The Arithmetic of Calabi–Yau Manifolds

17 Mar 2025, 08:00 → 28 Mar 2025, 16:00 Europe/Berlin

2413/2-430 - MITP Seminar Room (MITP - Mainz Institute for Theoretical Physics, Johannes Gutenberg University Mainz)

In the research area of string compactifications and, more recently, in the context of quantum field theory scattering amplitudes, geometric properties of Calabi–Yau varieties and their moduli spaces have played an important role. In the past these Calabi–Yau varieties have often been studied over the field of complex numbers. Considering such varieties over the field of rational numbers, or over finite fields, is not only a modern research topic in arithmetic geometry and number theory, but also offers novel perspectives and new insights in physics. For instance: the arithmetic of Calabi–Yau manifolds is
linked to supersymmetric black holes and supersymmetric flux vacua of string theory, the asymptotic growth of Gopakumar–Vafa invariants is determined by the L-values of modular Calabi–Yau varieties, and similar arithmetic methods have been applied in the context of quantum field theory Feynman integrals.

Monday 17 March

10:15 → 11:15 Arithmetic of hypergeometric Calabi-Yau families

Algebraic cycles on Calabi-Yau 3-folds played a starring role in demonstrating the inequivalence of algebraic and homological equivalence, opening mirror symmetry on the quintic, and describing asymptotics of local Gromov-Witten invariants. In this talk, they play the role of a new testing ground for connections between transcendental and arithmetic geometry.

On the transcendental side, we have the extensions of mixed Hodge structure that arise from the motivic cohomology / higher Chow / algebraic K-theory groups of a variety over Q. Their periods define the regulators and height pairings which, for a family of varieties, produce solutions of inhomogeneous Picard-Fuchs equations. Thinking in terms of families deforming the original variety turns out to be the key to going beyond “classical” cases, both for constructing such cycles and computing their transcendental invariants.

The arithmetic of a variety over Q is described in part by the L-function attached to the Galois representation on its middle cohomology. The Bloch-Beilinson conjectures, which posit a relationship between its special values and the generalized periods above, remain one of the deepest open problems in mathematics. In this talk, based on recent work with Vasily Golyshev, I’ll explain how to use hypergeometric families and Hadamard convolutions to produce new numerical evidence for these conjectures.

Speaker: Matt Kerr

11:15 → 11:45 Break

11:45 → 12:45 Topological strings on Q-factorial nodal Calabi-Yau with topologically non-trivial B-fields

To start the talk, I will briefly recall the classical relationship between topological strings, enumerative geometry and variations of Hodge structure via mirror symmetry.
I will then discuss a recent proposal for the interpretation of the A-model topological string partition function on nodal Calabi-Yau threefolds that carry a flat but topologically non-trivial B-field.
From an enumerative perspective, this is conjectured to encode a refinement of the usual Gopakumar-Vafa invariants with respect to a torsion curve class that only exists in small resolutions that have a trivial canonical class but are not Kaehler.
After illustrating the phenomenon at the hand of the quintic, my focus will be on the relation under mirror symmetry to variations of Hodge structure that have an atypical integral structure.
This integral structure can be interpreted in terms of the singular geometry and the non-trivial B-field topology.
I will then present an irrational Calabi-Yau operator that nonetheless corresponds to an integral variation of Hodge structure and encodes integral enumerative invariants.

Speaker: Thorsten Schimannek

12:45 → 14:00 Lunch

14:00 → 15:00 Invariants of Calabi-Yau Hybrid Models

Hybrid models are a class of Landau-Ginzburg orbifolds fibred over some base manifold. They naturally arise in limiting regions of the stringy Kahler moduli spaces of Calabi-Yaus and as phases of the associated gauged linear sigma models (GLSMs). We compute genus zero correlators for hybrids with one and two Kahler parameters. These correlators are generalisations of GW/FJRW invariants. We use GLSM techniques, combined with the physics formulation of hybrids, to extract the invariants from I/J-functions. This is joint work with D. Erkinger.

