Schedule for: 25w5318 - Recent Developments in Logarithmic Conformal Field Theory

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In the construction of the tensor product of two (untwisted) modules for a vertex operator algebra, the main result is that the subspace of the dual space of the vector space tensor product of the two modules consisting of elements satisfying a $P(z)$-compatibility condition and a $P(z)$-local-grading-restriction condition is the contragredient of the tensor product module. This $P(z)$-compatibility condition plays a crucial role in the proof of the (logarithmic) operator product expansion (associativity) of intertwining operator and the construction of the associativity isomorphism. It was formulated and proved by Lepowsky and me using an algebraic approach. But in the case of twisted modules associated to noncommuting automorphisms, the algebraic formulation does not work anymore. In this talk, I will give an analytic P(z)-compatibility condition formulated by Jishen Du and me. Then I will discuss our main result that the subspace of the dual space of the vector space tensor product of two twisted modules consisting of elements satisfying the analytic P(z)-compatibility condition and a P(z)-local-grading-restriction condition is the contragredient of the tensor product of the two twisted modules.

We discuss the notion of strongly interlocked modules for vertex operator algebras and the properties of the resulting graded pseudo-traces. Several examples of this context will be presented as well.

$W$-algebras form a broad family of vertex algebras built from a simple Lie algebra and parametrised by its nilpotent orbits. They are constructed by applying quantum Hamiltonian reductions to affine vertex algebras, a sophisticated process that remains largely mysterious. Partial and inverse quantum hamiltonian reductions have been introduced to address the intricacies of general reductions. They rely respectively on splitting and inverting the reductions. In this talk, we will explore partial and inverse reductions of a particular type, known as the Virasoro type. Such reductions can be applied to a substantial family of $W$-algebras associated with height-two nilpotent orbits. Moreover, they can be lifted to the universal W-algebra of $\mathfrak{sp}(2)$ type constructed by Creutzig, Kovalchuk and Linshaw. The talk is based on a work in collaboration with V. Kovalchuk and S. Nakatsuka.

The triplet algebra is an extension of the $(1,p)$-model of Virasoro algebra, which is a famous example of $C_2$-cofinite but irrational VOA. Feigin and Tipunin gave a construction and generalization of this algebra to the simply-laced principal $W$-algebras by using VOA bundles over flag varieties. In this talk, we'll generalize their construction for all the W-algebras together with some basic conjectural properties. Then I will explain the case of affine $\mathfrak{sl}2$. The talk is based on my joint work with Thomas Creutzig and Shoma Sugimoto (arXiv: 2306.13568).

In recent years there have been numerous instances where vertex operator algebras have been constructed from the perspective of an auxiliary geometry or topology, which are in turn related to certain quantum field theories in three dimensions. One such example is of that vertex operator algebras supported on the boundary of 3D supersymmetric quantum field theories and which are defined by the chiralisation of a quiver variety. This geometry is physically a moduli space of vacua of the corresponding 3D theory, where the global and local chiralisation procedures reveal hidden structures and relations among the resulting vertex operator algebras. Another intriguing instance occurs in connection to 3D manifolds and a type of q-series topological invariants, which allow associating to these manifolds Logarithmic or Cone VOAs depending on the precise modular structure of the q-series; these vertex operator algebras therefore appear within a complex network of relations. This talk will highlight some recent results from [2312.13363], [2403.14920], as well as upcoming work, together with a series of open questions.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A supersymmetric (SUSY) $W$-algebra, the minimal SUSY vertex algebra containing an ordinary $W$-algebra, can be understood as a quantum Hamiltonian reduction of a SUSY affine vertex algebra. Each SUSY $W$-algebra has a superconformal vector induced from the Kac-Todorov vector of the corresponding SUSY affine algebra and this vector allows to consider the Zhu algebra of SUSY $W$-algebra. This associative algebra is called a finite SUSY W-algebra. In other words, a finite SUSY $W$-algebra is a reduction of a Takiff superalgebra which is the Zhu algebra of a SUSY affine vertex algebra. To investigate the representation theory of SUSY $W$-algebras, one needs to study finite SUSY $W$-algebras and it is closely related to the Whittaker module theory of Takiff algebras. In this talk, I will briefly introduce finite SUSY $W$-algebras and explain about finitely generated module category of principal finite SUSY $W$-algebras using the Whittaker module theory.

