YOUNGST@RS - Physics and Number Theory

20 Jan 2025, 09:10 → 23 Jan 2025, 18:00 Europe/Berlin

Mainz Institute for Theoretical Physics, Johannes Gutenberg University

The aim of this workshop is to bring together young researchers interested in applying tools and techniques from number theory to problems in physics and vice-versa. For example, periods often play a central role in arithmetic geometry and the very same periods are important in the study of topological string theory and Feynman integrals.

Other fields of common interest to both communities include the study of modular forms, variations of Hodge structure, complex multiplication and conformal field theory, etc. We hope to hear from experts on all of these topics and more!

Monday 20 January

Welcome Greeting by MITP Directors

Abhiram Kidambi: Complex Multiplication: History, Theory, Computation, and Applications in Physics

11:00 Break

Sören Kotlewski: Modularity of Calabi-Yau Fourfolds and Applications to Physics

Modularity describes a topic in the framework of arithmetic geometry which is the interplay of algebraic geometry and number theory. If a variety suits the requirements of being modular, it determines a modular form. From a physics perspective, modularity turns out to play an important role for the investigation of Calabi-Yau compactifications of string, M- or F-theory.
In this talk, I present an introduction to the framework of modularity and discuss its applications in the context of string-compactifications. In particular, I focus on the search for supersymmetric flux vacua of 3-dimensional M-theory compactified on Calabi-Yau fourfolds, arguing that modularity may serve as a sufficient condition for non-trivial fluxes. To strengthen its applicability, I demonstrate this arithmetic approach by presenting concrete examples.

Modularity of Calabi-Yau Manifolds.pdf

12:30 Break

Nutsa Gegelia: Paramodular forms from Calabi-Yau operators

We report on the conjectural identification of paramodular forms from Calabi-Yau motives of Hodge type (1,1,1,1) of moderately low conductor. The identifications are done by calculating Euler factors from Calabi-Yau operators from the AESZ list, seeking a match with Hecke eigenvalues provided in the paramodular forms database and checking the approximate functional equation for the Euler product numerically.

15:00 Break

Max Wiesner: Counting of States in Quantum Gravity and the Weak Gravity Conjecture

In this talk, I will discuss properties of the spectrum of charged states in theories of gravity with minimal supersymmetry arising as compactifications of M-/F- and string theory to 6-,5-, and 4-dimensions. According to the Weak Gravity Conjecture, the spectrum of charged states has to contain charged states for which the charge-to-mass ratio exceeds that of extremal black holes. In theories with minimal supersymmetry, explicit checks of this conjecture require considering states in the non-BPS secotr of the theory arising as excitations of strings. Exploiting the properties of the geometry of the compactification manifold, I will demonstrate that the full theory of quantum gravity contains exactly those states required to satisfy the weak gravity conjecture. In particular, using the modular properties of the elliptic genera of weakly coupled strings arising from wrapped branes, I will show that for gauge theories, which can become weakly coupled in quantum gravity, a full tower of super-extremal states exists, ensuring consistency of the Weak Gravity Conjecture under dimensional reduction.

16:30 Break

Damian van de Heisteeg: Charting F-theory landscapes

In this talk we explore the landscape of F-theory compactifications from two directions. In the first approach we consider the simplest complex structure moduli space, the thrice-punctured sphere, and enumerate all corresponding Calabi-Yau fourfolds by monodromy considerations. In the second we explore the landscape of flux vacua with a vanishing superpotential, both from a general Hodge-theoretic point of view and by considering explicit examples. Based on 2404.03456 and 2404.12422.

Youngstars_Damian_van_de_Heisteeg.pdf

Tuesday 21 January

Robert Pryor: B-branes in N=(2,2) Hybrid Models

B-twisted N=(2,2) hybrid models represent a class of superconformal field theories of string theoretic relevance. They can be understood as a fibration of a Landau-Ginzburg model over a geometric base. The bulk theory of these hybrid models is relatively well understood, but much less is known about the boundary sector and the associated categories of D-branes. In this talk, I will introduce these hybrid models, as well as some of their key features. I will then present a formulation for the defining data of a hybrid B-brane, as well as a method for explicitly constructing such branes. This construction will be illustrated with several explicit examples. Finally, I will discuss the lifting of hybrid B-branes to the GLSM, and use the resulting construction to identify our example branes as analytic continuations of more familiar geometric B-branes.

NT_seminar.pdf

11:00 Break

Thorsten Schimannek: Counting curves on non-Kaehler Calabi-Yau 3-folds with topological strings

In general, a Kaehler Calabi-Yau threefold with nodal singularities does not admit a Kaehler small resolution.
This happens in particular if the exceptional curves are torsion in homology.
However, the presence of torsion also leads to the possibility of turning on a flat, topologically non-trivial B-field that stabilizes the singularities.
Using conifold transitions, we will describe a large family of examples for this phenomenon and explain how the resulting backgrounds can be studied using hybrid phases of gauged linear sigma models.
Using the sphere partition function, we can then extract periods of the mirror Calabi-Yaus, which allows us to study the topological string partition functions.
We argue that the latter encode Gopakumar-Vafa invariants associated to BPS states with discrete charges and that the invariants capture the enumerative geometry of the non-Kaehler small resolutions.
On the other hand, it turns out that the periods correspond to well known Q-variations of Hodge structure but with a new integral structure.

