# Snowmass Whitepaper: Physical Mathematics 2021

Ibrahima Bah1, Daniel Freed 2, Gregory W. Moore3,
Nikita Nekrasov4, Shlomo S. Razamat5, Sakura Schäfer-Nameki6

###### Abstract

This is a Snowmass whitepaper on physical mathematics. It briefly summarizes
and highlights some of the key questions drawn from a much more extensive
essay, by the same authors, entitled, “A Panorama Of Physical Mathematics
2021.” Version: August 27, 2024.

1 Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD
21218, USA
2 Department of Mathematics, University of Texas at Austin
3 NHETC and Department of Physics and Astronomy, Rutgers University
4 Simons Center for Geometry and Physics,
Stony Brook University, Stony Brook, NY 11794-3636, USA
5 Department of Physics, Technion, Haifa, 32000, Israel
6 Mathematical Institute,University of Oxford, Oxford OX2 6GG, United Kingdom

##  1 General Remarks

The history of the interactions between physics and mathematics is old and
venerable. Physics cannot flourish without mathematics and much of mathematics
takes its inspiration from physics. The inward bound trajectory of twentieth
century physics towards the discovery of the most fundamental laws of physics
resulted in the creation of quantum field theory and string theory, hereafter
abbreviated as QFT/ST. But, while QFT/ST is a revelation of twentieth century
scientists, well into the twenty-first century it is widely recognized as
being far from fully understood. The implications and ramifications of QFT/ST
leave much room for contributions by twenty-first century scientists. The
investigations into QFT/ST have made use of ever more sophisticated
mathematics, including cutting edge mathematics at the focus of present day
research. Conversely, many developments in QFT/ST have also led to profound
new insights, constructions, and even entire subfields of mathematics (such as
vertex operator algebra theory, or homological mirror symmetry – to choose but
two examples, out of very many). A community of scientists, involving both
mathematicians and physicists, is vigorously engaged in the pursuit of QFT/ST
and its relation to mathematics. In this community there has been an important
shift in viewpoint and emphasis from more traditional deparmentalizations.
There is a dual and equal emphasis on both the discovery of the fundamental
laws of nature as well as on mathematical discovery. This field of
intellectual endeavor has, on occasion, been called _physical mathematics_ and
we will adopt that term here for lack of a better designation. “Physical
mathematics” is a subfield of the much broader field of mathematical physics.

The essay “A Panorama Of Physical Mathematics 2021,” – hereafter referred to
simply as “the Essay” – by same authors as those of this whitepaper describes
a partial snapshot, or aerial view, or panorama of the subject as it stands in
2021. The Essay is forward-looking: After recalling the status of the subject,
some open problems that have the potential to lead to future progress have
been identified. In what follows we briefly summarize some of the main points
from the Essay. The interested reader is urged to consult the more extended
text, where a small sampling of relevant references may be found. Here we
highlight just some of the main areas of progress, and some promising future
directions. A more comprehensive – but still quite incomplete – discussion can
be found in the Essay.

##  2 Foundational Aspects of QFT

###  2.1 Overview

Quantum Field Theory (QFT) is the most successful tool in existence for
describing the fundamental laws of nature. The success - in both high energy
physics and in condensed matter theory - has been nothing short of
spectacular, leading to some of the most precise, experimentally verified,
predictions in all of science. It would appear that not only is the Book of
Nature written in the language of mathematics, but that the dialect is that of
Quantum Field Theory. Nevertheless, the vast array of phenomena QFT describes
is nowhere close to being fully fathomed. Moreover, from a mathematical
viewpoint, even giving a fully satisfying definition of what a QFT is remains
open.

Some classes of quantum field theories can be investigated with full rigor.
These include topological quantum field theories (TQFTs), for which there is
now a well-developed mathematical theory. Nevertheless, even some important
and standard quantum field theories that are generally regarded as TQFTs,
namely, those of “cohomological type,” often do not fit the rigid mathematical
framework proposed by mathematicians. 111A TQFT is said to be of
_cohomological type_ if it is a subsector of a metric-dependent QFT defined by
taking invariants with respect to an odd nilpotent symmetry. These are to be
distinguished from theories such as Chern-Simons theories, where the classical
action is metric-independent. Yet the cohomological type TQFT’s often lead to
some of the most interesting mathematical applications including the discovery
of new invariants of 3- and 4-dimensional manifolds and new viewpoints and
methods in the Geometric Langlands Program.

