arXiv research

We study the problem of finding good gauges for connections in higher gauge theories. We find that, for $2$-connections in strict $2$-gauge theory and $3$-connections in $3$-gauge theory, there are local "Coulomb gauges" that are more canonical than in classical gauge theory. In particular, they are essentially unique,…

Transforms classical connections using pushforwards and gauge transformations.

The paper connects calculus, gauge theory, and noncommutative worlds.

Survey on advanced gauge theory concepts.

Given a principal bundle on an orientable closed surface with compact connected structure group, we endow the space of based gauge equivalence classes of smooth connections relative to smooth based gauge transformations with the structure of a Fréchet manifold. Using Wilson loop holonomies and a certain characteristic …

In four dimensions one can use the chiral part of the spin connection as the main object that encodes geometry. The metric is then recovered algebraically from the curvature of this connection. We address the question of how isometries can be identified in this "pure connection" formalism. We show that isometries are r…

Study connections on Lie and Courant algebroids, defining basic curvature and Atiyah cocycle.

Many physical theories, including notably string theory, require non-abelian higher gauge fields defining higher holonomy. Previous approaches to such higher connections on categorified principal bundles require these to be fake flat. This condition, however, renders them locally gauge equivalent to connections on abel…

Develops a reduction theory for covariant field theories with gauge symmetries.

Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1-dimensional objects (e.g. strings) using 2-connections on 2-bundles. A 2-bundle is a categorified version of a bundle: that is, one where the fiber is not a ma…

Study gauge theory of real and quaternionic parabolic bundles over real curves.

Extends Coulomb gauge existence to non-associative gauge theory.

Defines a new conformally invariant Yang-Mills type energy for 6-manifolds.

Study Hom-Lie algebroid connections on complex manifolds.

We give a brief introduction to the Gauge Theory of Arbitrage. Treating a calculation of Net Present Values (NPV) and currencies exchanges as a parallel transport in some fibre bundle, we give geometrical interpretation of the interest rate, exchange rates and prices of securities as a proper connection components. Thi…

Paper introduces Atiyah sequence for Lie groupoids and studies gauge transformations.

A kinematics of the motion of a car is reformulated in terms of the theory of gauge potentials (connection on principal bundle). E(2)-connection originates in the no-slipping contact of the car with a road.

Study of real and quaternionic Lie algebroid connections on manifolds.

We propose a general notion of algebraic gauge theory obtained via extracting the main properties of classical gauge theory. Building on a recent work on transferring curved $A_{\infty}$-structures we show that, under certain technical conditions, algebraic gauge theories can be transferred along chain contractions. Sp…

Gauging procedure constructs lagrangians for carrollian gravity.

Smooth Yang-Mills fields proved in supercritical dimensions.

In four and higher dimensions, we show that any stationary admissible Yang-Mills field can be gauge transformed to a smooth field if the $L^2$ norm of the curvature is sufficiently small. There are three main ingredients. The first is Price's monotonicity formula, which allows us to assert that the curvature is small n…

Quantum cohomology gives a finite dimensional integrable system via the Dubrovin connection. Motivated by Givental's work on mirror symmetry, we use gauge theory techniques and the Frobenius Integrability Theorem to find flat sections for the Dubrovin connection. An explicit calculation is given for projective space.

Bundle gerbes are a higher version of line bundles, we present nonabelian bundle gerbes as a higher version of principal bundles. Connection, curving, curvature and gauge transformations are studied both in a global coordinate independent formalism and in local coordinates. These are the gauge fields needed for the con…

The Plebanski formulation of complex general relativity is given in terms of variables valued in the complexification of the $so(3)$ Lie algebra. Therefore, it is genuinely a gauge theory that is also diffeomorphism-invariant. For this reason, the way that the Levi-Civita connection emerges from this formulation is not…

The paper shows connections can be uniquely determined by their boundary data.

