Wilson loops in terms of color invariants

The theory of Wilson loops for gauge theories with unitary gauge groups is formulated in the language of symmetric functions. The main objects in this theory are two generating functions, which are related to each other by the involution that exchanges an irreducible representation with its conjugate. Both of them contain all information about the Wilson loops in arbitrary representations as well as the correlators of multiply-wound Wilson loops. This general framework is combined with the results of the Gaussian matrix model, which calculates the expectation values of 1/2-BPS circular Wilson loops in N = 4 Super-Yang-Mills theory. General, explicit, formulas for the connected correlators of multiply-wound Wilson loops in terms of the traces of symmetrized matrix products are obtained, as well as their inverses. It is shown that the generating functions for Wilson loops in mutually conjugate representations are related by a duality relation whenever they can be calculated by a Hermitian matrix model. * localization techniques [8,9] have provided a wealth of new solutions, in particular for highly symmetric loop configurations in supersymmetric gauge theories. A paradigmatic case is the case of 1 2 -BPS circular Wilson loops in N = 4 Super-Yang-Mills theory with gauge group U(N ) or SU(N ). On the one hand, the holographic dual fully captures the planar approximation in the limit of large 't Hooft coupling λ [10-15], and a lot of effort has been dedicated to obtain corrections in 1/λ [16][17][18][19][20][21][22][23][24][25][26][27][28][29]. On the other hand, localization reduces the calculation of the Wilson loops to the solution of a Gaussian matrix model [30][31][32][33][34], which, in principle, is exact in both, λ and N and provides an easier path to an asymptotic 1/N -expansion [35][36][37][38][39][40][41]. Wilson loops in N = 4 Super-Yang-Mills theory with fewer symmetries have been studied, e.g., in [42][43][44][45][46].Localization can be applied more generally in N = 2 Super-Yang-Mills theories. Interested readers are referred to the recent paper [47] and references therein.The purpose of this paper is to further develop a recent result [48], which relates the connected correlators of multiply-wound Wilson loops in N = 4 Super-Yang-Mills theory [40] to the exact solution of the corresponding Hermitian matrix model. This relation was worked out explicitly in [40] for the connected n-loop correlators with n ≤ 4 and was generalized in [48] to any n by recognizing the combinatorical pattern. In [48], it was conjectured that the relation has a deeper, group-theoretical, origin. It will be shown here that this is indeed true. To achieve this goal, it turns out to be most natural and effective to employ the framework of symmetric functions. Symmetric functions have already been used in the matrix model solution of the 1/2-BPS Wilson loops in general representations [34]. However, the generality of this framework does not seem to be widely appreciated. In fact, it allows to define the generating function(s) for ...

Section: Discussionmentioning

confidence: 82%

Combinatorics of Wilson loops in $$ \mathcal{N} $$ = 4 SYM theory

Mück

2019

“…He conjectured that this relation is exact in λ. In [31], it was shown that this is actually an example of a general relation between the Wilson loops in representations corresponding to transpose Young diagrams, in this case,…”

Section: Introductionmentioning

confidence: 89%

“…The proof in [31] was based on an expansion in terms of color invariants and is formally perturbative in powers of g 2 Y M , or, equivalently, λ. An expansion in 1/N requires further work.…”

Section: Introductionmentioning

confidence: 99%

Note on generating functions and connected correlators of 1/2-BPS Wilson loops in $$ \mathcal{N} $$ = 4 SYM theory

Garay

Faraggi

Mück

2019

The generating functions for the Wilson loops in the symmetric and antisymmetric representations of the gauge group U (N ) are expressed in terms of the connected correlators of multiply-wound Wilson loops, using ingredients from the representation theory of the symmetric group. This provides a proof of a recent observation by Okuyama. As a by-product, we present a new calculation of the connected 2-point correlator of multiply-wound Wilson loops at leading order in 1/N .

“…where B is called the Bremsstrahlung function 7 . For Lagrangian CFTs with N = 2 supersymmetry this function can be computed using supersymmetric localization 6,8,9 . For N = 2 SCFTs it was argued 8,10 and then proved 11 that B = 3h.…”

Section: Introductionmentioning

confidence: 99%

On scalar radiation

Fiol

Martínez-Montoya

2020

Self Cite

We discuss radiation in theories with scalar fields. Even in flat spacetime, the radiative fields depend qualitatively on the coupling of the scalar field to the Ricci scalar: for non-minimally coupled scalars, the radiative energy density is not positive definite, and the radiated power is not Lorentz invariant. We explore implications of this observation for radiation in conformal field theories. We find evidence that for a probe coupled to N = 4 super Yang-Mills, and following an arbitrary trajectory, the angular distribution of radiated power is independent of the Yang-Mills coupling.