Another important set of examples of quantum field theory are the two-
dimensional conformal field theories. At least in the case of two-dimensional
rational conformal field theories, these are mathematically completely
rigorous, thanks to the well-developed mathematics of vertex operator algebras
and the representation theory of loop groups.

Looking beyond the above examples, there are relatively new mathematical axiom
systems for Wick-rotated quantum field theories on Riemannian manifolds,
roughly corresponding to the Heisenberg and Schrödinger pictures. Much remains
to be done to develop these ideas and connect to physical theories of
interest. One crucial aspect of QFT, that mathematicians have yet to come to
grips with in a fully satisfactory way, is that many QFTs have infinitely many
degrees of freedom. Related to this, as has been clear from the very
beginnings of the subject in the 1930’s, is the need to deal with apparent
infinities that arise in computations.

Other important QFTs, often those connected to self-dual theories, are not
satisfactorily defined even by the standards of textbook QFT. Again, these
theories are often bound up with some of the most interesting new
developments. These developments include algebraic structures generalizing the
theory of vertex operator algebras, new categorical structures, connections to
the geometry of moduli spaces of general (anti-)self-dual gauge/Higgs fields,
Hitchin systems, hyperkähler and quaternionic kähler geometry, remarkable
aspects of integrable systems, deeper insights into – and generalizations of –
exact WKB theory, and more. Many of these developments are connected with the
formulation and discovery of exact results that hold even in the context of
difficult, strongly coupled, interacting theories. Often supersymmetry is
involved, and often the results involve quantities such as BPS states, that
preserve some of the supersymmetry.

The full extent of the possibilities offered by quantum field theories is
mind-boggling. Naturally, one wishes to impose some order on the apparent
chaos, and understand the laws that govern the possible laws of nature.
Ultimately one would like to understand something like the “space of quantum
field theories.” We are a long way from doing that, but some interesting
progress on this difficult topic is being made. Some of the issues regarding
classification of the kinds of theories that have been at the heart of
physical mathematics are addressed in the Essay. Given the depth of the
mathematics involved in TQFT and two-dimensional rational conformal field
theory we have every reason to expect that a full understanding of QFT will
lead to novel and profound mathematical structures.

###  2.2 Recent Progress And Future Directions

#### What Is A QFT?

As we have noted, one of the most pressing challenges is to understand what,
precisely, is the definition of a QFT. No universally applicable systematic or
axiomatic definition exists. Then, one would like to understand what
properties QFTs can have. For example, what is a suitably general notion of a
symmetry of a QFT? Furthermore, we would like to be able to do explicit
computations of the spectrum of physically relevant operators and of
correlation functions and partition functions, including in the presence of
boundaries and defects. Moreover, the connection with string theory has shown
that traditional textbook views of what constitutes a QFT vastly
underestimated the collection of QFTs which are expected to exist. The
geometric approach, discussed later on, will provide some hints about how one
might develop a classification program in tandem with string theory.

#### Topological Quantum Field Theory.

As noted above, the most mathematically developed QFT’s are the _topological_
quantum field theories (TQFTs). In recent years there has been a vigorous
development of this topic, and mathematicians have extended these ideas in
very sophisticated ways. There are deep applications to disparate problems in
topology, symplectic geometry, higher algebra, geometric representation
theory, number theory, and beyond. Although the subject has matured, there
remain many important open questions. For example, the study of defects and
boundary conditions has much future potential, as does the development of
partially defined TQFT’s. (An $n$-dimensional TQFT is said to be _fully
extended_ or _fully local_ if $(n-k)$ categories can be associated to all
compact $k$-manifolds with corners. In the partially defined case this is only
possible for a certain range of $k$. Some of the most important TQFT’s are in
fact only partially defined.)

TQFT’s have many applications in QFT, string theory, and condensed matter
theory. Motivated (in part) by these applications there has been recent
progress in the theory of anomalies and the related classification of
invertible field theories (including nontopological theories). Novel anomalies
constrain the dynamics of QFT’s, and the topological methods arising from the
classification of invertible phases are being used in general QFT as well.
Much remains to be done in these directions, and there are also many important
classification questions which persist.

