Welcome!
From October 4 to 8, 2021, the Mathematical Physics group in the Mathematisches Institut at Universität Heidelberg
will host a workshop on topics at the intersection of physics and pure mathematics, with a special
emphasis on algebra, representation theory, and complex and algebraic geometry.
Join
This is a hybrid event. Local participants and onsite speakers meet in the Hörsaal on the ground floor of the
Mathematikon.
Remote speakers and external participants may
Join the Zoom Meeting (ID: 656 720 7054, Passcode: 4AS82e), or
follow the live Stream.
If all else fails, you may
send us an email
Speakers

Drazen Adamovic (University of Zagreb)

Tomoyuki Arakawa (RIMS  Kyoto)

Laurent Baulieu (Sorbonne University and CNRS  Paris)

Nathan Berkovits (UNESP  São Paulo)

Martin Cederwall (Chalmers University  Göteborg)

Cyril Closset (University of Birmingham)

Thomas Creutzig (University of Alberta)

Thomas Dumitrescu (UCLA)

Richard Eager (Yokohama)

Konstantin Eder (Erlangen  Nürnberg)

Matthias Gaberdiel (ETH  Zürich)

Naoki Genra (Kavli IMPU)

Antonio Grassi (Università del Piemonte Orientale)

Owen Gwilliam (UMASS, Amherst)

Thorsten Heidersdorf (University of Bonn)

John Huerta (University of Lisbon)

Branislav Jurco (Charles University  Prague)

Axel Kleinschmidt (MPI  Potsdam)

Thomas Krämer (Humboldt Universität  Berlin)

Pavel Mnëv (University of Notre Dame)

Jakob Palmkvist (Örebro University)

Pavel Safronov (University of Edinburgh)

Volker Schomerus (DESY  Hamburg)

Vera Serganova (UC Berkeley)

Brian Williams (University of Edinburgh)
Description
The notion of a superalgebra has become a central part of both mathematics and physics. It provides a
helpful unifying perspective on many important mathematical constructions, such as polynomial and
exterior or Weyl and Clifford algebras; modern perspectives on Koszul duality and derived geometry
are grounded in the philosophy of replacing naive linear or algebraic objects by differential graded
analogues. A similar perspective took root independently among physicists; the BRST construction
implements gauge symmetries cohomologically using a variant of the ChevalleyEilenberg complex,
whereas the BV formalism goes further and rephrases the sheaf of classical solutions to equations of
motion as a derived critical locus equipped with shifted symplectic structure. In addition,
since a Z/2Z grading by fermion parity is present in any physical system, the study of symmetry
algebras in physics admits natural extensions to Lie superalgebras.
Supersymmetry algebras are a particularly interesting class of such Lie superalgebras, which extend
the spacetime symmetries of a quantum field theory (either the affine or conformal group) by
odd elements transforming in spin representations of the Lorentz group. Supersymmetric field
theories admit natural deformations of the differential, known as twists, by any squarezero
odd element of the supersymmetry algebra. Such twists have been the subject of intense study
for many years, for a variety of reasons: They produce myriads of interesting examples of
holomorphic and topological field theories, which stand in a precise relationship to protected
or BPS quantities in the original field theory. They are naturally derived objects, since
their sheaves of classical solutions (for example, holomorphic sections of bundles or locally
constant sheaves) are resolved in smooth functions. Their algebras of local operators admit
interesting higher operations, controlled by the geometry and topology of analogues of the
little disks operad. In addition, it has been pointed out in recent work that holomorphic
field theories naturally admit actions of centrally extended, infinitedimensional dg Lie
algebras, which enhance flavor and conformal symmetry in the twisted theory and generalize
KacMoody and Virasoro symmetry to higher dimensions.
A central object connecting representation theory of infinite dimensional superalgebras
with quantum field theory and exciting topics of pure mathematics are Vertex operator
algebras (VOAs). VOAs are a mathematical notion of the symmetry algebra of a two dimensional
conformal quantum field theory and as such they always have an action of the Virasoro algebra.
VOAs and their representation categories appear not only as meaningful invariants of higher
dimensional supersymmetric quantum field theories but they are also central in the quantum
geometric Langlands program and as new invariants of three and fourmanifolds. The relevant
VOAs are socalled Wsuperalgebras, which are obtained as certain semiinfinite cohomologies
from the VOAs of affine Lie superalgebras. Their representation theory is far more complicated
than the VOAs usually studied by researchers in the area and in the recent years new effective
methods for their study have been introduced. This is due to a very fruitful interaction
between physicists and mathematicians in many different directions of pure mathematics.
The last couple of years have seen a convergence of interest from various lines of research
related to superalgebras and supersymmetry. For one thing, it was realized that “twists”
are not only powerful tools for understanding sectors of specific supersymmetric theories,
but can also be used to “reconstruct” such theories when considered globally in families
over the space of squarezero elements, or “nilpotence variety.” These spaces are sometimes,
but not always, related to the space of Cartan pure spinors, which lends its name to
the purespinor formalism in physics. It turns out that these techniques can be applied
in any dimension and with any amount of supersymmetry, and Koszul duality predicts close
relations between the derived category of equivariant sheaves on the nilpotence variety and
a category of supermultiplet representations of the supersymmetry algebra, which has yet to
be properly understood. One expects that this geometric perspective can develop further
unifying power, and that wellknown folk theorems in physics (such as the absence of
auxiliaryfield formalisms for maximal supersymmetry) can perhaps be understood or
proved in terms of the algebraic geometry of the relevant nilpotent variety.
This should also offer a new perspective on the representation theory of superconformal
algebras. Unitary representations of the higher Virasoro algebra and other new
infinitedimensional dg Lie algebras remain completely unexplored territory.
Our goal is to bring together experts in quantum field theory, pure spinor techniques,
Lie superalgebras and representations, and derived geometry, and to foster exchange,
collaboration, and the development of shared language between these diverse groups.
Program
All times are given in Central European Time (UTC+1). Talks are divided into morning / afternoon
sessions. We have opted for a long (2.5 hours) lunch break to make it possible for participants
from the Americas to plausibly attend most of the talks in the afternoon session and, at the
same time, to allow enough time to discuss, share ideas but also rest to the in person participants.
Monday, October 4
Time 
Speaker 
Title, Abstract 
9:30  10:20am 
Baulieu 
The power of the 2d Beltrami parametization in gravity and twisted supergravity and its generalization for dimensions larger than 2. 