Speaker: Johanna Knapp

Knapp.pdf

15:00 → 15:30 Break

Tuesday 18 March

09:30 → 10:30 Non-perturbative topological strings and arithmetic

In this talk I will discuss some implications of non-perturbative topological string
theory for arithmetic properties in special geometry. I will mostly discuss the TS/ST correspondence, which provides a non-perturbative definition of topological string theory on toric Calabi-Yau manifolds.
In this correspondence, the quantization of the curve leads to predictions for the values of the periods at the so-called maximal conifold point. If time permits, I will make some comments on the relation between arithmetic and resurgence.

Speaker: Marcos Mariño

Marino.pdf

10:30 → 11:15 Break

11:15 → 12:15 Resurgence, number theory, and quantum mirror curves

Resurgence provides a powerful toolbox to access the non-perturbative sectors hidden within the divergent asymptotic series of quantum theories. Under some special assumptions, the non-perturbative data extracted via resurgent methods possess intrinsic number-theoretic properties that are deeply rooted in the symmetries and arithmetic of the geometry underlying the quantum theory. The framework of modular resurgence aims to formalise this observation. In this talk, after introducing the basics of modular resurgence, I will consider the TS/ST correspondence for toric Calabi-Yau threefolds and focus on the fermionic spectral traces of quantum mirror curves. Here, a complete realisation of the modular resurgence paradigm is found in the spectral theory of local P^2—where the bridge between non-perturbative physics and the arithmetic properties of the geometry takes the form of an exact strong-weak symmetry—and is now being generalised to all local weighted projective spaces. This talk is based on arXiv:2212.10606, 2404.10695, 2404.11550, and work in progress.

Speaker: Claudia Rella

12:15 → 14:00 Lunch

14:00 → 14:30 Hodge Structures of Complex Multiplication Type from Rational Conformal Field Theories

Two-dimensional conformal field theories (CFT) play an important role in the study of statistical mechanics, condensed matter systems and worldsheet descriptions of string theory. Even though two-dimensional CFTs possess an infinite dimensional symmetry algebra (Virasoro algebra), it is generically not possible to solve the theory exactly. Exceptions are so-called rational CFTs which have extended symmetry algebras that allow for complete solvability in all known examples. Therefore it is a natural problem to determine their distribution in the moduli space of all two-dimensional CFTs. For those rational CFTs for which one can construct a Hodge structure, there is a conjectured equivalence between the presence of an extended symmetry algebra of the CFT and existence of an enlarged endomorphism algebra of the Hodge structure (Hodge structures of complex multiplication type). If this relation turns out to be true, one could formulate the distribution of rational CFTs in terms of the distribution of Hodge structures of complex multiplication type. In this talk I will provide further evidence for the conjectured relation between rational CFTs and Hodge structures of complex multiplication type. In particular I will show how the Galois group naturally associated to any rational CFT provides under some technical assumptions an enlarged endomorphism algebra of the associated Hodge structure. This construction uses only CFT data and does not rely on a geometric realization of the associated Hodge structure.

This is based on upcoming work with Hans Jockers and Pyry Kuusela.

Speaker: Maik Sarve

14:30 → 15:00 Counting Points Over F_p and Periods of Calabi-Yau Manifolds

For the one-parameter mirror quintic Calabi–Yau threefold, we derive an explicit formula for the point count over F_p in terms of a p-adic expansion involving only the periods of the holomorphic (3,0)-form. The period basis is obtained via the Frobenius method, and after a suitable change of basis, the expansion naturally incorporates p-adic zeta values. This result establishes a direct arithmetic–geometric link between point counting and period integrals. If time permits, we will discuss extensions to F_q and the five-parameter Hulek–Verrill manifold.

Speaker: Eleonora Svanberg

15:00 → 15:30 Break

Wednesday 19 March

09:30 → 10:30 Partition ranks and mock Eisenstein traces

Recently certain traces that are related to quasimodular forms gained attention. These are related to cranks of partitions. Jointly with Pandey and van Ittersum we study partition ranks which are related to mock modular forms.

