Wilson Loop Observables Research Articles

Wilson Lines in the Abelian Lattice Higgs Model

AbstractLattice gauge theories are lattice approximations of the Yang–Mills theory in physics. The abelian lattice Higgs model is one of the simplest examples of a lattice gauge theory interacting with an external field. In a previous paper (Forsström et al. in Math Phys 4(2):257–329, 2023), we calculated the leading order term of the expected value of Wilson loop observables in the low-temperature regime of the abelian lattice Higgs model on $${\mathbb {Z}}^4,$$ Z 4 , with structure group $$G = {\mathbb {Z}}_n$$ G = Z n for some $$n \ge 2.$$ n ≥ 2 . In the absence of a Higgs field, these are important observables since they exhibit a phase transition which can be interpreted as distinguishing between regions with and without quark confinement. However, in the presence of a Higgs field, this is no longer the case, and a more relevant family of observables are so-called open Wilson lines. In this paper, we extend and refine the ideas introduced in Forsström et al. (Math Phys 4(2):257–329, 2023) to calculate the leading order term of the expected value of the more general Wilson line observables. Using our main result, we then calculate the leading order term of several natural ratios of expected values and confirm the behavior predicted by physicists.

Wilson loops and wormholes

We analyse the properties of Wilson loop observables for holographic gauge theories, when the dual bulk geometries have a single and/or multiple boundaries (Euclidean spacetime wormholes). Such observables lead to a generalisation and refinement of the characterisation in [1] based on the compressibility of cycles and the pinching limit of higher genus Riemann surfaces, since they carry information about the dynamics and phase structure of the dual gauge theory of an arbitrary dimensionality. Finally, we describe how backreacting correlated observables such as Wilson loops can lead to wormhole saddles in the dual gravitational path integral, by taking advantage of a representation theoretic entanglement structure proposed in [13, 15].

A State Space for 3D Euclidean Yang–Mills Theories

It is believed that Euclidean Yang–Mills theories behave like the massless Gaussian free field (GFF) at short distances. This makes it impossible to define the main observables for these theories—the Wilson loop observables—in dimensions greater than two, because line integrals of the GFF do not exist in such dimensions. Taking forward a proposal of Charalambous and Gross, this article shows that it is possible to define Euclidean Yang–Mills theories on the 3D unit torus as ‘random distributional gauge orbits’, provided that they indeed behave like the GFF in a certain sense. One of the main technical tools is the existence of the Yang–Mills heat flow on the 3D torus starting from GFF-like initial data, which is established in a companion paper. A key consequence of this construction is that under the GFF assumption, one can define a notion of ‘regularized Wilson loop observables’ for Euclidean Yang–Mills theories on the 3D unit torus.

Quantization conditions and the double copy

We formulate Wilson loop observables as products of eikonal Wilson lines given in terms of on-shell scattering amplitudes. We derive the eikonal phases for dyons in both gauge theory and gravity, which we use to derive the Dirac-Schwinger-Zwanziger quantization condition and its relativistic gravitational (Taub-NUT) counterpart via the double copy. We also compute the Wilson loop for an anyon-anyon system, obtaining a relativistic generalisation of the Aharonov-Bohm phase for gravitational anyons.

Langevin dynamic for the 2D Yang–Mills measure

We define a natural state space and Markov process associated to the stochastic Yang–Mills heat flow in two dimensions.To accomplish this we first introduce a space of distributional connections for which holonomies along sufficiently regular curves (Wilson loop observables) and the action of an associated group of gauge transformations are both well-defined and satisfy good continuity properties. The desired state space is obtained as the corresponding space of orbits under this group action and is shown to be a Polish space when equipped with a natural Hausdorff metric.To construct the Markov process we show that the stochastic Yang–Mills heat flow takes values in our space of connections and use the “DeTurck trick” of introducing a time dependent gauge transformation to show invariance, in law, of the solution under gauge transformations.Our main tool for solving for the Yang–Mills heat flow is the theory of regularity structures and along the way we also develop a “basis-free” framework for applying the theory of regularity structures in the context of vector-valued noise – this provides a conceptual framework for interpreting several previous constructions and we expect this framework to be of independent interest.

