Supersymmetric Wilson Loops Research Articles

More fermionic supersymmetric Wilson loops in four dimensions

We construct supersymmetric fermionic Wilson loops along general curves in four-dimensional N=4 super Yang-Mills theory and along general planar curves in N=2 superconformal SU(N)×SU(N) quiver theory. These loops are generalizations of the Zarembo loops and are cohomologically equivalent to them. In N=4 super Yang-Mills theory, we compute their expectation values and verify the cohomological equivalence relation up to the order g4 in perturbation theory.

Remarks on BPS Wilson loops in non-conformal N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 gauge theories and localization

We consider 1/2 BPS supersymmetric circular Wilson loops in four-dimensional N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SU(N) SYM theories with massless matter content and non-vanishing β-function. Following Pestun’s approach, we can use supersymmetric localization on the sphere S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathbbm{S} $$\\end{document}4 to map these observables into a matrix model, provided that the one-loop determinants are consistently regularized. Employing a suitable procedure, we construct the regularized matrix model for these theories and show that, at order g4, the predictions for the 1/2 BPS Wilson loop match standard perturbative renormalization based on the direct evaluation of Feynman diagrams on S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathbbm{S} $$\\end{document}4. Despite conformal symmetry begin broken at the quantum level, we also demonstrate that the matrix model approaches perfectly captures the expression of the renormalized observable in flat space at this perturbative order. Moreover, we revisit in detail the difference theory approach, showing that when the β-function is non-vanishing, this method does not account for evanescent terms which are made finite by the renormalization procedure and participate to the corrections at order g6.

Wilson-loop one-point functions in ABJM theory

In this paper we initiate the study of correlation functions of a single trace operator and a circular supersymmetric Wilson loop in ABJM theory. The single trace operator is in the scalar sector and is an eigenstate of the planar two-loop dilatation operator. The Wilson loop is in the fundamental representation of the gauge group or a suitable (super-)group. Such correlation functions at tree level can be written as an overlap of the Bethe state corresponding to the single trace operator and a boundary state which corresponds to the Wilson loop. There are various type of supersymmetric Wilson loops in ABJM theory. We show that some of them correspond to tree-level integrable boundary states while some are not. For the tree-level integrable ones, we prove their integrability and obtain analytic formula for the overlaps. For the non-integrable ones, we give examples of non-vanishing overlaps for Bethe states which violate selection rules.

A network of hyperloops

In this paper we complete the exploration of connected components of the space of BPS Wilson loops in three-dimensional mathcal{N} = 4 Chern-Simons-matter theory on S3. The algorithm is to start with a supersymmetric Wilson loop, choose a preserved supercharge, and look for BPS deformations built out of the matter fields in the proper representations. Using this, we discover many new moduli spaces of nonconformal BPS Wilson loops preserving a single or two supercharges, which are subsets of the symmetries of the 1/4 and 3/8 BPS operators. Along with the those previously found in [1, 2] and [3], the total moduli spaces are closed under this formalism.

Supersymmetric Wilson loops on the lattice in the large N limit

Supersymmetric Wilson loops on the lattice in the large N limit

Exact 1/N expansion of Wilson loop correlators in mathcal{N} = 4 Super-Yang-Mills theory

Supersymmetric circular Wilson loops in mathcal{N} = 4 Super-Yang-Mills theory are discussed starting from their Gaussian matrix model representations. Previous results on the generating functions of Wilson loops are reviewed and extended to the more general case of two different loop contours, which is needed to discuss coincident loops with opposite orientations. A combinatorial formula representing the connected correlators of multiply wound Wilson loops in terms of the matrix model solution is derived. Two new results are obtained on the expectation value of the circular Wilson loop, the expansion of which into a series in 1/N and to all orders in the ’t Hooft coupling λ was derived by Drukker and Gross about twenty years ago. The connected correlators of two multiply wound Wilson loops with arbitrary winding numbers are calculated as a series in 1/N. The coefficient functions are derived not only as power series in λ, but also to all orders in λ by expressing them in terms of the coefficients of the Drukker and Gross series. This provides an efficient way to calculate the 1/N series, which can probably be generalized to higher-point correlators.

Circular Wilson loop in mathcal{N} = 2* super Yang-Mills theory at two loops and localization

We present a two-loop calculation of the supersymmetric circular Wilson loop in the mathcal{N} = 2* super Yang-Mills theory on the four-sphere. We develop an efficient framework for computing contributing Feynman graphs that relies on using the embedding coordinates combined with the Mellin-Barnes techniques for propagator-like integrals on the sphere. Our results exactly match predictions of supersymmetric localization providing a nontrivial consistency check for the latter in non-conformal settings.

