Mock modular forms have found applications in numerous branches of mathematical sciences since they were first introduced by Ramanujan nearly a century ago. In this proceeding we highlight a new area where mock modular forms start to play an important role, namely the study of three-manifold invariants. For a certain c…
On-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
168,695 papers · 148 categories
Trend · papers per month
48 results for “mock modular forms”
3d Modularity Revisitedhep-th
Researchers clarify modular group representations and vertex operator algebras for 3d invariants.
problem Understanding the full set of 3d invariants and their modular properties.
method Introducing supersymmetric defects and constructing cone vertex operator algebras.
result The full vector-valued quantum modular form for \(\widetilde{
m SL}_2(\mathbb{Z})\) captures all \(\hat Z\)-invariants of a given three-manifold.
3d Modularityhep-th
We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d theories where such structures a priori are not manifest. These modular structures include: mock modular forms, Weil representations, quantum mo…
We compute the Moore-Witten regularized u-plane integral on CP^2, and we confirm their conjecture that it is the generating function for the SO(3)-Donaldson invariants of CP^2. We prove this conjecture using the theory of mock theta functions and harmonic Maass forms. We also derive further such generating functions fo…
Revisits SYM theory to compute Donaldson invariants using mock modular forms.
problem Computing Donaldson invariants in topological SYM theory.
method Uses mock modular forms and indefinite theta functions to evaluate correlation functions.
result Explicit evaluation of correlation functions leading to modular data predictions.
We compute the Moore-Witten regularized u-plane integral on CP^1 x CP^1 directly in a chamber where the elliptic unfolding technique fails to work. This allows us to determine explicit formulas for its SU(2) and SO(3)-Donaldson invariants in terms of Mock modular forms.
In this paper, we study modularity of several functions which naturally arose in a recent paper of Lau and Zhou on open Gromov-Witten potentials of elliptic orbifolds. They derived a number of examples of indefinite theta functions, and we provide modular completions for several such functions which involve more compli…
Study compares two methods to extend invariants, finding incompatibility for Brieskorn spheres.
problem Comparing two methods to extend invariants for 3-manifolds.
method Two prescriptions: regularized -surgery combined with false-mock modular conjecture, and resurgence-based construction.
result Incompatibility found between the two prescriptions for some Brieskorn spheres.
Study topological correlators for SYM on four-manifolds, deriving explicit formulae and confirming S-duality.
problem Topological correlation functions of , SYM on four-manifolds.
method Coupling to a Spin^c structure, deriving explicit formulae, and confirming S-duality.
result Topological correlators are mock modular forms for .
The u-plane integral is the contribution of the Coulomb branch to correlation functions of N=2 gauge theory on a compact four-manifold. We consider the u-plane integral for correlators of point and surface observables of topologically twisted theories with gauge group SU(2), for an arbitrary four-manifold with (b1,b2+)…
In this paper, we consider natural geometric objects coming from Lagrangian Floer theory and mirror symmetry. Lau and Zhou showed that some of the explicit Gromov-Witten potentials computed by Cho, Hong, Kim, and Lau are essentially classical modular forms. Recent work by Zwegers and two of the authors determined modul…
New resurgent analysis reveals dual -series for Chern-Simons theory crossing natural boundaries.
problem Understanding crossing natural boundaries in Chern-Simons theory.
method Resurgent analysis and Mordell integrals to identify dual -series.
result Practical numerical algorithm generates dual -series.
New unoriented algebraic concordance group defined using mock Seifert matrices.
problem Understanding unoriented algebraic concordance of knots in thickened surfaces.
method Introducing mock Seifert matrices and using them to define unoriented algebraic concordance.
result The unoriented algebraic concordance group is abelian and infinitely generated.
Clock theorem extended to knotoids and linkoids.
problem Generalizing the Clock Theorem to knotoids and linkoids.
method Extending the Clock Theorem to knotoids and linkoids.
result Clock states of knotoid diagrams form a lattice under transpositions.
Eguchi, Ooguri, and Tachikawa recently conjectured a new moonshine phenomenon. They conjecture that the coefficients of a certain mock modular form H(tau), which arises from the K3 surface elliptic genus, are sums of dimensions of irreducible representations of the Mathieu group M24. We prove that H(tau) surprisingly a…
Researchers extend Alexander polynomial to knotoids and linkoids.
problem Defining and studying Alexander polynomial extensions for knotoids and linkoids.
method Developed and proved conjecture on mock Alexander polynomial for knotoids and linkoids.
result Proved conjecture on mock Alexander polynomial for knotoids and linkoids.
