arXiv research

This article provides an account of the functorial correspondence between irreducible singular $G$-monopoles on $S^1\times Σ$ and $\vec{t}$-stable meromorphic pairs on $Σ$. The main theorem of [1] is thus generalized here from unitary to arbitrary compact, connected gauge groups. The required distinctions and similarit…

We discuss a general procedure for using characteristic classes to study the components of the gauge group for a principal G-bundle. To illustrate this, we work out the case where G is the projective unitary group.

Paper fills in technical details for Hitchin's self-duality equations proof.

We construct a 2+1 dimensional classical gauge theory on manifolds with spin structure whose action is a refinement of the Atiyah-Patodi- Singer eta-invariant for twisted Dirac operators. We investigate the properties of the Lagrangian field theory for closed, spun 3-manifolds and compact, spun 3-manifolds with boundar…

In this paper we discuss the mechanism of spontaneous symmetry breaking from the point view of vacuum pairs, considered as ground states of a Yang-Mills-Higgs gauge theory. We treat a vacuum as a section in an appropriate bundle that is naturally associated with a minimum of a (general) Higgs potential. Such a vacuum s…

New method for directed graphs using learnable spectral positional encodings.

3D topological order linked to Seifert manifolds and gauge groups.

We show that for a simple surface with boundary the attenuated ray transform in the presence of a unitary connection and a skew-Hermitian Higgs field is injective modulo the natural obstruction for functions and vector fields. We also show that the connection and the Higgs field are uniquely determined by the scatterin…

New algorithm constructs characters of rational VOAs from knot complements.

Let E be a Real or Quaternionic Hermitian vector bundle over a Klein surface M. We study the action of the gauge group of E on the space of Galois-invariant unitary connections and we show that the closure of a semi-stable orbit contains a unique unitary orbit of projectively flat, Galois-invariant connections. We then…

Paper proves injectivity of non-abelian X-ray transform on certain spaces.

Study of Dirac operator with chiral boundary conditions on spin manifolds.

We show that representations of the loop braid group arise from Aharonov-Bohm like effects in finite 2-group (3+1)-dimensional topological higher gauge theory. For this we introduce a minimal categorification of biracks, which we call W-bikoids (welded bikoids). Our main example of W-bikoids arises from finite 2-groups…

The paper shows how to uniquely determine a connection up to gauge.

This work extends Chern correspondence to higher gauge theory.

This paper develops a geometric framework for Wilson surfaces in higher gauge theory.

We give a construction of the abelian Chern-Simons gauge theory from the point of view of a 2+1 dimensional topological quantum field theory. The definition of the quantum theory relies on geometric quantization ideas which have been previously explored in connection to the nonabelian Chern-Simons theory [JW,ADW]. We f…

We develop a graphical representation of polynomial invariants of unitary gauge groups, and use it to find the algebraic curve corresponding to a hyperkahler quotient of a linear space. We apply this method to four dimensional ALE spaces, and for the A_k, D_k, and E_6 cases, derive the explicit relation between the def…

We study the Morse theory of the Yang-Mills-Higgs functional on the space of pairs $(A,Φ)$, where $A$ is a unitary connection on a rank 2 hermitian vector bundle over a compact Riemann surface, and $Φ$ is a holomorphic section of $(E, d_A")$. We prove that a certain explicitly defined substratification of the Morse str…

We revisit the computation of the phase of the Dirac fermion scattering operator in external gauge fields. The computation is through a parallel transport along the path of time evolution operators. The novelty of the present paper compared with the earlier geometric approach by Langmann and Mickelsson, [LM], is that w…

Given an invariant gauge potential and a periodic scalar potential \tilde{V} on a Riemannian manifold \tilde{M} with a discrete symmetry group Γ, consider a Γ-periodic quantum Hamiltonian \tilde{H}=-\tildeΔ_{B}+\tilde{V} where \tildeΔ_{B} is the Bochner Laplacian. Both the gauge group and the symmetry group Γcan be non…

We consider integral geometry inverse problems for unitary connections and skew-Hermitian Higgs fields on manifolds with negative sectional curvature. The results apply to manifolds in any dimension, with or without boundary, and also in the presence of trapped geodesics. In the boundary case, we show injectivity of th…

Let $(M,g)$ be a closed oriented negatively curved surface. A unitary connection on a Hermitian vector bundle over $M$ is said to be transparent if its parallel transport along the closed geodesics of $g$ is the identity. We study the space of such connections modulo gauge and we prove a classification result in terms …

We prove that refined analytic torsion on a manifold with boundary is an analytic section of the determinant line bundle over the representation variety. As a fundamental application we establish a gluing formula for refined analytic torsion on connected components of the complex representation space which contain a un…

In this paper, the moduli space of singular unitary Hermitian--Einstein monopoles on the product of a circle and a Riemann surface is shown to correspond to a moduli space of stable pairs on the Riemann surface. These pairs consist of a holomorphic vector bundle on the surface and a meromorphic automorphism of the bund…

A new method for spectral positional encodings in directed graphs using Hermitian block Krylov subspaces.

