2023 and the spring of 2024. See this page for some older versions of the lecture notes.
These are the lecture notes for my course Topics in Operator Algebras: Algebraic Conformal Field Theory given at Yau Mathematical Sciences Center in 2024 Fall.
These are the lecture notes for my course Vertex Operator Algebras, Conformal Blocks, and Tensor Categories given at Yau Mathematical Sciences Center in 2022 Spring.
We give a detailed and self-contained exposition of the topics indicated in the title of this monograph. I origianally planned to give a course āUnbounded Operators in Conformal Field Theoriesā at Yau Mathematical Sciences Center in 2022 Spring, but have since changed the course topic to VOAs and conformal blocks. This monograph is the outcome of the preparation of that course.
In this monograph we give a complex-analytic approach to the theory of conformal blocks for VOAs. Part of this note has been adapted to form the main body of my article Convergence of Sewing Conformal Blocks. The section on (multi) propagation of conformal blocks has been expanded and explained in more details in the article Sewing and Propagation of Conformal Blocks.