12:30 Break

Jarod Hattab: Non-Perturbative Topological Strings from M-theory

The embedding of Topological String Theory as a topological subsector in Type IIA String Theory enables, upon lifting to M-theory, the computation of its free energy by integrating out M2-branes wrapping two-cycles within a Calabi-Yau manifold. Revisiting the Gopakumar-Vafa calculation, we identify a subtlety related to poles that come from considering an imaginary field strength background. This allows us to derive non-perturbative corrections to the free energy, revealing deeper aspects of the theory's structure.

15:00 Break

Claudia Rella: The arithmetic of resurgent topological strings

Quantising the mirror curve of a toric Calabi-Yau threefold gives rise to quantum operators whose fermionic spectral traces produce factorially divergent series in the Planck constant and its inverse. These are captured by the Nekrasov-Shatashvili and standard topological strings via the TS/ST correspondence. In this talk, I will discuss the resurgence of these dual asymptotic series and present an exact solution for the spectral trace of local P^2. A full-fledged strong-weak symmetry exchanges the perturbative/nonperturbative contributions to the holomorphic and anti-holomorphic blocks in the factorisation of the spectral trace. This exact symmetry builds upon the interplay of the L-functions with coefficients given by the Stokes constants and the q-series acting as their generating functions. Guided by this pivotal example, I will propose a new perspective on the resurgence of a particular class of formal power series conjectured to possess specific summability and quantum modularity properties, leading to the general paradigm of modular resurgence. This talk is based on arXiv:2212.10606, 2404.10695, and 2404.11550.

rella-21012025.pdf

16:30 Break

Michael Lathwood: Zeta Functions via Periods of Calabi-Yau Hypersurfaces in Non-Fano Toric Varieties

We explore the interplay between non-Fano toric varieties and their mirror Landau-Ginzburg (LG) superpotentials. The mirror LG model is described by the Hori-Vafa potential, modified by corrections arising from tropical disks. These corrections are such that the classical period of the LG superpotential is equal to the quantum period of the non-Fano toric variety, extending mirror symmetry to this broader setting (arXiv:2404.16782). The corrected LG superpotential satisfies a new Picard-Fuchs system, offering new insights into the arithmetic properties of non-Fano toric varieties. We introduce ToricZeta, a software package in development for computing zeta functions of Calabi-Yau hypersurfaces in (possibly non-Fano) toric varieties. Finally, we outline new research directions in arithmetic and modularity made possible by these advancements.

NonFanoZeta.pdf

Wednesday 22 January

Sebastian Pögel: Calabi-Yaus and Curves: A correspondence from Feynman Integrals

Feynman Integrals are closely linked to periods of non-trivial geometries such as Calabi-Yau manifolds, elliptic, as well as higher genus curves. Starting from the archetypical Calabi-Yau Feynman integral, the four-loop equal-mass Banana integral, we construct an associated genus-two hyperelliptic curve by performing a matching on the level of (intermediate) Jacobians. We thus show that the geometry associated to a Feynman integral is not unique, and that it is possible to explicitly construct such equivalent geometries.

11:00 Break

Miroslava Mosso Rojas: Gromow-Witten invariants in quantum cohomology and quantum K-theory

A method to calculate instanton corrections to the non-perturbative effects of the topological string is by finding the Gromov-Witten (GW) invariants. They are rational numbers that appear in the enumerative geometry of Calabi-Yau 3-folds and have applications in string theory. They arise in the topological A-model as counts of worldsheet instantons. GW invariants are also present in quantum K-theory; however, in this context these numbers are integers. The invariants in K-theory have an interpretation of BPS objects in 3 dimensions (on the 3d world volume). In this work, we use the Atiyah-Bott localization method to compute the GW invariants in both contexts: quantum cohomology and quantum K-theory, for local Calabi-Yau manifolds. The final goal is to compare the results in both theories and provide an interpretation within string theory.

12:30 Break

Daniel Bryan: BPS structures and attractor flow in 4d N=2 theories

We study BPS structures in a class of 4d N=2 theories that can be
realised in type IIB string theory on a Calabi-Yau 3-fold.
The number of BPS states in these theories jumps discontinuously at
special loci in the moduli space called walls of marginal stability.
We use attractor flow methods to determine the BPS spectrum in each
chamber bounded by these walls in several cases including the
Argyres-Douglas A_2 model and pure SU(2) Seiberg-Witten theory. This is
done by looking at existence conditions on the endpoints
of the gradient flow lines of the BPS central charges and by considering
that flow lines can split at the walls where the composite
BPS states decay into their constituents. This involves first finding
the central charges of the BPS states as linear combinations
of periods associated to an elliptic curve embedded in the Calabi-Yau
3-fold. These are periods of a meromorphic differential and
are derived by solving the appropriate Picard-Fuchs equations.

