Wilson Loops Research Articles

Topological skyrmion phases of matter

We introduce topological phases of matter defined by skyrmions in the ground state spin—or pseudospin—expectation value textures in the Brillouin zone, the chiral and helical topological skyrmion phases of matter. These phases are protected by a symmetry present in centrosymmetric superconductors. We consider a tight-binding model for spin-triplet superconductivity in transition metal oxides and find it realizes each of these topological skyrmion phases. The chiral phase is furthermore realized for a parameter set characterizing Sr2RuO4 with spin-triplet superconductivity. We also find two types of topological phase transitions by which the skyrmion number can change. The second type occurs without the closing of energy gaps in a system described by a quadratic Hamiltonian without breaking the protecting symmetries when atomic spin–orbit coupling is non-negligible and there is a suitable additional degree of freedom. This contradicts the ‘flat band’ limit assumption important in use of entanglement spectrum and Wilson loops, and in construction of the ten-fold way classification scheme of topological phases of matter. We furthermore predict two kinds of bulk-boundary correspondence signatures—one for measurements which execute a partial trace over degrees of freedom other than spin, which yields quantized transport signatures—and a second resulting from skyrmions trapping defects with their own non-trivial topology that is discussed in a second work, which yields generalizations of unpaired Majorana zero-modes.

Bootstrap for lattice Yang-Mills theory

We study the $SU(\infty)$ lattice Yang-Mills theory at the dimensions $D=2,3,4$ via the numerical bootstrap method. It combines the Makeenko-Migdal loop equations, with a cut-off $L_{\mathrm{max}}$ on the maximal length of loops, and positivity conditions on certain matrices of Wilson loops. Our algorithm is inspired by the pioneering paper of P.Anderson and M.Kruczenski but it is significantly more efficient, as it takes into account the symmetries of the lattice theory and uses the relaxation procedure in line with our previous work on matrix bootstrap. We thus obtain rigorous upper and lower bounds on the plaquette average at various couplings and dimensions. For $D=4$, the lower bound data appear to be close to the MC data in the strong coupling phase and the upper bound data in the weak coupling phase reproduce well the 3-loop perturbation theory. Our results suggest that this bootstrap approach can provide a tangible alternative to the, so far uncontested, Monte Carlo approach.

Velocity Laws for Bound States in Asymptotically AdS Geometries

AbstractWe study the behavior of heavy quark bound states in moving plasmas that are dual to theories with generic non‐trivial renormalization group flows interpolating between an AdS geometry in the ultraviolet and infrared fixed points with broken symmetries. We investigate analytically the observables associated with the bound state and find their scaling exponents with respect to the Lorentz factor for ultrarelativistic motion. Despite having asymptotically an AdS geometry, the scaling is not universal and depends on geometric conditions of the Fefferman‐Graham expansion in the near boundary regime, or equivalently on the order of the asymptotic background expansion that provides the leading contributions to the Wilson loops. We finally discuss the qualitative implications on the dissociation of the bound states with respect to the relative angle of motion.

Deciphering colour building blocks of massive multiparton amplitudes at 4-loops and beyond

The soft function in non-abelian gauge theories exponentiate, and their logarithms can be organised in terms of the collections of Feynman diagrams called Cwebs. The colour factors that appear in the logarithm are controlled by the web mixing matrices. Direct construction of the diagonal blocks of Cwebs using the new concepts of Normal ordering, basis Cweb and Fused-Web was recently carried out in [1]. In this article we establish correspondence between the boomerang webs introduced in [2] and non-boomerang Cwebs. We use this correspondence together with Uniqueness theorem and Fused web formalism introduced in [1] to obtain the diagonal blocks of four general classes of Cwebs to all orders in perturbation theory which also cover all the four loop Boomerang Cwebs connecting four Wilson lines. We also fully construct the mixing matrix of a special Cweb to all orders in perturbation theory.

Constructing static quark-antiquark creation operators from Laplacian eigenmodes

We investigate static quark-antiquark operators based on trial states formed from eigenvectors of the covariant three-dimensional lattice Laplace operator. We test the method by computing the static quark-antiquark potential and comparing results to standard Wilson loop measurements. The new method is efficient not only for on-axis, but also for many off-axis quark-antiquark separations when a fine spatial resolution is required. We further improve the ground-state overlap by using multiple eigenvector pairs, weighted with Gaussian profile functions of the eigenvalues, providing a variational basis. The method presented here can be applied to potential functions for all possible excitations of a gluonic string with fixed ends, hybrid or tetraquark potentials, as well as static-light systems and allows visualization of the spatial distribution of the Laplace trial states.