A suggestive subfoliation of the ArnowittDeserMisner leafs of ddimensional
Lorentzian manifolds is presented, Sigma^{ADM}_{d1}= Sigma_{d3} x Sigma_2.
It defines an interesting covariant "ddimensional Beltrami parametrization”
for the dbein and the dmetric. The "Beltrami dbein" is parametrized by
d(d+1)/2 independent fields belonging to different categories,
each one with a specific interpretation. The Weyl invariant sector
beautifully selects the d(d3)/2 physical local degrees of freedom of
ddimensional gravity.

Coffee break

11:00  11:50am 
Krämer 
The Lie Algebra of Perverse Sheaves on Elliptic Curves 

Perverse sheaves appear in many parts of mathematics and physics such as in geometric
representation theory or in the Langlands program. In the case of abelian varieties,
they form a tensor category which is abstractly equivalent to the representation
category of some affine group scheme. This "Tannaka group of perverse sheaves"
captures a lot of information, but it is huge and hard to approach explicitly,
even for elliptic curves. I will discuss a new way to construct elements in its Lie
algebra via a global enrichment of vanishing cycles (work in progress with Amelie Flatt).

12:00  12:50am 
Kleinschmidt 
The most complicated way of writing D=11 supergravity


Exceptional field theory based on the exceptional group E_n is a way of combining aspects
of supergravity in a duality covariant way. It can make the CremmerJulia E_n symmetry
in 11n dimensions manifest but can also be seen as a way of rewriting the theory
in eleven dimensions, breaking the symmetry in the course of doing so.
I will review recent results where this idea is taken to the limiting case with
KacMoody symmetry E11, highlighting interesting representationtheoretic aspects.
Based on work with Guillaume Bossard and Ergin Sezgin.

Lunch break 
3:30  4:30pm 
Palmkvist 
Nonlinear realisations of Lie superalgebras 

The talk is based on 2012.10954. For any decomposition of a Lie superalgebra G into a
direct sum G=H+E of a subalgebra H and a subspace E, without any further resctrictions
on H and E, we construct a nonlinear realisation of G on E. The result generalises a
theorem by Kantor from Lie algebras to Lie superalgebras.
When G is a differential graded Lie algebra, we show that it gives a
construction of an associated Linfinityalgebra.