#### Algebraic Structures.

The study of the operator product expansion in QFT has led to the discovery of
many important algebraic structures. One of the most famous is vertex operator
algebra theory, which can be traced to the operator product in two-dimensional
conformal field theory. Applying techniques of topological, and holomorphic-
topological twists to higher dimensional theories has led to many remarkable
generalizations of vertex operator algebras. Another formalization of current
interest is the entire subject of factorization algebras.

#### Generalizations Of Symmetries.

Another important development of the past few years is an increasingly
improved understanding of various generalizations of the notion of symmetry.
One physical application of these generalizations has been a set of new
insights into the phase structure of certain strongly coupled QFTs, together
with new constraints on the IR dynamics of these theories. These applications
make use of anomalies for these generalized symmetries. One generalization of
global symmetries which is being intensively studied are the higher-form
symmetries, where charged objects can be associated to positive dimension
submanifolds of spacetime. The higher-form symmetries might not form a
traditional group structure but rather might involve generalizations of groups
known as “higher groups,” or – even more generally – “categorical symmetries.”

This multitude of generalized notions of symmetries are the subject of intense
current scrutiny and are expected to play a fundamental role in future
formulations of, and applications of, QFT. One notable application of higher-
form symmetries has been to confinement in non-abelian Yang-Mills theory.
Another, possibly related, fruitful direction of research continues to be the
study of both $\infty$-structures and BV geometry. But the full scope of the
physical implications of generalized symmetries remains to be seen, and
promises to be far-reaching.

#### Integrability.

There are several points of contact between QFTs and integrable systems.
Starting with the integrable quantum field theories in $1+1$ dimensions, a
novel type of integrability emerges in the planar limit of 4d $\mathcal{N}=4$
Super-Yang Mills theory. This connection led to a vast generalization of the
types of integrable systems and techniques to solve them. Another point of
contact with quantum integrable systems arises in the study of supersymmetric
vacua of gauge theories with lower supersymmetry and in various spacetime
dimensions. The search for a universal quantum algebra acting between these
vacua might bring yet another generalization of the notion of symmetry in QFT.

More broadly, integrability and solvability can be used as a fundamental
organizing theme for the study of all of physical mathematics, and this
viewpoint suggests an astonishing number of new directions for research. Many
of these are discussed in some detail in section 10 of the Essay.

##  3 Interactions Between Condensed Matter Physics, QFT, and Higher
Mathematics

###  3.1 Overview

An amazing aspect of physical mathematics is that, while much of the initial
impetus was tied to string theory and high energy particle physics of the
highest energies imaginable, some of the same mathematical structures turn out
to be of use in questions involving condensed matter physics at the lowest
energies imaginable. An old example of this is elementary homotopy theory.
This turned out to be of use in discussions of solitons in QFT and that
application inspired the use of homotopy theory in describing point, line, and
surface defects in real materials. Conversely, mathematics used to describe
crystals and crystallographic groups, and even quasicrystals, found useful
applications in perturbative string compactifications such as toroidal
compactifications and orbifolds thereof. In the past several decades there has
been an explosion of interest in the condensed matter community in
applications of topology to condensed matter physics. For example, the
(fractional) quantum Hall effect has been an enduring source of amazing
physical phenomena related to deep mathematical constructions involving Chern
classes, Chern-Simons theory, lattice theory, and noncommutative geometry.
Applications of modular tensor categories to the study of two dimensional
conformal field theory led to applications to anyon physics, quantum
computing, and quantum information theory.

###  3.2 Recent Progress And Future Directions

A more recent example of applications of topological ideas in condensed matter
physics is the study of topological insulators, topological superconductors,
and Weyl semimetals. For example the classification of crystalline band
structure insulators involves twisted equivariant K-theory, and nontrivial
K-theoretic torsion invariants associated to topological insulators have led
to striking experimental predictions, which have been dramatically confirmed
in the laboratory. Relations between the topological invariants of an
insulator with properties of its boundary modes, when it has a boundary, have
been an important theme of research, but more needs to be done here. An
interesting question for the future is whether other twistings of K-theory, or
the related subject of “coarse geometry,” will find a natural home in
condensed matter theory.

