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Connected correlator of 1/2 BPS Wilson loops in $\mathcal{N}=4$ SYM

Kazumi Okuyama

Abstract

We study the connected correlator of 1/2 BPS winding Wilson loops in $\mathcal{N}=4$ $U(N)$ super Yang-Mills theory, where those Wilson loops are on top of each other along the same circle. We find the exact finite $N$ expression of the connected correlator of such Wilson loops. We show that the $1/N$ expansion of this exact result is reproduced from the topological recursion of Gaussian matrix model. We also study the exact finite $N$ expression of the generating function of 1/2 BPS Wilson loops in the symmetric representation.

Connected correlator of 1/2 BPS Wilson loops in $\mathcal{N}=4$ SYM

Abstract

We study the connected correlator of 1/2 BPS winding Wilson loops in super Yang-Mills theory, where those Wilson loops are on top of each other along the same circle. We find the exact finite expression of the connected correlator of such Wilson loops. We show that the expansion of this exact result is reproduced from the topological recursion of Gaussian matrix model. We also study the exact finite expression of the generating function of 1/2 BPS Wilson loops in the symmetric representation.

Paper Structure

This paper contains 11 sections, 66 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Connected correlator of winding Wilson loops at finite $N$
  3. $1/N$ expansion from topological recursion
  4. Two-point function
  5. Three-point and four-point functions
  6. Comparison between the exact result and the $1/N$ expansion
  7. Small $\lambda$ regime
  8. Finite $\lambda$ regime
  9. Comment on large winding number
  10. Generating function of symmetric representations
  11. Conclusion

Figures (2)

  • Figure 1: Plot of two-point function $\mathcal{C}_{g,2}$ as a function of $\lambda$ for \ref{['sfig:C02']}$g=0$, \ref{['sfig:C12']}$g=1$, and \ref{['sfig:C22']}$g=2$. The red dots are the exact values of the right hand side of \ref{['eq:C-exact']} at $N=200$. The blue solid curves are the analytic result of $\mathcal{C}_{g,2}$ obtained from the topological recursion.
  • Figure 2: \ref{['sfig:C03']} and \ref{['sfig:C04']} are the plots of genus-zero three-point function $\mathcal{C}_{0,3}$ and four-point function $\mathcal{C}_{0,4}$, respectively. The red dots are the exact values of the right hand side of \ref{['eq:C-exact']} at $N=200$, while the blue solid curves are the analytic result of $\mathcal{C}_{0,3}$ and $\mathcal{C}_{0,4}$ obtained from the topological recursion.