Coffee break

5:00  5:50pm 
Mnëv 
Twodimensional BF theory as a conformal field theory 

We study topological BF theory on the complex plane in Lorenz gauge.
In the abelian case, we find that the gaugefixed theory is a Btwisted N=(2,2)
superconformal theory  Witten's Bmodel with a parityreversed target.
The BV algebra structure on 0observables is constructed explicitly using
operator product expansions with the superpartner of the stressenergy tensor.
In the nonabelian case, the theory becomes a logarithmic CFT with correlators
given by convergent integrals (e.g., 4point functions are expressed in terms
of dilogarithms). We find vertex operators in the nonabelian theory,
receiving a quantum correction to conformal dimension. This is a report on a
joint work with Andrey Losev and Donald Youmans, arXiv:1712.01186, arXiv:1902.02738

6:00  6:50pm 
Creutzig 
Nonsemisimple 3dimensional topological field theories 

A true highlight of modern mathematical physics is the construction of invariants of
3manifolds and links using modular tensor categories associated to a compact
Lie group G and a level k. Such a category has three realizations: the category of
Wilson lines in ChernSimons theory, the category of integrable modules of the WZW
2dimensional conformal field theory (or better its underlying vertex algebra),
a category of quantum group modules at root of unity.
Interesting "ChernSimonslike" 3dimensional topological field theories can be
constructed from corner considerations in 4dimensional N=4 super YangMills theory.
These theories have two modern features: they are non semisimple and they can be deformed by
flat connections. As in the semisimple case, the category of line operators of this theory can
also be realized by some vertex algebra and some quantum group.
Both can also be deformed by connections. I will try to explain this emerging picture.
This is a report on joint work with T. Dimofte. N. Garner and N. Geer.

Tuesday, October 5
Time 
Speaker 
Title, Abstract 
9:30  10:20am 
Genra 
FeiginSemikhatov duality 

Feigin and Semikhatov conjectured dualities between subregular Walgebras of sl_n and principal Walgebras of sl_{n1}.
In this talk, we prove these conjectures and study the structure of module categories by means of relative semiinfinite cohomology techniques.
This is a report on a joint work with Thomas Creutzig, Shigenori Nakatsuka and Ryo Sato.

Coffee break

11:00  11:50am 
Adamovic 
Realizations of affine vertex algebras, logarithmic vertex algebras and beyond


We shall first discuss our realization of affine vertex algebra $L_k(sl(2))$
and present some applications to the representation theory. Next, we present some generalizations.
We will study new realizations of certain logarithmic affine vertex algebras appearing in physical
theories (jointly with T. Creutzig, N. Genra and J. Yang). We shall also study a duality
between N=4 superconformal vertex algebra with central charge c=9 and the affine vertex
algebra $L_k(osp(1,2))$ at the critical level (jointly with Q. Wang).

12:00  12:50am 
Cederwall 
SL(5) supersymmetry


The talk is based on arXiv:2107.09037. We consider supersymmetry in five dimensions, where
the fermionic parameters are a 2form under SL(5). Supermultiplets are investigated using
the pure spinor superfield formalism, and are found to be closely related to
infinitedimensional extensions of the supersymmetry algebra: the Borcherds
superalgebra B(E_4), the tensor hierarchy algebra S(E_4) and the exceptional
superalgebra E(5,10). A theorem relating B(E_4) and E(5,10) to all levels is given.

Lunch break 
3:30  4:30pm 
Berkovits 
Manifest Spacetime Supersymmetry and the Superstring 

The algebra of spacetime supersymmetry generators in the RNS formalism for the
superstring closes only up to a picturechanging operation. After adding nonminimal
variables and working in the "large" Hilbert space, the algebra closes without
picturechanging and spacetime supersymmetry can be made manifest.
The resulting nonminimal version of the RNS formalism is related by a field
redefinition to the pure spinor formalism.

Coffee break

5:00  5:50pm 
Eager 
Maximally twisted elevendimensional supergravity 

TBA

6:00  6:50pm 
Dumitrescu 
2Group Global Symmetries in Six Dimensions 

I will review the notion of (continuous) 2group global symmetries,
which mix ordinary and higherform global symmetries, and explain why they
naturally arise in many 6d gauge theories. Most 6d SCFTs and little string
theories have weaklycoupled infrared phases with gauge fields. While nontrivial
2group symmetries are often present in little string theories, I will explain why
they can never arise in unitary SCFTs. This observation can be used to
establish a previously conjectured algorithm for computing ’t Hooft anomalies
of such SCFTs. Finally, I will use this understanding to shed light on
the atheorem in six dimensions. In particular, I will show that all unitary
6d SCFTs have positive aanomaly.

