Vertex algebras, factorization algebras and applications

17 Jul 2018, 08:00 → 21 Jul 2018, 18:45 Asia/Tokyo

Lecture Hall(1F), Kavli IPMU

Vertex algebras, factorization algebras and applications

Dates: July 17-July 21, 2018

Venue: Lecture Hall, Kavli IPMU

Vertex algebras are fundamental algebraic structures underlying conformal field theories in two dimensions. With the impetus given by the AGT conjecture, the recent couple of years have seen intense activity relating field theories in 2 and 4 dimensions. In particular, there emerged deep relations between vertex algebras and the geometry of smooth 4-dimensional manifolds. For example, classical constructions related to vertex algebras (such as coset models) obtained a new meaning from the 4-dimensional topology point of view (plumbing, Kirby moves etc.).

At the same time, vertex algebras are particular cases of factorization algebras, a concept that makes sense in any number of dimensions and provides a mathematical descriptions of quantum field theories.. Factorization structures and factorization homology also serve as a mathematical language for a local-to-global “integration” formalism in pure mathematics (Non-abelian Poincar ́e duality, recent work of Gaitsgory-Lurie on geometrization of the Weil conjectures for Tamagawa numbers and others). They have also been  crucial tools in representation theory of affile  Lie algebras and quantum groups.

 The emerging relation of factorization structures in various dimensions is an extremely promising area of development, of importance both in mathematics and physics. The conference aims to bring together experts in these related areas as well as younger researchers from Japan and overseas.

 

Invited Speakers:

Alexander Braverman (Univ. of Toronto, Canada)

Andrew Linshaw (Denver Univ. USA)

Benjamin Hennion (Univ. of Paris-Sud, France)

Boris Feigin (Higher School of Economics, Moscow / Kyoto Univ., Japan)  

David Yang (Harvard Univ., USA) 

Dennis Gaitsgory (Harvard Univ., USA)

Emily Cliff (Univ. of Illinois at Urbana-Champaign, USA)

Eric Vasserot (Univ. of Paris Diderot, France)

John Francis (Northwestern Univ., USA)

Kazuya Kawasetsu（Univ. of Melbourne, Australia)

Lin Chen (Harvard Univ., USA)

Quoc Ho (Institute of Science and Technology, Austria)

Sam Raskin (Univ. of Texas at Austin, USA)

Sergei Gukov (Caltech U., USA)

Takahiro Nishinaka (Ritsumeikan Univ., Japan)

Thomas Creutzig (Univ. of Alberta, Canada/ Kyoto Univ.)

Tomoyuki Arakawa(RIMS Kyoto Univ., Japan)

Toshiro Kuwabara (Tsukuba Univ., Japan)

Vadim Schechtman (Univ. of Toulouse, France)

Yakov Kremnitzer (Univ. of Oxford, England)

Yifei Zhao (Harvard Univ., USA)

 

 

Organizers:

B. Feigin (HSE Moscow / Kyoto University),
M. Kapranov (Kavli IPMU),
H. Nakajima (RIMS Kyoto University and Kavli IPMU)

 

Contact : seminar@ipmu.jp
Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), the University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa City, Chiba 277-8583, Japan

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Group photo2.jpg

Tuesday, 17 July

10:00 → 11:00 "Gluing manifolds and Vertex Algebras” By Sergei Gukov (Caltech Univ.)

Given a smooth 4-manifold M4, what algebras act on (generalized) cohomology of instantons on M4? One can tackle this question either directly or take an equivalent route suggested by the (conjectural) existence of super-conformal theory in six dimensions. The latter does not have enough supersymmetry to be defined on an arbitrary 6-manifold, but can be made fully topological on a 4-manifold and holomorphic in the resulting two dimensions. Such topological+holomorphic twist of 6d fivebrane theory leads to a large class of vertex algebras labeled by smooth 4-manifolds, in such a way that different cutting and gluing operations on 4-manifolds lead to their counterparts in VOA[M4] and equivalent ways of constructing 4-manifolds manifest themselves as equivalences of VOAs. This talk is based on the recent and ongoing work with Boris Feigin.

11:00 → 11:30 Break

11:30 → 12:30 "Vertex algebras describing the blowing-up of the 4-manifold" by Boris Feigin (HSE Moscow / Kyoto Univ.)

We describe some strange class of vertex algebras and formulate conjectures about the corresponding tensor categories of representations of quantum groups.

