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Bin Gui

Publications and source records attributed to Bin Gui.

At least 19 recordsLinked to original sources
arXiv2026

Minkowskian open/closed conformal field theory possibly without vacuum: the Cardy case

For any conformal net, not necessarily rational, we construct the associated Cardy-type conformal field theory on the Minkowski spacetimes $(\mathbb R/2\pi\mathbb Z)\times\mathbb R$ for closed strings and $[0,\pi]\times\mathbb R$ for open strings within the framework of algebraic quantum field theory. In addition to verifying some of their basic properties, we prove three forms of Haag duality for multi-double-cones and boundary intervals, interpreted respectively as the Minkowskian versions of modular invariance, the Cardy consistency condition, and the Morita equivalence of boundary field algebras.

math-ph↗
arXiv2025

How are pseudo-$q$-traces related to (co)ends?

Let $\mathbb V$ be an $\mathbb N$-graded $C_2$-cofinite vertex operator algebra (VOA), not necessarily rational or self-dual. Using a special case of the sewing-factorization theorem from [GZ25a], we show that the end $\mathbb E=\int_{\mathbb M\in\mathrm{Mod}(\mathbb V)}\mathbb M\otimes_{\mathbb C}\mathbb M'$ in $\mathrm{Mod}(\mathbb{V}^{\otimes2})$ (where $\mathbb{M}'$ is the contragredient module of $\mathbb{M}$) admits a natural structure of associative $\mathbb C$-algebra compatible with its $\mathbb{V}^{\otimes2}$-module structure. Moreover, we show that a suitable category $\mathrm{Coh}_{\mathrm{L}}(\mathbb E)$ of left $\mathbb E$-modules is isomorphic, as a linear category, to $\mathrm{Mod}(\mathbb V)$, and that the space of vacuum torus conformal blocks is isomorphic to the space $\mathrm{SLF}(\mathbb E)$ of symmetric linear functionals on $\mathbb E$. Combining these results with the main theorem of [GZ25b], we prove a conjecture of Gainutdinov-Runkel: For any projective generator $\mathbb G$ in $\mathrm{Mod}(\mathbb V)$, the pseudo-$q$-trace construction yields a linear isomorphism from $\mathrm{SLF}(\mathrm{End}_{\mathbb V}(\mathbb{G})^{\mathrm{opp}})$ to the space of vacuum torus conformal blocks of $\mathbb V$. In particular, if $A$ is a unital finite-dimensional $\mathbb C$-algebra such that the category of finite-dimensional left $A$-modules is equivalent to $\mathrm{Mod}(\mathbb V)$, then $\mathrm{SLF}(A)$ is linearly isomorphic to the space of vacuum torus conformal blocks of $\mathbb V$. This confirms a conjecture of Arike-Nagatomo.

math.QA↗
arXiv2025

Pseudotraces on Almost Unital and Finite-Dimensional Algebras

We introduce the notion of almost unital and finite-dimensional (AUF) algebras, which are associative $\mathbb C$-algebras that may be non-unital or infinite-dimensional, but have sufficiently many idempotents. We show that the pseudotrace construction, originally introduced by Hattori and Stallings for unital finite-dimensional algebras, can be generalized to AUF algebras. Let $A$ be an AUF algebra. Suppose that $G$ is a projective generator in the category $\mathrm{Coh}_{\mathrm{L}}(A)$ of finitely generated left $A$-modules that are quotients of free left $A$-modules, and let $B = \mathrm{End}_{A,-}(G)^{\mathrm{opp}}$. We prove that the pseudotrace construction yields an isomorphism between the spaces of symmetric linear functionals $\mathrm{SLF}(A)\xrightarrow{\simeq} \mathrm{SLF}(B)$, and that the non-degeneracies on the two sides are equivalent.

math.QA↗
arXiv2025

Comparison of Extensions of Unitary Vertex Operator Algebras and Conformal Nets

Let $V$ be one of the following unitary strongly-rational VOAs: unitary WZW models, discrete series W-algebras of type ADE, even lattice VOAs, parafermion VOAs, their tensor products, and their strongly-rational cosets. Let $U$ be a (unitary) VOA extension of $V$, described by a Q-system $Q$. We prove that $U$ is strongly local. Let $\mathcal A_V,\mathcal A_U$ be the conformal nets associated to $V,U$ in the sense of Carpi-Kawahigashi-Longo-Weiner (CKLW). We prove that $\mathcal A_U$ is canonically isomorphic to the conformal net extension of $\mathcal A_V$ defined by the Q-system $Q$. We prove that all unitary $U$-modules are strongly integrable in the sense of Carpi-Weiner-Xu (CWX). We show that the CWX $*$-functor from the $C^*$-category of unitary $U$-modules to the $C^*$-category of finite-index $\mathcal A_U$-modules is naturally isomorphic to $*$-functor defined by $Q$.

math.QA↗
arXiv2025

Analytic Conformal Blocks of $C_2$-cofinite Vertex Operator Algebras III: The Sewing-Factorization Theorems