Speaker: Kathrin Bringmann

10:30 → 11:15 Break

11:15 → 12:15 Certain families of K3 surfaces and their modularity

Speaker: Noriko Yui

12:15 → 14:00 Lunch

15:00 → 15:30 Break

Thursday 20 March

09:30 → 10:30 Modularity of enumerative invariants on Calabi-Yau threefolds

Once put into suitable generating series, various enumerative invariants on Calabi-Yau threefolds are expected to possess modular properties, allowing to determine them uniquely from a few data points and giving powerful control on their asymptotic growth. This includes Gopakumar-Vafa invariants in presence of a genus one fibration, Noether-Lefschetz invariants in presence of a K3 fibration, as well as Donalson-Thomas invariants supported on divisors (without requiring any particular fibration). While modularity has a clear mathematical origin in the first two cases, it is more mysterious in the last case, corresponding physically to BPS indices counting D4-D2-D0 black holes. I will review recent progress in testing the modularity (or more generally mock modularity) of D4-D2-D0 indices in one-parameter Calabi-Yau threefolds such as the quintic threefold, and in leveraging these results to compute Gopakumar-Vafa invariants at higher genus than hitherto possible. Time permitting, I will also discuss modularity of GV and DT invariants in K3-fibered two-parameter models. Based on work in collaboration with Sergey Alexandrov, Soheyla Feyzbakhsh, Albrecht Klemm and Thorsten Schimannek.

Speaker: Boris Pioline

10:30 → 11:15 Break

11:15 → 12:15 Integration on higher-genus Riemann surfaces

This goal of this talk is to review two equivalent constructions of polylogarithms on Riemann surfaces of arbitrary genus. In both cases, iterated integrals of explicitly known flat connections are combined into function spaces that close under integration over marked points. A first, string-theory inspired flat connection is built from single-valued but non-holomorphic one-forms on a genus-h surface that transform as tensors under the modular group Sp(2h, Z). An alternative flat connection comprising meromorphic but multivalued differentials was introduced in mathematical work of Enriquez through its functional properties and recently found explicit realizations. These two types of differentials exhibit striking parallels not only in their construction from convolution integrals but also in their algebraic relations and moduli variations.

Speaker: Oliver Schlotterer

Schlotterer.pdf

12:15 → 14:00 Lunch

14:00 → 15:00 Hadamard Products and BPS Networks

I will speak about fibred products of elliptic surfaces. We can sometimes identify 3-cycles on such geometries. We may also insist that the holomorphic 3-form has constant phase when restricted to three dimensional submanifolds. This leads to a construction that is reminiscent of spectral/exponential networks.

Speaker: Mohamed Elmi

15:00 → 15:30 Break

15:30 → 16:30 Modularity of Calabi-Yau Fourfolds and Applications to M-Theory Fluxes

Modularity describes a phenomenon, where it is possible to assign a unique modular form to an algebraic variety by considering its number theoretical properties. As it turns out, modularity plays an important role for many physical applications involving Calabi-Yau geometries.

Based on the well-known results for flux compactifications of type IIB string theory compactified on modular Calabi-Yau threefolds, we discuss in this talk the application of modularity in the context of supersymmetric flux vacua of M-theory compactified on Calabi-Yau fourfolds. In particular, we argue that modularity may serve as a sufficient condition for non-trivial fluxes admitting suitable vacua.

Modularity can be tested by investigating the local zeta-function of the given variety. Introducing a deformation method for the efficient computation of (parts of) the zeta-function, we provide a technique to systematically search for modular points within a given family of fourfolds. To demonstrate the applicability of this method, we discuss an explicit modular example

This talk is based on joint work with H. Jockers and P. Kuusela [2312.07611].

Speaker: Sören Kotlewski

Kotlewski.pdf

Friday 21 March

09:30 → 10:30 Refined BPS numbers on compact Calabi-Yau 3-folds from Wilson loops

We relate the counting of refined BPS numbers on compact elliptically fibred Calabi-Yau 3-folds $\hat X$ to Wilson loop expectations values in the gauge theories that emerge in various rigid local limits of the 5d supergravity theory defined by M-theory compactification on $\hat X$. In these local limits $X_*$ the volumes of curves in certain classes go to infinity, the corresponding very massive M2-brane states can be treated as Wilson loop particles and the refined topological string partition function on $\hat X$ becomes a sum of terms proportional to associated refined Wilson loop expectation values. The resulting ansatz for the complete refined topological partition function on $\hat X$ is written in terms of the proportionality coefficients which depend only on the $\epsilon$ deformations and the Wilson loop expectations values which satisfy holomorphic anomaly equations. Since the ansatz is quite restrictive and can be further constrained by the one-form symmetries and $E$-string type limits for large base curves, we can efficiently evaluate the refined BPS numbers on $\hat X$, which we do explicitly for local gauge groups up to rank three and $h_{11}(\hat X)=5$. These refined BPS numbers pass an impressive number of consistency checks imposed by the direct counting of these numbers using the moduli space of one dimensional stable sheaves on $\hat X$ and give us numerical predictions for the complex structure dependency of the refined BPS numbers.