Glueball spectroscopy in lattice QCD using gradient flow

Removing ultraviolet noise from the gauge fields is necessary for glueball spectroscopy in lattice QCD. It is known that the Yang-Mills gradient flow method is an alternative approach instead of link smearing or link fuzzing in various aspects. In this work we study the application of the gradient flow technique to the construction of the extended glueball operators. We examine a simple application of the gradient flow method, which has some problems in glueball mass calculations at large flow time because of its nature of diffusion in space-time. To avoid this problem, the spatial links are evolved by the ``spatial gradient flow'', that is defined to restrict the diffusion to spatial directions only. We test the spatial gradient flow in calculations of glueball two-point functions and Wilson loops as a new smearing method, and then discuss its efficiency in comparison with the original gradient flow method and the conventional method. Furthermore, to demonstrate the feasibility of our proposed method, we determine the masses of the three lowest-lying glueball states, corresponding to the $0^{++}$, $2^{++}$ and $0^{-+}$ glueballs, in the continuum limit in the pure Yang-Mills theory.

A dispersion relation for defect CFT

We present a dispersion relation for defect CFT that reconstructs two-point functions in the presence of a defect as an integral of a single discontinuity. The main virtue of this formula is that it streamlines explicit bootstrap calculations, bypassing the resummation of conformal blocks. As applications we reproduce known results for monodromy defects in the epsilon-expansion, and present new results for the supersymmetric Wilson line at strong coupling in mathcal{N} = 4 SYM. In particular, we derive a new analytic formula for the highest R-symmetry channel of single-trace operators of arbitrary length.

Hadronic structure on the light front. II. QCD strings, Wilson lines, and potentials

This is the second paper on hadronic wave functions in the light-front formulation. We first consider only confinement effects, using the classical Nambu-Gotto string with massive end points in the light-cone gauge. We derive the light-front Hamiltonian and show how to solve it, using an expansion in suitable basis functions. We next discuss the correlators of Wilson lines on the light front, leading to an effective Hamiltonian that includes spin effects. At the end, we consider a separate problem of instanton-induced interactions, at the origin of the pion as a massless Goldstone mode on the light front.

Dynamical emergence of SUq(2) from the regularization of (2+1)D gravity with a cosmological constant

The quantization of the reduced phase-space of the Einstein-Hilbert action for gravity in $2+1D$ has been shown to bring about the emergence, at the quantum level, of a topological quantum field theory endowed with an $SU_q(2)$ quantum group symmetry structure. We hereby tackle the same problem, but start from the kinematical $SU(2)$ (quantum) Hilbert space of the theory of $2+1D$ gravity with non-zero cosmological constant in the Palatini formalism, and subsequently impose the constraints. We hence show the dynamical emergence of the $SU_q(2)$ quantum group at the quantum level within the spin-foam framework. The regularized curvature constraint is responsible for the effective representations of $SU_q(2)$ that are recovered for any Wilson loop evaluated at the $SU(2)$ group element that encodes the discretization of the space-time curvature induced by the cosmological constant. The extension to the spin-network basis, and consequently to any transition amplitude between its generic states, enables us to derive in full generality the recoupling theory of $SU_q(2)$. We provide constructive examples for the scalar product of two loop states and spin-networks encoding trivalent vertices. We further comment on the diffeomorphism symmetry generated by the implementation of the curvature constraint, and finally derive explicitly the partition function amplitude of the Turaev-Viro model.

Interpolating boundary conditions on AdS2

We consider two instances of boundary conditions for massless scalars on AdS2 that interpolate between the Dirichlet and Neumann cases while preserving scale invariance. Assessing invariance under the full SL(2; ℝ) conformal group is not immediate given their non-local nature. To further clarify this issue, we compute holographically 2- and 4-point correlation functions using the aforementioned boundary conditions and study their transformation properties. Concretely, motivated by the dual description of some multi-parametric families of Wilson loops in ABJM theory, we look at the excitations of an open string around an AdS2 ⊂ AdS4 × ℂℙ3 worldsheet, thus obtaining correlators of operators inserted along a 1-dimensional defect in mathcal{N} = 6 super Chern-Simons-matter theory at strong coupling. Of the two types of boundary conditions analyzed, only one leads to the expected functional structure for conformal primaries; the other exhibits covariance under translations and rescalings but not under special conformal transformations.