Wednesday, October 6
Time 
Speaker 
Title, Abstract 
9:30  10:20am 
Arakawa 
Rfiltration and Vogan filtration 

The 4D/2D duality discovered by Beem et al. associates a VOA to any 4D N=2 SCFT, whose associated
variety is the Higgs branch of the 4D theory. Beem and Rastelli further states that such a
VOA should carry a filtration, called the Rfiltration, induced from the Rcharge of the 4D
theory. In this talk we present a possible mathematical definition of the Rfiltration.
This is a joint work with Anne Moreau.

Coffee break

11:00  11:50am 
Jurco 
Homological perturbation and homotopy transfer 

We will review the relation between the homologial perturbation, homotopy transfer for
quantum homotopy algberas and the BV path integral. Time permitting we will touch on
the notion of a category of quantum BV field theories suggested by this relation.

12:00  12:50am 
Grassi 
New Cohomologies for Lie Superalgebras


In the talk, we present new results in the computations of cohomologies for Lie Superalgebras.
In particular, we will give the first explicit example of an invariant pseudoform representing a cohomology
class for the ChevalleyEilenberg cohomology for osp(22) superalgebra emerging in N=2 string theory.

Lunch break & Social Activity 
Thursday, October 7
Time 
Speaker 
Title, Abstract 
9:30  10:20am 
Eder 
Super Cartan geometry and applications to (quantum) supergravity 

This talk is devoted to the geometric approach to supergravity. We interpret supergravity
in terms of a super Cartan geometry which provides a link between supergravity and YangMills gauge theory.
To this end, we first review important aspects of the theory of supermanifolds and we establish a link between
various different approaches. We then introduce super Cartan geometries
using the concept of socalled enriched categories. Studying these categories turns out to be mandatory to model
anticommuting classical fermionic fields in mathematical physics.
Then applications of these methods in the context of supergravity will be discussed. For this purpose, we derive
the socalled HolstMacDowellMasouri action of D=4 AdS supergravity for N=1,2 which is a 1parameter
family of deformed supergravity actions. We will show that these actions provide unique boundary terms that ensure
local supersymmetry invariance at boundaries. For certain values of this parameter, we show that the action can be
recast in the form of a constrained super BF theory and the boundary theory is a super ChernSimons theory.
Finally, we will give an outlook on possible applications of these results in the context of quantum supergravity.

Coffee break

11:00  11:50am 
Huerta 
Bundle gerbes on Lie supergroups 

Bundle gerbes are analogues of line bundles important for conformal field theory,
anomalies, and obstruction theory. Among bundle gerbes, a central role is played
by the basic bundle gerbe, an essentially unique gerbe on any compact,
simple and simplyconnected Lie group. In this talk, we describe our work
constructing the basic bundle gerbe for a large family of simple Lie supergroups,
and show how the basic gerbe on a Lie supergroup decomposes into a tensor product
of gerbes on the underlying Lie group and an auxiliary 2form.

12:00  12:50am 
Closset 
The Uplane of rankone 4d N=2 KK theories


I will revisit the SeibergWitten description of the Coulomb branch of rankone 4d N=2
supersymmetric QFTs, including aspects of the global symmetry,
from the point of view of rational elliptic surfaces. This will include,
in particular, a detailed study of the 5d superconformal field theories with E_n symmetry,
compactified on a circle. I will also sketch how to derive BPS quivers from
that perspective. Interesting modular properties of the SW geometries will play a key role.

Lunch break 
3:30  4:30pm 
Williams 
Exceptional Lie algebras from twisted supergravity 

Nontopological twists of supersymmetric gauge theories have played an increasingly
important role in math and physics in part due to relationships to vertex algebras and
quantum groups. On the other hand, motivated by the higher genus Bmodel, twists of
10dimensional theories of supergravity have been characterized.
In this talk, we give a complete description of the maximally nontopological
twist of 11dimensional supergravity, the low energy limit of Mtheory.
I will explain the unexpected result that the global symmetry algebra of
the model is equivalent to an infinitedimensional exceptional super Lie
algebra known as E(5,10). I will also explain the relationship between other
exceptional algebras and extended objects such as M2 and M5 branes in the twisted setting.

Coffee break

5:00  5:50pm 
Gwilliam 
Higher deformation quantization for twists of N=4 supersymmetric YangMills theory 

In work with Elliott and Williams, we constructed the BV quantization of the
holomorphic and topological twists of N=4 SYM in four dimensions,
which include the KapustinWitten theories. The observables of these theories
have interesting mathematical structure: for the topological twists, we showed
they encode algebras over the framed little 4disks operad and hence determine
fully extended TFTs in the sense of BaezDolanLurie. We will discuss the
construction and questions it raises.

