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

Starting with some known localization (matrix model) representations for correlators involving 1/2 BPS circular Wilson loop $$ \mathcal{W} $$ W in $$ \mathcal{N} $$ N = 4 SYM theory we work out their 1/N expansions in the limit of large ’t Hooft coupling λ. Motivated by a possibility of eventual matching to higher genus corrections in dual string theory we follow arXiv:2007.08512 and express the result in terms of the string coupling $$ {g}_{\mathrm{s}}\sim {g}_{\mathrm{YM}}^2\sim \lambda /N $$ g s ∼ g YM 2 ∼ λ / N and string tension $$ T\sim \sqrt{\lambda } $$ T ∼ λ . Keeping only the leading in 1/T term at each order in gs we observe that while the expansion of $$ \left\langle \mathcal{W}\right\rangle $$ W is a series in $$ {g}_{\mathrm{s}}^2/T $$ g s 2 / T , the correlator of the Wilson loop with chiral primary operators $$ {\mathcal{O}}_J $$ O J has expansion in powers of $$ {g}_{\mathrm{s}}^2/{T}^2 $$ g s 2 / T 2 . Like in the case of $$ \left\langle \mathcal{W}\right\rangle $$ W where these leading terms are known to resum into an exponential of a “one-handle” contribution $$ \sim {g}_{\mathrm{s}}^2/T $$ ∼ g s 2 / T , the leading strong coupling terms in $$ \left\langle {\mathcal{WO}}_J\right\rangle $$ WO J sum up to a simple square root function of $$ {g}_{\mathrm{s}}^2/{T}^2 $$ g s 2 / T 2 . Analogous expansions in powers of $$ {g}_{\mathrm{s}}^2/T $$ g s 2 / T are found for correlators of several coincident Wilson loops and they again have a simple resummed form. We also find similar expansions for correlators of coincident 1/2 BPS Wilson loops in the ABJM theory.

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confidence: 99%

On the structure of non-planar strong coupling corrections to correlators of BPS Wilson loops and chiral primary operators

Beccaria

Tseytlin

2021

“…The generating functions for higher rank Wilson loops introduced in [31] are contained in this language as special cases. In particular, the connected correlators of multiply would Wilson loops turn out to be a natural basis to work with and are a key ingredient in the proof of an interesting involution property [38,40,45,46]. In the case of N = 4 SYM theory with gauge group U(N ), the expression of these correlators in terms of the matrix model solution has been worked out in [38,45,46], and it will be one of the aims of this paper to further elaborate on this relation.…”

Section: Jhep07(2021)001mentioning

confidence: 99%

“…The structure of the paper is as follows. We will start by reviewing, in section 2, the general theory of Wilson loop generating functions in the language of symmetric functions [44,46]. This review will end with a generalization to the case of two independent contours, which is necessary for treating coincident circular loops winding with opposite orientations.…”

Section: Jhep07(2021)001mentioning

confidence: 99%

“…generalizing the results of [38,46]. Section 4 is dedicated to the manipulation of the matrix model result using harmonic oscillator algebra.…”

Section: Jhep07(2021)001mentioning

confidence: 99%

“…Generating functions for Wilson loops in arbitrary representations of the gauge group can be formulated elegantly [44,46] using the language of symmetric functions. We will restrict our treatment to unitary gauge groups.…”

Section: Wilson Loop Generating Functionsmentioning

confidence: 99%

See 2 more Smart Citations

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

Mück

2021

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Supersymmetric circular Wilson loops in $$ \mathcal{N} $$ 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.

Wilson loops in large symmetric representations through a double-scaling limit

Rodrı́guez-Gómez

Russo

2022

We derive exact formulas for circular Wilson loops in the $$ \mathcal{N} $$ N = 4 and $$ \mathcal{N} $$ N = 2* theories with gauge groups U(N) and SU(N) in the k-fold symmetrized product representation. The formulas apply in the limit of large k and small Yang-Mills coupling g, with fixed effective coupling κ ≡ g2k, and for any finite N. In the SU(2) and U(2) cases, closed analytic formulas are obtained for any k, while the 1/k series expansions are asymptotic. In the N ≫ 1 limit, with N ≪ k, there is an overlapping regime where the formulas can be confronted with results from holography. Simple formulas for correlation functions between the k-symmetric Wilson loops and chiral primary operators are also given.