Mock Alexander Polynomialsmath.GT
The paper defines new polynomials for links and linkoids.
problem No specific problem stated; focus on new polynomials.
method Defined as sums over states of link or linkoid diagrams with .
result Constructed new polynomials for starred links and linkoids.
MOCK learns complex systems from trajectories efficiently.
problem Learning nonparametric differential equations from high-dimensional data.
method MOCK uses multivariate occupation kernel functions to learn vector fields linearly.
result MOCK outperforms other methods on various datasets.
Study modular forms over Γ^0(2) and anomaly cancellation formulas.
problem Anomaly cancellation formulas for modular forms over Γ^0(2).
method Study and analysis of modular forms over Γ^0(2).
result Anomaly cancellation formulas derived for modular forms over Γ^0(2).
New modular forms for anomaly cancellation formulas on any dimensional manifolds.
problem Constructing new modular forms for anomaly cancellation formulas.
method Using bundles, constructing modular forms on any dimensional manifolds.
result Derived new anomaly cancellation formulas and applications.
The paper introduces elliptic quasi-modular forms via moduli spaces.
problem Developing a theory of elliptic quasi-modular forms.
method Using moduli spaces and the Gauss-Manin connection.
result Presented a succinct theory of elliptic quasi-modular forms.
Constructs modular forms and proves divisibility results for odd-dimensional manifolds.
problem Constructing modular forms over specific groups and proving divisibility results.
method SL(2, Z) modular forms and Witten genus in odd dimensions.
result Obtained divisibility results of index of Toeplitz operators on spin and spin^c manifolds.
New formulas derived for anomaly cancellation using modular forms and E8 bundles.
problem Anomaly cancellation in odd dimensions.
method Systematic twisting and generalization of SL(2,Z) modular forms to define new forms associated with E8 and E8*E8 bundles.
result Derivation of a new series of anomaly cancellation formulas.
New formulas derived from modular forms for manifold indices.
problem Calculating indices of twisted Dirac operators on manifolds.
method SL(2, Z) modular forms and Γ^0(2), Γ_0(2) forms.
result New anomaly cancellation formulas for spin and spin^c manifolds.
A new method integrates forms on Riemann surfaces, leading to modular forms.
problem Integrating differential forms with poles on Riemann surfaces.
method Simple procedure to integrate differential forms with arbitrary holomorphic poles, establishing an analytic theory for integrals over configuration spaces.
result Regularized graph integrals on elliptic curves are almost-holomorphic modular forms.
We introduce a method in differential geometry to study the derivative operators of Siegel modular forms. By determining the coefficients of the invariant Levi-Civita connection on a Siegel upper half plane, and further by calculating the expressions of the differential forms under this connection, we get a non-holomor…
Making mock simulated catalogs is an important component of astrophysical data analysis. Selection criteria for observed astronomical objects are often too complicated to be derived from first principles. However the existence of an observed group of objects is a well-suited problem for machine learning classification.…
The paper defines new modular forms from almost complex manifolds and derives anomaly cancellation formulas.
problem Anomaly cancellation in almost complex manifolds.
method Defined a generalized elliptic genus and derived SL(2,Z) modular forms.
result Derived anomaly cancellation formulas and divisibility results for holomorphic Euler characteristic.
New anomaly formulas derived from bundles.
problem Anomaly cancellation in characteristic forms.
method Constructing modular forms over specific groups using bundles.
result Derived new anomaly cancellation formulas.
SL(2,Z) forms lead to new anomaly formulas.
problem Anomaly cancellation in physics.
method SL(2,Z) modular forms.
result New anomaly cancellation formulas derived.
Abstract: Generalizes modular forms to family case and finds new anomaly cancellation formulas.
problem Finding anomaly cancellation formulas for determinant line bundles and index gerbes.
method Family index theory applied to modular forms.
result Obtains new anomaly cancellation formulas for determinant line bundles and index gerbes.
New anomaly cancellation formulas for E8*E8*E8 gauge group.
problem Anomaly cancellation for E8*E8*E8 gauge group.
method Constructed modular forms over SL2(Z) to get new anomaly cancellation formulas.
result New anomaly cancellation formulas for characteristic forms.
