Number Theory in Field and String Theory

4 Sept 2025, 09:00 → 6 Sept 2025, 18:00 Asia/Taipei

This interdisciplinary workshop explores emerging connections between theoretical physics—particularly string theory and quantum field theory—and advanced topics in number theory, such as automorphic forms, mock and quantum modular forms, and representation-theoretic structures. Recent developments in scattering amplitudes, topological strings, and counting functions in 2D and 3D conformal field theories underscore the role of a shared number-theoretic framework across physics and low-dimensional topology. The workshop brings together mathematicians and physicists—experts in vertex operator algebras, low-dimensional topology, automorphic forms, and related areas—to spark new conversations and foster cross-disciplinary collaborations.

Organizer:

• Miranda C. N. Cheng
• Ioana Coman
• Daniele Dorigoni
• Martí Rosselló

 

Sponsors: LeCosPA, NSTC, Max Planck-IAS-NTU Center, Institute of Mathematics Academia Sinica. 

Thursday 4 September

09:00 → 10:00 Modular Features of Superstring Scattering Amplitudes: Generalised Eisenstein Series and Theta Lifts

We briefly review how superstring scattering amplitudes produce infinite families of real-analytic modular forms with respect to $SL(2,\mathbb{Z})$. In particular, we focus our attention towards generalised Eisenstein series which are real-analytic modular invariant solutions to certain inhomogeneous Laplace-eigenvalues equations.
We conjecture that string theory selects only rational linear combinations of generalised Eisenstein series which are expressible by lattice-sums and which can be written as theta-lifts of Maass-lifted versions of the modular local polynomials introduced by Bringmann and Kane. This talk is based on joint work with Michael B. Green and Congkao Wen.

Speaker: Daniele Dorigoni

Daniele Dorigoni - Modular Features of Superstrings Scattering Amplitudes.pdf

10:00 → 11:00 Resurgent Methods for Integrated Correlators

Recently it has become possible to analyse certain observables in quantum field theories exactly by using the method of supersymmetric localization. These calculations give us unprecedented possibilities of doing precision tests for the AdS/CFT correspondence and checking the behaviour of supersymmetric field theories under weak-strong duality. In this talk I will be interested in looking at integrated correlators in N=4 Yang-Mills theory and showing how a modular-invariant perturbation theory can be formulated for them. Interestingly, the asymptotic series that emerge are factorially divergent and the theory of resurgence provides a unifying framework to understand them. The required non-perturbative effects have a very interesting interpretation in the gravitational dual picture.

Speaker: Rūdolfs Treilis

Rudolfs Treilis - Resurgent methods for integrated correlators.pdf

11:00 → 11:30 Break & Registration

11:30 → 12:30 String Amplitudes and Automorphic Representations

Automorphic forms attached to certain “minimal” representations of Lie groups play a central role in many different contexts, both in mathematics and in physics. In particular, they appear in theta correspondences as well as in string amplitudes. Such minimal automorphic forms are characterized by having very few non-vanishing Fourier coefficients. In these lectures I will present an overview of these mathematical structures, with comments on their role in string theory. In the second lecture talk I will discuss some ongoing work on understanding the representation theoretic structure of certain generalizations of automorphic forms that appear in string theory.

Speaker: Daniel Persson

Daniel Persson - String Amplitudes and Automorphic Representations.pdf

12:30 → 14:30 Lunch Break

14:30 → 15:30 Elliptic Modular Graph Forms as Equivariant Iterated Integrals (Online)

The low-energy expansion of string scattering amplitudes introduces infinite families of non-holomorphic modular forms dubbed "modular graph forms" with multiple zeta values among their Fourier coefficients. At genus one, modular graph forms can be expressed as Brown’s equivariant iterated integrals of holomorphic Eisenstein series which expose their algebraic and differential relations. As a main result of this talk, a similar link will be proposed for "elliptic modular graph forms" which additionally depend on a marked point on the torus. Elliptic modular graph forms generalize Zagier’s single-valued elliptic polylogarithms beyond depth one and arise in non-separating degenerations of genus-two string amplitudes. We reproduce elliptic modular graph forms from equivariant iterated integrals of Kronecker-Eisenstein Jacobi forms which (i) organize the contributing multiple zeta values, (ii) pinpoint bases under their algebraic relations and (iii) can be alternatively expressed in terms of elliptic polylogarithms and their complex conjugates.