Let $\mathbb V=\bigoplus_{n\in\mathbb N}\mathbb V(n)$ be a $C_2$-cofinite VOA, not necessarily rational or self-dual. In this paper, we establish various versions of the sewing-factorization (SF) theorems for conformal blocks associated to grading-restricted generalized modules of $\mathbb V^{\otimes N}$ (where $N\in\mathbb N$). In addition to the versions announced in the Introduction of [GZ23], we prove the following coend version of the SF theorem: Let $\mathfrak F$ be a compact Riemann surface with $N$ incoming and $R$ outgoing marked points, and let $\mathfrak G$ be another compact Riemann surface with $K$ incoming and $R$ outgoing marked points. Assign $\mathbb W\in\mathrm{Mod}(\mathbb V^{\otimes N})$ and $\mathbb X\in\mathrm{Mod}(\mathbb V^{\otimes K})$ to the incoming marked points of $\mathfrak F$ and $\mathfrak G$ respectively. For each $\mathbb{M} \in \mathrm{Mod}(\mathbb{V}^{\otimes R})$, assign $\mathbb{M}$ and its contragredient $\mathbb M'$ to the outgoing marked points of $\mathfrak F$ and $\mathfrak G$ respectively. Denote the corresponding spaces of conformal blocks by $\mathscr T_{\mathfrak F}^*(\mathbb M\otimes\mathbb W)$ and $\mathscr T_{\mathfrak{G}}^*(\mathbb M'\otimes\mathbb X)$. Let the $\mathfrak X$ be the $(N+K)$-pointed surface obtained by sewing $\mathfrak F$, $\mathfrak G$ along their outgoing marked points. Then the sewing of conformal blocks-proved to be convergent in [GZ25a]-yields an isomorphism of vector spaces $$\int^{\mathbb{M}\in\mathrm{Mod}(\mathbb V^{\otimes R})}\mathscr T_{\mathfrak F}^*(\mathbb M\otimes\mathbb{W})\otimes_{\mathbb C} \mathscr T_{\mathfrak G}^*(\mathbb M'\otimes \mathbb X)\simeq\mathscr T_{\mathfrak X}^*(\mathbb W\otimes \mathbb X)$$ We also discuss the relationship between conformal blocks and the modular functors defined using Lyubashenko's coend/construction.

math.QA↗
arXiv2024

Analytic Conformal Blocks of $C_2$-cofinite Vertex Operator Algebras II: Convergence of Sewing and Higher Genus Pseudo-$q$-traces

Let $\mathbb V=\bigoplus_{n\in\mathbb N}\mathbb V(n)$ be a $C_2$-cofinite vertex operator algebra. We prove the convergence of Segal's sewing of conformal blocks associated to analytic families of pointed compact Riemann surfaces and grading-restricted generalized $\mathbb V^{\otimes N}$-modules (where $N=1,2,\dots$) that are not necessarily tensor products of $\mathbb V$-modules, generalizing significantly the results on convergence in [Gui24]. We show that ``higher genus pseudo-$q$-traces" (called pseudo-sewing in this article) can be recovered from the above generalization of Segal's sewing to $\mathbb{V}^{\otimes N}$-modules. Therefore, our result on the convergence of the generalized Segal's sewing implies the convergence of pseudo-sewing, and hence covers both the convergence of genus-$0$ sewing in [Hua05a,HLZ12] and the convergence of pseudo-$q$-traces in [Miy04] and [Fio16]. Using a similar method, we also prove the convergence of Virasoro uniformization, i.e., the convergence of conformal blocks deformed by non-automomous meromorphic vector fields near the marked points. The local freeness of the analytic sheaves of conformal blocks is a consequence of this convergence. It will be used in the third paper of this series to prove the sewing-factorization theorem.

math.QA↗
arXiv2023

Geometric Positivity of the Fusion Products of Unitary Vertex Operator Algebra Modules

A unitary and strongly rational vertex operator algebra (VOA) $V$ is called strongly unitary if all irreducible $V$-modules are unitarizable. A strongly unitary VOA $V$ is called completely unitary if for each unitary $V$-modules $W_1$, $W_2$ the canonical nondegenerate Hermitian form on the fusion product $W_1\boxtimes W_2$ is positive. It is known that if $V$ is completely unitary, then the modular category of unitary $V$-modules is unitary [Gui19b], and all simple VOA extensions of V are automatically unitary and moreover completely unitary [Gui22, CGGH23]. In this paper, we give a geometric characterization of the positivity of the Hermitian product on $W_1$ and $W_2$, which helps us prove that the positivity is always true when the fusion product $W_1\boxtimes W_2$ is an irreducible and unitarizable $V$-module. We give several applications: (1) We show that if $V$ is a unitary (strongly rational) holomorphic VOA with a finite cyclic unitary automorphism group $G$, and if $V^G$ is strongly unitary, then $V^G$ is completely unitary. This result applies to the cyclic permutation orbifolds of unitary holomophic VOAs. (2) We show that if $V$ is unitary and strongly rational, and if $U$ is a simple current extension which is unitarizable as a $V$-module, then $U$ is a unitary VOA.

math.QA↗
arXiv2023

Analytic Conformal Blocks of $C_2$-cofinite Vertex Operator Algebras I: Propagation and Dual Fusion Products