We consider a gauge invariant one parameter family of families of fiberwise twisted Dirac type operators on a fiberation with the typical fiber an even dimensional compact manifold with boundary, i.e., a family $\{D_u\}, u\in [0,1]$ with $D_1=gD_0g^{-1}$ for a suitable unitary automorphism $g$ of the twisted bundle. Su…

The most general gauge-invariant marginal deformation of four-dimensional abelian BF-type topological field theory is studied. It is shown that the deformed quantum field theory is topological and that its observables compute, in addition to the usual linking numbers, smooth intersection indices of immersed surfaces wh…

The moduli space of stable vector bundles on a Riemann surface is smooth when the rank and degree are coprime, and is diffeomorphic to the space of unitary connections of central constant curvature. A classic result of Newstead and Atiyah-Bott asserts that its rational cohomology ring is generated by the universal clas…

We formulate the unitary rational orbifold conformal field theories in the algebraic quantum field theory framework. Under general conditions, we show that the orbifold of a given unitary rational conformal field theories generates a unitary modular category. Many new unitary modular categories are obtained. We also sh…

Contact group retracts to unitary subgroup.

We prove a convergence result for a family of Yang-Mills connections over an elliptic $K3$ surface $M$ as the fibers collapse. In particular, assume $M$ is projective, admits a section, and has singular fibers of Kodaira type $I_1$ and type $II$. Let $Ξ_{t_k}$ be a sequence of $SU(n)$ connections on a principal $SU(n)$…

Zeta functions for non-unitary twists are shown to have analytic continuation.

We study a quantum system in a Riemannian manifold M on which a Lie group G acts isometrically. The path integral on M is decomposed into a family of path integrals on a quotient space Q=M/G and the reduced path integrals are completely classified by irreducible unitary representations of G. It is not necessary to assu…

Recurrent neural networks are powerful models for processing sequential data, but they are generally plagued by vanishing and exploding gradient problems. Unitary recurrent neural networks (uRNNs), which use unitary recurrence matrices, have recently been proposed as a means to avoid these issues. However, in previous …

Study on determinants of unitary Brownian motion and their asymptotic laws.

We compute the homotopy type of the moduli space of flat, unitary connections over aspherical surfaces, after stabilizing with respect to the rank of the underlying bundle. Over the orientable surface M^g, we show that this space has the homotopy type of the infinite symmetric product of M^g, generalizing a well-known …

We construct a new family of toric manifolds generating the unitary bordism ring. Each manifold in the family is the complex projectivisation of the sum of a line bundle and a trivial bundle over a complex projective space. We also construct a family of special unitary quasitoric manifolds which contains polynomial gen…

New structure on unitary group of Hilbert space.

Study circle actions on unitary manifolds with discrete fixed points.

A major challenge in the training of recurrent neural networks is the so-called vanishing or exploding gradient problem. The use of a norm-preserving transition operator can address this issue, but parametrization is challenging. In this work we focus on unitary operators and describe a parametrization using the Lie al…

The study shows ergodicity of unitary frame flows on Kähler manifolds with specific curvature conditions.

With the usual definition of a super Hilbert space and a super unitary representation, it is easy to show that there are lots of super Lie groups for which the left-regular representation is not super unitary. I will argue that weakening the definition of a super Hilbert space (by allowing the super scalar product to b…

Study asymptotics of unitary matrix elements in quantum mechanics.

Convexity proven for sums of angles of unitary paths.

Study circumcenters in Finsler unitary groups with optimal convexity bounds.

This paper introduces a submanifold of the moduli space of unitary representations of the fundamental group of a punctured sphere with fixed local monodromy. The submanifold is defined via products of involutions through Lagrangian subspaces. We show that the moduli space of Lagrangian representations is a Lagrangian s…

We consider the problem of identifying a unitary Yang-Mills connection $\nabla$ on a Hermitian vector bundle from the Dirichlet-to-Neumann (DN) map of the connection Laplacian $\nabla^*\nabla$ over compact Riemannian manifolds with boundary. We establish uniqueness of the connection up to a gauge equivalence in the cas…