arXiv research

Sharp lower bounds for modular invariants and Dehn twist coefficients in genus 2 and 3.

Proposes a topological framework to study modular invariants and related concepts.

Researchers clarify modular group representations and vertex operator algebras for 3d invariants.

Infinite families of quantum modular invariants for 3-manifolds are discovered.

New invariants derived from a modular category for links and 3-manifolds.

Study connects surface twists to curve invariants.

Study Alexander polynomials of modular knots, revealing finite and infinite coefficient properties.

Unified approach to conformal and modular invariants on surfaces.

New invariants from quantum group theory for hyperbolic 3-manifolds.

The paper proves a quantum modularity conjecture for 3-manifolds.

E-string theory reveals modular properties of 4-manifold invariants.

This paper proves a conjecture linking quantum modular forms and WRT invariants for specific graphs.

Modular categories are a well-known source of quantum 3-manifold invariants. In this paper we study structures on modular categories which allow to define refinements of quantum 3-manifold invariants involving cohomology classes or generalized spin and complex spin structures. A crucial role in our construction is play…

Quantum invariants of three-manifolds linked to mock theta functions.

We study the SU(2) Witten--Reshetikhin--Turaev invariant for the Seifert fibered homology spheres with M-exceptional fibers. We show that the WRT invariant can be written in terms of (differential of) the Eichler integrals of modular forms with weight 1/2 and 3/2. By use of nearly modular property of the Eichler integr…

The article provides formulas for homological blocks of Seifert fibered homology 3-spheres.

Introduces modular $q$-holonomic modules to solve $q$-difference equations.

We consider an asymptotic expansion of Kashaev's invariant or the colored Jones function for the torus link T(2,2m). We shall give q-series identity related to these invariants, and show that the invariant is regarded as a limit of q being N-th root of unity of the Eichler integral of the modular form of weight 3/2.

Holomorphic torsion invariant for log-Enriques surfaces derived from Borcherds products.

Paper constructs new identities linking quantum invariants and modular forms.

We construct modular invariants on the moduli space of quantum vacua of N=2 SYM with gauge group SU(2). We also introduce a nonchiral function K which is expressed in terms of the Seiberg-Witten and Poincare' metrics. It turns out that K has all the expected properties of the next to leading term in the Wilsonian effec…

Mathematical study supports connection between 3D manifolds and modular tensor categories.

We construct modular categories from Hecke algebras at roots of unity. For a special choice of the framing parameter, we recover the Reshetikhin-Turaev invariants of closed 3-manifolds constructed from the quantum groups U_q sl(N) by Reshetikhin-Turaev and Turaev-Wenzl, and from skein theory by Yokota. We then discuss …

By studying modular invariance properties of some characteristic forms, we obtain twisted anomaly cancellation formulas. We apply these twisted cancellation formulas to study divisibilities on spin manifolds and congruences on spin$^c$ manifolds. Especially, we get twisted Rokhlin congruences for $8k+4$ dimensional spi…

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…

Quantum theory uses modular group representations to assign invariants to 3-manifolds.

Quantum invariants of 3-manifolds and links reviewed, with connections to other topological invariants.

Study modular class of Lie ∞-algebroids and their adjoint actions.

A p-periodic 3-manifold is a 3-manifold that admits a Z_{p}-action whose fixed point set is a circle. We give a congruence relates the quantum invariant of a p-periodic 3-manifold associated to any modular category over an integrally closed ground ring and the corresponding quantum invariant of its orbit space.

Proof confirms Witten's conjecture for a class of 3-spheres.

Study shows quantum modularity in figure-eight knot's colored Jones polynomial.

Study finds modular classes help in proving Berezin volumes for supersymmetric theories.

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

By studying modular invariance properties of some characteristic forms, we get some new anomaly cancellation formulas on $(4r-1)$ dimensional manifolds. As an application, we derive some results on divisibilities of the index of Toeplitz operators on $(4r-1)$ dimensional spin manifolds and some congruent formulas on ch…

Researchers calculate $(t,q)$-series invariants for Seifert manifolds.

We conjecture a formula for the refined $\mathrm{SU}(3)$ Vafa-Witten invariants of any smooth surface $S$ satisfying $H_1(S,\mathbb{Z}) = 0$ and $p_g(S)>0$. The unrefined formula corrects a proposal by Labastida-Lozano and involves unexpected algebraic expressions in modular functions. We prove that our formula satisfi…

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…

Abstract: Generalizes modular forms to family case and finds new anomaly cancellation formulas.

Linking numbers of modular knots derived from geometric and algebraic properties.

Skew algebroid is a natural generalization of the concept of Lie algebroid. In this paper, for a skew algebroid E, its modular class mod(E) is defined in the classical as well as in the supergeometric formulation. It is proved that there is a homogeneous nowhere-vanishing 1-density on E* which is invariant with respect…

Recent work extends Turaev's modular categories to non-semisimple settings.

Revisits SYM theory to compute Donaldson invariants using mock modular forms.

We focus on two supervised visual reasoning tasks whose labels encode a semantic relational rule between two or more objects in an image: the MNIST Parity task and the colorized Pentomino task. The objects in the images undergo random translation, scaling, rotation and coloring transformations. Thus these tasks involve…

We compute the class of arithmetic genus two Teichmueller curves in the Picard group of pseudo-Hilbert modular surfaces, distinguished according to their torsion order and spin invariant. As an application, we compute the number of genus two square-tiled surfaces with these invariants. The main technical tool is the co…

We study the asymptotic behaviour of the quantum representations of the modular group in the large level limit. We prove that each element of the modular group acts as a Fourier integral operator. This provides a link between the classical and quantum Chern-Simons theories for the torus. From this result we deduce the …

New invariant from Viro's gl(1|1) polynomial distinguishes lens spaces.

The f-invariant is a higher version of the e-invariant that takes values in the divided congruences between modular forms; it can be formulated as an elliptic genus of manifolds with corners of codimension two. In this thesis, we develop a geometrical interpretation of the f-invariant in terms of index theory, thereby …

The framed little 2-discs operad is homotopy equivalent to a cyclic operad. We show that the derived modular envelope of this cyclic operad (i.e., the modular operad freely generated in a homotopy invariant sense) is homotopy equivalent to the modular operad made from classifying spaces of diffeomorphism groups of 3-di…