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

Google Scholar

arXiv webpage

Biography

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:

• 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).

• 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 😪

• (2022 Spring) Vertex Operator Algebras, Conformal Blocks, and Tensor Categories

• (2023 Fall & 2024 Spring) Analysis I & II

• (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.

• Unitarity of The Modular Tensor Categories Associated to Unitary Vertex Operator Algebras, I, Comm. Math. Phys., (2019) 366(1), pp.333-396. arXiv:1711.02840 Preprint Errata

• Unitarity of The Modular Tensor Categories Associated to Unitary Vertex Operator Algebras, II, Comm. Math. Phys., (2019) 372: 893-950. arXiv:1712.04931 Preprint

  In this series of two papers, I initiated the systematic study of the unitarity of VOA representation tensor categories.

• 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:1809.07003 Preprint

• Categorical Extensions of Conformal Nets, Comm. Math. Phys., 383, 763-839 (2021). arXiv:1812.04470 Preprint (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.

• Q-systems and extensions of completely unitary vertex operator algebras, Int. Math. Res. Not. IMRN, Vol 2022, Issue 10, 7550–7614 arXiv:1908.03484 Preprint

• Bisognano-Wichmann Property for Rigid Categorical Extensions and Non-local Extensions of Conformal Nets, Ann. Henri Poincaré, 22, 4017-4062 (2021). arXiv:1912.10682 Preprint

• Unbounded Field Operators in Categorical Extensions of Conformal Nets, submitted. 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.

• Regular Vertex Operator Subalgebras and Compressions of Intertwining Operators, J. Algebra, (2020) 564. 32-48. arXiv:2003.02921 Preprint

• Polynomial Energy Bounds for Type F_4 WZW-models, Internat. J. Math., Vol. 31, No. 12 (2020). arXiv:2004.02064 Preprint

• Convergence of Sewing Conformal Blocks, Commun. Contemp. Math., Vol. 26, No. 03 (2024). arXiv:2011.07450 Preprint Errata

• Sewing and Propagation of Conformal Blocks, New York J. Math., 30 (2024) 187–230. arXiv:2110.04774 Preprint

• Genus-zero Permutation-twisted Conformal Blocks for Tensor Product Vertex Operator Algebras: The Tensor-factorizable Case. arXiv:2111.04662 Preprint Slides

• On a Connes Fusion Approach to Finite Index Extensions of Conformal Nets. arXiv:2112.15396 Preprint

• (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

• Geometric Positivity of the Fusion Products of Unitary Vertex Operator Algebra Modules, Comm. Math. Phys., Vol. 405 (2024). arXiv:2306.11856, Preprint Slides

Notes 🥱

• Qiuzhen Lectures on Analysis

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.

• Lectures on Vertex Operator Algebras and Conformal Blocks

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.

• Spectral Theory for Strongly Commuting Normal Closed Operators

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.

• Conformal Blocks: Vector Bundle Structures, Sewing, and Factorization

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.

Learning notes 😧

• See this page