Speaker: Albrecht Klemm

10:30 → 11:15 Break

11:15 → 12:15 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, but 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.

Speaker: Richard Nally

12:15 → 14:00 Lunch

14:00 → 15:00 TBA

Speaker: Abhiram Kidambi

Monday 24 March

09:30 → 10:30 Frobenius structure and p-adic zeta values

For an ordinary linear differential equation a p-adic Frobenius structure is an equivalence between the local system of its solutions and its pullback under the map t -> t^p which is realized over the field of p-adic analytic elements. Its existence is a strong property, we only expect it for differential equations arising from the Gauss-Manin connection in algebraic geometry. In a vicinity of a singular point such a structure can be described by a bunch of p-adic constants. We will show examples of families of Calabi-Yau hypersurfaces for which these constants turn out to be p-adic zeta values. This is joint work with Frits Beukers.

Speaker: Masha Vlasenko

10:30 → 11:15 Break

11:15 → 12:15 TBA

Speaker: Nutsa Gegelia

12:15 → 14:00 Lunch

14:00 → 15:00 Mirror symmetry and arithmetic for non-Fano varieties

The arithmetic properties of a Calabi-Yau manifold are encoded in the periods of its mirror. The mirror to a sigma model with Fano toric target is given by the Landau-Ginzburg model obtained from the Hori-Vafa construction. In this talk, we explain how this story generalizes to Calabi-Yau hypersurfaces in non-Fano toric varieties. We provide some preliminary calculations with our new Python package, ToricZeta. This talk is based on recent work with Per Berglund and Tim Gräfnitz (arXiv:2404.16782), and work in progress with Pyry Kuusela and Michael Stepniczka.

Speaker: Michael Lathwood

15:00 → 15:30 Break

Tuesday 25 March

09:30 → 10:30 Motives, classical modular forms, and rigid surfaces

We consider geometric realizations of motives in middle dimension with Hodge vector (n, 0, ..., 0, n) and having Hodge endomorphisms by a number field K of degree n; this includes some Calabi-Yau motives, taking n = 1. We then focus on the case (2,0,2) and its relation to classical modular forms of weight 3 with coefficients in a quadratic field, finding realizations in rigid surfaces.

Speaker: John Voight

10:30 → 11:15 Break

11:15 → 12:15 Computing paramodular forms

The goal of this talk is to explain how to compute weight 3 paramodular forms. After a brief introduction to paramodular forms, I will explain the relation between paramodular forms and orthogonal modular forms on definite quinary lattices, and how to compute the latter

(joint work with Neil Dummigan, Ariel Pacetti, Gustavo Rama)

Speaker: Gonzalo Tornaría

12:15 → 14:00 Lunch

14:00 → 15:00 TBA

Speaker: Emanuel Scheidegger

15:00 → 15:30 Break

15:30 → 16:00 From Quantum Cohomology to Quantum K-Theory: Insights into Gromov-Witten Invariants

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. In this work, we use the Atiyah-Bott localization method to compute the GW invariants in both contexts: quantum cohomology and quantum K-theory, or local Calabi-Yau manifolds. The final goal is to compare the results in both theories and provide an interpretation within string theory.

Speaker: Miroslava Mosso Rojas

Mosso_Rojas.pdf

16:00 → 16:30 TBA

Speaker: Michael Stepniczka

Wednesday 26 March

09:30 → 10:30 Canonical Differential Equations for Calabi-Yau Geometries

In recent years, it has been observed that certain Feynman integrals are linked to Calabi-Yau varieties. These integrals arise in various high-precision computations, including two-point functions in QED, scattering processes in the Standard Model, and the scattering of two black holes in general relativity. In my presentation, I will begin by outlining the general framework for performing precision computations. The main focus will be on calculating Feynman integrals, which we will approach using differential equations. These equations will be solved by transforming them into what is known as canonical form. I will describe our approach, which involves dividing the period matrix—also referred to as the Wronskian matrix—into semi-simple and unipotent parts. In the end, I will present three new and intriguing examples that go beyond the single Calabi-Yau geometry case.