Supersymmetric(SUSY) $W$-algebras are vertex algebras obtained via SUSY Hamiltonian reduction based on Lie superalgebras with $\mathfrak{osp}(1|2)$ embeddings. By construction, they possess a supersymmetric structure that naturally couples their free generators. The relationship between the SUSY $W$-algebras and ordinary $W$-algebras was suggested in a conjecture by Madsen and Ragoucy. In the ongoing work, we could have clarified the relationship between the two. Namely, we showed that SUSY $W$-algebras are isomorphic to the $W$-algebras up to a tensor product with certain free field algebras for any non-critical level. In particular, it implies the supersymmetric structure of principal $W$-algebras. In this talk, I will give a brief introduction to SUSY $W$-algebras and the notion of supersymmetries inside vertex algebras. Then, I’ll discuss key properties of SUSY $W$-algebras, a part of which contributes to the proof of the conjecture. This talk is based on the recent joint work with Genra and Suh, as well as the ongoing work with Kac, Linshaw, and Suh.

I will discuss ongoing joint work with Hao Li and Jinwei Yang on the structure of tensor categories of representations for the Virasoro Lie algebra at positive rational central charges $c>25$, and their relations to the quantum group of $\mathfrak{sl}_2$ at roots of unity. As an application, we construct a triplet vertex algebra at any rational central charge $c>25$ which contains the Virasoro vertex algebra of central charge $c$ as a subalgebra and has automorphism group $PSL(2,\mathbb{C})$. We expect that this triplet vertex algebra will have a non-semisimple modular tensor category of representations, and thus may be of interest for logarithmic conformal field theory.

Universal 2-parameter algebras $W_{\infty}$, $W^{ev}_{\infty}$, and $W^{sp_2}_{\infty}$ are classifying objects for vertex algebras of types W(2,3,…), W(2,4,…), and W(1^3,2,3^3,4,…), respectively. In this talk, we present two new such universal structures. The first is W^{so_2}_{\infty}, which has type W(1,2^3,3,4^3,…). We present its 1-parameter quotients, and exhibit it as a gluing of two W^{ev}_{\infty} algebras. The second structure we present is the superconformal W^{N=2}_{\infty} algebra, which has type W(1,2^2,3^3,…;(3/2)^2,(5/2)^2,…). Similarly to W^{so_2}_{\infty}, it is a gluing of two W_{\infty} algebras. However, unlike W^{so_2}_{\infty}, it enjoys two families of Feigin-Frenkel type dualities. In particular, we are able to show Ito’s conjecture, and give a coset realization of W^k(pls_{n|n},f_{n|n-1})^{U(1)}.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

There is a dark side to mathematics: it only celebrates the power of the abstract, shunning the concrete. Great advances can be made, but at the cost of hypotheses that might seem impossible to verify in interesting cases. It is a seductive path, fully capable of launching careers. But, it is not a healthy one. There is a balance that healthy mathematics must seek if it is to stay relevant. I speak not only of examples, but also of applications. This talk will focus on this side, outlining a little of where we are on the road to understanding and where we hope to meet the other side.

I will discuss joint work with Agustina Czenky, where we produce cochain valued TQFTs for surfaces with markings by cochains over a fixed modular tensor category, and corresponding ribbon bordisms. As a primary example, we can consider cochains of modules over a $C_2$-cofinite vertex operator algebra. I will explain how this cochain valued TQFT furthermore localizes to produce a TQFT for derived categories of such modules. If time allows I will remark on relations with Schweigert and Woike's derived conformal blocks, and the possible introduction of local systems over the automorphism groups of such VOA.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

It has been previously noticed that quantum invariants of some Seifert 3-manifolds can be identified with characters of certain logarithmic VOAs. This is closely related to the statement about quantum modularity of the 3-manifold invariants. On the other hand, quantum modularity at roots of unity for hyperbolic 3-manifolds has been also presented as the stronger version of the Volume Conjecture. In the talk I will present a version of the latter statement which universally applies to 3-manifolds modeled by different Thurston geometries.

