Schedule for: 21w5247 - Subfactors, Vertex Operator Algebras, and Tensor Categories (Online)

We will give an introduction to subfactors and highlight the Fourier duality from different perspectives.

We will talk about Fourier analysis on subfactors and a newly developing subject Quantum Fourier Analysis. We will review its early applications on subfactors and recent applications on tensor categories.

A well-known open problem is whether there exists an integral fusion category which is not weakly group-theoretical. A way to investigate this problem is to look for simple integral fusion rings, and see whether a non group-like one can be categorified. In joint works with Zhengwei Liu, Yunxiang Ren and Jinsong Wu, we developed several categorification criteria, coming from quantum Fourier analysis or localization strategies of the pentagon equations, and we applied them as efficient filters for above investigation.

A bi-unitary connection producing a subfactor of finite depth gives a 4-tensor appearing in a recent work on 2-dimensional topological order and anyons. Physicists have a special finite dimensional projection called a "projector matrix product operator" in this setting. We prove that the range of this projection of length k is naturally identified with the k-th higher relative commutant of the subfactor arising from the bi-unitary connection. This gives a further connection between 2-dimensional topological order and subfactor theory.

We show that every conformal net has an associated vertex algebra, thus identifying the class of conformal nets with a sub-class of the class of unitary vertex algebras. We also characterise those unitary vertex algebras that arise from a conformal net. (We conjecture that every unitary vertex algebras arises in this way, and hence that there is a bijective correspondence between conformal nets and unitary vertex algebras.) To construct the correspondence between conformal nets and unitary vertex algebras, we introduce a new notion of "field localised in a segment embedded in a Riemann surface", which could be of independent interest. This is joint work with James Tener.

In a recent work Carpi, Kawahigashi, Longo and Weiner defined a general correspondence between a class of unitary VOAs, called "strongly local", and conformal nets. Although it is believed that every unitary VOA is strongly local and hence gives rise to a conformal net, all known examples rely directly or indirectly on the so-called linear energy bounds. The latter method appears to be not sufficiently general to prove strong locality for various interesting unitary VOAs. In this talk I will describe a new method to prove strong locality based on local energy bounds which avoids the use of linear energy bounds. The simple VOAs generated by the W_3 algebras with c> 2 have been recently shown to be unitary and it turns out that they satisfy the needed local energy bounds. Accordingly, all simple W_3 VOAs with c>2 are strongly local. This talk is based on recent works with Y. Tanimoto and M. Weiner.

Orbifold theory studies a vertex operator algebra V under the action of a finite automorphism group G. The main objective is to understand the module category of fixed point vertex operator subalgebra $V^G$. We prove a conjecture by Dijkgraaf-Pasquier-Roche on $V^G$-module category if V is holomorphic. We also establish a connection between rational orbifold theory and minimal modular extensions. Our work is based on the previous results on modular extensions by Drinfeld-Gelaki-Nikshych-Ostrik and Lan-Kong-Wen. This is a joint work with Richard Ng and Li Ren.

In this talk, we will talk about the lattice structure of the integral form for the lattice vertex algebra and show that divided powers of general vertex operators preserve the integral lattice spanned by Schur functions indexed by partition-valued functions. We also show that the Garland operators, counterparts of divided powers of Heisenberg elements in affine Lie algebras, also preserve the integral form. These construe analogs of the Kostant Z-forms for the enveloping algebras of simple Lie algebras and the algebraic affine Lie groups in the situation of the lattice vertex algebras.

Let V be a vertex operator superalgebra and G a finite automorphism group of V containing the canonical automorphism such that $V^G$ is regular. We classify the irreducible $V^G$-modules appearing in twisted V-modules and prove that these are all the irreducible $V^G$-modules. Moreover, the quantum dimensions of irreducible $V^G$-modules are determined, a global dimension formula for V in terms of twisted modules is obtained and a super quantum Galois theory is established. In addition, the S-matrix of $V^G$ is computed. This is a joint work with Chongying Dong and Meiling Yang.

While regular W-algebras have enjoyed many years of study and attention, recent developments in physics have the less popular subregular W-algebras playing an important role. Moreover, these subregular W-algebras appear at levels where the corresponding conformal field theory is likely non-rational. This necessitates a deeper understanding of the structure and representation theory of such vertex operator algebras at non-rational levels. In type A_n, only the n=1 (sl_2) and n=2 (Bershadsky-Polyakov algebra) cases are particularly well-understood. In both cases an 'inverse reduction-by-stages' approach, first described for sl_2 in vertex operator algebra language by D. Adamovic, relates much of the representation theory to that of the corresponding regular W-algebra. The most important ingredient in this approach is an embedding of the type A_1 or A_2 subregular W-algebra into the tensor product of a certain lattice vertex algebra and the regular W-algebra of the same type. In this talk, I will describe how to generalise this approach to all type A_n subregular W-algebras using screening operators developed by N. Genra.