We prove the non-abelian Poincare lemma in higher gauge theory in two different ways. The first method uses a result by Jacobowitz which states solvability conditions for differential equations of a certain type. The second method extends a proof by Voronov and yields the explicit gauge parameters connecting a flat loc…

Develops a new geometric formulation for abelian gauge theories.

This work extends Chern correspondence to higher gauge theory.

New gauge-theoretic construction of 4D hyperkähler ALE spaces.

By resorting to Noether's Second Theorem, we relate the generalized Bianchi identities for Lagrangian field theories on gauge-natural bundles with the kernel of the associated gauge-natural Jacobi morphism. A suitable definition of the curvature of gauge-natural variational principles can be consequently formulated in …

A geometric construction for obtaining a prolongation of a connection to a connection of a bundle of connections is presented. This determines a natural extension of the notion of canonical energy-tensor which suits gauge and gravitational fields, and shares the main properties of the energy-tensor of a matter field in…

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…

Study of gauge theory and parallel transport in Lie 2-group bundles over Lie groupoids.

In the $3$-gauge theory, a $3$-connection is given by a $1$-form $A$ valued in the Lie algebra $\mathfrak g$, a $2$-form $B$ valued in the Lie algebra $\mathfrak h$ and a $3$-form $C$ valued in the Lie algebra $\mathfrak l$, where $(\mathfrak g,\mathfrak h, \mathfrak l)$ constitutes a differential $2$-crossed modu…

In this review we present some of the fundamental mathematical structures which permit to define noncommutative gauge field theories. In particular, we emphasize the theory of noncommutative connections, with the notions of curvatures and gauge transformations. Two different approaches to noncommutative geometry are co…

A projective geometry is an equivalence class of torsion free connections sharing the same unparametrised geodesics; this is a basic structure for understanding physical systems. Metric projective geometry is concerned with the interaction of projective and pseudo-Riemannian geometry. We show that the BGG machinery of …

Class lecture notes at a beginning graduate level on the mathematical background needed to understand classical gauge theory. Covers group actions, fiber bundles, principal bundles, connections, gauge transformations, parallel transport, curvature, covariant derivatives, pseudo-riemannian manifolds, lagrangians, cliffo…

Narasimhan and Ramadas showed that the Gribov ambiguity was maximal for the product SU(2) bundle over S^3. Specifically they showed that the holonomy group of the Coulomb connection is dense in the connected component of the identity of the gauge group. Instead of base manifold S^3, we consider here a base manifold wit…

We review the basic elements of the geometrical formalism for description of gauge fields and the theory of invariant connections, and their applications to the coset space dimensional reduction of Yang-Mills theories. We also discuss the problem of classification of principal fibre bundles, which is important for the …

The homotopy theory of gauge groups has received considerable attention in recent decades. In this work, we study the homotopy theory of gauge groups over some high dimensional manifolds. To be more specific, we study gauge groups of bundles over $(n-1)$-connected closed $2n$-manifolds, the classification of which was …

We consider dimensional reduction of gauge theories with arbitrary gauge group in a formalism based on equivariant principal bundles. For the classical gauge groups we clarify the relations between equivariant principal bundles and quiver bundles, and show that the reduced quiver gauge theories are all generically buil…

This is the first of a series of two technical papers devoted to the analysis of holonomy invariants in strict higher gauge theory with end applications in higher Chern--Simons theory. For a flat 2--connection, we define the 2-holonomy of surface knots of arbitrary genus and determine its covariance properties under 1-…

A previous paper of the authors' contained an error in the proof of a key claim, that Rasmussen's knot-invariant s(K) is equal to its gauge-theory counterpart. The original paper is included here together with a corrigendum, indicating which parts still stand and which do not. In particular, the gauge-theory counterpar…

Survey of Floer theories and their connections.

Investigates connections in Lie group bundles, focusing on geometric reduction.

Researchers calculate the Ray-Singer Torsion for $S^{1}$ bundles.

Introduces generalized principal bundles and connections, linking them to standard gauge theories.