Speaker: Oliver Schlotterer

15:30 → 16:00 Break

16:00 → 17:00 3-Manifolds, 3D N=4 Rank-0 SCFTs, and Rational VOAs

I will discuss the following two topics:
i) the construction of 3D N=4 rank-0 SCFTs from certain 3-manifolds via the 3D–3D correspondence, and
ii) the 2D rational VOAs (and their characters) at the boundaries of these 3D N=4 rank-0 SCFTs.

Speaker: Dongmin Gang

Dongming Gang - 3-manifolds, rank 0 SCFTs and rational VOAs.pdf

17:00 → 18:00 Reception

Friday 5 September

09:00 → 10:00 Quantum Modularity in Topological String on Local Weighted Projective Surfaces

In the framework of the Topological String/Spectral Theory correspondence of Grassi–Hatsuda–Mariño, we show that the modularity of certain non-perturbative corrections is governed by quantum modular functions. More precisely, studying the resurgence structure of the perturbative expansion of the first fermionic trace of local weighted projective surfaces, we uncover new arithmetic properties of the associated Stokes constants.
In this talk, I will discuss the resurgent structure of these perturbative expansions and the quantum modularity of their non-perturbative corrections. These results are joint projects with C. Rella, arXiv:2404.10695, arXiv:2506.08265.

Speaker: Veronica Fantini

Veronica Fantini - Quantum Modularity in Topological String on Local Weighted Projective Spaces.pdf

10:00 → 11:00 The Habiro Ring and Quantum Invariants

Work of Habiro and Lê on quantum invariants led to a definition of the Habiro ring——a natural home for the invariants associated to the trivial connection for certain manifolds. The Habiro ring has strong integrality properties that connect to work of Ohtsuki. I will discuss a vast generalisation of this ring that includes perturbative invariants labelled by arbitrary flat connections. This is based on joint work with Garoufalidis, Scholze, and Zagier.

Speaker: Campbell Wheeler

11:00 → 11:30 Break

11:30 → 12:30 Quantum Invariant and Double Affine Hecke Algebra

The double affine Hecke algebra (DAHA) is introduced by Cherednik, and it is
useful techniques in studies of the Macdonald polynomials. We will discuss the topological role of DAHA to construct the quantum invariant.

Speaker: Kazuhiro Hikami

Kazuhiro Hikami - Quantum invariant and DAHA.pdf

12:30 → 14:30 Lunch Break

14:30 → 15:30 A Framework for Proving Quantum Modularity: Application to Witten’s Asymptotic Expansion Conjecture

I address two linked problems at the interface of quantum topology and number theory: deriving asymptotic expansions of the Witten–Reshetikhin–Turaev invariants for 3-manifolds and establishing quantum modularity of false theta functions. Previous progress covers Seifert homology 3-spheres for the former and rank-one cases for the latter, both relying on single-variable integral representations. I extend these results to negative definite plumbed 3-manifolds and to general false theta functions respectively. I address this limitation by developing two techniques: a Poisson summation formula with signature and a framework of modular series, both of which enable a precise and explicit analysis of multivariable integral representations.

Speaker: Yuya Murakami

Yuya Murakami - A Framework for Proving Quantum Modularity.pdf

15:30 → 16:00 Break

16:00 → 17:00 Transcendental Numbers in One-Loop Closed String Amplitudes (Online)

Scattering amplitudes in string theory naturally give rise to interesting classes of numbers. At tree level, the low-energy expansion of closed-string amplitudes involves multiple zeta values (MZVs), whereas in field theory one only encounters algebraic numbers. At one loop, the amplitudes are expressed through iterated integrals of Eisenstein series. Because of divergences, these integrals lead to numbers that go beyond MZVs and perhaps even beyond the usual notion of periods. In this talk, I will explain how such numbers appear in one-loop closed-string amplitudes and how they can be evaluated numerically.

