Welcome!
From October 4 to 8, 2021, the Mathematical Physics group in the Mathematisches Institut at Universität
Heidelberg will host a fiveday workshop on recent topics at the intersection of Physics and Geometry. The
workshop is intended to serve as an unofficial satellite workshop of
String Math 2017, held in Hamburg July 24–29.
Confirmed speakers

Matthew Bullimore (Oxford)

Mykola Dedushenko (Pasadena)

Ori Ganor (Berkeley)

Owen Gwilliam (Bonn)

Justin Hilburn (Philadelphia)

Kentaro Hori (Tokyo)

Si Li (Beijing)

Hyenho Lho (Zurich)

Mauricio Romo (Princeton)

Piotr Sułkowski (Warsaw)

Brian Williams (Evanston)

Piljin Yi (Seoul)

Michele del Zotto (Stony Brook)
Participants
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
Thursday, July 20
Time 
Speaker 
Title, Abstract 
9:30am 
Ganor 
Quadratic Reciprocity and Janus Configurations 

Compactification of N=4 SuperYangMills theory on a circle with various
SL(2,Z) duality twists will be discussed. A technique for counting
ground states on a torus will be presented, and simple Wilson loops will
be analyzed as well. The partition function on a mapping torus,
calculated in two different ways, leads to the LandsbergSchaar identity
for quadratic Gauss sums, and generalizations.

Coffee break

11:00am 
Sułkowski 
Knots are quivers? 

I will present a surprising relation between knot invariants
and quiver representation theory, motivated by various string theory
constructions involving BPS states. Consequences of this relation
include the proof of the famous LabastidaMarinoOoguriVafa
conjecture, explicit (and unknown before) formulas for colored HOMFLY
polynomials for various knots, new viewpoint on knot homologies, new
dualities between quivers, and many others. The crucial role in this
relation is play by Apolynomials  which represent certain flat
connections in ChernSimons theory  and their quantization.

Lunch break 
1:30pm 
Bullimore 
The Twisted Hilbert Space of 3d N = 2 Theories


I will discuss a description of 3d N = 2 gauge theories on R x
C with topological twist on a Riemann surface C as a supersymmetric
quantum mechanics on R. I will focus on a mathematical description of
the Hilbert space of supersymmetric ground states, which in general
contains more information than the twisted partition function on S^1 x
C. In particular, I will consider some simple abelian examples and
demonstrate invariance of the Hilbert space under threedimensional
mirror symmetry.

3:00pm 
Yi 
Index, Partition Functions, and Rational Invariants 

In recent years, varieties of indexlike quantitites
have been computed by exact path integral, a.k.a.
the localization. For gauge theories, the path integral
reduces to contour integrals, albeit with many subtleties.
However, it turns out that interpretation of the result,
which should be really called the twisted partition
function rather than the refined index, requires even
more care.
After a cursory review of the recent derivations and
accompanying subtleties, we consider theories with
noncompact Coulomb phases. The rational nature
of the twisted partition function is observed and
physically explained, which also connects to the
rational invariant of the KontsevichSoibelman
wallcrossing algebra. This gives us a nontrivial
tool for extracting the refined and integral index
out of such rational twisted partition functions.
An application to pure YangMills quantum mechanics
solves a decadesold problem of counting D0 bound
states with an Orientifold point. Along the way, we
also resolve a critical conflict between Kac/Smilga
and Staudacher/Pestun, circa 1999~2002, by isolating
the notion of Hsaddles. The latter also proves to
be a universal feature of partition functions in the
high "temperature" limit.

4:15pm 
del Zotto 
Discrete Integrable Systems, Supersymmetric Quantum Mechanics, and
Framed BPS States 

It is possible to understand whether a given BPS spectrum is generated
by a relevant deformation of a 4D N=2 SCFT or of an asymptotically free
theory from the periodicity properties of the corresponding quantum
monodromy.
With the aim of giving a better understanding of the above conjecture,
we revisit the description of framed BPS states of fourdimensional
relativistic quantum field theories with eight conserved supercharges in
terms of supersymmetric quantum mechanics.
We unveil aspects of the deep interrelationship in between the
Seibergdualities of the latter, the discrete symmetries of the theory
in the bulk, and quantum discrete integrable systems.

Friday, July 21
Time 
Speaker 
Title, Abstract 
9:30am 
Williams 
Symmetries of higher dimensional holomorphic field theories 

The most well known examples of holomorphic field theories come from chiral CFTs in complex dimension one. Of the symmetries of these theories, a special role is played by both the KacMoody and Virasoro vertex algebras. In this talk I will discuss generalizations of these symmetries to higher dimensions in the language of factorization algebras. For example, holomorphic gauge theories in complex dimension d (theories depending on the data of a holomorphic connection) generically have the symmetry of a certain dg Lie algebra studied recently by Faonte, Hennion, and Kapranov. My talk will focus on this higher KacMoody algebra and I will relate it to a shadow of a richer OPE algebra. Time permitting, I will discuss a similar picture for the higher Virasoro algebras.

