arXiv research

New indefinite false theta functions match homological blocks for a specific 3-manifold.

Formula for arborescent link tails using theta functions.

New techniques prove quantum modularity for various functions.

Hikami observed a discontinuity in a WRT invariant at roots of unity.

The article extends a knot polynomial algorithm for singular knots.

The tail of the colored Jones polynomial of an alternating link is a $q$-series invariant whose first $n$ terms coincide with the first $n$ terms of the $n$-th colored Jones polynomial. Recently, it has been shown that the tail of the colored Jones polynomial of torus knots give rise to Ramanujan type identities. In th…

Researchers calculate BPS invariants for Seifert manifolds using Witten-Reshetikhin-Turaev invariants.

The paper calculates colored Jones polynomials for specific link configurations.

Researchers found a $q$-series identity for a specific knot using $\mathfrak{sl}_3$ representations.

The tail of a sequence $\{P_n(q)\}_{n \in \mathbb{N}}$ of formal power series in $\mathbb{Z}[[q]]$ is the formal power series whose first $n$ coefficients agree up to a common sign with the first $n$ coefficients of $P_n$. This paper studies the tail of a sequence of admissible trivalent graphs with edges colored $n$ o…

Study compares two methods to extend $\widehat{Z}$ invariants, finding incompatibility for Brieskorn spheres.

Conditional probabilities modeled using Riemann-Theta Boltzmann Machines.

Proves existence of flat connection on theta functions for G-bundles.

Study theta functions and adiabatic curvature on Abelian varieties.

The paper proves rigidity of certain Dirac operators using theta functions.

Introduces a Boltzmann machine with Riemann-Theta functions for continuous and discrete states.

New geometric proofs and interpretations of scattering diagrams and theta functions.

The Kodaira--Thurston M manifold is a compact, 4-dimensional nilmanifold which is symplectic and complex but not Kaehler. We describe a construction of theta-functions associated to M which parallels the classical theory of theta-functions associated to the torus (from the point of view of representation theory and geo…

We construct an analogue of the classical theta-function on an Abelian variety for closed 4-dimensional symplectic manifolds which are T^2-bundles over T^2 with the zero Euler class. We use our theta-functions for a canonical symplectic embedding of these manifolds into complex projective spaces (an analogue of the Lef…

We study the Selberg zeta and the theta function associated to bundles over even-dimensional locally symmetric spaces of rank one.

Study on planar graph braid groups' second homology.

New resurgent analysis reveals dual $q$-series for Chern-Simons theory crossing natural boundaries.

Bracelets and theta bases match in various cluster algebras.

The Riemann-Theta Boltzmann machine's visible sector is sampled using a discrete multi-variate Gaussian.

This paper is a continuation of our work on theta and zeta functions In the previous papers we considered the case of even dimensional rank one symmetric spaces of non-compact type. The present is concerned with the odd-dimensional case, i.e. with odd-dimensional real hyperbolic manifolds. It is the natural appearence …

For a Riemann surface with cusps we define a theta function using the eigenvalues of the Laplacian and the singularities of the scattering determinant. We provide its meromorphic continuation and discuss its singularities.

In this paper, Hamiltonian monodromy is studied from the point of view of geometric quantization abd theta functions, and various differential geometric aspects thereof are dealt with, all related to holonomies of suitable flat connections.

This paper outlines an approach to the non-abelian theta functions of the $SU(2)$-Chern-Simons theory with the methods used by A. Weil for studying classical theta functions. First we translate in knot theoretic language classical theta functions, the action of the finite Heisenberg group, and the discrete Fourier tran…

In this paper we construct the quantum group, at roots of unity, of abelian Chern-Simons theory. We then use it to model classical theta functions and the actions of the Heisenberg and modular groups on them.

Geometrically connects theta functions and WZNW blocks.

In this thesis we study asymptotic behavior of projective embeddings of abelian varieties and their amoebas. The projective embeddings are given by theta functions. It is known that a Lagrangian fibration of the abelian variety determines a basis of theta functions. After reviewing the relation from the viewpoint of ge…

We define analogue of theta-functions on the Kodaira--Thurston manifold which is a compact 4-dimensional symplectic manifold and use them to construct canonical symplectic embedding of the Kodaira--Thurston manifold into the complex projective space (analogue of the Lefshetz theorem).

Proves conjecture linking WRT invariants and homological blocks for plumbed 3-manifolds.

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…

Improved Kriging model reduces prediction errors.

New method calculates growth rates of Artin-Tits monoids, linking to partial theta function.

New formulas connect knot invariants with theta functions.

Study theta-curves on torus in 3-sphere, classifying them.

Classifies instantons on a specific gravitational instanton and computes partition functions.

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…

Determinants of theta curves and symmetric graphs are studied.

Proves prime theta-curves for knots on minimal genus surfaces.

We prove a folklore theorem of W. Thurston which provides necessary and sufficient conditions for primality of a certain class of theta-curves. Namely, a theta-curve in the 3-sphere with an unknotted constituent knot U is prime if and only if lifting the third arc of the theta-curve to the double branched cover over U …

The objective of this paper is to present some geometric aspects of surfaces associated with theta function solutions of the periodic 2D-Toda lattice. For this purpose we identify the $(N^2-1)$-dimensional Euclidean space with the ${\frak su}(N)$ algebra which allows us to construct the generalized Weierstrass formula …

The Kinoshita graph is the most famous example of a Brunnian theta graph, a nontrivial spatial theta graph with the property that removing any edge yields an unknot. We produce a new family of diagrams of spatial theta graphs with the property that removing any edge results in the unknot. The family is parameterized by…

We show that any cyclically symmetric monopole is gauge equivalent to Nahm data given by Sutcliffe's ansatz, and so obtained from the affine Toda equations. Further the direction (the Ercolani-Sinha vector) and base point of the linearising flow in the Jacobian of the spectral curve associated to the Nahm equations ari…

A new machine learning model uses score matching to estimate probability densities efficiently.

Constructs unique bases for CY varieties over valued fields.