In this talk, I will discuss recent developments in the theory of $\mathbb{N}$-graded vertex algebras. I will focus on the interplay between rational $\mathbb{N}$-graded vertex operator algebras of the form $\bigoplus_{n\in\mathbb{N}} V_n$ with $\dim V_0 \geq 2$, and rational vertex operator algebras of CFT-type (i.e., those with $\dim V_0 = 1$ and $V_n = 0$ for $n < 0$). This comparison sheds light on the structural and representation-theoretic aspects of $\mathbb{N}$-graded vertex operator algebras $\bigoplus_{n\in\mathbb{N}} V_n$ when $\dim V_0 \geq 2$.

The homological block is a topological invariant of a 3-manifold, which is a certain partition function of analytically continued Chern-Simons theory, and it is closely related to characters of logarithmic conformal field theories. In this talk, we discuss an interpretation of homological blocks in various contexts as half-indices in 3d N=2 supersymmetric quantum field theories through the 3d-3d correspondence.

I'll discuss recent work (with Wenjun Niu and Victor Py) on the representation theory of "meromorphic tensor categories": categories with a coherent family of tensor products (x)_z over the punctured disc, generalizing the notion of OPE in vertex algebras. Physically, these are line operators in 3d holomorphic-topological theories. Alternatively, module categories for vertex algebras that don't have a stress tensor (conformal vector) are expected to have this structure. I'll explain an analogue of a "Kazhdan-Lusztig correspondence" for these categories, relating them to representations of new objects that we call dg-shifted Yangians.

Rigidity is a much studied duality structure which objects in monoidal categories may or may not have. For example in the category of finite dimensional vector spaces over a field k the rigid dual of a vector spaces V is just its standard linear algebra dual space: V^*=Hom(V,k). However, there are many interesting monoidal categories where objects need not admit rigid duals. For example, the category of finite dimensional bimodules over some finite dimensional associative k-algebra. In this talk, I will present recent work on a more general duality structure that goes beyond rigidity called Grothendieck-Verdier or *-autonomous duality and why it is the natural duality structure for categories of vertex operator algebra modules.

I will reveal a relation between (a class of) 2d conformal field theories (CFTs) and symmetric quivers. One manifestation of this relation is equality of characters of CFTs in question, and certain specializations of generating series associated to corresponding quivers, which take the form of the Nahm sums. This equality follows from a correspondence between 2d CFTs and 4d N=2 theories, and specifically from equality of 2d CFT characters and Schur indices of 4d N=2 theories (computed as traces of Kontsevich-Soibelman wall-crossing invariant operators assigned to 4d BPS states). As symmetric quivers encode 3d N=2 theories, all this leads to a web of connections between 2d, 3d, and 4d theories.

The Monster Lie algebra is a quotient of the physical space of the vertex algebra tensor product of a specific rank-2 lattice vertex algebra with the Moonshine module of Frenkel, Lepowsky, and Meurman. In this talk, I will describe elements in the tensor product vertex algebra that project onto generators of gl2 subalgebras corresponding to each imaginary simple root of the Monster Lie algebra. Furthermore, for a fixed imaginary simple root, I will illustrate how the action of the Monster simple group on the Moonshine module induces an orbit of each gl2 subalgebra of the Monster Lie algebra constructed in this way. We conjecture that this Monster action is non-trivial. This talk is based on joint work with Darlayne Addabbo, Lisa Carbone, Elizabeth Jurisich, and Scott H. Murray.

In this work we studied algebraic extensions of the affine $\mathfrak{sl}2$ Kac-Moody Lie algebra. The extension is inspired by the category of weight modules at admissible level of the vertex algebra $V_{k}(\mathfrak{sl}2)$.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We will have a panel with a few researchers at different career stages sharing information that they think can be helpful to junior participants.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.