We will talk about some recent results on varieties of affine VOAS with non-addmisible levels for the $sl_3$ case.

It is an important and well-known result that the category of modules of a rational vertex operator algebra is a semisimple modular tensor category. Such categories define three-dimensional topological field theories together with their associated projective representations of surface mapping class groups. One way of stating the famous Verlinde formula is that on the torus, this action agrees with the behaviour of characters of VOA-modules under modular transformations. In this talk I will present some results and conjectures on how the ingredients presented above generalise to a finitely non-semisimple setting.

I was asked by the organizers of this workshop to give two introductory talks on the application of category theory in the study of topological phases in physics. In my first talk, I will use 2+1D toric code lattice model to explain the physical meanings of (braided) fusion categories and Drinfeld center.

In my second talk, I will use 3+1D toric code model and 1+1D Ising model to explain the physical meanings of braided fusion 2-categories and enriched fusion categories, respectively.

In 2012, Yi-Zhi Huang constructed two cohomology theories for grading-restricted vertex algebras and its modules, which are analogous to Hochschild and Harryson cohomologies for associative and commutative associative algebras. In both theories, the first cohomology $H^1(V, W)$ is given by grading-preserving derivations from the vertex algebra $V$ to the $V$-module $W$. In this work we compute the first cohomologies for the affine, Virasoro and lattice VOAs that are strongly rational. We show that for every $\mathbb{N}$-graded $V$-module $W$, $H^1(V, W)$ is given by zero-mode derivations, i.e., linear maps of the form $v\mapsto w_0 v$ for some $w$ of weight 1. This agrees with the conjecture by Huang and the author in 2018.

Group graded extensions of fusion categories arise naturally in the study of rational conformal field theories with finite symmetry G. They appear as categories of G-twisted modules of VOAs. We will explain a categorical perspective for computing the fusion rules of a G-graded extension of a fusion category C in terms of the associated action of G on its center Z(C) constructed by Etingof, Nikshych and Ostrik. This can be applied to find fusion rules for the extensions arising from permutation symmetry in CFT. Based on joint work with Marcel Bischoff.

Conformal nets and (unitary) vertex operator algebras are two separate mathematical structures which are supposed to encode the same physical idea: unitary chiral conformal field theory. As a result, the two structures should be equivalent up to minor technical hypotheses. In practice the two definitions looks very different, and provide easy access to different aspects of the conformal field theories they describe. In the first part of this introductory series of talks, I will present the two frameworks and describe how to translate between them, and how this correspondence can be used to gain new insight into mathematical problems.

In the second part of the talk I will introduce Segal CFT, a third formalism for studying chiral conformal field theories. I’ll give a comparison of Segal CFT with the two frameworks introduced in the first talk, and describe mathematical structures which unify all three perspectives.

The connection between intertwining operators and $A(V)$-module maps (where $A(V)$ is the Zhu algebra associated to a vertex operator algebra $V$) is needed in the last step of my proof of the Moore-Seiberg conjecture on the modular invariance of intertwining operators for rational conformal field theories. In this talk, I will discuss my recent work proving that the spaces of (logarithmic) intertwining operators are isomorphic to suitable spaces of module maps between suitable modules for the associative algebras $A^\infty(V)$ and $A^N(V)$ that I introduced last year. The main motivation of this work is to give the last step of the proof of the modular invariance of (logarithmic) intertwining operators when V is $C_2$-cofinite but might not be reductive.

I will report on a joint work with K.Coulembier and P.Etingof. We give a characterization of symmetric tensor categories over fields of positive characteristic which admit an exact tensor functor to the Verlinde category; in particular we give a characterization of Tannakian categories. A crucial ingredient of this characterization is exactness of the Frobenius twist functor which mimics the Frobenius twist for representations of algebraic groups.

