Wilson Loops Research Articles

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.

Improved renormalization scheme for nonlocal operators

In this paper we present an improved regularization-independent (RI)-type prescription appropriate for the nonperturbative renormalization of gauge-invariant nonlocal operators. In this prescription, the nonperturbative vertex function is improved by subtracting unwanted finite lattice spacing ($a$) effects, calculated in lattice perturbation theory. The method is versatile and can be applied to a wide range of fermion and gluon actions, as well as types of nonlocal operators. The presence of operator mixing can also be accommodated. Compared to the standard ${\mathrm{RI}}^{\ensuremath{'}}$ prescription, this variant can be recast as a supplementary finite renormalization, whose coefficients bring about corrections of higher order in $a$; consequently, it coincides with standard ${\mathrm{RI}}^{\ensuremath{'}}$ as $a\ensuremath{\rightarrow}0$---however, it can afford us a smoother and more controlled extrapolation to the continuum limit. In this proof-of-concept calculation we focus on nonlocal fermion bilinear operators containing a straight Wilson line. In the numerical implementation we use Wilson clover fermions and Iwasaki improved gluons. The finite-$a$ terms were calculated to one-loop level in lattice perturbation theory, and to all orders in $a$, using the same action as the nonperturbative vertex functions. We find that the method leads to significant improvement in the perturbative region indicated by small and intermediate values of the length of the Wilson line. This results in a robust extraction of the renormalization functions in that region. We have also applied the above method to operators with stout-smeared links. We show how to perform the perturbative correction for any number of smearing iterations and evaluate its effect on the power divergent renormalization coefficients.

Intermediate-Distance String Effects in Wilson Loops via Boundary Action

Intermediate-Distance String Effects in Wilson Loops via Boundary Action

The Asymptotic Statistics of Random Covering Surfaces

AbstractLet$\Gamma _{g}$be the fundamental group of a closed connected orientable surface of genus$g\geq 2$. We develop a new method for integrating over the representation space$\mathbb {X}_{g,n}=\mathrm {Hom}(\Gamma _{g},S_{n})$, where$S_{n}$is the symmetric group of permutations of$\{1,\ldots ,n\}$. Equivalently, this is the space of all vertex-labeled,n-sheeted covering spaces of the closed surface of genusg.Given$\phi \in \mathbb {X}_{g,n}$and$\gamma \in \Gamma _{g}$, we let$\mathsf {fix}_{\gamma }(\phi )$be the number of fixed points of the permutation$\phi (\gamma )$. The function$\mathsf {fix}_{\gamma }$is a special case of a natural family of functions on$\mathbb {X}_{g,n}$called Wilson loops. Our new methodology leads to an asymptotic formula, as$n\to \infty $, for the expectation of$\mathsf {fix}_{\gamma }$with respect to the uniform probability measure on$\mathbb {X}_{g,n}$, which is denoted by$\mathbb {E}_{g,n}[\mathsf {fix}_{\gamma }]$. We prove that if$\gamma \in \Gamma _{g}$is not the identity andqis maximal such that$\gamma $is aqthpower in$\Gamma _{g}$, then$$\begin{align*}\mathbb{E}_{g,n}\left[\mathsf{fix}_{\gamma}\right]=d(q)+O(n^{-1}) \end{align*}$$as$n\to \infty $, where$d\left (q\right )$is the number of divisors ofq. Even the weaker corollary that$\mathbb {E}_{g,n}[\mathsf {fix}_{\gamma }]=o(n)$as$n\to \infty $is a new result of this paper. We also prove that$\mathbb {E}_{g,n}[\mathsf {fix}_{\gamma }]$can be approximated to any order$O(n^{-M})$by a polynomial in$n^{-1}$.

A study in $\mathbb{G}_{\mathbb{R}, \geq 0}(2,6)$: from the geometric case book of Wilson loop diagrams and SYM $N=4$

We study the geometry underlying the Wilson loop diagram approach to calculating scattering amplitudes in Supersymmetric Yang Mills (SYM) $N=4$. In particular, we study the smallest non-trivial multi-propagator case, consisting of 2 propagators on 6 vertices. We do this by translating the integrals of the theory to the combinatorics of the positive geometry each diagram represents, specifically identifying the positroid cells defined by each diagram and the homology of the subcomplex they collectively generate in $\mathbb{G}\_{\mathbb{R}, \geq 0}(2,6)$. We verify the conjecture that the spurious singularities of the volume functional do all cancel on the codimension 1 boundaries of these cells, in this case. We also show that how the spurious singularities cancel is actually much more complicated than previously understood. The direct calculation laid out in this paper identifies many intricacies and artifacts of the geometry of Wilson loop diagram that need further study.

A Stochastic Analysis Approach to Lattice Yang–Mills at Strong Coupling

We develop a new stochastic analysis approach to the lattice Yang--Mills model at strong coupling in any dimension $d>1$, with t' Hooft scaling $\beta N$ for the inverse coupling strength. We study their Langevin dynamics, ergodicity, functional inequalities, large $N$ limits, and mass gap. Assuming $|\beta| < \frac{N-2}{32(d-1)N}$ for the structure group $SO(N)$, or $|\beta| < \frac{1}{16(d-1)}$ for $SU(N)$, we prove the following results. The invariant measure for the corresponding Langevin dynamic is unique on the entire lattice, and the dynamic is exponentially ergodic under a Wasserstein distance. The finite volume Yang--Mills measures converge to this unique invariant measure in the infinite volume limit, for which Log-Sobolev and Poincar\'e inequalities hold. These functional inequalities imply that the suitably rescaled Wilson loops for the infinite volume measure has factorized correlations and converges in probability to deterministic limits in the large $N$ limit, and correlations of a large class of observables decay exponentially, namely the infinite volume measure has a strictly positive mass gap. Our method improves earlier results or simplifies the proofs, and provides some new perspectives to the study of lattice Yang--Mills model.