6:00  6:50pm 
Serganova 
Volumes of supergrassmannians and splitting subgroups 

The supergrassmannians are compact homogeneous supermanifolds with invariant
volume forms. We determine in which cases the volumes of supergrassmannians are not zero.
Our main tool is the SchwartzZaboronsky localization formula for Berezin integral.
We also discuss applications of this calculation to representations of
superalgebras. In particular, we generalize Green correspondence, well known theorem
in representation theory of finite groups in positive characteristics, to general linear supergroups.

Friday, October 8
Time 
Speaker 
Title, Abstract 
9:30  10:20am 
Heidersdorf 
On recent results in the representation theory of supergroups 

I will give a (biased) survey talk about recent developments in the representation theory of
supergroups and Lie superalgebras. The focus is here on "elementary" questions such as the
computation of dimensions, superdimensions, character formulas and fusion rules of irreducible
representations. It turns out that these have very nonelementary answers (if they are known at all).

Coffee break

11:00  11:50am 
Safronov 
Virtual fundamental classes and BatalinVilkovisky quantization from supersymmetric twists 

Supersymmetric localization allows one to reduce the computation of the partition
function of a supersymmetric theory to a finitedimensional integral.
In this talk I will explain how virtual fundamental classes of (2)shifted
symplectic schemes recently introduced by BorisovJoyce, Pridham and OhThomas arise
from such a supersymmetric localization in the presence of extended (i.e. 0d N=2)
supersymmetry. For instance, this gives a fieldtheoretic origin of the
DT invariants of CY4 manifolds. Similarly, I will explain that spaces of
states in the presence of extended (i.e. 1d N=4) supersymmetry may be
computed in terms of the cohomology of a certain perverse sheaf associated
to (1)shifted symplectic schemes. This is a report on joint work with Brian Williams.

12:00  12:50am 
Gaberdiel 
The string dual of free N=4 SYM


A proposal for the worldsheet string theory that is dual to free N=4 SYM
in 4d is made. It is described by a free field sigma model on the twistor
space of AdS5 x S5, and it exhibits a psu(2,24)_1 affine symmetry.
The theory is a natural generalisation of the corresponding model for
tensionless string theory on AdS3 x S3 whose description involves a free
field realisation of psu(1,12)_1. I will explain how our proposal fits
into the general framework of AdS/CFT, and review the various checks
that have been performed.

Lunch break 
3:30  4:30pm 
Schomerus 
Supergroup ChernSimons Theory 

ChernSimons gauge theory is being studied for a wide range of profound applications in
mathematics and physics. It is of significant interest to consider extensions in which the
gauge field takes values in a Lie superalgebra. Such supergroup ChernSimons theories
appear, for example, by twisting the low energy effective action for an intersection D3 and
NS5 branes. In my talk I will review the framework of combinatorial quantization of Chern
Simons theory and explain how this framework can be adapted for applications to super
algebras. This will give rise to interesting new observables which can be computed by
exploiting the rich representation theory of Lie superalgebras.

Practical Information and Links
Venue, Zoom and YouTube Links
This is a hybridformat event: participation is available both online and in person in Heidelberg.
Talks that will be given in person will be be held in the 'Hörsaal', on the groundfloor of the
Mathematikon and livestreamed.
Zoom: https://us02web.zoom.us/j/6567207054?pwd=TFphTnpBRVJodHlRV3MvSDl4V1ZUZz09
Meeting ID: 656 720 7054
Passcode: 4AS82e
YouTube Channel: https://www.youtube.com/channel/UCsi29QiswqpMuCf1CVN4spw
Accommodation and Provisioning
The Vier Jahreszeiten
hotel is located just next to the Old Bridge,
one of the landmark of the city of Heidelberg, and few steps away from the main attractions of the Old Town.
For a fast connection to the university campus, board route 31
at Universität Platz, headead to Kopfklinik and get off at Bunsengymnasium. The main entrance of the Mathematikon is
accross the street.
Alternatively, cross the Old Bridge and board route 37 at Alte Brücke Nord, headed
to Sportzentrum Nord and get off at Technologiepark. Cross the street and head south,
toward the southern wing of the Mathematikon: the main entrance is right at the next crossroad.
The pleasant walk, crossing the Old Bridge and heading East down the Neckar river, and up Berliner Strasse
takes about 40/45 minutes at a chill pace.
Organizers
For any further information, please send us an
Email
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