BPS structures and attractor flow in 4d N=2 theories.pdf

15:00 Break

Michael Stepniczka: Computing Zeta Functions of Calabi-Yau Threefolds

We discuss the importance of computing zeta functions from both a mathematical perspective, relating to modularity, as well as a physics perspective, seeing where these zeta functions can arise. Our focus lies in the computation of these local zeta functions, and we highlight the Dwork deformation method in which these local factors can be computed for families of varieties rather than a single variety. We discuss steps of the computation, including the relevance of computing periods, as well as an implementation currently being built in Python to be compatible with software such as CYTools.

CY3_Zeta_Functions-4.pdf

16:30 Break

Richard Nally: Landscaping with Number Theory: Counting Calabi-Yau Threefolds

Batyrev constructed a large but finite list of Calabi-Yau threefolds (CY3s) obtained from suitable triangulations of four-dimensional reflexive polytopes; all such polytopes were classified by Kruezer and Skarke. Working with these CY3s has been extremely fruitful, and has enabled significant recent progress in the explicit construction of flux vacua with interesting physical properties. Nevertheless, it also comes with practical and computational challenges: the Kreuzer-Skarke list hosts large redundancies, and the same CY3 can be realized from many different triangulations. In recent work (arXiv:2310.06820, with N. Gendler, N. MacFadden, L. McAllister, J. Moritz, A. Schachner, and M. Stillman), we enumerated CY3s obtained in this way, obtaining exact irredundant counts of CY3s with 1<= h^{1,1} <=5. In this talk, I will emphasize the key role techniques from arithmetic geometry played in this classification. I will also comment on ongoing work (with F. Abbasi and W. Taylor) on classifying elliptic fibrations in this class of CY3s and the search for elliptic curves of high rank therein.

Thursday 23 January

Giulia Gugiatti: Hypergeometric local systems over Q with Hodge vector (1,1,1,1)

A hypergeometric local system is the space of solutions to a hypergeometric differential equation. Under suitable assumptions, hypergeometric local systems have a geometric origin: they arise from the variation of cohomology of the fibers of a morphism. In this talk, we classify all hypergeometric local systems that support a rational variation of Hodge structure with Hodge numbers (1,1,1,1). We show that all such local systems are associated to families of generically smooth threefolds and we discuss the geometry and the arithmetic of the family at the conifold point. This is joint work with Fernando Rodriguez Villegas.

GG-Youngst@rs.pdf

11:00 Break

Raphael Senghaas: Exponential Networks for Linear Partitions

Exponential Networks are a useful tool to count BPS states in local Calabi-Yau 3-folds. In this talk we apply Exponential Networks to D0-D4 bound states in flat space. This leads us to an explicit correspondence between torus fixed points of the Hilbert scheme of points on C2 and anomaly free finite webs attached to the quadratically framed pair of pants. This can be viewed as an A-model description of the ADHM construction. We will highlight some relation of our construction to Donaldson-Thomas theory.

12:30 Break

Franziska Porkert: (Hyperelliptic) Feynman Integrals from Differential Equations

Feynman integrals are the building blocks of multi-loop scattering amplitudes and have fascinating connections to various areas of mathematics. For example, beyond one loop, large classes of Feynman integrals are related to (hyper)elliptic curves or Calabi-Yau surfaces. In this talk, I will provide a brief review of computing Feynman integrals using canonical differential equations and discuss recent results regarding the associated forms. In particular, we will focus on hyperelliptic Feynman integrals (as examples).

Talk_Jan_2025.pdf

15:00 Break

Ilia Gaiur: Higher Bessel Product Formulas: Explicit Examples of Multiplication Kernels

Higher Bessel functions are the solutions to the quantum differential equations for $\mathbb{P}^{N-1}$. These functions are connected to the periods of the Dwork families via the Laplace transform, and the functions themselves are exponential integrals. In my talk, I will show how product formulas for these irregular special functions lead to other geometric differential equations associated with higher-dimensional families of algebraic varieties. I will discuss the geometric and algebraic properties of the periods for these families and later provide further perspectives on these correspondences.

slides_Ph&NT4.pdf

16:30 Break

Simon Douaud: Borel singularities and Stokes constants of the topological string free energy on one-parameter Calabi-Yau threefolds

After having summarized the relevant resurgence and BPS counting notions, I will discuss the Borel plane of the topological string free energy on all hypergeometric
one-parameter Calabi-Yau models, close to singular points in moduli space.
I will focus on the location of Borel singularities and the value of the associated Stokes constants. We found in particular that in
models which exhibit massless D-branes at a singular point, the central charge of the D-brane close
to the singular point coincides with the location of the leading Borel singularity, and the generalized
Donaldson-Thomas invariant associated to the charge of the D-brane, in as far as its value is known,
coincides with the Stokes constant associated to the Borel singularity.

MTP slides Simon Douaud.pdf