12:30 → 14:30 Lunch time

14:30 → 15:30 "Universal chiral algebras and universal factorization algebras" by Emily Cliff (Univ. of Illinois at Urbana-Champaign)

In this talk we study bridges between the different perspectives offered by vertex algebras, chiral algebras, and factorization algebras. I introduce a notion of etale pullback for factorization spaces and algebras, which allows me to define categories of universal factorization spaces/algebras in any dimension. These families of factorization algebras are equivalent to universal chiral algebras in the same dimension. In particular, when working over curves, both notions are equivalent to quasi-conformal vertex algebras. I discuss examples of universal families of dimension one already appearing in the literature, and also new examples in higher dimensions coming from Hilbert schemes.

15:30 → 16:00 Break

16:00 → 17:00 "Differentiable Chiral Algebras" by Yakov Kremnitzer (Univ. of Oxford)

I will explain how to define O-modules and D-modules
in the setting of differentiable pre-stacks. Using this I will describe a
theory of differentiable chiral and factorization algebras and how to use a differentiable
version of the Beilinson-Drinfeld Grassmannian in order to construct chiral algebras.
This is joint work with Dennis Borisov and Jack Kelly.

Wednesday, 18 July

10:00 → 11:00 By Tomoyuki Arakawa (RIMS, Kyoto Univ.)

11:00 → 11:30 Break

11:30 → 12:30 "Vertex algebras defined over commutative rings and W_{\infty}-algebras" by Andrew Linshaw (Denver Univ.)

I will discuss vertex algebras defined over commutative rings, and as special cases the universal W_{\infty}-algebras of types W(2,3,4,...) and W(2,4,6,...), which are defined over the polynomial ring in two variables. The existence and uniqueness of these algebras was conjectured in the physics literature, and was recently established in my papers arXiv:1710.02275 and arXiv:1805.11031 (joint with S. Kanade). All one-parameter vertex algebras of type W(2,3,...,N) or W(2,4,...,2N) for some N satisfying some mild hypotheses, can be obtained as quotients of these algebras. This includes the principal W-algebras of types A, B, and C, as well as many others arising as cosets of affine vertex algebras inside larger structures. Each of these one-parameter vertex algebras corresponds to a certain curve in the plane, and the intersection points of these curves give rise to nontrivial isomorphisms between these vertex algebras. Finally, we will describe some remarkable infinite families of such curves whose singular points and pairwise intersection points are all rational.

12:30 → 14:30 Lunch time

14:30 → 15:30 By Alexander Braverman (Univ. of Toronto)

15:30 → 16:00 Break

16:00 → 17:00 ”Equivalences at admissible level” By Thomas Creutzig (Univ. of Alberta)

Equivalences of braided tensor categories of modules of W-algebras of simply-laced Lie algebras at admissible levels can be deduced using what we call a W-algebra translation functor together with a coset realization of the W-algebra. I want to explain how this goes and how the findings relate to physics and quantum geometric Langlands.

Thursday, 19 July

09:30 → 10:30 "Vertex algebras associated with hypertoric varieties" by Toshiro Kuwabara (Tsukuba Univ.)

Hypertoric varieties are known as an example of conical symplectic singularities
and their resolutions. Using BRST reduction, we construct a sheaf of (h-adic)
vertex algebras over a family of Poisson deformations of a hypertoric variety.
A conformal vector is constructed explicitly, and we obtain a vertex operator
algebra as a vertex algebra of global sections.
As certain special cases, the construction gives localization of affine W-algebras
of subregular type A of level -N+1, and one of simple affine VOA of type A of level -1.

10:30 → 11:00 Break

11:00 → 12:00 "Relaxed highest-weight modules over affine vertex operator algebras" by Kazuya Kawasetsu （Univ. of Melbourne)

In this talk, we classify and compute characters of N-gradable simple weight modules over non-integrable affine vertex operator algebras using theory of relaxed highest-weight modules
and Mathieu's coherent families. The results have significant applications in the Creutzig-Ridout Verlinde formula
of non-integrable affine vertex operator algebras. As an example, we compute (Grothendieck) fusion rules
for affine sl(3) vertex operator algebra of level -3/2 using the Verlinde type formula.
This is based on joint works with David Ridout and Simon Wood.

12:00 → 12:10 Group photo

12:10 → 14:00 Lunch time

14:00 → 15:00 "Densities and stability via factorization homology" by Quoc Ho (IST Austria)

Using factorization homology, we develop a uniform and conceptual approach for treating homological stability, homological densities, and arithmetic densities for configuration spaces (and generalizations thereof) in algebraic geometry. This categorifies and generalizes the coincidences appearing in the work of Farb-Wolfson-Wood, and in fact, provides a conceptual understanding of these coincidences. Our computation of the stable homological densities also yields rational homotopy types which answer a question posed by Vakil-Wood. Our approach hinges on the study of homological stability of cohomological Chevalley complexes, which is of independent interest.