On topological recursion for Wilson loops in mathcal{N} = 4 SYM at strong coupling

We consider U(N) mathcal{N} = 4 super Yang-Mills theory and discuss how to extract the strong coupling limit of non-planar corrections to observables involving the frac{1}{2} -BPS Wilson loop. Our approach is based on a suitable saddle point treatment of the Eynard-Orantin topological recursion in the Gaussian matrix model. Working directly at strong coupling we avoid the usual procedure of first computing observables at finite planar coupling λ, order by order in 1/N, and then taking the λ ≫ 1 limit. In the proposed approach, matrix model multi-point resolvents take a simplified form and some structures of the genus expansion, hardly visible at low order, may be identified and rigorously proved. As a sample application, we consider the expectation value of multiple coincident circular supersymmetric Wilson loops as well as their correlator with single trace chiral operators. For these quantities we provide novel results about the structure of their genus expansion at large tension, generalising recent results in arXiv:2011.02885.

Lattice mathcal{N} = 4 super Yang-Mills at strong coupling

In this paper we present results from numerical simulations of mathcal{N} = 4 super Yang-Mills for two color gauge theory over a wide range of ’t Hooft coupling 0 < λ ≤ 30 using a supersymmetric lattice action [1]. Numerical study of this lattice theory has been stymied until recently by both sign problems and the occurrence of lattice artifact phases at strong coupling. We have recently developed a new action that appears capable of solving both problems. The resulting action possesses just SU(2) rather than U(2) gauge symmetry. By explicit computations of the fermion Pfaffian we present evidence that the theory possesses no sign problem and exists in a single phase out to arbitrarily strong coupling. Furthermore, preliminary work shows that the logarithm of the supersymmetric Wilson loop varies as the square root of the ’t Hooft coupling λ for large λ in agreement with holographic predictions.

On three-point functions in ABJM and the latitude Wilson loop

I consider three-point functions of twist-one operators in ABJM at weak coupling. I compute the structure constant of correlators involving one twist-one un-protected operator and two protected ones for a few finite values of the spin, up to two-loop order. As an application I enforce a limit on the gauge group ranks, in which I relate the structure constant for three chiral primary operators to the expectation value of a supersymmetric Wilson loop. Such a relation is then used to perform a successful five-loop test on the matrix model conjectured to describe the supersymmetric Wilson loop.

3d mathcal{N} = 2 Chern-Simons-matter theory, Bethe ansatz, and quantum K -theory of Grassmannians

We study a correspondence between 3d mathcal{N} = 2 topologically twisted Chern-Simons-matter theories on S1× Σg and quantum K -theory of Grassmannians. Our starting point is a Frobenius algebra depending on a parameter β associated with an algebraic Bethe ansatz introduced by Gorbounov-Korff. They showed that the Frobenius algebra with β = −1 is isomorphic to the (equivariant) small quantum K -ring of the Grassmannian, and the Frobenius algebra with β = 0 is isomorphic to the equivariant small quantum cohomology of the Grassmannian. We apply supersymmetric localization formulas to the correlation functions of supersymmetric Wilson loops in the Chern-Simons-matter theory and show that the algebra of Wilson loops is isomorphic to the Frobenius algebra with β = −1. This allows us to identify the algebra of Wilson loops with the quantum K - ring of the Grassmannian. We also show that correlation functions of Wilson loops on S1× Σg satisfy the axiom of 2d TQFT. For β = 0, we show the correspondence between an A-twisted GLSM, the Frobenius algebra for β = 0, and the quantum cohomology of the Grassmannian. We also discuss deformations of Verlinde algebras, omega-deformations, and the K -theoretic I -functions of Grassmannians with level structures.

Supersymmetric Wilson loops via integral forms

We study supersymmetric Wilson loops from a geometrical perspective. To this end, we propose a new formulation of these operators in terms of an integral form associated to the immersion of the loop into a supermanifold. This approach provides a unifying description of Wilson loops preserving different sets of supercharges, and clarifies the flow between them. Moreover, it allows to exploit the powerful techniques of super- differential calculus for investigating their symmetries. As remarkable examples, we discuss supersymmetry and kappa-symmetry invariance.

Exact SUSY Wilson loops on S3 from q-Virasoro constraints

Using the ideas from the BPS/CFT correspondence, we give an explicit recur- sive formula for computing supersymmetric Wilson loop averages in 3d mathcal{N} = 2 Yang-Mills-Chern-Simons U(N) theory on the squashed sphere {S}_b^3 with one adjoint chiral and two antichiral fundamental multiplets, for specific values of Chern-Simons level κ2 and Fayet- Illiopoulos parameter κ1. For these values of κ1 and κ2 the north and south pole turn out to be completely independent, and therefore Wilson loop averages factorize into answers for the two constituent D2× S1 theories. In particular, our formula provides results for the theory on the round sphere when the squashing is removed.

Solving q-Virasoro constraints

We show how q-Virasoro constraints can be derived for a large class of (q, t)-deformed eigenvalue matrix models by an elementary trick of inserting certain q-difference operators under the integral, in complete analogy with full-derivative insertions for beta -ensembles. From free field point of view, the models considered have zero momentum of the highest weight, which leads to an extra constraint T_{-1} mathcal {Z} = 0. We then show how to solve these q-Virasoro constraints recursively and comment on the possible applications for gauge theories, for instance calculation of (supersymmetric) Wilson loop averages in gauge theories on D^2 times S^1 and S^3.