Coffee break

11:00am 
Gwilliam 
ChernSimons theory and Koszul duality 

The observables of perturbative ChernSimons theory naturally form an algebra over the
little 3disks operad, as I will explain. In consequence, line operators form a braided
monoidal category. We then use higher abstract nonsense, notably Koszul duality, to recover
a quantum group with formal parameter. This work in progress is joint with K. Costello and
J. Francis.

Lunch break 
1:30pm 
Lho 
Stable quotient and holomorphic anomaly equations 

I will prove holomorphic anomaly equations for the stable quotient invariant of
Local P^2 in precise form predicted by Bmodel Physics. After that I will also
discuss about the holomorphic anomaly equation for [C3/Z3] and formal quintic invariants.
This talk is based on joint work with Rahul Pandharipande.

3:00pm 
Li 
Quantum master equation and integrable hierarchy 

We establish the BatalinVilkovisky quantization theory for chiral deformation of free
conformal field theories. As an application, we describe a universal approach to intebrable
hierarchies associated to topological Bmodel on CalabiYau geometry. The talk is based on
arXiv: 1612.01292[math.QA] and a joint work in progress with Weiqiang He and Philsang Yoo.

4:15pm 
Hilburn 
Sduality of boundary conditions
for 4d N=4 and ring objects in the geometric Satake category 

In a recent paper Braverman, Finkelberg, and Nakajima showed how one can associate
a ring object in the category of equivariant Dmodules on the affine Grassmannian
Gr_G to a 3d N=4 gauge theory whose cohomology gives the ring of functions on the
Coulomb branch. I will explain a physical context for this result in terms of boundary
conditions to 4d N=4 super YangMills and use this to give a mathematical formulation
of some ideas of Gaiotto.

Saturday, July 22
Time 
Speaker 
Title, Abstract 
9:30am 
Hori 
2d Seiberg duality, with boundary 

I will talk about 2d Seiberg duality and discuss how it may
act on the Btype boundary conditions. The grade restriction
rule, obtained from the hemisphere partition function, plays
an important role. Based on a joint work in progress with Richard Eager.

Coffee break

11:00am 
Dedushenko 
Fourmanifold invariants and 2d CFT 

Recently, it has been conjectured that there exists a class of 2d N=(0,2) SCFTs T[M_4, G],
labeled by a 4manifold M_4 and a Lie group G, which could provide a new type of smooth structure
invariants for M_4. Such theories are given by twisted compactifications of the 6d (2,0) theory on M_4.
We study one of the simplest cases when G=U(1). Even though such T[M_4, U(1)] looks trivial
(in particular, it is free), from the string theory point of view, it is natural to include
certain local defect operators in this theory. They are naturally defined in the UV, while in
the IR description they explicitly depend on the SeibergWitten (SW) invariants of
M_4. T[M_4, U(1)] perspective leads to the equivariant multimonopole generalization of
SW invariants. We study them from several points of view, including the 4d gauge theoretic
approach, and identify the structure of the VOA T[M_4, U(1)]. We also propose some further
directions for generalizations, including the case of nonabelian G.

Lunch break 
1:30pm 
Romo 
AllOrder Volume Conjecture for
Closed 3Manifolds from Complex ChernSimons Theory 

I will review some general aspects of complex ChernSimons theory on hyperbolic
3manifolds, focusing on the case of gauge group G=SL(2,C). After a brief introduction
to the Volume Conjecture (VC), for knot complements and, a very recent mathematical proposal,
for closed hyperbolic 3manifolds, I'll show how complex ChernSimons theory is related with
them and how this connection leads to a novel generalization of the most recently proposed
VC for closed 3manifolds.

Practical Information
Venue
All talks will be held in the 5th floor Conference Room (5/104) in the Southern wing
of the
MATHEMATIKON.
Morning and afternoon coffee/tea and lunches will be served in the 5th floor common room.
Accommodation and Provisioning
The Heidelberger Hof
hotel is conveniently located midway
between the Neuenheimer Feld and the main attractions of old town Heidelberg.
For a fast bus connection to Campus, board route 31 at Bismarckplatz, headed
to Kopfklinik
(not Universitätsplatz!) and get off 5 stops later at Bunsengymnasium. The main
entrance of the
MATHEMATIKON is across the street.
Shops and restaurants are in the middle and Northern wing.
The pleasant walk from Bismarckplatz across the bridge, down the Neckar river, and up Berliner
Strasse takes about 25 minutes.
Registration and Workshop Dinner
There is no registration fee, however we ask that all those interested in attending
the talks send us an
Email ahead of time, so we may have an idea of the number of prospective participants.
(We reserve the right to raise a guard in the unlikely event of a general stampede.)
In your message, please indicate whether or not you are interested in joining us for the
workshop dinner on Friday night. You will be asked for a € 20 contribution for the dinner;
further information to follow.
Organizers
For any further information, please send us an
Email
Sponsors