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.
We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d N=2 theories where such structures a priori are not manifest. These modular structures include: mock modular forms, SL(2,Z) Weil representations, quantum mo…
We build a connection between topology of smooth 4-manifolds and the theory of topological modular forms by considering topologically twisted compactification of 6d (1,0) theories on 4-manifolds with flavor symmetry backgrounds. The effective 2d theory has (0,1) supersymmetry and, possibly, a residual flavor symmetry. …
E-string theory reveals modular properties of 4-manifold invariants.
problem Understanding the topology of 4-dimensional manifolds.
method Computing partition function on M4imesT2 and verifying its modular properties. result The partition function of the E-string theory is modular and can be lifted to a topological modular form.
We review the polynomial structure of the topological string partition functions as solutions to the holomorphic anomaly equations. We also explain the connection between the ring of propagators defined from special Kähler geometry and the ring of almost-holomorphic modular forms defined on modular curves.
We show that the Atiyah-Patodi-Singer reduced η-invariant of the twisted Dirac operator on a closed 4m−1 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 2m up to an integral q-series. We prove this resu…
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 SL2(Z)-invariants. result Alexander polynomials of modular knots have both finite and infinite coefficient properties.
Researchers found the global topology of the Eisenstein-Picard modular surface.
problem Understanding the global topology of the Eisenstein-Picard modular surface.
method Quotient space of the complex hyperbolic plane by the modular group.
result Determined the global topology of the Eisenstein-Picard modular surface as a 4-orbifold.
Study topological correlators for SU(2) SYM on four-manifolds, deriving explicit formulae and confirming S-duality.
problem Topological correlation functions of SU(2), N=2∗ 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 b2+=1. Proposes a topological framework to study modular invariants and related concepts.
problem Exploring modular invariants and related concepts in topological quantum field theory.
method Topological paradigm in alterfold topological quantum field theory.
result Establishes a novel integral identity for modular invariance across multiple Morita contexts.
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.
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.
ETQFTs created from non-semisimple modular categories.
problem Constructing ETQFTs from non-semisimple modular categories.
method Explicitly identify linear categories and functors in the image of ETQFTs constructed from modular categories.
result The circle category of ETQFTs is equivalent to the full subcategory of projective objects of the underlying modular category, which need not be semisimple.
Recent work extends Turaev's modular categories to non-semisimple settings.
problem Building TQFTs beyond semisimplicity.
method Generalized modular categories.
result Success in extending Turaev's construction to non-semisimple settings.
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 E8 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.
The general method of Reshetikhin and Turaev is followed to develop topological invariants of closed, connected, orientable 3-manifolds from a new class of algebras called pseudo-modular Hopf algebras. Pseudo-modular Hopf algebras are a class of Z_2-graded ribbon Hopf algebras that generalise the concept of a modular H…
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.
Mathematical study supports connection between 3D manifolds and modular tensor categories.
problem Connecting geometric topology and quantum topology using Chern-Simons invariants and Reidemeister torsions.
method Developed an algorithm to generate modular T-matrices and quantum dimensions from Seifert fibered spaces and torus bundles over the circle. result Mathematically constructed premodular categories from Seifert fibered spaces and torus bundles over the circle, conjecturing their modularity under specific conditions.
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…
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 E8 bundles.
problem Anomaly cancellation in characteristic forms.
method Constructing modular forms over specific groups using E8 bundles. result Derived new anomaly cancellation formulas.
We consider the topological and geometric structures associated with cohomological and homological objects in M-theory. For the latter, we have M2-branes and M5-branes, the analysis of which requires the underlying spacetime to admit a String structure and a Fivebrane structure, respectively. For the former, we study h…
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 SL(2,Z) 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.
In this paper we provide descriptions of the Whitehead groups with coefficients in a ring of the Hilbert modular group and its reduced version, as well as for the topological K-theory of C∗-algebras, after tensoring with Q, by computing the source of the assembly maps in the Farrell-Jones and the Baum-Con…
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.
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…
Modular knots follow Chebotarev law from surgeries on hyperbolic fibered links.
problem Understanding knots that behave like prime numbers.
method Analyzing planetary links and surgeries on hyperbolic fibered links in S3. result Modular knots around torus knots Ka,b follow the Chebotarev law. The paper finds infinite families of exotic spheres with free actions.
problem Detecting smooth free S1 and S3 actions on exotic spheres. method Topological modular forms.
result Infinite families of very exotic spheres with free actions.
Researchers create projective representations of Hecke groups using TQFT.
problem Constructing projective representations of Hecke groups.
method Using Witten-Reshetikhin-Turaev topological quantum field theory of higher genus surfaces.
result The representation's image group is infinite at low levels in genus 2.
New techniques prove quantum modularity for various functions.
problem Proving quantum modularity of false theta functions and related series.
method Developed techniques including Poisson summation formula and modular series framework.
result Unified approach to proving quantum modularity for various functions.
Motivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifold M to be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that e…
Constructs dg categories from surfaces using Khovanov homology.
problem Categorify quantum topology using surfaces and Khovanov homology.
method Constructs dg categories from surfaces using structures in Khovanov homology.
result Unified perspective on various categorified quantum topology constructions.
Homotopy Quantum Field Theories (HQFTs) generalize more familiar Topological Quantum Field Theories (TQFTs). In generalization of the surgery construction of 3-dimensional TQFTs from modular categories, we use surgery to derive 3-dimensional HQFTs from G-modular categories.
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.
We classify, in terms of topology of highest arcs, low height non-simple geodesics on the modular hyperbolic punctured sphere with three elliptic fixed points of order two. Of eight possible types, exactly one consists of geodesics that form a bigon about the cusp; we express all such geodesics in terms of Markoff trip…
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…
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…
A modular functor is constructed from non-semisimple 3d TFTs.
problem Constructing modular functors from non-semisimple 3d topological field theories.
method Using a 3d TFT defined in [arXiv:1912.02063], a symmetric monoidal 2-functor is constructed from a 2-category of bordisms to a 2-category of finite linear categories.
result A modular functor is explicitly described as a symmetric monoidal 2-functor.