15:00 → 15:30 Break

15:30 → 17:00 MS Seminar -----"Gelfand--Fuks cohomology for algebraic varieties" by Sam Raskin (Univ. of Texas in Autstin)

This is joint work with Mikhail Kapranov and Anton Koroshkin.

Given a smooth affine algebraic variety over the complex numbers, we prove that the Chevalley-Eilenberg cohomology of its Lie algebra of global vector fields is a topological invariant of the underlying complex manifold and is finite dimensional in every degree. The proof uses methods from factorization homology.

In this talk, we will first explain the case of smooth real manifolds as studied in the 70's (Gelfand, Fuks, Bott--Segal, Haefliger, Guillemin, ...). We will show how to transpose those methods to complex algebraic varieties.

Friday, 20 July

09:30 → 10:30 "C_2-cofinite VOAs from BRST reduction" by Takahiro Nishinaka (Ritsumeikan Univ.)

I will talk about an infinite series of conjecturally C_2-cofinite conformal VOAs with negative central charge, which are obtained by a BRST reduction of the tensor product of simple affine vertex algebras. The simplest example in the series is conjectured to be isomorphic to one of the doublet algebras studied by B. Feigin, E. Feigin and I. Tipunin. I will also give a conjectural formula for their characters. All these conjectures arise from the study of Argyres-Douglas theories in physics.

10:30 → 11:00 Break

11:00 → 12:00 "Gelfand--Fuks cohomology for algebraic varieties" by Benjamin Hennion (Univ. of Paris)

This is joint work with Mikhail Kapranov and Anton Koroshkin.

Given a smooth affine algebraic variety over the complex numbers, we prove that the Chevalley-Eilenberg cohomology of its Lie algebra of global vector fields is a topological invariant of the underlying complex manifold and is finite dimensional in every degree. The proof uses methods from factorization homology.

In this talk, we will first explain the case of smooth real manifolds as studied in the 70's (Gelfand, Fuks, Bott--Segal, Haefliger, Guillemin, ...). We will show how to transpose those methods to complex algebraic varieties.

12:00 → 14:00 Lunch time

14:00 → 15:00 ""The factorization algebra that encodes the quantum group"" by Dennis Gaitsgory (Harvard Univ.)

The fundamental local equivalence for quantum geometric Langlands is
a conjecture that states that the Kazhdan-Lusztig category (for a group G
at level \kappa) is equivalent to the twisted Whittaker category on the affine
Grassmannian (for the Langlands dual group G^L and the dual level \kappa^L).
Since the relationship between G and G^L is expressed combinatorially, in
order to prove this conjecture one has to introduce a combinatorial object
that encodes (or at least approximates) both categories. In this talk we will
describe such combinatorial object. It comes in the guise of a
factorization algebra, denoted \Omega_q. This factorization algebra has
many remarkable features. On the one hand, it encodes the quantum group
(attached to G, with quantum parameter q expressible in terms of \kappa):
namely U_q(n^+)^{Lus} identifies with hyperbolic cohomology of \Omega_q.
On the other hand, \Omega_q, viewed as a geometric object, can be decsribed
explicitly in terms of the Cartan matrix and \kappa, which makes it amenable
for quantum Langlands type comparison.

14:00 Break (Tea)

15:00 → 15:30 Break

15:30 → 16:00 "Semi-inifinite cohomology vs quantum group cohomology" by Lin Chen (Harvard Univ.)

Via the Kazhdan-Lusztig equivalence, we relate various semi-inifinite cohomology functors to cohomology of various versions of quantum Borels.

16:00 → 16:30 "The master chiral algebra and Langlands duality" by David Yang (Harvard Univ.)

Following Gaitsgory, we define the master chiral algebra and discuss some evidence that it provides the equivalence expected in geometric Langlands.

16:30 → 17:00 By Yifei Zhao (Harvard Univ.)

Saturday, 21 July

09:30 → 10:30 By Eric Vasserot (Univ. of Paris Diderot)

10:30 → 10:45 Break

10:45 → 11:45 "Three lines and a bialgebra" by Vadim Schechtman (Univ. of Toulouse)

We will discuss Laplacians and a Lefschetz type decomposition of linear algebra data for perverse sheaves over hyperplane arrangements.
In the factorizable case these objects are closely related to braided bialgebras.

Joint work in progress with M.Kapranov.

11:45 → 12:00 Break

12:00 → 13:00 By John Francis (Northwestern Univ.)