Yang\u2013Mills Measure on the Two-Dimensional Torus as a Random Distribution

We introduce a space of distributional 1-forms Omega ^1_alpha on the torus mathbf {T}^2 for which holonomies along axis paths are well-defined and induce Hölder continuous functions on line segments. We show that there exists an Omega ^1_alpha -valued random variable A for which Wilson loop observables of axis paths coincide in law with the corresponding observables under the Yang–Mills measure in the sense of Lévy (Mem Am Math Soc 166(790), 2003). It holds furthermore that Omega ^1_alpha embeds into the Hölder–Besov space mathcal {C}^{alpha -1} for all alpha in (0,1), so that A has the correct small scale regularity expected from perturbation theory. Our method is based on a Landau-type gauge applied to lattice approximations.

Chern-Simons theory and Wilson loops in the Brillouin zone

Berry connection is conventionally defined as a static gauge field in the Brillouin zone. Here we show that for three-dimensional (3d) time-reversal invariant superconductors, a generalized Berry gauge field behaves as a fluctuating field of a Chern-Simons gauge theory. The gapless nodal lines in the momentum space play the role of Wilson loop observables, while their linking and knot invariants modify the gravitational theta angle. This angle induces a topological gravitomagnetoelectric effect where a temperature gradient induces a rotational energy flow. We also show how topological strings may be realized in the 6 dimensional phase space, where the physical space defects play the role of topological D-branes.

Chern-Simons Path Integrals in S2 × S1

Using torus gauge fixing, Hahn in 2008 wrote down an expression for a Chern-Simons path integral to compute the Wilson Loop observable, using the Chern-Simons action \(S_{CS}^\kappa\), \(\kappa\) is some parameter. Instead of making sense of the path integral over the space of \(\mathfrak{g}\)-valued smooth 1-forms on \(S^2 \times S^1\), we use the Segal Bargmann transform to define the path integral over \(B_i\), the space of \(\mathfrak{g}\)-valued holomorphic functions over \(\mathbb{C}^2 \times \mathbb{C}^{i-1}\). This approach was first used by us in 2011. The main tool used is Abstract Wiener measure and applying analytic continuation to the Wiener integral. Using the above approach, we will show that the Chern-Simons path integral can be written as a linear functional defined on \(C(B_1^{\times^4} \times B_2^{\times^2}, \mathbb{C})\) and this linear functional is similar to the Chern-Simons linear functional defined by us in 2011, for the Chern-Simons path integral in the case of \(\mathbb{R}^3\). We will define the Wilson Loop observable using this linear functional and explicitly compute it, and the expression is dependent on the parameter \(\kappa\). The second half of the article concentrates on taking \(\kappa\) goes to infinity for the Wilson Loop observable, to obtain link invariants. As an application, we will compute the Wilson Loop observable in the case of \(SU(N)\) and \(SO(N)\). In these cases, the Wilson Loop observable reduces to a state model. We will show that the state models satisfy a Jones type skein relation in the case of \(SU(N)\) and a Conway type skein relation in the case of \(SO(N)\). By imposing quantization condition on the charge of the link \(L\), we will show that the state models are invariant under the Reidemeister Moves and hence the Wilson Loop observables indeed define a framed link invariant. This approach follows that used in an article written by us in 2012, for the case of \(\mathbb{R}^3\).

Theory of intersecting loops on a torus

We continue our investigation into intersections of closed paths on a torus, to further our understanding of the commutator algebra of Wilson loop observables in 2+1 quantum gravity, when the cosmological constant is negative. We give a concise review of previous results, e.g. that signed area phases relate observables assigned to homotopic loops, and present new developments in this theory of intersecting loops on a torus. We state precise rules to be applied at intersections of both straight and crooked/rerouted paths in the covering space $\mathbb{R}^2$. Two concrete examples of combinations of different rules are presented.

CHERN–SIMONS THEORY AND THE QUANTUM RACAH FORMULA

We generalize several results on Chern–Simons models on Σ × S1in the so-called "torus gauge" which were obtained in [A. Hahn, An analytic approach to Turaev's shadow invariant, J. Knot Theory Ramifications17(11) (2008) 1327–1385] (= arXiv:math-ph/0507040) to the case of general (simply-connected simple compact) structure groups and general link colorings. In particular, we give a non-perturbative evaluation of the Wilson loop observables corresponding to a special class of simple but non-trivial links and show that their values are given by Turaev's shadow invariant. As a byproduct, we obtain a heuristic path integral derivation of the quantum Racah formula.