Holomorphic quantum modular forms linked to knot volumes.
problem Understanding algebraic properties of quantum modular forms.
method Analyzing descendant state integrals for specific knots.
result Illustrated algebraic properties for the (-2,3,7)-pretzel knot.
The dynamics of an ideal fluid or plasma is constrained by topological invariants such as the circulation of (canonical) momentum or, equivalently, the flux of the vorticity or magnetic fields. In the Hamiltonian formalism, topological invariants restrict the orbits to submanifolds of the phase space. While the coadjoi…
Fuchsian groups with a modular embedding have the richest arithmetic properties among non-arithmetic Fuchsian groups. But they are very rare, all known examples being related either to triangle groups or to Teichmueller curves. In Part I of this paper we study the arithmetic properties of the modular embedding and deve…
The abstract discusses fiber sum formulas for 4-manifolds using topological modular forms.
problem Understanding fiber sum formulas for 4-manifolds.
method Using the connection between 4-manifolds and topological modular forms from 6d (1,0) SCFTs.
result Even free theories exhibit nontrivial fiber sum formulas, sensitive to individual theories and parameters.
Study linking numbers in hyperbolic 3-folds, linking to Siegel modular forms.
problem Linking numbers between geodesics in arithmetic hyperbolic 3-folds.
method Analyzing integral of Kudla--Millson theta series over Seifert surfaces.
result Series converges to genus 2 Siegel modular forms of weight 2.
Quantum modularity proved for SU(2) TQFT signature on genus 2 surfaces.
problem Proving quantum modularity of SU(2) TQFT signature for genus 2 surfaces.
method Using quantum modularity of generalized Dedekind sums associated with modular forms and trigonometric sum expressions.
result Quantum modularity of SU(2) TQFT signature on genus 2 surfaces proved.
Study evaluates LLMs like ChatGPT and GPT-4 on financial analysis tasks.
problem Assessing financial reasoning capabilities of large language models.
method Mock CFA exam questions, Zero-Shot, Chain-of-Thought, Few-Shot scenarios.
result LLMs perform well on financial analysis tasks, but have limitations.
New elliptic genera defined for spin manifolds.
problem Defining new elliptic genera for spin manifolds.
method Defined new two-variable elliptic genera using Jacobi forms and modular forms.
result Anomaly cancellation formulas derived from modular forms.
Note on new cancellation formulas for manifolds.
problem Generalizing anomaly cancellation formulas to manifolds.
method Proving new (a, b) type cancellation formulas and using transgression.
result Obtained characteristic forms with modularity properties.
We examine the relationship between nonabelian Hodge theory for Riemann surfaces and the theory of vector valued modular forms. In particular, we explain how one might use this relationship to prove a conjectural three-term inequality on the weights of free bases of vector valued modular forms associated to complex, fi…
By studying modular invariance properties of some characteristic forms, we prove some new anomaly cancellation formulas which generalize the Han-Zhang and Han-Liu-Zhang anomaly cancellation formulas
Study Alexander polynomials of modular knots, revealing finite and infinite coefficient properties.
problem Investigate Alexander polynomials of modular knots.
method Use Burau representation and geometric -invariants.
result Alexander polynomials of modular knots have both finite and infinite coefficient properties.
Proves modular functors for SO(3) have integral Hodge structures.
problem Proving modular functors have integral Hodge structures.
method Based on homological models and geometric identification.
result Geometric construction of Hodge structures on SO(3) modular functors.
ROCK: A variational formulation for occupation kernel methods in Reproducing Kernel Hilbert Spacesstat.ML
ROCK method generalizes MOCK for learning dynamical systems efficiently.
problem Learning dynamical systems from data efficiently.
method Variational formulation in Reproducing Kernel Hilbert Spaces.
result ROCK method is more computationally efficient and performs better on benchmarks.
-invariant and Modular Formsmath.DG
We show that the Atiyah-Patodi-Singer reduced -invariant of the twisted Dirac operator on a closed dimensional spin manifold, with the twisted bundle being the Witten bundle appearing in the theory of elliptic genus, is a meromorphic modular form of weight up to an integral -series. We prove this resu…
Origami structures are enumerated and shown to be quantum modular.
problem Counting and understanding origami structures with real structures.
method Using combinatorics of zonal polynomials and Schur polynomials, and relating to quantum modular forms and double Hurwitz numbers.
result The generating functions of certain origami structures are quantum modular forms.