I will discuss joint work with Terry Gannon. We show that there is an equivalence of ribbon tensor categories between the category of representations for small quantum SL(2) and the category of modules over the triplet vertex operator algebra. We provide similar equivalences for big quantum SL(2) and the Virasoro VOA, and a torus extension of small quantum SL(2) and the singlet VOA. Here our quantum groups are taken at a parameter q of order 2p, and the corresponding VOAs are of central charge 1-6(p-1)^2/p. Our results verify certain conjectures of Gainutdinov-Semikhatov-Tipunin-Feigin and Creutzig-Gainutdinov-Runkel.

Among those who study modular tensor categories, or the representation theory of rational vertex operator algebras, unitarity is often assumed because unitary categories align best with their theoretical applications. But from an abstract point of view, there is no inherent reason to make this assumption. In fact, unitarity is destroyed by very simple operations on any modular tensor category. Even the (potentially) weaker condition of pseudounitarity, the coincidence of Frobenius-Perron and categorical (global) dimension, is destroyed by things like Galois conjugacy. We will discuss a recent result which proves the existence of infinitely many fusion categories (modular tensor categories, even) whose fusion rules have no pseudounitary categorifications. These examples are as classical as one could imagine, coming from the representation theory of quantum groups at roots of unity (affine Lie algebras, WZW, etc).

Representation theory of W-algebras, in particular exceptional W-algebras, has been developed in the last thirty years and fundamental results like the levels for regularity, classification of irreducible modules, and fusions rules among them, have been established. Compared with this, relatively little is known for W-superalgebras. These superalgebras are not just super analogues but contain important objects like N=1,2 supercomformal algebras. Moreover, recent interaction between higher dimensional quantum field theories and vertex superalgebras appearing at boundary leads to a conjectural correspondence between W-algebras and W-superalgebras: beyond the Feigin-Frenkel duality between principal W-algebras, appear dualities between affine cosets of W-algebras and W-superalgebras. The very beginning case is the one between affine $\mathfrak{sl}_2$ and $\mathcal{N}=2$ superconformal algebra whose connection is obtained by Kazama-Suzuki coset construction and its inverse due to Feigin-Semikhatov-Tipunin. This coset construction has been used by Adamovic to study the representation theory of $\mathcal{N}=2$ superconformal algebra. In this talk, we report its generalization to the pair of the subregular W-algebras and the principal W-superalgebras which was originally conjectured by Feigin-Semikhatov. We present the correspondence of their module categories together with intertwining operators via several approach. Based on this example, we then present a general picture on the correspondence between hook type W-algebras and W-superalgebras. This talk is based on joint works (some on-going) with T. Creutzig, N. Genra, A. Linshaw and R. Sato.

De Sole and Kac established an isomorphism between twisted Zhu algebras of W-algebras and finite W-algebras. We will talk about the case of principal W-algebras of osp(1|2n). We show that the (non-twisted) Zhu algebras of principal W-algebras of osp(1|2n) coincide with the even part of the twisted Zhu algebras, and is also isomorphic to the center of the universal enveloping algebra of osp(1|2n).

Let $V$ be a vertex operator algebra and $g=(1, 2,\ldots, k)$ be a $k$-cycle which is viewed as an automorphism of the vertex operator algebra $V^{\otimes k}$. In this talk, I will talk about permutation orbifolds and the associative algebras $A_g(V^{\otimes k})$.

Given a VOA V, the permutation-twisted modules for the tensor product $V^{\otimes k}$ have been systematically studied by Barron-Dong-Mason. However, the study of the intertwining operators (or more generally, the genus-0 conformal blocks) among these twisted modules seems to be more difficult. In this talk, we report an ongoing work to understand these permutation-twisted conformal blocks. Motivated by a work of Barmeier-Schweigert, 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 can be calculated using (1) the fusion rules for untwisted modules (2) the factorization property of untwisted conformal blocks (recently proved by Damiolini-Gibney-Tarasca) (3) the Riemann-Hurwitz formula, which calculates the genus of the branched covering C. If time permits, we also discuss its application to a high genus VOA-conformal net correspondence in the sense of Carpi-Kawahigashi-Longo-Weiner and Tener.

We consider the W* tensor category $\widetilde{\chi}(M)$ of approximately inner and centrally trivial bimodules over a $\rm II_1$ factor $M$, which generalizes the usual notions for automorphisms in Connes' definition of $\chi(M)$. We construct a unitary braiding on $\widetilde{\chi}(M)$ extending Jones' $\kappa$ invariant on $\chi(M)$, solving a problem posed by Popa from 1994. For a non-Gamma finite depth $\rm II_1$ subfactor $N\subset M$, we prove that the unitary braided tensor category $\widetilde{\chi}_{\fus}(M_\infty)\subset \tilde{\chi}(M_\infty)$ of finite depth objects is the Drinfeld center $Z(\mathcal{C})$, where $\mathcal{C}$ is the standard invariant of $N\subset M$, and $M_\infty$ is the inductive limit $\rm II_1$ factor of $N\subset M$. This is joint work with Corey Jones and Quan Chen.