Semiclassical correlation functions of Wilson loops and local vertex operators

We analyze correlation functions of Wilson loop observables and local vertex operators within the strong-coupling regime of the AdS/CFT correspondence. When the local operator corresponds to a light string state with finite conserved charges the correlation function can be evaluated in the semiclassical approximation of large string tension, where the contribution from the light vertex can be neglected. We consider the cases where the Wilson loops are described by two concentric surfaces and the local vertices are the superconformal chiral primary scalar or a singlet massive scalar operator.

Unquenched flavor and tropical geometry in strongly coupled Chern-Simons-matter theories

We study various aspects of the matrix models calculating free energies and Wilson loop observables in supersymmetric Chern-Simons-matter theories on the three-sphere. We first develop techniques to extract strong coupling results directly from the spectral curve describing the large N master field. We show that the strong coupling limit of the gauge theory corresponds to the so-called tropical limit of the spectral curve. In this limit, the curve degenerates to a planar graph, and matrix model calculations reduce to elementary line integrals along the graph. As an important physical application of these tropical techniques, we study $ \mathcal{N} = 3 $ theories with fundamental matter, both in the quenched and in the unquenched regimes. We calculate the exact spectral curve in the Veneziano limit, and we evaluate the planar free energy and Wilson loop observables at strong coupling by using tropical geometry. The results are in agreement with the predictions of the AdS duals involving tri-Sasakian manifolds.

From Weak to Strong Coupling in ABJM Theory

The partition function of N=6 supersymmetric Chern-Simons-matter theory (known as ABJM theory) on S^3, as well as certain Wilson loop observables, are captured by a zero dimensional super-matrix model. This super-matrix model is closely related to a matrix model describing topological Chern-Simons theory on a lens space. We explore further these recent observations and extract more exact results in ABJM theory from the matrix model. In particular we calculate the planar free energy, which matches at strong coupling the classical IIA supergravity action on AdS_4 x CP^3 and gives the correct N^{3/2} scaling for the number of degrees of freedom of the M2 brane theory. Furthermore we find contributions coming from world-sheet instanton corrections in CP^3. We also calculate non-planar corrections, both to the free energy and to the Wilson loop expectation values. This matrix model appears also in the study of topological strings on a toric Calabi-Yau manifold, and an intriguing connection arises between the space of couplings of the planar ABJM theory and the moduli space of this Calabi-Yau. In particular it suggests that, in addition to the usual perturbative and strong coupling (AdS) expansions, a third natural expansion locus is the line where one of the two 't Hooft couplings vanishes and the other is finite. This is the conifold locus of the Calabi-Yau, and leads to an expansion around topological Chern-Simons theory. We present some explicit results for the partition function and Wilson loop observables around this locus.

Wilson loops, geometric operators and fermions in 3d group field theory

Abstract Group field theories whose Feynman diagrams describe 3d gravity with a varying configuration of Wilson loop observables and 3d gravity with volume observables at each vertex are defined. The volume observables are created by the usual spin network grasping operators which require the introduction of vector fields on the group. We then use this to define group field theories that give a previously defined spin foam model for fermion fields coupled to gravity, and the simpler “quenched” approximation, by using tensor fields on the group. The group field theory naturally includes the sum over fermionic loops at each order of the perturbation theory.

Composite representation invariants and unoriented topological string amplitudes

Sinha and Vafa [1] had conjectured that the SO Chern–Simons gauge theory on S 3 must be dual to the closed A-model topological string on the orientifold of a resolved conifold. Though the Chern–Simons free energy could be rewritten in terms of the topological string amplitudes providing evidence for the conjecture, we needed a novel idea in the context of Wilson loop observables to extract cross-cap c = 0 , 1 , 2 topological amplitudes. Recent paper of Marino [2] based on the work of Morton and Ryder [3] has clearly shown that the composite representation placed on the knots and links plays a crucial role to rewrite the topological string cross-cap c = 0 amplitude. This enables extracting the unoriented cross-cap c = 2 topological amplitude. In this paper, we have explicitly worked out the composite invariants for some framed knots and links carrying composite representations in U ( N ) Chern–Simons theory. We have verified generalised Rudolph's theorem, which relates composite invariants to the invariants in SO ( N ) Chern–Simons theory, and also verified Marino's conjectures on the integrality properties of the topological string amplitudes. For some framed knots and links, we have tabulated the BPS integer invariants for cross-cap c = 0 and c = 2 giving the open-string topological amplitude on the orientifold of the resolved conifold.