Speaker: Mehregan Doroudiani

18:00 → 20:00 Speakers Dinner

Saturday 6 September

09:00 → 10:00 Symmetry-Resolved Asymptotics of 2d CFTs in the Presence of Boundaries

Conformal field theories exhibit universality in their high energy data. One celebrated instantiation of this is the Cardy formula, which controls the asymptotic density of states of a 2d CFT in the "closed string channel" (i.e. states of the theory when it is placed on a spatial circle). There is also an "open string" analog of the Cardy formula (sometimes called the Affleck-Ludwig-Cardy formula) which controls the asymptotic density of states of the theory when it is placed on a spatial interval. I will explain how this open string formula can be refined when the theory possesses a finite noninvertible symmetry. Much of the work involves carefully defining what it even means for a noninvertible symmetry to act on open string states, which leads to rich algebraic structures. Time permitting, I will describe an application of the resulting symmetry-resolved Affleck-Ludwig-Cardy formula to entanglement entropy. Based on 2409.02159 and 2409.02806 with Yichul Choi and Yunqin Zheng.

Speaker: Brandon Rayhaun

10:00 → 11:00 Vector-Valued Jacobi Forms and Conformal Field Theory

In this talk I'll explain the relevance of vector-valued Jacobi forms to (rational and nonrational) CFT, sketch some of the effective tools available, and use this to probe the (non)existence of certain exotic CFT.

Speaker: Terry Gannon

Terry Gannon - Vector-Valued Jacobi Forms and CFT.pdf

11:00 → 11:30 Break

11:30 → 12:30 Modular Bootstrap for Calabi-Yau Threefolds

BPS indices encoding the entropy of supersymmetric black holes in compactifications of Type II string theory on Calabi-Yau threefolds, known in mathematics as generalized Donaldson-Thomas invariants, possess remarkable (mock) modular properties. I'll explain the physical origin of mock modularity and show, for a set of one-parameter threefolds, how it can be used, together with wall-crossing and direct integration of topological string, to compute the BPS indices and other topological invariants. As a result, one obtains explicit (mock) modular
functions encoding infinite sets of D4-D2-D0 BPS indices as well as new boundary conditions for the holomorphic anomaly equation of the topological string partition function allowing to overcome the limitations of the direct integration method. In the end, I'll present preliminary results on the asymptotic growth of DT invariants hinting for the existence of some phase transitions.

Speaker: Sergei Alexandrov

Sergei Alexandrov - Modular Bootstrap for Calabi-Yau manifolds.pdf

12:30 → 14:30 Lunch Break

14:30 → 15:30 Scalar Partition Functions of 2d CFTs

I will discuss the spectrum of scalar primary operators in any two-dimensional conformal field theory. I will show that the scalars alone obey a nontrivial crossing equation, and that at high temperature, the difference between the true scalar partition function and the one predicted from a semiclassical gravity calculation is controlled by: the modular integral of the partition function, the light states of the theory, and an infinite series terms directly related to the nontrivial zeros of the Riemann zeta function.

Speaker: Nathan Benjamin

Nathan Benjamin - Scalar Partition Functions of 2d CFTs.pdf

15:30 → 16:00 Break

16:00 → 17:00 Z-hat with Defects and Equivalence Classes

For a large class of three-manifolds, the Z-hat invariant is a component of a vector-valued quantum modular form. Other components of the vector can be obtained by inserting supersymmetric line defects into the half-index of the corresponding T[M] theories. Infinitely many different half-indices with supersymmetric line defects give the same q-series (component of the vector-valued quantum modular form) up to the addition of a polynomial in q. This suggests a notion of an equivalence class of line defects in T[M] theories. In this talk, I'll discuss these equivalence classes of defects and a surprising relation between them and equivalence classes of certain meromorphic functions.

Speaker: Mrunmay Jagadale

17:00 → 18:00 Drinks