Supersymmetric solitons in gauged mathcal{N} = 8 supergravity

We consider soliton solutions in AdS4 with a flat slicing and Wilson loops around one cycle. We study the phase structure and find the ground state and identify supersymmetric solutions as a function of the Wilson loops. We work in the context of a scalar field truncation of gauged mathcal{N} = 8 supergravity, where all the dilatons are equal and all the axions vanish in the STU model. In this theory, we construct new soliton solutions parameterized by two Wilson lines. We find that there is a degeneracy of supersymmetric solutions. We also show that, for alternate boundary conditions, there exists a non-supersymmetric soliton solution with energy lower than the supersymmetric one.

Gauge Invariance of Non-Abelian Field Strength Correlators: The Axial Gauge Puzzle.

Many transport coefficients of the quark-gluon plasma and nuclear structure functions can be written as gauge invariant correlation functions of non-Abelian field strengths dressed with Wilson lines. We discuss the applicability of axial gauge n·A=0 to calculate them. In particular, we address issues that appear when one attempts to trivialize the Wilson lines in the correlation functions by gauge fixing. We find it is always impossible to completely remove the gauge fields n·A in Wilson lines that extend to infinity in the n direction by means of gauge transformations. We show how the obstruction appears in an explicit example of a perturbative calculation, and we also explain it more generally from the perspective of the path integral that defines the theory. Our results explain why the two correlators that define the heavy quark and quarkonium transport coefficients, which are seemingly equal in axial gauge, are actually different physical quantities of the quark-gluon plasma and have different values. Furthermore, our findings provide insights into the difference between two inequivalent gluon parton distribution functions.

Constant primary operators and where to find them: the strange case of BPS defects in ABJ(M) theory

We investigate the one-dimensional defect SCFT defined on the 1/2 BPS Wilson line/loop in ABJ(M) theory. We show that the supermatrix structure of the defect imposes a covariant supermatrix representation of the supercharges. Exploiting this covariant formulation, we prove the existence of a long multiplet whose highest weight state is a constant supermatrix operator. At weak coupling, we study this operator in perturbation theory and confirm that it acquires a non-trivial anomalous dimension. At strong coupling, we conjecture that this operator is dual to the lowest bound state of fluctuations of the fundamental open string in AdS4 × ℂℙ3 around the classical 1/2 BPS solution. Quite unexpectedly, this operator also arises in the cohomological equivalence between bosonic and fermionic Wilson loops. We also discuss some regularization subtleties arising in perturbative calculations on the infinite Wilson line.

Fermionic Bogomolǹyi-Prasad-Sommerfield Wilson loops in four-dimensional $\mathcal{N}=2$ superconformal gauge theories

We construct for the first time Drukker-Trancanelli (DT) type fermionic Bogomolǹyi-Prasad-Sommerfield (BPS) Wilson loops in four-dimensional \mathcal{N}=2𝒩=2 superconformal SU(N)\times SU(N)SU(N)×SU(N) quiver theory and \mathcal{N}=4𝒩=4 super Yang-Mills theory. The connections of these fermionic BPS Wilson loops have a supermatrix structure. We construct timelike BPS Wilson lines in Minkowski spacetime and circular BPS Wilson loops in Euclidean space. These Wilson loops involve dimensionful parameters. For generic values of parameters, they preserve one real (complex) supercharge in Lorentzian (Euclidean) signature. Supersymmetry enhancement for Wilson loops happens when the parameters satisfy certain constraints. The nature of such loops is quite different from the Wilson loop operators involving fermions constructed previously in the literature on four-dimensional gauge theories. We hope that further investigations of such new Wilson loops will explore deep structures in both the gauge theories and gauge/gravity dualities.

Supersymmetric Wilson loops on the lattice in the large N limit

Supersymmetric Wilson loops on the lattice in the large N limit

Local observables in SUq(2) lattice gauge theory

We consider a deformation of 3D lattice gauge theory in the canonical picture, first classically, based on the Heisenberg double of $\mathrm{SU}(2)$, then at the quantum level. We show that classical spinors can be used to define a fundamental set of local observables. They are invariant quantities that live on the vertices of the lattice and are labeled by pairs of incident edges. Any function on the classical phase space, e.g., Wilson loops, can be rewritten in terms of these observables. At the quantum level, we show that spinors become spinor operators. The quantization of the local observables then requires the use of the quantum $\mathcal{R}$ matrix, which we prove to be equivalent to a specific parallel transport around the vertex. We provide the algebra of the local observables, as a Poisson algebra classically, then as a $q$ deformation of ${\mathfrak{so}}^{*}(2n)$ at the quantum level. This formalism can be relevant to any theory relying on lattice gauge theory techniques such as topological models, loop quantum gravity or of course lattice gauge theory itself.