Infrared Yang-Mills wave functional due to percolating center vortices

Inspired by the center-vortex dominance in the infrared sector of $SU(N)$ Yang-Mills theory observed on the lattice, we propose a vacuum wave functional localized on an ensemble of correlated center vortices endowed with stiffness and magnetic monopoles that change the orientation of the vortex flux. In the electric-field representation, this wave functional becomes an effective partition function for N complex scalar fields. The inclusion of both oriented and nonoriented vortices as well as so-called N-vortex matchings leads to an effective potential that has only a center symmetry left. In the center-vortex condensed phase, this symmetry is spontaneously broken. In this case, the Wilson loop average can be approximated by a solitonic saddle point localized around the minimal surface. The asymptotic string tension thus obtained displays Casimir scaling.

BPS invariants for a Knot in Seifert manifolds

We calculate homological blocks for a knot in Seifert manifolds when the gauge group is SU(N). We obtain the homological blocks with a given representation of the gauge group from the expectation value of the Wilson loop operator by analytically continuing the Chern-Simons level. We also obtain homological blocks with the analytically continued level and representation for a knot in the Seifert integer homology spheres.

Lattice Chern-Simons model for FQHE

Motivated by [60][61] and using topological invariance and cycle homology, we construct an effective gauge invariant lattice Chern-Simons (CS) model for the fractional quantum Hall effect (FQHE) with applications to (i) the Laughlin state on 3D crystals, (ii) Haldane hierarchy, and (iii) composite fermions. First, we give a new derivation of the lattice CS action and demonstrate its gauge symmetry and invariance under lattice duality. Then, we use this lattice construction to study the filling factors of the above mentioned QH states. We show amongst others that the νlattL of Laughlin state on lattice is given by the ratio Φ0/ΦWL with ΦWL designating the flux of CS link variables through Wilson loops WL. For composite fermions, we use the adb formalism to show that the factor νlattCF is given by the charge Qb of the dumbbell field under the dual gauge link variables. Our model gives explicitly the effective filling fraction of composite fermions on the lattice as νlatt⁎=νlattCF/(1−2nνlattCF) with νlattCF being the bare lattice filling factor in agreement with the continuum limit result.

Detecting topological order from modular transformations of ground states on the torus

The ground states encode the information of the topological phases of a 2-dimensional system, which makes them crucial in determining the associated topological quantum field theory (TQFT). Most numerical methods for detecting the TQFT relied on the use of minimum entanglement states (MESs), extracting the anyon mutual statistics and self statistics via overlaps and/or the entanglement spectra. The MESs are the eigenstates of the Wilson loop operators, and are labeled by the anyons corresponding to their eigenvalues. Here we revisit the definition of the Wilson loop operators and MESs. We derive the modular transformation of the ground states purely from the Wilson loop algebra, and as a result, the modular $S$ and $T$ matrices naturally show up in the overlap of MESs. Importantly, we show that due to the phase degree of freedom of the Wilson loop operators, the MES-anyon assignment is not unique. This ambiguity obstructs our attempt to detect the topological order, that is, there exist different TQFTs that cannot be distinguished solely by the overlap of MESs. In this paper, we provide the upper limit of the information one may obtain from the overlap of MESs without other additional structure. Finally, we show that if the phase is enriched by rotational symmetry, there may be additional TQFT information that can be extracted from overlap of MESs.

On continuous 2-category symmetries and Yang-Mills theory

We study a 4d gauge theory U(1)N−1 ⋊ SN obtained from a U(1)N−1 theory by gauging a 0-form symmetry SN. We show that this theory has a global continuous 2-category symmetry, whose structure is particularly rich for N > 2. This example allows us to draw a connection between the higher gauging procedure and the difference between local and global fusion, which turns out to be a key feature of higher categorical symmetries. By studying the spectrum of local and extended operators, we find a mapping with gauge invariant operators of 4d SU(N) Yang-Mills theory. The largest group-like subcategory of the non-invertible symmetries of our theory is a {mathbb{Z}}_N^{(1)} 1-form symmetry, acting on the Wilson lines in the same way as the center symmetry of Yang-Mills theory does. Supported by a path-integral argument, we propose that the U(1)N−1 ⋊ SN gauge theory has a relation with the ultraviolet limit of SU(N) Yang-Mills theory in which all Gukov-Witten operators become topological, and form a continuous non-invertible 2-category symmetry, broken down to the center symmetry by the RG flow.

Wilson lines and boundary operators of BCFW shifts

Boundary operators are gauge invariant operators whose form factors correspond to boundary contributions of BCFW shifts. In gauge theory, the boundary operators contain infinite series, which are constrained by gauge symmetry. We compute the boundary operators of all possible BCFW shifts in Yang-Mills theory and QCD, and show that the infinite series can be elegantly organized into Wilson lines, which are natural building blocks for non-local gauge invariant operators. We comment on their connection to jet functions and gauge invariant off-shell amplitudes. We also verify our results by studying various BCFW shifts of four and five-point amplitudes.

Classifying BPS bosonic Wilson loops in 3d mathcal{N} = 4 Chern-Simons-matter theories

We study the possible BPS Wilson loops in three-dimensional mathcal{N} = 4 Chern-Simons-matter theory which involve only the gauge field and bilinears of the scalars. Previously known examples are the analogues of the Gaiotto-Yin loops preserving four supercharges and “latitude” loops preserving two. We carry out a careful classification and find, in addition, loops preserving three supercharges, further inequivalent classes of loops preserving two supercharges and loops preserving a single supercharge. For each of the classes of loops, we present a representative example and analyse their full orbit under the broken symmetries.