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

Wolfgang Mück

doi:10.1007/jhep11(2019)096

Wolfgang Mück

Open Access

https://doi.org/10.1007/jhep11(2019)096

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Abstract

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 multiplywound Wilson loops. This general framework is combined with the results of the Gaussian matrix model, which calculates the expectation values of frac{1}{2} -BPS circular Wilson loops in mathcal{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.

Highlights

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
The theory of Wilson loops for gauge theories with unitary gauge groups has been formulated in the language of symmetric functions

Summary

Wilson loop generating functions

This section will follow the account of [49]. Consider a gauge theory with gauge group U(N ) or SU(N ). Irreducible representation of the gauge group, expressed in terms of symmetric polynomials. Where we recognize in the power-sum functions pλ(u) the products of multiply-wound Wilson loops l(λ) l(λ) pλ(u) = pλi (u) = Tr U λi. Expanded in the Schur basis, they generate the Wilson loops in the irreducible representations conjugate to each other. Another observation is that they contain all information on the Wilson loops, because the Schur functions (or the power-sum funtions or the monomial functions) form a complete basis of symmetric functions. We define the connected correlators of multiply-wound Wilson loops by taking the logarithms of Z and Z′5.

Involution property

Conclusions