Supersymmetric Wilson loops in two dimensions and duality

We classify bosonic $\mathcal{N}=(2,2)$ supersymmetric Wilson loops on arbitrary backgrounds with vector-like R-symmetry. These can be defined on any smooth contour and come in two forms which are universal across all backgrounds. We show that these Wilson loops, thanks to their cohomological properties, are all invariant under smooth deformations of their contour. At genus zero they can always be mapped to local operators and computed exactly with supersymmetric localisation. Finally, we find the precise map, under two-dimensional Seiberg-like dualities, of correlators of supersymmetric Wilson loops.

Topologically twisted SUSY gauge theory, gauge-Bethe correspondence and quantum cohomology

We calculate the partition function and correlation functions in A-twisted 2d mathcal{N} = (2, 2) U(N) gauge theories and topologically twisted 3d mathcal{N} = 2 U(N) gauge theories containing an adjoint chiral multiplet with particular choices of R-charges and the magnetic fluxes for flavor symmetries. According to the Gauge-Bethe correspondence, they correspond to the Heisenberg XXX1/2 and XXZ1/2 spin chain models, respectively. We identify the partition function with the inverse of the norm of the Bethe eigenstate. Correlation functions are identified to coefficients of the expectation value of Baxter Q-operator. In addition, we consider correlation functions of 2d mathcal{N} = (2, 2)* theories and their relations to the equivariant integration of the equivariant quantum cohomology classes of the cotangent bundle of Grassmann manifolds and the equivariant quantum cohomology ring. Also, we study the twisted chiral ring relations of supersymmetric Wilson loops in 3d mathcal{N} = 2* theories and the Bethe subalgebra of the XXZ1/2 spin chain models.

N=3 harmonic supersymmetric Wilson loop

We study supersymmetric Wilson loops in $d=3$, $\mathcal{N}=3$ harmonic superspace, leading to a construction of a supersymmetrized generalization of the $\frac{1}{3}$-BPS Wilson loop for $\mathcal{N}=3$ gauge theories. This also includes a generalization of the $\frac{1}{6}$-BPS loop for ABJM theory. We perform a 'one-loop' computation of the vacuum expectation value of this operator directly in superspace and compare with the known $\mathcal{N}=2$ localization results at large $N$. This comparison also lets us identify certain fermionic contributions that do not receive any subleading corrections.

Precision matching of circular Wilson loops and strings in AdS5 \xd7 S5

Previous attempts to match the exact mathcal{N}=4 super Yang-Mills expression for the expectation value of the frac{1}{2} -BPS circular Wilson loop with the semiclassical AdS5 × S5 string theory prediction were not successful at the first subleading order. There was a missing prefactor ∼ λ−3/4 which could be attributed to the unknown normalization of the string path integral measure. Here we resolve this problem by computing the ratio of the string partition functions corresponding to the circular Wilson loop and the special frac{1}{4} supersymmetric latitude Wilson loop. The fact that the latter has a trivial expectation value in the gauge theory allows us to relate the prefactor to the contribution of the three zero modes of the “transverse” fluctuation operator in the 5-sphere directions.

Precision holography for mathcal{N}={2}^{ast } on S4 from type IIB supergravity

We find a new supersymmetric solution of type IIB supergravity which is holographically dual to the planar limit of the four-dimensional mathcal{N}={2}^{ast } supersymmetric Yang-Mills theory on S4. We study a probe fundamental string in this background which is dual to a supersymmetric Wilson loop in the mathcal{N}={2}^{ast } theory. Using holography we calculate the expectation value of this line operator to leading order in the ’t Hooft coupling. The result is a non-trivial function of the mass parameter of the mathcal{N}={2}^{ast } theory that precisely matches the result from supersymmetric localization.

Non-supersymmetric Wilson loop in mathcal{N} = 4 SYM and defect 1d CFT

Following Polchinski and Sully (arXiv:1104.5077), we consider a generalized Wilson loop operator containing a constant parameter ζ in front of the scalar coupling term, so that ζ = 0 corresponds to the standard Wilson loop, while ζ = 1 to the locally supersymmetric one. We compute the expectation value of this operator for circular loop as a function of ζ to second order in the planar weak coupling expansion in mathcal{N} = 4 SYM theory. We then explain the relation of the expansion near the two conformal points ζ = 0 and ζ = 1 to the correlators of scalar operators inserted on the loop. We also discuss the AdS5 × S5 string 1-loop correction to the strong-coupling expansion of the standard circular Wilson loop, as well as its generalization to the case of mixed boundary conditions on the five-sphere coordinates, corresponding to general ζ. From the point of view of the defect CFT1 defined on the Wilson line, the ζ-dependent term can be seen as a perturbation driving a RG flow from the standard Wilson loop in the UV to the supersymmetric Wilson loop in the IR. Both at weak and strong coupling we find that the logarithm of the expectation value of the standard Wilson loop for the circular contour is larger than that of the supersymmetric one, which appears to be in agreement with the 1d analog of the F-theorem.