This is the first paper of a three-part series in which we develop a theory of conformal blocks for $C_2$-cofinite vertex operator algebras (VOAs) that are not necessarily rational. The ultimate goal of this series is to prove a sewing-factorization theorem (and in particular, a factorization formula) for conformal blocks over holomorphic families of compact Riemann surfaces, associated to grading-restricted (generalized) modules of $C_2$-cofinite VOAs. In this paper, we prove that if $\mathbb V$ is a $C_2$-cofinite VOA, if $\mathfrak X$ is a compact Riemann surface with $N$ incoming marked points and $M$ outgoing ones, each equipped with a local coordinate, and if $\mathbb W$ is a grading-restricted $\mathbb V^{\otimes N}$-modules, then the ``dual fusion product" exists as a grading-restricted $\mathbb V^{\otimes M}$-module. Indeed, we prove a more general version of this result without assuming $\mathbb V$ to be $C_2$-cofinite. Our main method is a generalization of the propagation of conformal blocks.

math.QA↗
arXiv2023

Lectures on Vertex Operator Algebras and Conformal Blocks

These are the lecture notes for a course taught at Tsinghua University in the spring of 2022. In these notes, we develop the basic theory of vertex operator algebras (VOAs) and their conformal blocks using complex-analytic methods. In particular, many well-known subtleties in VOA theory (about formal variables and, e.g., delta-functions) are presented in the form of proving the absolute and locally uniform (a.l.u.) convergence of certain series of complex analytic functions. We also provide many motivations from the perspective of Segal CFT.

math.QA↗
arXiv2021

Genus-zero Permutation-twisted Conformal Blocks for Tensor Product Vertex Operator Algebras: The Tensor-factorizable Case

For a vertex operator algebra $V$, we construct an explicit isomorphism between the space of genus-0 conformal blocks associated to permutation-twisted $V^{\otimes n}$-modules and the space of conformal blocks associated to untwisted $V$-modules and a branched covering C of the Riemann sphere. As a consequence, when V is CFT-type, rational, and C2 cofinite, the fusion rules for permutation-twisted modules are determined. We also relate the sewing and factorization of permutation-twisted $V^{\otimes n}$-conformal blocks and untwisted $V$-conformal blocks. Various applications are discussed. Note the differences in theorem and equation numbering between the arXiv version and the published version. Some terminology also varies: See Def. 2.2.1 (Def. 2.20 of the published version) for a slight difference in the meanings of $\mathbb U$. The term "Analytic Jacobi identity" in the arXiv version is called the "duality property" in the published version.

math.QA↗
arXiv2021

Sewing and Propagation of Conformal Blocks

We clarify the relations between sewing and propagating conformal blocks. In particular, we show that sewing and propagation and commuting procedures. As an application, we give a geometric construction of permutation-twisted modules for tensor product VOAs. Their first (algebraic) construction is due to Barron-Dong-Mason. The results and the point of view in this article are crucial for relating the (genus-0) permutation-twisted conformal blocks associated to a tensor product VOA $V^{\otimes k}$ and the untwisted conformal blocks (of possibly higher genera) associated to $V$, which will be discussed in an upcoming work.

math.QA↗
arXiv2021

Bisognano-Wichmann property for rigid categorical extensions and non-local extensions of conformal nets

Given an (irreducible) Mobius covariant net $\mathcal A$, we prove a Bisognano-Wichmann theorem for its categorical extension $\mathscr E^{\textrm{d}}$ associated to the braided $C^*$-tensor category $\textrm{Rep}^{\textrm{d}}(\mathcal A)$ of dualizable (more precisely "dualized") Mobius covariant $\mathcal A$-modules. As a closely related result, we prove a (modified) Bisognano-Wichmann theorem for any (possibly) non-local extension of $\mathcal A$ obtained by a $C^*$-Frobenius algebra $Q$ in $\textrm{Rep}^{\textrm{d}}(\mathcal A)$. As an application, we discuss the relation between the domains of modular operators and the preclosedness of certain unbounded operators in $\mathscr E^{\textrm{d}}$.

math-ph↗
arXiv2020

Categorical extensions of conformal nets

An important goal in studying the relations between unitary VOAs and conformal nets is to prove the equivalence of their ribbon categories. In this article, we prove this conjecture for many familiar examples. Our main idea is to construct new structures associated to conformal nets: the categorical extensions.

math.QA↗
arXiv2020

Convergence of Sewing Conformal Blocks

In recent work, Damiolini-Gibney-Tarasca showed that for a $C_2$-cofinite rational CFT-type vertex operator algebra $\mathbb V$, sheaves of conformal blocks are locally free and satisfy the factorization property. In this article, we use analytic methods to prove that sewing conformal blocks is convergent, solving a conjecture proposed by Zhu and Huang.

math.QA↗
arXiv2020

Polynomial energy bounds for type $F_4$ WZW-models

We prove that sufficiently many intertwining operators of type $F_4$ unitary affine VOAs satisfy polynomial energy bounds. This finishes the Wassermann type analysis of intertwining operators for all WZW-models.

math.QA↗