The tensor category of a rational and C_2-cofinite vertex operator algebra is a semi-simple modular tensor category. However most VOAs are neither rational nor C_2-cofinite. These VOAs then have usually an uncountable number of inequivalent simple objects and modules often lack nice finiteness conditions. I want to present the current status of our understanding. I plan to illustrate this in a few instructive examples (they will be associated to affine Lie (super)algebras and W-algebras).

Corresponding to a vertex operator algebra $V$, one can attach an associative algebra called Zhu algebra $A(V)$. There is also another commutative algebra $R(V)$ called $C_2$-algebra. The commutative algebra $R(V)$ is in fact a Poisson algebra. Under a certain finite generation assumption, the $C_2$-algebra $ R(V)$ defines an affine Poisson variety. These Poisson varieties for special vertex operator algebras have been studied by Arakawa and his collaborators. When $V$ is $C_2$-cofinite, the $C_2$ algebra is nilpotent (and thus local) and the corresponding Poisson variety is a single point. In this talk we well study the Yeneda cohomology algebra $ Ext^*_{R(V)}(C,C)$. The motivation of studying the cohomologocal variety arises from Quillen's work on cohomological varieties of finite groups over a field of positive characteristic. Unlike in the Hopf algebra case, the cohomology algebra under Yoneda product is not graded commutative in general. We will give a definition of what the cohomological variety is. For affine rational VOAs, we determine when the algebra $R(V)$ is complete intersections. By approximating the algebra $R(V)$ algebras of complete intersections, we give lower bounds for the dimensions of cohomological varieties of affine rational VOAs. . This is a joint work with Antoine Caradot and Cuipo Jiang.

Quantum subgroups are module categories, which encode the ``higher representation theory'' of the Lie algebras. They appear naturally in mathematical physics, where they correspond to extensions of the Wess-Zumino-Witten models. The classification of these quantum subgroups has been a long-standing open problem. The main issue at hand being the possible existence of exceptional examples. Despite considerable attention from both physicists and mathematicians, full results are only known for sl_2 and sl_3. In this talk I will discuss recent progress in the classification of type II quantum subgroups for sl_n. Our results finish off the classification for n = 4,5,6,7, and pave the way for higher ranks. In particular we discover several exceptional examples.

We will give an overview of the zesting construction on pre-modular categories and explain how it gives rise to different modular categories with the same modular data.

The logarithmic Kazhdan-Lusztig correspondence is a series of conjectural equivalences between quantum groups and logarithmic vertex algebras. In this talk I will discuss joint work with Thomas Creutzig and Simon Lentner where we develop tools which constrain the form of quasi-bialgebras appearing in the logarithmic Kazhdan-Lusztig correspondence and how they apply to the triplet vertex algebra in particular.

For any congruence subgroup $\Gamma$, we study the vertex operator algebra $\Omega^{ch}(\mathbb H,\Gamma)$ constructed from the $\Gamma$-invariant global sections of the chiral de Rham complex on the upper half plane, which are holomorphic at all the cusps. We first present the relations between the invariant chiral differential operators on the upper half plane and modular forms. We then construct a basis of $\Omega^{ch}(\mathbb H,\Gamma)$ in terms of the liftings of modular forms into it, and show that the vertex operations can be expressed via a modification of the Rankin-Cohen brackets of modular forms.

Rigidity of tensor categories plays an important role, in quantum topology and in the representation theory of many algebraic objects, in particular of Hopf algebras and vertex algebras. In this talk, we discuss inherent restrictions of the notion of rigidity. We then explain why rigidity is so useful in the study of bulk fields of conformal field theories.

The definition of a Relative Modular Category (RMC) was first defined by Marco De Renzi. This definition was motivated by the non-semisimple TQFTs defined by myself with Christian Blanchet, Francesco Costantino and Bertrand Patureau-Mirand. This talk will start with an overview of some recent results within the area of RMCs including some relationships with VOAs and QFTs. Then I will recall the definition of a RMC and explain why it includes the definition of a modular category. I will end the talk with some examples and conjectures coming from the Lie superlagebra sl(2,1).