Constructing modular categories from orbifold data

The notion of an orbifold datum \operatorname{\mathbb{A}} in a modular fusion category \mathcal{C} was introduced as part of a generalised orbifold construction for Reshetikhin–Turaev TQFTs by Carqueville, Runkel, and Schaumann in 2018. In this paper, given a simple orbifold datum \operatorname{\mathbb{A}} in \mathcal{C} , we introduce a ribbon category \mathcal{C}_{\operatorname{\mathbb{A}}} and show that it is again a modular fusion category. The definition of \mathcal{C}_{\operatorname{\mathbb{A}}} is motivated by properties of Wilson lines in the generalised orbifold. We analyse two examples in detail: (i) when \operatorname{\mathbb{A}} is given by a simple commutative \Delta -separable Frobenius algebra A in \mathcal{C} ; (ii) when \operatorname{\mathbb{A}} is an orbifold datum in \mathcal{C} = \operatorname{Vect} , built from a spherical fusion category \mathcal{S} . We show that, in case (i), \mathcal{C}_{\operatorname{\mathbb{A}}} is ribbon-equivalent to the category of local modules of A , and, in case (ii), to the Drinfeld centre of \mathcal{S} . The category \mathcal{C}C_{\operatorname{\mathbb{A}}} thus unifies these two constructions into a single algebraic setting.

Improved Spin-Wave Estimate for Wilson Loops in U(1) Lattice Gauge Theory

Abstract In this paper, we obtain bounds on the Wilson loop expectations in 4D $U(1)$ lattice gauge theory, which quantify the effect of topological defects. In the case of a Villain interaction, by extending the non-perturbative technique introduced in [24], we obtain the following estimate for a large loop $\gamma $ at low temperatures: $ |\langle W_\gamma \rangle _{\beta }|\leq \exp \Big (-\frac {C_{GFF}} {2\beta }(1+C \beta e^{- 2\pi ^2 \beta } )(|\gamma |+o(|\gamma |)) \Big )\,.$ Our result is in line with recent works [4, 9, 13, 15] which analyze the case where the gauge group is discrete. In the present case where the gauge group is continuous and Abelian, the fluctuations of the gauge field decouple into a Gaussian part, related to the so-called free electromagnetic wave [11, 23], and a gas of topological defects. As such, our work gives new quantitative bounds on the fluctuations of the latter which complement the works by Guth and Frölich-Spencer [17, 27]. Finally, we improve, also in a non-perturbative way, the correction term from $e^{-2\pi ^2\beta }$ to $e^{-\pi ^2\beta }$ in the case of the free-energy of the system. This provides a matching lower-bound with the prediction of Guth [27] based on renormalization group techniques.

Exact strong coupling results in mathcal{N} = 2 Sp(2N) superconformal gauge theory from localization

We apply the localization technique to compute the free energy on four-sphere and the circular BPS Wilson loop in the four-dimensional mathcal{N} = ∈ superconformal Sp(2N) gauge theory containing vector multiplet coupled to four hypermultiplets in fundamental representation and one hypermultiplet in rank-2 antisymmetric representation. This theory is unique among similar mathcal{N} = ∈ superconformal models that are planar-equivalent to mathcal{N} = ∆ SYM in that the corresponding localization matrix model has the interaction potential containing single-trace terms only. We exploit this property to show that, to any order in large N expansion and an arbitrary ’t Hooft coupling λ, the free energy and the Wilson loop satisfy Toda lattice equations. Solving these equations at strong coupling, we find remarkably simple expressions for these observables which include all corrections in 1/N and 1/ sqrt{lambda } . We also compute the leading exponentially suppressed mathcal{O}left({e}^{-sqrt{lambda }}right) corrections and consider a generalization to the case when the fundamental hypermultiplets have a non-zero mass. The string theory dual of this mathcal{N} = ∈ gauge theory should be a particular orientifold of AdS5 × S5 type IIB string and we discuss the string theory interpretation of the obtained strong-coupling results.

Quantum field theoretic representation of Wilson surfaces. Part II. Higher topological coadjoint orbit model

This is the second of a series of two papers devoted to the partition function realization of Wilson surfaces in strict higher gauge theory. A higher 2-dimensional counterpart of the topological coadjoint orbit quantum mechanical model computing Wilson lines is presented based on the derived geometric framework, which has shown its usefulness in 4-dimensional higher Chern-Simons theory. Its symmetries are described. Its quantization is analyzed in the functional integral framework. Strong evidence is provided that the model does indeed underlie the partition function realization of Wilson surfaces. The emergence of the vanishing fake curvature condition is explained and homotopy invariance for a flat higher gauge field is shown. The model’s Hamiltonian formulation is further furnished highlighting the model’s close relationship to the derived Kirillov-Kostant-Souriau theory developed in the companion paper.