Speaker: Christoph Nega

10:30 → 11:15 Break

11:15 → 12:15 Bananas with three loops and three equal masses

The three-loop banana integral with three equal masses in two dimensions is related to a two-parameter family of K3 surfaces. First, I will show that the periods and the mirror map can be expressed in terms of ordinary modular forms and functions. As a result, its maximal cuts can be written as a product of two copies of the maximal cuts of the two-loop equal-mass sunrise integral. Secondly, I will show how to get an eps-factorised differential equation for integrals with un underlying K3 surface. Finally, combining these two steps, I will indicate how to derive an analytic solution for this three-loop integral dependent on two kinematic scales.

Speaker: Sara Maggio

12:15 → 14:00 Lunch

14:00 → 15:00 Modular properties of some families of K3 surfaces

Speaker: Claude Duhr

15:00 → 15:30 Break

Thursday 27 March

09:30 → 10:30 Point counts, diagonal coefficients, and Apery limits

The theme of this talk is to use a combinatorial approach, namely the diagonal coefficients of powers of a given polynomial, to infer arithmetic and analytic properties of the corresponding hypersurface; in the case where the hypersurface has degree equal to the number of variables (Calabi-Yau condition). Firstly, I will recall how the number of points over Fp can be read off, modulo p, from a single diagonal coefficient. I will then discuss a generalization to prime powers (joint work with Francis Brown). In the second part of the talk, I will discuss how the Apery limits of the sequence of diagonal coefficients furthermore encode the periods of the complement of the hypersurface relative to the coordinate simplex. I will illustrate all these phenomena with examples from Feynman integrals, i.e. hypersurfaces associated to graphs, using previous joint work with Karen Yeats.

Speaker: Erik Panzer

10:30 → 11:15 Break

11:15 → 12:15 TBA

Speaker: Pierre Vanhove

12:15 → 14:00 Lunch

14:00 → 15:00 Ansatzing ε-factorized differential equations for Feynman Integrals

In this talk, I will discuss an approach to derive epsilon-factorized differential equations for Feynman integrals. This approach is based on minimal information, namely the choice of a seed integral and an appropriate Ansatz for the differential equation. Previously developed for simple Calabi-Yau integrals, I will show how to deal with complications arising in integrals in physical applications, particularly the appearance of apparent singularities in the Picard-Fuchs operator of the seed integral. I will demonstrate this approach on the example of a four-loop Calabi-Yau integral appearing in the scattering of black holes.

Speaker: Sebastian Pögel

Poegel.pdf

15:00 → 15:30 Break

15:30 → 16:30 TBA

Speaker: Eric Pichon-Pharabod

Friday 28 March

09:30 → 10:30 Harmonic weak Maass forms and periods

We show that the algebraicity of Fourier coefficients of harmonic weak Maass forms of negative half-integral weight is related to the algebraicity of the coefficients of certain canonical meromorphic modular forms of positive even weight with poles at Heegner divisors. We also present a conjecture relating these results to the vanishing of central L-derivatives of integral weight cusp forms. Moreover, we give an explicit formula for the coefficients of harmonic Maass forms in terms of periods of certain meromorphic modular forms with algebraic coefficients. This is joint work with Jan Bruinier and Markus Schwagenscheidt.

Speaker: Claudia Alfes

10:30 → 11:15 Break

11:15 → 12:15 The i-epsilon Prescription for String Amplitudes and Regularized Modular Integrals

I will discuss one-loop amplitudes in string theory, and in particular their analytic continuation based on a string theoretic analog of the i-epsilon prescription of quantum field theory. For various zero- and two-point one-loop amplitudes of both open and closed strings, I will explain that this analytic continuation is equivalent to a regularization using generalized exponential integrals. Our approach provides exact expressions in terms of the degeneracies at each mass level. Based on joint work with Zhi-Zhen Wang.

Speaker: Jan Manschot

12:15 → 14:30 Lunch