The study of TQFT and its relation to bordism theory has proven to be of great
use in the study of short-range entangled phases of matter and quantum
information theory. Much remains to be done in this program of classifying
phases of matter. Another fertile area for investigation is the detailed
relation between topological phases associated to discrete models (such as
lattice models) and continuum field theories (which are possibly topological).
Recently fracton models have been under intense study since they pose some
interesting challenges in this area. These fracton models raise interesting
questions about relations of foliation theory to QFT.

While these topics constitute applications to condensed matter physics they
should also be considered applications to mathematics, because the
applications make use of very sophisticated mathematics and the questions
inspired by the applications are in turn driving further developments in the
relevant mathematics.

##  4 Low Dimensional Topology And Manifold Invariants

###  4.1 Overview

Some of the most dramatic developments in physical mathematics have been
connected to both differential geometry and low dimensional topology. Here we
highlight some of the most important developments.

One of the most surprising and notable developments is the entire subject of
(homological) mirror symmetry in the context of enumerative algebraic geometry
and symplectic topology. This important area of mathematics has its roots in
the computation of string worldsheet instanton effects in Calabi-Yau
compactification of string theory. Another dramatic and important development
is the relation between intersection theory on the moduli space of curves and
matrix models. This development too has roots in physics. It came as a
synthesis of ideas of topological field theory with the matrix model
formulations of two-dimensional quantum gravity. More recently, there have
been numerous physically-inspired generalizations of (homological) invariants
of knots and links in three-manifolds, along with new invariants of three-
manifolds. Finally, in one of the paradigmatic tales of the development of
physical mathematics the physical insights of instantons in gauge theory
together with topological field theory have led to dramatic developments in
the understanding of the differential topology of four-manifolds.

Remarkable progress has been made in Thurston’s geometrization conjecture (now
a theorem) for three-manifolds. This progress has points of contact with
string theory, the renormalization group flow of sigma models, and the use of
functionals with monotonic behavior under the renormalization group.

Finally, the constructions of manifolds of special (including exceptional)
holonomy, and the exploration of their moduli spaces of such structures, has
seen dramatic progress, e.g. with concrete constructions of families of
compact exceptional holonomy spaces.

###  4.2 Recent Progress And Future Directions

All of the above subjects continue to be a source of new ideas and progress:

The relation of matrix models to intersection theory on moduli spaces has
interesting relations to “geometric recursion” and “topological recursion,” as
well as to models of two-dimensional gravity. One would expect that there are
analogs of these relations for super-Riemann surfaces. These are all topics of
current research.

(Homological) mirror symmetry has been given rigorous proofs in special
examples but remains conjectural in many cases. The full scope of the
conjecture continues to be a source of research. The case of three-dimensional
“mirror symmetry” is a very vigorous area of mathematical research. Via the
connection with the Geometric Langlands Program there are many important
connections to geometric representation theory.

Homological invariants of knots and links have remarkable connections to
conformal field theories, Landau-Ginzburg theories, monopole moduli spaces,
integrable models, noncompact Chern-Simons theory, and more. The intricate
connections between all these topics continue to be actively pursued.

Although the progress in Thurston’s geometrization conjecture makes use of
many ideas from physics a truly satisfying synthesis of the subject with
physical models is still a topic of current research. There is still no really
workable classification of three-manifolds. It remains to be seen if physical
ideas can change the situation.

The differential topology of four-manifolds remains one of the most
challenging subjects in low dimensional topology. While supersymmetric gauge
theory has led to astonishing progress in this area important problems, such
as the 11/8 conjecture and the smooth Poincaré conjecture remain open. It is
natural to wonder if the physics of supersymmetric QFT/ST will lead to yet
more invariants of the diffeomorphism type of four manifolds. It is also
natural to wonder if QFT/ST can inform us of the structure of ${\rm BDiff}(X)$
for a four-manifold $X$. Finally, there is much work in the mathematics
community on various flavors of Floer theory, but this work has drifted from
the main attention of physicists, and it would be desirable to see more
interaction between the physics and math communities in this subject.

The progress related to manifolds of special holonomy and their moduli spaces
in particular in the exceptional holonomy case remains in general a challenge.
These questions are closely connected to the geometrization of QFTs and is
covered in the next section below.

