Workshop Report

Vertex Algebras, Factorization Algebras, andApplications
Mikhail Kapranov
Kavli IPMU Principal Investigator

Hiraku Nakajima
Kavli IPMU Principal Investigator

From July 17 to July 21, 2018, Kavli
IPMU hosted a workshop on Vertex
algebras, factorization algebras, andapplications, organized by B. Feigin(HSE Moscow and Kyoto University),
M. Kapranov (Kavli IPMU), and H.
Nakajima (Kavli IPMU).
Vertex algebras are fundamental
algebraic structures underlying
conformal field theories in twodimensions. With the impetus givenby 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 ofsmooth 4-dimensional manifolds .
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 mathematicaldescriptions of quantum field
theories . Factorization structures andfactorization homology also serve as amathematical language for a local-toglobal integration formalism in puremathematics. They have also served as
crucial tools in representation theoryof affine Lie algebras and quantum

groups. The emerging relation offactorization structures in 2 and 4
dimensions is an extremely promisingarea of development , of importanceboth in mathematics and physics. The
conference brought together experts
in these related areas as well as
younger researchers from Japan andoverseas. The talks of S. Gukov and B.
Feigin were devoted to the exciting
connections between vertex algebra
and smooth 4-manifolds. In particular,classical constructions related tovertex algebras (such as coset models)obtained a new meaning from the4-dimensional topology point of view(plumbing , Kirby moves, etc.).
The talk of T. Arakawa wasdevoted to a new geometrization
approach to vertex algebras basedon the concept of associatedvarieties, which provide naturalvertex Poisson algebras and ordinary
Poisson manifolds. This approachalso establishes a connection to themathematical analysis of Coulomb
branches of 4-dimensional field
theories . A . Braverman, in histalk, explained a wider and moreconceptual categorical framework forsome of these questions. The talk of T.

Kuwabara, devoted to chiralizationof hypertoric manifolds, made several
steps in the opposite direction, goingfrom certain Poisson manifolds tovertex algebras.
D. Gaitsgory presented a vertex
algebra approach to the quantum
Langlands correspondence,
understood as equivalence of
categories of twisted D-modules.
In the talks of S. Raskin, L. Chen,
and D. Yang, several aspects ofthe connection between vertex
algebras, affine Lie algebras, andquantum groups were investigated,with applications to the Langlands
correspondence playing an importantrole.
The talks of T. Creutzig, K.
Kawasetsu and T. Nishinaka weredevoted to various aspects ofvertex algebras coming from affine
Lie algebras at admissible level.
A . Linshaw talked about realizing
W-algebras as specializations of vertex
algebras depending on parameters.
Various applications of factorization
structures (including those in higher
dimensions) were given in the talks of
E. Cliff, J. Francis, B. Hennion, Q. Ho,
Y. Kremnitzer, and V. Schechtman.
Relations with Hall algebras (whichplay a role in some mathematicalaspects of the AGT conjectures) werediscussed by E. Vasserot.

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Workshop Report

IGM2018: Revealing Cosmology and Reionization
History with the Intergalactic Medium
Khee-Gan Lee
Kavli IPMU Assistant Professor

On September 18−21, 2018 ,the 4-day workshop titled
IGM2018: Revealing Cosmology
and Reionization History with theIntergalactic Medium was heldat the Kavli IPMU. The goal of thisworkshop was to bring theorists
and observers working on theintergalactic medium (IGM) acrossa range of cosmic history andobservational methods to discuss
the interplay between cosmology
and reionization history. Our invited
speakers ranged from experts on thecosmic microwave background, highredshift galaxy observations, quasarobservations, Lyman-alpha forest
cosmology, and 21cm studies. Over100 participants showed up fromboth within Japan and also all over
the world including Asia, Europe andNorth America.
An intriguing development inrecent years is that the optical depth
to electron scattering as measured
through the cosmic microwave
background has been creeping lower,
implying that hydrogen reionization
occurred later than previously
assumed, with a midpoint aroundz = 7.7. This means that the most
distant quasars, and the foreground
IGM absorption seen in their spectra,
are now directly probing the epochof reionization! Indeed, Fred Davies(UCSB) demonstrated direct neutral
fraction constraints from thehydrogen damping wing in the most
distant (z = 7.5) quasar spectrum,

which bodes well for the exciting
possibility of directly measuring theevolution of the neutral fraction
across the epoch of reionization
with future samples of high-redshiftquasars. Xiaohui Fan (Arizona), in
particular described ambitious efforts
to find more high-redshift quasars atz > 7.
At slightly lower redshift, we
heard about efforts to understandthe nature of optically-thick GunnPeterson troughs at z ~ 5.5, whenhydrogen reionization should havebeen completed. George Becker(UC Riverside) showed narrow-band
imaging results from Subaru Hyper
Suprime-Cam indicating a deficit ofLyman-alpha emitters near these dark
troughs, which could be interpreted
as being due to UV backgroundfluctuations or even pockets of latereionization as advocated by Laura
Keating (CITA).
The effects of reionization persist
to lower redshifts especially in thetemperature and pressure evolutionof the IGM. Elisa Boera (UC Riverside)
discussed constraints on the thermal
properties of the IGM using theLyman-alpha forest power spectrum

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Kavli IPMU News No. 43 September 2018at z ~ 4−5, while Michael Walther(UCSB) showed impressive forest
power spectrum constraints over 12
Gigayears of cosmic time fromz ~ 0−5. We also had presentations
on cosmological constraints using
the IGM, first via a comprehensive
review by Andreu Font-Ribera (UCL)on the use of Lyman-alpha forest
baryon oscillation signal to early
expansion history of the universe.
Nathalie Palanque-Delabrouille (CEASaclay) then presented 1D powerspectrum constraints on neutrino
masses (from z ~ 2−3 data) as wellas warm dark matter and sterile
neutrino masses from z ~ 4–5
data. During the discussions it was
clear that the understanding ofreionization and thermal history of
the IGM is becoming a high priority
for mitigating this systematic in
Lyman-alpha forest cosmology.
Overall it was a successful
workshop, and the participants
remarked on the multi-disciplinary
nature which allowed for discussions
between scientists who would notusually encounter each other atmeetings.