Welcome to my homepage
About Me 😼
I am now an assistant professor at Tsinghua University, Yau Mathematical Sciences Center.
My name in Chinese: 归斌/歸斌
Email:
binguimath(at)gmail(dot)com
bingui(at)tsinghua(dot)edu(dot)cn
Research Interests 🧐
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 😪
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 😵💫
The following articles are listed in the order they were finished and submitted to arXiv. The arXiv indentifiers (YYMM.NNNNN) indicate the time of submission. Note that the preprints on this website might be more updated than the arXiv versions.
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Unitarity of The Modular Tensor Categories Associated to Unitary Vertex Operator Algebras, I, Comm. Math. Phys., (2019) 366(1), pp.333-396.
arXiv:1711.02840Preprint Errata - Unitarity of The Modular Tensor Categories Associated to Unitary Vertex Operator Algebras, II, Comm. Math. Phys., (2019) 372: 893-950.
arXiv:1712.04931PreprintIn this series of two papers, I initiated the systematic study of the unitarity of VOA representation tensor categories.
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Energy Bounds Condition for Intertwining Operators of Type B, C, and G_2 unitary affine vertex operator algebras, Trans. Amer. Math. Soc., 372 (2019), 7371-7424.
arXiv:1809.07003Preprint - Categorical Extensions of Conformal Nets, Comm. Math. Phys., 383, 763-839 (2021).
arXiv:1812.04470Preprint 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.
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Q-systems and extensions of completely unitary vertex operator algebras, Int. Math. Res. Not. IMRN, Vol 2022, Issue 10, 7550–7614
arXiv:1908.03484Preprint -
Bisognano-Wichmann Property for Rigid Categorical Extensions and Non-local Extensions of Conformal Nets, Ann. Henri Poincaré, 22, 4017-4062 (2021).
arXiv:1912.10682Preprint - Unbounded Field Operators in Categorical Extensions of Conformal Nets, submitted.
arXiv:2001.03095PreprintThis 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.
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Regular Vertex Operator Subalgebras and Compressions of Intertwining Operators, J. Algebra, (2020) 564. 32-48.
arXiv:2003.02921Preprint -
Polynomial Energy Bounds for Type F_4 WZW-models, Internat. J. Math., Vol. 31, No. 12 (2020).
arXiv:2004.02064Preprint -
Convergence of Sewing Conformal Blocks, Commun. Contemp. Math., Vol. 26, No. 03 (2024).
arXiv:2011.07450Preprint Errata -
Sewing and Propagation of Conformal Blocks, New York J. Math., 30 (2024) 187–230.
arXiv:2110.04774Preprint -
Genus-zero Permutation-twisted Conformal Blocks for Tensor Product Vertex Operator Algebras: The Tensor-factorizable Case, To appear in Commun. Contemp. Math.,
arXiv:2111.04662Preprint Slides -
On a Connes Fusion Approach to Finite Index Extensions of Conformal Nets.
arXiv:2112.15396Preprint -
(Joint with Hao Zhang) Analytic Conformal Blocks of C2-cofinite Vertex Operator Algebras I: Propagation and Dual Fusion Products.
arXiv:2305.10180Preprint Slides -
Geometric Positivity of the Fusion Products of Unitary Vertex Operator Algebra Modules, Comm. Math. Phys., Vol. 405 (2024).
arXiv:2306.11856, Preprint Slides -
(Joint with Hao Zhang) Analytic Conformal Blocks of C2-cofinite Vertex Operator Algebras II: Convergence of Sewing and Higher Genus Pseudo-q-traces.
arXiv:2411.07707Preprint. (A talk with slides) - (Joint with Hao Zhang) Analytic Conformal Blocks of C2-cofinite Vertex Operator Algebras III: The Sewing-Factorization Theorems.
arXiv:2503.23995Preprint.
Notes 🥱
These are the lecture notes for the course Analysis I & II offered to undergraduates at Qiuzhen College (求真书院) 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 “Unbounded Operators in Conformal Field Theories” 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.