##  5 Geometrization Of QFTs

###  5.1 Overview: String Theory And M-Theory

Like QFT, string theory and M-theory are subjects about which enough is known
to be recognized as mathematical entities which we would like to define
precisely. But a suitable definition would seem to be a long way off.
Nevertheless, we know enough now to engage in remarkable dialogues with pure
mathematics.

Even the traditional perturbative formulation of string theories involves
extremely subtle aspects of the algebraic geometry of supermanifolds, and open
problems in this field remain the focus of modern research. In a different
direction, some reasonable-looking formulations of QFT and string theory turn
out to be mathematically inconsistent due to a precisely defined notion of an
anomaly. This insight of the 1970’s and 1980’s has blossomed into a vigorous
mathematical study of the geometrical formulation of anomalies and its
connection to invertible field theories. The subject is being developed for
its own sake. It also has important applications to the question of the
consistency of string theory compactifications. Moreover, as noted above,
there are applications to the dynamics of QFT and the classification of phases
of matter.

Closely related to this, some modern ideas regarding quantum gravity have led
to new mathematical conjectures about moduli spaces of e.g. Calabi-Yau
manifolds. They have also raised some interesting questions related to summing
over topologies in quantum gravity, yielding new insights into the relation of
matrix models to moduli spaces of Riemann surfaces. Some of these developments
have raised fundamental issues about what the proper meaning of holography
should be. For example, should string theory be dual to a single QFT or an
ensemble average of QFTs ?

###  5.2 Recent Progress And Future Directions

In addition to giving proper definitions of QFTs one would like to compute.
Therefore, one would like to develop mathematical tools to study strong-
coupling regimes quantitatively. Recent years have seen dramatic progress in
numerous approaches to tackling this question, many of which have an
underlying connection with geometry and the subject of geometric engineering.
Broadly speaking, geometric engineering is the realization of QFTs in the
context of string compactifications on non-compact spaces with special
holonomy, (after taking suitable limits). The mathematical fruits of geometric
engineering include new enumerative invariants, as well as new differential
geometric results following from holography and string compactification.

#### Classification Program Of SCFTs.

String theory predicts the existence of superconformal field theories in 6d
and 5d, which are intrinsically strongly-coupled UV fixed points – and thus
not accessible using standard perturbative QFT. In some sense string theory is
an amazing laboratory in which we can construct QFTs and CFTs, whose existence
would be missed in the textbook view of QFT. This “experimental” construction
complements the axiomatic approaches to defining and understanding QFTs and
CFTs. The full classification of 6d, 5d, and 4d SCFTs is probably out of reach
in the near future, but the more narrowly defined class of those which can be
geometrically engineered is probably susceptible to a rigorous classification
program. Even this more narrowly defined classification remains a great
challenge, but one which is reasonable to pursue, and which will likely lead
to rapid progress. The realization of 6d and 5d CFTs in F-theory and M-theory,
respectively, gives a direct connection between QFTs and so-called canonical
singularities in algebraic geometry. In 6d this program has been completed,
but a mathematically precise and detailed analysis of the Mori minimal model
program in algebraic geometry, applied to canonical three-fold singularities,
has yet to be carried out to achieve a classification of 5d SCFTs. Much of
this interconnects with the structure of generalized symmetries and
enumerative invariants, which play an essential role in characterizing the
physical properties of these strongly-coupled QFTs. Another direction that has
seen substantial progress in the past years is the charaterization of the
(quantum) moduli spaces of theories with 8 supercharges, and their relation to
so-called magnetic quivers and hyperkähler singularities.

A huge remaining challenge is the classification of 4d SCFTs, in particular
those with $\mathcal{N}=1$ supersymmetry. Geometrically their realization is
either in terms of so-called $G_{2}$ manifolds in M-theory or elliptic Calabi-
Yau four-folds in F-theory. In contrast to the singular Calabi-Yau three-fold
geometries underlying 5d and 6d SCFTs, these geometries provide a far larger
challenge in mathematics. This applies both to the Mori minimal model program,
but also the challenges that exceptional holonomy spaces pose in differential
geometry. Interconnected with that are the enumerative invariants, which
determine the low energy effective theory of the string theory compactified on
such spaces. This, in turn, opens up challenging questions regarding
enumerative invariants related to – for example – the counting of associative
three-cycles in manifolds of $G_{2}$ holonomy.

