arXiv research

New modular forms for anomaly cancellation formulas on any dimensional manifolds.

SL(2,Z) forms lead to new anomaly formulas.

New formulas derived from modular forms for manifold indices.

The paper defines new modular forms from almost complex manifolds and derives anomaly cancellation formulas.

Constructs modular forms and proves divisibility results for odd-dimensional manifolds.

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

New elliptic genera defined for spin manifolds.

New anomaly formulas derived from $E_8$ bundles.

We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d $\mathcal{N}=2$ theories where such structures a priori are not manifest. These modular structures include: mock modular forms, $SL(2,\mathbb{Z})$ Weil representations, quantum mo…

New formulas derived for anomaly cancellation using modular forms and E8 bundles.

Defines new two-variable elliptic genera for manifolds and derives modular forms.

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

The paper extends elliptic genus and proves anomaly cancellation formulas for almost complex manifolds.

New anomaly formulas from E8 bundles.

É.Ghys proved that the linking numbers of modular knots and the "missing" trefoil $K_{2,3}$ in $S^3$ coincide with the values of a highly ubiquitous function called the Rademacher symbol for ${\rm SL}_2\mathbb{Z}$. In this paper, we replace ${\rm SL}_2\mathbb{Z}=Γ_{2,3}$ by the triangle group $Γ_{p,q}$ for any coprime …

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

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

Describes representations of modular group into SL(3,R)/SO(3).

Researchers prove continuity of knot invariant under modular transformations.

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

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

Zagier's conjecture on knot invariants is proven for irrationals, with applications to quantum modular forms.

Examples of SL(2, Z) actions on differential graded categories are defined and explored.

Research classifies quadratic forms over various fields.

We calculate the Euler characteristics of all of the Teichmuller curves in the moduli space of genus two Riemann surfaces which are generated by holomorphic one-forms with a single double zero. These curves can all be embedded in Hilbert modular surfaces and our main result is that the Euler characteristic of a Teichmu…

In this article we construct link invariants and 3-manifold invariants from the quantum group associated with Lie superalgebra $\mathfrak{sl}(2|1)$. This construction based on nilpotent irreducible finite dimensional representations of quantum group $\mathcal{U}_ξ\mathfrak{sl}(2|1)$ where $ξ$ is a root of unity of odd …

In [AU2] we constructed the vacua modular functor based on the sheaf of vacua theory developed in [TUY] and the abelian analog in [AU1]. We here provide an explicit isomorphism from the modular functor underlying the skein-theoretic model for the Witten-Reshetikhin-Turaev TQFT due to Blanchet, Habbeger, Masbaum and Vog…

Formula for colored invariants of torus knots linked to $\mathcal{W}_r$ algebras.

The paper identifies a component of representations mapping modular group elements to isometries with unique fixed points.

Study modular forms over Γ^0(2) and anomaly cancellation formulas.

We study modular fibers of elliptic differentials, which are roughly spaces of torus-coverings over a fixed base torus. For genus 2 torus covers with fixed degree we show, that the modular fibers F_d(1,1) are itself connected torus covers with Veech group SL_2(Z). Using results of Eskin, Masur and Schmoll we calculate …

The paper introduces elliptic quasi-modular forms via moduli spaces.

A new method integrates forms on Riemann surfaces, leading to 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…

We construct a Poincaré section for the horocycle flow on the modular surface $SL(2, \R)/SL(2, \Z)$, and study the associated first return map, which coincides with a transformation (the {\it BCZ map}) defined by Boca-Cobeli-Zaharescu. We classify ergodic invariant measures for this map and prove equidistribution of pe…

The integrability of the geodesic flow on the three-folds $\mathcal M^3$ admitting $SL(2,\mathbb R)$-geometry in Thurston's sense is investigated. The main examples are the quotients $\mathcal M^3_Γ=Γ\backslash PSL(2,\mathbb R)$, where $Γ\subset PSL(2,\mathbb R)$ is a cofinite Fuchsian group. We show that the correspon…

New patterns deform Farey triangulation in symmetric space.

The paper classifies topological properties of specific geometric forms on manifolds.

New anomaly cancellation formulas for E8*E8*E8 gauge group.

Holomorphic quantum modular forms linked to knot volumes.

The paper finds a contact form on SL(2p) for p > 1.

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…

Constructs dg categories from surfaces using Khovanov homology.

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 …

The abstract discusses fiber sum formulas for 4-manifolds using topological modular forms.

Proves the relative h-principle for SL(3,R)^2 3-forms on 6-manifolds.

Study linking numbers in hyperbolic 3-folds, linking to Siegel modular forms.

Quantum modularity proved for SU(2) TQFT signature on genus 2 surfaces.