Geometry and observables in (2+1)-gravity

We review the geometrical properties of vacuum spacetimes in (2+1)-gravity with vanishing cosmological constant. We explain how these spacetimes are characterised as quotients of their universal cover by holonomies. We explain how this description can be used to clarify the geometrical interpretation of the fundamental physical variables of the theory, holonomies and Wilson loops. In particular, we discuss the role of Wilson loop observables as the generators of the two fundamental transformations that change the geometry of (2+1)-spacetimes, grafting and earthquake. We explain how these variables can be determined from realistic measurements by an observer in the spacetime.

Cosmological measurements, time and observables in (2+1)-dimensional gravity

We investigate the relation between measurements and the physical observables for vacuum spacetimes with compact spatial surfaces in (2+1)-gravity with vanishing cosmological constant. By considering an observer who emits lightrays that return to him at a later time, we obtain explicit expressions for several measurable quantities as functions on the physical phase space of the theory: the eigentime elapsed between the emission of a lightray and its return to the observer, the angles between the directions into which the light has to be emitted to return to the observer and the relative frequencies of the lightrays at their emission and return. This provides a framework in which conceptual questions about time, observables and measurements can be addressed. We analyse the properties of these measurements and their geometrical interpretation and show how they allow an observer to determine the values of the Wilson loop observables that parametrize the physical phase space of (2+1)-gravity. We discuss the role of time in the theory and demonstrate that the specification of an observer with respect to the spacetime's geometry amounts to a gauge-fixing procedure yielding Dirac observables.

AN ANALYTIC APPROACH TO TURAEV'S SHADOW INVARIANT

In the present paper, we extend the "torus gauge fixing" approach by Blau and Thompson, which was developed in [10] for the study of Chern–Simons models with base manifolds M of the form M = Σ × S1, in a suitable way. We arrive at a heuristic path integral formula for the Wilson loop observables associated to general links in M. We then show that the right-hand side of this formula can be evaluated explicitly in a non-perturbative way and that this evaluation naturally leads to the face models in terms of which Turaev's shadow invariant is defined.

Chern–Simons models on [formula omitted], torus gauge fixing, and link invariants II

The present paper is the continuation of a previous paper [cf. A. Hahn, Chern–Simons models on S 2 × S 1 , torus gauge fixing, and link invariants I, J. Geom. Phys. 53 (3) (2005) 275–314], in which we derived a heuristic path integral formula for the Wilson loop observables (WLOs) of Non-Abelian Chern–Simons theory on manifolds M of the form M = Σ × S 1 using “torus gauge fixing”. In order to obtain this formula we had to make the assumption that the surface Σ is non-compact. In the present paper we will generalize the treatment of the previous paper in such a way that also the case of compact Σ is included.

Geometrical (2+1)-Gravity and the Chern-Simons Formulation: Grafting, Dehn Twists, Wilson Loop Observables and the Cosmological Constant

We relate the geometrical and the Chern-Simons description of (2+1)-dimensional gravity for spacetimes of topology \({\mathbb{R}}\times S_g\) , where Sg is an oriented two-surface of genus g > 1, for Lorentzian signature and general cosmological constant and the Euclidean case with negative cosmological constant. We show how the variables parametrising the phase space in the Chern-Simons formalism are obtained from the geometrical description and how the geometrical construction of (2+1)-spacetimes via grafting along closed, simple geodesics gives rise to transformations on the phase space. We demonstrate that these transformations are generated via the Poisson bracket by one of the two canonical Wilson loop observables associated to the geodesic, while the other acts as the Hamiltonian for infinitesimal Dehn twists. For spacetimes with Lorentzian signature, we discuss the role of the cosmological constant as a deformation parameter in the geometrical and the Chern-Simons formulation of the theory. In particular, we show that the Lie algebras of the Chern-Simons gauge groups can be identified with the (2+1)-dimensional Lorentz algebra over a commutative ring, characterised by a formal parameter ΘΛ whose square is minus the cosmological constant. In this framework, the Wilson loop observables that generate grafting and Dehn twists are obtained as the real and the ΘΛ-component of a Wilson loop observable with values in the ring, and the grafting transformations can be viewed as infinitesimal Dehn twists with the parameter ΘΛ.