Geometric constructions of QFTs in string theory, relate naturally with other
(inter)faces of physical mathematics, e.g., as discussed below, enumerative
invariants from counting BPS states are related to analytic number theory. But
they are also related to algebraic geometry and the theory of generalized
symmetries. In this way physical insights can lead to surprising
interconnections within pure mathematics itself.

#### Compactification From 6d.

Starting from 6d, compactification results in new, often strongly coupled,
theories in lower dimensions. The Class ${\cal S}$ (and its less
supersymmetric counsins) construction is the main paradigm, where the geometry
of a Riemann surface determines the properties of the 4d supersymmetric QFT
(SQFT). Similar connections between three-manifolds and four-manifolds and
their invariants have been developed in the context of reductions to 3d and
2d. For example, the 3d-3d correspondence has led to a wealth of insights into
new constructions of 3d supersymmetric field theories and their relations to
3-manifold topology. Using the geometric constructions one can systematically
understand and classify complicated strong coupling phenomena in lower
dimensions such as dualities of different kinds and the emergence of symmetry.
A goal of this program can be stated as an attempt to find the most general
lower dimensional QFT. An example of a question one can try to address is
whether any (S)CFT in dimensions lower than five can be constructed by
deforming a Gaussian fixed point in the UV and whether all conceivable (S)CFTs
can be found in one of these geometric setups.

#### Holography And Geometry.

Holography is yet another central framework for exploring and engineering QFTs
beyound the Lagrangian paradigm. In holography, the classification question
can be made precise and mathematically rigorous by reformulating it as the
space of AdS solutions, of various dimensions, in string theory. Such spaces
can be defined as the set of solutions of some PDEs such as Monge-Ampère
equations obtained from the Einstein equations of M-theory. An ambitious goal
is to study the different classes of PDEs from Einstein equations in
supergravity that can define the space of QFTs, and understand the various
mathematical tools that either exist or must be developed to characterize
their solution spaces. Often results in holography provides complementary
points of view to the geometric engineering perspectives above. It is also
important to characterize how various systems of PDEs encode the data of QFTs
whether it is in their Lagrangian formulations, in their geometric engineering
in ST or from compactifications from 6d.

##  6 Relations To Number Theory

###  6.1 Overview

Naively, one might imagine that number theory has little to do with QFT/ST.
Nevertheless, there turn out to be many aspects of number theory that appear
to be closely related to questions in QFT/ST. One of the most common ways in
which number theoretic questions arise comes about when one studies partition
functions, or enumerates protected operators, or BPS states. Then aspects of
analytic number theory, and in particular the theory of automorphic forms,
become quite relevant. Two examples are, first, expressions of geometric
symmetry, or duality, of (supersymmetric) theories through the automorphic
properties of their partition functions, and, second, connections of BPS state
counting functions to Poincaré series, Rademacher summability, and Eichler
cohomology. Some of these connections to analytic number theory have led to
generalized notions of automorphy such as mock modularity and quantum
modularity, as discussed below.

There are other ways in which ideas of number theory appear to be relevant.
One example is the relation between the attractor mechanism for the
construction of supersymmetric black holes (or flux vacua) and Calabi-Yau
manifolds with special Hodge structures. Related to this are curious relations
of the attractor mechanism, as well as generalizations of string theory over
finite fields, to some arithmetic aspects of Calabi-Yau manifolds. Perhaps the
most compelling of the bridges to number theory has been the relation of
supersymmetric quantum field theory with the Geometric Langlands Program. In
the past few years other intriguing ideas for connections between number
theory and QFT have been proposed including “arithmetic QFT” and “motivic
QFT.”

###  6.2 Recent Progress and Future Directions

#### Enumerative Invariants.

Counting curves, or more generally calibrated cycles, in string
compactifications sometimes results in partition functions that have
intriguing number theoretic properties. A good example is given by K3
surfaces. A great deal of work is being done on enumerative algebraic geometry
related to physical models. For example the study of Donaldson-Thomas
invariants, which count certain wrapped brane states in string theory, is a
very active field in mathematics that has seen a lot of recent progress.
Recent progress includes generalizations to higher-dimensional and/or non-
compact Calabi-Yau spaces. Sometimes the counting functions exhibit
interesting automorphic properties. In the future we can expect progress on
the enumeration of other calibrated cycles in special holonomy manifolds,
where the mathematical technology is still being developed. Insights from
string theory might well be crucial for future progress. It remains to be seen
if these new counting functions have interesting arithmetic properties.

#### Mock Modularity And Quantum Modularity.

Important generalizations of modularity turn out to be closely connected to
physical phenomena. For example mock modularity has made an appearance in the
context of partition functions of topologically twisted Yang-Mills theory,
related to four-manifolds, and three-manifolds. But it has also shown up in
other contexts such as the AdS/CFT correspondence, BPS state counting, the
elliptic genus of noncompact sigma models, indices of supersymmetric gauge
theories, and umbral Moonshine. Some understanding of the origin of mock
modularity (and its relation in general to noncompactness of field space) has
been achieved, but a deeper understanding, that ties together all the above
instances, would be more desirable. An example of an important concrete
question in this subject is the formulation of a suitable representation-
theoretic interpretation of mock modularity.

Some of the new three-manifold invariants, and some BPS state counting
functions have intriguing relations to “quantum modular forms,” a mathematical
concept that is still under development. It is possible that physical ideas
could help guide the theory of quantum modular forms.

#### Geometric Langlands Program.

As mentioned above, one the deepest and most intensively studied connections
to number theory is the large body of work related to the geometric Langlands
program and its connection to boundaries, defects, and duality symmetries of
supersymmetric Yang-Mills. While much studied, many important open questions
remain and are being actively pursued. The Essay describes several of these
questions and potentially fruitful directions these questions have inspired.

##  7 Many Omissions

The authors of the Essay have made some effort to cast a wide net, but many of
the big ones got away. Even the broadest panorama will miss shining wonders
that lie beyond the observer’s horizon. Three examples are, fortunately,
covered by other Snowmass documents. The first example is the splendid topic
of Moonshine. The second is the collection of remarkable mathematical
developments in the theory of perturbative QFT/ST amplitudes. The third is the
important and vast topic of mathematical aspects of general relativity. The
list of topics which should have been covered in the Essay – but are not –
remains long, and surely contains some jewels that will be of great importance
to future generations.

### Acknowledgements

We would like to thank David Ben-Zvi, Lakshya Bhardwaj, Daniel Brennan,
Nikolay Bobev, Miranda Cheng, Mykola Dedushenko, Emanuel Diaconescu, Tudor
Dimofte, Lance Dixon, Yakov Eliashberg, Michael Freedman, Arthur Jaffe, Jeff
Harvey, Mike Hopkins, Dominic Joyce, Anton Kapustin, Ahsan Khan, Alexei
Kitaev, Zohar Komargodski, Maxim Kontsevich, Craig Lawrie, Andrei Losev,
Marcos Marino, Dave Morrison, Andy Neitzke, Natalie Paquette, Sara Pasquetti,
Peter E. Pushkar, Pavel Safronov, Nathan Seiberg, Shu-Heng Shao, Yuji
Tachikawa, Constantin Teleman, Thomas Walpuski, Maxim Zabzine for useful
correspondence and discussions.

The work of IB is supported in part by NSF grant PHY-2112699 and in part by
the “Simons Collaboration on Global Categorical Symmetry”. DF is partially
supported by the National Science Foundation under Grant Number DMS-2005286
and partially by the “Simons Collaboration on Global Categorical Symmetries”.
The work of GM is supported by US Department of Energy under grant DE-
SC0010008. Part of the work of NN was done while he was a visiting professor
at the CAS (Skoltech), and Kharkevich IPPI in Moscow, Russia, as well as
visiting the Isla San Francisco in Puccalpa, Peru. He thanks these (non-
government, public-sponsored) institutions for hospitality and support. SSN is
supported in part by the European Union’s Horizon 2020 Framework: ERC grant
682608 and in part by the “Simons Collaboration on Special Holonomy in
Geometry, Analysis and Physics”. The research of SSR was supported in part by
Israel Science Foundation under grant no. 2289/18, by a Grant No.
I-1515-303./2019 from the GIF, the German-Israeli Foundation for Scientific
Research and Development, by BSF grant no. 2018204.